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\chapter*{Short Summary}
\subsubsection{Functional RG for Quantum Impurity Systems: Status Report}
In this Thesis, we study transport properties of quantum impurity systems using the functional renormalization group (FRG). The latter is an RG-based diagrammatic tool to treat Coulomb interactions in a more flexible (but less accurate) way than, e.g., by virtue of the numerical renormalization group approach. It was first applied to quantum dot systems, where electronic correlations lead to interesting strong-coupling effects, roughly five years ago. The employed approximation scheme, which can be viewed as a kind of RG enhanced Hartree Fock theory not suffering from typical mean-field artifacts, succeeds in accurately describing \textsl{linear transport properties} (such as the conductance) of various single- as well as multi-level spinful and spin-polarised quantum dot geometries at \textsl{zero temperature} and even captures aspects of Kondo physics [1-5].
\subsubsection{Functional RG for Quantum Impurity Systems: Goals}
In a nutshell, advance in this Thesis is three-fold. First, we introduce a \textsl{frequency-dependent second-order truncation scheme} in order to eventually address \textsl{finite-energy linear-response transport} properties of quantum dot systems. Secondly, a generalisation of the Hartree-Fock-like FRG approximation to Keldysh space allows for computing \textsl{non-linear steady-state transport} properties. Thirdly, we investigate the physics of a quantum dot Josephson junction as well as the charging of a single narrow level, (mainly) using the frequency-independent approach.
\subsubsection{Method Development, Vol.~I: Finite-Frequency Properties}
As mentioned above, the FRG was mainly used to compute equilibrium zero-energy properties of quantum dot systems (such as the linear conductance) in the $T=0$ -- limit. In order to treat finite temperatures and to address energy-dependent observables (such as the density of states), one needs to account for an additional higher-order class of functional RG flow equations -- which is technically involved. We demonstrate for two distinct problems (namely the single impurity Anderson as well as the interacting resonant level model) that this turns out to be possible in principle and leads to systematic improvements for small to intermediate Coulomb interactions [6,10]. In general, however, calculating energy-dependent properties needs for an ill-controlled analytic continuation of numerical Matsubara data which can only be circumvented in certain special situations [12]. More severely, aspects of Kondo physics contained in the simple Hartree-Fock-like functional RG approximation scheme can no longer be described by the -- \textsl{a priori} more elaborate -- higher-order approach. Thus, it is still an altogether open issue how to reliably compute energy-dependent properties (e.g., the density of states) in the strong-coupling limit using the functional RG.
\subsubsection{Method Development, Vol.~II: Towards Non-Equilibrium}
Treating systems in non-equilibrium requires a fundamental extension of the method to Keldysh space. This can be done straightforward in the long-time (steady-state) limit, and even the most simple (Hartree-Fock-like) FRG approximation scheme shows satisfying agreement with time-dependent density matrix renormalization group (DMRG) data published for the interacting resonant level model [10]. The latter provides a reasonable basis for a study of non-equilibrium transport through a quantum dot dominated by charge fluctuations. In the so-called scaling limit of large bandwidths (which cannot be addressed, e.g., by the DMRG), it features universal power laws which can be described analytically by the functional renormalization group scheme in complete agreement with real-time RG data [11].
\subsubsection{The Quantum Dot Josephson Junction}
The Josephson current through a quantum dot coupled to superconductors is governed by a singlet-doublet quantum phase transition. Experimental progress in realising such systems has triggered a lot of interest in modelling quantum impurities attached to BCS leads. In this line, the functional RG allows for calculating both the phase boundary and the supercurrent in good agreement with exact results as well as with numerical RG reference data [5]. Whereas the latter is accurate for arbitrarily large values of $U$ but limited to highly symmetric problems, any system parameters -- particularly the experimentally most relevant case of finite gate voltages [9] -- can be treated by the FRG approach. Placing the quantum dot in an interferometric Aharonov-Bohm geometry leads to multiple singlet-doublet transitions, and the model exhibits re-entrance behaviour [7].
\subsubsection{Charging of Narrow Quantum Dot Levels}
A quantum dot which comprises of one level (labelled by $\sigma=+$) contacted to a higher-dimensional bath by tunnel barriers of height $\Gamma_+$ as well as a second level ($\sigma=-$) that couples to the system via a Coulomb repulsion features a `quantum phase transition' as the energy of the latter crosses the chemical potential. In presence of small tunnelling elements to some bath ($\Gamma_-$) or to the first level ($t'$) -- which might be an overall generic scenario within various experimental situations -- the charging transition acquires a finite width scaling as a power law of the bare coupling strength $t',\Gamma_-\ll\Gamma_+$. This can be shown analytically by mapping the system to the anisotropic Kondo model using bosonisation (and exploiting well-known results for the latter). We confirm power-law variations using the functional and numerical renormalization group frameworks [5].
\vspace*{3ex}
\begin{enumerate}\setlength{\itemsep}{0.5em}
\item[{[1]}]
C.~Karrasch, T.~Enss, and V.~Meden\\[0.5ex]
\textsl{`A functional renormalization group approach to transport through correlated quantum dots'}\\[0.5ex]
\href{http://prb.aps.org/abstract/PRB/v73/i23/e235337}{Phys.~Rev.~B {\bfseries 73}, 235337 (2006)}
\item[{[2]}]
C.~Karrasch, T.~Hecht, A.~Weichselbaum, Y.~Oreg, J.~von Delft, and V.~Meden\\[0.5ex]
\textsl{`Mesoscopic to Universal Crossover of the Transmission Phase of Multilevel Quantum Dots'}\\[0.5ex]
\href{http://prl.aps.org/abstract/PRL/v98/i18/e186802}{Phys.~Rev.~Lett.~{\bfseries 198}, 186802 (2007)}
\item[{[3]}]
C.~Karrasch, T.~Hecht, A.~Weichselbaum, Y.~Oreg, J.~von Delft, and V.~Meden\\[0.5ex]
\textsl{`Phase lapses in transmission through interacting two-level quantum dots'}\\[0.5ex]
\href{http://dx.doi.org/10.1088/1367-2630/9/5/123}{New.~J.~Phys.~{\bfseries 9}, 123 (2007)}
\item[{[4]}]
S.~Andergassen, T.~Enss, C.~Karrasch, and V.~Meden\\[0.5ex]
\textsl{`A gentle introduction to the functional renormalization group: The Kondo effect in quantum dots'}\\[0.5ex]
in ``Quantum Magnetism'' (Springer, New York, 2008)
\item[{[5]}]
C.~Karrasch, A.~Oguri, and V.~Meden\\[0.5ex]
\textsl{`Josephson current through a single Anderson impurity coupled to BCS leads'}\\[0.5ex]
\href{http://prb.aps.org/abstract/PRB/v77/i2/e024517}{Phys.~Rev.~B {\bfseries 77}, 024517 (2008)}
\item[{[6]}]
C.~Karrasch, R.~Hedden, R.~Peters, Th.~Pruschke, K.~Sch\"onhammer, and V.~Meden\\[0.5ex]
\textsl{`A finite-frequency functional renormalization group approach to the single impurity Anderson model'}\\[0.5ex]
\href{http://dx.doi.org/10.1088/0953-8984/20/34/345205}{J.~Phys.: Condensed Matter {\bfseries 20}, 345205 (2008)}
\item[{[7]}]
C.~Karrasch and V.~Meden\\[0.5ex]
\textsl{`Supercurrent and multiple singlet-doublet phase transitions of a quantum dot Josephson junction inside an Aharonov-Bohm ring'}\\[0.5ex]
\href{http://link.aps.org/abstract/PRB/v79/e045110}{Phys.~Rev.~B {\bfseries 79}, 045110 (2009)}
\item[{[8]}]
V.~Kashcheyevs, C.~Karrasch, T.~Hecht, A.~Weichselbaum, V.~Meden, and A.~Schiller\\[0.5ex]
\textsl{`Quantum Criticality Perspective on the Charging of Narrow Quantum-Dot Levels'}\\[0.5ex]
\href{http://prl.aps.org/abstract/PRL/v102/i13/e136805}{Phys.~Rev.~Lett.~{\bfseries 102}, 136805 (2009)}
\item[{[9]}]
A.~Eichler, R.~Deblock, M.~Weiss, C.~Karrasch, V.~Meden, C.~Sch\"onenberger, and H.~Bouchiat\\[0.5ex]
\textsl{`Tuning the Josephson current in carbon nanotubes with the Kondo effect'}\\[0.5ex]
\href{http://prb.aps.org/abstract/PRB/v79/i16/e161407}{Phys.~Rev.~B {\bfseries 79}, 161407(R) (2009)}
\item[{[10]}]
C.~Karrasch, M.~Pletyukhov, L.~Borda, and V.~Meden\\[0.5ex]
\textsl{`Functional renormalization group study of the interacting resonant level model in and out of equilibrium'}\\[0.5ex]
\href{http://prb.aps.org/abstract/PRB/v81/i12/e125122}{Phys.~Rev.~B {\bfseries 81}, 125122 (2010)}
\item[{[11]}]
C.~Karrasch, S.~Andergassen, M.~Pletyukhov, D.~Schuricht, L.~Borda, V.~Meden, and H.~Schoeller\\[0.5ex]
\textsl{`Non-equilibrium current and relaxation dynamics of a charge-fluctuating quantum dot'}\\[0.5ex]
\href{http://dx.doi.org/10.1209/0295-5075/90/30003}{Eur.~Phys.~Lett.~{\bfseries 90}, 30003 (2010)}
\item[{[12]}]
C.~Karrasch, V.~Meden, and K.~Sch\"onhammer\\[0.5ex]
\textsl{`Finite-temperature linear conductance from the Matsubara Green function without analytic continuation to the real axis'}\\[0.5ex]
\href{http://link.aps.org/doi/10.1103/PhysRevB.82.125114}{Phys.~Rev.~B {\bfseries 82}, 125114 (2010)}
\end{enumerate}
\vspace*{0.5cm}
\Large
For the complete Thesis, please go to
\vspace*{0.3cm}
\large
\href{http://www.theorie.physik.uni-goettingen.de/~karrasch/publications/thesis_karrasch.pdf}{www.theorie.physik.uni-goettingen.de/$\sim$karrasch/publications/thesis\_karrasch.pdf}
\vspace*{0.3cm}
\normalsize
(sorry for the inconvenience)
\end{document}
|
{
"timestamp": "2010-09-21T02:03:42",
"yymm": "1009",
"arxiv_id": "1009.3852",
"language": "en",
"url": "https://arxiv.org/abs/1009.3852"
}
|
\section{INTRODUCTION}
The low- and intermediate-temperature parts of the interstellar medium (ISM)
constitute a thermally bistable medium that results from the balance between
radiative heating and cooling as well as heating by cosmic rays \citep{FGH69} and
photoelectric heating from PAHs \citep{W95}, the dominant heating
source. The two stable phases are referred to as the cold neutral medium (CNM),
having $T_{\rm CNM} \sim 10^{1-2}$ K, and the warm neutral medium (WNM), with
$T_{\rm WNM} \sim 10^{3-4}$ K. The degree to which magnetic fields are frozen
into this interstellar gas is parameterized by the magnetic Reynolds number,
$Re_{\rm M}$, and the ambipolar Reynolds number, $Re_{\rm AD}$ \citep{ZB97}. The magnetic
Reynolds number is given by the ratio of the Ohmic diffusion time to the
dynamical time, and for ISM parameters is of order $10^{15} - 10^{21}$. The
ambipolar Reynolds number, given by the ratio of the ion-neutral drift time to
the dynamical time, is many orders of magnitude smaller and may approach unity in
dense molecular gas. Based on these estimates, one would expect that magnetic
fields should be well coupled to both the ionized part of the gas and to the
neutrals for all but the most dense or low column density clouds.
Under ideal magnetohydrodynamic (MHD) conditions, such as those indicated above,
one might expect a strong correlation between magnetic field strength and
density. If the relationship is expressed as $B \propto \rho^\chi$ and we ignore
diffusion, for flows directed transverse to the field we have $\chi=1$, whereas
for field aligned flows $\chi=0$. The median magnetic field strength in the CNM
has been measured at $B \sim 6 \hspace{0.1cm} \mu$G \citep{HT05}. If the field
was frozen in we might expect to detect much smaller field strengths in warmer,
lower density gas, but instead it is found that the field strength in other ISM
components is similar to that of the CNM. This was demonstrated by measurements
of the Zeeman effect over the density range $0.1$ cm$^{-3} < n < 100$ cm$^{-3}$
that yielded a flat magnetic field strength-gas density ($B-\rho$) relation
\citep{TH86}. The most obvious explanation for this relation is that motions are
aligned with the magnetic field. However, this has been argued against in two
ways. First, in order for a magnetic field to collimate a flow in this manner it
must dominate the turbulent energy density, but the field strength is less than
or equal to equipartition \citep{HZ04}. Second, the accumulation length for the
formation of giant molecular clouds is of order a kiloparsec and may be too large a
scale over which to expect coherent flows \citep{M85}.
Thus, the flat $B-\rho$ relation may be indicative of magnetic diffusion.
Among the mechanisms that have been proposed to account for the flat $B-\rho$
relation in the diffuse ISM are turbulent ambipolar diffusion (Zweibel 2002;
Heitsch et al. 2004), decorrelation due to MHD waves \citep{PVS03}, and turbulent
magnetic reconnection (e.g. Santos-Lima et al. 2010). These dynamical studies
argued that ambipolar diffusion alone was not sufficiently fast to transport
magnetic flux over the large scales under consideration and so invoked turbulence
to enhance transport. However, a 1-D two-fluid dynamical study of the thermal
instability as a formation mechanism for diffuse clouds showed ambipolar
diffusion to efficiently transport magnetic field such that the observed $B-\rho$
relation could be reproduced \citep{IIK07}.
The work presented here complements those findings
but is also a significant departure from that and the other cited examples as we
shall consider the actual transitions from one phase to another.
Our approach is advantageous in that we control the diffusive processes and are
not hampered by numerical diffusion.
The hydrodynamic structure of CNM/WNM
transitions has already been presented (Inoue et al. 2006; hereafter IIK06), but
the effects of a magnetic field have not previously been studied. In the case of
a magnetic field orthogonal to a transition layer, the field has no effect on
the structure as it does not exert any force or modify thermal conduction in the
direction of the temperature gradient, although it has a large effect on
stability \citep{SZ09}. In this work, we consider
the case of a magnetic field tangential to a transition layer in a simple 1-D
geometry and include ion-neutral drift as the magnetic diffusion mechanism.
Our paper is organized as follows: in \S2 we present our numerical method for
calculating the structure of a phase transition layer for a given initial
density and magnetic field strength. In \S3 we discuss the effect of ambipolar
drift heating on the two-phase structure of the neutral ISM. In \S4 we show a
selection of our ambipolar diffusion-mediated front solutions, and include a
brief discussion of the flux-freezing approximation. In \S 5 we discuss the
physical significance of our results, and in \S 6 we summarize our findings.
\section{METHOD}
We consider the scenario of a phase transition layer, or front, separating two
uniform media of different densities and temperatures in a simple
one-dimensional geometry with $x$ as the direction of variation. A uniform
magnetic field is tangential to the front such that
$\boldsymbol B = B(x)\hat{z}$. The geometry is illustrated in Figure
\ref{geometry}. We assume a steady-state and ionization equilibrium, and work in
the reference frame of the front. These assumptions shall be justified in \S 5.
In order to calculate the physical structure of a front we consider six
variables: pressure ($p$), density ($\rho$), bulk velocity ($v$), plasma
velocity ($v_p$), magnetic field strength ($B$), and temperature ($T$), that are
described by the following five equations, namely the equation of state:
\begin{equation}
\label{eos}
p = \frac{R \rho T}{\mu} ,
\end{equation}
the continuity equation:
\begin{equation}
\label{cont}
\frac{\partial \rho}{\partial t} + \frac{\partial}{\partial x} \rho v = 0 ,
\end{equation}
the momentum equation:
\begin{equation}
\label{momeq}
\frac{\partial \rho v}{\partial t} + \frac{\partial}{\partial x}
\Bigg(\rho v^2 + p +\frac{B^2}{8 \pi}\Bigg) = 0 ,
\end{equation}
the induction equation:
\begin{equation}
\frac{\partial B}{\partial t} = - \frac{\partial}{\partial x}(v_p B) ,
\label{induct}
\end{equation}
and the energy equation:
\begin{equation}
\frac{\gamma}{\gamma - 1}\frac{R}{\mu}\rho\frac{dT}{dt} - \frac{dp}{dt}=
\frac{\partial}{\partial x} \kappa \frac{\partial T}{\partial x} -
\rho \cal{L} ,
\label{nrg}
\end{equation}
where $\gamma$ is the adiabatic index, R is the molar gas constant, $\mu$ is the
mean molecular weight, $\kappa$ is the thermal conductivity, $\rho \cal{L}$ is
the cooling function (which includes ambipolar drift heating), and $d/dt \equiv \partial/\partial t + v \hspace{0.1cm}
\partial/ \partial x$. In the approximation that the plasma and neutral fluids
are well coupled, and the neutral density dominates, the plasma velocity may be
written as the sum of the drift velocity, $v_D = v_i - v_n$, and center of mass
velocity, $v \approx v_n$, such that $v_p \approx v + v_D$, where the drift
velocity is given by \citep{Shu83}:
\begin{equation}
\label{drift}
\boldsymbol v_D = \frac{ \boldsymbol J \times \boldsymbol B}{c \rho_i \rho_n
\gamma_{AD}} ,
\end{equation}
with the drag coefficient for collisions between ions and neutrals given by
$\gamma_{AD} = <\sigma v>_{in}/(m_i + m_n)$ cm$^3$ s$^{-1}$ g$^{-1}$, where
$<\sigma v>_{in} = 2 \times 10^{-9}$ cm$^3$ s$^{-1}$
\citep{DRD83}.
Assuming a steady-state (and having already dropped the $\hat{y}$ and $\hat{z}$
dimensions), integrating Equations (\ref{cont}), (\ref{momeq}), and (\ref{nrg})
with respect to $x$ yields the following conservation laws:
\begin{equation}
j \equiv \rho v ,
\end{equation}
\begin{equation}
M_B \equiv \rho v^2 + p + \frac{B^2}{8 \pi}
\label{MB}
\end{equation}
\begin{equation}
\frac{\gamma}{\gamma - 1}\frac{R}{\mu}j\frac{dT}{dx} - v\frac{dp}{dx}=
\frac{\partial}{\partial x} \kappa \frac{\partial T}{\partial x} -
\rho \cal{L} ,
\label{nrg2}
\end{equation}
where $j$ is the mass flux and $M_B$ is the total energy density. To solve
Equation (\ref{nrg2}) we require expressions describing the evolution of the
flow speed and magnetic field strength, including the process of ambipolar
diffusion. An equation for the flow speed is obtained by taking the derivative
of the total energy density, Equation (\ref{MB}), to obtain:
\begin{equation}
\frac{dv}{dx} = \frac{(\mu v^2 B \frac{dB}{dx} + 4 \pi R j v \frac{dT}{dx})}
{4 \pi j (RT - \mu v^2)} .
\label{vev}
\end{equation}
An equation describing the magnetic field strength is derived by substituting
$\boldsymbol B = B(x)\hat{z}$ into Faraday's law in 1-D and using Equation
(\ref{induct}) to yield:
\begin{equation}
\frac{\partial B_z}{\partial t} = -c \frac{\partial E_y}{\partial x}
\Rightarrow c E_y = -(v_p \times B)_y = v_{p x} B_z = \rm constant .
\label{vv}
\end{equation}
Substituting the plasma velocity, with the drift velocity given by Equation
(\ref{drift}), into Equation (\ref{vv}) we obtain:
\begin{equation}
vB - \frac{B^2}{4 \pi \rho_i \rho_n \gamma_{AD}}\frac{d B}{d x} =
cE .
\label{magev}
\end{equation}
This is a first-order ODE with one parameter, $cE$. Mathematically, $cE$ can
take any value since it is a constant of integration. However, we will argue at
the end of this section that physical considerations of the magnetic field
strength and ambipolar heating across a front serve to greatly reduce the $cE$
parameter space.
Equations (\ref{nrg2}), (\ref{vev}), and (\ref{magev}), and the definition
$z \equiv dT/dx$ yield a system of four ODEs for $T$, $B$, and $v$ that apply
for any functional form of conductivity and cooling function. In the gas states
studied here, conductivity is dominated by neutral atoms such that
$\kappa = 2.5 \times 10^3 T^{1/2}$ ergs s$^{-1}$ K$^{-1}$ cm$^{-1}$ \citep{P53}.
The cooling function is written in full as:
\begin{equation}
\rho \cal{L} = \rm n [n \Lambda - (\Gamma_{PAH} + \Gamma_{AD})] .
\label{TE}
\end{equation}
We take the simple functional forms used by IIK06 for Ly$\alpha$ and [C II]
radiative cooling such that:
\begin{equation}
\Lambda = 7.3 \times 10^{-21} \rm{exp} \Bigg(\frac{-118400 \hspace{0.1cm} K}
{T+1500 \hspace{0.1cm} K}\Bigg)+
7.9 \times 10^{-27} \rm{exp} \Bigg(\frac{-92 \hspace{0.1cm} K}{T}\Bigg)
\rm ergs \hspace{0.2cm} s^{-1} cm^{-3} ,
\label{lambdacool}
\end{equation}
and for photoelectric heating:
\begin{equation}
\Gamma_{\rm PAH} = 2 \times 10^{-26} \rm ergs \hspace{0.2cm} s^{-1} .
\label{gammapah}
\end{equation}
Heating by ion-neutral friction is represented by $\Gamma_{AD}$:
\begin{equation}
\label{adheat}
n \Gamma_{\rm AD} = \rho_i \rho_n \gamma_{\rm AD} v_D^2 =
\frac{1}{\rho_i \rho_n \gamma_{\rm AD}} \Bigg(\frac{B}{4 \pi}
\frac{d B}{d x}\Bigg)^2 \rm ergs \hspace{0.2cm} s^{-1} cm^{-3}
\end{equation}
(Scalo 1977, Padoan et al. 2000), where the density of neutrals is given by
$\rho_n = \mu_n m_H n$, and the density of ions by $\rho_i = \mu_i m_p n_e$. We
compute the ionization fraction using:
\begin{eqnarray}
\frac{n_e}{n_H} & = & \Bigg(1.19 \times 10^{-4} - 1.36 \times 10^{-8}
\frac{T^{0.845}}{n_H}\Bigg) + \nonumber \\
& &
\Bigg(1.42 \times 10^{-8} + 2.72 \times 10^{-8} \frac{T^{0.845}}{n_H}
+ 1.85 \times 10^{-16} \frac{T^{1.69}}{n_H^2}\Bigg)^{1/2}
\end{eqnarray}
\citep{FZS88}.
In solving this system we choose initial values for the density and magnetic
field strength and impose the following boundary conditions:
\begin{equation}
\label{Tbc}
T(x=x_1) = T_1, \hspace{1in} T(x=x_2) = T_2,
\end{equation}
\begin{equation}
\label{dTbc}
\frac{dT}{dx} \Bigg |_{x_1,x_2} = 0 ,
\end{equation}
where $x_1$ and $x_2$ represent the left- and right-hand boundaries,
respectively, and $T_1$ and $T_2$ satisfy thermal equilibrium at these
boundaries. $T_1$ is found by solving Equation (\ref{TE}) for the chosen initial
value of the density at $x_1$, and $T_2$ is the temperature obtained by
integrating as far as $x_2$, where the size of domain is chosen such that $T_2$
will also satisfy thermal equilibrium. For our third and fourth conditions,
given by Equation (\ref{dTbc}), we impose zero temperature gradient at both
boundaries. Finally, we set the value of the initial magnetic field strength
gradient, $|dB/dx|_{x_1}$, as this controls the amount of ambipolar heating in a
given front model. As we will show in \S4, the choice of the initial field
strength gradient affects the structure of the front. Given that we set five
boundary values but have a system of only four ODEs, we thus set up an
eigenvalue problem in which the mass flux, $j$, is the parameter to be adjusted
to find a self-consistent solution.
The numerical method we employ is that of shooting, in which the integration is
performed with an initial guess for $j$, the resulting boundary values compared
to the desired conditions, and $j$ adjusted accordingly so that the integration
can be repeated as necessary until the right-hand boundary conditions are
satisfied to within some chosen tolerance. We find that the degree to which
thermal equilibrium is satisfied at the right-hand boundary depends on the size
of the domain, which should be adjusted to achieve optimum results. For cases in
which ambipolar drift heating does not dominate it is possible to satisfy
thermal equilibrium to better than one part in $10^5$. We use a 5th-order
adaptive Runge-Kutta scheme \citep{Numrec} and adjust the eigenvalue according to the secant method.
When appropriate bounds are chosen our method converges to a solution quickly,
requiring of order ten iterations. Note that we always integrate from the cold
medium to the warmer one.
We close this section with a brief discussion of the initial magnetic field
strength gradient boundary condition and the parameter $cE$. In setting up our
initial conditions, instead of choosing the value of $cE$ directly we instead
set the initial value of the magnetic field strength gradient, $|dB/dx|_{x_1}$.
This implies the value of $cE$, which is kept constant across the domain, as we
may evaluate it by substituting our initial conditions into Equation
(\ref{magev}). Note that the value of $cE$ will change with each iteration of
the shooting method because it depends on the bulk velocity, which is adjusted
according to the secant method. For all density and magnetic field strength
initial conditions there is some minimum value of $cE$ below which the magnetic
field strength gradient is positive throughout the domain. The outcome for
choosing an initial field gradient that yields a value of $cE$ below this
minimum would be a larger magnetic field strength in the warm medium than in the
colder one. However, if one imagines an evaporating cool cloud with the
assumption of frozen-in magnetic field lines, this does not seem like a
physically reasonable scenario as the field lines will become further apart as
the cloud expands. Furthermore, if the value of cE is too large, ambipolar drift
heating may dominate over photoelectric heating making it increasingly difficult to
satisfy thermal equilibrium at the far boundary, implying that a front can no
longer exist. We demonstrate quantitatively the effects of $|dB/dx|_{x_1}$, and
hence $cE$, in \S 4, but do not refer to $cE$ explicitly in the rest of the
paper.
\section{EFFECTS OF AMBIPOLAR DRIFT HEATING ON TWO-PHASE STRUCTURE}
The two neutral phases of the ISM are enabled by the balance of radiative
cooling and heating by, in this work, photoelectric heating and ion-neutral friction. We
present the equilibrium state of the cooling function, $\rho \cal{L} \rm (n, T)
= 0$, in Figure \ref{te}, with $\rho \cal{L} \rm$ given by Equation (\ref{TE}).
The solid line shows the case in which there is no ambipolar drift heating, for
which IIK06 report that a two-phase structure is possible for $10^{2.8}$ K
cm$^{-3} < p/k_B < 10^{4.1}$ K cm$^{-3}$. The other lines illustrate the effects
of increasing the ambipolar drift heating rate at a fixed magnetic field
strength of $3 \hspace{0.1cm} \mu$G. Although we have already shown
$\Gamma_{AD}$ to be a function of the density and field strength, for the
purposes of this plot we have set it to be a constant fraction of the photoelectric
heating rate, $\Gamma_{\rm PAH}$. Increasing the total heating rate serves to
increase the pressure at which two phases can co-exist: the minimum pressure at
which the cold phase can exist, and the maximum pressure at which the warmer
phase can exist, both increase. In fact, the pressure range over which two
phases can exist becomes larger as the total heating is increased. For example,
for the $\Gamma_{\rm AD}/\Gamma_{\rm PAH} = 0.50$ case plotted in Figure \ref{te},
two-phase structure is possible for $10^{3.0}$ K cm$^{-3} < p/k_B < 10^{4.3}$ K
cm$^{-3}$, and for the $\Gamma_{\rm AD}/\Gamma_{\rm PAH} = 1.00$ case the pressure range
is $10^{3.1}$ K cm$^{-3} < p/k_B < 10^{4.4}$ K cm$^{-3}$. We can understand the
shift towards lower densities as follows: increasing the heating increases the
temperature, so it must decrease the density. The upshift of the equilibrium to
higher pressures also reflects the increased heating.
\section{FRONT SOLUTIONS}
The characteristics of a front are determined by its thermal pressure. There
exists a ``saturation pressure'' at which heating and cooling are balanced
within a front (Zel'dovich \& Pikel'ner 1969, Penston \& Brown 1970). If
$\Lambda$ and $\Gamma$ can be written as functions of pressure and temperature
(where $\Gamma$ is the total heating rate), this pressure may be calculated by
solving the integral
\citep{IIK06}:
\begin{equation}
\int_{T_1}^{T_2} \kappa \rho \cal{L} \rm dT =
\int_{T_1}^{T_2} \kappa n (n \Lambda - \Gamma) dT = 0
\label{psat}
\end{equation}
and substituting for $n$ using the equation of state, Equation (\ref{eos}).
For the hydrodynamic case ($\Gamma_{AD} = 0$) IIK06 obtain $p_{sat}/k_B = 2612$ K
cm$^{-3}$ (where $k_B$ is the Boltzmann constant), which, by solving Equation (\ref{TE}), implies an initial density of
$n = 106.08$ cm$^{-3}$ and hence an initial temperature of $T = 24.63$ K. If the
thermal pressure exceeds this value of $p_{sat}$ a fluid element passing through
the front experiences net cooling, so we have a condensation front. If instead
the thermal pressure is less than the saturation value a fluid element
experiences net heating, so we have an evaporation front.
In this section we demonstrate the effect of ambipolar diffusion on the
saturation pressure and present our ambipolar diffusion-mediated front
solutions. We also argue that the flux-freezing approximation is not accurate
for steady-state thermal fronts.
\subsection{Effects of Ambipolar Drift Heating on Saturation Pressure}
The saturation pressure is altered in the presence of a magnetic field due to
ambipolar drift heating. The integral given by Equation (\ref{psat}) cannot be
solved analytically when $\Gamma_{\rm AD}$ is non-zero, so instead we use our
shooting method, as discussed in \S2, to find the initial density that yields a
static solution as a function of the initial magnetic field strength gradient.
The results for initial magnetic field strengths of $1$, $3$, and $5
\hspace{0.1cm} \mu$G are shown in Figure \ref{Bpsat}. Note that the field
gradients are actually negative, as we anticipate the magnetic field strength to
decrease with increasing temperature, and we refer to the absolute magnitude of
the quantity, which we give in units of $\mu$G pc$^{-1}$.
As $|dB/dx|_{x_1}$ is
increased the saturation density and pressure for all magnetic field strengths
initially decrease until a sufficiently large value of $|dB/dx|_{x_1}$ is
reached, after which the density and pressure both increase. Therefore it is
possible to have two different fronts at the same saturation pressure. This
nonmonotonic behavior may be understood by solving Equation (\ref{psat}) for the saturation
pressure using the equation of state, Equation (\ref{eos}), to obtain:
\begin{equation}
\frac{p_{sat}}{k_B} = \frac{\Gamma \int_{T_1}^{T_2} \frac{\kappa}{T} dT}
{\int_{T_1}^{T_2} \frac{\kappa \Lambda}{T^2} dT} .
\label{psat2}
\end{equation}
This shows that increasing the total heating rate, $\Gamma$,
tends to increase the saturation pressure. But increased heating also tends to
drive up the temperature, which for CNM temperatures greatly increases the
cooling rate, $\Lambda$, and according to Equation (\ref{psat2}) this decreases the
saturation pressure. For example, Figure \ref{Bpsat} shows that if $B_0 = 5
\hspace{0.1cm} \mu$G and $|dB/dx|_{x_1} = 300 \hspace{0.1cm} \mu$G pc$^{-1}$ the CNM temperature
is increased from $24$ to $28$ K. According to Equation (\ref{lambdacool}), this
results in a greater than $70 \%$ increase in the cooling rate, $\Lambda$.
Such a large increase
in cooling requires a lower density and a lower saturation pressure for
equilibrium to be maintained. This effect dominates as long as the heating and
cooling rates, $\Gamma$ and $\Lambda$, are not too large and is the reason for
the dip in the saturation pressure seen in Figure \ref{Bpsat}.
An inflection point is not observed in the saturation pressure in the $1
\hspace{0.1cm} \mu$G case, the reason being that at higher values of
$|dB/dx|_{x_1}$ (and hence larger ambipolar heating rates) the magnetic field
profile is so steep that a thermal equilibrium phase cannot be reached before
the magnetic field strength becomes negative. In such instances the temperature
on the cold side is still well within the range of CNM values, so it is not the
medium being overheated that prohibits physical front solutions.
We present example saturation fronts having a thermal pressure of $p_{th}/k_B =
2500$ K cm$^{-3}$ and an initial field strength of $3 \hspace{0.1cm} \mu$G, but
with different initial values of $|dB/dx|_{x_1}$, in Figure \ref{doublestatic}.
Note that the values of $|dB/dx|_{x_1}$ given represent the largest gradients at
any point throughout the front. The magnetic field strength gradients of all the
fronts we present quickly relax to become much smaller than the initial values
that we impose. The front having the larger value of $|dB/dx|_{x_1}$ has a
higher ambipolar drift heating rate and connects a lower density, higher
temperature CNM with a higher density, cooler WNM than the static front with the
lower heating rate. The front with the lower ambipolar drift heating rate is the
most diffusive, which is illustrated by its flatter magnetic field strength
profile. This is also indicated by the ratio of the field strength to the number
density, which shows a larger variation across the domain than the same quantity
for the static front with the higher heating rate.
\subsection{Ambipolar Diffusion-Mediated Front Solutions}
As stated at the beginning of \S 4, in the hydrodynamic case a static front has
a thermal pressure of $p_{sat}/k_B = 2612$ K cm$^{-3}$, which corresponds to an
initial density and temperature of $n = 106.08$ cm$^{-3}$ and $T = 24.63$ K,
respectively. To demonstrate the effects of ambipolar diffusion we present
several front models having this same initial density and an initial magnetic
field strength of $3 \hspace{0.1cm} \mathrm{\mu G}$ in Figure \ref{prof1}. The
initial temperature changes according to the ambipolar drift heating rate, so is
not the same as in the hydrodynamic case. The different models correspond to
various initial magnetic field strength gradients, $|dB/dx|_{x_1}$, where a
larger gradient corresponds to increased heating. The properties of the phases
connected by these fronts are listed in Table 1.
The overall shapes of the temperature profiles, shown in the top left panel of
Figure \ref{prof1}, are fairly similar with the main differences being the
temperature gradients on small scales and the final temperatures of the warm
phases becoming lower as $|dB/dx|_{x_1}$ is increased. The main effect of
increasing $|dB/dx|_{x_1}$ is that the size of the integration domain required
to reach thermal equilibrium at the right-hand boundary becomes smaller, due to
the increased ambipolar drift heating. In fact, the lowest $|dB/dx|_{x_1}$
profile shown here is very similar to the hydrodynamic solution of IIK06. The
top right panel of Figure \ref{prof1} shows that the density varies by more than
two orders of magnitude across the front for all heating rates. As
$|dB/dx|_{x_1}$ is increased the density of the warm phase at the far boundary
increases and hence the temperature decreases.
For insight into the actual nature of fronts, one may begin by looking at the
bulk velocity profiles, shown in the bottom left panel of Figure \ref{prof1}. The
effect of the initial magnetic field strength gradient on the velocity profile
of a front is not straightforward. For the lowest initial $|dB/dx|_{x_1}$ case
shown the velocity profile is flat and close to zero, as should be the case for
a static front. As $|dB/dx|_{x_1}$ is increased the velocity at first becomes
larger and negative. This is because the saturation pressure is altered from the
original hydrodynamic value of $p_{sat}/k_B = 2612$ K cm$^{-3}$, as we discussed
in \S4.1. The models shown here with negative velocity profiles are actually
condensation fronts. However, as $|dB/dx|_{x_1}$ is further increased there
comes a point when the velocity no longer becomes increasingly negative, and
instead begins to increase. Eventually the front transitions from being a
condensation front to an evaporation front, which is illustrated by the positive
velocity profile of the largest $|dB/dx|_{x_1}$ model shown in Figure
\ref{prof1}. This is expected because of the nonmonotonic behavior of the
saturation pressure as the ambipolar heating rate is increased (see Figure
\ref{Bpsat}).
The magnetic field strength profile of the lowest value $|dB/dx|_{x_1}$ model,
given in the lower right panel of Figure \ref{prof1}, is extremely flat. As
$|dB/dx|_{x_1}$ is increased the field strength decreases across the domain in
an almost linear fashion; for sufficiently large values the profile becomes
nonlinear. Given that the change in density across a front is much more dramatic
than that of the magnetic field strength, the ratio of the magnetic field
strength to the number density of neutrals, $B/n$, changes markedly throughout
the transition layer.
In Figure \ref{driftfig} we plot the plasma velocity profile, given by $v_p
\approx v + v_D$ and Equation (\ref{drift}), of each of the front models of
Figure \ref{prof1}. The shapes of the profiles are governed by the behavior of
the magnetic field strength. The plasma velocity is
almost constant across the lowest $|dB/dx|_{x_1}$ model since the magnetic field
strength profile is close to flat, whereas the larger $|dB/dx|_{x_1}$ models show more
variation in their plasma velocity profiles due to the presence of significant
gradients in the magnetic field. In all cases the drift velocity is positive and
larger than the bulk velocity of the front, such that the plasma velocity is also
positive.
In Figure \ref{heat} we compare the ion-neutral drift heating rate, given by
Equation (\ref{adheat}), to that of photoelectric heating, given by Equation
(\ref{gammapah}), for each of the front models of Figure \ref{prof1}. The lowest
$|dB/dx|_{x_1}$ model has a much smaller ambipolar drift heating rate than
photoelectric heating rate which is why the structure of that front is barely
different from the hydrodynamic case. The three larger $|dB/dx|_{x_1}$ fronts
have larger ambipolar heating rates that are comparable to the photoelectric
heating rate. These fronts depart more noticeably from the hydrodynamic
solution and are less diffusive.
We also investigate the effect of magnetic field strength on front profiles at
fixed initial $|dB/dx|_{x_1}$. Figure \ref{Bfig} shows a variety of front
characteristics for inital field strengths of $1$, $3$, and $5 \hspace{0.1cm}
\mu$G and $|dB/dx|_{x_1} = 308.6 \hspace{0.1cm} \mu$G pc$^{-1}$. The temperature
profiles are very similar, with the effect of increasing the field strength
being larger temperature gradients at small scales and thinner fronts. The effect on
the density
profile is that the higher magnetic field strength fronts connect warm phases with higher
densities. The velocity profiles are slightly negative, which implies that these
are actually condensation fronts, and the departure from a static solution seems
to increase with increasing field strength. Higher field strength solutions have
flatter magnetic profiles because the efficiency of ambipolar diffusion
increases with magnetic field strength.
In Figure \ref{Bdriftfig} we plot the plasma velocity profiles of each of the front
models of Figure \ref{Bfig}. The size of the plasma velocity increases with
increasing magnetic field strength, and the shape of the profile becomes
flatter. This is also due to the higher efficiency of ambipolar diffusion at
larger magnetic field strengths.
In Figure \ref{Bheat} we compare the
photoelectric and ambipolar heating rates for the front models shown in Figure
\ref{Bfig}. For these particular cases the photoelectric heating rate is larger than
the ambipolar heating rate for all field strengths, and the ambipolar heating
rate increases with magnetic field strength.
\subsection{Flux-Freezing Approximation}
For completeness, we also present the flux-freezing approximation, in which the
behavior of the magnetic field is tied to the density such that $B/\rho$ is
constant in 1-D. This result can be obtained by computing the total derivative
of the quantity $B/\rho$ using the continuity and induction equations. To
calculate the structure of a front in this approximation, we solve Equations
(\ref{nrg2}) and (\ref{vev}) and everywhere replace $B$ by $\rho\cal{C}$, where
$\cal{C}$ is a constant. Including the definition $z \equiv dT/dx$, we have a
system of three ODEs, which we solve using our shooting method, with the mass
flux, $j$, the parameter to be adjusted. We impose the boundary conditions given
by equations (\ref{Tbc}) and (\ref{dTbc}), with no need for a condition on the
magnetic field strength since its behavior is governed by that of the density.
Figure \ref{ff} shows solutions for an initial density of $n = 106.08$ cm$^{-3}$
at various initial magnetic field strengths. As the field strength is increased
the front becomes thinner and the transition reaches a progressively lower
temperature, higher density final state at the right-hand boundary. Both the
density and magnetic field strength span more than two orders of magnitude from
one phase to the other. While such a range of densities is routinely observed in
the neutral ISM, such widely varying magnetic field strengths are not (e.g.
Troland \& Heiles 1986), and this provides the first indication that the
flux-freezing approximation is not suitable for our problem.
We go on to use these results to calculate ambipolar drift velocities, using
Equation (\ref{drift}), and heating rates, given by Equation (\ref{adheat}),
throughout the front. These are plotted in the lower two panels of Figure
\ref{ff}. For the most extreme case shown, a saturated front with an initial
density of $106.08$ cm$^{-3}$ and an initial magnetic field strength of $5
\hspace{0.1cm} \mu G$, we obtain a maximum drift velocity of $19.4$ km s$^{-1}$
and a maximum heating rate of $1.9 \times 10^{-21}$ ergs \hspace{0.1cm} s$^{-1}$
cm$^{-3}$, three orders of magnitude greater than the photoelectric heating rate. Although
the equation for drift velocity breaks down for cases in which it is supersonic
we may still employ it to show that if the flux-freezing approximation held, the
drift velocities and heating rates would be enormous. Such an outcome is not
self-consistent with the rest of the model, and allows us to argue that the
solutions must be closer to what we have already presented, with the magnetic
field strength almost constant over the extent of the front for cases in which
ambipolar drift heating does not dominate. We thus suggest that by the time
steady-state fronts are established in the neutral ISM the flux-freezing
approximation does not apply.
\section{DISCUSSION}
We have shown the magnetic field strength profiles of fronts having ion-neutral
drift heating rates much smaller than the photoelectric heating rate to be almost
flat. In this section we argue that it is the thin extent of these fronts that
mediates the leakage of the magnetic field by ambipolar diffusion. We begin by
using our results to justify our steady-state and ionization equilibrium
assumptions. The minimum flow time through a front is of order $\tau_{flow} \sim
0.01$ km s$^{-1} / 0.1$ pc $\sim 10$ Myr (refer to Figure \ref{prof1}). This
should be compared to the ion-neutral collision time, given by $\tau_{in} \sim
(\rho_n \gamma_{AD})^{-1} \sim 15.8 / n_n$ yr. Thus, we have
$\tau_{in}/\tau_{flow} \ll 1$ so are safe in our steady-state formulation of
ambipolar diffusion. The
assumption of ionization equilibrium is scrutinized by comparing $\tau_{flow}$
to the recombination time for hydrogen, given by $\tau_{rec} \sim
1/\alpha^{(2)} n$, where $\alpha^{(2)} \sim 2.06 \times 10^{-11} T^{-1/2}$
cm$^{3}$ s$^{-1}$ \citep{Spitzer}. For a front with an initial density of
$106.08$ cm$^{-3}$ we calculate a recombination time of $\sim 70$ yr on the cold
side, and on the warm side we obtain $\sim 5000$ yr. For all our other front
models we also find $\tau_{rec}/\tau_{flow} \ll 1$, so for this work our simple
single-fluid treatment of ambipolar diffusion will suffice.
We now present a diffusive description of fronts in which we compare the thermal
and ambipolar diffusivities. Taking $U = n k_B T$ to be the energy density, we
write the thermal timescale as $\tau_{th} = U/\rho \cal{L} \rm$, and the thermal
diffusivity as $\lambda_{th} = \kappa T/U$, such that the characteristic length
scale of the problem, the Field length, is given by $l_F = \sqrt{\lambda_{th}
\tau_{th}}$ (Field 1965; Begelman \& McKee 1990). Hence, the thermal timescale
and flow velocity may be written in terms of the thermal diffusivity, such that
$\tau_{th} \sim l_F^2/\lambda_{th}$ and $v_{th} \sim \lambda_{th}/l_F$,
respectively. In the magnetic field case, the field is redistributed diffusively,
with ambipolar diffusivity, $\lambda_{AD} = v_A^2 \tau_{ni}$, where $\tau_{ni}$
is the neutral-ion collision time, approximated by $\tau_{ni} \sim 1.58 \times
10^3/n_i$ yr \citep{PZN00}. Comparing the thermal and ambipolar diffusivities we
obtain:
\begin{equation}
\frac{\lambda_{\rm th}}{\lambda_{\rm AD}} = \frac{\kappa} {n k_B v_A^2 \tau_{ni}}
\sim 10^{-2} \frac{n_i T^{1/2}}{B_{\mu}^2} ,
\label{diffuse}
\end{equation}
where $B_{\mu}$ is the field strength in units of $\mu$G. We compute Equation
(\ref{diffuse}) at both boundaries of our front models and for all cases obtain
$\tau_{AD}/\tau_{th} \ll 1$. For example, for a front with an initial density
and magnetic field strength of $106.08$ cm$^{-3}$ and $5 \hspace{0.1cm} \mu$G,
respectively, and $|dB/dx|_{x_1} = 308.6 \hspace{0.1cm} \mu$G pc$^{-1}$, we
obtain $\tau_{AD}/\tau_{th} \sim 4.9 \times 10^{-5}$ on the cold side, and
$\tau_{AD}/\tau_{th} \sim 4.0 \times 10^{-4}$ on the warm side. This means the
drift time is always much smaller than the time to flow through the front,
suggesting that the field has time to become close to uniform \footnote[1]{Note
that the value of $|dB/dx|_{x_1}$ enters into this estimate only insofar as it
affects the equilibrium temperature and front structure.}.
Our results show that increasing the ambipolar heating rate changes the
structure of our front solutions. By balancing the ambipolar and photoelectric
heating rates, Equations (\ref{gammapah}) and (\ref{adheat}), and approximating
the magnetic field strength gradient as $B_0/L_{B crit}$, we can estimate the
critical length scale at which the magnetic field becomes important in
determining structure:
\begin{equation}
L_{B crit}=\Bigg(\frac{\lambda_{\rm AD} B_0^2}{4 \pi n \Gamma_{\rm PAH}}\Bigg)^{1/2} .
\end{equation}
The magnetic length scale is given by $L_B \sim B/|\nabla B|$, so if $L_B >
L_{B crit}$ the effect of the field on the structure of a front is small. We
compare $L_B$ and $L_{B crit}$ in Figure \ref{lb} for a front with an initial
density of $n \sim 106.08$ cm$^{-3}$ and initial field strengths of $B = 1$,
$3$, and $5 \hspace{0.1cm} \mu$G, with an initial field strength gradient of
$|dB/dx|_{x_1} = 308.6 \hspace{0.1cm} \mu$G pc$^{-1}$. For the $5 \hspace{0.1cm}
\mu$G case we obtain $L_B \sim 1.6 \times 10^{-2}$ pc and $L_{B crit} \sim 5.4
\times 10^{-3}$ pc on the cold side, and on the warm side we find $L_B \sim
16.3$ pc and $L_{B crit} \sim 2.9$ pc. The magnetic length scale is larger than
the critical scale throughout the front, thus ambipolar drift heating does not
have a dramatic effect on the structure of a front.
Previous dynamical studies have claimed that ion-neutral drift is not a
sufficiently fast diffusion process for transporting magnetic energy, and
instead invoked turbulent ambipolar drift \citep{HZ04} or turbulent magnetic
reconnection \citep{SL10} to explain the $B-\rho$ relation. However, these
studies were on larger scales than the fronts considered here. Our results
suggest that for this simple scenario in which the phase transitions are thin,
ambipolar diffusion alone is a sufficient mechanism for redistributing the
magnetic field energy, without the need for turbulence. Our work directly
complements a study of the thermal instability as a formation mechanism for
diffuse \ion{H}{1} clouds \citep{IIK07}. In that work it was shown that ambipolar diffusion
is a necessary and sufficient ingredient for the formation of a two-phase medium. Once that
medium is established the methods discussed in this paper may be applied to calculate its
structure.
\section{SUMMARY AND CONCLUSIONS}
In this work we have investigated the effect of magnetic fields on two-phase
structure in the neutral ISM. We have presented a numerical method for
calculating the 1-D structure of fronts separating the cold neutral medium from
the warm neutral medium, including the effects of ambipolar diffusion. We showed
that the pressure range over which two-phase structure is permitted becomes
larger, by as much as a factor of two, due to the contribution of ambipolar drift
heating,
with both the minimum and maximum pressures increasing from their hydrodynamic
values. We find our magnetized front profiles to be very similar to the
hydrodynamic solutions, and, in cases where photoelectric heating dominates
ambipolar drift heating, to have close to flat magnetic field strength profiles.
We also showed that the flux-freezing assumption yields unphysically large drift
velocities and frictional heating rates. Our method is generic and, by including
the appropriate physics, may be extended to other astrophysical multi-phase
systems.
Although the 1-D picture discussed in this work is fairly simple, if the
magnetic field strength and density were related we would have expected to see a
correlation. Our results are consistent with the observational evidence that
there is no relationship between magnetic field strength and density in
interstellar atomic gas, which suggests that ambipolar diffusion is an efficient
transport mechanism in the neutral ISM. The effect of ambipolar diffusion on the
stability properties of thermal fronts will be the subject of forthcoming
publications.
\acknowledgments
We acknowledge support from NASA ATP Grant NNGO5IGO9G and NSF Grant PHY0821899.
Our work has benefited from useful discussions with J. E. Everett, F. Heitsch,
K. M. Hess, A. S. Hill, N. A. Murphy, L. M. Nigra, and R. H. D. Townsend. We
thank the referee for useful comments that enabled us to improve our manuscript.
\newpage
|
{
"timestamp": "2010-09-22T02:00:14",
"yymm": "1009",
"arxiv_id": "1009.3926",
"language": "en",
"url": "https://arxiv.org/abs/1009.3926"
}
|
\section{Introduction}
In this paper, we consider the global existence for small data for a semilinear equation with null condition on a Kerr spacetime. Kerr spacetimes are stationary axisymmetric asymptotically flat black hole solutions to the vacuum Einstein equations
$$R_{\mu\nu}=0$$
in $3+1$ dimensions. They are parametrized by two parameters $\left(M, a\right)$, representing respectively the mass and the angular momentum of a black hole. We study semilinear equations on a Kerr spacetime with $a\ll M$ of the form
$$\Box_{g_K}\Phi=F(D \Phi),$$
where $\Box_{g_K}$ is the Laplace-Beltrami operator for the Kerr metric $g_K$, and $F$ denotes nonlinear terms that are at least quadratic and satisfy the null condition that we will define in Section \ref{theorem}.
The corresponding problem on Minkowski spacetime has been well studied. In 4+1 or higher dimension, the decay of the linear wave equation is sufficiently fast for one to prove global existence of small data for nonlinear wave equations with any quadratic nonlinearity \cite{Kl}. However, in 3+1 dimensions, which is also the dimension of physical relevance, the decay rate is only sufficient to prove the almost global existence of solutions \cite{JoK}. Indeed, a counter-example is known \cite{John} for the equation
$$\Box_m\Phi=(\partial_t\Phi)^2.$$
Nevertheless, if the quadratic nonlinearity satisfies the null condition defined by Klainerman, it has been proved independently by Christodoulou \cite{Chr} and Klainerman \cite{Knull} that any solutions to sufficiently small initial data are global in time. There have been an extensive literature on extensions and variations of the original results, including the cases of the multiple-speed system and the exterior domains (\cite{ST}, \cite{Sogge}, \cite{MNS}, \cite{MSo}).
The decay rate of the solutions to the linear wave equation on Kerr spacetimes with $a\ll M$ have been proved in \cite{DRL}, \cite{AB}, \cite{Ta} and \cite{LKerr}. The known decay outside the set $\{ct^*\leq r\leq Ct^*\}$ is sufficiently strong and the proof of them (in \cite{DRL}, \cite{AB} and \cite{LKerr}) is sufficiently robust that one expects the main obstacle from proving a small data global existence result (if it indeed holds) would come from quantities in the set $\{ct^*\leq r\leq Ct^*\}$. This set, however, approaches the same set in Minkowski spacetime as $t^*\to\infty$ due to the asymptotic flatness of Kerr spacetimes. Therefore, one expects that with a null condition similar to that on the Minkowski spacetime, a similar global existence result holds. Indeed, we have (see the precise version in Section \ref{theorem})
\begin{maintheorem}
Consider $\Box_{g_K}\Phi= F(D\Phi)$ where $F$ satisfies the null condition (see Section \ref{theorem}). Then for any initial data that are sufficiently small, the solution exists globally in time.
\end{maintheorem}
Our major motivation in studying the null condition on a Kerr spacetime is the problem of the stability of the Kerr spacetime. It is conjectured that Kerr spacetimes are stable. In the framework of the initial value problem, the stability of Kerr would mean that for any solution to the vacuum Einstein equations with initial data close to the initial data of a Kerr spacetime, its maximal Cauchy development has an exterior region that approaches a nearby, but possibly different, Kerr spacetime. In the case of the Minkowski spacetime, the null condition has served as a good model problem for the study of the stability of the Minkowski spacetime. We hope that this work will find relevance to the problem of the stability of the Kerr spacetime.
\subsection{Some Related Known Results}
We turn to some relevant work on linear and nonlinear scalar wave equations on Kerr spacetimes. The decay of solutions to the linear wave equation on Kerr spacetimes has received considerable attention. We mention some results on Kerr spacetimes with $a>0$ here and refer the readers to \cite{DRL}, \cite{LS} for references on the corresponding problem on Schwarzschild spacetimes. There has been a large literature on the mode stability and non-quantitative decay of azimuthal modes (See for example \cite{PT}, \cite{HW}, \cite{Wh}, \cite{FKSY}, \cite{FKSY2} and references in \cite{DRL}). The first global result for the Cauchy problem was obtained by Dafermos-Rodnianski in \cite{DRK}, in which they proved that for a class of small, axisymmetric, stationary perturbations of Schwarzschild spacetime, which include Kerr spacetimes that rotate sufficiently slowly, solutions to the wave equation are uniformly bounded. Similar results were obtained later using an integrated decay estimate on slowly rotating Kerr spacetimes by Tataru-Tohaneanu \cite{TT}. Using the integrated decay estimate, Tohaneanu also proved Strichartz estimates \cite{To}.
Decay for general solutions to the wave equation on sufficiently slowly rotating Kerr spacetimes was first proved by Dafermos-Rodnianski \cite{DRL} with a quantitative rate of $|\Phi|\leq C(t^*)^{-1+Ca}$. A similar result was later obtained by \cite{AB} using a physical space construction to obtain an integrated decay estimate. In all of \cite{TT}, \cite{DRL} and \cite{AB}, the integrated decay estimate is proved and plays an important role. All proofs of such estimates rely heavily on the separability of the wave equation, or equivalently, the existence of a Killing tensor on Kerr spacetime. In a recent work \cite{DRNPS}, Dafermos-Rodnianski prove the non-degenerate energy decay and the pointwise decay assuming the integrated local energy decay estimate and boundedness for the wave equation on an asymptotically flat spacetime. Their work shows a decay rate of $|\Phi|\leq Ct^{-1}$ and improves the rates in \cite{DRL} and \cite{AB}. In a similar framework, but assuming in addition exact stationarity, Tataru \cite{Ta} proved a local decay rate of $(t^*)^{-3}$ using Fourier-analytic methods. This applies in particular to sufficiently slowly rotating Kerr spacetimes. Dafermos and Rodnianski have recently announced a proof for the decay of solutions to the wave equation on the full range of sub-extremal Kerr spacetimes $a< M$.
For nonlinear equations, global existence for the equation with power nonlinearity $\Box_{g_k}\Phi=\pm |\Phi|^p \Phi$ was initiated in \cite{NiS} and \cite{NiK}, in which the large data subcritical defocusing case of $p=2$ is studied. Later, there have been much work on the small data problem in which the sign of the nonlinearity is not important, and that the dispersive properties of the linear equation plays a crucial role. Global existence was proved for small radial data for $p>3$ on Reissner-Nordstrom spacetime \cite{DRNL} and for general small data vanishing on the bifurcate sphere for $p>2$ \cite{BSt} on Schwarzschild spacetime. Global existence was also proved for $p=4$ on Schwarzschild spacetime with general data that has small non-degenerate energy \cite{MMTT}. This was extended to the case of sufficiently slowly rotating Kerr spacetime in \cite{To}. Counterexample is known for the case $0<p<\sqrt{2}$ \cite{CG}. To our knowledge, the present work is the first work on semilinear equations with derivatives on black hole spacetimes.
\subsection{The Statement of the Main Theorem}\label{theorem}
Before introducing the null condition and stating the precise version of the Main Theorem, we briefly introduce the necessary concepts and notations on Kerr geometry and the vector field method. See \cite{LKerr} for more details.
The Kerr metric in the Boyer-Lindquist coordinates takes the following form:
\begin{equation}\label{kerrmetric}
\begin{split}
g_K=&-\left(1-\frac{2M}{r\left(1+\frac{a^2 \cos ^2{\theta}}{r^2}\right)}\right)dt^2+\frac{1+\frac{a^2\cos ^2 \theta}{r^2}}{1-\frac{2M}{r}+\frac{a^2}{r^2}}dr^2+r^2\left(1+\frac{a^2\cos ^2 \theta}{r^2}\right)d\theta^2 \\
&+r^2\left(1+\frac{a^2}{r^2}+\left(\frac{2M}{r}\right)\frac{a^2\sin ^2\theta}{r^2\left(1+\frac{a^2\cos ^2\theta}{r^2}\right)}\right)\sin ^2\theta d\phi^2
-4M \frac{a \sin ^2\theta}{r\left(1+\frac{a^2\cos ^2\theta}{r^2}\right)}dtd\phi.
\end{split}
\end{equation}
Let $r_+$ be the larger root of $\Delta=r^2-2Mr+a^2$. $r=r_+$ is the event horizon. In this paper, we will use the coordinate system $(t^*,r,\theta,\phi^*)$ defined by
$$t^*=t+\chi(r)h(r),\quad \mbox{where }\frac{dh(r)}{dr}=\frac{2Mr}{r^2-2Mr+a^2},$$
$$\phi^*=\phi+\chi(r)P(r),\quad \mbox{where }\frac{dP(r)}{dr}=\frac{a}{r^2-2Mr+a^2},$$
where
$$\chi(r)=\left\{\begin{array}{clcr}1&r\le r^-_Y-\frac{r^-_Y-r_+}{2}\\0&r\ge r^-_Y-\frac{r^-_Y-r_+}{4}\end{array}\right.,$$
where $r_+$, as above, is the larger root of $\Delta=r^2-2Mr+a^2$ and $r^-_Y>r_+$ is a fixed constant very close to $r_+$, the value of which can be determined from the proof of the energy estimates in \cite{LKerr}. Following the notation in \cite{LKerr}, we will use $t^*=\tau$ to denote the $t^*$ slice on which we want to prove estimates and $t^*=\tau_0$ to denote the $t^*$ slice on which the initial data is posed.
In \cite{LKerr}, following \cite{DRK}, various quantities are defined via an explicit identification of the Kerr spacetime with the corresponding Schwarzschild spacetime with the same mass. We recall the identification:
$$r_S^2-2Mr_S=r^2-2Mr+a^2,$$
$$t_S+\chi(r_S)2M\log\left(r_S-2M\right)=t^*,$$
$$\theta_S=\theta,$$
$$\phi_S=\phi^*,$$
where $\chi$ is as above.
Define
$$r^*_S=r_S+2M\log(r_S-2M)-3M-2M\log M,$$
$$\mu=\frac{2M}{r_S},$$
$$u=\frac{1}{2}(t_S-r^*_S),$$
$$v=\frac{1}{2}(t_S+r^*_S).$$
We note that the variable $u$ will also be used to quantify decay.
We define in coordinates
$$\underline{L}=\partial_u\mbox{, in the $(u,v,\theta_S,\phi_S)$ coordinates},$$
$$L=2\partial_{t^*}+\chi(r)\frac{a}{Mr_+}\partial_{\phi^*}-\underline{L}$$
We can now define the ``good'' and ``bad'' derivatives. Define
$${\nabla} \mkern-13mu /\,\in\{\frac{1}{r}\partial_{\theta},\frac{1}{r}\partial_\phi\},$$
$$\overline{D}\in\{L,\frac{1}{r}\partial_{\theta},\frac{1}{r}\partial_\phi\},$$
$${D}\in\{\frac{1}{1-\mu}\underline{L}, L,\frac{1}{r}\partial_{\theta},\frac{1}{r}\partial_\phi\}.$$
Notice that $D$ spans the whole tangent space and that we always have $[D,\partial_{t^*}]=0$.
We now define the null condition. On Minkowski spacetime, the classical null condition can be defined geometrically by requiring the nonlinearity to have the form
$$A^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi,$$
where $A$ satisfies $A^{\mu\nu}\xi_\mu\xi_\nu=0$ whenever $\xi$ is null. On Kerr spacetime, we would like to define a notion of the null condition that includes this geometric notion. This is also because many physically relevant semilinear equations satisfy this condition. On the other hand, in order to prove the global existence result, we need to use the vector fields that capture the good derivative. We would therefore like to define the null condition using the vector fields defined in \cite{DRL}, \cite{LKerr}, i.e., using $D$ and $\overline{D}$. In particular, we want the nonlinearity to have at least one good, i.e., $\overline{D}$, derivative. This on its own is however inconsistent with the geometric null condition. We therefore allow a term in the quadratic nonlinearity that does not have a good derivative but decays in $r$.
\begin{definition}
Consider the nonlinearity $F(\Phi,D\Phi,t^*,r,\theta,\phi^*).$ We say that $F$ satisfies the null condition if
$$F=\Lambda_0(\Phi,t^*,r,\theta,\phi^*)D\Phi\overline{D}\Phi+\Lambda_1(\Phi,t^*,r,\theta,\phi^*)D\Phi{D}\Phi+\mathcal C(\Phi,D\Phi,t^*,r,\theta,\phi^*),$$
where $$ |D_\Phi^{i_1}\partial_{t^*}^{i_2}\partial_r^{i_3}\partial_\theta^{i_4}\partial_{\phi^*}^{i_5}\Lambda_j|\leq C(t^*)^{-i_2}r^{-i_3} \quad\mbox{for } i_1+i_2+i_3+i_4+i_5\leq 16, j=0,1$$
and
$$|D_\Phi^{i_1}\partial_{t^*}^{i_2}\partial_r^{i_3}\partial_\theta^{i_4}\partial_{\phi^*}^{i_5}\Lambda_1|\leq C(t^*)^{-i_2}r^{-1-i_3} \quad\mbox{for } i_1+i_2+i_3+i_4+i_5\leq 16\mbox{ and }r\geq\frac{9t^*}{10}$$
and $\mathcal C$ denotes a polynomial that is at least cubic in $D\Phi$ (with coefficients in $\Phi,t^*,r,\theta,\phi^*$) satisfying
$$|D_\Phi^{i_1}\partial_{t^*}^{i_2}\partial_r^{i_3}\partial_\theta^{i_4}\partial_{\phi^*}^{i_5}\mathcal C|\leq C(t^*)^{-i_2}r^{-i_3}\sum_{s= 3}^S|D\Phi|^s \quad\mbox{for } i_1+i_2+i_3+i_4+i_5\leq 16$$
\end{definition}
\begin{remark}
The null condition is a special structure for the quadratic nonlinearity. We note that in our case, the restriction is necessary only for $r\geq \frac{9t^*}{10}$. Moreover, higher order terms should give better estimates and do not need any special structure.
\end{remark}
Under this definition of the null condition, global existence holds for small data. Moreover, the solution $\Phi$ satisfies pointwise decay estimates. In order to appropriately describe smallness, we introduce the language of compatible currents. Define the energy-momentum tensor
$$T_{\mu\nu}=\partial_\mu\Phi\partial_\nu\Phi-\frac{1}{2}g_{\mu\nu}\partial^\alpha\Phi\partial_\alpha\Phi.$$
By virtue of the wave equation, $T$ is divergence-free,
$$\nabla^\mu T_{\mu\nu}=0.$$
For a vector field $V$, define the compatible currents
$$J^V_{\mu}\left(\Phi\right)=V^{\nu}T_{\mu\nu}\left(\Phi\right),$$
$$K^V\left(\Phi\right)=\pi^V_{\mu\nu}T^{\mu\nu}\left(\Phi\right),$$
where $\pi^V_{\mu\nu}$ is the deformation tensor defined by
$$\pi^V_{\mu\nu}=\frac{1}{2}\left(\nabla_{\mu}V_{\nu}+\nabla_{\nu}V_{\mu}\right).$$
In particular, $K^V\left(\Phi\right)=\pi^V_{\mu\nu}=0$ if $V$ is Killing.
Since the energy-momentum tensor is divergence-free,
$$\nabla^{\mu}J^{V}_{\mu}\left(\Phi\right)=K^V\left(\Phi\right).$$
We also define the modified currents $$J^{V,w}_{\mu}\left(\Phi\right)=J^{V}_{\mu}\left(\Phi\right)+\frac{1}{8}\left(w\partial_{\mu}\Phi^2-\partial_{\mu}w\Phi^2\right),$$
$$K^{V,w}\left(\Phi\right)=K^V\left(\Phi\right)+\frac{1}{4}w\partial^{\nu}\Phi\partial_{\nu}\Phi-\frac{1}{8}\Box_g w\Phi^2.$$
Then $$\nabla^{\mu}J^{V,w}_{\mu}\left(\Phi\right)=K^{V,w}\left(\Phi\right).$$
In \cite{LKerr}, we have used the currents corresponding to $N$ and $(Z,w^Z)$ defined by
$$N=\partial_{t^*}+e\left(y_1\left(r\right)\hat{Y}+y_2\left(r\right)\hat{V}\right)$$
$$Z=u^2\underline{L}+v^2L,$$
$$w^Z=\frac{8tr^*_S\left(1-\frac{2M}{r_S}\right)}{r},$$
where
$$y_1\left(r\right)=1+\frac{1}{(\log(r-r_+))^3},$$
$$y_2\left(r\right)=\frac{1}{(\log(r-r_+))^3},$$
$$\Delta=r^2-2Mr+a^2,$$
and $\hat{Y}$ and $\hat{V}$ are compactly supported vector fields in a neighborhood of $\{r_+\leq r\leq r^-_Y\}$ and are null in $\{r_+\leq r\leq r^-_Y\}$ and $e$ is an appropriately small constant depending only on $a$ (See \cite{LKerr}).
Since $N$ is future-directed, we have the pointwise inequality $$J^N_\mu(\Phi) n^\mu_{\Sigma_{t^*}}\geq 0.$$ In \cite{LKerr} we have shown that there exists a constant $C$ such that
$$\int_{\Sigma_{t^*}}J^{Z,w^Z}_\mu(\Phi) n^\mu_{\Sigma_{t^*}}+C(t^*)^2\int_{\Sigma_{t^*}\cap\{r\leq r^-_Y\}}J^{N}_\mu(\Phi) n^\mu_{\Sigma_{t^*}}\geq 0.$$
These energy quantities will be used for $\Phi$ as well as derivatives of $\Phi$. We now define the commutators that we will used. $\partial_{t^*}$ is a Killing vector field that is defined as the coordinate vector field with respect to the $(t^*,r,\theta,\phi^*)$ coordinate system. Near the event horizon, we use the commutator $\hat{Y}$ which is compactly supported in $\{ r\leq r^+_Y\}$ (where $r^+_Y>r^-_Y$ is an explicit constant in \cite{LKerr}), null in $\{r_+\leq r\leq r^-_Y\}$ and is transverse to the event horizon (See \cite{LKerr}). $\hat{Y}$ has good positivity property that reflects the celebrated red-shift effect. In the region of large $r$, we use the commutators $\tilde{\Omega}$. Let $\Omega_i$ be a basis of vector fields of rotations in Schwarzschild spacetimes. An explicit realization can be $\Omega=\partial_\phi, \sin\phi\partial_\theta\pm \frac{\cos\phi\cos\theta}{\sin\theta}\partial_\phi$. Define $\tilde{\Omega}_i=\chi(r)\Omega_i$ to be cutoff so that it is supported in $\{r>R_\Omega\}$ and equals $\Omega_i$ for $r>R_\Omega+1$ for some large $R$. We also use the commutator $S$ that would provide an improved decay rate of the solution. It is defined as
$$S=t^*\partial_{t^*}+h(r_S)\partial_{r},$$ where $h(r_S)= \left\{\begin{array}{clcr}r_S-2M &r_S\sim 2M\\r_S^*(1-\mu )&r\ge R\end{array}\right.$, for some large $R$ and is interpolated so that it is smooth and non-negative. For the commutators, we also use the notation that
$$\Gamma\in\{\partial_{t^*},\tilde{\Omega}\}.$$
We are now in a position to state our Main Theorem precisely.
\begin{theorem}
Consider the equation
\begin{equation}\label{equation}
\Box_g\Phi=F(\Phi,D\Phi,t^*,r,\theta,\phi^*),
\end{equation}
where $F$ satisfies the null condition.
There exists an $\epsilon$ such that if the initial data of $\Phi$ satisfies
\begin{equation*}
\begin{split}
\sum_{i+j+k=16}\int_{\Sigma_{\tau_0}} J^{Z+CN,w^Z}_\mu\left(\hat{Y}^k\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_{\tau_0}} +\sum_{i+j+k=16}\int_{\Sigma_{\tau_0}} J^{Z+CN,w^Z}_\mu\left(\hat{Y}^k S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_{\tau_0}}\leq \epsilon
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
\sum_{\ell=0}^{13} r|D^\ell\Phi(\tau_0)|+ r|D^\ell S\Phi(\tau_0)|\leq \epsilon.
\end{split}
\end{equation*}
Then $\Phi$ exists globally in time.
Moreover, for all $\eta>0$, we can take $a$ sufficiently small such that the solution $\Phi$ obeys the decay estimate
$$|\Phi|\leq C\epsilon r^{-1}u^{-\frac{1}{2}}(t^*)^\eta, |D\Phi|\leq C\epsilon r^{-1}u^{-1}(t^*)^{\eta}, |\overline{D}\Phi|\leq C\epsilon r^{-1}(t^*)^{-1+\eta}\mbox{ for }r\geq R,\mbox{ and}$$
$$|\Phi|\leq C_\delta\epsilon (t^*)^{-\frac{3}{2}+\eta}r^{\delta}, |D\Phi|\leq C_\delta\epsilon (t^*)^{-\frac{3}{2}+\eta}r^{-\frac{1}{2}+\delta}\mbox{ for }r\leq \frac{t^*}{4}.$$
\end{theorem}
We specialize to a particular case which resembles better the classical null condition \cite{Knull}.
\begin{theorem}
Consider the equation
\begin{equation}\label{geometric null}
\Box_g\Phi=\Gamma(\Phi)A^{\mu\nu}\partial_{\mu}\Phi\partial_{\nu}\Phi,
\end{equation}
where $A$ satisfies $A^{\mu\nu}\xi_\mu\xi_\nu=0$ whenever $\xi\in TK$ is null. If the initial data of $\Phi$ satisfies \begin{equation*}
\begin{split}
\sum_{i+j+k=16}\int_{\Sigma_{\tau_0}} J^{Z+CN,w^Z}_\mu\left(\hat{Y}^k\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_{\tau_0}} +\sum_{i+j+k=16}\int_{\Sigma_{\tau_0}} J^{Z+CN,w^Z}_\mu\left(\hat{Y}^k S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_{\tau_0}}\leq \epsilon
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
\sum_{\ell=0}^{13} r|D^\ell\Phi(\tau_0)|+ r|D^\ell S\Phi(\tau_0)|\leq \epsilon.
\end{split}
\end{equation*}
Then $\Phi$ exists globally in time.
Moreover, for all $\eta>0$, we can take $a$ sufficiently small such that the solution $\Phi$ obeys the decay estimate
$$|\Phi|\leq C\epsilon r^{-1}u^{-\frac{1}{2}}(t^*)^\eta, |D\Phi|\leq C\epsilon r^{-1}u^{-1}(t^*)^{\eta}, |\overline{D}\Phi|\leq C\epsilon r^{-1}(t^*)^{-1+\eta}\mbox{ for }r\geq R,\mbox{ and}$$
$$|\Phi|\leq C_\delta\epsilon (t^*)^{-\frac{3}{2}+\eta}r^{\delta}, |D\Phi|\leq C_\delta\epsilon (t^*)^{-\frac{3}{2}+\eta}r^{-\frac{1}{2}+\delta}\mbox{ for }r\leq \frac{t^*}{4}.$$
\end{theorem}
The above formulation is geometric and independent of the choice of coordinates. We note that this condition is obviously satisfied by the wave map equation in the intrinsic formulation.
\subsection{The Case of Minkowski Spacetime}
We now turn to the outline of the proof of the main theorem. In the original proof in \cite{Knull}, many symmetries of Minkowski spacetime are captured and exploited using the vector field method. Kerr spacetime, on the other hand, lacks symmetries and this limits the set of vector fields that is at our disposal. In view of this, we would like to re-examine the proof of the small data global existence result for the nonlinear wave equation with a null condition on Minkowski spacetime, using only the vector fields whose analogues in Kerr spacetimes have been established in previous works. In particular, we would have to avoid using the Lorentz boost.
We first study the decay properties of the solutions to the linear wave equation on Minkowski spacetime. Since the vector field $T=\partial_t$ is Killing and $Z=u^2\partial_u+v^2\partial_v$ is conformally Killing, we have for $w=8t$ that
$$\int_{\Sigma_t}J^T_\mu(\Phi)n^\mu_{\Sigma_t}, \int_{\Sigma_t}J^{Z,w^Z}_\mu(\Phi)n^\mu_{\Sigma_t}$$
are conserved in time.
Decay can be proved using the above conserved quantities for $V\Phi$ for appropriate vector fields $V$. It is proved separately for $r\geq\frac{t}{2}$ and $r\leq\frac{t}{2}$. In the former case, we use the fact that $\Omega_{ij}=x_i\partial_{x_j}+x_j\partial_{x_i}$ is Killing on Minkowski spacetime and hence $\Box_m(\Omega^k\Phi)=0$. Since $\Omega$ has a weight in $r$, it can be proved that
$$|D\Phi|^2\leq Cr^{-2}\sum_{k=0}^2\int_{\Sigma_t} J^T_\mu(\Omega^k\Phi)n^\mu_{\Sigma_t}.$$
Notice that in this region $r^{-2}\leq Ct^{-2}$. It is known, for example by the representation formula, that this decay rate cannot be improved. In the region $r\leq\frac{t}{2}$, however, the decay rate is better. One can consider the conformal energy
$$\int J^{Z,w^Z}_\mu(\Phi)n^\mu_{\Sigma_t}\geq \int_{\Sigma_t} u^2\left(\underline{L}\Phi\right)^2+v^2\left(L\Phi\right)^2+\left(u^2+v^2\right)|{\nabla} \mkern-13mu /\,\Phi|^2+\left(\frac{u^2+v^2}{r^2}\right)\Phi^2,$$
where $u=\frac{1}{2}(t-r)$, $v=\frac{1}{2}(t+r)$.
In particular, we have $$|D\Phi|^2\leq t^{-2}\int_{\Sigma_t\cap\{r\leq\frac{t}{2}\}} \tau^2(D\Phi)^2\leq t^{-2}\int J^{Z,w^Z}_\mu(\Phi)n^\mu_{\Sigma_t}.$$
To improve the decay rate in this region, we can consider the equation for $S\Phi=(t\partial_t+r\partial_r)\Phi$ and use the integrated decay estimates as in \cite{LS}, \cite{LKerr}. This approach allows us to avoid the use of Lorentz boost in \cite{Knull} and the global elliptic estimates in \cite{KS}, both of which have no clear analogue in Kerr spacetimes. On Minkowski spacetime, a local energy decay estimate can be proved using the vector field $\left(1-\frac{1}{(1+r^2)^{\frac{1+\delta}{2}}}\right)\partial_r$ for the linear wave equation \cite{StR} which together with the conformal energy yields:
$$\int_{t}^{(1.1)t}\int_{\Sigma_{t'}\cap\{r\leq\frac{t'}{2}\}} r^{-1-\delta} J^T_\mu(\Phi)n^\mu_{\Sigma_{t'}} dt'\leq C\int_{\Sigma_\tau\cap\{r\leq\frac{t}{2}\}} (D\Phi)^2\leq Ct^{-2}\int J^{Z,w^Z}_\mu(\Phi)n^\mu_{\Sigma_t}.$$
This would imply that there exists a "dyadic" sequence $t_i\sim (1.1)^i t_0$ on which there is better decay
$$\int_{\Sigma_{t_i}\cap\{r\leq\frac{t}{2}\}} r^{-1-\delta} J^T_\mu(\Phi)n^\mu_{\Sigma_{t_i}}\leq Ct_i^{-3}\int J^{Z,w^Z}_\mu(\Phi)n^\mu_{\Sigma_t}.$$
Since $S$ is Killing on Minkowski spacetime, $\Box_m\Phi=0$ implies $\Box_m(S\Phi)=0$. Then the above argument would give
$$\int_{t}^{(1.1)t}\int_{\Sigma_{t}\cap\{r\leq\frac{t}{2}\}} r^{-1-\delta} J^T_\mu(S\Phi)n^\mu_{\Sigma_t} dt\leq t^{-2}\int J^{Z,w^Z}_\mu(S\Phi)n^\mu_{\Sigma_t}.$$
Since $S=t\partial_t+r\partial_r$ has a weight in $t$, we can integrate along the integral curves of $S$ from the "good" $t_i$ slice and get
$$\int_{\Sigma_{t}\cap\{r\leq\frac{t}{2}\}} r^{-1-\delta} J^T_\mu(\Phi)n^\mu_{\Sigma_{t}}\leq Ct^{-3}\int J^{Z,w^Z}_\mu(\Phi)n^\mu_{\Sigma_{t_0}}.$$
Together with the use of $\Omega$, we have the pointwise estimate
$$|D\Phi|^2\leq Cr^{-1+\delta}\sum_{k=0}^2\int_{\Sigma_{t}\cap\{r\leq\frac{t}{2}\}} r^{-1-\delta} J^T_\mu(\Omega^k\Phi)n^\mu_{\Sigma_{t}}\leq Cr^{-1+\delta}t^{-3}\sum_{k=0}^2\int J^{Z,w^Z}_\mu(\Omega^k\Phi)n^\mu_{\Sigma_{t_0}}.$$
We now study how this decay rate can be used for the nonlinear problem. The main idea is to prove the above conservation and decay estimates in a bootstrap setting, showing that the decay to the linear wave equation is sufficiently strong that the nonlinear terms can be treated as error. In this framework, the decay of $t^{-1}$ is borderline and since the decay rate is better when $r\leq\frac{t}{2}$, the difficulty arises when dealing with terms in the region $r\geq\frac{t}{2}$. Furthermore, in order to achieve this decay of $t^{-1}$ it is imperative to show that $\int_{\Sigma_t} J^T_\mu(\Phi)n^\mu_{\Sigma_t}$ is uniformly bounded in time. We now show a heuristic argument. With the inhomogeneous term, the conservation law for the energy now has the error term:
$$\int_{\Sigma_t} J^T_\mu(\Phi)n^\mu_{\Sigma_t}\leq \int_{\Sigma_{t_0}} J^T_\mu(\Phi)n^\mu_{\Sigma_{t_0}}+\left(\int_{t_0}^t\left(\int_{\Sigma_{t}} \left(\Box_m\Phi\right)^2\right)^{\frac{1}{2}}dt\right)^2,$$
and that for the conformal energy has the error term:
$$\int_{\Sigma_t} J^{Z,w^Z}_\mu(\Phi)n^\mu_{\Sigma_t}\leq \int_{\Sigma_{t_0}} J^{Z,w^Z}_\mu(\Phi)n^\mu_{\Sigma_{t_0}}+\left(\int_{t_0}^t\left(\int_{\Sigma_{t}} (t^2+r^2)\left(\Box_m\Phi\right)^2\right)^{\frac{1}{2}}dt\right)^2$$
Since $\Box_m\Phi$ is quadratic in $D\Phi$, we can use Holder's inequality on the inside integral to control one term in $L^2$ and one in $L^\infty$. However, since on the linear level $D\Phi$ is bounded in $L^2$ and decays as $t^{-1}$ in $L^\infty$, the inhomogeneous term for the estimate for the energy is controlled by
$$\left(\int_{t_0}^t (t)^{-1}(\int_{\Sigma_{t}} J^{T}_\mu(\Phi)n^\mu_{\Sigma_{t}})^{\frac{1}{2}} dt\right)^2.$$
This is barely insufficient to show that the energy is bounded. We therefore need to make use of the null condition. The null condition would allow one to prove
\begin{equation}\label{null}
\int (D\Phi\overline{D}\Phi)^2\leq C t^{-2}\int_{\Sigma_{t}} J^{Z,w^Z}_\mu(\partial^k\Phi)n^\mu_{\Sigma_{t}}
\end{equation}
In order to prove this estimate, we observe that in the conformal energy, the good derivatives ($\partial_v$, ${\nabla} \mkern-13mu /\,$) has better decay rates. In order to use this, we then need to control the conformal energy. Using again the null condition, the inhomogeneous term in the conservation law for the conformal energy can be bounded by
$$\left(\int_{t_0}^t t^{-1}(\int_{\Sigma_{t}} J^{Z,w^Z}_\mu(\Phi)n^\mu_{\Sigma_{t}})^{\frac{1}{2}} dt\right)^2.$$
This would not be sufficient to prove that the conformal energy is bounded, but is sufficient to prove that it grows no faster than $t^{\eta}$ for sufficiently small data. This in turn would be sufficient to prove the boundedness of the energy and obtain all the necessary decay rates. In practice, the argument is more complicated as we need to control the higher order energy and conformal energy in order to obtain the decay rates.
\subsection{The Case of Kerr Spacetime}
In \cite{DRL} and \cite{LKerr}, all the analogues of the above estimates have been proved in the linear setting in Kerr spacetimes. However, it is apparent from the linear case that several issues arise as we apply a similar strategy to the nonlinear problem on Kerr spacetime.
Among other issues, two difficulties loom large. The first of these is the lack of symmetries in Kerr spacetimes. While Kerr spacetimes possess the Killing vector field $\partial_{t^*}$, it is spacelike in a neighborhood of the event horizon and thus does not give a non-negative conserved quantities. The works \cite{DRK}, \cite{DRL} suggest that we can instead use the vector fields $N$ and $Z$ on Kerr spacetime as substitutes for $T$ and $Z$ on Minkowski spacetime. $N$ in constructed as the Killing vector field $\partial_{t^*}$ added to a small amount of the red-shift vector field near the event horizon. The red-shift vector field, first introduced in \cite{DRS}, takes advantage of the geometry of the event horizon and has been used crucially to obtain decay rates in \cite{DRS}, \cite{DRK}, \cite{DRL}, \cite{LS}, and \cite{LKerr}. It is one of the few stable features of the Schwarzschild spacetime. The vector field $Z$ approaches the corresponding $Z$ on Minkowski spacetime at the asymptotically flat end and has the weights in $r$ and $t^*$ from which we can prove decay. These vector fields, however, do not correspond to any symmetries of Kerr spacetimes, and therefore, as already is apparent in the linear scenario, the energy estimates would contain error terms that need to be controlled. One consequence is that even in the linear setting, the conformal energy is not bounded. Similar issues arise for the vector field commutators $\Omega$ and $S$, which are crucial in obtaining pointwise decay estimates, whose corresponding error terms at the linear level have been studied in \cite{DRL}, \cite{LS}, \cite{LKerr}. A further issue that arises in the case of the Kerr spacetime is the lack of good vector field commutator that are useful to obtain control of higher order derivatives. This has been treated in the linear setting in \cite{DRK} and \cite{DRL} using $\partial_{t^*}$ and the red-shift vector field as commutators and retrieving all other derivatives via elliptic estimates. In the nonlinear setting, we again use elliptic estimates, noting however that the proof of the elliptic estimates now couples with that of the energy estimates in a bootstrap argument.
Secondly, Kerr spacetimes contain trapped null geodesics. As a consequence, any decay results at the linear level must involve a loss of derivatives. This is manifested in the degeneracy of the integrated decay estimate near $r=3M$. We note, however, that on the linear level the non-degenerate energy can be proved to be bounded without any loss of derivatives. We therefore prove energy bounds that is consistent with the linear scenario. We would try to prove on the highest level of derivatives only a boundedness result and begin to prove decay results on the level of fewer derivatives. However, as we will see, the nonlinear effect comes into play and it is not possible to prove even the boundedness of the non-degenerate energy at the highest level of derivatives. We can nevertheless show that it is bounded by $(t^*)^\eta$. On the level of one less derivative, we can prove that the conformal energy grows no faster than $\tau^{1+\eta}$. Using this fact as we prove the estimates for the non-degenerate energy, we can show that at this level of derivatives, the non-degenerate energy is bounded. This is crucial for obtaining the necessary borderline decay of $(t^*)^{-1}$ in $r\geq\frac{t^*}{2}$, thus allowing us to close the bootstrap argument. Trapping would also cause a loss in derivatives when controlling the error terms arising from the commutation with $S$. To tackle this problem, we would commute with $S$ only once. With this approach, we would not have an improved decay for $DS\Phi$ in $r\leq\frac{t^*}{2}$. Nevertheless, we can show that the bootstrap can be closed. Here we make use of the fact that as we close the assumptions for $S\Phi$, we are at a level of derivatives of $\Phi$ such that the local energy flux decays.
In the next section, we will introduce the energy quantities on Kerr spacetimes that can be thought of as analogues of the energy, conformal energy and the integrated local energy. In section \ref{estimates}, we will state the energy estimates that they satisfy. In section \ref{sectionelliptic}, we will state the elliptic estimates that will be used. Then in section \ref{sectionpointwise}, we prove the necessary $L^\infty$ estimates. With all this preparation, we then prove all the estimates using a bootstrap argument in section \ref{bootstrap}. This then easily implies the main theorem in \ref{pfmaintheorem}.
\section{The Energy Quantities}\label{estimates}
We use three kinds of energy quantities, following the notation in \cite{LKerr}. The represent the non-degenerate energy, the conformal energy and the energy norm for the integrated decay estimate. The nondegnerate energy controls all derivatives:
\begin{proposition}
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau} (D\Phi)^2
\leq C\int_{\Sigma_\tau} J^{N}_\mu\left(\Phi\right)n^\mu_{\Sigma_\tau}.
\end{split}
\end{equation*}
\end{proposition}
The conformal energy gives different weights to different derivatives and this will be crucially used to capture the null condition:
\begin{proposition}\label{Zlowerbound}
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau\cap\{r\geq r^-_Y\}} u^2\left(\underline{L}\Phi\right)^2+v^2\left(L\Phi\right)^2+\left(u^2+v^2\right)|{\nabla} \mkern-13mu /\,\Phi|^2+\left(\frac{u^2+v^2}{r^2}\right)\Phi^2\\
\leq &C\int_{\Sigma_\tau} J^{Z+N,w^Z}_\mu\left(\Phi\right)n^\mu_{\Sigma_\tau}+C^2 \tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}} J^{N}_\mu\left(\Phi\right)n^\mu_{\Sigma_\tau}.
\end{split}
\end{equation*}
\end{proposition}
We use the following notations even though they do not correspond to any vector fields:
\begin{definition}
$$K^{X_0}\left(\Phi\right)=r^{-1-\delta}\mathbbm{1}_{\{|r- 3M|\geq \frac{M}{8}\}}J^N_\mu\left(\Phi\right)n^\mu_{\Sigma_\tau}+r^{-1-\delta}\left(\partial_r\Phi\right)^2+r^{-3-\delta}\Phi^2,\quad\mbox{and}$$
$$K^{X_1}\left(\Phi\right)=r^{-1-\delta}J^N_\mu\left(\Phi\right)n^\mu_{\Sigma_\tau}+r^{-3-\delta}\Phi^2.$$
\end{definition}
\section{The Energy Estimates}
We have proved in \cite{LKerr} the energy estimates for the energy quantities define in the last section for $\Box_{g_K}\Phi=G$. We have boundedness for the non-degenerate energy:
\begin{proposition}\label{bddcom}
Let $G=G_1+G_2$ be any way to decompose the function $G$. Then
\begin{equation*}
\begin{split}
&\int_{\Sigma_{\tau}} J^{N}_\mu\left(\Phi\right)n^\mu_{\Sigma_{\tau}} +\int_{\mathcal H(\tau',\tau)} J^{N}_\mu\left(\Phi\right)n^\mu_{\mathcal H^+} +\iint_{\mathcal R(\tau',\tau)\cap\{r\leq r^-_Y\}}K^{N}\left(\Phi\right)+\iint_{\mathcal R(\tau',\tau)}K^{X_0}\left(\Phi\right)\\
\leq &C\left(\int_{\Sigma_{\tau'}} J^{N}_\mu\left(\Phi\right)n^\mu_{\Sigma_{\tau'}}+\left(\int_{\tau'-1}^{\tau+1}\left(\int_{\Sigma_{t^*}} G_1^2\right)^{\frac{1}{2}}dt^*\right)^2+\iint_{\mathcal R(\tau'-1,\tau+1)}G_1^2\right.\\
&\left.+\sum_{m=0}^{1}\iint_{\mathcal R(\tau'-1,\tau+1)}r^{1+\delta}\left(\partial_{t^*}^{m}G_2\right)^2+\sup_{t^*\in [\tau'-1,\tau+1]}\int_{\Sigma_{t^*}\cap\{|r-3M|\leq\frac{M}{8}\}} G_2^2\right).
\end{split}
\end{equation*}
\end{proposition}
We need an extra derivative for the inhomogeneous term because of trapping. If we know a priori that $G$ is supported away from the trapped region, this loss in derivative is unnecessary.
\begin{proposition}\label{bddcom2}
Let $G=G_1+G_2$ be any way to decompose the function $G$. Suppose $G_2$ is supported away from $\{r:|r-3M|\leq\frac{M}{8}\}$. Then
\begin{equation*}
\begin{split}
&\int_{\Sigma_{\tau}} J^{N}_\mu\left(\Phi\right)n^\mu_{\Sigma_{\tau}} +\int_{\mathcal H(\tau',\tau)} J^{N}_\mu\left(\Phi\right)n^\mu_{\mathcal H^+} +\iint_{\mathcal R(\tau',\tau)\cap\{r\leq r^-_Y\}}K^{N}\left(\Phi\right)+\iint_{\mathcal R(\tau',\tau)}K^{X_0}\left(\Phi\right)\\
\leq &C\left(\int_{\Sigma_{\tau'}} J^{N}_\mu\left(\Phi\right)n^\mu_{\Sigma_{\tau'}}+\left(\int_{\tau'-1}^{\tau+1}\left(\int_{\Sigma_{t^*}} G_1^2\right)^{\frac{1}{2}}dt^*\right)^2+\iint_{\mathcal R(\tau'-1,\tau+1)}G_1^2\right.\\
&\left.+\iint_{\mathcal R(\tau'-1,\tau+1)}r^{1+\delta}G_2^2\right).
\end{split}
\end{equation*}
\end{proposition}
The estimates for $K^{X_1}$ were also proved. It is be estimated in the same way as $K^{X_0}$ except with an extra derivative.
\begin{proposition}\label{bddcom1}
\begin{equation*}
\begin{split}
&\iint_{\mathcal R(\tau',\tau)}K^{X_1}\left(\Phi\right)\\
\leq& C\left(\sum_{m=0}^1\int_{\Sigma_{\tau'}} J^{N}_\mu\left(\partial_{t^*}^{m}\Phi\right)n^\mu_{\Sigma_{\tau'}}+\sum_{m=0}^1\left(\int_{\tau'-1}^{\tau+1}\left(\int_{\Sigma_{t^*}} \left(\partial_{t^*}^{m}G_1\right)^2\right)^{\frac{1}{2}}dt^*\right)^2+\sum_{m=0}^1\iint_{\mathcal R(\tau'-1,\tau+1)}\left(\partial_{t^*}^{m}G_1\right)^2\right.\\
&\left.+\sum_{m=0}^{2}\iint_{\mathcal R(\tau'-1,\tau+1)}r^{1+\delta}\left(\partial_{t^*}^{m}G_2\right)^2+\sup_{t^*\in [\tau'-1,\tau+1]}\sum_{m=0}^1\int_{\Sigma_{t^*}\cap\{|r-3M|\leq\frac{M}{8}\}} \left(\partial_{t^*}^{m}G_2\right)^2\right).
\end{split}
\end{equation*}
\end{proposition}
As before, if the inhomogeneous term is supported away from the trapped set, we can save a derivative:
\begin{proposition}\label{bddcom3}
Let $G=G_1+G_2$ be any way to decompose the function $G$. Suppose $G_2$ is supported away from $\{r:|r-3M|\leq\frac{M}{8}\}$. Then
\begin{equation*}
\begin{split}
&\iint_{\mathcal R(\tau',\tau)}K^{X_1}\left(\Phi\right)\\
\leq& C\left(\sum_{m=0}^1\int_{\Sigma_{\tau'}} J^{N}_\mu\left(\partial_{t^*}^{m}\Phi\right)n^\mu_{\Sigma_{\tau'}}+\sum_{m=0}^1\left(\int_{\tau'-1}^{\tau+1}\left(\int_{\Sigma_{t^*}} \left(\partial_{t^*}^{m}G_1\right)^2\right)^{\frac{1}{2}}dt^*\right)^2+\sum_{m=0}^1\iint_{\mathcal R(\tau'-1,\tau+1)}\left(\partial_{t^*}^{m}G_1\right)^2\right.\\
&\left.+\sum_{m=0}^{1}\iint_{\mathcal R(\tau'-1,\tau+1)}r^{1+\delta}\left(\partial_{t^*}^{m}G_2\right)^2\right).
\end{split}
\end{equation*}
\end{proposition}
The conformal energy satisfies the following estimates:
\begin{proposition}\label{conformalenergy}
For $\delta, \delta'>0$ sufficiently small and $0\leq\gamma< 1$, there exist $c=c(\delta,\gamma)$ and $C=C(\delta,\gamma)$ such that the following estimate holds for any solution to $\Box_{g_K}\Phi=G$:
\begin{equation*}
\begin{split}
&c\int_{\Sigma_{\tau}} J^{Z,w^Z}_\mu\left(\Phi\right)n^\mu_{\Sigma_{{\tau}}}+\tau^2\int_{\Sigma_{\tau}\cap\{r\leq \gamma\tau\}} J^{N}_\mu\left(\Phi\right)n^\mu_{\Sigma_{\tau}}\\
\leq &C\int_{\Sigma_{\tau_0}} J^{Z+CN,w^Z}_\mu\left(\Phi\right)n^\mu_{\Sigma_{{\tau_0}}}+C\iint_{\mathcal R(\tau_0,\tau)} t^*r^{-1+\delta}K^{X_1}\left(\Phi\right)\\
& +C\delta'\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq \frac{t^*}{2}\}} (t^*)^2K^{X_0}\left(\Phi\right)+C\left(\delta'+a\right)\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq r^-_Y\}}(t^*)^2K^N\left(\Phi\right)\\
&+C(\delta')^{-1}\left(\int_{\tau_0}^{\tau}\left(\int_{\Sigma_{t^*}\cap\{r\geq \frac{t^*}{2}\}}r^2 G^2 \right)^{\frac{1}{2}}dt^*\right)^2+C(\delta')^{-1}\sum_{m=0}^1\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq\frac{9t^*}{10}\}} (t^*)^2r^{1+\delta}\left(\partial_{t^*}^m G\right)^2\\
&+C(\delta')^{-1}\sup_{t^*\in [\tau_0,\tau]}\int_{\Sigma_{t^*}\cap\{r^-_Y\leq r\leq \frac{25M}{8}\}} (t^*)^2 G^2.
\end{split}
\end{equation*}
\end{proposition}
\begin{remark}
As in Proposition \ref{bddcom2}, we can save a derivative if we know that the inhomogeneous term is supported away from the trapped region. More precisely, let $G=G_1+G_2$ be any way to decompose the function $G$. Suppose $G_2$ is supported away from $\{r:|r-3M|\leq\frac{M}{8}\}$. Then we can replace
$$\sum_{m=0}^1\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq\frac{9t^*}{10}\}} (t^*)^2r^{1+\delta}\left(\partial_{t^*}^m G\right)^2+\sup_{t^*\in [\tau_0,\tau]}\int_{\Sigma_{t^*}\cap\{r^-_Y\leq r\leq \frac{25M}{8}\}} (t^*)^2 G^2$$
by $$\sum_{m=0}^1\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq\frac{9t^*}{10}\}} (t^*)^2r^{1+\delta}\left(\partial_{t^*}^m G_1\right)^2+\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq\frac{9t^*}{10}\}} (t^*)^2r^{1+\delta} G_2^2+\sup_{t^*\in [\tau_0,\tau]}\int_{\Sigma_{t^*}\cap\{r^-_Y\leq r\leq \frac{25M}{8}\}} (t^*)^2 G_1^2.$$
in Proposition \ref{conformalenergy}. This follows from a straight forward modification of the proof in \cite{LKerr}.
\end{remark}
The estimates for $K^{X_0}$ and $K^{X_1}$ can be localized to $r\leq\frac{t^*}{2}$ if we control it by the conformal energy:
\begin{proposition}\label{X0}
\begin{enumerate}
\item Localized Estimate for $X_0$
\begin{equation*}
\begin{split}
&\iint_{\mathcal R(\tau',\tau)\cap\{r\leq\frac{t^*}{2}\}}K^{X_0}\left(\Phi\right)\\
\leq& C\left(\tau^{-2}\int_{\Sigma_{\tau'}} J^{Z+N,w^Z}_\mu\left(\Phi\right)n^\mu_{\Sigma_{\tau'}}+ C\int_{\Sigma_{\tau'}\cap\{r\leq r^-_Y\}} J^{N}_\mu\left(\Phi\right)n^\mu_{\Sigma_{\tau'}}\right)\\
&+C\left(\sum_{m=0}^{1}\iint_{\mathcal R(\tau'-1,\tau+1)\cap\{r\leq\frac{9t^*}{10}\}}r^{1+\delta'}\left(\partial_{t^*}^{m}{G}\right)^2+\sup_{t^*\in [\tau'-1,\tau+1]}\int_{\Sigma_{t^*}\cap\{|r-3M|\leq\frac{M}{8}\}\cap\{r\leq\frac{9t^*}{10}\}} G^2\right).
\end{split}
\end{equation*}
\item Localized Estimate for $X_1$
\begin{equation*}
\begin{split}
&\iint_{\mathcal R(\tau',\tau)\cap\{r\leq\frac{t^*}{2}\}}K^{X_1}\left(\Phi\right)\\
\leq& C\left(\tau^{-2}\sum_{m=0}^{1}\int_{\Sigma_{\tau'}} J^{Z+N,w^Z}_\mu\left(\partial_{t^*}^m\Phi\right)n^\mu_{\Sigma_{\tau'}}+ C\sum_{m=0}^{1}\int_{\Sigma_{\tau'}\cap\{r\leq r^-_Y\}} J^{N}_\mu\left(\partial_{t^*}^m\Phi\right)n^\mu_{\Sigma_{\tau'}}\right)\\
&+C\left(\sum_{m=0}^{2}\iint_{\mathcal R(\tau'-1,\tau+1)\cap\{r\leq\frac{9t^*}{10}\}}r^{1+\delta}\left(\partial_{t^*}^{m}{G}\right)^2\right.\\
&\left.+\sup_{t^*\in [\tau'-1,\tau+1]}\sum_{m=0}^{1}\int_{\Sigma_{t^*}\cap\{|r-3M|\leq\frac{M}{8}\}\cap\{r\leq\frac{9t^*}{10}\}} \left(\partial_{t^*}^m G\right)^2\right).
\end{split}
\end{equation*}
\end{enumerate}
\end{proposition}
\begin{remark}
As before, if $G=G_1+G_2$ and $G_2$ is supported outside $\{|r-3M|\leq \frac{M}{8}\}$, we can replace, in Proposition \ref{X0}.1,
$$\sum_{m=0}^{1}\iint_{\mathcal R(\tau'-1,\tau+1)\cap\{r\leq\frac{9t^*}{10}\}}r^{1+\delta}\left(\partial_{t^*}^{m}{G_2}\right)^2+\sup_{t^*\in [\tau'-1,\tau+1]}\int_{\Sigma_{t^*}\cap\{|r-3M|\leq\frac{M}{8}\}\cap\{r\leq\frac{9t^*}{10}\}} G^2$$
by $$\iint_{\mathcal R(\tau'-1,\tau+1)\cap\{r\leq\frac{9t^*}{10}\}}r^{1+\delta}G_2^2;$$
and replace in Proposition \ref{X0}.2,
$$\sum_{m=0}^{2}\iint_{\mathcal R(\tau'-1,\tau+1)\cap\{r\leq\frac{9t^*}{10}\}}r^{1+\delta}\left(\partial_{t^*}^{m}{G_2}\right)^2+\sum_{m=0}^1\sup_{t^*\in [\tau'-1,\tau+1]}\int_{\Sigma_{t^*}\cap\{|r-3M|\leq\frac{M}{8}\}\cap\{r\leq\frac{9t^*}{10}\}} \left(\partial_{t^*}^m G_2\right)^2$$
by $$\sum_{m=0}^{1}\iint_{\mathcal R(\tau'-1,\tau+1)\cap\{r\leq\frac{9t^*}{10}\}}r^{1+\delta}\left(\partial_{t^*}^{m}{G_2}\right)^2.$$
\end{remark}
\section{The Elliptic Estimates and Hardy Inequality}\label{sectionelliptic}
We have also proved in \cite{LKerr} the following elliptic estimates:
\begin{proposition}\label{elliptic}
Suppose $\Box_{g_K}\Phi=G$. For $m\geq 1$ and for any $\alpha$,
\begin{enumerate}
\item Boundedness of Weighted Energy
\begin{equation*}
\begin{split}
\int_{\Sigma_{\tau}\cap\{r\geq r^-_Y\}} r^\alpha\left(D^m\Phi\right)^2 \leq C_\alpha\left(\sum_{j=0}^{m-1}\int_{\Sigma_{\tau}} r^\alpha J^N_\mu\left(\partial_{t^*}^j\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{j=0}^{m-2}\int_{\Sigma_{\tau}} r^\alpha\left(D^jG\right)^2\right).
\end{split}
\end{equation*}
\item Boundedness of Local Energy\\
For any $0< \gamma<\gamma'$,
$$\int_{\Sigma_{\tau}\cap\{r^-_Y\leq r\leq \gamma t^*\}} r^\alpha\left(D^m\Phi\right)^2 \leq C_\alpha\left(\sum_{j=0}^{m-1}\int_{\Sigma_{\tau}\cap\{r\leq \gamma' t^*\}} r^\alpha J^N_\mu\left(\partial_{t^*}^j\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{j=0}^{m-2}\int_{\Sigma_{\tau}} r^\alpha\left(D^jG\right)^2\right).$$
\end{enumerate}
\end{proposition}
We need a Hardy-type inequality that is improves the analogous one in \cite{LKerr}:
\begin{proposition}\label{Hardy}
For $R>R'$,
$$\int_{\Sigma_\tau\cap\{r\geq R\}} r^{\alpha-2}\Phi^2 \leq C\int_{\Sigma_\tau\cap\{r\geq R'\}} r^{\alpha}J^N_\mu\left(\Phi\right)n^\mu_{\Sigma_\tau}.$$
\end{proposition}
\begin{proof}
Let $k(r)$ be defined by solving $$k'(r,\theta,\phi)=r^{\alpha-2}vol,$$ in the region $r\geq R'$, where $vol=vol\left(r,\theta,\phi\right)$ is the volume density on $\Sigma_\tau$ with $r, \theta,\phi$ coordinates, with boundary condition $k(R',\theta,\phi)=0$.
Now
\begin{equation*}
\begin{split}
\int_{\Sigma_\tau\cap\{r\geq R\}} r^{\alpha-2}\Phi^2=& \iiint_{R'}^{\infty} k'(r)\Phi^2 dr d\theta d\phi\\
\leq& -2\iiint k(r)\Phi\partial_r\Phi dr d\theta d\phi\\
\leq &2\left(\iiint_{R'}^{\infty} \frac{1+k(r)^2}{1+k'(r)}\left(\partial_r \Phi\right)^2 dr d\theta d\phi\right)^{\frac{1}{2}}\left(\iiint_{R'}^{\infty} (1+k'(r))\Phi^2 dr d\theta d\phi\right)^{\frac{1}{2}}
\end{split}
\end{equation*}
Notice that $vol \sim r^2$, $k(r)\sim r^{\alpha+1}$ and $1+k'(r)\sim r^{\alpha}$. Hence $\frac{1+k(r)^2}{1+k'(r)}\sim r^{\alpha}vol$. The lemma follows.
\end{proof}
With the help of this Hardy inequality, we are able to ``localize'' the elliptic estimates for $r\geq R$.
\begin{proposition}\label{ellipticoutside}
Suppose $\Box_{g_K}\Phi=G$. For $m\geq 1$ and for any $\alpha$,
For any $R>R'$,
$$\int_{\Sigma_{\tau}\cap\{r\geq R\}} r^\alpha\left(D^m\Phi\right)^2 \leq C_{\alpha,R,R'}\left(\sum_{j=0}^{m-1}\int_{\Sigma_{\tau}\cap\{r\geq R'\}} r^\alpha J^N_\mu\left(\partial_{t^*}^j\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{j=0}^{m-2}\int_{\Sigma_{\tau}} r^\alpha\left(D^jG\right)^2\right).$$
\end{proposition}
Near the event horizon, elliptic estimates have been proved to control all the derivatives if we have control the $\partial_{t^*}$ and the $\hat{Y}$ derivatives \cite{DRK}, \cite{DRL}, \cite{LKerr}:
\begin{proposition}\label{elliptichorizon}
Suppose $\Box_{g_K}\Phi=G$. For every $m\geq 1$,
\begin{equation*}
\begin{split}
\int_{\Sigma_{\tau}\cap\{r\leq r^-_Y\}} \left(D^m\Phi\right)^2 \leq C\left(\sum_{j+k\leq m-1}\int_{\Sigma_{\tau}\cap\{r\leq r^-_Y\}} J^N_\mu\left(\partial_{t^*}^j\hat{Y}^k\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{j=0}^{m-2}\int_{\Sigma_{\tau}\cap\{r\leq r^-_Y\}}\left(D^jG\right)^2\right).
\end{split}
\end{equation*}
\end{proposition}
This is useful together with the follow control for the equation commuted with $\hat{Y}$:
\begin{proposition}\label{commYcontrol}
Suppose $\Box_{g_K}\Phi=G$. For every $k\geq 0$,
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau\cap\{r\leq r^+_Y\}} J^{N}_\mu\left(\hat{Y}^{k}\Phi\right)n^\mu_{\Sigma_\tau} +\int_{\mathcal H(\tau',\tau)} J^{N}_\mu\left(\hat{Y}^{k}\Phi\right)n^\mu_{\Sigma_\tau}+ \iint_{\mathcal R(\tau',\tau)\cap\{r\leq r^-_Y\}}J_\mu^{N}\left(\hat{Y}^{k}\Phi\right)n^\mu_{\Sigma_{t^*}}\\
\leq &C\left(\sum_{j+m\leq k}\int_{\Sigma_{\tau'}\cap\{r\leq r^+_Y\}} J^N_\mu\left(\partial_{t^*}^j\hat{Y}^m\Phi\right)n^\mu_{\Sigma_{\tau'}}+\sum_{j=0}^k\int_{\Sigma_{\tau}\cap\{r\leq r^+_Y\}} J^N_\mu\left(\partial_{t^*}^j\Phi\right)n^\mu_{\Sigma_{\tau}}\right.\\
&\left.+\sum_{j=0}^k\iint_{\mathcal R(\tau',\tau)\cap\{ r\leq \frac{23M}{8}\}} J^N_\mu\left(\partial_{t^*}^j\Phi\right)n^\mu_{\Sigma_{t^*}}+\sum_{j=0}^{k}\iint_{\mathcal R(\tau',\tau)\cap\{r\leq\frac{23M}{8}\}}\left(D^jG\right)^2\right).
\end{split}
\end{equation*}
\end{proposition}
\section{Pointwise Estimates}\label{sectionpointwise}
We prove pointwise estimates using Sobolev embedding. We will have different estimates in the region $\{r\geq\frac{t^*}{4}\}$ and $\{r\leq\frac{t^*}{4}\}$. We first consider $\{r\geq\frac{t^*}{4}\}$. For this region, we will prove five different pointwise estimates. First, we prove a boundedness result for $D\Phi$ (Proposition \ref{SE}) using only standard Sobolev Embedding and the elliptic estimates in Proposition \ref{elliptic}. Then we prove decay estimates of $r^{-1}$ for $D^\ell\Phi$ using the $r$ weight in the vector field commutator $\tilde{\Omega}$ and the non-degenerate energy (Proposition \ref{r}). It is crucial that this depends only on the non-degenerate energy but not the conformal energy because we will not be able to prove boundedness of the conformal energy (which already is the case in the \emph{linear} situation, see \cite{DRL}, \cite{AB}, \cite{LKerr}). Notice that Proposition \ref{SE} does not follow from Proposition \ref{r} because the latter requires an extra derivative. This save in derivatives is strictly speaking not necessary for the bootstrap if we have instead assumed an extra derivative of regularity in the initial data. Thirdly, using similar ideas, we will prove the decay of $r^{-1}$ for $\Phi$ using $\tilde{\Omega}$ and the conformal energy (Proposition \ref{rnoderivatives}). Then we prove an extra decay rate of $D\Phi$ using the conformal energy. For any derivatives, we will have an extra decay in the $u$ variable, which degenerates in the wave zone (Proposition \ref{ru}). For the good derivatives, we will have an extra decay in the $v$ variable (Proposition \ref{rv}). This decay rate will be crucial in capturing the good derivative in the null condition.
\begin{proposition}\label{SE}
For $r\geq\frac{t^*}{4}$\, we have
\begin{equation*}
\begin{split}
|D\Phi|^2
\leq& C\left(\sum_{k=0}^2\int_{\Sigma_\tau} J^{N}_\mu\left(\partial_{t^*}^k\Phi\right) n^\mu_{\Sigma_\tau}+\sum_{k=0}^1\int_{\Sigma_\tau} \left(D^k\Box_{g_K}\Phi\right)^2\right).
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
By standard Sobolev Embedding in three dimensions and Proposition \ref{elliptic},
\begin{equation*}
\begin{split}
|D\Phi|^2
\leq& C\sum_{k=1}^3\int_{\Sigma_\tau\cap\{r\geq r^-_Y\}} (D^k\Phi)^2\\
\leq& C\left(\sum_{k=0}^2\int_{\Sigma_\tau} J^{N}_\mu\left(\partial_{t^*}^k\Phi\right) n^\mu_{\Sigma_\tau}+\sum_{k=0}^1\int_{\Sigma_\tau} \left(D^k\Box_{g_K}\Phi\right)^2\right).
\end{split}
\end{equation*}
\end{proof}
We then prove the decay rate of $r^{-1}$ for $D^\ell\Phi$. The idea here is standard: Making use of the commutator $\tilde{\Omega}$, we use the Sobolev Embedding on the 2-sphere and then integrate along the $r$ direction.
\begin{proposition}\label{r}
For $r\geq\frac{t^*}{4}$ and $\ell\geq 1$, we have
\begin{equation*}
\begin{split}
&|D^\ell\Phi|^2\\
\leq& Cr^{-2}\left(\sum_{m=0}^{\ell}\sum_{k=0}^2\int_{\Sigma_\tau} J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}+\sum_{j=0}^{\ell-1}\sum_{k=0}^2\int_{\Sigma_\tau\cap\{u'\sim u\}\cap\{r\geq\frac{\tau}{2}\}} \left(D^j\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2\right).
\end{split}
\end{equation*}
\begin{proof}
\begin{equation*}
\begin{split}
&r^{2}|D^\ell\Phi|^2\\\leq &C\int_{\mathbb S^2}\left(\left(D^\ell\Phi\right)^2+\left(\tilde{\Omega}D^\ell\Phi\right)^2+\left(\tilde{\Omega}^2 D^\ell\Phi\right)^2\right)r^{2} dA\\
\leq&C\left(\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(\tilde{\Omega}^kD^\ell\Phi\right)^2\tilde{r}^{2} dA+\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}|\partial_r\tilde{\Omega}^kD^\ell\Phi\tilde{\Omega}^kD^\ell\Phi|(r')^{2}+\left(\tilde{\Omega}^kD^\ell\Phi\right)^2r' dAdr'\right).
\end{split}
\end{equation*}
Noticing that $|[D,\tilde{\Omega}]\Phi|\leq C|D\Phi|$, we have
\begin{equation*}
\begin{split}
&r^{2}|D^\ell\Phi|^2\\
\leq&C\left(\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(\tilde{\Omega}^kD^\ell\Phi\right)^2\tilde{r}^{2} dA+\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}|\partial_r D^\ell\tilde{\Omega}^k\Phi D^\ell\tilde{\Omega}^k\Phi|(r')^{2}+\left(\tilde{\Omega}^kD^\ell\Phi\right)^2r' dAdr'\right)\\
\leq&C\left(\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(D^\ell\tilde{\Omega}^k\Phi\right)^2\tilde{r}^{2} dA+\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}|D^{\ell+1}\tilde{\Omega}^k\Phi D^\ell\tilde{\Omega}^k\Phi|(r')^{2}+\left(D^\ell\tilde{\Omega}^k\Phi\right)^2r' dAdr'\right)\\
\leq&C\left(\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(D^\ell\tilde{\Omega}^k\Phi\right)^2\tilde{r}^{2} dA+\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}\left(D^{\ell+1}\tilde{\Omega}^k\Phi \right)^2(r')^{2}+\left(D^\ell\tilde{\Omega}^k\Phi\right)^2(r')^2 dAdr'\right).\\
\end{split}
\end{equation*}
Take $r\leq\tilde{r}\leq r+1$. By Proposition \ref{elliptic},
\begin{equation*}
\begin{split}
&\sum_{k=0}^2\int^{r+1}_r\int_{\mathbb S^2(r')}\left(D^{\ell+1}\tilde{\Omega}^k\Phi \right)^2(r')^{2}+\left(D^\ell\tilde{\Omega}^k\Phi\right)^2(r')^2 dAdr'\\
\leq&C\left(\sum_{m=0}^{\ell}\sum_{k=0}^2\int_{\Sigma_\tau}J^N_\mu\left(\partial_{t^*}^m\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{j=0}^{\ell-1}\sum_{k=0}^2\int_{\Sigma_\tau\cap\{u'\sim u\}\cap\{r\geq\frac{\tau}{2}\}} \left(D^j\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2\right).
\end{split}
\end{equation*}
By pigeonholing on this we also get that for some $\tilde{r}$,
\begin{equation*}
\begin{split}
&\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(D^\ell\Omega^k\Phi\right)^2\tilde{r}^{2} dA\\
\leq&C\left(\sum_{m=0}^{\ell}\sum_{k=0}^2\int_{\Sigma_\tau}J^N_\mu\left(\partial_{t^*}^m\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{j=0}^{\ell-1}\sum_{k=0}^2\int_{\Sigma_\tau\cap\{u'\sim u\}\cap\{r\geq\frac{\tau}{2}\}} \left(D^j\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2\right).
\end{split}
\end{equation*}
\end{proof}
\end{proposition}
We would also like to prove the pointwise decay in $r$ for $\Phi$. However, we need to use the conformal energy as well as the non-degenerate energy. We note that only the decay in $r$ will be used in the bootstrap argument, the decay in $u$ is proved to achieved the decay rate asserted in Theorem 1.
\begin{proposition}\label{rnoderivatives}
Consider $\Box_{g_K}\Phi=G$. For $r\geq\frac{t^*}{4}$, we have
\begin{equation*}
\begin{split}
|\Phi|^2
\leq&Cr^{-2}(1+|u|)^{-1}\left(\sum_{k=0}^2\int_{\Sigma_\tau}J^{Z+N,w^Z}_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}+C\tau^2\sum_{k=0}^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^N_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}\right).
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
Following the proof of Proposition \ref{r}, we have
\begin{equation*}
\begin{split}
r^2|\Phi|^2
\leq&C\left(\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(\tilde{\Omega}^k\Phi\right)^2\tilde{r}^{2} dA+|\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}|\tilde{\Omega}^k\Phi D\tilde{\Omega}^k\Phi| (r')^{2}+\left(\tilde{\Omega}^k\Phi\right)^2r' dAdr'|\right).\\
\end{split}
\end{equation*}
We will treat separately the cases $|u|\leq 1$, $u\geq 1 $, $u\leq 1$. For $|u|\leq 1$,
take $r\leq\tilde{r}\leq r+1$. By Proposition \ref{Zlowerbound},
\begin{equation*}
\begin{split}
&\sum_{k=0}^2\int^{r+1}_r\int_{\mathbb S^2(r')}\left(D\tilde{\Omega}^k\Phi \right)^2(r')^{2}+\left(\tilde{\Omega}^k\Phi\right)^2(r')^2 dAdr'\\
\leq&C\left(\sum_{k=0}^2\int_{\Sigma_\tau}J^{Z+N,w^Z}_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}+C\tau^2\sum_{k=0}^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^N_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}\right).
\end{split}
\end{equation*}
By pigeonholing on this we also get that for some $\tilde{r}$,
\begin{equation*}
\begin{split}
&\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(\Omega^k\Phi\right)^2\tilde{r}^{2} dA\\
\leq&C\left(\sum_{k=0}^2\int_{\Sigma_\tau}J^{Z+N,w^Z}_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}+C\tau^2\sum_{k=0}^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^N_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}\right).
\end{split}
\end{equation*}
For $u\geq 1$, pick a fixed $R$ and let $\tilde{r}\in [R,R+1]$. Then by a pigeonhole argument, there is some $\tilde{r}$ such that
$$\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(\tilde{\Omega}^k\Phi\right)^2\tilde{r}^{2} dA\leq Cu^{-2}\left(\sum_{k=0}^2\int_{\Sigma_\tau}J^{Z+N,w^Z}_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}+C\tau^2\sum_{k=0}^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^N_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}\right).$$
By Proposition \ref{Zlowerbound},
\begin{equation*}
\begin{split}
&\sum_{k=0}^2\int^{r}_R\int_{\mathbb S^2(r')}|\tilde{\Omega}^k\Phi D\tilde{\Omega}^k\Phi|(r')^2+\left(\tilde{\Omega}^k\Phi\right)^2r' dAdr'\\
\leq&C\left(rt^*u^{-2}+ru^{-2}\right)\left(\sum_{k=0}^2\int_{\Sigma_\tau}J^{Z+N,w^Z}_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}+C\tau^2\sum_{k=0}^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^N_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}\right).
\end{split}
\end{equation*}
Using the fact that $t^*\leq Cu$ in this region, we have the desired bound in this region.
Finally, for $u\leq 1$, pick $\tilde{r}\in [-2u,-3u]$. Then by a pigeonhole argument, there is an $\tilde{r}$ such that
$$\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(\tilde{\Omega}^k\Phi\right)^2\tilde{r}^{2} dA\leq Cu^{-2}\left(\sum_{k=0}^2\int_{\Sigma_\tau}J^{Z+N,w^Z}_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}+C\tau^2\sum_{k=0}^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^N_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}\right).$$
By Proposition \ref{Zlowerbound},
\begin{equation*}
\begin{split}
&\sum_{k=0}^2\int^{\infty}_r\int_{\mathbb S^2(r')}|\tilde{\Omega}^k\Phi D\tilde{\Omega}^k\Phi|(r')^2+\left(\tilde{\Omega}^k\Phi\right)^2r' dAdr'\\
\leq&C|u|^{-1}\left(\sum_{k=0}^2\int_{\Sigma_\tau}J^{Z+N,w^Z}_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}+C\tau^2\sum_{k=0}^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^N_\mu\left(\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}\right),
\end{split}
\end{equation*}
which gives the desired bound.
\end{proof}
We would like to use the conformal energy and elliptic estimates to prove decay in the $u$ variable. However, we need to be careful when applying the localized version of the elliptic estimates. In particular, we need to perform a dyadic decomposition in the variable $u$. We remark that we can prove this for any number of derivatives by iterating the cutoff procedure in the proof of the following Proposition. However, as this will not be necessary in the sequel, we will be content with the following Proposition:
\begin{proposition}\label{energyu}
Suppose $\Box_{g_K}\Phi=G$. Let $r\geq\frac{t^*}{4}$, $\ell=1$ or $2$ and $u_0$ be the $u$- coordinate corresponding to the two sphere $(\tau,r_0)$.
\begin{equation*}
\begin{split}
&\int_{r_0}^{r_0+1}\int_{\mathbb S^2(r')} \left(D^\ell\Phi\right)^2 (r')^2 dA dr'\\
\leq& C\left(1+|u_0|\right)^{-2}\sum_{j=0}^{\ell-1}\left(\int_{\Sigma_\tau}J^{Z+CN}_\mu\left(\partial^j_{t^*}\Phi\right) n^\mu_{\Sigma_\tau}+C\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\partial^j_{t^*}\Phi\right) n^\mu_{\Sigma_\tau}\right)\\
&+C\sum_{j=0}^{\ell-2}\int_{\Sigma_\tau\cap\{u\sim u_0\}\cap\{r\geq\frac{\tau}{2}\}} \left(D^j G\right)^2.
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
The $\ell=1$ case is trivial. For $\ell=2$, we consider separately the cases: Case $0$: $|u_0|\leq C_*$, Case $1_k$: $2^k\leq u_0\leq 2^{k+1}$, Case $2_k$: $-2^{k+1}\leq u_0\leq -2^k$, $k \geq \frac{\log C_*}{\log 2}$ for some sufficiently large but fixed $C_*$. In Case $0$, we have $|u|\leq C$ for the range $[r_0,r_0+1]$ and hence the Proposition is obvious as we have $1\leq C\left(1+|u|\right)^{-2}$.
For the other cases, we consider a cutoff function $\chi:\mathbb R \to \mathbb R_{\geq 0}$ which is compactly supported in $[-2, 2]$ and identically $1$ in $[-1,1]$. In case $1_k$ (resp. $2_k$), we consider $\tilde{\Phi}$ to be defined by $\tilde{\Phi}(\tau,r,\theta,\phi)=\chi\left(2^{-k+3}\left(r-r_0\right)\right)\Phi\left(\tau,r,\theta,\phi\right)$. Then $\tilde{\Phi}$ is supported in $[r_0-2^{k-2}, r_0+2^{k-2}]$ and equals $\Phi$ in $[r_0-2^{k-3}, r_0+2^{k-3}]$. On the support of $\tilde{\Phi}$, $|\Box_{g_K}\tilde{\Phi}-G|\leq C\displaystyle\sum_{j=0}^1 2^{-(2-j)k}|D^j\Phi|$. We also have that on the support of $\tilde{\Phi}$, $|u-u_0|\leq \frac{1}{2}|r^*_S-(r^*_0)_S|\leq \frac{1}{2}|r-r_0|+\frac{M}{2}|\log \frac{r-2M}{r_0-2M}|\leq |r-r_0|\leq 2^{k-1}$ for $r_0$ sufficiently large (which we can assume for otherwise $\tau$ and $r$ must both be bounded, in which case we must be in Case $0$ for appropriately chosen $C_*$). Hence $u\sim 2^k$ (resp. $u\sim -2^k$).
Therefore, by Proposition \ref{elliptic}.1 applied twice, first to $\tilde{\Phi}$ then to $\Phi$, we have,
\begin{equation*}
\begin{split}
&\int_{r_0}^{r_0+1}\int_{\mathbb S^2(r')} \left(D^2\Phi\right)^2 (r')^2 dA dr'\leq \int_{\Sigma_\tau\cap\{r_0\leq r\leq r_0+1\}} \left(D^2\tilde{\Phi}\right)^2 \\
\leq& C\sum_{j=0}^{1}\int_{\Sigma_\tau\cap\{r_0-2^{k-3}\leq r\leq r_0+2^{k-3}\}}J^{N}_\mu\left(\partial_{t^*}^j\tilde{\Phi}\right) n^\mu_{\Sigma_\tau}+C\sum_{j=0}^1\int_{\Sigma_\tau\cap\{r_0-2^{k-2}\leq r\leq r_0+2^{k-2}\}}2^{-(2-j)2k}\left(D^j\Phi\right)^2\\
&+C\int_{\Sigma_\tau\cap\{r_0-2^{k-2}\leq r\leq r_0+2^{k-2}\}} G^2\\
\leq& C\sum_{j=0}^{1}\int_{\Sigma_\tau\cap\{r_0-2^{k-2}\leq r\leq r_0+2^{k-2}\}}\left(2^{-2k}\Phi^2+J^{N}_\mu\left(\partial_{t^*}^j\Phi\right) n^\mu_{\Sigma_\tau}\right)+C\int_{\Sigma_\tau\cap\{r_0-2^{k-2}\leq r\leq r_0+2^{k-2}\}} G^2\\
\leq& C\left(1+|u_0|\right)^{-2}\sum_{j=0}^{1}\left(\int_{\Sigma_\tau}J^{Z+CN}_\mu\left(\partial^j_{t^*}\Phi\right) n^\mu_{\Sigma_\tau}+C\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\partial^j_{t^*}\Phi\right) n^\mu_{\Sigma_\tau}\right)+C\int_{\Sigma_\tau\cap\{u\sim u_0\}} G^2.
\end{split}
\end{equation*}
\end{proof}
Using this we can prove more decay in the $u$ variable:
\begin{proposition}\label{ru}
Suppose $\Box_{g_K}\Phi=G$. For $r\geq\frac{t^*}{4}$ and $\ell\geq 1$, we have
\begin{equation*}
\begin{split}
&|D\Phi|^2\\
\leq& Cr^{-2}\left(1+|u|\right)^{-2}\sum_{j=0}^{1}\sum_{k=0}^2\left(\int_{\Sigma_\tau}J^{Z+CN}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}+C\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}\right)\\
&+Cr^{-2}\sum_{k=0}^2\int_{\Sigma_\tau\cap\{u'\sim u\}\cap\{r\geq\frac{\tau}{2}\}} \left(\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2.
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
Following the proof of Proposition \ref{r}, we have
\begin{equation*}
\begin{split}
&r^{2}|D\Phi|^2\\
\leq&C\left(\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(D\Omega^k\Phi\right)^2\tilde{r}^{2} dA+\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}\left(D^{2}\Omega^k\Phi\right)\left(D\Omega^k\Phi\right)(r')^{2}+\left(D\Omega^k\Phi\right)^2r' dAdr'\right)\\
\end{split}
\end{equation*}
Take $r\leq\tilde{r}\leq r+1$. Then by Proposition \ref{energyu},
\begin{equation*}
\begin{split}
&\sum_{k=0}^2\int^{r+1}_r\int_{\mathbb S^2(r')}\left(D^{2}\Omega^k\Phi\right)\left(D\Omega^k\Phi\right)(r')^{2}+\left(D\Omega^k\Phi\right)^2r' dAdr'\\
\leq& C\left(1+|u|\right)^{-2}\sum_{j=0}^{1}\sum_{k=0}^2\left(\int_{\Sigma_\tau}J^{Z+CN}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}+C\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}\right)\\
&+C\sum_{k=0}^2\int_{\Sigma_\tau\cap\{u'\sim u\}\cap\{r\geq\frac{\tau}{2}\}}\left(\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2.
\end{split}
\end{equation*}
By pigeonholing on this we also get that for some $\tilde{r}$,
\begin{equation*}
\begin{split}
&\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(D^\ell\Omega^k\Phi\right)^2\tilde{r}^{2} dA\\
\leq& C\left(1+|u|\right)^{-2}\sum_{j=0}^{1}\sum_{k=0}^2\left(\int_{\Sigma_\tau}J^{Z+CN}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}+C\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}\right)\\
&+C\sum_{k=0}^2\int_{\Sigma_\tau\cap\{u'\sim u\}\cap\{r\geq\frac{\tau}{2}\}} \left(\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2.
\end{split}
\end{equation*}
\end{proof}
We have a better pointwise decay for a ``good'' derivative:
\begin{proposition}\label{rv}
For $r\geq\frac{t^*}{4}$, we have
\begin{equation*}
\begin{split}
|\bar{D}\Phi|^2\leq&C r^{-4}\sum_{k=0}^2\sum_{i+j\leq 1}\left(\int_{\Sigma_{\tau}} J^N_\mu\left(S^i\partial_{t^*}^j\Phi\right)n^\mu_{\Sigma_\tau}+\int_{\Sigma_\tau}J^{Z+CN}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}\right.\\
&\left.\quad\quad\quad\quad\quad\quad+C\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}+\int_{\Sigma_{\tau}} \left(\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2\right)\\
&+Cr^{-2}\sum_{k=0}^2\int_{\Sigma_\tau\cap\{r\geq\frac{\tau}{2}\}} \left(\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2.
\end{split}
\end{equation*}
\begin{proof}
\begin{equation*}
\begin{split}
&r^{2}|\bar{D}\Phi|^2\\\leq &C\int_{\mathbb S^2}\left(\left(\bar{D}\Phi\right)^2+\left(\tilde{\Omega}\bar{D}\Phi\right)^2+\left(\tilde{\Omega}^2 \bar{D}\Phi\right)^2\right)r^{2} dA\\
\leq&C\left(\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(\tilde{\Omega}^k\bar{D}\Phi\right)^2\tilde{r}^{2} dA+\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}|\partial_r\tilde{\Omega}^k\bar{D}\Phi\tilde{\Omega}^k\bar{D}\Phi|(r')^{2}+\left(\tilde{\Omega}^k\bar{D}\Phi\right)^2r' dAdr'\right).
\end{split}
\end{equation*}
Noticing that $|[D,\tilde{\Omega}]\Phi|\leq C|D\Phi|$, $|[\bar{D},\tilde{\Omega}]\Phi|\leq C\left(|\bar{D}\Phi|+r^{-1}|D\Phi|\right)$ and $|\bar{D},\partial_r\Phi|\leq Cr^{-1}|D\Phi|$, we have
\begin{equation}\label{rv1}
\begin{split}
&r^{2}|\bar{D}\Phi|^2\\
\leq&C\sum_{k=0}^2\left(\int_{\mathbb S^2(\tilde{r})}\left(\tilde{\Omega}^k\bar{D}\Phi\right)^2\tilde{r}^{2} dA\right.\\
&\left.+\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}|\partial_r\bar{D}\tilde{\Omega}^k\Phi\bar{D}\tilde{\Omega}^k\Phi|(r')^{2}+\left(\tilde{\Omega}^k\bar{D}\Phi\right)^2r' dAdr'\right)\\
\leq&C\sum_{k=0}^2\left(\int_{\mathbb S^2(\tilde{r})}\left(\left(\bar{D}\tilde{\Omega}^k\Phi\right)^2+\tilde{r}^{-2}\left(D\tilde{\Omega}^k\Phi\right)^2\right)\tilde{r}^{2} dA\right.\\
&\left.+\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}\left(\left(\bar{D}D\tilde{\Omega}^k\Phi\right)^2+\left(\bar{D}\tilde{\Omega}^k\Phi\right)^2+(r')^{-2}\left(D\tilde{\Omega}^k\Phi\right)^2\right)(r')^2 dAdr'\right).
\end{split}
\end{equation}
The last term already exhibits better decay rate:
\begin{equation*}
\begin{split}
\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}(r')^{-2}\left(D\tilde{\Omega}^k\Phi\right)^2(r')^2 dAdr'\leq &Cr^{-2}\int_{\Sigma_\tau}J^{N}_\mu\left(\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}
\end{split}
\end{equation*}
We will now show that the energy quantities involving $\bar{D}$ obey better decay rates. This is immediate for the term $\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}\left(\bar{D}\tilde{\Omega}^k\Phi\right)^2(r')^2 dAdr'$ using the conformal energy:
\begin{equation*}
\begin{split}
&\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}\left(\bar{D}\tilde{\Omega}^k\Phi\right)^2(r')^2 dAdr'\\
\leq&Cv^{-2}\left(\int_{\Sigma_\tau}J^{Z+CN}_\mu\left(\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}+C\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}\right).
\end{split}
\end{equation*}
However, we note that this cannot be shown directly for the term $\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}\left(\bar{D}D\tilde{\Omega}^k\Phi\right)^2(r')^2 dAdr'$ with the conformal energy because we cannot commute $\Box_{g_K}$ with derivatives of every direction. In order to remedy this, we use the non-degenerate energy for $S\Phi$. In particular, we use the fact that for $r\geq r^-_Y$, $|\bar{D}\Phi|\leq Cv^{-1}\left(|S\Phi|+u|D\Phi|+vr^{-1}|D\Phi|\right)$.
\begin{equation*}
\begin{split}
&\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}\left(\bar{D}D\tilde{\Omega}^k\Phi\right)^2(r')^2dAdr'\\
\leq&C\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}\left((v')^{-2}\left(SD\tilde{\Omega}^k\Phi\right)^2+(u')^2(v')^{-2}\left(D^{2}\tilde{\Omega}^k\Phi\right)+(r')^{-2}\left(D^{2}\tilde{\Omega}^k\Phi\right)^2 \right)(r')^2dAdr'\\
\leq&C\int^{\tilde{r}}_r\int_{\mathbb S^2(r')}\left((v')^{-2}\left(DS\tilde{\Omega}^k\Phi\right)^2+(v')^{-2}\left(D\tilde{\Omega}^k\Phi\right)^2\right.\\
&\left.\quad\quad\quad\quad\quad\quad\quad+(u')^2(v')^{-2}\left(D^{2}\tilde{\Omega}^k\Phi\right)+(r')^{-2}\left(D^{2}\tilde{\Omega}^k\Phi\right)^2 \right)(r')^2dAdr'\\
\end{split}
\end{equation*}
Take $r\leq\tilde{r}\leq r+1$. We have, for the first two terms,
\begin{equation*}
\begin{split}
&\int^{r+1}_r\int_{\mathbb S^2(r')}(v')^{-2}\left(\left(DS\tilde{\Omega}^k\Phi\right)^2+\left(D\tilde{\Omega}^k\Phi\right)^2\right)(r')^2dAdr'\leq Cv^{-2}\sum_{j=0}^1\int_{\Sigma_\tau}J^{N}_\mu\left(S^j\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}.
\end{split}
\end{equation*}
The third term can be estimated by Proposition \ref{energyu},
\begin{equation*}
\begin{split}
&\int^{r+1}_r\int_{\mathbb S^2(r')}(u')^2(v')^{-2}\left(D^{2}\tilde{\Omega}^k\Phi\right)(r')^2dAdr'\\
\leq& Cv^{-2}\sum_{j=0}^{1}\left(\int_{\Sigma_\tau}J^{Z+CN}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}+C\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}\right)\\
&+C\int_{\Sigma_\tau\cap\{u'\sim u\}\cap\{r\geq\frac{\tau}{2}\}} \left(\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2.
\end{split}
\end{equation*}
The fourth term can be estimated elliptically by Proposition \ref{elliptic}:
\begin{equation*}
\begin{split}
&\int^{r+1}_r\int_{\mathbb S^2(r')}(r')^{-2}\left(D^{2}\tilde{\Omega}^k\Phi\right)^2(r')^2dAdr'\\
\leq& C r^{-2}\left(\sum_{j=0}^{1}\int_{\Sigma_{\tau}} J^N_\mu\left(\partial_{t^*}^j\tilde{\Omega}^k\Phi\right)n^\mu_{\Sigma_\tau}+\int_{\Sigma_{\tau}} \left(\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2\right).
\end{split}
\end{equation*}
Collecting all the above estimates and noting that $r\geq\frac{\tau}{2}$, we get
\begin{equation}\label{rv2}
\begin{split}
&\sum_{k=0}^2\int^{r+1}_r\int_{\mathbb S^2(r')}\left(\left(\bar{D}D\tilde{\Omega}^k\Phi\right)^2+\left(\bar{D}\tilde{\Omega}^k\Phi\right)^2+(r')^{-2}\left(D\tilde{\Omega}^k\Phi\right)^2\right)(r')^2 dAdr'\\
\leq&C v^{-2}\sum_{k=0}^2\sum_{i+j\leq 1}\left(\int_{\Sigma_{\tau}} J^N_\mu\left(S^i\partial_{t^*}^j\Phi\right)n^\mu_{\Sigma_\tau}+\int_{\Sigma_\tau}J^{Z+CN}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}\right.\\
&\left.\quad\quad\quad\quad\quad\quad+C\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}+\int_{\Sigma_{\tau}} \left(\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2\right)\\
&+C\sum_{k=0}^2\int_{\Sigma_\tau\cap\{r\geq\frac{\tau}{2}\}} \left(\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2.
\end{split}
\end{equation}
By pigeonholing on this we also get that for some $\tilde{r}$,
\begin{equation}\label{rv3}
\begin{split}
&\sum_{k=0}^2\left(\int_{\mathbb S^2(\tilde{r})}\left(\bar{D}\tilde{\Omega}^k\Phi\right)^2+\tilde{r}^{-2}\left(D\tilde{\Omega}^k\Phi\right)^2\right)\tilde{r}^{2} dA\\
\leq&C v^{-2}\sum_{k=0}^2\sum_{i+j\leq 1}\left(\int_{\Sigma_{\tau}} J^N_\mu\left(S^i\partial_{t^*}^j\Phi\right)n^\mu_{\Sigma_\tau}+\int_{\Sigma_\tau}J^{Z+CN}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}\right.\\
&\left.\quad\quad\quad\quad\quad\quad+C\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\partial^j_{t^*}\tilde{\Omega}^k\Phi\right) n^\mu_{\Sigma_\tau}+\int_{\Sigma_{\tau}} \left(\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2\right)\\
&+C\sum_{k=0}^2\int_{\Sigma_\tau\cap\{r\geq\frac{\tau}{2}\}} \left(\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2.
\end{split}
\end{equation}
(\ref{rv1}), (\ref{rv2}) and (\ref{rv3}) together imply the Proposition.
\end{proof}
\end{proposition}
We now turn to the region $r\leq\frac{t^*}{4}$. We first show a simple Sobolev embedding result.
\begin{proposition}\label{sSobolev}
Suppose $\Box_{g_K}\Phi=G$. For $\ell\geq 1$ and $r\leq\frac{t^*}{4}$,
\begin{equation*}
|D^\ell\Phi|^2\leq C\left(\sum_{j+m\leq \ell+1}\int_{\Sigma_\tau\cap\{r\leq\frac{t^*}{2}\}}J^{N}_\mu\left(\partial_{t^*}^m\hat{Y}^j\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{j=0}^{\ell}\int_{\Sigma_\tau}(D^jG)^2\right)
\end{equation*}
\end{proposition}
We can capture better estimates in $r$ if we use an extra derivative.
\begin{proposition}\label{SEDinside}
For $\ell\geq 1$ and $r\leq\frac{t^*}{4}$,
\begin{equation*}
|D^\ell\Phi|^2\leq Cr^{-2}\left(\sum_{j+m+k\leq \ell+2}\int_{\Sigma_\tau\cap\{r\leq\frac{t^*}{2}\}}J^{N}_\mu\left(\partial_{t^*}^m\hat{Y}^j\tilde{\Omega}^k\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{j=0}^{\ell+1-k}\sum_{k=0}^2\int_{\Sigma_\tau}\left(D^j\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2\right)
\end{equation*}
\end{proposition}
\begin{proof}
We only need to consider the situation when $r\geq R_\Omega +C$. For otherwise, this Proposition is implied by Proposition \ref{sSobolev} since $r$ is finite. We assume from now on that $r\geq R_\Omega +C$.
Following the proof of Proposition \ref{r}, we have
\begin{equation*}
\begin{split}
&r^2|D^\ell\Phi|^2\\
\leq&C\left(\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(D^\ell\tilde{\Omega}^k\Phi\right)^2\tilde{r}^{2} dA+\int_{\tilde{r}}^r\int_{\mathbb S^2(r')}\left(D^{\ell+1}\tilde{\Omega}^k\Phi \right)^2(r')^{2}+\left(D^\ell\tilde{\Omega}^k\Phi\right)^2(r')^2 dAdr'\right)\\
\end{split}
\end{equation*}
Take $r-1\leq\tilde{r}\leq r$. By Proposition \ref{elliptic}.2 and \ref{elliptichorizon},
\begin{equation*}
\begin{split}
&\sum_{k=0}^2\int^{r}_{r-1}\int_{\mathbb S^2(r')}\left(D^{\ell+1}\tilde{\Omega}^k\Phi \right)^2(r')^{2}+\left(D^{\ell}\tilde{\Omega}^k\Phi\right)^2(r')^2 dAdr'\\
\leq&C\left(\sum_{j+m\leq \ell}\sum_{k=0}^2\int_{\Sigma_\tau\cap\{r\leq\frac{t^*}{2}\}}J^{N}_\mu\left(\partial_{t^*}^m\hat{Y}^j\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{j=0}^{\ell-1}\sum_{k=0}^2\int_{\Sigma_\tau}\left(D^j\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2\right).
\end{split}
\end{equation*}
By pigeonholing on this we also get that for some $\tilde{r}$ with $r-1\leq\tilde{r}\leq r$,
\begin{equation*}
\begin{split}
&\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(D\Omega^k\Phi\right)^2\tilde{r}^{2} dA\\
\leq&C\left(\sum_{j+m\leq \ell}\sum_{k=0}^2\int_{\Sigma_\tau\cap\{r\leq\frac{t^*}{2}\}}J^{N}_\mu\left(\partial_{t^*}^m\hat{Y}^j\Omega^k\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{j=0}^{\ell-1}\sum_{k=0}^2\int_{\Sigma_\tau}\left(D^j\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2\right).
\end{split}
\end{equation*}
\end{proof}
We also have pointwise estimates for $\Phi$ instead of $D\Phi$ if we use the conformal energy.
\begin{proposition}\label{SEinside}
Suppose $\Box_{g_K}\Phi=0$. For $r\leq\frac{t^*}{4}$,
\begin{equation*}
\begin{split}
|\Phi|^2\leq&C\tau^{-2}\left(\sum_{i+j\leq 2}\int_{\Sigma_\tau}J^{Z+N,w^Z}_\mu\left(\hat{Y}^i\partial_{t^*}^j\Phi\right)n^\mu_{\Sigma_\tau}+C\tau^2\sum_{i+j\leq 2}\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^N_\mu\left(\hat{Y}^i\partial_{t^*}^j\Phi\right)n^\mu_{\Sigma_\tau}\right).
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
By Sobolev Embedding in three dimensions, for $r\leq\frac{t^*}{4}$,
\begin{equation*}
\begin{split}
|\Phi|^2\leq&C\sum_{k=0}^{2}\int_{\Sigma_\tau\cap\{r\leq\frac{t^*}{4}\}} \left(D^k\Phi\right)^2.
\end{split}
\end{equation*}
Then, using the elliptic estimates in Propositions \ref{elliptic}.2 and \ref{elliptichorizon}, we have
\begin{equation*}
\begin{split}
|\Phi|^2\leq&C\sum_{i+j\leq 2}\int_{\Sigma_\tau\cap\{r\leq\frac{t^*}{2}\}} \Phi^2+J^N_\mu\left(\hat{Y}^i\partial_{t^*}^j\Phi\right)n^\mu_{\Sigma_\tau}.
\end{split}
\end{equation*}
Using Proposition \ref{Zlowerbound}, we can conclude the Proposition.
\end{proof}
We proceed to show that the pointwise estimate is better if we use the vector field commutator $S$. To this end, we first show that we can control a fixed $t^*$ quantity by an integrated quantity. The proof follows ideas in \cite{LS}, \cite{LKerr} and applies an integration in the direction of $S$.
\begin{proposition}\label{extradecay}
For any sufficiently regular $\Phi$, not necessarily satisfying any differential equations, and $\alpha_0$ a constant,
\begin{equation*}
\begin{split}
&\int_{\Sigma_{\tau}\cap\{r \leq\frac{\tau}{4}\}} r^{\alpha_0-2}\Phi^2\\
\leq &C\tau^{-1}\left(\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{\alpha_0-2}\Phi^2 +\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{\alpha_0-2} \left(S\Phi\right)^2\right).
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
To use the estimates for $S\Phi$, we need to integrate along integral curves of $S$. The following argument imitates that for proving improved decay for the homogeneous equation in \cite{LKerr}. We first find the integral curves by solving the ordinary differential equation
$$\frac{dr_S}{dt^*_S}=\frac{h(r_S)}{t^*_S}$$
where $h(r_S)$ is as in the definition of $S$. Hence the integral curves are given by
$$\frac{\exp\left(\int_{(r_S)_0}^{r_S} \frac{dr'_S}{h(r'_S)}\right)}{t^*_S}=\mbox{constant},$$
where $(r_S)_0>2M$ can be chosen arbitrarily.
Let $\sigma=t^*$, $\rho=\frac{\exp\left(\int_{(r_S)_0}^{r_S} \frac{dr_S'}{h(r_S')}\right)}{t_S^*}$ and consider $(\sigma,\rho, x^A, x^B)$ as a new system of coordinates. Notice that
$$\partial_\sigma=\frac{h(r_S)}{t_S^*}\partial_{r_S}+\partial_{t_S^*}=\frac{1}{t_S^*}S.$$
Now for each fixed $\rho$, we have
$$\Phi^2(\tau)\leq \Phi^2(\tau')+|\int_{\tau'}^\tau \frac{1}{\sigma}S(\Phi^2) d\sigma|.$$
Multiplying by $\rho^{\alpha}$ and integrating along a finite region of $\rho$, we get:
$$\int_{\rho_1}^{\rho_2} \Phi^2(\tau)\rho^{\alpha} d\rho\leq \int_{\rho_1}^{\rho_2} \Phi^2(\tau')\rho^{\alpha} d\rho+\int_{\rho_1}^{\rho_2} \int_{\tau'}^\tau |\frac{2\rho^{\alpha}}{\sigma}\Phi S\Phi |d\sigma d\rho.$$
We choose $\alpha$ so that $\alpha=0$ for $r\leq r^-_Y$ and $\alpha=\alpha_0$ for $r \geq R$ and smooth depending on $r$ in between.
We would like to change coordinates back to $(t^*_S,r_S, x_S^A, x_S^B)$. Notice that since $h(r_S)$ is everywhere positive, $(\rho,\tau)$ would correspond to a point with a larger value of $r_S$ than $(\rho,\tau')$. Therefore,
\begin{equation*}
\begin{split}
&\int_{2M}^{(r_S)_2} \Phi^2(\tau)\frac{\exp\left((1+\alpha)\int_{2M}^{r_S} \frac{dr_S'}{h(r_S')}\right)}{\tau h(r_S)}dr_S\\
\leq & \int_{2M}^{(r_S)_2} \Phi^2(\tau')\frac{\exp\left((1+\alpha)\int_{(r_S)_0}^{r_S} \frac{dr_S'}{h(r_S')}\right)}{\tau' h(r_S)}dr_S+\int_{\tau'}^\tau \int_{2M}^{(r_S)_2} |\frac{2}{\sigma}\Phi S\Phi |\frac{\exp\left((1+\alpha)\int_{(r_S)_0}^{r_S} \frac{dr_S'}{h(r_S')}\right)}{t^* h(r_S)}dr_S dt^*.
\end{split}
\end{equation*}
We have to compare $\frac{\exp\left((1+\alpha(r_S))\int_{(r_S)_0}^{r_S} \frac{dr'_S}{h(r'_S)}\right)}{h(r_S)}$ with the volume form. Very close to the horizon, $h(r_S)=r_S-2M$ and $\alpha(r)=0$. Hence
$$\frac{\exp\left((1+\alpha)\int_{(r_S)_0}^{r_S} \frac{dr'_S}{h(r'_S)}\right)}{h(r_S)}=e^{\int_{(r_S)_0}^{r_S}\frac{dr'_S}{h(r'_S)}}\left(\frac{1}{r_S-2M}\right)\sim 1.$$
On the other hand, for $r\geq R$, $h(r_S)=(r_S+2M\log(r_S-2M)-3M-2M \log M )(1-\mu )$ and $\alpha(r_S)=\alpha_0$. In particular, for a sufficiently large choice of $R$, $h(r_S)\sim r_S$. Hence
$$\frac{\exp\left((1+\alpha)\int_{(r_S)_0}^{r_S} \frac{dr'_S}{h(r'_S)}\right)}{h(r_S)}\sim \frac{\exp\left((1+\alpha)\int_{(r_S)_0}^{r_S} \frac{dr'_S}{h(r'_S)}\right)}{r_S}\sim\left(\frac{r_S^{\alpha_0}}{R}\right)\sim r^{\alpha_0-2}.$$
The corresponding expression on the compact set $[r^-_Y,R]$ is obviously bounded. Hence, since the volume density both on a slice and on a spacetime region is $\sim r^2$, we have
\begin{equation*}
\begin{split}
&\int_{\Sigma_{\tau}\cap\{r <r_2\}} \frac{\Phi^2(\tau)}{\tau}r^{\alpha_0-2}\leq C\left(\int_{\Sigma_{\tau'}\cap\{r <r_2\}} \frac{\Phi^2(\tau')}{\tau'}r^{\alpha_0-2}+\iint_{\mathcal R(\tau',\tau)\cap\{r <r_2\}} r^{\alpha_0-2}|\frac{2}{(t^*)^2}\Phi S\Phi |\right).
\end{split}
\end{equation*}
This easily implies the following improved decay for the non-degenerate energy for $\tau'\in[(1.1)^{-1}\tau,\tau]$:
\begin{equation}\label{intS}
\begin{split}
&\int_{\Sigma_{\tau}\cap\{r <\frac{\tau}{4}\}} r^{\alpha_0-2}\Phi^2\leq C\tau^{-1}\left(\int_{\Sigma_{\tau'}\cap\{r <\frac{\tau'}{3}\}} r^{\alpha_0-2}\Phi^2 +\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r <\frac{t^*}{3}\}} r^{\alpha_0-2} \left(S\Phi\right)^2\right).
\end{split}
\end{equation}
By choosing an appropriate $\tilde{\tau}$, we have
$$\int_{\Sigma_{\tilde{\tau}}\cap\{r <\frac{\tilde{\tau}}{3}\}} r^{\alpha_0-2}\Phi^2\leq C\tau^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r <\frac{t^*}{3}\}} r^{\alpha_0-2}\Phi^2.$$
Now, apply (\ref{intS}) with $\tau'=\tilde{\tau}$, we have
\begin{equation*}
\begin{split}
&\int_{\Sigma_{\tau}\cap\{r \leq\frac{\tau}{4}\}} r^{-1-\delta}\Phi^2\\
\leq &C\tau\left(\int_{\Sigma_{\tilde{\tau}}\cap\{r <\frac{\tilde{\tau}}{3}\}} \frac{\Phi^2}{\tilde{\tau}}r^{\alpha_0-2}+\iint_{\mathcal R(\tilde{\tau},\tau)\cap\{r <\frac{t^*}{3}\}} r^{\alpha_0-2}|\frac{2}{(t^*)^2}\Phi S\Phi |\right)\\
\leq &C\tau^{-1}\left(\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{\alpha_0-2}\Phi^2 +\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{\alpha_0-2} \left(S\Phi\right)^2\right),
\end{split}
\end{equation*}
using Cauchy-Schwarz for the second term.
\end{proof}
By Sobolev Embedding, this would give an improved decay estimate in $t^*$ in the region $\{r\leq\frac{t^*}{4}\}$. For the application, we also need an improved decay in $r$, which we get by commuting with the angular momentum $\tilde{\Omega}$.
\begin{proposition}\label{inside1}
Suppose $\Box_{g_K}\Phi=G$. For $r\leq\frac{t^*}{4}$ and $\ell\geq 1$, we have
\begin{equation*}
\begin{split}
|D^\ell\Phi|^2\leq &C(t^*)^{-1}r^{-1+\delta}\sum_{i+j\leq \ell-1}\sum_{k=0}^2\iint_{\mathcal R((1.1)^{-1}t^*,t^*)\cap\{r\leq\frac{t^*}{2}\}}\left(K^{X_1}\left(\hat{Y}^i\partial_{t^*}^{j}\tilde{\Omega}^k\Phi\right)+K^{X_1}\left(S\hat{Y}^i\partial_{t^*}^{j}\tilde{\Omega}^k\Phi\right)\right)\\
&+C(t^*)^{-1}r^{-1+\delta}\sum_{j=0}^{\ell-1}\sum_{k=0}^2\iint_{\mathcal R((1.1)^{-1}t^*,t^*)\cap\{r \leq\frac{t^*}{2}\}}r^{-1-\delta}\left(D^j\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2.\\
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
Using a similar argument as before, except for choosing $\tilde{r}\leq r$, we have
\begin{equation*}
\begin{split}
&r^{1-\delta}|D^\ell\Phi|^2\\
\leq &C\sum_{k=0}^{2}\int_{\mathbb S^2}\left(\tilde{\Omega}^kD^\ell\Phi\right)^2r^{1-\delta}dA\\
\leq&C\left(\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(D^\ell\Omega^k\Phi\right)^2\tilde{r}^{1-\delta} dA+\int_{\tilde{r}}^r\int_{\mathbb S^2(r')}\left(\left(D^{\ell+1}\Omega^k\Phi\right)^2+\left(D^\ell\Omega^k\Phi\right)^2\right)(r')^{1-\delta} dAdr'\right).\\
\end{split}
\end{equation*}
Using Proposition \ref{extradecay}, we have
\begin{equation*}
\begin{split}
&\int_{\tilde{r}}^r\int_{\mathbb S^2(r')}\left(D^\ell\Omega^k\Phi\right)^2(r')^{1-\delta} dAdr'\\
\leq&C\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}r^{-1-\delta}\left(D^\ell\Omega^k\Phi\right)^2\\
\leq&C\tau^{-1}\left(\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{-1-\delta}\left(D^\ell\Omega^k\Phi\right)^2 +\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{-1-\delta} \left(SD^\ell\Omega^k\Phi\right)^2\right)
\end{split}
\end{equation*}
By first commuting $[D,S]$ and then using Proposition \ref{elliptic}.2 and \ref{elliptichorizon} on each fixed $t^*$ slice in the integral, we have
\begin{equation*}
\begin{split}
&\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{-1-\delta}\left(D^\ell\Omega^k\Phi\right)^2 +\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{-1-\delta} \left(SD^\ell\Omega^k\Phi\right)^2\\
\leq&C\sum_{i+j\leq\ell-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{2}\}}r^{-1-\delta}\left(J^N_\mu\left(Y^i\partial_{t^*}^{j-i}\tilde{\Omega}^k\Phi\right)n^\mu_{\Sigma_\tau}+J^N_\mu\left(SY^i\partial_{t^*}^{j-i}\tilde{\Omega}^k\Phi\right)n^\mu_{\Sigma_\tau}\right)\\
&+C\sum_{j=0}^{\ell-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{2}\}}r^{-1-\delta}\left(D^j\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2\\
\leq&C\sum_{i+j\leq\ell-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{2}\}}\left(K^{X_1}\left(Y^i\partial_{t^*}^{j-i}\tilde{\Omega}^k\Phi\right)+K^{X_1}\left(SY^i\partial_{t^*}^{j-i}\tilde{\Omega}^k\Phi\right)\right)\\
&+C\sum_{j=0}^{\ell-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{2}\}}r^{-1-\delta}\left(D^j\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2.\\
\end{split}
\end{equation*}
Therefore, we have
\begin{equation*}
\begin{split}
&r^{1-\delta}|D^\ell\Phi|^2\\
\leq&C\tau^{-1}\sum_{i+j\leq \ell-1}\sum_{k=0}^2\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}\left(K^{X_1}\left(\hat{Y}^i\partial_{t^*}^{j}\tilde{\Omega}^k\Phi\right)+K^{X_1}\left(S\hat{Y}^i\partial_{t^*}^{j}\tilde{\Omega}^k\Phi\right)\right)\\
&+C\tau^{-1}\sum_{j=0}^{\ell-1}\sum_{k=0}^2\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{2}\}}r^{-1-\delta}\left(D^j\Box_{g_K}\left(\tilde{\Omega}^k\Phi\right)\right)^2.\\
\end{split}
\end{equation*}
\end{proof}
Similar ideas can be used to prove decay of $\Phi$ without derivatives, except for a loss in powers of $r$. This will not be used for the bootstrap argument, but will be used to prove the decay for $\Phi$ in the statement of Theorem 1.
\begin{proposition}\label{inside2}
Suppose $\Box_{g_K}\Phi=G$. For $r\leq\frac{t^*}{4}$, we have
\begin{equation*}
\begin{split}
|\Phi|^2\leq &C(t^*)^{-1}r^{\delta}\sum_{k=0}^2\iint_{\mathcal R((1.1)^{-1}t^*,t^*)\cap\{r\leq\frac{t^*}{3}\}}\left(K^{X_1}\left(\tilde{\Omega}^k\Phi\right)+K^{X_1}\left(S\tilde{\Omega}^k\Phi\right)\right).
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
Fix $R$. Take $\tilde{r}\in [R, \frac{\tau}{5}]$, we have
\begin{equation*}
\begin{split}
&r^{-\delta}|\Phi|^2\\
\leq &C\sum_{k=0}^{2}\int_{\mathbb S^2}\left(\tilde{\Omega}^k\Phi\right)^2r^{-\delta}dA\\
\leq&C\left(\sum_{k=0}^2\int_{\mathbb S^2(\tilde{r})}\left(\Omega^k\Phi\right)^2\tilde{r}^{-\delta} dA+|\int_{\tilde{r}}^r\int_{\mathbb S^2(r')}|\Omega^k\Phi D\Omega^k\Phi|^2(r')^{-\delta}+\left(\Omega^k\Phi\right)^2(r')^{-1-\delta} dAdr'|\right).\\
\end{split}
\end{equation*}
There exists $\tilde{r}\in [R, \frac{\tau}{5}]$ such that
$$\int_{\mathbb S^2(\tilde{r})}\left(\Omega^k\Phi\right)^2\tilde{r}^{-\delta} dA\leq \tau^{-1}\int_{r_+}^{\frac{\tau}{4}}\int_{\mathbb S^2(r')}\left(\Omega^k\Phi\right)^2(r')^{-\delta} dAdr'.$$
Using Proposition \ref{extradecay}, we have
\begin{equation*}
\begin{split}
&\int_{r_+}^{\frac{\tau}{4}}\int_{\mathbb S^2(\tilde{r})}\left(\Omega^k\Phi\right)^2(r')^{-\delta} dAdr'\\
\leq&C\tau\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}r^{-3-\delta}\left(\Omega^k\Phi\right)^2\\
\leq&C\left(\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{-3-\delta}\left(\Omega^k\Phi\right)^2 +\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{-3-\delta} \left(S\Omega^k\Phi\right)^2\right) \\
\leq&C\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}}\left(K^{X_1}\left(\tilde{\Omega}^k\Phi\right)+K^{X_1}\left(S\tilde{\Omega}^k\Phi\right)\right).
\end{split}
\end{equation*}
Using Proposition \ref{extradecay}, we also have
\begin{equation*}
\begin{split}
&|\int_{\tilde{r}}^r\int_{\mathbb S^2(r')}|\Omega^k\Phi D\Omega^k\Phi|^2(r')^{-\delta}+\left(\Omega^k\Phi\right)^2(r')^{-1-\delta} dAdr'|\\
\leq&C\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}r^{-3-\delta}\left(\Omega^k\Phi\right)^2+\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}r^{-1-\delta}\left(D\Omega^k\Phi\right)^2\\
\leq&C\tau^{-1}\left(\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{-3-\delta}\left(\Omega^k\Phi\right)^2 +\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{-3-\delta} \left(S\Omega^k\Phi\right)^2\right) \\
&+C\tau^{-1}\left(\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{-1-\delta}\left(D\Omega^k\Phi\right)^2 +\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}} r^{-1-\delta} \left(SD\Omega^k\Phi\right)^2\right)\\
\leq&C\tau^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{3}\}}\left(K^{X_1}\left(\tilde{\Omega}^k\Phi\right)+K^{X_1}\left(S\tilde{\Omega}^k\Phi\right)\right).
\end{split}
\end{equation*}
Therefore, we have
\begin{equation*}
\begin{split}
|\Phi|^2
\leq&C\tau^{-1}r^{\delta}\sum_{k=0}^2\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}\left(K^{X_1}\left(\tilde{\Omega}^k\Phi\right)+K^{X_1}\left(S\tilde{\Omega}^k\Phi\right)\right).\\
\end{split}
\end{equation*}
\end{proof}
\section{Bootstrap}\label{bootstrap}
Bootstrap assumptions (J): We first introduce the bootstrap assumptions corresponding to energy quantities on a fixed $t^*$ slice.
\begin{equation}\label{BA1}
\sum_{i+j=16}A_j^{-1}\int_{\Sigma_\tau}J^{N}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}+\sum_{i+k=16}A_Y^{-1}\int_{\Sigma_\tau}J^{N}_\mu\left(\hat{Y}^k\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau}\leq \epsilon \tau^{\eta_{16}}.
\end{equation}
\begin{equation}\label{BA2}
\begin{split}
\sum_{i+j=15}A_j^{-1}\left(\int_{\Sigma_\tau} J^{Z+N,w^Z}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau} +C\tau^2\int_{\Sigma_\tau\cap\{r\leq \frac{9\tau}{10}\}} J^{N}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}\right)&\\
+\sum_{i+k=15}A_{Y}^{-1}\tau^2\int_{\Sigma_\tau\cap\{r\leq r^+_Y\}}J^{N}_\mu\left(\hat{Y}^k\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau}&\leq \epsilon \tau^{1+\eta_{15}}.
\end{split}
\end{equation}
\begin{equation}\label{BA3}
\begin{split}
\sum_{i+j\leq 14}A_j^{-1}\left(\int_{\Sigma_\tau} J^{Z+N,w^Z}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau} +C\tau^2\int_{\Sigma_\tau\cap\{r\leq \frac{9\tau}{10}\}} J^{N}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}\right)&\\
+\sum_{i+k\leq 14}A_{Y}^{-1}\tau^2\int_{\Sigma_\tau\cap\{r\leq r^+_Y\}}J^{N}_\mu\left(\hat{Y}^k\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau}&\leq \epsilon \tau^{\eta_{14}}.
\end{split}
\end{equation}
\begin{equation}\label{BA4}
\sum_{i+j\leq 15}A_j^{-1}\int_{\Sigma_\tau} J^{N}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}\leq \epsilon .
\end{equation}
\begin{equation}\label{BA5}
\sum_{i+j=13}A_{S,j}^{-1}\int_{\Sigma_\tau} J^{N}_\mu\left(S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}+\sum_{i+k=13}A_{S,Y}^{-1}\int_{\Sigma_\tau}J^{N}_\mu\left(\hat{Y}^kS\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau}\leq \epsilon \tau^{\eta_{S,13}}.
\end{equation}
\begin{equation}\label{BA6}
\begin{split}
\sum_{i+j=12}A_{S,j}^{-1}\left(\int_{\Sigma_\tau} J^{Z+N,w^Z}_\mu\left(S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau} +C\tau^2\int_{\Sigma_\tau\cap\{r\leq \frac{9\tau}{10}\}} J^{N}_\mu\left(S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}\right)&\\
+\sum_{i+k=12}A_{S,Y}^{-1}\tau^2\int_{\Sigma_\tau\cap\{r\leq r^+_Y\}}J^{N}_\mu\left(\hat{Y}^kS\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau}&\leq \epsilon \tau^{1+\eta_{S,12}}.
\end{split}
\end{equation}
\begin{equation}\label{BA7}
\begin{split}
\sum_{i+j\leq 11}A_{S,j}^{-1}\left(\int_{\Sigma_\tau} J^{Z+N,w^Z}_\mu\left(S\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\right) n^{\mu}_{\Sigma_\tau} +C\tau^2\int_{\Sigma_\tau\cap\{r\leq \frac{9\tau}{10}\}} J^{N}_\mu\left(S\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\right) n^{\mu}_{\Sigma_\tau}\right)&\\
+\sum_{i+k\leq 12}A_{S,Y}^{-1}\tau^2\int_{\Sigma_\tau\cap\{r\leq r^+_Y\}}J^{N}_\mu\left(\hat{Y}^kS\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau}&\leq \epsilon \tau^{\eta_{S,11}}
\end{split}
\end{equation}
\begin{equation}\label{BA8}
A_{S,j}^{-1}\sum_{j=0}^{12}\int_{\Sigma_\tau} J^{N}_\mu\left(S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}\leq \epsilon .
\end{equation}
Bootstrap Assumptions (K): We also need bootstrap assumptions for the energy quantities in a spacetime slab.
\begin{equation}\label{BAK1}
\begin{split}
\sum_{i+j=16}A_{X,j}^{-1}\iint_{\mathcal R(\tau_0,\tau)} \left(K^{X_0}\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)+K^{N}\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\right)\leq \epsilon\tau^{\eta_{16}}.
\end{split}
\end{equation}
\begin{equation}\label{BAK1.4}
\sum_{i+j=15}A_{X,j}^{-1}\iint_{\mathcal R(\tau_0,\tau)} K^{X_1}\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\leq \epsilon\tau^{\eta_{16}}.
\end{equation}
\begin{equation}\label{BAK1.5}
\sum_{i+j\leq 15}A_{X,j}^{-1}\iint_{\mathcal R(\tau_0,\tau)} K^{X_0}\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\leq \epsilon.
\end{equation}
\begin{equation}\label{BAK1.6}
\sum_{i+j\leq 14}A_{X,j}^{-1}\iint_{\mathcal R(\tau_0,\tau)} K^{X_1}\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\leq \epsilon.
\end{equation}
\begin{equation}\label{BAK2}
\begin{split}
\sum_{i+j\leq 15}A_{X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}\left(K^{X_0}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)+K^{N}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\right)\leq\epsilon\tau^{-1+\eta_{15}}.
\end{split}
\end{equation}
\begin{equation}\label{BAK3}
\sum_{i+j\leq 14}A_{X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}} K^{X_1}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\leq\epsilon\tau^{-1+\eta_{15}}.
\end{equation}
\begin{equation}\label{BAK4}
\begin{split}
\sum_{i+j\leq 14}A_{X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}\left(K^{X_0}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)+K^{N}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\right)\leq\epsilon\tau^{-2+\eta_{14}}.
\end{split}
\end{equation}
\begin{equation}\label{BAK5}
\sum_{i+j\leq 13}A_{X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}K^{X_1}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\leq\epsilon\tau^{-2+\eta_{14}}.
\end{equation}
\begin{equation}\label{BAK6}
\begin{split}
\sum_{i+j=13}A_{S,X,j}^{-1}\iint_{\mathcal R(\tau_0,\tau)} K^{X_0}\left(S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\leq \epsilon\tau^{\eta_{S,13}}.
\end{split}
\end{equation}
\begin{equation}\label{BAK6.5}
\begin{split}
\sum_{i+j\leq 12}A_{S,X,j}^{-1}\iint_{\mathcal R(\tau_0,\tau)} K^{X_0}\left(S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\leq \epsilon.
\end{split}
\end{equation}
\begin{equation}\label{BAK7}
\begin{split}
\sum_{i+j+k\leq 12}A_{S,X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}K^{X_0}\left(S\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\leq\epsilon\tau^{-1+\eta_{S,12}}.
\end{split}
\end{equation}
\begin{equation}\label{BAK8}
\sum_{i+j\leq 11}A_{S,X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}} K^{X_1}\left(S\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\leq\epsilon\tau^{-1+\eta_{S,12}}.
\end{equation}
\begin{equation}\label{BAK9}
\begin{split}
\sum_{i+j\leq 11}A_{S,X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}K^{X_0}\left(S\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\leq\epsilon\tau^{-2+\eta_{S,11}}.
\end{split}
\end{equation}
\begin{equation}\label{BAK10}
\sum_{i+j\leq 10}A_{S,X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}K^{X_1}\left(S\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\leq\epsilon\tau^{-2+\eta_{S,11}}.
\end{equation}
Bootstrap Assumptions (P): We also introduce bootstrap assumptions for the pointwise behavior.
For $r\geq\frac{t^*}{4}$,
\begin{equation}\label{BAP1}
\sum_{j=0}^{13}|\Gamma^j\Phi|^2\leq BA\epsilon r^{-2}(t^*)^{1+\eta_{14}}.
\end{equation}
\begin{equation}\label{BAP1.5}
\sum_{j=0}^{13}|D\Gamma^j\Phi|^2\leq BA\epsilon.
\end{equation}
\begin{equation}\label{BAP2}
\sum_{\ell=1}^{13-j}\sum_{j=0}^{12}|D^\ell\Gamma^j\Phi|^2\leq BA\epsilon r^{-2}.
\end{equation}
\begin{equation}\label{BAP3}
\sum_{j=0}^8|D\Gamma^j\Phi|^2\leq BA\epsilon r^{-2}(t^*)^{\eta_{14}}(1+|u|)^{-2}.
\end{equation}
\begin{equation}\label{BAP4}
\sum_{j=0}^8|\bar{D}\Gamma^{j}\Phi|^2\leq BA\epsilon r^{-2}(t^*)^{-2+\eta_{14}}.
\end{equation}
\begin{equation}\label{BAP5}
\sum_{j=0}^{6}|S\Gamma^j\Phi|^2\leq B_SA\epsilon r^{-2}(t^*)^{\eta_{S,11}}.
\end{equation}
\begin{equation}\label{BAP6}
\sum_{j=0}^8|DS\Gamma^j\Phi|^2\leq B_SA\epsilon r^{-2}.
\end{equation}
\begin{equation}\label{BAP7}
\sum_{j=0}^6|DS\Gamma^j\Phi|^2\leq B_SA\epsilon r^{-2}(t^*)^{\eta_{S,11}}(1+|u|)^{-2}.
\end{equation}
For $r\leq\frac{t^*}{4}$,
\begin{equation}\label{BAPI1}
\sum_{j=0}^{13}|\Gamma^j\Phi|^2\leq BA\epsilon (t^*)^{-1+\eta_{14}}.
\end{equation}
\begin{equation}\label{BAPI1.1}
\sum_{\ell=1}^{14-j}\sum_{j=0}^{13}|D^\ell\Gamma^j\Phi|^2\leq BA\epsilon (t^*)^{-1+\eta_{14}}.
\end{equation}
\begin{equation}\label{BAPI1.2}
\sum_{\ell=1}^{13-j}\sum_{j=0}^{12}|D^\ell\Gamma^j\Phi|^2\leq BA\epsilon (t^*)^{-2+\eta_{14}}.
\end{equation}
\begin{equation}\label{BAPI2}
\sum_{\ell=1}^{9-j}\sum_{j=0}^8|D^\ell\Gamma^j\Phi|^2\leq BA\epsilon (t^*)^{-3+\eta_{S,11}}r^{-1+\delta}.
\end{equation}
\begin{equation}\label{BAPI3}
\sum_{j=0}^{6}|S\Gamma^j\Phi|^2\leq B_SA\epsilon (t^*)^{-2+\eta_{S,11}}.
\end{equation}
\begin{equation}\label{BAPI4}
\sum_{\ell=1}^{7-j}\sum_{j=0}^{6}|D^\ell S\Gamma^j\Phi|^2\leq B_SA\epsilon r^{-2}(t^*)^{-2+\eta_{S,11}}.
\end{equation}
\begin{remark}
Notice that in general, for most of the bootstrap assumptions on $\Phi$, there is a corresponding one on $S\Phi$. The argument to retrieve these assumptions are quite similar, we only have to estimate the commutator term (in a manner similar to \cite{LS}, \cite{LKerr}) and track the appropriate constants.
\end{remark}
\begin{remark}
Notice that all this assumption are satisfied initially by the assumption of the Theorem.
\end{remark}
\begin{remark}
We will bootstrap to improve the constants $A_j$, $A_{X,j}$, $A_S$, $A_{S,X,j}$, $A_Y$, $A_{S,Y}$ and $B$. The constant $B$ is only used for the bootstrap of the pointwise estimates. The constants satisfy
$$1\ll B\sim B_S\ll A_0\ll A_{X,0}\ll A_1\ll ...\ll A_{16}\ll A_{X,16}\ll A_Y\ll A_{S,0}\ll A_{S,X,0}\ll ... \ll A_{S,X,13}\ll A_{S,Y}.$$
We will use $A$ as a shorthand to denote the maximum of all these constants, i.e., $A_{S,Y}$. We will always assume by taking $\epsilon$ small that
$$A\epsilon\ll 1.$$
Moreover, we set the constants so that
$$\frac{A_{j-1}}{A_j}\ll\frac{A_{X,j-1}}{A_j}\ll\delta'\eta^{-1}.$$
$$\delta\sim\delta'\ll\frac{A_j}{A_{X,j}}.$$
The $\eta$'s, on the other hand, satisfy
$$\delta\sim\delta'\ll\eta_{16}\ll\eta_{15}\ll\eta_{14}\ll\eta_{S,13}\ll\eta_{S,12}\ll\eta_{S,11}.$$
The $\eta$'s are chosen so that
$$\frac{A_j}{A_{X,j}}\ll\eta_{14}\ll 1\quad\mbox{for all }j.$$
$\epsilon$ will be much smaller than any combinations of the other constants.
\end{remark}
We will use energy estimates and decay estimates to eventually close the bootstrap. In order to derive the estimates, we consider equations for $\Gamma^k\Phi$. We now introduce the notations that will facilitate the discussion below.
\begin{definition}
Define $G_k=\displaystyle\sum_{|j|=k}|\Box_{g_K}\left(\Gamma^j\Phi\right)|$, $U_k=\displaystyle\sum_{|j|=k}|[\Box_{g_K},\Gamma^j]\Phi|$ and $N_k=\displaystyle\sum_{|j|=k}|\Gamma^j\left(\Box_{g_K}\Phi\right)|$.
\end{definition}
\begin{definition}
Define $G_{\leq k}=\displaystyle\sum_{|j|\leq k}|\Box_{g_K}\left(\Gamma^j\Phi\right)|$, $U_{\leq k}=\displaystyle\sum_{|j|\leq k}|[\Box_{g_K},\Gamma^j]\Phi|$ and $N_{\leq k}=\displaystyle\sum_{|j|\leq k}|\Gamma^j\left(\Box_{g_K}\Phi\right)|$.
\end{definition}
In order to keep track of the constants, we define also
\begin{definition}
Define $U_{k,j}=|[\Box_{g_K},\partial_{t^*}^{k-j}\tilde{\Omega}^j]\Phi|$ and $U_{\leq k,\leq j}=\displaystyle\sum_{j'\leq j, k'\leq k} U_{k',j'}$.
\end{definition}
\begin{remark}
We will refer to $G$ as the inhomogeneous term, $U$ as the commutator term and $N$ as the nonlinear term. Clearly we have $G_k\leq U_k+N_k$ and $G_{\leq k}\leq U_{\leq k}+N_{\leq k}$
\end{remark}
We now estimate the inhomogeneous terms that will appear in the analysis several times below. It is necessary to study the commutator terms and the nonlinear terms together because the estimates for each depend on the estimates for the other when we use elliptic estimates.
\begin{proposition}\label{Uestprop}
$U_k$ satisfies the following estimates:
\begin{equation}\label{Uest}
\begin{split}
&\int_{\Sigma_\tau}r^{\alpha}\left(D^\ell U_{k,j}\right)^2\\
\leq& C\left(\sum_{m=0}^{k+\ell-i}\sum_{i=0}^{j-1}\int_{\Sigma_\tau\cap\{r\geq R_\Omega-1\}}r^{\alpha-4}J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{m=0}^{k+\ell-i-1}\sum_{i=0}^{j-1}\int_{\Sigma_\tau}r^{\alpha-4}\left(D^m N_i\right)^2\right)\quad\mbox{for }\alpha\leq 4,
\end{split}
\end{equation}
where it is understood that $\displaystyle\sum_{i=0}^{-1}=0$.
\end{proposition}
\begin{proof}
The commutator terms are estimated in \cite{LS}. Notice that since $\tilde{\Omega}$ is supported away from the trapped set, there is no loss of derivatives in using the integrated decay estimate. We have that $U_{k,j}$ supported in $\{r\geq R_\Omega\}$ and that
$$|U_{k,j}|\leq C\sum_{i=0}^j|\partial_{t^*}^{k-j}[\Box_{g_K},\tilde{\Omega}^i]\Phi|\leq C\sum_{i=0}^{j-1}r^{-2}\left(|D^2\partial_{t^*}^{k-j}\tilde{\Omega}^i\Phi|+|D\partial_{t^*}^{k-j}\tilde{\Omega}^i\Phi|\right),$$
and therefore
$$|D^\ell U_{k,j}|\leq C\sum_{m=1}^{\ell+2}\sum_{i=0}^{j-1}r^{-2}|D^m\partial_{t^*}^{k-j}\tilde{\Omega}^i\Phi|\leq C\sum_{m=1}^{k+\ell-j+2}\sum_{i=0}^{j-1}r^{-2}|D^m\tilde{\Omega}^i\Phi|,$$
where, as in the statement of the Proposition it is understood that the sum vanishes if $j=0$.
Hence, using the elliptic estimate for $\{r\geq R_\Omega\}$, i.e., Proposition \ref{ellipticoutside},
\begin{equation}\label{Uinduct}
\begin{split}
&\int_{\Sigma_\tau}r^{\alpha}\left(D^\ell U_{k,j}\right)^2\\
\leq&C\sum_{m=1}^{k+\ell-j+2}\sum_{i=0}^{j-1}\int_{\Sigma_\tau\cap\{r\geq R_\Omega\}}r^{\alpha-4}\left(D^m\tilde{\Omega}^i\Phi\right)^2\\
\leq&C\sum_{m=0}^{k+\ell-i}\sum_{i=0}^{j-1}\int_{\Sigma_\tau\cap\{r\geq R_\Omega-1\}}r^{\alpha-4}J^N_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_{\tau}}\\
&+C\sum_{m=0}^{k+\ell-i-1}\sum_{i=0}^{j-1}\int_{\Sigma_\tau\cap\{r\geq R_\Omega-1\}}r^{\alpha-4}\left(\left(D^{m}U_{i,i}\right)^2+\left(D^{m}N_{i}\right)^2\right).\\
\end{split}
\end{equation}
Now we can estimate $U_k$ by induction: Fix any $k$ and we will induct on $j$. By definition, $U_{k,0}=0$. Now, assume that for all $k+\ell\leq 16$ and for some $j_0\geq 1$, we have
\begin{equation*}
\begin{split}
&\sum_{k+\ell\leq M, j\leq min\{j_0-1,k\}}\int_{\Sigma_\tau}r^{\alpha}\left(D^\ell U_{k,j}\right)^2\\
\leq& C\left(\sum_{m=0}^{M-i}\sum_{i=0}^{j_0-2}\int_{\Sigma_\tau\cap\{r\geq R_\Omega-1\}}r^{\alpha-4}J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{m=0}^{M-1-i}\sum_{i=0}^{j_0-1}\int_{\Sigma_\tau}r^{\alpha-4}\left(D^m N_i\right)^2\right).
\end{split}
\end{equation*}
for all $\alpha\leq 4$. Then, using (\ref{Uinduct}), we have that for $k+\ell\leq 16$, and $j_0\leq k$,
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau}r^{\alpha}\left(D^\ell U_{k,j_0}\right)^2\\
\leq& C\left(\sum_{m=0}^{k+\ell-i}\sum_{i=0}^{j_0-1}\int_{\Sigma_\tau\cap\{r\geq R_\Omega-1\}}r^{\alpha-4}J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{m=0}^{k+\ell-i-1}\sum_{i=0}^{j_0-1}\int_{\Sigma_\tau}r^{\alpha-4}\left(D^m N_i\right)^2\right).
\end{split}
\end{equation*}
Hence, (\ref{Uest}) holds.
\end{proof}
We now estimate the nonlinear term $N_k$. Since $N_k$ is at least quadratic, we do not need to be precise about the constants $A$ and we will always estimate with the maximum $A$.
\begin{proposition}\label{Nestprop}
$N_k$ satisfies the following estimates for fixed $t^*=\tau$:
\begin{equation*}
\begin{split}
\sum_{k+\ell= 16}\int_{\Sigma_\tau}\left(D^\ell N_k\right)^2\leq CBA^2\epsilon^2\tau^{-2+\eta_{16}},
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\sum_{k+\ell\leq 15}\int_{\Sigma_\tau}\left(D^\ell N_k\right)^2\leq CBA^2\epsilon^2\tau^{-2}.
\end{split}
\end{equation*}
$N_k$ also satisfy the following estimates for when integrated over $t^*\in [(1.1)^{-1}\tau,\tau]$:
\begin{equation*}
\begin{split}
\sum_{k+\ell= 15}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{-1-\delta}\left(D^\ell N_k\right)^2\leq CBA^2\epsilon^2\tau^{-2+\eta_{16}},
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\sum_{k+\ell\leq 14}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{-1-\delta}\left(D^\ell N_k\right)^2\leq CBA^2\epsilon^2\tau^{-2}.
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
Here, there is no need to distinguish between the good and bad derivatives.
$$|D^\ell N_k|\leq |D^\ell\Gamma^k\left(\Lambda_i D\Phi D\Phi\right)|+|\Gamma^k \mathcal C|.$$
We claim that the most important terms will be those that are quadratic in $D^{j+1}\Gamma^i\Phi$ or cubic of the form $$\left(D^{j_1+1}\Gamma^{i_1}\Phi \right)\left(D^{j_2+1}\Gamma^{i_2}\Phi\right)\left(\Gamma^{i_3}\Phi\right)$$ with $i_1+j_1,i_2+j_2\leq 8$. For by the assumptions every term has the form
$$\left(D^{j_1}\Gamma^{i_1}\Phi \right)\left(D^{j_2}\Gamma^{i_2}\Phi\right)\left(D^{j_3}\Gamma^{i_3}\Phi\right)\left(D^{j_4}\Gamma^{i_4}\Phi\right)...\left(D^{j_r}\Gamma^{i_r}\Phi\right),$$
with $r\geq 2$, at least two $j$'s $\geq 1$ and $i+j\leq 9$ for all but at most one factor. If all factors $i+j\leq 9$ or the factor that $i+j>9$ has $i\geq 1$, we can reduce to the case $D^{j_1+1}\Gamma^{i_1}\Phi D^{j_2+1}\Gamma^{i_2}\Phi$ by putting all other factors in $L^\infty$ using bootstrap assumptions (\ref{BAP1}), (\ref{BAP1.5}), (\ref{BAPI1}) and (\ref{BAPI1.1}). If the factor that $i+j>9$ has $i=0$ we reduce to
$$\left(D^{j_1+1}\Gamma^{i_1}\Phi \right)\left(D^{j_2+1}\Gamma^{i_2}\Phi\right)\left(\Gamma^{i_3}\Phi\right)$$
again by putting all other factors in $L^\infty$ using bootstrap assumptions (\ref{BAP1}), (\ref{BAP1.5}), (\ref{BAPI1}) and (\ref{BAPI1.1}).
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau}\left(D^\ell N_k\right)^2\\
\leq &C\left(\sup\sum_{i_1+j_1\leq 8}|D^{j_1+1}\Gamma^{i_1}\Phi|^2\right)\sum_{j_2=0}^\ell\sum_{i_2=0}^k\int_{\Sigma_\tau}| D^{j_2+1}\Gamma^{i_2}\Phi|^2\\
&+C\left(\sup\sum_{i_1+j_1\leq 8}|D^{j_1+1}\Gamma^{i_1}\Phi|^2\right)\left(\sup\sum_{i_2+j_2\leq 8}r^2| D^{j_2+1}\Gamma^{i_2}\Phi|^2\right)\sum_{i_3=0}^k\int_{\Sigma_\tau}r^{-2}| \Gamma^{i_3}\Phi|^2\\
\leq&CBA\epsilon\tau^{-2}\sum_{i=1}^{\ell+1}\sum_{j=0}^k\int_{\Sigma_\tau} \left(D^i\Gamma^j\Phi\right)^2\\
&\quad\mbox{using Hardy's inequality Proposition \ref{Hardy}}\\
\leq&CBA\epsilon\tau^{-2}\sum_{i+m=0}^{\ell}\sum_{j=0}^k\int_{\Sigma_\tau} J^N_\mu\left(\partial_{t^*}^m\Gamma^j\hat{Y}^i\Phi\right)n^\mu_{\Sigma_{\tau}}+CBA\epsilon\tau^{-2}\sum_{i+j\leq k+\ell-1}\int_{\Sigma_\tau}\left(\left(D^iU_{\leq j}\right)^2+\left(D^iN_{\leq j}\right)^2\right)\\
&\quad\mbox{using the elliptic estimates in Propositions \ref{elliptic}, \ref{elliptichorizon}}\\
\leq&CBA\epsilon\tau^{-2}\sum_{i=0}^{\ell}\sum_{j=0}^k\int_{\Sigma_\tau} J^N_\mu\left(\partial_{t^*}^m\Gamma^j\hat{Y}^i\Phi\right)n^\mu_{\Sigma_{\tau}}+CBA\epsilon\tau^{-2}\sum_{i+j\leq k+\ell-1}\int_{\Sigma_\tau}\left(D^iN_{\leq j}\right)^2,\\
\end{split}
\end{equation*}
where we have used Proposition \ref{Uestprop} in the last step.
Now, a simple induction would show that
\begin{equation*}
\begin{split}
\sum_{k+\ell= 16}\int_{\Sigma_\tau}\left(D^\ell N_k\right)^2\leq CBA^2\epsilon^2\tau^{-2+\eta},\quad\mbox{and}
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\sum_{k+\ell\leq 15}\int_{\Sigma_\tau}\left(D^\ell N_k\right)^2\leq CBA^2\epsilon^2\tau^{-2}.
\end{split}
\end{equation*}
We now move on the the terms integrated over $t^*\in [(1.1)^{-1}\tau,\tau]$. Arguing as before, and noticing that the elliptic estimate in Proposition \ref{elliptic} also allows weights in $r$, we have
\begin{equation*}
\begin{split}
&\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{-1-\delta}\left(D^\ell N_k\right)^2\\
\leq&CBA\epsilon\tau^{-2}\sum_{i+m=0}^{\ell}\sum_{j=0}^k\iint_{\mathcal R((1.1)^{-1}\tau,\tau)} r^{-1-\delta}J^N_\mu\left(\partial_{t^*}^m\Gamma^j\hat{Y}^i\Phi\right)n^\mu_{\Sigma_{\tau}}\\
&+CBA\epsilon\tau^{-2}\sum_{i+j\leq k+\ell-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{-1-\delta}\left(\left(D^iU_{\leq j}\right)^2+\left(D^iN_{\leq j}\right)^2\right)\\
\leq&CBA\epsilon\tau^{-2}\sum_{i=0}^{\ell}\sum_{j=0}^k\iint_{\mathcal R((1.1)^{-1}\tau,\tau)} r^{-1-\delta}J^N_\mu\left(\partial_{t^*}^m\Gamma^j\hat{Y}^i\Phi\right)n^\mu_{\Sigma_{\tau}}\\
&+CBA\epsilon\tau^{-2}\sum_{i+j\leq k+\ell-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{-1-\delta}\left(D^iN_{\leq j}\right)^2\\
\leq&CBA\epsilon\tau^{-2}\sum_{i=0}^{\ell}\sum_{j=0}^k\iint_{\mathcal R((1.1)^{-1}\tau,\tau)} K^{X_1}\left(\partial_{t^*}^m\Gamma^j\hat{Y}^i\Phi\right)\\
&+CBA\epsilon\tau^{-2}\sum_{i+j\leq k+\ell-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{-1-\delta}\left(D^iN_{\leq j}\right)^2\\
\end{split}
\end{equation*}
Now, the bootstrap assumptions (\ref{BAK1.4}), (\ref{BAK1.6}) and an induction on $k+\ell$ would conclude the Proposition.
\end{proof}
Now, the estimates for $N_k$ will also give improved estimates for $U_k$ via Proposition \ref{Uestprop}:
\begin{proposition}\label{U}
The following estimates for $U_k$ on a fixed $t^*$ slice hold for $\alpha\leq 2$:
$$\sum_{k+\ell=16}\int_{\Sigma_\tau}r^{\alpha}\left(D^\ell U_{k,j}\right)^2\leq CA_{j-1}\epsilon\tau^{\eta_{16}},$$
$$\sum_{k+\ell=15}\int_{\Sigma_\tau}r^{\alpha}\left(D^\ell U_{k,j}\right)^2\leq CA_{j-1}\epsilon\tau^{-1+\eta_{15}},$$
$$\sum_{k+\ell\leq 14}\int_{\Sigma_\tau}r^{\alpha}\left(D^\ell U_{k,j}\right)^2\leq CA_{j-1}\epsilon\tau^{-2+\eta_{14}}.$$
The following estimates for $U_k$ integrated on $[(1.1)^{-1}\tau,\tau]$ also hold for $\alpha\leq 1+\delta$:
$$\sum_{k+\ell=16}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{\alpha}\left(D^\ell U_{k,j}\right)^2\leq CA_{X,j-1}\epsilon\tau^{\eta_{16}},$$
$$\sum_{k+\ell=15}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{\alpha}\left(D^\ell U_{k,j}\right)^2\leq CA_{X,j-1}\epsilon\tau^{-1+\eta_{15}},$$
$$\sum_{k+\ell\leq 14}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{\alpha}\left(D^\ell U_{k,j}\right)^2\leq CA_{X,j-1}\epsilon\tau^{-2+\eta_{14}}.$$
\end{proposition}
\begin{proof}
We first prove the Proposition for the estimates for the constant in $\tau$ terms. By Proposition \ref{Uestprop},
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau}r^{\alpha}\left(D^\ell U_{k,j}\right)^2\\
\leq& C\left(\sum_{m=0}^{k+\ell-i}\sum_{i=0}^{j-1}\int_{\Sigma_\tau\cap\{r\geq R_\Omega-1\}}r^{\alpha-4}J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{m=0}^{k+\ell-1-i}\sum_{i=0}^{j-1}\int_{\Sigma_\tau}r^{\alpha-4}\left(D^m N_i\right)^2\right)\quad\mbox{for }\alpha\leq 4.
\end{split}
\end{equation*}
The second term satisfy the required estimate by Proposition \ref{Nestprop}.
We estimate the first term. By (\ref{BA1}),
$$\sum_{m=0}^{16-i}\sum_{i=0}^{j-1}\int_{\Sigma_\tau\cap\{r\geq R_\Omega-1\}}r^{\alpha-4}J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}\leq CA_{j-1}\epsilon\tau^{\eta_{16}}.$$
By (\ref{BA2}) and Proposition \ref{Zlowerbound},
\begin{equation*}
\begin{split}
&\sum_{m=0}^{15-i}\sum_{i=0}^{j-1}\int_{\Sigma_\tau\cap\{r\geq R_\Omega-1\}}r^{\alpha-4}J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}\\
\leq&C\left(\sum_{m=0}^{15-i}\sum_{i=0}^{j-1}\int_{\Sigma_\tau\cap\{r\leq \frac{\tau}{2}\}}J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{m=0}^{15-i}\sum_{i=0}^{j-1}\tau^{-2}\int_{\Sigma_\tau\cap\{r\geq \frac{\tau}{2}\}}J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}\right)\\
\leq&CA_{j-1}\epsilon\tau^{-1+\eta_{15}}.
\end{split}
\end{equation*}
By (\ref{BA3}) and Proposition \ref{Zlowerbound},
\begin{equation*}
\begin{split}
&\sum_{m=0}^{14-i}\sum_{i=0}^{j-1}\int_{\Sigma_\tau\cap\{r\geq R_\Omega-1\}}r^{\alpha-4}J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}\\
\leq&C\left(\sum_{m=0}^{14-i}\sum_{i=0}^{j-1}\int_{\Sigma_\tau\cap\{r\leq \frac{\tau}{2}\}}J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{m=0}^{14-i}\sum_{i=0}^{j-1}\tau^{-2}\int_{\Sigma_\tau\cap\{r\geq \frac{\tau}{2}\}}J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}\right)\\
\leq&CA_{j-1}\epsilon\tau^{-2+\eta_{14}}.
\end{split}
\end{equation*}
For the integrated terms, we similarly have, by Proposition \ref{Uestprop},
\begin{equation*}
\begin{split}
&\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{\alpha}\left(D^\ell U_{k,j}\right)^2\\
\leq& C\left(\sum_{m=0}^{k+\ell-i}\sum_{i=0}^{j-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\geq R_\Omega-1\}}r^{\alpha-4}J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}\right.\\
&\left.\quad\quad+\sum_{m=0}^{k+\ell-1-i}\sum_{i=0}^{j-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{\alpha-4}\left(D^m N_i\right)^2\right)\quad\mbox{for }\alpha\leq 4.
\end{split}
\end{equation*}
The second term can be estimated by Proposition \ref{Nestprop}. Notice that $r^{-1+\delta}J^N_\mu\left(\Phi\right)n^\mu_{\Sigma_\tau}\leq K^{X_0}\left(\Phi\right)$. Hence, following the argument above for the fixed $\tau$ case, we would have proved the Proposition for the case $\alpha\leq 1-\delta$. Nevertheless, with more care, we can improve to $\alpha\leq 1+\delta$.
\begin{equation*}
\begin{split}
&\sum_{m=0}^{k+\ell-i}\sum_{i=0}^{j-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\geq R_\Omega\}}r^{\alpha-4}J^{N}_\mu\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_{t^*}}\\
\leq&C\sum_{m=0}^{k+\ell-i}\sum_{i=0}^{j-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq \frac{t^*}{2}\}}K^{X_0}\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right) \\
&+C\sum_{m=0}^{k+\ell-i}\sum_{i=0}^{j-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\geq \frac{t^*}{2}\}}r^{-2+2\delta}K^{X_0}\left(\partial_{t^*}^m\tilde{\Omega}^i\Phi\right).
\end{split}
\end{equation*}
For $k+\ell=16$, this is bounded by $CA_{X,j-1}\epsilon\tau^{\eta_{16}}$ by (\ref{BAK1}). For $k+\ell=15$, this is bounded by $CA_{X,j-1}\epsilon\tau^{-1+\eta_{15}}$ by (\ref{BAK2}) and (\ref{BAK1.5}). For $k+\ell\leq 14$, this is bounded by $CA_{X,j-1}\epsilon\tau^{-2+\eta_{14}}$ by (\ref{BAK4}) and (\ref{BAK1.5}) since $2\delta\leq \eta_{14}$.
\end{proof}
While the above is sufficient to recover the bootstrap assumptions for the pointwise bounds, we will need improvements to achieve the energy bounds. For the improvements, we study separately the region $r\leq\frac{t^*}{4}$, $\frac{t^*}{4}\leq r\leq\frac{9t^*}{10}$ and $r\geq\frac{9t^*}{10}$. For the region $r\geq\frac{t^*}{4}$, we will only show the improvement for $N_k$ instead of the derivatives of $N_k$. Various complications would arise in estimating the derivatives of $N_k$. For the region $r\leq\frac{t^*}{4}$, however, we will estimate also the derivatives of $N_k$ as they will be necessary to estimate the error terms arising from commuting with the red-shift vector field.
\begin{proposition}
\begin{equation*}
\sum_{\ell+k=16}\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}r^{1-\delta}\left(D^\ell N_k\right)^2 \leq CBA^2\epsilon^2\tau^{-3+\eta_{S,{11}}+\eta_{16}}.
\end{equation*}
\begin{equation*}
\sum_{\ell+k=15}\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}r^{1-\delta}\left(D^\ell N_k\right)^2 \leq CBA^2\epsilon^2\tau^{-4+\eta_{S,{11}}+\eta_{15}}.
\end{equation*}
\begin{equation*}
\sum_{\ell+k\leq 14}\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}r^{1-\delta}\left(D^\ell N_k\right)^2 \leq CBA^2\epsilon^2\tau^{-5+\eta_{S,{11}}+\eta_{14}}.
\end{equation*}
\end{proposition}
\begin{proof}
As before, we only have to estimate terms quadratic in $D^j\Gamma^i\Phi$ with $j\geq 1$ or cubic of the form $$\left(D^{j_1+1}\Gamma^{i_1}\Phi \right)\left(D^{j_2+1}\Gamma^{i_2}\Phi\right)\left(\Gamma^{i_3}\Phi\right)$$ with $i_1+j_1,i_2+j_2\leq 8$.
\begin{equation}\label{Nlocal}
\begin{split}
&\int_{\Sigma_\tau}r^{1-\delta}\left(D^\ell N_k\right)^2\\
\leq &C\left(\sup_{r\leq\frac{\tau}{4}}\sum_{i_1+j_1\leq 8}r^{1-\delta}|D^{j_1+1}\Gamma^{i_1}\Phi|^2\right)\sum_{j_2=0}^\ell\sum_{i_2=0}^k\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}| D^{j_2+1}\Gamma^{i_2}\Phi|^2\\
&+C\left(\sup_{r\leq\frac{\tau}{4}}\sum_{i_1+j_1\leq 8}r^{1-\delta}|D^{j_1+1}\Gamma^{i_1}\Phi|^2\right)^2\sum_{i_3=0}^k\tau^{2\delta}\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}r^{-2}| \Gamma^{i_3}\Phi|^2\\
\leq&CBA\epsilon\tau^{-3+\eta_{S,11}}\sum_{i=1}^{\ell+1}\sum_{j=0}^k\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}} \left(D^i\Gamma^j\Phi\right)^2+CBA^2\epsilon^2\tau^{-6+2\eta_{S,11}+2\delta}\sum_{i_3=0}^k\int_{\Sigma_\tau}\left( D\Gamma^{i_3}\Phi\right)^2\\
&\mbox{by Hardy's inequality. Notice now that the second term has more decay than we need,}\\
&\mbox {so we will drop it from now on.}\\
\leq&CBA\epsilon\tau^{-3+\eta_{S,11}}\sum_{i+m=0}^{\ell}\sum_{j=0}^k\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{2}\}} J^N_\mu\left(\partial_{t^*}^m\Gamma^j\hat{Y}^i\Phi\right)n^\mu_{\Sigma_{\tau}}\\
&+CBA\epsilon\tau^{-3+\eta_{S,11}}\sum_{i=0}^{\ell-1}\sum_{j=0}^k\int_{\Sigma_\tau}\left(\left(D^iU_{j}\right)^2+\left(D^iN_{j}\right)^2\right)\\
\leq&CBA\epsilon\tau^{-3+\eta_{S,11}}\sum_{i+j\leq k+\ell}\int_{\Sigma_\tau} J^N_\mu\left(\Gamma^j\hat{Y}^i\Phi\right)n^\mu_{\Sigma_{\tau}}+CBA\epsilon\tau^{-3+\eta_{S,11}}\sum_{i+j\leq k+\ell-1}\int_{\Sigma_\tau}\left(D^iN_{j}\right)^2,\\
\end{split}
\end{equation}
The Proposition would follow from an induction on $k+\ell$ and the bootstrap assumptions (\ref{BA1}), (\ref{BA2}), (\ref{BA3}). The $k+\ell=0$ case also follows from the above computation as we have adopted the notation that $\displaystyle\sum_{i+j\leq -1} =0$.
\end{proof}
We now move to the region $\{\frac{t^*}{4}\leq r\leq\frac{9t^*}{10}\}$. In this region, $u\sim t^*$, and therefore we can exploit the decay in the variable $u$ given by the estimates from the conformal energy.
\begin{proposition}
$$\int_{\Sigma_\tau\cap\{\frac{\tau}{4}\leq r\leq\frac{9\tau}{10}\}}N_{16}^2\leq CBA^2\epsilon^2\tau^{-4+\eta_{14}+\eta_{16}},$$
$$\int_{\Sigma_\tau\cap\{\frac{\tau}{4}\leq r\leq\frac{9\tau}{10}\}}N_{15}^2\leq CBA^2\epsilon^2\tau^{-5+\eta_{14}+\eta{15}},$$
and
$$\sum_{j=0}^{14}\int_{\Sigma_\tau\cap\{\frac{\tau}{4}\leq r\leq\frac{9\tau}{10}\}}N_{j}^2\leq CBA^2\epsilon^2\tau^{-6+2\eta_{14}}.$$
\end{proposition}
\begin{proof}
Arguing as before, we see that the main terms for the nonlinearity are those that are quadratic in $D\Phi$ or those that are cubic with the form $\Gamma^{i_3}\Phi D\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi$ with $i_1, i_2\leq 8$. The quadratic terms can be estimated:
\begin{equation*}
\begin{split}
&\sum_{i_1=0}^{\lfloor\frac{k}{2}\rfloor}\sum_{i_2=0}^{k}\int_{\Sigma_\tau\cap\{\frac{\tau}{4}\leq r\leq\frac{9\tau}{10}\}}|{D}\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi|^2\\
\leq &C \left(\sum_{i_1=0}^8 \sup_{\frac{\tau}{4}\leq r\leq\frac{9\tau}{10}}|{D}\Gamma^{i_1}\Phi|^2\right)\sum_{i_2=0}^{k}\int_{\Sigma_\tau\cap\{\frac{\tau}{4}\leq r\leq\frac{9\tau}{10}\}}| D\Gamma^{i_2}\Phi|^2\\
\leq&CAB\epsilon\tau^{-4+\eta_{14}}\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{\frac{\tau}{4}\leq r\leq\frac{9\tau}{10}\}}| D\Gamma^{i}\Phi|^2.
\end{split}
\end{equation*}
The particular cubic term can be estimated as follows:
\begin{equation*}
\begin{split}
&\sum_{i_1, i_2=0}^{\lfloor\frac{k}{2}\rfloor}\sum_{i_3=0}^{k}\int_{\Sigma_\tau\cap\{\frac{\tau}{4}\leq r\leq\frac{9\tau}{10}\}}|\Gamma^{i_3}\Phi D\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi|^2\\
\leq &C \left(\sum_{i_1=0}^8 \sup_{\frac{\tau}{4}\leq r\leq\frac{9\tau}{10}}|{D}\Gamma^{i_1}\Phi|^2\right)^2\sum_{i_3=0}^{k}\int_{\Sigma_\tau\cap\{\frac{\tau}{4}\leq r\leq\frac{9\tau}{10}\}}| \Gamma^{i_3}\Phi|^2\\
\leq&CA^2B^2\epsilon^2\tau^{-8+2\eta_{14}}\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{\frac{\tau}{4}\leq r\leq\frac{9\tau}{10}\}}| \Gamma^{i}\Phi|^2.
\end{split}
\end{equation*}
In principle, for $k\leq 15$, we can then control the last term using the conformal energy. For $k=16$, however, conformal energy is not available, and we need to use Hardy's inequality.
\begin{equation*}
\begin{split}
&A\epsilon\tau^{-8+2\eta_{14}}\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{\frac{\tau}{4}\leq r\leq\frac{9\tau}{10}\}}| \Gamma^{i}\Phi|^2\\
\leq &CA\epsilon\tau^{-6+2\eta_{14}}\sum_{i=0}^{k}\int_{\Sigma_\tau}r^{-2}| \Gamma^{i}\Phi|^2\\
\leq &CA\epsilon\tau^{-6+2\eta_{14}}\sum_{i=0}^{k}\int_{\Sigma_\tau} J^N_\mu\left(\Gamma^i\Phi\right) n^\mu_{\Sigma_\tau}.
\end{split}
\end{equation*}
The estimates now follow from the Bootstrap Assumptions (\ref{BA1}), (\ref{BA2}), (\ref{BA3}).
\end{proof}
For many applications, we only need a much weaker estimate on $N_k$. We write down the following Proposition which corresponds to the estimates that will be proved for the quantities involving $S$. This would allow a unified approach in dealing with many estimates with or without $S$.
\begin{proposition}\label{NI}
\begin{equation*}
\begin{split}
\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}r^{1-\delta}N_{16}^2\leq CA^2\epsilon^2\tau^{-3+\eta_{S,11}+\eta_{16}}
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}r^{1-\delta}N_{\leq 15}^2\leq CA^2\epsilon^2\tau^{-4+\eta_{S,11}+\eta_{15}}
\end{split}
\end{equation*}
\end{proposition}
We now move to the estimates for $N_k$ in the region $\{r\geq \frac{9t^*}{10}\}$. Here, we need to exploit the null condition:
\begin{proposition}\label{NO}
For $\alpha=0$ or $2$,
$$\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}N_{16}^2\leq CBA^2\epsilon^2\tau^{-2+\eta_{16}},$$
$$\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha}N_{15}^2\leq CBA^2\epsilon^2\tau^{-3+\alpha+\eta_{15}},$$
and
$$\sum_{j=0}^{14}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha}N_j^2\leq CBA^2\epsilon^2\tau^{-4+\alpha+\eta_{14}}.$$
\end{proposition}
\begin{proof}
Following the argument before, we reduce to quadratic and cubic terms. This time, however, the null condition plays a crucial role. For the quadratic terms, we need to consider
$$\left(\bar{D}\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi\right),\left(D\Gamma^{i_1}\Phi \bar{D}\Gamma^{i_2}\Phi\right),r^{-1} \left(D\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi\right),$$
where $i_1\geq i_2$.
For the cubic terms, we need to consider
$$\left({D}\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi D\Gamma^{i_3}\Phi\right), \left(\bar{D}\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi \Gamma^{i_3}\Phi\right).$$
Notice that for the first cubic term can be dominated pointwise by quadratic terms of the third type listed above using the bootstrap assumptions (\ref{BAP2}) and (\ref{BAPI2}). The second cubic term can also be dominated by the first two types of quadratic terms if $i_3\leq 13$ by (\ref{BAP1}) and (\ref{BAPI1}). We can thus assume $i_3>13$ and hence $i_1, i_2\leq 8$. We now estimate the quadratic terms
\begin{equation*}
\begin{split}
&\sum_{i_2=0}^{\lfloor\frac{k}{2}\rfloor}\sum_{i_1=0}^{k-j}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha}\left(|\bar{D}\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi|^2+|D\Gamma^{i_1}\Phi \bar{D}\Gamma^{i_2}\Phi|^2+r^{-2} |D\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi|^2\right)\\
\leq&C\left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_2=0}^{8}r^2|D\Gamma^{i_2}\Phi|^2\right)\left(\sum_{i_1=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} r^{\alpha-2}|\bar{D}\Gamma^{i_1}\Phi|^2\right)\\
&+C\left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_2=0}^{8}r^2|\bar{D}\Gamma^{i_2}\Phi|^2\right)\left(\sum_{i_1=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha-2}| D\Gamma^{i_1}\Phi|^2\right)\\
&+C\tau^{-2}\left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_2=0}^{8}r^2|D\Gamma^{i_2}\Phi|^2\right)\left(\sum_{i_1=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} r^{\alpha-2}|{D}\Gamma^{i_1}\Phi|^2\right)\\
\leq &CAB\epsilon\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} r^{\alpha-2}|\bar{D}\Gamma^{i}\Phi|^2\right)+CAB\epsilon\tau^{-2+\eta_{14}}\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} r^{\alpha-2}|{D}\Gamma^{i}\Phi|^2\right).
\end{split}
\end{equation*}
We then estimate the particular cubic term:
\begin{equation*}
\begin{split}
&\sum_{i_3=0}^{k}\sum_{i_1, i_2=0}^{8}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha}\left(\bar{D}\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi \Gamma^{i_3}\Phi\right)^2\\
\leq&C\left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_1=0}^{8}r^2|D\Gamma^{i_2}\Phi|^2\right)\left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_2=0}^{8}r^2|\bar{D}\Gamma^{i_2}\Phi|^2\right)\left(\sum_{i_3=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} r^{\alpha-4}|\Gamma^{i_1}\Phi|^2\right)\\
\leq &CA^2B^2\epsilon^2\tau^{-2+\eta_{14}}\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} r^{\alpha-4}|\Gamma^{i}\Phi|^2\right).
\end{split}
\end{equation*}
Therefore,
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}N_{16}^2\\
\leq &CAB\epsilon\tau^{-2}\left(\sum_{i=0}^{16}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} |\bar{D}\Gamma^{i}\Phi|^2\right)+CAB\epsilon\tau^{-4+\eta_{14}}\left(\sum_{i=0}^{16}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} |{D}\Gamma^{i}\Phi|^2\right)\\
&+CA^2B^2\epsilon^2\tau^{-4+\eta_{14}}\left(\sum_{i=0}^{16}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} r^{-2}|\Gamma^{i}\Phi|^2\right)\\
\leq &\left(CAB\epsilon\tau^{-2}+\left(CAB\epsilon+CA^2B^2\epsilon^2\right)\tau^{-4+\eta_{14}}\right)\left(\sum_{i=0}^{16}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} |{D}\Gamma^{i}\Phi|^2\right)\\
\leq &CA^2B\epsilon^2\tau^{-2+\eta_{16}}.
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}N_k^2\\
\leq &CAB\epsilon\tau^{-4}\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} \tau^2|\bar{D}\Gamma^{i}\Phi|^2\right)+CAB\epsilon\tau^{-4+\eta_{14}}\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} |{D}\Gamma^{i}\Phi|^2\right)\\
&+CA^2B^2\epsilon^2\tau^{-4+\eta_{14}}\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} r^{-2}|\Gamma^{i}\Phi|^2\right)\\
\leq &CAB\epsilon\tau^{-4}\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} \tau^2|\bar{D}\Gamma^{i}\Phi|^2\right)+\left(CAB\epsilon+CA^2B^2\epsilon^2\right)\tau^{-4+\eta_{14}}\left(\sum_{i=0}^{k}\int_{\Sigma_\tau} |{D}\Gamma^{i}\Phi|^2\right)\\
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{2}N_k^2\\
\leq &CAB\epsilon\tau^{-2}\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} \tau^2|\bar{D}\Gamma^{i}\Phi|^2\right)+CAB\epsilon\tau^{-2+\eta_{14}}\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} |{D}\Gamma^{i}\Phi|^2\right)\\
&+CA^2B^2\epsilon^2\tau^{-2+\eta_{14}}\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} r^{-2}|\Gamma^{i}\Phi|^2\right)\\
\leq &CAB\epsilon\tau^{-2}\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} \tau^2|\bar{D}\Gamma^{i}\Phi|^2\right)+\left(CAB\epsilon+CA^2B^2\epsilon^2\right)\tau^{-2+\eta_{14}}\left(\sum_{i=0}^{k}\int_{\Sigma_\tau} |{D}\Gamma^{i}\Phi|^2\right)\\
\end{split}
\end{equation*}
The conclusion follows from Proposition \ref{Zlowerbound} and the bootstrap assumptions (\ref{BA1}), (\ref{BA2}), (\ref{BA3}).
\end{proof}
From the the estimates for $U_k$ and $N_k$ and the $L^2-L^\infty$ estimates in the last section, we get the following pointwise bounds:
\begin{proposition}\label{pointwise}
For $r\geq\frac{t^*}{4}$,
\begin{equation}\label{P1}
\sum_{j=0}^{13}|\Gamma^j\Phi|^2\leq \frac{B}{2}A\epsilon r^{-2}(t^*)^{1+\eta_{15}}.
\end{equation}
\begin{equation}\label{P1.5}
\sum_{j=0}^{13}|D\Gamma^j\Phi|^2\leq \frac{B}{2}A\epsilon.
\end{equation}
\begin{equation}\label{P2}
\sum_{\ell=1}^{13-j}\sum_{j=0}^{12}|D^\ell\Gamma^j\Phi|^2\leq \frac{B}{2}A\epsilon r^{-2}.
\end{equation}
\begin{equation}\label{P3}
\sum_{j=0}^8|D\Gamma^j\Phi|^2\leq \frac{B}{2}A\epsilon r^{-2}(t^*)^{\eta_{14}}(1+|u|)^{-2}.
\end{equation}
\begin{equation}\label{P4}
\sum_{j=0}^8|\bar{D}\Gamma^{j}\Phi|^2\leq \frac{B}{2}A\epsilon r^{-2}(t^*)^{-2+\eta_{14}}.
\end{equation}
For $r\leq\frac{t^*}{4}$,
\begin{equation}\label{PI1}
\sum_{j=0}^{13}|\Gamma^j\Phi|^2\leq \frac{B}{2}A\epsilon (t^*)^{-1+\eta_{15}}.
\end{equation}
\begin{equation}\label{PI1.1}
\sum_{\ell=1}^{14-j}\sum_{j=0}^{13}|D^\ell\Gamma^j\Phi|^2\leq BA\epsilon (t^*)^{-1+\eta_{15}}.
\end{equation}
\begin{equation}\label{PI1.2}
\sum_{\ell=1}^{13-j}\sum_{j=0}^{12}|D^\ell\Gamma^j\Phi|^2\leq BA\epsilon (t^*)^{-2+\eta_{14}}.
\end{equation}
\begin{equation}\label{PI2}
\sum_{\ell=1}^{9-j}\sum_{j=0}^8|D^\ell\Gamma^j\Phi|^2\leq \frac{B}{2}A\epsilon r^{-1+\delta}(t^*)^{-3+\eta_{S,11}}.
\end{equation}
\end{proposition}
\begin{proof}
(\ref{P1}) is immediate from Proposition \ref{rnoderivatives} and the bootstrap assumptions (\ref{BA2}) and (\ref{BA3}).
By Proposition \ref{SE},
\begin{equation*}
\begin{split}
\sum_{j=0}^{13}|D\Gamma^j\Phi|^2
\leq& C\left(\sum_{k=0}^{15}\int_{\Sigma_\tau} J^{N}_\mu\left(\Gamma^k\Phi\right) n^\mu_{\Sigma_\tau}+\sum_{k=0}^1\int_{\Sigma_\tau} \left(D^kG_{\leq 13}\right)^2\right).
\end{split}
\end{equation*}
Hence we get (\ref{P1.5}) by bootstrap assumption (\ref{BA4}) and Propositions \ref{Nestprop} and \ref{U}. The constant is improved since $A\epsilon\ll 1$ and $C\ll B$.
By Proposition \ref{r},
\begin{equation*}
\begin{split}
\sum_{\ell=1}^{13-j}\sum_{j=0}^{12}|D^\ell\Gamma^j\Phi|^2\leq &Cr^{-2}\left(\sum_{m=0}^{13-j}\sum_{k=0}^2\sum_{j=0}^{12}\int_{\Sigma_\tau} J^{N}_\mu\left(\partial_{t^*}^m\Omega^k\Gamma^j\Phi\right) n^\mu_{\Sigma_\tau}+\sum_{m+k\leq 10}\int_{\Sigma_\tau}\left(D^m G_{\leq k}\right)^2\right)\\
\leq&Cr^{-2}\left(\sum_{j=0}^{11}\int_{\Sigma_\tau} J^{N}_\mu\left(\Gamma^j\Phi\right) n^\mu_{\Sigma_\tau}+\sum_{m+k\leq 10}\int_{\Sigma_\tau}\left(D^m G_{\leq k}\right)^2\right)\\
\end{split}
\end{equation*}
We hence get (\ref{P2}) by bootstrap assumption (\ref{BA4}) and Propositions \ref{Nestprop} and \ref{U}. The constant is improved since $A\epsilon\ll 1$ and $C\ll B$.
By Proposition \ref{ru},
for $r\geq\frac{t^*}{4}$, we have
\begin{equation*}
\begin{split}
&\sum_{j=0}^8|D\Gamma^j\Phi|^2\\
\leq& Cr^{-2}\left(1+|u|\right)^{-2}\sum_{m=0}^{1}\sum_{k=0}^2\sum_{j=0}^8\left(\int_{\Sigma_\tau}J^{Z+CN}_\mu\left(\partial^m_{t^*}\tilde{\Omega}^k\Gamma^j\Phi\right) n^\mu_{\Sigma_\tau}+C\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\partial^m_{t^*}\tilde{\Omega}^k\Gamma^j\Phi\right) n^\mu_{\Sigma_\tau}\right)\\
&+Cr^{-2}\sum_{k=0}^2\sum_{j=0}^8\int_{\Sigma_\tau\cap\{u'\sim u\}\cap\{r\geq\frac{\tau}{4}\}} \left(\Box_{g_K}\left(\tilde{\Omega}^k\Gamma^j\Phi\right)\right)^2\\
\leq& CA\epsilon\tau^{\eta_{14}}r^{-2}\left(1+|u|\right)^{-2}+Cr^{-2}\int\int_{\Sigma_\tau\cap\{u'\sim u\}\cap\{r\geq\frac{\tau}{4}\}} G_{\leq 10}^2\\
&\quad\mbox{by bootstrap assumption (\ref{BA3})}\\
\leq& CA\epsilon\tau^{\eta_{14}}r^{-2}\left(1+|u|\right)^{-2}+CA\epsilon(t^*)^{-2+\eta_{14}}r^{-2}+CA^2B\epsilon^2(t^*)^{-2+\eta_{14}}r^{-2}\\
&\quad\mbox{by Propositions \ref{NI}, \ref{NO} and \ref{U}}\\
\leq &\frac{B}{2}A\epsilon r^{-2}(t^*)^{\eta_{14}}(1+|u|)^{-2}.
\end{split}
\end{equation*}
Hence we have proved (\ref{P3}).
By Proposition \ref{rv}, for $r\geq\frac{t^*}{4}$, we have
\begin{equation*}
\begin{split}
\sum_{j=0}^8|\bar{D}\Gamma^j\Phi|^2\leq&C r^{-4}\sum_{k=0}^2\sum_{j=0}^8\sum_{i+m\leq 1}\left(\int_{\Sigma_{\tau}} J^N_\mu\left(S^i\partial_{t^*}^m\Gamma^j\Phi\right)n^\mu_{\Sigma_\tau}+\int_{\Sigma_\tau}J^{Z+CN}_\mu\left(\partial^m_{t^*}\tilde{\Omega}^k\Gamma^j\Phi\right) n^\mu_{\Sigma_\tau}\right.\\
&\left.\quad\quad\quad\quad\quad\quad+C\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\partial^m_{t^*}\tilde{\Omega}^k\Gamma^j\Phi\right) n^\mu_{\Sigma_\tau}+\int_{\Sigma_{\tau}} \left(\Box_{g_K}\left(\tilde{\Omega}^k\Gamma^j\Phi\right)\right)^2\right)\\
&+Cr^{-2}\sum_{k=0}^2\sum_{j=0}^8\int_{\Sigma_\tau\cap\{r\geq\frac{\tau}{2}\}} \left(\Box_{g_K}\left(\tilde{\Omega}^k\Gamma^j\Phi\right)\right)^2\\
\leq &CA\epsilon r^{-4}(t^*)^{\eta_{14}}+CA\epsilon(t^*)^{-2+\eta_{14}}r^{-2}+CA^2B\epsilon^2(t^*)^{-2+\eta_{14}}r^{-2}\\
\leq &\frac{B}{2}A\epsilon r^{-2}(t^*)^{-2+\eta_{14}}.
\end{split}
\end{equation*}
Hence we have proved (\ref{P4}) and completed the proof for $r\geq\frac{t^*}{4}$. We now move to the pointwise estimates in the region $r\leq\frac{t^*}{4}$. (\ref{PI1}) follows directly from Proposition \ref{SEinside} and the bootstrap assumptions (\ref{BA2}) and (\ref{BA3}). By Proposition \ref{sSobolev},
\begin{equation*}
\begin{split}
\sum_{\ell=1}^{14-j}\sum_{j=0}^{13}|D^\ell\Gamma^j\Phi|^2\leq C\left(\sum_{i+j\leq 15}\int_{\Sigma_\tau\cap\{r\leq\frac{t^*}{2}\}}J^{N}_\mu\left(\hat{Y}^i \Gamma^j\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{\ell=1}^{14-j}\sum_{j=0}^{13}\int_{\Sigma_\tau}(D^\ell G_{\leq j})^2\right),
\end{split}
\end{equation*}
where we have used the fact that $[\hat{Y},\Gamma]=0$. Hence (\ref{PI1.1}) follows from the bootstrap assumptions (\ref{BA2}) and (\ref{BA3}) and Propositions \ref{Nestprop} and \ref{U}. The proof of (\ref{PI1.2}) follows similarly as (\ref{PI1.1}): by Proposition \ref{sSobolev},
\begin{equation*}
\begin{split}
\sum_{\ell=1}^{13-j}\sum_{j=0}^{12}|D^\ell\Gamma^j\Phi|^2\leq C\left(\sum_{i+j\leq 14}\int_{\Sigma_\tau\cap\{r\leq\frac{t^*}{2}\}}J^{N}_\mu\left(\hat{Y}^i \Gamma^j\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{\ell=1}^{13-j}\sum_{j=0}^{13}\int_{\Sigma_\tau}(D^\ell G_{\leq j})^2\right),
\end{split}
\end{equation*}
Hence (\ref{PI1.2}) follows from the bootstrap assumptions (\ref{BA3}) and Propositions \ref{Nestprop} and \ref{U}.
Finally, by Proposition \ref{extradecay}, for $r\leq\frac{t^*}{4}$, we have
\begin{equation*}
\begin{split}
\sum_{\ell=1}^{9-j}\sum_{j=0}^8|D^\ell\Gamma^j\Phi|^2\leq &C(t^*)^{-1}r^{-1+\delta}\sum_{i+j\leq 10}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}\left(K^{X_1}\left(Y^i\Gamma^j\Phi\right)+K^{X_1}\left(SY^i\Gamma^j\Phi\right)\right)\\
&+C(t^*)^{-1}r^{-1+\delta}\sum_{i+j\leq 10}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r \leq\frac{t^*}{2}\}}r^{-1-\delta}\left(D^iG_j\right)^2\\
\leq&CA\epsilon\tau^{-3+\eta_{S,11}}r^{-1+\delta}+CA^2\epsilon^2\tau^{-3+\eta_{14}}r^{-1+\delta}\\
\leq&\frac{B}{2}A\epsilon\tau^{-3+\eta_{S,11}}r^{-1+\delta},
\end{split}
\end{equation*}
where in the third line we have used the bootstrap assumptions (\ref{BAK5}) and (\ref{BAK10}) and Propositions \ref{NI}, \ref{NO} and \ref{U}. The only caveat is that when using (\ref{BAK10}), the vector fields $\hat{Y}$ and $S$ are in different order. However, since $[S,\hat{Y}]\sim D$, we can estimate the commutator term by (\ref{BAK5}).
\end{proof}
Now, we have proved the $L^{\infty}$ bounds, we will replace the constant $B$ in the bootstrap assumption (\ref{BAP1}), (\ref{BAP1.5}), (\ref{BAP2}), (\ref{BAP3}), (\ref{BAP4}), (\ref{BAPI1}), (\ref{BAPI2}) by $C$ in the sequel. Notice that we have originally assumed $B\ll A_0$ and therefore $C\ll A_0$ still holds. We now proceed to recover the bootstrap assumptions (K) that do not involve the commutators $Y$ or $S$. We first retrieve (\ref{BAK2})-(\ref{BAK5}). Notice also that we will retrieve (\ref{BAK1}) and (\ref{BAK1.5}) later together with (\ref{BA1}) and (\ref{BA4}).
\begin{proposition}\label{K}
\begin{equation}\label{K2}
\sum_{i+j\leq 15}A_{X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}\left(K^{X_0}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)+K^{N}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\right)\leq\frac{\epsilon}{2}\tau^{-1+\eta_{15}}.
\end{equation}
\begin{equation}\label{K3}
\sum_{i+j\leq 14}A_{X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}} K^{X_1}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\leq\frac{\epsilon}{2}\tau^{-1+\eta_{15}}.
\end{equation}
\begin{equation}\label{K4}
\sum_{i+j\leq 14}A_{X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}\left(K^{X_0}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)+K^{N}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\right)\leq\frac{\epsilon}{2}\tau^{-2+\eta_{14}}.
\end{equation}
\begin{equation}\label{K5}
\sum_{i+j\leq 13}A_{X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}K^{X_1}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\leq\frac{\epsilon}{2}\tau^{-2+\eta_{14}}.
\end{equation}
\end{proposition}
\begin{proof}
We first prove the estimates involving $X_0$, i.e., (\ref{K2}) and (\ref{K4}). By Proposition \ref{X0}.1 and the Remark following it and the fact that $|\partial_{t^*}^mN_k|\leq |N_{\leq k+m}|$, we have
\begin{equation*}
\begin{split}
& \sum_{i+j\leq 15}A_{X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}\left(K^{X_0}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)+K^{N}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\right)\\
\leq& C\sum_{i+j\leq 15}A_{X,j}^{-1}\left(\tau^{-2}\int_{\Sigma_{(1.1)^{-1}\tau}} J^{Z+N,w^Z}_\mu\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)n^\mu_{\Sigma_{(1.1)^{-1}\tau}}+ C\int_{\Sigma_{(1.1)^{-1}\tau}\cap\{r\leq r^-_Y\}} J^{N}_\mu\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)n^\mu_{\Sigma_{(1.1)^{-1}\tau}}\right)\\
&+C\sum_{i+j\leq 15}A_{X,j}^{-1}\left(\iint_{\mathcal R((1.1)^{-1}\tau-1,\tau+1)\cap\{r\leq\frac{9t^*}{10}\}}r^{1+\delta}N_{\leq 16}^2+\sup_{t^*\in [(1.1)^{-1}\tau-1,\tau+1]}\int_{\Sigma_{t^*}\cap\{|r-3M|\leq\frac{M}{8}\}} N_{\leq 16}^2\right.\\
&\left.+\iint_{\mathcal R((1.1)^{-1}\tau-1,\tau+1)\cap\{r\leq\frac{9t^*}{10}\}}r^{1+\delta}\left( U_{\leq 15,\leq j}\right)^2\right)\\
\leq& C\sum_{i+j\leq 15}\left(A_{X,j}^{-1}A_j\epsilon\tau^{-1+\eta_{15}}+A_{X,j}^{-1}A^2\epsilon^2\tau^{-2+\eta_{S,11}+\eta_{16}+2\delta}+CA_{X,j}^{-1}A_{X,j-1}\epsilon\tau^{-1+\eta_{15}}\right)\\
\leq&\frac{\epsilon}{2}\tau^{-1+\eta_{15}},
\end{split}
\end{equation*}
by Propositions \ref{NI}, \ref{NO} and \ref{U}. Notice that our integrated estimates for $U$ in Proposition \ref{U} is only for $[(1.1)^{-1}\tau,\tau]$. Nevertheless, for the region $[(1.1)^{-1}\tau-1,(1.1)^{-1}\tau]\cap[\tau,\tau+1]$, we can use the integrate over the fixed $\tau$ estimate in the same Proposition. By Proposition \ref{X0}.1 and the Remark following it and the fact that $|\partial_{t^*}^mN_k|\leq |N_{\leq k+m}|$, we have
\begin{equation*}
\begin{split}
& \sum_{i+j\leq 14}A_{X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}\left(K^{X_0}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)+K^{N}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\right)\\
\leq& C\sum_{i+j\leq 14}A_{X,j}^{-1}\left(\tau^{-2}\int_{\Sigma_{(1.1)^{-1}\tau}} J^{Z+N,w^Z}_\mu\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)n^\mu_{\Sigma_{(1.1)^{-1}\tau}}+ C\int_{\Sigma_{(1.1)^{-1}\tau}\cap\{r\leq r^-_Y\}} J^{N}_\mu\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)n^\mu_{\Sigma_{(1.1)^{-1}\tau}}\right)\\
&+C\sum_{i+j\leq 14}A_{X,j}^{-1}\left(\iint_{\mathcal R((1.1)^{-1}\tau-1,\tau+1)\cap\{r\leq\frac{9t^*}{10}\}}r^{1+\delta}N_{\leq 15}^2+\sup_{t^*\in [(1.1)^{-1}\tau-1,\tau+1]}\int_{\Sigma_{t^*}\cap\{|r-3M|\leq\frac{M}{8}\}} N_{\leq 15}^2\right.\\
&\left.+\iint_{\mathcal R((1.1)^{-1}\tau-1,\tau+1)\cap\{r\leq\frac{9t^*}{10}\}}r^{1+\delta}\left(U_{\leq 14,\leq j}\right)^2\right)\\
\leq& C\sum_{i+j\leq 14}\left(A_{X,j}^{-1}A_j\epsilon\tau^{-2+\eta_{14}}+A_{X,j}^{-1}A^2\epsilon^2\tau^{-3+\eta_{S,11}+\eta_{15}+2\delta}+A_{X,j}^{-1}A_{X,j-1}\epsilon\tau^{-2+\eta_{14}}\right)\\
\leq&\frac{\epsilon}{2}\tau^{-2+\eta_{14}}.
\end{split}
\end{equation*}
The proof of (\ref{K3}) and (\ref{K5}) proceeds in an identical manner. Notice that using Proposition \ref{X0}.2, the right hand side when we estimate (\ref{K3}) (respectively (\ref{K5})) is identical to that when we estimate (\ref{K2}) (respectively (\ref{K4})).
\end{proof}
Now we move on to retrieving the bootstrap assumptions (J) with better constants:
\begin{proposition}\label{J}
\begin{equation}\label{J1}
\sum_{i+j=16}A_j^{-1}\int_{\Sigma_\tau}J^{N}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}\leq \frac{\epsilon}{4} \tau^{\eta_{16}}.
\end{equation}
\begin{equation}\label{J4}
\sum_{i+j\leq 15}A_j^{-1}\int_{\Sigma_\tau} J^{N}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}\leq \frac{\epsilon}{2}.
\end{equation}
\begin{equation}\label{K1.0.1}
\sum_{i+j=16}A_{X,j}^{-1} \left(\iint_{\mathcal R(\tau_0,\tau)}K^{X_0}\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)+\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq r^-_Y\}}K^{N}\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\right)\leq \frac{\epsilon}{2}\tau^{\eta_{16}}.
\end{equation}
\begin{equation}\label{K1.4}
\sum_{i+j=15}A_{X,j}^{-1}\iint_{\mathcal R(\tau_0,\tau)} K^{X_1}\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\leq \frac{\epsilon}{2}\tau^{\eta_{16}}.
\end{equation}
\begin{equation}\label{K1.5}
\sum_{i+j\leq 15}A_{X,j}^{-1}\iint_{\mathcal R(\tau_0,\tau)} K^{X_0}\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\leq \frac{\epsilon}{2}.
\end{equation}
\begin{equation}\label{K1.6}
\sum_{i+j\leq 14}A_{X,j}^{-1}\iint_{\mathcal R(\tau_0,\tau)} K^{X_1}\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\leq \epsilon.
\end{equation}
\end{proposition}
\begin{proof}
We will prove the slightly stronger statements with $A_{X,j}$ replaced by $A_j$. Using Proposition \ref{bddcom} and \ref{bddcom2}, we have
\begin{equation*}
\begin{split}
&\sum_{i+j=16}A_{j}^{-1}\left(\int_{\Sigma_\tau}J^{N}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau} +\iint_{\mathcal R(\tau_0,\tau)} K^{X_0}\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)+\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq r^-_Y\}}K^{N}\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\right)\\
\leq &C\sum_{i+j=16}A_{j}^{-1}\left(\int_{\Sigma_{\tau_0}} J^{N}_\mu\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)n^\mu_{\Sigma_{\tau'}}+\left(\int_{\tau_0-1}^{\tau+1}\left(\int_{\Sigma_{t^*}} N_{16}^2\right)^{\frac{1}{2}}dt^*\right)^2+\iint_{\mathcal R(\tau_0-1,\tau+1)}N_{16}^2\right.\\
&\left.+\iint_{\mathcal R(\tau_0-1,\tau+1)}r^{1+\delta}U_{16,j}^2+\sup_{t^*\in [\tau_0-1,\tau+1]}\int_{\Sigma_{t^*}\cap\{|r-3M|\leq\frac{M}{8}\}} U_{16,j}^2\right)\\
\leq &C\sum_{i+j=16}A_{j}^{-1}\left(\epsilon+A^2\epsilon^2\eta_{16}^{-1}\tau^{\eta_{16}}+A_{X,j-1}\epsilon\tau^{\eta_{16}}\right)\\
\leq &\frac{\epsilon}{4}\tau^{\eta_{16}}.
\end{split}
\end{equation*}
We now turn to the estimates for $\displaystyle\sum_{j=0}^{15}|\Gamma^j\Phi|$. We have
\begin{equation*}
\begin{split}
&\sum_{i+j\leq 15}A_{j}^{-1}\left(\int_{\Sigma_\tau}J^{N}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau} +\iint_{\mathcal R(\tau_0,\tau)} K^{X_0}\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\right)\\
\leq &C\sum_{i+j\leq 15}A_j^{-1}\left(\int_{\Sigma_{\tau_0}} J^{N}_\mu\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)n^\mu_{\Sigma_{\tau_0}}+\left(\int_{\tau_0-1}^{\tau+1}\left(\int_{\Sigma_{t^*}} N_{\leq 15}^2\right)^{\frac{1}{2}}dt^*\right)^2+\iint_{\mathcal R(\tau_0-1,\tau+1)}N_{\leq 15}^2\right.\\
&\left.+\iint_{\mathcal R(\tau_0-1,\tau+1)}r^{1+\delta}U_{\leq 15,\leq j}^2+\sup_{t^*\in [\tau_0-1,\tau+1]}\int_{\Sigma_{t^*}\cap\{|r-3M|\leq\frac{M}{8}\}} U_{\leq 15,\leq j}^2\right)\\
\leq& C\sum_{i+j\leq 15}A_j^{-1}\left(\epsilon+A^2\epsilon^2+CA_{X,j-1}\epsilon\right)\\
\leq& \frac{\epsilon}{2}.
\end{split}
\end{equation*}
It now remains to show (\ref{K1.4}) and (\ref{K1.6}). By Proposition \ref{bddcom3} they can be estimated by exactly the same terms as (\ref{K1.0.1}) and (\ref{K1.5}) respectively. The Proposition hence follows.
\end{proof}
We now move on to control the conformal energy and close the part of the bootstrap assumption (\ref{BA2}) without $\hat{Y}$.
\begin{proposition}\label{Z1}
\begin{equation}\label{J2}
\begin{split}
\sum_{i+j=15}A_j^{-1}\left(\int_{\Sigma_\tau} J^{Z+N,w^Z}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau} +C\tau^2\int_{\Sigma_\tau\cap\{r\leq \frac{9\tau}{10}\}} J^{N}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}\right)\leq \frac{\epsilon}{4} \tau^{1+\eta_{15}}.
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
By Proposition \ref{conformalenergy},
\begin{equation*}
\begin{split}
&\sum_{i+j= 15}A_j^{-1}\left(\int_{\Sigma_{\tau}} J^{Z,w^Z}_\mu\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)n^\mu_{\Sigma_{{\tau}}}+C\tau^2\int_{\Sigma_{\tau}\cap\{r\leq \frac{9\tau}{10}\}} J^{N}_\mu\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)n^\mu_{\Sigma_{\tau}}\right)\\
\leq &C\sum_{i+j= 15}A_j^{-1}\left(\int_{\Sigma_{\tau_0}} J^{Z+CN,w^Z}_\mu\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)n^\mu_{\Sigma_{{\tau_0}}}+\delta'\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq \frac{t^*}{2}\}} (t^*)^2K^{X_0}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\right.\\
&\left. +\left(\delta'+a\right)\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq r^-_Y\}}(t^*)^2K^N\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)+(\delta')^{-1}\iint_{\mathcal R(\tau_0,\tau)} t^* r^{-1+\delta}K^{X_1}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\right.\\
&\left.+(\delta')^{-1}\sum_{m=0}^1\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq\frac{9t^*}{10}\}} (t^*)^2r^{1+\delta}\left(\partial_{t^*}^m N_{ 15}\right)^2+(\delta')^{-1}\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq\frac{9t^*}{10}\}} (t^*)^2r^{1+\delta} U_{ 15, j}^2\right. \\
&\left.+(\delta')^{-1}\left(\int_{\tau_0}^{\tau}\left(\int_{\Sigma_{t^*}\cap\{r\geq \frac{t^*}{2}\}}r^2 G_{ 15, j}^2 \right)^{\frac{1}{2}}dt^*\right)^2+(\delta')^{-1}\sup_{t^*\in [\tau_0,\tau]}\int_{\Sigma_{t^*}\cap\{r^-_Y\leq r\leq \frac{25M}{8}\}} (t^*)^2 N_{15}^2\right). \\
\end{split}
\end{equation*}
We will estimate the terms one by one. First, the term with the initial data, i.e., the very first term, is clearly bounded by $C(\sum_j A_j^{-1})\epsilon$. Second, we consider the two $(t^*)^2K$ terms on the second line. To this end, we define as before $\tau_0\leq \tau_1\leq ...\leq \tau_n=\tau$ with $\tau_{i+1}\leq (1.1)\tau_i$ and $n\sim\log(\tau-\tau_0)$ is the minimum such that this can be done. Thus, these two terms can be bounded, using the bootstrap assumption (\ref{BAK2}),
\begin{equation*}
\begin{split}
&\delta'\sum_{i+j=15}A_j^{-1}\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq \frac{t^*}{2}\}} (t^*)^2K^{X_0}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)+\left(\delta'+a\right)\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq r^-_Y\}}(t^*)^2K^N\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\\
\leq &C\sum_{i+j=15}A_j^{-1}\sum_{k=0}^{n-1}\left(\delta'\tau_k^2\iint_{\mathcal R(\tau_k,\tau_{k+1})\cap\{r\leq \frac{t^*}{2}\}} K^{X_0}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)+\left(\delta'+a\right)\tau_k^2\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq r^-_Y\}}K^N\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\right)\\
\leq& C\left(\sum_j\frac{A_{X,j}}{A_j}\right)\epsilon\left(2\delta'+a\right)\tau^{1+\eta_{15}}.
\end{split}
\end{equation*}
This is acceptable since $a,\delta'\ll\frac{A_j}{A_{X,j}}$.
Third, the term with $t^* r^{-1+\delta}K$ can be bounded using the bootstrap assumption (\ref{BAK1.4}),
\begin{equation*}
\begin{split}
&(\delta')^{-1}\sum_{i+j=15}A_j^{-1}\iint_{\mathcal R(\tau_0,\tau)} t^* r^{-1+\delta}K^{X_1}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right) \\
\leq & C(\delta')^{-1}\sum_{i+j=15}A_j^{-1}\sum_{k=0}^{n-1}\tau_k\iint_{\mathcal R(\tau_k,\tau_{k+1})} K^{X_1}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)
\leq C(\delta')^{-1}\left(\sum_j\frac{A_{X,j}}{A_j}\right)\tau^{1+\eta_{16}}.
\end{split}
\end{equation*}
This is acceptable since $\eta_{16}\ll\eta_{15}$ and therefore the constant can be improved for $\tau$ large.
Fourth, the integrals involving $N_{15}$ can be bounded using Propositions \ref{NI} and \ref{NO}:
\begin{equation*}
\begin{split}
&C(\delta')^{-1}A_0^{-1}\sum_{m=0}^1\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq\frac{9t^*}{10}\}} (t^*)^2r^{1+\delta}\left(\partial_{t^*}^m N_{15}\right)^2 \\
\leq &CA^2A_0^{-1}\epsilon^2(\delta')^{-1}\int_{\tau_0}^\tau \left(t^*\right)^{-1+\eta_{S,11}+\eta_{15}+2\delta}dt^*\leq CA^2A_0^{-1}\epsilon^2(\delta')^{-1}\tau^{\eta_{S,11}+\eta_{15}+2\delta},
\end{split}
\end{equation*}
$$C(\delta')^{-1}A_0^{-1}\left(\int_{\tau_0}^{\tau}\left(\int_{\Sigma_{t^*}\cap\{r\geq \frac{t^*}{2}\}}r^2 N_{15}^2 \right)^{\frac{1}{2}}dt^*\right)^2\leq CA^2A_0^{-1}\epsilon^2(\delta')^{-1}\tau^{1+\eta_{15}},$$
$$C(\delta')^{-1}A_0^{-1}\sup_{t^*\in [\tau_0,\tau]}\int_{\Sigma_{t^*}\cap\{r^-_Y\leq r\leq \frac{25M}{8}\}} (t^*)^2 N_{15}^2\leq CA^2A_0^{-1}\epsilon^2(\delta')^{-1}.$$
These are all acceptable since $\epsilon$ would beat all the constants.
Fifth, for the commutator terms $U_{15,j}^2$, we estimate by Proposition \ref{U},
$$(\delta')^{-1}A_j^{-1}\iint_{\mathcal R(\tau_0,\tau)} (t^*)^2r^{1+\delta}\left(U_{15,j}\right)^2\leq C\sum_{i=0}^{n-1}(\delta')^{-1}\iint_{\mathcal R(\tau_i,\tau_{i+1})} \tau_i^2r^{1+\delta}\left(U_{ 15, j}\right)^2\leq C(\delta')^{-1}\frac{A_{X,j-1}}{A_j}\tau^{1+\eta_{15}},$$
$$(\delta')^{-1}A_j^{-1}\left(\int_{\tau_0}^{\tau}\left(\int_{\Sigma_{t^*}\cap\{r\geq \frac{t^*}{2}\}}r^2 U_{ 15,j}^2 \right)^{\frac{1}{2}}dt^*\right)^2\leq C(\delta')^{-1}\frac{A_{j-1}}{A_j}\tau.$$
Since
$$\frac{A_{j-1}}{A_j}\ll\frac{A_{X,j-1}}{A_j}\ll\delta',$$
all terms are acceptable.
\end{proof}
With $14$ or less derivatives, the conformal energy behaves better. We now close the part of the bootstrap assumption (\ref{BA3}) without $\hat{Y}$.
\begin{proposition}\label{Z2}
\begin{equation}\label{J3}
\begin{split}
\sum_{i+j\leq 14}A_j^{-1}\left(\int_{\Sigma_\tau} J^{Z+N,w^Z}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau} +C\tau^2\int_{\Sigma_\tau\cap\{r\leq \frac{9\tau}{10}\}} J^{N}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}\right)\leq \frac{\epsilon}{4} \tau^{\eta_{14}}.
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
By Proposition \ref{conformalenergy}, and noticing that $U$ is supported away from $\{|r-3M|\leq\frac{M}{8}\}$, we have
\begin{equation*}
\begin{split}
&\sum_{i+j\leq 14}A_j^{-1}\left(\int_{\Sigma_\tau} J^{Z+N,w^Z}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau} +C\tau^2\int_{\Sigma_\tau\cap\{r\leq \frac{9\tau}{10}\}} J^{N}_\mu\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}\right)\\
\leq &C\sum_{i+j\leq 14}A_j^{-1}\left(\int_{\Sigma_{\tau_0}} J^{Z+CN,w^Z}_\mu\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)n^\mu_{\Sigma_{{\tau_0}}}+\delta'\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq \frac{t^*}{2}\}} (t^*)^2K^{X_0}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\right.\\
&\left. +\left(\delta'+a\right)\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq r^-_Y\}}(t^*)^2K^N\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)+(\delta')^{-1}\iint_{\mathcal R(\tau_0,\tau)} t^* r^{-1+\delta}K^{X_1}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\right.\\
&\left.+(\delta')^{-1}\sum_{m=0}^1\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq\frac{9t^*}{10}\}} (t^*)^2r^{1+\delta}\left(\partial_{t^*}^m N_{\leq 15}\right)^2+(\delta')^{-1}\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq\frac{9t^*}{10}\}} (t^*)^2r^{1+\delta} U_{\leq 15, \leq j}^2\right. \\
&\left.+(\delta')^{-1}\left(\int_{\tau_0}^{\tau}\left(\int_{\Sigma_{t^*}\cap\{r\geq \frac{t^*}{2}\}}r^2 G_{\leq 15,\leq j}^2 \right)^{\frac{1}{2}}dt^*\right)^2+(\delta')^{-1}\sup_{t^*\in [\tau_0,\tau]}\int_{\Sigma_{t^*}\cap\{r^-_Y\leq r\leq \frac{25M}{8}\}} (t^*)^2 N_{\leq 15}^2\right). \\
\end{split}
\end{equation*}
As before, we estimate each term one by one. First, the term with initial data is clearly bounded by $C\sum_jA_j\epsilon$. Second, the $(t^*)^2K$ terms can be bounded, using (\ref{BAK4}), and dividing the interval into $\tau_0< \tau_1<...<\tau_n=\tau$ as before by
$$C\frac{A_{X,j}}{A_j}\epsilon\left(2\delta'+a\right)\sum_{i=0}^{n-1}\tau_i^{\eta_{14}}\leq C\frac{A_{X,j}}{A_j}\epsilon\eta_{14}^{-1}\left(2\delta+a\right)\tau^{\eta_{14}}.$$
This is acceptable since $a,\delta'\ll\frac{A_j}{A_{X,j}}$. Third, the $t^*r^{-1+\delta}K$ term can be bounded, using the bootstrap assumptions (\ref{BAK1.6}) and (\ref{BAK3}), by
\begin{equation*}
\begin{split}
&(\delta')^{-1}\sum_{i+j\leq 14}A_j^{-1}\iint_{\mathcal R(\tau_0,\tau)} t^* r^{-1+\delta}K^{X_1}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right) \\
\leq & (\delta')^{-1}\sum_{i+j\leq 14}A_j^{-1}\left(\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq\frac{t^*}{2}\}} t^* K^{X_1}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)+\iint_{\mathcal R(\tau_0,\tau)\cap\{r\geq \frac{t^*}{2}\}} t^* r^{-1+\delta}K^{X_1}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\right) \\
\leq & C(\delta')^{-1}\sum_{i+j\leq 14}A_j^{-1}\sum_{k=0}^{n-1}\left(\tau_k\iint_{\mathcal R(\tau_k,\tau_{k+1})\cap\{r\leq\frac{t^*}{2}\}} K^{X_1}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)+\tau_k^\delta\iint_{\mathcal R(\tau_k,\tau_{k+1})\cap\{r\geq \frac{t^*}{2}\}} K^{X_1}\left(\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right) \right) \\
\leq& C(\delta')^{-1}\sum_j\frac{A_{X,j}}{A_j}\eta_{15}^{-1}\tau^{\eta_{15}},
\end{split}
\end{equation*}
which is acceptable for $\tau$ large since $\eta_{15}\ll\eta_{14}$.
Fourth, the integrals involving $N_{\leq 14}$ can be bounded using Propositions \ref{NI} and \ref{NO} by noticing that $|\partial_{t^*}N_{\leq 14}|\leq CN_{\leq 15}$:
\begin{equation*}
\begin{split}
&C(\delta')^{-1}A_0^{-1}\sum_{m=0}^1\iint_{\mathcal R(\tau_0,\tau)\cap\{r\leq\frac{9t^*}{10}\}} (t^*)^2r^{1+\delta}\left(\partial_{t^*}^m N_{\leq 14}\right)^2 \\
\leq &CA^2A_0^{-1}\epsilon^2(\delta')^{-1}\int_{\tau_0}^\tau \left(t^*\right)^{-2+\eta_{S,11}+\eta_{15}+2\delta}dt^*\leq CA^2A_0^{-1}\epsilon^2(\delta')^{-1},
\end{split}
\end{equation*}
$$C(\delta')^{-1}A_0^{-1}\left(\int_{\tau_0}^{\tau}\left(\int_{\Sigma_{t^*}\cap\{r\geq \frac{t^*}{2}\}}r^2 N_{15}^2 \right)^{\frac{1}{2}}dt^*\right)^2\leq CA^2A_0^{-1}\epsilon^2(\delta')^{-1}\eta_{15}^{-1}\tau^{\eta_{15}},$$
$$C(\delta')^{-1}A_0^{-1}\sup_{t^*\in [\tau_0,\tau]}\int_{\Sigma_{t^*}\cap\{r^-_Y\leq r\leq \frac{25M}{8}\}} (t^*)^2 N_{15}^2\leq CA^2A_0^{-1}\epsilon^2(\delta')^{-1},$$
which is acceptable since $\epsilon\ll A\delta\eta_{15}^{-1}$.
Fifth, for the commutator terms $U_{\leq 14,\leq j}^2$, we estimate by Proposition \ref{U},
$$(\delta')^{-1}A_j^{-1}\iint_{\mathcal R(\tau_0,\tau)} (t^*)^2r^{1+\delta}\left(U_{\leq 14,\leq j}\right)^2\leq C\sum_{i=0}^{n-1}(\delta')^{-1}A_{X,j-1} \tau_i^2\tau_i^{-2+\eta_{14}}\leq C(\delta')^{-1}\eta_{14}^{-1}\frac{A_{X,j-1}}{A_j}\tau^{\eta_{14}},$$
$$(\delta')^{-1}A_j^{-1}\left(\int_{\tau_0}^{\tau}\left(\int_{\Sigma_{t^*}\cap\{r\geq \frac{t^*}{2}\}}r^2 U_{\leq 14,j}^2 \right)^{\frac{1}{2}}dt^*\right)^2\leq C(\delta')^{-1}\frac{A_{j-1}}{A_{j}}\eta_{14}^{-1}\tau.$$
Since
$$\frac{A_{j-1}}{A_j}\ll\frac{A_{X,j-1}}{A_j}\ll\delta'\eta_{14}^{-1},$$
all terms are acceptable.
\end{proof}
We now consider terms involving commutation with $\hat{Y}$ and recover the bootstrap assumptions (\ref{BA1}), (\ref{BA2}) and (\ref{BA3}).
\begin{proposition}\label{Yest}
\begin{equation*}
\sum_{i+k=16}\int_{\Sigma_\tau}J^{N}_\mu\left(\hat{Y}^k\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau}\leq \frac{A_Y}{4}\epsilon \tau^{\eta_{16}},
\end{equation*}
and
\begin{equation*}
\begin{split}
&\sum_{i+k=15}\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\hat{Y}^k\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau} \leq \frac{A_Y}{4}\epsilon \tau^{1+\eta_{15}},
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
\sum_{i+k\leq 14}\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\hat{Y}^k\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau}&\leq \frac{A_Y}{4}\epsilon \tau^{\eta_{14}}.
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
The idea is to use Proposition 13 and use the fact that it gives the control of an integrated in time quantity. From this we can extract a good slice to improve the constant.
By Proposition \ref{commYcontrol},
\begin{equation*}
\begin{split}
&\sum_{i+k=16}\left(\int_{\Sigma_\tau\cap\{r\leq r^+_Y\}} J^{N}_\mu\left(\hat{Y}^{k}\partial_{t^*}^i\Phi\right)n^\mu_{\Sigma_\tau} + \iint_{\mathcal R(\tau',\tau)\cap\{r\leq r^-_Y\}}J_\mu^{N}\left(\hat{Y}^{k}\partial_{t^*}^i\Phi\right)n^\mu_{\Sigma_{t^*}}\right)\\
\leq &C\left(\sum_{j+m\leq 16}\int_{\Sigma_{\tau'}\cap\{r\leq r^+_Y\}} J^N_\mu\left(\partial_{t^*}^j\hat{Y}^m\Phi\right)n^\mu_{\Sigma_{\tau'}}+\sum_{j=0}^{16}\int_{\Sigma_{\tau}\cap\{r\leq r^+_Y\}} J^N_\mu\left(\partial_{t^*}^j\Phi\right)n^\mu_{\Sigma_{\tau}}\right.\\
&\left.+\sum_{j=0}^{16}\iint_{\mathcal R(\tau',\tau)\cap\{ r\leq \frac{23M}{8}\}} J^N_\mu\left(\partial_{t^*}^j\Phi\right)n^\mu_{\Sigma_{t^*}}+\sum_{i+j\leq 16}\iint_{\mathcal R(\tau',\tau)\cap\{r\leq\frac{23M}{8}\}}\left(D^iG_{\leq j,0}\right)^2\right)\\
\leq&CA_Y(\tau')^{\eta_{16}}+CA_0\tau^{\eta_{16}}+CA^2\epsilon^2(\tau')^{-1+\eta_{16}},\\
\end{split}
\end{equation*}
by (\ref{BA1}), (\ref{BAK1}) and Proposition \ref{Nestprop}. Take $\tau'=\tau-A_0$. Then
\begin{equation*}
\begin{split}
&\sum_{i+k=16}\iint_{\mathcal R(\tau-A_0,\tau)\cap\{r\leq r^-_Y\}}J_\mu^{N}\left(\hat{Y}^{k}\partial_{t^*}^i\Phi\right)n^\mu_{\Sigma_{t^*}}\leq CA_Y\tau^{\eta_{16}}+CA_0\tau^{\eta_{16}}+CA^2\epsilon^2\tau^{-1+\eta_{16}}.\\
\end{split}
\end{equation*}
Hence there is some $\tilde{\tau}\in [\tau-A_0,\tau]$ such that
\begin{equation*}
\begin{split}
\sum_{i+k=16}\int_{\Sigma_{\tilde{\tau}}\cap\{r\leq r^-_Y\}} J^{N}_\mu\left(\hat{Y}^{k}\partial_{t^*}^i\Phi\right)n^\mu_{\Sigma_{\tilde{\tau}}}\leq CA_YA_0^{-1}\tau^{\eta_{16}}+C\tau^{\eta_{16}}+CA^2\epsilon^2\tau^{-1+\eta_{16}}.\\
\end{split}
\end{equation*}
We also have by (\ref{BA1}) and the elliptic estimates in Proposition \ref{elliptic},
\begin{equation*}
\begin{split}
&\sum_{i+k=16}\int_{\Sigma_{\tilde{\tau}}\cap\{r^-_Y\leq r\leq r^+_Y\}} J^{N}_\mu\left(\hat{Y}^{k}\partial_{t^*}^i\Phi\right)n^\mu_{\Sigma_{\tilde{\tau}}} \\
\leq&C \sum_{i=0}^{16}\int_{\Sigma_{\tilde{\tau}}\cap\{r^-_Y\leq r\leq \frac{t^*}{2}\}} J^{N}_\mu\left(\partial_{t^*}^i\Phi\right)n^\mu_{\Sigma_{\tilde{\tau}}}+\sum_{i+j\leq 15}\int_{\Sigma_{\tilde{\tau}}}(D^iG_{\leq j,0})^2\\
\leq &CA_0\tau^{\eta_{16}}+CA^2\epsilon^2\tau^{-1+\eta_{16}}.\\
\end{split}
\end{equation*}
Now reapply Proposition \ref{commYcontrol}, from $\tilde{\tau}$ to $\tau$, we get
\begin{equation*}
\begin{split}
&\sum_{i+k=16}\int_{\Sigma_\tau\cap\{r\leq r^+_Y\}} J^{N}_\mu\left(\hat{Y}^{k}\partial_{t^*}^i\Phi\right)n^\mu_{\Sigma_\tau}\\
\leq &C\sum_{j+m\leq 16}\int_{\Sigma_{\tilde{\tau}}\cap\{r\leq r^+_Y\}} J^N_\mu\left(\partial_{t^*}^j\hat{Y}^m\Phi\right)n^\mu_{\Sigma_{\tilde{\tau}}}+CA_0\tau^{\eta_{16}}+CA^2\epsilon^2\\
\leq&CA_YA_0^{-1}\tau^{\eta_{16}}+CA_0\tau^{\eta_{16}}+CA^2\epsilon^2\tau^{-1+\eta_{16}}.\\
\end{split}
\end{equation*}
Since $C\ll A_0\ll A_Y$, we get the first statement in the Proposition.
The derivations for the other bounds are identical, with the constants and exponents replaced appropriately.
\end{proof}
From this we can also derive some integrated estimates for $Y^k\Gamma^j\Phi$. This will be useful in controlling the commutator $[\Box_{g_K},S]$.
\begin{proposition}\label{Yintest}
\begin{equation*}
\sum_{i+k=16}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\hat{Y}^k\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau}\leq A_Y\epsilon \tau^{\eta_{16}},
\end{equation*}
and
\begin{equation*}
\begin{split}
&\sum_{i+k=15}\tau^2\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\hat{Y}^k\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau} \leq A_Y\epsilon \tau^{1+\eta_{15}},
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
\sum_{i+k\leq 14}\tau^2\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\hat{Y}^k\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau}&\leq A_Y\epsilon \tau^{\eta_{14}}.
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
This is a direct consequence of Proposition \ref{commYcontrol}, \ref{Nestprop}, \ref{Yest}, as well as the bootstrap assumptions (\ref{BA1}), (\ref{BA2}) and (\ref{BA3}).
\end{proof}
We will finally proceed to the quantities associated to the vector field $S$. Recall from \cite{LKerr} that for large values of $r$
\begin{equation*}
\begin{split}
&|[\Box_{g_K},S]\Phi-\left(2+\frac{r^*\mu}{r}\right)\Box_g \Phi -\frac{2}{r}\left(\frac{r^*}{r}-1-\frac{2r^*\mu}{r}\right)\partial_{r^*}\Phi-2\left(\left(\frac{r^*}{r}-1\right)-\frac{3r^*\mu}{2r}\right){\Delta} \mkern-13mu /\,\Phi|\\
\leq &Ca r^{-2}(\sum_{k=1}^2|D^k\Phi|).
\end{split}
\end{equation*}
and that for finite values of $r$, we have
\begin{equation*}
\begin{split}
&|[\Box_{g_K},S]\Phi|\leq C(\sum_{k=1}^2|D^k\Phi|).
\end{split}
\end{equation*}
Moreover, all the coefficients in the commutator term obey the same estimates (with a different constant) upon differentiation.
Therefore,
$$\Box_{g_K}\left(S\Gamma^k\Phi\right)=V_k+S(U_k)+S(N_k),$$
where $$\left(D^\ell V_k\right)^2\leq C r^{-4}\left(\log r\right)^2\left(\sum_{j=1}^{\ell+1}\left(D^j\Gamma^{k+1}\Phi\right)^2+ \sum_{j=1}^{\ell+2} \left(D^j\Gamma^k\Phi\right)^2\right).$$
We would now estimate these three terms separately. We first estimate the $V_k$ terms:
\begin{proposition}\label{V}
For $\alpha\leq 2$,
\begin{equation*}
\begin{split}
\sum_{\ell+k\leq 13}\int_{\Sigma_\tau}r^\alpha \left(D^\ell V_{\leq k}\right)^2\leq CA_{Y}\epsilon\tau^{-2+\eta_{14}+\delta}.
\end{split}
\end{equation*}
For $\alpha\leq 1+\delta$,
\begin{equation*}
\begin{split}
\sum_{\ell+k=13}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^\alpha \left(D^\ell V_{\leq k}\right)^2\leq CA_{Y}\epsilon\tau^{-1+\eta_{15}+\delta}.
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\sum_{\ell+k\leq 12}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^\alpha \left(D^\ell V_{\leq k}\right)^2\leq CA_{Y}\epsilon\tau^{-2+\eta_{14}+\delta}.
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
By the elliptic estimates in Propositions \ref{elliptic} and \ref{elliptichorizon}, we have, for $\alpha\leq 2$,
\begin{equation*}
\begin{split}
&\sum_{\ell+k\leq 13}\int_{\Sigma_\tau}r^\alpha \left(D^\ell V_{\leq k}\right)^2\\
\leq &C \sum_{i+j\leq 12}\int_{\Sigma_\tau}r^{\alpha-4+\delta}J^N_\mu\left(\hat{Y}^i\Gamma^{j}\Phi\right)n^\mu_{\Sigma_\tau}+C\sum_{i+j\leq 11}\int_{\Sigma_\tau}r^{\alpha-4+\delta} \left(D^i G_{\leq j}\right)^2\\
\leq&C\left(A_{Y}\epsilon+A^2\epsilon^2\right)\tau^{-2+\eta_{14}+\delta},
\end{split}
\end{equation*}
where we have used Propositions \ref{Nestprop} and \ref{U}, the bootstrap assumptions (\ref{BA3}) (for $r\leq \frac{9t^*}{10}$) and (\ref{BA4}) (for $r\geq \frac{9t^*}{10}$).
By the elliptic estimates in Propositions \ref{elliptic} and \ref{elliptichorizon}, we have
\begin{equation*}
\begin{split}
&\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^\alpha \left(D^\ell V_{\leq k}\right)^2\\
\leq &C\sum_{i+j=0}^{\ell+k+1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{\alpha-4+\delta}J^N_\mu\left(\hat{Y}^i\Gamma^{j}\Phi\right)n^\mu_{\Sigma_\tau}+\sum_{i+j=0}^{\ell+k}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{\alpha-4+\delta} \left(D^i G_{\leq j}\right)^2\\
\end{split}
\end{equation*}
We first consider the case $\ell+k=13$. For the first term, we divide into $r\leq\frac{t^*}{2}$ (which we estimate by (\ref{BAK3}) and Proposition \ref{Yintest}) and $r\geq\frac{t^*}{2}$ (which we estimate using the extra decay in $r$ by (\ref{BAK1.5})). The second term contains the $U_k$ and the $N_k$ part. The $U_k$ part can be estimated by Proposition \ref{U}. The $N_k$ part can be estimated by Proposition \ref{Nestprop}. The $\ell+k\leq 12$ case in completely analogous, replacing the bootstrap assumption (\ref{BAK3}) by (\ref{BAK5}).
\end{proof}
We then proceed to the estimates for $S(U_k)$. Notice that when we prove the estimates for the derivatives for $S(U_k)$, the derivatives for $S(N_k)$ will be involved. Like the proof of the estimates for $U_k$, we will first prove estimates for the derivatives of $S(U_k)$ depending on $S(N_k)$, and close the estimates after we control $S(N_k)$.
\begin{proposition}\label{SUprop}
The following estimates for $S(U_k)$ on a fixed $t^*$ slice hold for $\alpha\leq 2$:
\begin{equation*}
\begin{split}
\sum_{k+\ell=13}\int_{\Sigma_\tau}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,j-1}\epsilon\tau^{\eta_{S,13}}+\sum_{m=1}^{13-j}\int_{\Sigma_\tau}(D^mS(N_{\leq j-1}))^2.
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\sum_{k+\ell=12}\int_{\Sigma_\tau}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,j-1}\epsilon\tau^{-1+\eta_{S,12}}+\sum_{m=1}^{12-j}\int_{\Sigma_\tau}(D^mS(N_{\leq j-1}))^2.
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\sum_{k+\ell\leq 11}\int_{\Sigma_\tau}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,j-1}\epsilon\tau^{-2+\eta_{S,11}}+\sum_{m=1}^{11-j}\int_{\Sigma_\tau}(D^mS(N_{\leq j-1}))^2.
\end{split}
\end{equation*}
The following estimates for $S(U_k)$ integrated on $[(1.1)^{-1}\tau,\tau]$ also hold for $\alpha\leq 1+\delta$:
\begin{equation*}
\begin{split}
\sum_{k+\ell=13}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,X,j-1}\epsilon\tau^{\eta_{S,13}}+\sum_{m=1}^{13-j}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{-2}(D^mS(N_{\leq j-1}))^2.
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\sum_{k+\ell=12}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,X,j-1}\epsilon\tau^{-1+\eta_{S,12}}+\sum_{m=1}^{12-j}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{-2}(D^mS(N_{\leq j-1}))^2.
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\sum_{k+\ell\leq 11}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,X,j-1}\epsilon\tau^{-2+\eta_{S,11}}+\sum_{m=1}^{11-j}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^{-2}(D^mS(N_{\leq j-1}))^2.
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
Notice that $D^\ell\left(S(U_{k,j})\right)$ is supported in $\{r\geq R_\Omega\}$ and satisfies
\begin{equation*}
\begin{split}
|D^\ell\left(S(U_{k,j})\right)|&\leq C\sum_{m=1}^{\ell+2}\sum_{i=0}^{j-1}r^{-2}\left(|D^m S\partial_{t^*}^{k-j}\tilde{\Omega}^i\Phi|+|D^m\partial_{t^*}^{k-j}\tilde{\Omega}^i\Phi|\right)\\
&\leq C\sum_{m=1}^{\ell+k-j+2}\sum_{i=0}^{j-1}r^{-2}\left(|D^m S\tilde{\Omega}^i\Phi|+|D^m\tilde{\Omega}^i\Phi|\right).
\end{split}
\end{equation*}
We can ignore the last term because it appears already in $D^\ell U_{k,j}$ and can be estimated by Proposition \ref{U}.
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\\
\leq&C\sum_{m=1}^{\ell+k-j+2}\sum_{i=0}^{j-1}\int_{\Sigma_\tau\cap\{r\geq R_\Omega\}}r^{\alpha-4}\left(D^m S\tilde{\Omega}^i\Phi\right)^2\\
\leq&C\sum_{m=1}^{\ell+k-j+1}\sum_{i=0}^{j-1}\int_{\Sigma_\tau\cap\{r\geq R_\Omega-1\}}r^{\alpha-4}J^N_\mu\left(\partial_{t^*}^m S\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}+C\sum_{m=1}^{\ell+k-j}\sum_{i=0}^{j-1}\int_{\Sigma_\tau}r^{-2}\left(D^m\Box_{g_K}(S\tilde{\Omega}^i\Phi)\right)^2\\
\leq&C\sum_{m=1}^{\ell+k-j+1}\sum_{i=0}^{j-1}\int_{\Sigma_\tau\cap\{r\geq R_\Omega-1\}}r^{\alpha-4}J^N_\mu\left(\partial_{t^*}^m S\tilde{\Omega}^i\Phi\right)n^\mu_{\Sigma_\tau}\\
&+C\sum_{m=1}^{\ell+k-j}\int_{\Sigma_\tau}r^{-2}\left((D^mS(U_{\leq j-1,\leq j-1}))^2+(D^mS(N_{\leq j-1}))^2+(D^mV_{\leq j-1})^2\right).
\end{split}
\end{equation*}
We now apply the bootstrap assumptions. By bootstrap assumption (\ref{BA5}) and Proposition \ref{V},
$$\sum_{k+\ell=13}\int_{\Sigma_\tau}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,j-1}\epsilon\tau^{\eta_{S,13}}+\sum_{m=1}^{13-j}\int_{\Sigma_\tau}\left((D^mS(U_{\leq j-1,\leq j-1}))^2+(D^mS(N_{\leq j-1}))^2\right).$$
By bootstrap assumption (\ref{BA4}) (for $r\geq\frac{t^*}{2}$), (\ref{BA6}) (for $r\leq\frac{t^*}{2}$) and Proposition \ref{V},
$$\sum_{k+\ell=12}\int_{\Sigma_\tau}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,j-1}\epsilon\tau^{-1+\eta_{S,12}}+\sum_{m=1}^{12-j}\int_{\Sigma_\tau}\left((D^mS(U_{\leq j-1,\leq j-1}))^2+(D^mS(N_{\leq j-1}))^2\right).$$
By bootstrap assumption (\ref{BA4}) (for $r\geq\frac{t^*}{2}$), (\ref{BA7}) (for $r\leq\frac{t^*}{2}$) and Proposition \ref{V},
$$\sum_{k+\ell\leq 11}\int_{\Sigma_\tau}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,j-1}\epsilon\tau^{-2+\eta_{S,11}}+\sum_{m=1}^{11-j}\int_{\Sigma_\tau}\left((D^mS(U_{\leq j-1,\leq j-1}))^2+(D^mS(N_{\leq j-1}))^2\right).$$
Noticing that $U_{k,0}=0$, we can conclude the first three statements in the Proposition using an induction in $j$. (See the proof of Proposition \ref{Uestprop}).
For the integrated in time estimate, we note that $r^{-1-\delta}J^N_\mu\left(\Gamma^i\Phi\right)\leq CK^{X_0}\left(\Gamma^i\Phi\right)$ and use the bootstrap assumptions (\ref{BAK6}), (\ref{BAK6.5}), (\ref{BAK7}) and (\ref{BAK9}) (See proof of Propositions \ref{Uestprop} and \ref{U}).
\end{proof}
We then move on to the $S(N_k)$ terms, first we will prove an estimate for the derivatives of $S(N_k)$. The decay rate here is not optimal, but would be sufficient to close the bootstrap argument. Our approach here is to prove the decay rate that is driven only by the pointwise decay of $D^\ell\Phi$ but not by that of $D^\ell S\Phi$. The latter can, in principle, be done by similar methods, but we will skip it since it will not be necessary. In subsequent propositions, we will then prove refined decay rate for $S(N_k)$ (without derivatives) as well as for $D^\ell S(N_k)$ restricted to the region $r\leq\frac{t^*}{4}$.
\begin{proposition}\label{SNestprop}
$S(N_{k})$ satisfies the following estimates for any fixed $t^*=\tau$:
\begin{equation*}
\sum_{k+\ell =13}\int_{\Sigma_\tau} (D^\ell S(N_{k}))^2 \leq CB_SA^2\epsilon^2 \tau^{-2+\eta_{S,11}}
\end{equation*}
\begin{equation*}
\sum_{k+\ell \leq 12}\int_{\Sigma_\tau} (D^\ell S(N_{k}))^2 \leq CA^2\epsilon^2 \tau^{-2+\eta_{14}+\delta}
\end{equation*}
$S(N_{k})$ also satisfies the following integrated estimates over $t^*\in [(1.1)^{-1}\tau,\tau]$:
\begin{equation*}
\sum_{k+\ell =12}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)} r^{-1-\delta}(D^\ell S(N_{k}))^2 \leq CB_SA^2\epsilon^2 \tau^{-2+\eta_{S,11}}
\end{equation*}
\begin{equation*}
\sum_{k+\ell \leq 11}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)} r^{-1-\delta}(D^\ell S(N_{k}))^2 \leq CA^2\epsilon^2 \tau^{-2+\eta_{14}+\delta}
\end{equation*}
\end{proposition}
\begin{proof}
We would like to do a reduction similar to how we estimated $N_k$. Clearly, only the quadratic and cubic terms matter and we only need to consider terms that contain $S$ for the other terms are already controlled by the estimates of $N_k$. We will denote terms that are already in $N_k$ by ``good terms''. The only cubic terms that are relevant are those which contain $S\Gamma^{i}\Phi$ since in the terms with $D^{j+1}S\Gamma^i\Phi$, we can put all but one other factors in $L^\infty$ using the bootstrap assumptions (\ref{BAP1}), (\ref{BAP2}), (\ref{BAPI1}) and (\ref{BAPI1.1}). Notice also that the conditions for $D_\Phi\Lambda_0$, $D_\Phi\Lambda_1$ and $D_\Phi\mathcal C$ in the definition of the null condition guarantee that the bounds do not deteriorate if $S$ acts on the coefficients. The relevant terms are
$$(D^{j_1}S\Gamma^{i_1}\Phi)(D^{j_2}\Gamma^{i_2}\Phi),\quad\mbox{and}$$
$$(D^{j_1}\Gamma^{i_1}\Phi)(D^{j_2}\Gamma^{i_2}\Phi)(S\Gamma^{i_3}\Phi).$$
We first treat the case that $k+\ell\leq 12$. In this case we will always put factors without $S$ in $L^\infty$.
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau} (D^\ell S(N_{k}))^2 \\
\leq&C\sum_{i_1+i_2+j_1+j_2\leq k+\ell+2, j_1,j_2\geq 1}\int_{\Sigma_\tau} (D^{j_1}S\Gamma^{i_1}\Phi)^2(D^{j_2}\Gamma^{i_2}\Phi)^2 \\
&+C\sum_{i_1+i_2+i_3+j_1+j_2\leq 14,j_1,j_2\geq 1}\int_{\Sigma_\tau} (D^{j_1}\Gamma^{i_1}\Phi)^2(D^{j_2}\Gamma^{i_2}\Phi)^2(S\Gamma^{i_3}\Phi)^2+\mbox{good terms} \\
\leq&C\left(\sum_{i+j\leq k+\ell+1, j\geq 1}\sup \left(D^{j}\Gamma^{i}\Phi\right)^2\right)\sum_{i+j\leq k+\ell+1, j\geq 1}\int_{\Sigma_\tau} (D^{j}S\Gamma^{i}\Phi)^2\\
&+C\left(\sum_{i+j\leq k+\ell+1, j\geq 1}\sup \left(D^{j}\Gamma^{i}\Phi\right)^2\right)\left(\sum_{i+j\leq k+\ell+1, j\geq 1}\sup r^2\left(D^{j}\Gamma^{i}\Phi\right)^2\right)\sum_{i\leq k+\ell}\int_{\Sigma_\tau} r^{-2}(S\Gamma^{i}\Phi)^2 \\
&+\mbox{good terms}\\
\leq&CA\epsilon\tau^{-2+\eta_{14}}\sum_{i+j\leq k+\ell+1, j\geq 1}\int_{\Sigma_\tau} (D^{j}S\Gamma^{i}\Phi)^2 +CA^2\epsilon^2\tau^{-2+\eta_{14}}\sum_{i\leq k+\ell}\int_{\Sigma_\tau} J^N_\mu(S\Gamma^{i}\Phi)n^\mu_{\Sigma_\tau}+\mbox{good terms}\\
&\quad\mbox{using the bootstrap assumption (\ref{BAP1.5}), (\ref{BAP2}), (\ref{BAP6}), (\ref{BAPI1.2}) and (\ref{BAPI4}) and Proposition \ref{Hardy}}\\
\leq&CA\epsilon\tau^{-2+\eta_{14}}\sum_{i\leq k+\ell}\int_{\Sigma_{\tau}} \left(J^N_\mu\left(S\Gamma^i\Phi\right)n^\mu_{\Sigma_\tau}+J^N_\mu\left(\Gamma^i\Phi\right)n^\mu_{\Sigma_\tau}\right) \\
&+CA\epsilon\tau^{-2+\eta_{14}}\sum_{i+j\leq k+\ell-1}\int_{\Sigma_{\tau}} \left((D^i U_{\leq j})^2+(D^i S(U_{\leq j}))^2+\left(D^i N_{\leq j}\right)^2+\left(D^i S(N_{\leq j})\right)^2+\left(D^i V_{\leq j}\right)^2\right).
\end{split}
\end{equation*}
We now apply the estimates for the inhomogeneous terms, i.e., Propositions \ref{Nestprop}, \ref{U}, \ref{V} \ref{SUprop}. Since $k+\ell\leq 12$,:
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau} (D^\ell S(N_{k}))^2 \\
\leq&CA\epsilon\tau^{-2+\eta_{14}}\sum_{i\leq 12}\int_{\Sigma_{\tau}} \left(J^N_\mu\left(S\Gamma^i\Phi\right)n^\mu_{\Sigma_\tau}+J^N_\mu\left(\Gamma^i\Phi\right)n^\mu_{\Sigma_\tau}\right) +CA^2\epsilon^2\tau^{-2+\eta_{14}+\delta} \\
&+CA\epsilon\tau^{-2+\eta_{14}}\sum_{i+j\leq k+\ell-1}\int_{\Sigma_{\tau}} \left(D^i S(N_{\leq j})\right)^2.
\end{split}
\end{equation*}
The desired estimates then follow from an induction, together with the bootstrap assumptions (\ref{BA4}) and (\ref{BA8}), since according to this notation $\displaystyle\sum_{i+j=0}^{-1}=0$.
We then treat the case that $k+\ell=13$. In this case it is possible to have $14$ derivatives falling on the factor with $\Phi$ and hence cannot be controlled in $L^\infty$. However, in this scenario, we must have
$$\sum_{i+j=14}(DS\Phi)(D^{j}\Gamma^{i}\Phi)$$
and therefore $DS\Phi$ can be controlled in $L^\infty$ by the bootstrap assumptions (\ref{BAP6}) and (\ref{BAPI4}). In short, we have
\begin{equation*}
\begin{split}
&\sum_{k+\ell =13}\int_{\Sigma_\tau} (D^\ell S(N_{k}))^2 \\
\leq&C\sum_{i_1+i_2+j_1+j_2\leq 15, j_1,j_2\geq 1}\int_{\Sigma_\tau} (D^{j_1}S\Gamma^{i_1}\Phi)^2(D^{j_2}\Gamma^{i_2}\Phi)^2 \\
&+C\sum_{i_1+i_2+i_3+j_1+j_2\leq 15,j_1,j_2\geq 1}\int_{\Sigma_\tau} (D^{j_1}\Gamma^{i_1}\Phi)^2(D^{j_2}\Gamma^{i_2}\Phi)^2(S\Gamma^{i_3}\Phi)^2 +\mbox{good terms}\\
\leq&C\left(\sum_{i+j\leq 14, j\geq 1}\sup \left(D^{j}\Gamma^{i}\Phi\right)^2\right)\sum_{i+j\leq 14, j\geq 1}\int_{\Sigma_\tau} (D^{j}S\Gamma^{i}\Phi)^2+ \left(\sup\left(DS\Phi\right)^2\right)\sum_{i+j=14, j\geq 1}\int_{\Sigma_\tau}(D^{j}\Gamma^{i}\Phi)^2\\
&+C\left(\sum_{i+j\leq 14, j\geq 1}\sup \left(D^{j}\Gamma^{i}\Phi\right)^2\right)\left(\sum_{i+j\leq 14, j\geq 1}\sup r^2\left(D^{j}\Gamma^{i}\Phi\right)^2\right)\sum_{i\leq 13}\int_{\Sigma_\tau} r^{-2}(S\Gamma^{i}\Phi)^2+\mbox{good terms}\\
\leq&CA\epsilon\tau^{-2+\eta_{14}}\sum_{i+j\leq 14, j\geq 1}\int_{\Sigma_\tau} (D^{j}S\Gamma^{i}\Phi)^2+ CB_SA\epsilon \tau^{-2+\eta_{S,11}}\sum_{i+j=14,j\geq 1}\int_{\Sigma_\tau}(D^{j}\Gamma^{i}\Phi)^2\\
&+CA^2\epsilon^2\tau^{-2+\eta_{14}}\sum_{i\leq 13}\int_{\Sigma_\tau} J^N_\mu(S\Gamma^{i}\Phi)n^\mu_{\Sigma_\tau}+\mbox{good terms}\\
&\quad\mbox{using the bootstrap assumption (\ref{BAP1.5}), (\ref{BAP2}), (\ref{BAP6}), (\ref{BAPI1.2}) and (\ref{BAPI4}) and Proposition \ref{Hardy}}\\
\leq&CA\epsilon\tau^{-2+\eta_{14}}\sum_{i\leq 13}\int_{\Sigma_{\tau}} J^N_\mu\left(S\Gamma^i\Phi\right)n^\mu_{\Sigma_\tau}+CB_SA\epsilon\tau^{-2+\eta_{S,11}}\sum_{i\leq 13}\int_{\Sigma_{\tau}}J^N_\mu\left(\Gamma^i\Phi\right)n^\mu_{\Sigma_\tau} \\
&+CA\epsilon\tau^{-2+\eta_{14}}\sum_{\ell+k\leq 12}\int_{\Sigma_{\tau}} \left((D^\ell U_{\leq k})^2+(D^iS(U_{\leq k}))^2+\left(D^\ell N_{\leq k}\right)^2+\left(D^\ell S(N_{\leq k})\right)^2+\left(D^\ell V_{\leq k}\right)^2\right).
\end{split}
\end{equation*}
We now apply the estimates for the inhomogeneous terms, i.e., Propositions \ref{Nestprop}, \ref{U}, \ref{V}, \ref{SUprop}:
\begin{equation*}
\begin{split}
&\sum_{k+\ell =13}\int_{\Sigma_\tau} (D^\ell S(N_{k}))^2 \\
\leq&CA\epsilon\tau^{-2+\eta_{14}}\sum_{i\leq 13}\int_{\Sigma_{\tau}} J^N_\mu\left(S\Gamma^i\Phi\right)n^\mu_{\Sigma_\tau}+CB_SA\epsilon\tau^{-2+\eta_{S,11}}\sum_{i\leq 13}\int_{\Sigma_{\tau}}J^N_\mu\left(\Gamma^i\Phi\right)n^\mu_{\Sigma_\tau} \\
&+CA\epsilon\tau^{-2+\eta_{14}}\sum_{\ell+k\leq 12}\int_{\Sigma_{\tau}} \left(D^\ell S(N_{\leq k})\right)^2,
\end{split}
\end{equation*}
which is acceptable.
The estimates for the integrated in $t^*$ terms are proved analogously, noting that the elliptic estimate in Proposition \ref{elliptic} would allow for weight in $r$ and use the second parts of Propositions \ref{Nestprop}, \ref{U}, \ref{V} and \ref{SUprop}.
\end{proof}
This would allow us to close the estimates for $S(U_k)$ from Proposition \ref{SUprop}.
\begin{proposition}\label{SU}
The following estimates for $S(U_k)$ on a fixed $t^*$ slice hold for $\alpha\leq 2$:
\begin{equation*}
\begin{split}
\sum_{k+\ell=13}\int_{\Sigma_\tau}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,j-1}\epsilon\tau^{\eta_{S,13}}.
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\sum_{k+\ell=12}\int_{\Sigma_\tau}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,j-1}\epsilon\tau^{-1+\eta_{S,12}}.
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\sum_{k+\ell\leq 11}\int_{\Sigma_\tau}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,j-1}\epsilon\tau^{-2+\eta_{S,11}}.
\end{split}
\end{equation*}
The following estimates for $S(U_k)$ integrated on $[(1.1)^{-1}\tau,\tau]$ also hold for $\alpha\leq 1+\delta$:
\begin{equation*}
\begin{split}
\sum_{k+\ell=13}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,X,j-1}\epsilon\tau^{\eta_{S,13}}.
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\sum_{k+\ell=12}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,X,j-1}\epsilon\tau^{-1+\eta_{S,12}}.
\end{split}
\end{equation*}
\begin{equation*}
\begin{split}
\sum_{k+\ell\leq 11}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)}r^\alpha \left(D^\ell\left(S(U_{k,j})\right)\right)^2\leq CA_{S,X,j-1}\epsilon\tau^{-2+\eta_{S,11}}.
\end{split}
\end{equation*}
\end{proposition}
\begin{proof}
This follows directly from Proposition \ref{SUprop} and \ref{SNestprop}.
\end{proof}
In the region $\{r\leq\frac{t^*}{4}\}$, we have refined decay rates for $D^\ell S(N_k)$:
\begin{proposition}\label{SNII}
\begin{equation*}
\sum_{k+\ell=13}\int_{\Sigma_\tau\cap\{r\leq\frac{t^*}{4}\}}r^{1-\delta} (D^\ell S(N_{k}))^2 \leq CA^2\epsilon^2 \tau^{-3+\eta_{S,11}}.
\end{equation*}
\begin{equation*}
\sum_{k+\ell\leq 12}\int_{\Sigma_\tau\cap\{r\leq\frac{t^*}{4}\}}r^{1-\delta} (D^\ell S(N_k))^2 \leq CA^2\epsilon^2 \tau^{-4+\eta_{S,12}+\eta_{S,11}}.
\end{equation*}
\end{proposition}
\begin{proof}
Take $k+\ell\leq 13$. Notice that $|[D,S]\Phi|\leq C|D\Phi|$.
We would like to do a reduction similar to how we estimated $N_k$. Clearly, only the quadratic and cubic terms matter and we only need to consider terms that contain $S$ for the other terms are already controlled by the estimates of $N_k$. Notice also as before that the conditions in the null condition guarantee that the bounds do not deteriorate if $S$ acts on the coefficients. The relevant terms are
$$(D^{j_1}S\Gamma^{i_1}\Phi)(D^{j_2}\Gamma^{i_2}\Phi)\quad j_1,j_2\geq 1,$$
$$(D^{j_1}S\Gamma^{i_1}\Phi)(D^{j_2}\Gamma^{i_2}\Phi)(\Gamma^{i_3}\Phi), \quad j_1,j_2\geq 1, i_3> 8\quad\mbox{and}$$
$$(D^{j_1}\Gamma^{i_1}\Phi)(D^{j_2}\Gamma^{i_2}\Phi)(S\Gamma^{i_3}\Phi)\quad j_1,j_2\geq 1, i_3 >8.$$
We first tackle the quadratic terms:
\begin{equation*}
\begin{split}
&\sum_{i_1+\ell_1\leq 7, \ell_1\geq 1}\sum_{i_2+j_2\leq k+\ell+1, j_2\geq 1}\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}r^{1-\delta} \left(|{D}^{j_1}S\Gamma^{i_1}\Phi D^{j_2}\Gamma^{i_2}\Phi|^2+|{D}^{j_1}\Gamma^{i_1}\Phi D^{j_2}S\Gamma^{i_2}\Phi|^2\right)\\
\leq &C\left(\sup_{r\leq\frac{\tau}{4}}\sum_{i+j \leq 7, j\geq 1}r^{1-\delta}|{D}^{j}S\Gamma^{i}\Phi|^2\right)\left(\sum_{i+j\leq k+\ell+1, j\geq 1}\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}| D^{j}\Gamma^{i}\Phi|^2\right)\\
&+C\left(\sup_{r\leq\frac{\tau}{4}}\sum_{i+j \leq 7, j\geq 1}r^{1-\delta}|{D}^j\Gamma^{i}\Phi|^2\right)\left(\sum_{i+j\leq k+\ell+1, j\geq 1}\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}| D^j S\Gamma^{i}\Phi|^2\right)\\
\leq&CA\epsilon\tau^{-2+\eta_{S,11}}\sum_{i+j\leq k+\ell}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}J^N_\mu(\hat{Y}^j\Gamma^{i}\Phi)+CA\epsilon\tau^{-3+\eta_{S,11}}\sum_{i+j\leq k+\ell}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}J^N_\mu(\hat{Y}^jS\Gamma^{i}\Phi) \\
&+CA\epsilon\tau^{-2+\eta_{S,11}}\sum_{i+j\leq k+\ell-1}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}} \left((D^i U_j)^2+(D^i N_j)^2\right) \\
&+CA\epsilon\tau^{-3+\eta_{S,11}}\sum_{i+j\leq k+\ell-1}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}} \left((D^iS( U_j))^2+(D^i S( N_j))^2+(D^iV_j)^2\right)
\end{split}
\end{equation*}
by the bootstrap assumptions (\ref{BAP2}) and (\ref{BAP4}) and the elliptic estimates Proposition \ref{elliptic} and \ref{elliptichorizon}. Since $k+\ell-1\leq 12$, the inhomogeneous terms can be bounded using Proposition \ref{Nestprop}, \ref{U}, \ref{V}, \ref{SNestprop} and \ref{SU} to be
$$\leq CA^2\epsilon^2\tau^{-4+\eta_{S,12}+\eta_{S,11}}.$$
We then move on to the cubic terms:
\begin{equation*}
\begin{split}
&\sum_{i_1+j_1\leq 7, j_1\geq 1}\sum_{i_2+j_2\leq 7, j_2\geq 1}\sum_{i_3=0}^{k}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}r^{1-\delta} \left((D^{j_1}S\Gamma^{i_1}\Phi D^{j_2}\Gamma^{i_2}\Phi \Gamma^{i_3}\Phi)^2+(D^{j_1}\Gamma^{i_1}\Phi D^{j_2}\Gamma^{i_2}\Phi S\Gamma^{i_3}\Phi)^2\right)\\
\leq&C\left(\sup_{r\leq\frac{\tau}{4}}\sum_{i+j\leq 7, j\geq 1}r^2\left(D^jS\Gamma^{i}\Phi\right)^2\right)\left(\sup_{r\leq\frac{\tau}{4}}\sum_{i+j\leq 7, j\geq 1} r^{1-\delta}\left(D^j\Gamma^{i}\Phi\right)^2\right)\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}r^{-2} (\Gamma^{i_3}\Phi)^2\\
&+C\left(\sup_{r\leq\frac{\tau}{4}}\sum_{i+j\leq 7, j\geq 1}r^{1-\delta}(D^j\Gamma^{i}\Phi)^2\right)^2\tau^{1+\delta}\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}r^{-2}( S\Gamma^{i}\Phi)^2\\
\leq&CA^2\epsilon^2\tau^{-5+2\eta_{S,11}}\sum_{i=0}^{k}\int_{\Sigma_\tau} (D\Gamma^{i}\Phi)^2+CA^2\epsilon^2\tau^{-5+2\eta_{S,11}+\delta}\sum_{i=0}^{k}\int_{\Sigma_\tau}( DS\Gamma^{i}\Phi)^2,
\end{split}
\end{equation*}
by the bootstrap assumptions (\ref{BAP2}) and (\ref{BAP4}), which now clearly decays better than we need by using the bootstrap assumptions (\ref{BA4}), (\ref{BA5}) and (\ref{BA8}).
Therefore,
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}r^{1-\delta} (D^\ell S(N_k))^2\\
\leq&CA\epsilon\tau^{-2+\eta_{S,11}}\sum_{i+j\leq k+\ell}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}J^N_\mu(\hat{Y}^j\Gamma^{i}\Phi)+CA\epsilon\tau^{-3+\eta_{S,11}}\sum_{i+j\leq k+\ell}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}J^N_\mu(\hat{Y}^jS\Gamma^{i}\Phi) \\
&+CA^2\epsilon^2 \tau^{-4+\eta_{S,12}+\eta_{S,11}}.
\end{split}
\end{equation*}
The Proposition follows from the Bootstrap Assumptions (\ref{BA3}), (\ref{BA5}), (\ref{BA6}) and (\ref{BA7}).
\end{proof}
A similar decay rate can be proved in the region $\{r\leq\frac{9t^*}{10}\}$, if we do not require the estimate for the derivatives:
\begin{proposition}\label{SNI}
\begin{equation*}
\int_{\Sigma_\tau\cap\{r\leq\frac{9t^*}{10}\}}r^{1-\delta} (S(N_{13}))^2 \leq CA^2\epsilon^2 \tau^{-3+\eta_{S,11}}
\end{equation*}
\begin{equation*}
\sum_{k=0}^{12}\int_{\Sigma_\tau\cap\{r\leq\frac{9t^*}{10}\}}r^{1-\delta} (S(N_k))^2 \leq CA^2\epsilon^2 \tau^{-4+\eta_{S,12}+\eta_{S,11}}
\end{equation*}
\end{proposition}
\begin{proof}
Take $k\leq 13$. The proof follows very closely from that of the previous Proposition, by noting that we have similar pointwise decay estimates in the region (without higher derivatives) by the bootstrap assumptions (\ref{BAP3}) and (\ref{BAP7}). As in the previous Proposition, the relevant terms are
$$(DS\Gamma^{i_1}\Phi)(D\Gamma^{i_2}\Phi),$$
$$(DS\Gamma^{i_1}\Phi)(D\Gamma^{i_2}\Phi)(\Gamma^{i_3}\Phi), \quad i_3> 8\quad\mbox{and}$$
$$(D\Gamma^{i_1}\Phi)(D\Gamma^{i_2}\Phi)(S\Gamma^{i_3}\Phi)\quad i_3 >8.$$
We first tackle the quadratic terms:
\begin{equation*}
\begin{split}
&\sum_{i_1=0}^{6}\sum_{i_2=0}^{k}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}r^{1-\delta} \left(|{D}S\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi|^2+|{D}\Gamma^{i_1}\Phi DS\Gamma^{i_2}\Phi|^2\right)\\
\leq &C\left(\sup_{r\leq\frac{9\tau}{10}}\sum_{i=0}^{6}r^{1-\delta}|{D}S\Gamma^{i}\Phi|^2\right)\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}| D\Gamma^{i}\Phi|^2\right)\\
&+C\left(\sup_{r\leq\frac{9\tau}{10}}\sum_{i=0}^{6}r^{1-\delta}|{D}\Gamma^{i}\Phi|^2\right)\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}| DS\Gamma^{i}\Phi|^2\right)\\
\leq&CA^2\epsilon^2\tau^{-4+\eta_{14}+\eta_{S,11}}+CA\epsilon\tau^{-3+\eta_{S,11}}\left(\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}| DS\Gamma^{i}\Phi|^2\right).
\end{split}
\end{equation*}
We then move on to the cubic terms:
\begin{equation*}
\begin{split}
&\sum_{i_1, i_2=0}^{6}\sum_{i_3=0}^{k}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}r^{1-\delta} \left((DS\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi \Gamma^{i_3}\Phi)^2+(D\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi S\Gamma^{i_3}\Phi)^2\right)\\
\leq&C\left(\sup_{r\leq\frac{9\tau}{10}}\sum_{i=0}^{6}r^2\left(DS\Gamma^{i}\Phi\right)^2\right)\left(\sup_{r\leq\frac{9\tau}{10}}\sum_{i=0}^6 r^{1-\delta}\left(D\Gamma^{i}\Phi\right)^2\right)\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}r^{-2} (\Gamma^{i}\Phi)^2\\
&+C\left(\sup_{r\leq\frac{9\tau}{10}}\sum_{i=0}^{6}r^{1-\delta}(D\Gamma^{i}\Phi)^2\right)^2\tau^{1+\delta}\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}r^{-2}( S\Gamma^{i}\Phi)^2\\
\leq&CA^2\epsilon^2\tau^{-5+2\eta_{S,,11}}\sum_{i=0}^{k}\int_{\Sigma_\tau} (D\Gamma^{i}\Phi)^2+CA^2\epsilon^2\tau^{-5+2\eta_{S,11}+\delta}\sum_{i=0}^{k}\int_{\Sigma_\tau}( DS\Gamma^{i}\Phi)^2\\
\leq&CA^3\epsilon^3\tau^{-5+2\eta_{S,11}}+CA^2\epsilon^2\tau^{-5+2\eta_{S,11}+\delta}\sum_{i=0}^{k}\int_{\Sigma_\tau}( DS\Gamma^{i}\Phi)^2.\\
\end{split}
\end{equation*}
Therefore,
\begin{equation*}
\begin{split}
&\int_{\Sigma_\tau\cap\{r\leq\frac{\tau}{4}\}}r^{1-\delta} (S(N_k))^2\\
\leq&CA^2\epsilon^2\tau^{-4+\eta_{14}+\eta_{S,11}}+CA\epsilon\tau^{-3+\eta_{S,11}}\sum_{i=0}^{k}\int_{\Sigma_\tau\cap\{r\leq\frac{9\tau}{10}\}}\left( DS\Gamma^{i}\Phi\right)^2+CA^2\epsilon^2\tau^{-5+2\eta_{S,11}+\delta}\sum_{i=0}^{k}\int_{\Sigma_\tau}( DS\Gamma^{i}\Phi)^2.
\end{split}
\end{equation*}
The Proposition follows from the Bootstrap Assumptions (\ref{BA5}), (\ref{BA6}) and (\ref{BA7}).
\end{proof}
We then move on to the region $\{r\geq\frac{9t^*}{10}\}$.
\begin{proposition}\label{SNO}
For $\alpha=0$ or $2$,
\begin{equation*}
\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} (S(N_{13}))^2 \leq CA^2\epsilon^2 \tau^{-2+\eta_{S,13}}
\end{equation*}
\begin{equation*}
\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha} (S(N_{12}))^2 \leq CA^2\epsilon^2 \tau^{-3+\alpha+\eta_{S,12}}
\end{equation*}
\begin{equation*}
\sum_{k=0}^{11}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{1-\delta} (S(N_k))^2 \leq CA^2\epsilon^2 \tau^{-4+\alpha+\eta_{S,11}}.
\end{equation*}
\end{proposition}
\begin{proof}
Take $k\leq 13$. Following the reduction before and noticing that $[D,S]\sim D$ and $[\bar{D},S]\sim \bar{D}$, we have to consider quadratic terms
$$(\bar{D}S\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi), (\bar{D}\Gamma^{i_1}\Phi DS\Gamma^{i_2}\Phi), ({D}S\Gamma^{i_1}\Phi \bar{D}\Gamma^{i_2}\Phi),$$
$$({D}\Gamma^{i_1}\Phi \bar{D}S\Gamma^{i_2}\Phi), r^{-1}({D}S\Gamma^{i_1}\Phi {D}\Gamma^{i_2}\Phi), r^{-1}({D}\Gamma^{i_1}\Phi {D}S\Gamma^{i_2}\Phi),$$
for $i_1\geq i_2$ and the cubic terms
$$ (\bar{D}\Gamma^{i_1}\Phi {D}\Gamma^{i_2}\Phi S\Gamma^{i_3}\Phi), ({D}\Gamma^{i_1}\Phi \bar{D}\Gamma^{i_2}\Phi S\Gamma^{i_3}\Phi), r^{-1}({D}\Gamma^{i_1}\Phi {D}\Gamma^{i_2}\Phi S\Gamma^{i_3}\Phi).$$
For these cubic terms, we can assume $i_1, i_2\leq 6$ for otherwise $i_3\leq 6$ and we can control the last factor in the sup norm and reduce to the quadratic terms above. The cubic terms
$$(\bar{D}S\Gamma^{i_1}\Phi D\Gamma^{i_2}\Phi\Gamma^{i_3}\Phi), (\bar{D}\Gamma^{i_1}\Phi DS\Gamma^{i_2}\Phi\Gamma^{i_3}\Phi), ({D}S\Gamma^{i_1}\Phi \bar{D}\Gamma^{i_2}\Phi\Gamma^{i_3}\Phi),$$
$$({D}\Gamma^{i_1}\Phi \bar{D}S\Gamma^{i_2}\Phi\Gamma^{i_3}\Phi), r^{-1}({D}S\Gamma^{i_1}\Phi {D}\Gamma^{i_2}\Phi\Gamma^{i_3}\Phi), r^{-1}({D}\Gamma^{i_1}\Phi {D}S\Gamma^{i_2}\Phi\Gamma^{i_3}\Phi).$$
are irrelevant here because $i_3\leq 13$ and we can thus control the last factor in the sup norm to reduce to the quadratic terms above. As before, we also have terms that do not have $S$ (from $S\Lambda$ or from the commutators $[D,S], [\bar{D},S]$), but they already appear in $N_k$ and we will use the estimates proved for $N_k$ in Proposition \ref{NO}. We first estimate the quadratic terms. The crucial technical point here is that we do not have an improved pointwise decay estimate for $\bar{D}S\Gamma^i\Phi$ because we have used $S$ in the proof of Proposition \ref{rv} and we are only commuting with $S$ once. Nevertheless, since $k\leq 13$, we can instead put $D\Gamma^i\Phi$ in $L^\infty$.
\begin{equation*}
\begin{split}
&\sum_{i_2=0}^{\lfloor\frac{k}{2}\rfloor}\sum_{i_1=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^\alpha \left(\mbox{Quadratic Terms}\right)^2\\
\leq&C \left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_2=0}^{6} r^2|D\Gamma^{i_2}\Phi|^2\right)\sum_{i_1=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha-2} |\bar{D}S\Gamma^{i_1}\Phi |^2\\
&+C\left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_2=0}^{6} r^2|DS\Gamma^{i_2}\Phi|^2\right)\sum_{i_1=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha-2} |\bar{D}\Gamma^{i_1}\Phi|^2\\
&+C\left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_2=0}^{6} r^2|\bar{D}\Gamma^{i_2}\Phi|^2\right)\sum_{i_1=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha-2} |{D}S\Gamma^{i_1}\Phi|^2\\
&+C \left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_1=0}^{k} r^2|D\Gamma^{i_1}\Phi|^2\right)\sum_{i_2=0}^{6}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha-2} |\bar{D}S\Gamma^{i_2}\Phi |^2\\
&+C\tau^{-2}\left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_2=0}^{6} r^2|{D}\Gamma^{i_2}\Phi|^2\right)\sum_{i_1=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha-2} |{D}S\Gamma^{i_1}\Phi|^2\\
&+C\tau^{-2}\left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_2=0}^{6} r^2|{D}S\Gamma^{i_2}\Phi|^2\right)\sum_{i_1=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha-2} |{D}\Gamma^{i_1}\Phi|^2\\
\leq&CA\epsilon\sum_{i_1=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha-2} \left(|\bar{D}S\Gamma^{i_1}\Phi |^2+|\bar{D}\Gamma^{i_1}\Phi|^2\right)+CA\epsilon\tau^{-2+\eta_{14}}\sum_{i_1=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha-2} |{D}S\Gamma^{i_1}\Phi |^2\\
&+CA\epsilon\tau^{-4+\alpha+\eta_{S,11}}\left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_1=0}^{k} r^2|D\Gamma^{i_1}\Phi|^2\right)+CA\epsilon\tau^{-4+\alpha}
\end{split}
\end{equation*}
We then estimate the cubic terms:
\begin{equation*}
\begin{split}
&\sum_{i_1, i_2=0}^{\lfloor\frac{k}{2}\rfloor}\sum_{i_3=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^\alpha \left(\mbox{Cubic Terms}\right)^2\\
\leq&C\left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_1=0}^6 \left(r^2\bar{D}\Gamma^{i_1}\Phi\right)^2\right)\left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_2=0}^6 r^2\left({D}\Gamma^{i_2}\Phi\right)^2\right)\sum_{i_3=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha-4}(S\Gamma^{i_3}\Phi)^2\\
&+C\tau^{-2}\left(\sup_{r\geq\frac{9\tau}{10}}\sum_{i_1=0}^6 r^2\left({D}\Gamma^{i_1}\Phi\right)^2\right)^2\sum_{i_3=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha-4}(S\Gamma^{i_3}\Phi)^2\\
\leq & CA^2\epsilon^2\tau^{-2+\eta_{14}}\sum_{i_3=0}^{k}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}}r^{\alpha-2}(DS\Gamma^{i_3}\Phi)^2,
\end{split}
\end{equation*}
which is better then the estimates obtained for the quadratic terms. We hence focus on the quadratic terms and spell out explicitly what the estimates amount to for different values of $k$ and $\alpha$.
\begin{equation*}
\begin{split}
&\sum_{i_2=0}^{6}\sum_{i_1=0}^{13}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} \left(\mbox{Quadratic Terms}\right)^2
\leq CA^2\epsilon^2\tau^{-2+\eta_{S,13}}+CA^2\epsilon^2\tau^{-4+\eta_{S,11}+\eta_{16}},
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
&\sum_{i_2=0}^{6}\sum_{i_1=0}^{12}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} \left(\mbox{Quadratic Terms}\right)^2
\leq CA^2\epsilon^2\tau^{-3+\eta_{S,12}}+CA^2\epsilon^2\tau^{-4+\eta_{S,11}},
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
&\sum_{i_2=0}^{6}\sum_{i_1=0}^{11}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} \left(\mbox{Quadratic Terms}\right)^2
\leq CA^2\epsilon^2\tau^{-4+\eta_{S,11}}+CA^2\epsilon^2\tau^{-4+\eta_{S,11}},
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
&\sum_{i_2=0}^{6}\sum_{i_1=0}^{12}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} r^2\left(\mbox{Quadratic Terms}\right)^2
\leq CA^2\epsilon^2\left(\tau^{-1+\eta_{S,12}}+\tau^{-2+\eta_{14}}+\tau^{-2+\eta_{S,11}}\right),
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
&\sum_{i_2=0}^{6}\sum_{i_1=0}^{11}\int_{\Sigma_\tau\cap\{r\geq\frac{9\tau}{10}\}} r^2\left(\mbox{Quadratic Terms}\right)^2
\leq CA^2\epsilon^2\left(\tau^{-2+\eta_{S,11}}+\tau^{-2+\eta_{14}}\right).
\end{split}
\end{equation*}
\end{proof}
With the estimates for the inhomogeneous terms for the equations involving $S$, we can now retrieve the bootstrap assumptions involving $S$. We will follow the order that we proved the estimates without $S$, namely, first proving the pointwise estimates, then the integrated estimates, then the energy estimates and finally the energy estimates involving also $\hat{Y}$. Noticing that $U_{k,j}$ (respectively $N_k$) and $S(U_{k,j})$ (respectively $S(N_k)$) satisfy similar estimates (see Propositions \ref{NI}, \ref{NO}, \ref{U}, \ref{SNI}, \ref{SNO} and \ref{SU}), we would focus on showing that the estimates for $V_k$ are enough to close the bootstrap assumptions. We now prove the pointwise estimates and retrieve the bootstrap assumptions (\ref{BAP6}), (\ref{BAP7}), (\ref{BAPI3}) and (\ref{BAPI4}).
\begin{proposition}
For $r\geq\frac{t^*}{4}$,
\begin{equation}\label{P6}
\sum_{j=0}^8|DS\Gamma^j\Phi|^2\leq \frac{B_S}{2}A\epsilon r^{-2}.
\end{equation}
\begin{equation}\label{P7}
\sum_{j=0}^6|DS\Gamma^j\Phi|^2\leq \frac{B_S}{2}A\epsilon r^{-2}(t^*)^{\eta_{S,11}}(1+|u|)^{-2}.
\end{equation}
For $r\leq\frac{t^*}{4}$,
\begin{equation}\label{PI3}
\sum_{j=0}^{6}|S\Gamma^j\Phi|^2\leq \frac{B_S}{2}A\epsilon (t^*)^{-2+\eta_{15}}.
\end{equation}
\begin{equation}\label{PI4}
\sum_{\ell=1}^{7-j}\sum_{j=0}^{6}|D^\ell S\Gamma^j\Phi|^2\leq \frac{B_S}{2}A\epsilon r^{-2}(t^*)^{-2+\eta_{15}}.
\end{equation}
\end{proposition}
\begin{proof}
The proof of the estimates for $r\geq\frac{t^*}{4}$ (i.e. (\ref{P6}) and (\ref{P7})) is completely analogous to Proposition \ref{pointwise}, with the use of Propositions \ref{U}, \ref{NI}, \ref{NO} replaced by Propositions \ref{V}, \ref{SU}, \ref{SNI}, \ref{SNO} appropriately. Notice especially that the estimates in Proposition \ref{V} for $V$ is better than that in Proposition \ref{SU} for $SU$ and are thus acceptable.
(\ref{PI3}) follows directly from Proposition \ref{SEinside} and the bootstrap assumptions (\ref{BA3}) and (\ref{BA7}). Here, we need to use also (\ref{BA3}) because we would need to commute $S$ with $\partial_{t^*}$ and would get terms that do not contain $S$.
(\ref{PI4}) follows directly from Proposition \ref{SEDinside}, the bootstrap assumptions (\ref{BA3}) and (\ref{BA7}), as well as Proposition \ref{U}, \ref{NI}, \ref{V}, \ref{SU} and \ref{SNII} to control the inhomogeneous terms. As before, (\ref{BA3}) and Proposition \ref{U}, \ref{NI} are used to control the terms arising from $[S,\partial_{t^*}]$. Notice here that the decay rate for $\displaystyle\sum_{\ell=1}^{7-j}\sum_{j=0}^{6}|D^\ell S\Gamma^j\Phi|^2$ is not as good as that for $\displaystyle\sum_{\ell=1}^{9-j}\sum_{j=0}^{8}|D^\ell \Gamma^j\Phi|^2$ because in proving the decay rate for $\displaystyle\sum_{\ell=1}^{9-j}\sum_{j=0}^{8}|D^\ell \Gamma^j\Phi|^2$, we have used the quantities associated to $S\Phi$, while we do not have estimates for $S^2\Phi$ at our disposal.
\end{proof}
As before, once we have proved the $L^\infty$ bounds, we will replace the constant $B_S$ by $C$.
\begin{proposition}
\begin{equation}\label{K7}
\begin{split}
\sum_{i+j+k\leq 12}A_{S,X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}K^{X_0}\left(S\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\leq\frac{\epsilon}{2}\tau^{-1+\eta_{S,12}}.
\end{split}
\end{equation}
\begin{equation}\label{K8}
\sum_{i+j\leq 11}A_{S,X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}} K^{X_1}\left(S\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\leq\frac{\epsilon}{2}\tau^{-1+\eta_{S,12}}.
\end{equation}
\begin{equation}\label{K9}
\begin{split}
\sum_{i+j\leq 11}A_{S,X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}K^{X_0}\left(S\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\leq\frac{\epsilon}{2}\tau^{-2+\eta_{S,11}}.
\end{split}
\end{equation}
\begin{equation}\label{K10}
\sum_{i+j\leq 10}A_{S,X,j}^{-1}\iint_{\mathcal R((1.1)^{-1}\tau,\tau)\cap\{r\leq\frac{t^*}{2}\}}K^{X_1}\left(S\partial_{t^*}^{i}\tilde{\Omega}^j\Phi\right)\leq\frac{\epsilon}{2}\tau^{-2+\eta_{S,11}}.
\end{equation}
\end{proposition}
\begin{proof}
This follows exactly as Proposition \ref{K} except for replacing the use of Propositions \ref{U}, \ref{NI} and \ref{NO} with Propositions \ref{V}, \ref{SU}, \ref{SNI} and \ref{SNO}.
\end{proof}
\begin{proposition}
\begin{equation}\label{5}
\sum_{i+j=13}A_{S,j}^{-1}\int_{\Sigma_\tau} J^{N}_\mu\left(S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}\leq \frac{\epsilon}{4} \tau^{\eta_{S,13}}.
\end{equation}
\begin{equation}\label{8}
A_{S,j}^{-1}\sum_{j=0}^{12}\int_{\Sigma_\tau} J^{N}_\mu\left(S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}\leq \frac{\epsilon}{2} .
\end{equation}
\begin{equation}\label{K6}
\begin{split}
\sum_{i+j=13}A_{S,X,j}^{-1}\iint_{\mathcal R(\tau_0,\tau)} K^{X_0}\left(S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\leq \frac{\epsilon}{2}\tau^{\eta_{S,13}}.
\end{split}
\end{equation}
\begin{equation}\label{K6.5}
\begin{split}
\sum_{i+j\leq 12}A_{S,X,j}^{-1}\iint_{\mathcal R(\tau_0,\tau)} K^{X_0}\left(S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\leq \frac{\epsilon}{2}.
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
This follows exactly as Proposition \ref{J} except for replacing the use of Propositions \ref{U}, \ref{NI} and \ref{NO} with Propositions \ref{V}, \ref{SU}, \ref{SNI} and \ref{SNO}.
\end{proof}
\begin{proposition}
\begin{equation}\label{6}
\begin{split}
\sum_{i+j=12}A_{S,j}^{-1}\left(\int_{\Sigma_\tau} J^{Z+N,w^Z}_\mu\left(S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau} +C\tau^2\int_{\Sigma_\tau\cap\{r\leq \frac{9\tau}{10}\}} J^{N}_\mu\left(S\partial_{t^*}^i\tilde{\Omega}^j\Phi\right) n^{\mu}_{\Sigma_\tau}\right)\leq \frac{\epsilon}{4} \tau^{1+\eta_{S,12}}.
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
This follows exactly as Proposition \ref{Z1} except for replacing the use of Propositions \ref{U}, \ref{NI} and \ref{NO} with Propositions \ref{V}, \ref{SU}, \ref{SNI} and \ref{SNO}.
\end{proof}
\begin{proposition}
\begin{equation}\label{7}
\begin{split}
\sum_{i+j\leq 11}A_{S,j}^{-1}\left(\int_{\Sigma_\tau} J^{Z+N,w^Z}_\mu\left(S\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\right) n^{\mu}_{\Sigma_\tau} +C\tau^2\int_{\Sigma_\tau\cap\{r\leq \frac{9\tau}{10}\}} J^{N}_\mu\left(S\left(\partial_{t^*}^i\tilde{\Omega}^j\Phi\right)\right) n^{\mu}_{\Sigma_\tau}\right)\leq \frac{\epsilon}{4} \tau^{\eta_{S,11}}
\end{split}
\end{equation}
\end{proposition}
\begin{proof}
This follows exactly as Proposition \ref{Z2} except for replacing the use of Propositions \ref{U}, \ref{NI} and \ref{NO} with Propositions \ref{V}, \ref{SU}, \ref{SNI} and \ref{SNO}.
\end{proof}
To close the bootstrap argument we need finally to consider energy quantities with both $S$ and $\hat{Y}$.
\begin{proposition}
$$\sum_{i+k=13}A_{S,Y}^{-1}\int_{\Sigma_\tau}J^{N}_\mu\left(\hat{Y}^kS\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau}\leq \frac{\epsilon}{4} \tau^{\eta_{S,13}}.$$
$$\sum_{i+k=12}A_{S,Y}^{-1}\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\hat{Y}^kS\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau}\leq \frac{\epsilon}{4} \tau^{1+\eta_{S,12}}$$
$$\sum_{i+k\leq 11}A_{S,Y}^{-1}\tau^2\int_{\Sigma_\tau\cap\{r\leq r^-_Y\}}J^{N}_\mu\left(\hat{Y}^kS\partial_{t^*}^i\Phi\right) n^{\mu}_{\Sigma_\tau}\leq \frac{\epsilon}{4} \tau^{\eta_{S,11}}$$
\end{proposition}
\begin{proof}
This follows exactly as Proposition \ref{Yest} except for replacing the use of Propositions \ref{NI} with Propositions \ref{V} and \ref{SNI}.
\end{proof}
\section{Proof of Theorem 1}\label{pfmaintheorem}
Now all the bootstrap assumptions are closed and all the estimates hold. The solution hence exists globally by a standard local existence argument that we omit here. The decay estimates of the the derivatives of $\Phi$ claimed in the Theorem are restatements of (\ref{P3}), (\ref{P4}), (\ref{BAP2}). The decay estimates follow from the use of Proposition \ref{rnoderivatives} and (\ref{Z2}) for $r\geq R$ and Proposition \ref{inside2} and (\ref{K5}) for $r\leq\frac{t^*}{4}$.
\section{Acknowledgment}
The author thanks his advisor Igor Rodnianski for his continual support and encouragement and for many enlightening discussions.
\bibliographystyle{hplain}
|
{
"timestamp": "2010-09-22T02:02:20",
"yymm": "1009",
"arxiv_id": "1009.4109",
"language": "en",
"url": "https://arxiv.org/abs/1009.4109"
}
|
\section{Introduction}\label{sec-counter-ex}
Among the most important theorems on the topology of (families of)
projective complex manifolds, there are the hard Lefschetz theorem,
degeneration of the Leray spectral sequence at $E_2,$ and Deligne's
semisimplicity theorem of cohomology local systems. They all fail
for algebraic varieties with singularities, but if we replace the
ordinary cohomology groups with the intersection cohomology groups,
introduced by Goresky and Mac Pherson, these results turn out to
hold, and what lies in the center of the story is the notion of
\textit{perverse sheaves} and the \textit{decomposition theorem}
for them, first proved by Beilinson, Bernstein, Deligne and Gabber
\cite{BBD}. For more discussion on these, see \cite{deC-M}.
Then the notion of perverse sheaves was generalized to spaces with
group actions (the so-called \textit{equivariant perverse sheaves}),
and then more generally, to algebraic stacks \cite{LO3}. The key
observation here is that perverse sheaves can be glued with respect
to the \textit{smooth topology.} It is interesting to know if the
decomposition theorem generalizes. The case for equivariant perverse
sheaves has been proved in (\cite{BL}, 5.3). Let us remark here that
for algebraic stacks, it does not follow directly from the case for
algebraic varieties via the proper base change, because the
decomposability of a complex of sheaves is \textit{not} local for
the smooth topology, as the following counter-example of Drinfeld shows.
Let $E$ be a complex elliptic curve, and let
$f:\text{pt = Spec }\bb C\to BE$ be the natural
projection; this is a representable proper smooth map. A perverse
sheaf on $BE$ is the same as a lisse sheaf (which turns out to be
constant), appropriately shifted. There is a
natural non-zero morphism $\underline{\bb C}_{BE}\to
Rf_*\underline{\bb C}_{\text{pt}},$ adjoint to the
isomorphism $f^*\underline{\bb C}_{BE}\simeq
\underline{\bb C}_{\text{pt}},$ but there is no
non-zero morphism in the other direction, because
$$
Hom(Rf_*\underline{\bb C}_{\text{pt}},\underline{\bb
C}_{BE})=Hom(\underline{\bb C}_{\text{pt}},f^!\underline{\bb
C}_{BE})=Hom(\underline{\bb C}_{\text{pt}},
\underline{\bb C}_{\text{pt}}[2])=0.
$$
Here the $Hom$'s are taken in the derived categories.
Similarly, the non-zero natural map
$Rf_*\underline{\bb C}_{\text{pt}}\to R^2f_*\underline
{\bb C}_{\text{pt}}[-2]=\underline{\bb C}_{BE}[-2]$ lies in
$$
Hom(Rf_*\underline{\bb C}_{\text{pt}},\underline{\bb
C}_{BE}[-2])=Hom(\underline{\bb C}_{\text{pt}},f^!\underline{\bb
C}_{BE}[-2])=Hom(\underline{\bb C}_{\text{pt}},\underline{\bb
C}_{\text{pt}})=\bb C,
$$
but the Hom set in the other direction is zero:
$$
Hom(\underline{\bb C}_{BE}[-2],Rf_*\underline{\bb
C}_{\text{pt}})=Hom(f^*\underline{\bb C}_{BE}[-2],
\underline{\bb C}_{\text{pt}})=Hom(\underline{\bb
C}_{\text{pt}}[-2],\underline{\bb C}_{\text{pt}})=0.
$$
Therefore, $Rf_*\underline{\bb C}$ is not semi-simple
(since it is not a direct sum of the
$(R^if_*\underline{\bb C})[-i]$'s). The
same argument applies to finite fields, with $\underline{\bb
C}$ replaced by $\overline{\bb Q}_{\ell}.$
\begin{remark}\label{source}
This example was first given by Drinfeld, who asked for
the reason of the failure of the usual argument for
schemes. Later, it was communicated by J. Bernstein to
Y. Varshavsky, who asked M. Olsson in an email
correspondence. Olsson kindly shared this email with me,
and explained to me that the reason is the failure of the
upper bound of weights in \cite{Del2} for $BE.$
\end{remark}
In the following we explain why the usual proof (as in
\cite{BBD}) fails for $f.$ The proof of the
decomposition theorem over $\bb C$ relies on the
decomposition theorem over finite fields (\textit{loc. cit.},
5.3.8, 5.4.5), so it suffices to explain why the proof of
(\textit{loc. cit.}, 5.4.5) fails for $f:\text{Spec }\bb F_q
\to BE,$ for an $\bb F_q$-elliptic curve $E.$
Let $K_0=Rf_*\overline{\bb Q}_{\ell}.$ The perverse $t$-structure
agrees with the standard $t$-structure on $\text{Spec }\bb F_q,$ and
by definition (\cite{LO3}, 4), we have $\leftexp{p}{\s H}^iK_0=\s
H^{i+1}(K_0)[-1]$ on $BE,$ and so
$$
\bigoplus_i(\leftexp{p}{\s H}^iK)[-i]=\bigoplus_i
(\s H^iK)[-i].
$$
Each $R^if_*\overline{\bb Q}_{\ell}[-i]$ is pure of
weight 0. In the proof of (\cite{BBD}, 5.4.5), the exact
triangles
$$
\xymatrix@C=.5cm{
\tau_{<i}K_0 \ar[r] & \tau_{\le i}K_0 \ar[r] & (\s
H^iK_0)[-i] \ar[r] &}
$$
would split geometrically, because $Ext^1((\s H^iK)[-i],
\tau_{<i}K)$ would have weights $>0.$ We will see that this
group is pure of weight 0, and in fact has 1 as a Frobenius eigenvalue.
For simplicity, we denote $H^i(\s X,\overline{\bb
Q}_{\ell})$ by $H^i(\s X).$
Let $\pi:BE\to\text{Spec }\bb F_q$ be the structural
map; then $\pi\circ f=\text{id}.$ Since $E$ is connected,
the sheaf $R^if_*\overline{\bb Q}_{\ell}$ is the
inverse image of some sheaf on $\text{Spec }\bb
F_q,$ namely $f^*R^if_*\overline{\bb Q}_{\ell}.$ By
smooth base change, it is isomorphic to $\pi^*H^i(E)$
as a $\text{Gal}(\overline{\bb F}_q/\bb F_q)$-module. In particular,
$R^0f_*\overline{\bb Q}_{\ell}=\overline{\bb
Q}_{\ell},\ R^1f_*\overline{\bb Q}_{\ell}\cong\pi^*
H^1(E)$ and $R^2f_*\overline{\bb Q}_{\ell}=\overline{
\bb Q}_{\ell}(-1).$ Then the exact triangle above becomes
\begin{gather*}
i=2:\qquad\xymatrix@C=.5cm{
\tau_{\le1}K_0 \ar[r] & K_0 \ar[r] & \overline{\bb
Q}_{\ell}(-1)[-2] \ar[r] &} \\
i=1:\qquad\xymatrix@C=.5cm{
\overline{\bb Q}_{\ell} \ar[r] & \tau_{\le1}K_0
\ar[r] & \pi^*H^1(E)[-1] \ar[r] &.}
\end{gather*}
Apply $Ext^*(\overline{\bb Q}_{\ell}(-1)[-2],-)$ to
the second triangle. One can compute $H^*(BE)$ by a
theorem of Borel (see (\cite{Sun}, 7.2)): $H^{2i-1}(BE)=0,$
and $H^{2i}(BE)=\text{Sym}^iH^1(E).$ Let $\alpha$ and
$\beta$ be the eigenvalues of the Frobenius $F$ on
$H^1(E).$ We have
$$
Ext^1(\overline{\bb Q}_{\ell}(-1)[-2],
\overline{\bb Q}_{\ell})=Ext^3(\overline{\bb
Q}_{\ell},\overline{\bb Q}_{\ell}(1))=H^3(BE)(1)=0,
$$
and
\begin{equation*}
\begin{split}
Ext^1(\overline{\bb Q}_{\ell}(-1)[-2],\pi^*H^1(E)[-1])
&=H^2(BE)\otimes H^1(E)(1) \\
&=H^1(E)\otimes H^1(E)(1)=End(H^1(E)),
\end{split}
\end{equation*}
which is 4-dimensional with eigenvalues
$\alpha/\beta,\beta/\alpha,1,1,$ and
$$
Ext^2(\overline{\bb Q}_{\ell}(-1)[-2],
\overline{\bb Q}_{\ell})=H^4(BE)(1),
$$
which is 3-dimensional with eigenvalues $\alpha/\beta,
\beta/\alpha,1.$ This implies that the kernel
\begin{equation*}
\begin{split}
Ext^1(\overline{\bb Q}_{\ell}(-1)[-2],\tau_{\le1}K)
&= \\
&\text{Ker}\Big(Ext^1(\overline{\bb Q}_{\ell}(-1)[-2],
\pi^*H^1(E)[-1])\to Ext^2(\overline{\bb
Q}_{\ell}(-1)[-2],\overline{\bb Q}_{\ell})\Big)
\end{split}
\end{equation*}
is non-zero, pure of weight 0, and has 1 as a Frobenius
eigenvalue. So the first exact triangle above does not
necessarily (in fact does not, as the argument in the
beginning shows) split geometrically. Also
$$
Ext^1(\pi^*H^1(E)[-1],\overline{\bb Q}_{\ell})=
Ext^2(\overline{\bb Q}_{\ell},\pi^*H^1(E)^{\vee})=
H^1(E)\otimes H^1(E)^{\vee}=End(H^1(E))
$$
is 4-dimensional and has eigenvalues
$\alpha/\beta,\beta/\alpha,1,1,$ hence the proof for the
geometric splitting of the second exact triangle fails
too.
\vskip.5truecm
In \cite{LO3}, Laszlo and Olsson generalized the theory of
perverse sheaves to Artin stacks locally of finite type
over some field. In \cite{Sun}, we proved that for Artin
stacks of finite type over a finite field, with affine
stabilizers (\ref{affine-stab}),
Deligne's upper bound of weights for the compactly
supported cohomology groups still applies. In this paper,
we will show that for such stacks, similar argument as in
\cite{BBD} gives the decomposition theorem. We state it as follows
(see (\ref{torsion}) for the notation).
\begin{theorem}\label{main-thm}
Let $f:\s X_0\to\s Y_0$ be a proper morphism of finite diagonal
between $\bb F_q$-algebraic stacks of finite type with affine
stabilizers (\ref{affine-stab}), and let $K_0$ be an $\iota$-pure
$\overline{\bb Q}_{\ell}$-complex on $\s X_0.$ Then we have
$$
Rf_*K\simeq\bigoplus_{i\in\bb Z}(\leftexp{p}{R}^if_*K)[-i],
$$
and each $\leftexp{p}{R}^if_*K$ is a semi-simple perverse sheaf on
$\s Y.$ Consequently, if $K_0$ is a semi-simple $\overline{\bb
Q}_{\ell}$-perverse sheaf on $\s X_0,$ the conclusion above also holds.
\end{theorem}
\textbf{Organization.} In $\S\ref{sec-prototype}$ we complete the
proof of the structure theorem for $\iota$-mixed sheaves on stacks,
as claimed in (\cite{Sun}, 2.7.1). In $\S\ref{sec-decomp-F_q},$ we
generalize the decomposition theorem for perverse sheaves on stacks
over finite fields, using weight theory. In the end, we mention the
decomposition theorem for stacks over $\bb C.$
\begin{notation-convention}\label{notat-conv}
\begin{anitem}\label{frob}
We fix an algebraic closure $\bb F$ of the finite
field $\bb F_q$ with $q$ elements. Let $F$ or $F_q$ be
the $q$-geometric Frobenius, namely the $q$-th root
automorphism on $\bb F.$ Let $\ell$ be a prime number,
$\ell\ne p,$ and fix an embedding of fields
$\overline{\bb Q}_{\ell}\overset{\iota}{\to}\bb
C.$ For $z\in\bb C,$ let $w_q(z)=2\log_q|z|.$
\end{anitem}
\begin{anitem}\label{ft}
For the definition of an Artin stack (or an algebraic stack), we
refer to (\cite{Ols2}, 1.2.22). We only consider algebraic stacks
\textit{of finite type} over the base.
\end{anitem}
\begin{anitem}\label{torsion}
Objects over $\bb F_q$ will be denoted with a subscript
$_0,$ and suppression of it means passing to $\bb F$ by extension
of scalars. For instance, if $K_0$ is a $\overline{\bb
Q}_{\ell}$-complex of sheaves on an $\bb F_q$-Artin stack $\s X_0,$
then $K$ denotes its inverse image
on $\s X:=\s X_0\otimes_{\bb F_q}\bb F.$ For $b\in\overline
{\bb Q}_{\ell}^*,$ let $\overline{\bb Q}_{\ell}^{(b)}$
be the lisse Weil sheaf of rank one on $\text{Spec }\bb
F_q$ corresponding to the character that sends $F_q$ to
$b$ (see (\cite{Sun}, 2.4)).
\end{anitem}
\begin{anitem}
For an algebraic stack $X$ over a field $k,$ we say it is
\textit{essentially smooth} if $(X_{\overline{k}})
_{\text{red}}$ is smooth over $\overline{k}.$
\end{anitem}
\begin{anitem}
For a map $f:X\to Y$ and a complex of sheaves $K$ on $Y,$ we
sometimes write $H^n(X,K)$ for $H^n(X,f^*K).$
\end{anitem}
\begin{anitem}
We will denote $Rf_*,Rf_!,Lf^*$ and $Rf^!$ by $f_*,f_!,f^*$ and
$f^!$ respectively. We use $Hom$ and $Ext$ (resp. $\s Hom$ and $\s
Ext$) for the global Hom and Ext (resp. sheaf Hom and Ext).
\end{anitem}
\begin{anitem}
We will only consider the middle perversity (\cite{BBD}, 4.0). We use
$\leftexp{p}{\s H}^i$
and $\leftexp{p}{\tau}_{\le i}$ to denote cohomology and truncations with
respect to this perverse $t$-structure.
\end{anitem}
\end{notation-convention}
\begin{flushleft}
\textbf{Acknowledgment.}
\end{flushleft}
I would like to thank my advisor Martin Olsson for
introducing this topic to me, and giving so many suggestions
during the writing. Yves Laszlo and Weizhe Zheng pointed out
some mistakes and gave many helpful comments. Many
people, especially Brian Conrad and Matthew Emerton, have
helped to answer my questions related to this paper on
mathoverflow. The revision of the paper was done during the stay in Ecole
polytechnique CMLS (UMR 7640) and Universit\'e Paris-Sud (UMR 8628), while
I was supported by ANR grant G-FIB.
\section{The prototype: the structure theorem of $\iota$-mixed
sheaves on stacks}\label{sec-prototype}
We generalize the structure theorem of $\iota$-mixed
sheaves (\cite{Del2}, 3.4.1) to stacks. This result has little
to do with the rest of this paper (except in (\ref{5.3.4})), but it is the
prototype of the corresponding
results (e.g. weight filtrations and the decomposition
theorem) for perverse sheaves. In this section, sheaves are
understood as Weil sheaves. See (\cite{Sun}, 2.4.3) for the definitions
of punctually $\iota$-pure sheaves and $\iota$-mixed sheaves.
\begin{theorem}\label{3.4.1}\emph{(stack version of
(\cite{Del2}, 3.4.1))}
Let $\s X_0$ be an $\bb F_q$-algebraic stack.
(i) Every $\iota$-mixed sheaf $\s F_0$ on $\s X_0$ has a unique
decomposition $\s F_0=\bigoplus_{b\in\bb{R/Z}}\s F_0(b),$
called the \emph{decomposition according to the weights
mod }$\bb Z,$ such that the punctual $\iota$-weights
of $\s F_0(b)$ are all in the coset $b.$ This
decomposition, in which almost all the $\s F_0(b)$'s
are zero, is functorial in $\s F_0.$
(ii) Every $\iota$-mixed lisse sheaf $\s F_0$ with
integer punctual $\iota$-weights on $\s X_0$ has a
unique finite increasing filtration $W$ by lisse
subsheaves, called the \emph{weight filtration}, such that
$\emph{Gr}_i^W$ is punctually $\iota$-pure of weight $i.$
Every morphism between such sheaves on $\s X_0$ is
strictly compatible with their weight filtrations.
(iii) If $\s X_0$ is a normal algebraic stack (i.e. it has
a normal presentation), and
$\s F_0$ is a lisse and punctually $\iota$-pure
sheaf on $\s X_0,$ then $\s F$ on $\s X$ is semi-simple.
\end{theorem}
\begin{proof}
(i) and (ii) are proved in (\cite{Sun}, 2.7.1), where
(iii) is claimed without giving a detailed proof.
Here we complete the proof of (iii).
First of all, note that we may make a finite extension of
the base field $\bb F_{q^v}/\bb F_q.$
From the proof of (\cite{LO3}, 8.3), we see that if
$\s U\subset\s X$ is an open substack, and
$\s G_{\s U}$ is a lisse subsheaf of $\s
F|_{\s U},$ then it extends to a unique lisse subsheaf
$\s G\subset\s F.$ Applying the full-faithfulness
in (\textit{loc. cit.}) we see that if $\s F|_{\s U}$
is semi-simple, so also is $\s F.$ Therefore, we may shrink
$\s X$ to a dense open substack $\s U,$ and
replace $\s X_0$ by some model of $\s U$ over a
finite extension $\bb F_{q^v}.$ We can then assume
$\s X_0$ is smooth and geometrically connected.
Following the proof (\cite{Del2}, 3.4.5), it suffices to
show (\cite{Del2}, 3.4.3) for stacks. We claim that, if
$\s F_0$ is lisse and punctually $\iota$-pure of
weight $w,$ then $H^1(\s{X,F})$ is $\iota$-mixed of
weights $\ge1+w.$ The conclusion follows from this claim.
Let $N=\dim\s X_0.$ By Poincar\'e duality, it
suffices to show that, for every lisse sheaf $\s
F_0,$ punctually $\iota$-pure of weight $w,\ H^{2N-1}_c
(\s{X,F})$ is $\iota$-mixed of weights $\le2N-1+w.$
To show this, we may shrink $\s X_0$ to assume that
the inertia $\s I_0\to\s X_0$ is flat, with rigidification
$\pi:\s X_0\to X_0$ (cf. (\cite{Ols2}, 1.5)). We have the spectral
sequence
$$
H^r_c(X,R^k\pi_!\s F)\Longrightarrow
H^{r+k}_c(\s{X,F}),
$$
so let $r+k=2N-1.$ Note that $k$ can only be of the form
$-2i-2d,$ for $i\ge0,$ where $d=\text{rel. dim}(\s
I_0/\s X_0).$ So we have $r=2\dim X_0+2i-1,$ and in
order for $H^r_c(X,-)$ to be non-zero, $i=0.$ Then
$$
H^{2N-1}_c(\s{X,F})=H^{2\dim X-1}_c(X,R^{-2d}\pi_!\s F).
$$
The claim follows from the fact that $R^{-2d}\pi_!\s
F_0$ is punctually $\iota$-pure of weight $w-2d.$ To see this
fact, it suffices to show that $H^{-2d}_c(BG,\s F)$ has
weight $w-2d,$ for any $\bb F_q$-algebraic group $G_0$ of
dimension $d,$ and any lisse punctually $\iota$-pure sheaf
$\s F_0$ on $BG_0$ of weight $w.$ By considering the Leray spectral
sequence for the natural map $BG_0\to B\pi_0(G_0),$ we reduce to
the case where $G_0$ is connected, and this case is clear. See
the proof of (\cite{Sun}, 1.4) for more details.
\end{proof}
\section{Decomposition theorem for stacks over $\bb
F_q$}\label{sec-decomp-F_q}
For an $\bb F_q$-algebraic stack $\s X_0,$ let $D_m(\s
X_0,\overline{\bb Q}_{\ell})$ be the full subcategory of mixed
complexes in $D_c(\s X_0,\overline{\bb Q}_{\ell})$ (see (\cite{LO3},
9.1)). It is stable under the six operations (\cite{Sun}, 2.11,
2.12) and the perverse truncations $\leftexp{p}{\tau}_{\le0}$ and
$\leftexp{p}{\tau}_{\ge0}.$ The latter can be checked smooth
locally, and hence follows from (\cite{BBD}, 5.1.6). The core of
$D_m (\s X_0,\overline{\bb Q}_{\ell})$ with respect to this induced
perverse $t$-structure is called the category of \textit{mixed
perverse sheaves} on $\s X_0,$ as defined in (\cite{LO3}, 9.1). This
is a Serre subcategory of all $\overline{\bb Q}_{\ell}$-perverse
sheaves $\text{Perv}(\s X_0),$ i.e. it is closed under sub-quotients
and extensions. For sub-quotients, see (\cite{LO3}, 9.3). For
extensions, note that a short exact sequence of perverse sheaves is
an exact triangle in $D_c^b,$ and we may apply (the mixed variant
of) (\cite{Sun}, 2.5 iii).
Nevertheless, in this paper, we will consider the more general notion of
\textit{$\iota$-mixed complexes} and in particular,
\textit{$\iota$-mixed perverse sheaves}. This weaker
condition will be sufficient for the purpose of proving the
decomposition theorem.
In fact, Lafforgue has proved the conjecture of Deligne that,
all (Weil) sheaves are $\iota$-mixed, for any $\iota.$ See (\cite{Lau},
1.3) and (\cite{Sun}, 2.8.1).
The following definition comes from (\cite{Del2}, 6.2.4).
\begin{definition}\label{D3.1}
Let $K_0\in D_c(\s X_0,\overline{\bb Q}_{\ell})$
and $w\in\bb R.$ We say that $K_0$ \emph{has $\iota$-weights
$\le w$} if for each $i\in\bb Z,$ the punctual $\iota$-weights
of $\s H^iK_0$ are $\le i+w,$ and we denote by
$D_{\le w}(\s X_0,\overline{\bb Q}_{\ell})$ the full
subcategory of such complexes. We say that \emph{$K_0$ has
$\iota$-weights $\ge w$} if its Verdier dual $DK_0$ has
$\iota$-weights $\le-w,$ and denote by $D_{\ge w}(\s
X_0,\overline{\bb Q}_{\ell})$ the subcategory of such
complexes. We say that \emph{$K_0$ is $\iota$-pure of weight
$w$} if it belongs to both $D_{\le w}$ and $D_{\ge w}.$
\end{definition}
\begin{lemma}\label{L-pres}
Let $P:X_0\to\s X_0$ be a presentation, and $K_0\in
D_c(\s X_0,\overline{\bb Q}_{\ell}).$ Then $K_0$
is $\iota$-mixed of weights $\le w$ (resp. $\ge w$) if and
only if $P^*K_0$ (resp. $P^!K_0$) is so.
\end{lemma}
\begin{proof}
The two statements are dual to each other, so
it suffices to consider only the case where $K_0$ has
weights $\le w.$ The ``only if" part is obvious, and the
``if" part follows from (\cite{Sun}, 2.8) and the assumption
that $P$ is surjective.
\end{proof}
\begin{lemma}\label{5.3.2}\emph{(stack version of (\cite{BBD},
5.3.1, 5.3.2))}
(i) For $\s F_0\in\emph{Perv}_{\le w}(\s X_0)$ (resp. $\s F_0
\in\emph{Perv}_{\ge w}(\s X_0)$), all of its sub-quotients
are $\iota$-mixed of weights $\le w$ (resp. $\ge w$).
(ii) Let $j:\s U_0\hookrightarrow\s X_0$ be an
immersion of algebraic stacks. Then for any real number
$w,$ the intermediate extension $j_{!*}$ (\cite{LO3}, 6)
respects $\emph{Perv}_{\ge w}$ and $\emph{Perv}_{\le w}.$
In particular, if $\s F_0$ is an $\iota$-pure
perverse sheaf on $\s U_0,$ then $j_{!*}\s
F_0$ is $\iota$-pure of the same weight.
\end{lemma}
\begin{proof}
(i) Note that the variant for mixed perverse sheaves on stacks is given
in (\cite{LO3}, 9.3). Recall that for a morphism $u:\s F_0\to\s G_0$ of
perverse sheaves, with cone $K_0$ in $D^b_c(\s X_0,\overline
{\bb Q}_{\ell}),$ we have $\text{Ker}(u)=\leftexp{p}{\s
H}^{-1}K_0$ and $\text{Coker}(u)=\leftexp{p}{\s H}^0K_0.$
Let $P:X_0\to\s X_0$ be a presentation of relative dimension
$d.$ Since $P^*[d]$ commutes with $\leftexp{p}{\s H}^i,$
we see that $P^*[d]:\text{Perv}(\s X_0)\to\text{Perv}(X_0)$
is an exact functor. If $\s F_0'$ is a sub-object (resp.
quotient object) of $\s F_0,$ then $D\s F_0'$ is a quotient
object (resp. sub-object) of $D\s F_0,$ so by duality it
suffices to prove the $``\le w"$ part. This follows from
the exactness of $P^*[d]$ and the $\iota$-mixed variant
of (\cite{BBD}, 5.3.1).
(ii) For a closed immersion $i,$ we see that $i_*$ respects
$D_{\ge w}$ and $D_{\le w},$ so we may assume that $j$ is
an open immersion. We only need to consider the case for
$\text{Perv}_{\le w},$ since the case for $\text{Perv}_{\ge
w}$ follows from $j_{!*}D=Dj_{!*}.$
Let $P:X_0\to\s X_0$ be a presentation, and let the
following diagram be 2-Cartesian:
$$
\xymatrix@C=.8cm @R=.6cm{
U_0 \ar[r]^-{j'} \ar[d]_-{P'} & X_0 \ar[d]^-P \\
\s U_0 \ar[r]_-j & \s X_0.}
$$
For $\s F_0\in\text{Perv}_{\le w}(\s U_0),$ by
(\ref{L-pres}) it suffices to show that $P^*j_{!*}\s
F_0\in D_{\le w}(X_0,\overline{\bb Q}_{\ell}).$ Let
$d$ be the relative dimension of $P.$ By (\cite{LO3}, 6.2)
we have
$$
P^*j_{!*}\s F_0=(P^*(j_{!*}\s F_0)[d])[-d]
=j'_{!*}(P'^*\s F_0[d])[-d].
$$
Since $P'^*\s F_0\in D_{\le w},\ P'^*\s
F_0[d]\in D_{\le w+d},$ and by (\cite{BBD}, 5.3.2),
$j'_{!*}(P'^*\s F_0[d])\in\text{Perv}_{\le w+d},$
and by definition $P^*j_{!*}\s F_0\in D_{\le w}.$
\end{proof}
\begin{corollary}\label{5.3.4}\emph{(stack version of
(\cite{BBD}, 5.3.4))}
Every simple perverse sheaf $\s F_0$ on an algebraic
stack $\s X_0$ is $\iota$-pure.
\end{corollary}
\begin{proof}
By (\cite{LO3}, 8.2ii), $\s F_0\simeq j_{!*}L_0[d]$ for
some essentially smooth irreducible
substack $j:\s U_0\hookrightarrow\s X_0$ of dimension $d,$
and a simple lisse sheaf $L_0$ on $\s U_0,$ which is punctually
$\iota$-pure by (\ref{3.4.1}ii).
The result follows from (\ref{5.3.2}ii).
\end{proof}
\begin{theorem}\label{5.4.1}\emph{(stack version of
(\cite{BBD}, 5.4.1, 5.4.4))}
Let $K_0\in D_c^b(\s X_0,\overline{\bb Q}_{\ell}).$ Then $K_0$
has $\iota$-weights $\le w$ (resp. $\ge w$) if and only if
$\leftexp{p}{\s H}^iK_0$ has $\iota$-weights $\le
w+i$ (resp. $\ge w+i$), for each $i\in\bb Z.$ In
particular, $K_0$ is $\iota$-pure of weight $w$ if and
only if each $\leftexp{p}{\s H}^iK_0$ is
$\iota$-pure of weight $w+i.$
\end{theorem}
\begin{proof}
The case of $``\ge"$ follows from the case of $``\le"$ and
$\leftexp{p}{\s H}^i\circ D=D\circ\leftexp{p}{\s H}^{-i}.$
So we only need to show the case of $``\le".$
Let $P:X_0\to\s X_0$ be a presentation of relative
dimension $d.$ Then $K_0$ has $\iota$-weights $\le w$ if
and only if (\ref{L-pres}) $P^*K_0$ has $\iota$-weights
$\le w,$ if and only if (\cite{BBD}, 5.4.1) each
$\leftexp{p}{\s H}^i(P^*K_0)$ has $\iota$-weights
$\le w+i.$ We have $\leftexp{p}{\s H}^i(P^*K_0)=
\leftexp{p}{\s H}^i(P^*(K_0[-d])[d])=P^*
\leftexp{p}{\s H}^i(K_0[-d])[d]=P^*(\leftexp{p}
{\s H}^{i-d}K_0)[d],$ so $P^*(\leftexp{p}{\s
H}^{i-d}K_0),$ and hence $\leftexp{p}{\s
H}^{i-d}K_0,$ has $\iota$-weights $\le w+i-d.$
\end{proof}
The category $\text{Perv}(\s X_0)$ is artinian
and noetherian (\cite{LO3}, 8.2i). By the \textit{irreducible
constituents} of a perverse sheaf we mean its Jordan-H\"older
components.
\begin{definition}\label{D3.2}
Let $\s F_0$ be a perverse sheaf on an
algebraic stack $\s X_0,$ and let $\beta\in\bb{R/Z}.$
We say that \emph{$\s F_0$ has $\iota$-weights in $\beta,$}
if all irreducible constituents of $\s F_0,$ which are
$\iota$-pure by (\ref{5.3.4}), have $\iota$-weights in
the coset $\beta$ (in the sense of (\ref{D3.1})).
\end{definition}
Now we give the perverse sheaf version of (\ref{3.4.1}i,
ii). For (ii), the variant for mixed perverse sheaves
(which is the stack version of (\cite{BBD}, 5.3.5)) is
given in (\cite{LO3}, 9.2).
\begin{theorem}\label{5.3.5}
Let $\s F_0$ be a perverse sheaf on $\s X_0.$
(i) $\s F_0$ has a unique decomposition $\s
F_0=\bigoplus_{\beta\in\bb{R/Z}}\s F_0(\beta)$ into
perverse subsheaves, called the \emph{decomposition
according to the weights mod} $\bb Z,$ such that for
each $\beta,$ the $\iota$-weights of $\s F_0(\beta)$ are in
$\beta$ (in the sense of (\ref{D3.2})). This decomposition,
in which almost all the $\s F_0(\beta)$'s are zero, is functorial in
$\s F_0.$
(ii) If the $\iota$-weights of $\s F_0$ are integers
(\ref{D3.2}), then there exists a unique finite increasing
filtration $W$ of $\s F_0$ by perverse subsheaves, called
the \emph{weight filtration,} such that $\emph{Gr}^W_i\s
F_0$ is $\iota$-pure of weight $i,$ for each $i.$ Every morphism
between such perverse sheaves on $\s X_0$ is
strictly compatible with their weight filtrations.
\end{theorem}
\begin{proof}
(i) By descent theory (\cite{LO3}, 7.1) we reduce to the case
where $\s X_0=X_0$ is a scheme. One can further replace $X_0$ by an open
affine covering, and assume $X_0$ is separated.
\begin{sublemma}
Let $K_0$ and $L_0$ be $\iota$-pure complexes in
$D^b_c(X_0,\overline{\bb Q}_{\ell})$ of $\iota$-weights $w$
and $w',$ respectively, and assume $w-w'\notin\bb Z.$ Then
$Ext^n(K_0,L_0)=0$ for all $n.$
\end{sublemma}
\begin{proof}
By (\ref{5.4.1}), $\leftexp{p}{\tau}_{\le i}K_0$ and
$\leftexp{p}{\s H}^iK_0$ are $\iota$-pure, of $\iota$-weights
$w$ and $w+i$ respectively. Since $RHom(K_0,L_0)$ is a
triangulated functor in both $K_0$ and $L_0,$ we may
assume they are both perverse sheaves, and hence simple
perverse sheaves (\ref{5.3.2}i).
To show $Ext^n(K_0,L_0)=0,$ it suffices to show, by (\cite
{BBD}, 5.1.2.5), that the $\iota$-weights of $Ext^n(K,L)$
are not integers, for all $n.$ Therefore, we may make a finite
extension of the base field $\bb F_q.$ By (\cite{BBD}, 4.3.1ii)
we have $K_0=j_{!*}F_0[d]$ (resp. $L_0=i_{!*}G_0[d']$) for
some irreducible smooth subscheme (since we can take a finite
base extension) $j:U_0\hookrightarrow X_0$ of dimension $d$ (resp.
$i:V_0\hookrightarrow X_0$ of dimension $d'$), and for some
irreducible lisse sheaf $F_0$ on $U_0$ (resp. $G_0$ on
$V_0$). The sheaf $F_0=j^*K_0[-d]$ is $\iota$-mixed (or use Lafforgue's
result), hence punctually $\iota$-pure by (\cite{Del2}, 3.4.1ii), and
therefore $\iota$-pure by (\cite{Del2}, 6.2.5b). By (\cite
{BBD}, 5.3.2), the punctual $\iota$-weight of $F_0$ is
$w-d.$ By (\cite{Del2}, 1.3.6), there exists a number $b\in\overline
{\bb Q}_{\ell}^*$ such that the determinant of the lisse
sheaf $F_0^{(b)}$ deduced from $F_0$ by twist (\ref{torsion}) has
finite order, therefore, by Lafforgue's result, $F_0^{(b)}$ is
punctually pure of weight 0. The same is true for $G_0^{(b')}$
for some $b'\in\overline{\bb Q}_{\ell}^*.$ We see that
$w_q(\iota b)=d-w$ and $w_q(\iota b')=d'-w'.$
Then we have
$$
R\s Hom(K_0,L_0)=R\s Hom(j_{!*}F_0^{(b)}[d],i_{!*}G_0^{(b')}
[d'])^{(b/b')}
$$
and, by the projection formula,
$$
a_*R\s Hom(K_0,L_0)=\big(a_*R\s
Hom(j_{!*}F_0^{(b)}[d],i_{!*}G_0^{(b')}[d'])\big)^{(b/b')},
$$
where $a:X_0\to\text{Spec }\bb F_q$ is the structural morphism. By
(\cite{Del2}, 6.1.11), the complex $a_*R\s
Hom(j_{!*}F_0^{(b)}[d],i_{!*}G_0^{(b')}[d'])$ is mixed, hence
$\s H^n(a_*R\s Hom(K_0,L_0)),$ whose underlying vector space is
$Ext^n(K,L),$ does not have integer punctual $\iota$-weights.
\end{proof}
For every $\beta\in\bb{R/Z},$ we apply (\cite{BBD}, 5.3.6) to
$\text{Perv}(X_0),$ taking $S^+$ (resp. $S^-$) to be the set of
isomorphism classes of simple perverse sheaves
(hence $\iota$-pure) of $\iota$-weights not in $\beta$ (resp.
in $\beta$). Then we get a unique sub-object $\s F_0(\beta)$
with $\iota$-weights in $\beta$ (\ref{D3.2}), such that $\s
F_0/\s F_0(\beta)$ has $\iota$-weights not in $\beta,$ and
$\s F_0(\beta)$ is functorial in $\s F_0.$ This extension
splits since $Ext^1=0.$ By induction on length we get the
decomposition, which is unique and functorial.
(ii) As in (\cite{LO3}, 9.2), we may assume $\s X_0=X_0$
is a scheme. The proof in (\cite{BBD}, 5.3.5) still applies.
Namely, by (\ref{5.1.15}ii), if $\s F_0$ and $\s G_0$ are
$\iota$-pure simple perverse sheaves on $X_0,$ of
$\iota$-weights $f$ and $g$ respectively, with $f>g,$ then
$Ext^1(\s G_0,\s F_0)=0.$ Then apply (\cite{BBD}, 5.3.6) for each integer $i,$ by
taking $S^+$ (resp. $S^-$) to be the set of isomorphism
classes of $\iota$-pure simple perverse sheaves on $X_0$ of
$\iota$-weights $>i$ (resp. $\le i$).
\end{proof}
\begin{blank}\label{affine-stab}
Let $k$ be a field and let $\mathcal X$ be a $k$-algebraic
stack. We say that $\mathcal X$ has \textit{affine
stabilizers} if for every $x\in\mathcal X(\overline{k}),$
the group scheme $\text{Aut}_x$ is affine. Since being
affine is fpqc local on the base, we see that for any finite
field extension $k'/k$ and any $x\in\mathcal X(k'),$ the group
scheme $\text{Aut}_x$ over $k'$ is affine.
\end{blank}
\begin{proposition}\label{5.1.14}\emph{(stack version of
(\cite{BBD}, 5.1.14))}
(i) The Verdier dualizing functor $D_{\s X_0}$ interchanges $D_{\le w}$
and $D_{\ge-w}.$
(ii) For every morphism $f$ of $\bb F_q$-algebraic
stacks, $f^*$ respects $D_{\le w}$ and $f^!$ respects
$D_{\ge w}.$
(iii) For every morphism $f:\s X_0\to\s Y_0,$ where $\s
X_0$ is an $\bb F_q$-algebraic stack with affine stabilizers,
$f_!$ respects $D_{\le w}^-$ and $f_*$ respects
$D_{\ge w}^+.$
(iv) $\otimes$ takes $D_{\le w}^-\times D_{\le w'}^-$
into $D_{\le w+w'}^-.$
(v) $R\s Hom$ takes $D_{\le w}^-\times D_{\ge
w'}^+$ into $D_{\ge w'-w}^+.$
\end{proposition}
\begin{proof}
(i), (ii) and (iv) are clear, and (v) follows from (iv).
For (iii), if $\mathscr X_0$ has affine stabilizers,
so are all fibers $f^{-1}(y),$ for $y\in\s
Y_0(\bb F_{q^v}),$ and the claim for $f_!$ follows
from the spectral sequence
$$
H^i_c(f^{-1}(\overline{y}),\s H^jK)\Longrightarrow
H^{i+j}_c(f^{-1}(\overline{y}),K)
$$
and (\cite{Sun}, 1.4), and the claim for $f_*$ follows
from the claim for $f_!.$
\end{proof}
\begin{corollary}\label{5.1.15}\emph{(stack version of
(\cite{BBD}, 5.1.15))}
Let $\s X_0$ be an $\bb F_q$-algebraic stack with affine
stabilizers, with structural map $a:\s X_0\to\emph{Spec
}\bb F_q.$ Let $K_0$ (resp. $L_0$) be in $D_{\le
w}^-(\s X_0,\overline{\bb Q}_{\ell})$ (resp.
$D_{>w}^+(\s X_0,\overline{\bb Q}_{\ell})$) for
some real number $w.$ Then
(i) $a_*R\s Hom(K_0,L_0)$ is in $D_{>0}^+(\emph{Spec }\bb
F_q,\overline{\bb Q}_{\ell}).$
(ii) $Ext^i(K_0,L_0)=0$ for $i>0.$
If $L_0\in D_{\ge w}^+,$ then $a_*R\s Hom(K_0,L_0)$
is in $D_{\ge0}^+,$ and we have
(iii) $Ext^i(K,L)^F=0$ for $i>0.$ Here $F$ is the Frobenius
(\ref{frob}). In particular, for
$i>0,$ the canonical morphism $Ext^i(K_0,L_0)\to Ext^i(K,L)$ is zero.
\end{corollary}
The proof is the same as (\cite{BBD}, 5.1.15), using the
above stability result for algebraic stacks with affine stabilizers.
The following is the perverse sheaf version of
(\ref{3.4.1}iii).
\begin{theorem}\label{5.3.8}\emph{(stack version of
(\cite{BBD}, 5.3.8))}
Let $\s X_0$ be an $\bb F_q$-algebraic stack
with affine stabilizers. Then every $\iota$-pure
perverse sheaf $\s F_0$ on $\s X_0$ is
geometrically semi-simple (i.e. $\s F$ is
semi-simple), hence $\s F$ is a direct sum of
perverse sheaves of the form $j_{!*}L[d_{\s U}],$
for inclusions $j:\s U\hookrightarrow\s X$ of
$d_{\s U}$-dimensional irreducible smooth
substacks, and for irreducible lisse sheaves $L$ on $\s U.$
\end{theorem}
\begin{proof}
Let $\s F'$ be the sum in $\s F$ of simple
perverse subsheaves; it is a direct sum, and is the
largest semi-simple perverse subsheaf of $\s F.$
Then $\s F'$ is stable under Frobenius, hence
descends to a perverse subsheaf $\s F_0'\subset
\s F_0$ ((\cite{BBD}, 5.1.2) holds for stacks also).
Let $\s F_0''=\s F_0/\s F_0'.$ By (\ref{5.1.15}iii), the extension
$$
\xymatrix@C=.5cm{
0 \ar[r] & \s F' \ar[r] & \s F \ar[r] & \s F'' \ar[r] & 0}
$$
splits, because $\s F_0'$ and $\s F_0''$ have
the same weight (\cite{LO3}, 9.3). Then $\s F''$
must be zero, since otherwise it contains a simple
perverse subsheaf, and this contradicts the maximality of
$\s F'.$ Therefore $\s F=\s F'$ is semi-simple. The other
claim follows from (\cite{LO3}, 8.2ii): we may replace
$\s U$ by $\s U_{\text{red}}$ and hence assume that it is smooth.
\end{proof}
\begin{theorem}\label{5.4.5}\emph{(stack version of
(\cite{BBD}, 5.4.5))}
Let $\s X_0$ be an $\bb F_q$-algebraic stack
with affine stabilizers, and let $K_0\in D_c^b(\s X_0,\overline{\bb
Q}_{\ell})$ be an $\iota$-pure complex. Then $K$ on $\s X$ is
isomorphic non-canonically to the direct sum of the shifted perverse
cohomology sheaves $(\leftexp{p}{\s H}^iK)[-i].$
\end{theorem}
\begin{proof}
By (\ref{5.4.1}), both $\leftexp{p}{\tau}_{<i}K_0$ and
$(\leftexp{p}{\s H}^iK_0)[-i]$ are $\iota$-pure of
the same weight as that of $K_0.$ Therefore, by
(\ref{5.1.15}iii), the exact triangle
$$
\xymatrix@C=.5cm{
\leftexp{p}{\tau}_{<i}K_0 \ar[r] & \leftexp{p}{\tau}_{\le
i}K_0 \ar[r] & (\leftexp{p}{\s H}^iK_0)[-i] \ar[r]
&}
$$
geometrically splits, i.e. we have
$$
\leftexp{p}{\tau}_{\le i}K\simeq\leftexp{p}{\tau}_{<i}K
\oplus(\leftexp{p}{\s H}^iK)[-i],
$$
and the result follows by induction.
\end{proof}
\begin{blank}
\textit{Proof of theorem (\ref{main-thm}).} For the second claim,
we may assume that $K_0$ is an irreducible perverse sheaf, hence is $\iota$-pure (\ref{5.3.4}), therefore it follows from the first one.
For the first claim, by (\cite{Ols1}, 5.17) and (\ref{5.1.14} iii), we see that $f_*K_0$ is $\iota$-pure, hence by (\ref{5.4.5}) we have
$$
f_*K\simeq\bigoplus_{i\in\bb Z}(\leftexp{p}{R}^if_*K)[-i].
$$
By (\ref{5.4.1}), each $\leftexp{p}{R}^if_*K_0$ is also $\iota$-pure,
therefore it is geometrically semi-simple by (\ref{5.3.8}).
\end{blank}
\begin{remark}
In order to generalize the decomposition theorem to algebraic
stacks over $\bb C,$ one needs some foundational results, such as
the generic base change theorem for $f_*$ on stacks, the comparison
between the adic derived category and the topological derived
category of a $\bb C$-algebraic stack, and so on. We only give the
statement of the decomposition theorem for $\bb C$-stacks here,
and we will publish the details of the proof somewhere else.
\end{remark}
\begin{theorem}\label{6.2.5}\emph{(stack version of
(\cite{BBD}, 6.2.5))}
Let $f:\mathcal X\to\mathcal Y$ be a proper morphism of
finite diagonal between $\bb C$-algebraic stacks with
affine stabilizers. If $K\in\s D_c^b(\mathcal X^{\emph{an}},
\bb C)$ is semi-simple of geometric origin, then $f^{\emph{an}}_*K$
is also bounded, and is
semi-simple of geometric origin on $\mathcal Y^{\text{an}}.$
\end{theorem}
|
{
"timestamp": "2012-03-13T01:02:09",
"yymm": "1009",
"arxiv_id": "1009.4398",
"language": "en",
"url": "https://arxiv.org/abs/1009.4398"
}
|
\section{Introduction}
\label{}
Suppose you want to analyse a system $\mathcal{A}$ whose number of states is finite.
This system reacts to inputs from the environment in
a probabilistic fashion: if $\mathcal{A}$ is in state $s$ and receives
$\alpha$ from the environment, the probability that $\mathcal{A}$ transitions
to state $s'$ is $\prob{s}{\alpha}{s'}$. Moreover, assume that the environment
cannot observe the state of $\mathcal{A}$ in order to choose the particular input
$\alpha$. The analysis you want to perform on this system is to calculate a
tight lower bound of the probability that the system achieves a certain goal,
no matter what the inputs are. For instance, inputs can model notifications
of the (un)availability of resources, and you might want to check that your
system sends a message with probability at least $0.8$, no matter what the
available resources are.
The problem in the paragraph above can be modelled using Probabilistic Finite
Automata (PFA)~\cite{DBLP:journals/iandc/Rabin63,DBLP:journals/ai/MadaniHC03}.
The assumption that inputs do not depend on the internal state of the
state of the input is central to assert that a PFA model adequately
reflects the behaviour of the system. In case the environment can observe
the state of $\mathcal{A}$ to choose the particular input $\alpha$, the problem
can be modelled using Markov Decision Processes (MDP)~\cite{BOOK:Puterman}.
The usual semantics for PFA rely on the concept of \emph{acceptance},
by considering the set of finite words ending in an acceptance state
with probability greater than a given cut-point $\eta$. In contrast,
we focus on the concept of \emph{reachability}, and we are interested
on the probability with which each infinite word reaches some of the states
in a given set $\mathcal{S}$. In the realm of MDPs, both the supremum and
the infimum probability can be calculated in polynomial
time~\cite{DBLP:conf/fsttcs/BiancoA95}. In contrast, in the PFA setting
the supremum problem is undecidable~\cite{DBLP:journals/ai/MadaniHC03} for
both finite and infinite words~\footnote{Here, we consider only infinite words,
as the infimum probability over finite words is either $1$, if the initial
state of the system is in $\mathcal{S}$, or $0$, if it is not.}.
In fact, the supremum probability that $\mathcal{A}$ reaches a state in
$\mathcal{S}$ cannot be even approximated algorithmically. This undecidability
result was the key to prove undecidability results for MDPs under partial
information~\cite{DBLP:conf/formats/GiroD07} as well as undecidability
for Probabilistic B\"uchi Automata~\cite{DBLP:conf/fossacs/BaierBG08}.
We present an algorithm to approximate the infimum probability that a PFA
$\mathcal{A}$ reaches a set of states $\mathcal{S}$. Moreover, the computed value
$v$ is a lower bound of the infimum and, by performing a
sufficient number of iterations, we can ensure that it is as close to the
infimum as desired. Using the value $v$, we can answer our motivating
problem by stating that ``the probability that the goal is achieved is at
least $v$, no matter what the inputs are''. The fact that the value
$v$ is close to the infimum implies that the bound we provide is tight.
\section{Algorithm}
For our algorithm, we use the following definitions: a Probabilistic Finite
Automata (PFA) is a quintuple
$\mathcal{A} = (S,\Sigma,\mathcal{P},\state^{i},\mathcal{S})$, where
$S$ is a finite set of states, $\Sigma$ is a set of symbols,
$\mathcal{P}$ is a set of probability distributions on $S$, comprising
one probability distribution $\prob{s}{\alpha}{\cdot}$ for each
pair $(s,\alpha)$ in $S \times \Sigma$. The state $\state^{i}$
is called the \emph{initial state} of $\mathcal{A}$, and $\mathcal{S}$ is a set of
\emph{hitting states}. We assume $\state^{i} \not\in \mathcal{S}$.
A finite path in $\mathcal{A}$ is a sequence
\[ \pi = \state^{i}.\alpha_{1}.s_{1}.\cdots
.\alpha_{n}.s_{n} \]
where $\alpha_{i} \in \Sigma$ and $s_{i} \in S$
for all $i$. Note that paths always start with the initial state
$\state^{i}$. We write $\len{\pi}$ for $n$ and $\last{\pi}$
for $s_{n}$. In an analogous way to finite paths, infinite paths
are infinite sequences alternating symbols and states. The set of all infinite
paths having the finite path $\pi$ as prefix is denoted by
$\extension{\pi}$.
Given a word $\psi$ over $\Sigma$, let $\psi[k]$ denote the $k$-th
symbol in $\psi$. For every infinite word $\psi$ over $\Sigma$,
for every finite path $\pi$, the probability
$\prWord\,\!^{\psi}(\extension{\pi})$ is defined as $1$
if $\pi = \state^{i}$; if $\psi[\len{\pi}+1] = \alpha$, we have
$\prWord\,\!^{\psi}(\extension{\pi.\alpha.s})
= \prWord\,\!^{\psi}(\extension{\pi})
\cdot \prob{\last{\pi}}{\alpha}{s}$; if
$\psi[\len{\pi}+1] \not= \alpha$, then
$\prWord\,\!^{\psi}(\extension{\pi.\alpha.s}) = 0$.
In the same way as for Markov chains and MDPs (namely, by resorting to the
Carath\'eodory extension theorem), the previous definition for sets of the form
$\extension{\pi}$ can be extended in such a way that, for all infinite
words $\psi$, the value $\prWord\,\!^{\psi}(\mathcal{Z})$ is defined for all
measurable sets $\mathcal{Z}$ of infinite paths.
Let $\mathcal{H}$ be the set of all infinite paths $\rho$ such that
some of the states in $\rho$ is in $\mathcal{S}$. The amount we want to
approximate is $I = \inf_{\psi} \prWord\,\!^{\psi}(\mathcal{H})$. Note that
$\mathcal{H}$ can be written as
\begin{equation}
\label{eq:def-phit}
\mathcal{H} = \biguplus_{\pi \in C} \extension{\pi} \; ,
\end{equation}
where $C$ is the set of all finite paths $\pi$ such that $\last{\pi}$
is the only state of $\pi$ in $\mathcal{S}$.
In order to approximate $I$, our algorithm iterates producing two
values in each iteration $r$. One of the values is a lower bound $l_{r}$
and the other one is an upper bound $u_{r}$. These bounds comply with:
\begin{eqnarray}
l_{r} & \leq & l_{r+1} \label{ineq:lb-increase} \\
l_{r} & \leq & I \label{ineq:lb-islower} \\
\lim_{r \to \infty} l_{r} & = & I \label{eq:lb-tends-inf} \\
u_{r} & \geq & u_{r+1} \label{ineq:ub-decrease} \\
u_{r} & \geq & I \label{ineq:ub-isupper} \\
\lim_{r \to \infty} u_{r} & = & I \label{eq:ub-tends-inf} \; .
\end{eqnarray}
To approximate $I$ with error at most $\epsilon$, the algorithm stops
when $u_{r} - l_{r} < \epsilon$ (this is guaranteed to occur as
both $u_{r}$ and $l_{r}$ converge to the
same limit), and then returns $l_{r}$. Note that $u_{r}$ is also a
value with error less than $\epsilon$ but, in order to give a safe lower bound
on the probability that a hitting state is reached, we use the pessimistic
value $l_{r} \leq I$.
In the next subsections, we show how to calculate upper and lower bounds
complying with the desired properties.
\subsection{Lower bounds}
Let $\mathcal{H}_{r} = \biguplus_{C_{r}} \extension{\pi}$ where
$C_{r}$ is the set of paths such that $\last{\pi}$ is the only state
of $\pi$ in $\mathcal{S}$ and $\len{\pi} \leq r$. By making the
same observation as for~\recallEquation{eq:def-phit}, we deduce that
$\mathcal{H}_{r}$ is the set of all infinite paths reaching $\mathcal{S}$
after at most $r$ symbols. We often profit from the inclusion
\[ \mathcal{H}_{r} \subseteq \mathcal{H}_{r+1} \; . \]
We take $l_{r} = \inf_{\psi} \prWord\,\!^{\psi}(\mathcal{H}_{r})$.
Next, we show that this number can be calculated by brute force.
Since only the first $r$ symbols are relevant, we need to consider each of
the finite words $w$ having exactly $r$ symbols. The truncation
operator $\wprefix{\psi}{r}$, that returns the prefix of $\psi$
having length $r$, will thus be quite useful in this subsection.
In addition, we use the notation $\prWord\,\!^{w}(\mathcal{H}_{r})$ to mean
$\prWord\,\!^{\psi}(\mathcal{H}_{r})$, where $\psi$ is any infinite word such
that $\wprefix{\psi}{r} = w$.
For each $w$ with $\len{w} = r$, we construct a finite Markov
chain $\mathcal{M}$. The procedure resembles the standard unfolding of a
probabilistic automaton (or an MDP) for a particular
adversary~\cite{thesis:segala}, and so we merely outline it. The states of
$\mathcal{M}$ are pairs $(s,k)$ with $s$ in $S$
and $0 \leq k \leq r$. To describe $\mathcal{M}$ briefly, let's say that
the path
$\state^{i}.\alpha_{1}.s_{1}.\cdots.\alpha_{n}.s_{n}$
in $\mathcal{A}$ maps to the path
$(\state^{i},0).(s_{1},1).(s_{2},2).
\cdots.(s_{n},n)$ in $\mathcal{M}$.
For all $0 \leq k < r$, the probability of transitioning
from $(s,k)$ to $(s',k+1)$ is $\prob{s}{w[k+1]}{s'}$
(note that these probabilities depend on $w$). For simplicity, the states
$(s,r)$ are stuttering. The initial state of $\mathcal{M}$ is
$(\state^{i},0)$. The previous definitions for $\mathcal{M}$ imply that the
probabilities of the paths in $\mathcal{A}$ having length at most $r$ coincide
with the probabilities of the corresponding paths in $\mathcal{M}$:
\[
\begin{array}{rcl}
& & \prWord\,\!^{w}_{\mathcal{A}}(\extension{\state^{i}.\alpha_{1}.s_{1}.
\cdots.\alpha_{n}.s_{n}}) \\
& = & \prob{\state^{i}}{\alpha_{1}}{s_{1}} \cdot
\prod_{k=1}^{n-1} \prob{s_{k}}{\alpha_{k+1}}{s_{k+1}}\\
& = & \prWord\,\!_{\mathcal{M}}( \, (\state^{i},0).(s_{1},1).(s_{2},2).
\cdots.(s_{n},n) \, ) \; .
\end{array}
\]
As a consequence, the probability that $w$ reaches $\mathcal{S}$ in
at most $r$ steps equals the probability that $\mathcal{M}$ reaches a state in
$\mathcal{S} \times \{ 0, \cdots, r \}$. The latter probability can be
calculated using standard techniques, as it poses a simple reachability
problem for finite Markov chains.
We have just showed that $l_{r}$ is computable. We still need to prove
that it complies with the properties we need so that our main algorithm works.
In order to prove \recallInequation{ineq:lb-increase}, we use the fact
that $l_{r} = \min_{w \in W_{r}} \prWord\,\!^{w}(\mathcal{H}_{r})$,
where $W_{r}$ is the set of all words of length $r$. Let $w^{*}$ be
$\arg \min_{w \in W_{r+1}} \prWord\,\!^{w}(\mathcal{H}_{r+1})$
and $w^{*-1}$ be $\wprefix{w^{*}}{r}$. The required inequality
$l_{r} \leq \prWord\,\!^{w^{*}}(\mathcal{H}_{r+1})$ follows since
$l_{r} = \min_{w \in W_{r}} \prWord\,\!^{w}(\mathcal{H}_{r})
\leq \prWord\,\!^{w^{*-1}}(\mathcal{H}_{r})
= \prWord\,\!^{w^{*}}(\mathcal{H}_{r})
\leq \prWord\,\!^{w^{*}}(\mathcal{H}_{r+1})$,
where the last inequality holds since
$\mathcal{H}_{r} \subseteq \mathcal{H}_{r+1}$.
Next, we prove \recallInequation{ineq:lb-islower}. Let
$\mu = ( \, \psi(m) \, )_{m=1}^{\infty}$ be a sequence of
infinite words such that
$\lim_{m \to \infty} \prWord\,\!^{\psi(m)}(\mathcal{H}_{r}) = I$
and the sequence $( \, \prWord\,\!^{\psi(m)}(\mathcal{H}_{r}) \, )_{m=1}^{\infty}$
is non-increasing (such a sequence exists by definition of infimum). Let
$w^{*}$ be a word of length $r$ that appears infinitely often in the
sequence $( \, \wprefix{\psi(m)}{r} \, )_{m=1}^{\infty}$ (this word
exists as the sequence is infinite, and there are finitely many words of
length $r$).
We prove \recallInequation{ineq:lb-islower} by proving
$\prWord\,\!^{w^{*}}(\mathcal{H}_{r}) \leq I$. Suppose, towards a
contradiction, that $\prWord\,\!^{w^{*}}(\mathcal{H}_{r}) > I$. Then,
by definition of $\mu$ there exists $\psi(p)$ in $\mu$ such that
$\prWord\,\!^{w^{*}}(\mathcal{H}_{r}) > \prWord\,\!^{\psi(p)}(\mathcal{H}_{r})
\geq I$.
Since $w^{*}$ appears infinitely often in
$( \, \wprefix{\psi(m)}{r} \, )_{m=1}^{\infty}$, there exists
$q > p$ such that $\wprefix{\psi(q)}{r} = w^{*}$. Since
the values $\psi(m)$ are non-increasing, we reach the following
contradiction:
$\prWord\,\!^{w^{*}}(\mathcal{H}_{r})
> \prWord\,\!^{\psi(p)}(\mathcal{H}_{r})
\geq \prWord\,\!^{\psi(q)}(\mathcal{H}_{r})
= \prWord\,\!^{w^{*}}(\mathcal{H}_{r})$.
It remains to prove \recallEquation{eq:lb-tends-inf}. In
other to prove this equality, let
$\Psi$ be the sequence
\[ ( \, \Psi_{r} = \arg \min_{w \in W_{r}}
\prWord\,\!^{w}(\mathcal{H}_{r})
\, )_{r=1}^{\infty} \]
(the set $W_{r}$ has been defined above). Note that
\begin{equation}
\label{eq:alt-def-lb}
l_{r} = \prWord\,\!^{\Psi_{r}}(\mathcal{H}_{r}) \; .
\end{equation}
Given $\Psi$, we construct an infinite \emph{limit} word\footnote{We use the word
\emph{limit} as it resembles the \emph{limit schedulers}
in~\cite{DBLP:journals/entcs/GiroD09}.} $\vec{\iword}$ having the property that,
for every $M$, the prefix $\wprefix{\vec{\iword}}{M}$ appears infinitely often in
the sequence $( \, \wprefix{\Psi_{r}}{M} \, )_{r=M}^{\infty}$.
We take the first symbol $\vec{\iword}[1]$ to be any symbol that appears
infinitely often in $( \, \wprefix{\Psi_{k}}{1} \, )_{k=1}^{\infty}$.
In order to obtain the second symbol $\vec{\iword}[2]$, we consider
the subsequence $\Psi^{1}$ of $\Psi$ containing all words in $\Psi$ whose
first symbol is $\vec{\iword}[1]$. Then, $\vec{\iword}[2]$ is any symbol
that appears infinitely often as the second symbol in
$( \, \wprefix{\Psi^{1}_{k}}{2} \, )_{k=2}^{\infty}$. In general, we can
describe the process to obtain $\Psi^{M}$ and $\vec{\iword}[M]$
in an inductive fashion, by stating that $\vec{\iword}[M]$ is any symbol that
appears infinitely often in $( \, \Psi^{M-1}_{k}[M] \, )_{k=M}^{\infty}$ and
$\Psi^{M}$ is an (infinite) subsequence of $\Psi^{M-1}$ complying with
$\Psi^{M}_{k}[M] = \vec{\iword}[M]$. The existence of the subsequence $\Psi^{M}$
ensures that $\wprefix{\vec{\iword}}{M}$ appears infinitely often in
$( \, \wprefix{\Psi_{r}}{M} \, )_{r=M}^{\infty} \:$, as desired.
As an auxiliary result, we prove $\prWord\,\!^{\vec{\iword}}(\mathcal{H}) = I$.
Suppose, towards a contradiction, that $\prWord\,\!^{\vec{\iword}}(\mathcal{H}) > I$.
Then, there exists $\psi'$ such that
$\prWord\,\!^{\vec{\iword}}(\mathcal{H}) > \prWord\,\!^{\psi'}(\mathcal{H}) \geq I$.
As\footnote{This equality is standard for reachability properties, and can
be deduced from
$\prWord\,\!^{\psi}(\mathcal{H})
= \prWord\,\!^{\psi}(\biguplus_{k=1}^{\infty} \mathcal{H}_{k} \setminus \mathcal{H}_{k-1})
= \sum_{k=1}^{\infty} \prWord\,\!^{\psi}(\mathcal{H}_{k} \setminus \mathcal{H}_{k-1})$.}
\begin{equation}
\label{eq:reach-by-limit}
\forall \psi : \prWord\,\!^{\psi}(\mathcal{H})
= \lim_{k \to \infty} \prWord\,\!^{\psi}(\mathcal{H}_{k}) \; ,
\end{equation}
there exists $K$ such that
\begin{equation}
\label{ineq:limword-gt-other-word}
\prWord\,\!^{\vec{\iword}}(\mathcal{H}_{K}) > \prWord\,\!^{\psi'}(\mathcal{H})
\geq \prWord\,\!^{\psi'}(\mathcal{H}_{M}) =
\prWord\,\!^{\wprefix{\psi'}{M}}(\mathcal{H}_{M})
\end{equation}
for all $M$.
By definition of $\vec{\iword}$, there exists $M > K$ such that
$\wprefix{\Psi_{M}}{K} = \wprefix{\vec{\iword}}{K}$. Then,
$\prWord\,\!^{\vec{\iword}}(\mathcal{H}_{K})
= \prWord\,\!^{\wprefix{\vec{\iword}}{K}}(\mathcal{H}_{K})
= \prWord\,\!^{\wprefix{\Psi_{M}}{K}}(\mathcal{H}_{K})
\leq \prWord\,\!^{\Psi_{M}}(\mathcal{H}_{M})
\leq \prWord\,\!^{\wprefix{\psi'}{M}}(\mathcal{H}_{M})$ (where the last inequality holds by
definition of $\Psi_{M}$) thus
contradicting~\recallInequation{ineq:limword-gt-other-word}.
Now we are ready to prove $\lim_{r \to \infty} l_{r} = I$.
Since $l_{r} \leq I$ for all $r$, we have
$\lim_{r \to \infty} l_{r} \leq I$. Suppose, towards a
contradiction, that $\lim_{r \to \infty} l_{r} < I$. Then, by
$\prWord\,\!^{\vec{\iword}}(\mathcal{H}) = I$ and~\recallEquation{eq:reach-by-limit},
there exists $K$ such that
\begin{equation}
\label{ineq:limit-lt-inf}
\lim_{r \to \infty} l_{r} < \prWord\,\!^{\vec{\iword}}(\mathcal{H}_{K})
= \prWord\,\!^{\wprefix{\vec{\iword}}{K}}(\mathcal{H}_{K}) \; .
\end{equation}
By definition of $\vec{\iword}$, there exists $M > K$ such that
$\wprefix{\Psi_{M}}{K} = \wprefix{\vec{\iword}}{K}$.
Then, by~\recallEquation{eq:alt-def-lb}, we have
$\lim_{r \to \infty} l_{r}
\geq \prWord\,\!^{\Psi_{M}}(\mathcal{H}_{M})
\geq \prWord\,\!^{\wprefix{\Psi_{M}}{K}}(\mathcal{H}_{K})
= \prWord\,\!^{\wprefix{\vec{\iword}}{K}}(\mathcal{H}_{K})$, which
contradicts~\recallInequation{ineq:limit-lt-inf}.
\subsection{Upper bounds}
For our upper bounds, we use \emph{lasso-shaped} words (LSW). A LSW\ is an
infinite word of the form
$\psi = \alpha_{1} \cdots \alpha_{K}
( \beta_{1} \cdots \beta_{M} )^{\omega}$, in which the last $M$
in which the sequence of symbols $\beta_{1} \cdots \beta_{M}$ is looped
infinitely many times. The name \emph{lasso-shaped} is borrowed from the
counterexamples for LTL properties of B\"uchi automata, this name being used,
for instance, in~\cite{DBLP:conf/tacas/SchuppanB05}. Such counterexamples also
consist of a finite stem and a sequence that is looped infinitely many times.
In this paper, we restrict to LSWs\ with $M \leq 2^{\card{S}}$ (recall
that $S$ is the set of states of the PFA), and we say that $K$ is the
$\emph{order}$ of $\psi$, denoted by $\order{\psi}$. Note that, because of
our restriction on the length of the loop, the amount of LSWs\ with order at
most $K$ is finite.
We denote by $\alllsw{r}$ the set of all LSW\ with order at most $r$.
The set of all infinite words is denoted by $\allwords$.
For upper bounds, we take
$u_{r} = \inf_{\psi \in \alllsw{r}} \prWord\,\!^{\psi}(\mathcal{H})$.
Inequalities~\ref{ineq:ub-decrease} and~\ref{ineq:ub-isupper} follow from
$\alllsw{r} \subseteq \alllsw{r+1} \subseteq \allwords$.
The computability of $u_{r}$ follows in a similar way to that of
$l_{r}$: the amount of LSWs\ having order at most $r$ is finite, and we
can explore the probabilities for each of these words. Similarly as for the
lower bounds, the probability for a word $w_{1}(w_{2})^{\omega}$ is calculated
by constructing a finite Markov chain. We just outline the construction. The
set of the states of the Markov chain is
\newcommand{\mathsf{S}}{\mathsf{S}}
\newcommand{\mathsf{L}}{\mathsf{L}}
\[ S
\times \{ \; 1, \, \cdots, \, \max \{ K, M \} \; \}
\times \{ \mathsf{S}, \mathsf{L} \} \]
(where $K = \len{w_{1}}$ and $M = \len{w_{2}}$).
The initial state is $(\state^{i},1,\mathsf{S})$. In the state $(s,n,\mathsf{S})$
($(s,n,\mathsf{L})$, resp.), the probability distribution for the next state is
determined by the $n$-th symbol in the stem (in the loop, resp.) In symbols, the
probability of transitioning from $(s,n,\mathsf{S})$ to
$(s',n+1,\mathsf{S})$ is $\prob{s}{w_{1}[n]}{s'}$ whenever
$n < K$. From $(s,K,\mathsf{S})$ to
$(s',K,\mathsf{L})$, the probability is
$\prob{s}{w_{1}[K]}{s'}$. The probabilities for the loop are
defined in a similar way: the only difference is that in a state
$(s,M,\mathsf{L})$ in the end of the loop, we have that
$\prob{s}{w_{2}[M]}{s'}$ is the probability of transitioning to
$(s,1,\mathsf{L})$ (that is, we return to the beginning of the loop). Note that
all the paths with positive probability are of the form
\[
\begin{array}{cl}
& (s_{1},1,\mathsf{S}) \cdots (s_{K},K,\mathsf{S}) \\
\cdots & (s_{K+1},1,\mathsf{L}) \cdots (s_{K+M},M,\mathsf{L}) \\
\cdots & (s_{K+iM+1},1,\mathsf{L}),\cdots,(s_{K+iM+M},M,\mathsf{L})
\quad \cdots \; .
\end{array}
\]
Is is easy to see that the probability
$\prWord\,\!^{w_{1}(w_{2})^{\omega}}(\mathcal{H})$ is the probability of reaching a state
$(s,n,l)$ such that $s \in \mathcal{S}$, and so the minimum probability for all
words of order at most $K$ can be obtained by constructing a Markov chain for
each of such words.
It remains to prove \recallEquation{eq:ub-tends-inf}.
If $I = 1$, then $u_{r} = 1$ for all $r$, and so the equation
is trivial. From now on, we concentrate on the case $I < 1$. In order to
prove that the limit is the infimum, it suffices to show that, for all
$\epsilon$, there exists $R$ such that
\begin{equation}
\label{ineq:desired-lim}
\inf_{\psi \in \alllsw{R}} \prWord\,\!^{\psi}(\mathcal{H})
< I+\epsilon \; .
\end{equation}
We can indeed restrict to $\epsilon$ such that
\begin{equation}
\label{ineq:restrict-epsilon}
\epsilon < 1-I \; .
\end{equation}
(Having proved the result for such values, the result also holds for the values
$\epsilon'$ such that $\epsilon' \geq 1-I$, by taking $\epsilon$ such
that $\epsilon = (1-I)/2 < 1-I \leq \epsilon'$ and hence
$\inf_{\psi \in \alllsw{R}} \prWord\,\!^{\psi}(\mathcal{H})
< I+\epsilon < I+\epsilon'$.)
We prove~\recallInequation{ineq:desired-lim} by showing that there exists
$\psi^{*} = w_{1}(w_{2})^{\omega}$ with
$\len{w_{2}} \leq 2^{S}$ such that $\prWord\,\!^{\psi^{*}}(\mathcal{H})
< I + \epsilon$,
By taking $R$ to be the order of $\psi^{*}$, we obtain
\recallInequation{ineq:desired-lim}, that is, the value $u_{R}$
is $\epsilon$-close to $I$.
Let $\iword^{\epsilon/2}$ be an infinite word such that
$\prWord\,\!^{\iword^{\epsilon/2}}(\mathcal{H}) < I + \epsilon/2$ (such a word exists by
definition of infimum). Using this word, we construct the word $\psi^{*}$
with the desired properties. For this construction, we focus on
the probability of \emph{not} reaching $\mathcal{S}$ (that is, the probability of
all infinite paths such that none of the states is in $\mathcal{S}$).
By definition of $\iword^{\epsilon/2}$, we know that $\iword^{\epsilon/2}$ does not reach
$\mathcal{S}$ with probability greater than $1 - \epsilon/2 - I$; in
symbols:
\begin{equation}
\label{ineq:iwordeps-gt-epsilon}
\prWord\,\!^{\iword^{\epsilon/2}}(\lnot\mathcal{H}) > 1 - \epsilon/2 - I \; ,
\end{equation}
where $\lnot\mathcal{H}$ is the complement of $\mathcal{H}$, that is, the set of all infinite
paths $\rho$ such that $\rho[k] \not\in \mathcal{S}$ for all $k$.
Using $\iword^{\epsilon/2}$, we define $\psi^{*}$ in such a way that
\begin{equation}
\label{ineq:iword-star-avoids-reach}
\prWord\,\!^{\psi^{*}}(\lnot\mathcal{H}) > 1 - \epsilon - I
\end{equation}
and so $\prWord\,\!^{\psi^{*}}(\mathcal{H}) < I + \epsilon$. The proof proceeds by
finding numbers $K$ and $M$ such that the first $K+M$ symbols of $\psi^{*}$
are the same as in $\iword^{\epsilon/2}$. We name these symbols
$\alpha_{1}, \alpha_{2}, \cdots, \alpha_{K}, \beta_{1}, \beta_{2},
\cdots, \beta_{M}$.
After these symbols, the word $\psi^{*}$ repeats
$\beta_{1}, \cdots, \beta_{M}$ indefinitely. This word is illustrated
in~\recallFigure{fig:counter-example-pfa}. The intuition behind the proof is
that there exists a set $Q_{1}$ of states such that, after exactly $K$ steps,
there is sufficiently high probability to be in $Q_{1}$, without hitting
$\mathcal{S}$ (in the figure, states in $\mathcal{S}$ are represented with crosses).
Moreover, if $Q_{i+1}$ ($Q_{1}$, respectively) is the set of all states that
can be reached after symbol $\beta_{i}$ ($\beta_{M}$, resp.) occurs in some
state in $Q_{i}$ ($Q_{M}$, resp.), then $Q_{i} \cap \mathcal{S} = \emptyset$ for
all $1 \leq i \leq M$.
\begin{figure}
\centering
\input{counter-example-pfa.pstex_t}
\caption{\label{fig:counter-example-pfa}Avoiding
$\mathcal{S}$ with high probability}
\end{figure}
We find $K$, $M$ and show that $\psi^{*}$ complies with
\recallInequation{ineq:iword-star-avoids-reach}.
In order to obtain the required $K$, $M$, we profit from the fact that a PFA
according to our definition can be seen as a particular case of an MDP. For the
sake of completeness, we show how our definition for PFA matches the definition
of MDP in~\cite{thesis:deAlfaro}. If the MDP underlying a PFA $\mathcal{A}$ is obvious
to the reader, then the rest of this paragraph can be safely skipped.
In~\cite{thesis:deAlfaro}, (\recallExternalDef{3.1}), an MDP $\Pi = (S,A,p)$
is defined by a set of states $S$, a set of actions $A(s)$ enabled at each
state $s$, and probabilities $p_{st}(a)$ of stepping from $s$ to $t$ using $a$,
for each $a \in A(s)$. When mapping a PFA $\mathcal{A}$ to an MDP $\Pi$, the set of
states $S$ of $\Pi$ is the same set of states as in $\mathcal{A}$; for each $s$ the
set $A(s)$ of actions enabled is the set $\Sigma$; the probabilities
$p_{st}(a)$ in~\cite{thesis:deAlfaro} are simply $\prob{s}{a}{t}$.
Using the MDP underlying $\mathcal{A}$, we can resort to the \emph{end-component
theorem} (\cite[\recallExternalThm{3.2}]{thesis:deAlfaro}). In terms of PFA,
the definition of an end component is as follows.
\begin{definition}
\label{def:end-component}
An end component is a set $E \subseteq S \times \Sigma$ such that for
every states $s_{1} \not= s_{n}$ in (a pair in) $E$ there
exists a path
$s_{1}.\alpha_{1}.s_{2}.\cdots.\alpha_{n-1}.s_{n}$
such that $(s_{k},\alpha_{k}) \in E$ and
$\prob{s_{k}}{\alpha_{k}}{s_{k+1}} > 0$ for all
$1 \leq k \leq n-1$. We write $\stec{E}$ for the set of states of $E$.
When no confusion arises, we simply write $s \in E$ instead of
$s \in \stec{E}$.
\end{definition}
Let $\mathcal{E}$ be the set of infinite paths
$s_{1}.\alpha_{1}.s_{2}.\cdots$ such that there exists $T$ such that
the set $\{ (s_{t},\alpha_{t}) \mid t > T \}$ is an end component. The
end-component theorem states that $\mathcal{E}$ has probability $1$ for all
words. The paths in $\mathcal{E}$ are said to \emph{end} in an end component.
Then, the set of paths that do not end in an end component (that is, the paths
for which no such $T$ exists) has probability $0$ for all words and, roughly
speaking, can thus be disregarded in probability calculations.
From now on, we are interested on the set $\mathcal{E}$ comprising all paths
ending in an end component. Now we show a partition for $\mathcal{E}$. For all
finite paths $\pi$, end components $E$, let $Z(\pi,E)$ be the set of all
infinite paths $\pi.\alpha_{1}.s_{2}.\alpha_{2}.\cdots$ such that
$(\alpha_{k},s_{k+1}) \in E$ for all $k$. Next, we prove that the set
$\mathcal{E}$ is equal to
$\mathcal{E}' = \biguplus_{(\pi,E) \in \mathcal{Z}} Z(\pi,E)$
where $\mathcal{Z}$ is the set of all pairs $(\pi,E)$ such that
$\pi$ is either the trivial path $\state^{i}$, and $\state^{i} \in E$; or
$\pi = \state^{i}.\cdots
.s_{n-1}.\alpha_{n}.s_{n}$
and $(s_{n-1},\alpha_{n}) \not\in E$ and $s_{n}$
in $E$. In words, the last state/symbol pair is not in $E$, but the last state
is. Clearly, the inclusion $\mathcal{E}' \subseteq \mathcal{E}$ holds as the
paths in $Z(\pi,E)$ end in $E$ for all $\pi$, $E$. In order to prove the
inclusion $\mathcal{E} \subseteq \mathcal{E}'$, we prove that any path
$\psi \in \mathcal{E}$ is also in $\mathcal{E}'$. Since
$\psi \in \mathcal{E}$, there exists $T$ as
in~\recallDefinition{def:end-component}. Let's consider the minimum such $T$.
The existence of $T$ ensures that $\rho$ has a prefix $\pi$ after which
all the pairs state/symbol are in $E$. Moreover, since we are considering the
minimum $T$, either $\pi$ is the trivial path $\state^{i}$, and $\state^{i}$
is in $E$; or the last state/symbol pair before $\last{\pi}$ is not in $E$.
In summary, the fact that $R$ is minimum ensures that there exists
$(\pi,E) \in \mathcal{Z}$ such that $\rho \in Z(\pi,E)$. It remains to
prove disjointness, that is, $Z(\pi,E) \cap Z(\pi',E') \not= \emptyset$
imply $(\pi,E) = (\pi',E')$. Suppose that there exists
$\rho \in Z(\pi,E) \cap Z(\pi',E')$. The set of all state/symbol pairs
that appear infinitely often in $\rho$ are all the pairs in $E$
(as $\rho \in Z(\pi,E)$), and the same goes for $E'$, thus yielding
$E = E'$. It remains to prove $\pi = \pi'$. We have that $\pi$ and
$\pi'$ are both a prefix of $\rho$. Moreover, since we consider only
finite paths in which the last state/symbol pair is not in $E$, we have that
$\pi$ is the smallest prefix such that after $\pi$ all the state/symbol
pairs are in $E$, and the same holds for $\pi'$. Then, both $\pi$ and
$\pi'$ have the same length, and so $\pi = \pi'$.
As a consequence of the partition we found, and the end-component theorem, for
all words $\psi$ we have
$ \prWord\,\!^{\psi}(\Omega)
= \prWord\,\!^{\psi}(\mathcal{E})
= \sum_{\pi} \sum_{\{ E \mid (\pi,E) \in \mathcal{Z} \}}
\: \prWord\,\!^{\psi}(\, Z(\pi,E) \,)$.
If a paths ends in an end component $E$ and does not hit $\mathcal{S}$, then no
prefix hits $\mathcal{S}$, and $E$ has no states in $\mathcal{S}$. Hence, for all
words $\psi$ we have
\[ \prWord\,\!^{\psi}(\lnot\mathcal{H}) =
\sum_{\{ \pi \mid \pi \cap \mathcal{S} = \emptyset \}}
\sum_{\{ E \mid (\pi,E) \in \mathcal{Z}
\land E \cap \mathcal{S} = \emptyset \}}
\prWord\,\!^{\psi}(\, Z(\pi,E) \,) \; . \]
The outer sum ranges over all finite paths such that no state is in $\mathcal{S}$
(which we denote as $\pi \cap \mathcal{S} = \emptyset$), and the inner sum
ranges over all end components $E$ such that the last state/action pair in
$\pi$ is not in $E$, the last state is in $E$, and no state of $E$ is in
$\mathcal{S}$ (denoted by $E \cap \mathcal{S} = \emptyset$). In particular, for the
word $\iword^{\epsilon/2}$ in~\recallInequation{ineq:iwordeps-gt-epsilon}, we have
$\prWord\,\!^{\iword^{\epsilon/2}}(\lnot\mathcal{H}) = \sum_{\pi \cap \mathcal{S} = \emptyset}
\sum_{(\pi,E) \in \mathcal{Z} \land
E \cap \mathcal{S} = \emptyset}
\prWord\,\!^{\iword^{\epsilon/2}}(\, Z(\pi,E) \,) > 1 - \epsilon/2 - I$.
Then, there exists a finite set
$\mathcal{B} \subseteq \{ \pi \mid \pi \not\in \mathcal{S} \}$ such that
$\sum_{\pi \cap \mathcal{S} = \emptyset}
\sum_{(\pi,E) \in \mathcal{Z}
\land E \cap \mathcal{S} = \emptyset}
\prWord\,\!^{\iword^{\epsilon/2}}(\, Z(\pi,E) \,) >
1 - \frac{3}{4}\epsilon - I$.
Let $B = \max_{\pi \in \mathcal{B}} \len{\pi}$. For the sake of brevity,
let $\mathcal{V}$ be the set of all pairs $(\pi,E)$ such that
$\pi \cap \mathcal{S} = \emptyset$, and $\len{\pi} \leq B$, and
$(\pi,E) \in \mathcal{Z}$, and $E \cap \mathcal{S} = \emptyset$, and
$\prWord\,\!^{\iword^{\epsilon/2}}(\, Z(\pi,E) \,) > 0$. Then,
\begin{equation}
\label{ineq:prob-bounded-non-reaching}
\sum_{(\pi,E) \in \mathcal{V}}
\prWord\,\!^{\iword^{\epsilon/2}}(\, Z(\pi,E) \,) > 1 - \frac{3}{4}\epsilon - I \; .
\end{equation}
Note that we can restrict to the pairs $(\pi,E)$ such that
$\prWord\,\!^{\iword^{\epsilon/2}}(\, Z(\pi,E) \,) > 0$, as the pairs with probability $0$
do not affect the sum.
In addition, by \recallInequation{ineq:restrict-epsilon}, we have
$1 - \epsilon - I > 0$, and so in the sum in
\recallInequation{ineq:prob-bounded-non-reaching} there is at least one
positive summand $\prWord\,\!^{\iword^{\epsilon/2}}(\, Z(\pi,E) \,)$.
\begin{figure}
\centering
\input{exiting-end-component.pstex_t}
\caption{\label{fig:exiting-end-component}$(s_{1},\alpha)$ is
in $E$, but $(s_{2},\alpha)$ is not}
\end{figure}
The desired $K$, $M$ are now obtained from $\iword^{\epsilon/2}$ and $B$. Note that,
although we restricted to the summands complying with
$\prWord\,\!^{\iword^{\epsilon/2}}(\, Z(\pi,E) \,) > 0$, it is still possible that $\mathcal{A}$
exits $E$ after $\pi$ with positive probability (as the same symbol might
be inside $E$ for a reachable state $s$, but outside $E$ for a state
$s'$ that is reachable after the same number of steps as $s$, see
\recallFigure{fig:exiting-end-component}). We
show that, by considering arbitrarily large paths, the probability that $E$ is
exited becomes arbitrarily small.
Let $\belief{w}{k}{s}$ be the probability that, after $k$ steps,
the state is reached is $s$. We generalize this notation to sets of states.
Formally:
$\belief{w}{k}{T} = \sum_{\len{\pi} = k, \last{\pi} \in T}
\prWord\,\!^{w}(\extension{\pi})$.
We call the distribution $\belief{w}{k}{\cdot}$ a \emph{belief state},
following the nomenclature for POMDPs~\cite{DBLP:journals/ai/KaelblingLC98}.
Since the set of states is finite, there exist two indices $x < y$ such that
$\supp(\beliefns{\iword^{\epsilon/2}}{x}) = \supp(\beliefns{\iword^{\epsilon/2}}{y})$
(where $\supp$ denotes the support of the distribution). Moreover, given any
two numbers $X$, $Y$, such that $Y > X+2^{\card{S}}$, we have
$X \leq x \leq y \leq Y$ and
$\supp(\beliefns{\iword^{\epsilon/2}}{x}) = \supp(\beliefns{\iword^{\epsilon/2}}{y})$ for some
$x$, $y$. Since the amount of sequences of the form
$T_{0} \gamma_{1} \cdots \gamma_{V} T_{V}$ with $V \leq 2^{\card{S}}$ is
finite (where each $T_{v}$ is a set of states), at least one of such finite
sequences appears infinitely many times in the infinite sequence
$\supp(\beliefns{\iword^{\epsilon/2}}{0}) \iword^{\epsilon/2}(1)
\supp(\beliefns{\iword^{\epsilon/2}}{1}) \iword^{\epsilon/2}(2) \cdots$. Suppose this
finite sequence is $\sigma = T_{0} \gamma_{1} \cdots \gamma_{V} T_{V}$.
We show that we can take
$\beta_{1}, \cdots, \beta_{M}
= \gamma_{1}, \cdots, \gamma_{V}$.
In addition, we take $K$ to be a number (defined below) greater than $B$, in
which an occurrence of $\sigma$ starts.
Given a component $E$ in a pair in $\mathcal{V}$ (defined before
\recallInequation{ineq:prob-bounded-non-reaching}), let
\begin{multline*}
Q^{E} =
\{ s' \in T_{0} \cap \stec{E} \mid \forall v \leq V : \\
\prWord\,\!^{\gamma_{1} \cdots \gamma_{v}}
(s'.\gamma_{1}.s_{1}.\cdots.\gamma_{v}.s_{v})
> 0
\implies s_{k} \in E \} \; .
\end{multline*}
In other words, $Q^{E}$ comprises the states in $T_{0} \cap \stec{E}$ from
which, when executing $\gamma_{1} \cdots \gamma_{V}$, we can only reach states
in $E$. Let $Q^{\lnot E}$ be $\stec{E} \setminus Q^{E}$.
Consider the infinite sequence
$e(1), e(2) \cdots$ of indices where $\sigma$ starts. We show that
$\lim_{v \to \infty} \belief{\iword^{\epsilon/2}}{e(v)}{s} = 0$ for all
$s \in Q^{\lnot E} > 0$. (As the number of states is finite, this implies
$\lim_{v \to \infty} \belief{\iword^{\epsilon/2}}{e(v)}{Q^{\lnot E}} = 0$.)
Suppose, towards a contradiction, that for some $s \in Q^{\lnot E}$, $l > 0$,
we have $\belief{\iword^{\epsilon/2}}{e(v)}{s} \geq l$ for all $v$. By definition of
$Q^{\lnot E}$, there exists $s' \not\in E$, $d > 0$, and $v$ such that
$\prWord\,\!^{\gamma_{1} \cdots \gamma_{v}}
(s.\gamma_{1} \cdots \gamma_{v}.s')
= d$. Then, the
probability of staying in $E$ after the $n$-th repetition of $\sigma$ is
less than or equal to $(1-(l \cdot d))^{n}$, for all $n$. This implies that the
probability of staying in $E$ indefinitely is $0$, thus contradicting the fact
that $\prWord\,\!^{\iword^{\epsilon/2}}(\, Z(\pi,E) \,) > 0$.
As a result, for all pairs $(\pi,E)$ in $\mathcal{V}$, there exists
$e(\pi,E) \in \{ e(1), e(2), \cdots \}$ such that
\begin{equation}
\label{ineq:escape-small}
\belief{\iword^{\epsilon/2}}{e(\pi,E)}{Q^{\lnot E}}
< \epsilon/(4 \cdot \card{\mathcal{V}}) \; .
\end{equation}
Define
$K = \max(B , \{ e(\pi,E) \mid (\pi,E) \in \mathcal{V} \})$
and
$Y(\pi,E) = Z(\pi,E) \setminus
\{ \rho \mid \rho[K] \in Q^{\lnot E} \}$.
We have
$Z(\pi,E) \subseteq Y(\pi,E) \cup
\{ \extension{\pi} \mid \pi[K] \in Q^{\lnot E}
\land \len{\pi} = K \}$.
Then,
\begin{equation}
\label{ineq:bound-Z}
\prWord\,\!^{\psi}(\, Z(\pi,E) \,) \leq \prWord\,\!^{\psi}(\, Y(\pi,E) \,)
+ \belief{\psi}{K}{Q^{\lnot E}}
\end{equation}
for all infinite words $\psi$.
We have
\begin{equation}
\label{ineq:star-better-than-eps-for-Y}
\prWord\,\!^{\psi^{*}}(\, Y(\pi,E) \,) \geq \prWord\,\!^{\iword^{\epsilon/2}}( \, Y(\pi,E) \,)
\end{equation}
as, under $\psi^{*}$, all paths of length $K$ ending in a state in
$Q^{E}$ do not reach states outside $E$ (because of our definition of $Q^{E}$
and the symbols $\gamma_{v}$). In fact, if $\len{\pi} \geq K$, the
scenario in \recallFigure{fig:exiting-end-component} is possible under
$\iword^{\epsilon/2}$, but not possible under $\psi^{*}$. Roughly speaking, after $K$
steps the word $\psi^{*}$ does not escape $E$, thus yielding higher
(or equal) probability for $Y(\pi, E)$ than any word $\psi$ such that
$\wprefix{\psi}{K} = \wprefix{\psi^{*}}{K}$ and, in particular, than
$\iword^{\epsilon/2}$. Then,
\[
\begin{array}{cl}
& \prWord\,\!^{\psi^{*}}(\lnot\mathcal{H}) \\
\geq & \explanation{$Y(\pi,E)\!\subseteq\!Z(\pi,E)$,
the sets $Z(\pi,E)$ partition $\lnot\mathcal{H}$} \\
& \sum_{(\pi,E) \in \mathcal{V}} \prWord\,\!^{\psi^{*}}(\, Y(\pi,E) \,) \\
\geq & \explanation{Ineq.~\ref{ineq:star-better-than-eps-for-Y}} \;
\sum_{(\pi,E) \in \mathcal{V}} \prWord\,\!^{\iword^{\epsilon/2}}(\, Y(\pi,E) \,) \\
\geq & \explanation{Ineq.~\ref{ineq:bound-Z}} \;
\sum_{(\pi,E) \in \mathcal{V}}
\prWord\,\!^{\iword^{\epsilon/2}}(\, Z(\pi,E) \,)
- \belief{\iword^{\epsilon/2}}{K}{Q^{\lnot E}} \\
> & \explanation{Inequations~\ref{ineq:prob-bounded-non-reaching},
\ref{ineq:escape-small}} \;
1 - \frac{3}{4}\epsilon - I - \card{\mathcal{V}} \cdot
\epsilon/(4\!\cdot\!\card{\mathcal{V}}) \\
\geq & 1 - \epsilon - I
\end{array}
\]
In conclusion, the word
$\psi^{*}
= \alpha_{1} \cdots \alpha_{K} (\beta_{1} \cdots \beta_{M})^{\omega}$
(where $\alpha_{1} \cdots \alpha_{K} \beta_{1} \cdots \beta_{M}$ are first
$K+M$ symbols in $\iword^{\epsilon/2}$) complies with
\recallInequation{ineq:iword-star-avoids-reach}. Since
$\order{\psi^{*}} = K$, we obtain
$\inf_{\psi \in \alllsw{K}} \prWord\,\!^{\psi}(\mathcal{H}) < I + \epsilon$. By
\recallInequation{ineq:ub-decrease}, this inequality implies
$\inf_{\psi \in \alllsw{k}} \prWord\,\!^{\psi}(\mathcal{H}) < I + \epsilon$
for all $k \geq K$, thus ensuring \recallEquation{eq:ub-tends-inf}.
\section{Discussion}
Our algorithm is nonprimitive recursive, and we have still nothing to say
about the complexity of the problem.
However, the fact that there exists an algorithm to approximate the value is
quite surprising considering similar problems for PFA, as shown in
\recallTable{tab:decidability}. The table indicates, for the problems of
reachability and acceptance, whether there exists an
algorithm to approximate and/or to compute extremal values.
\begin{table}
\centering
\footnotesize
\begin{tabular}{|c|c|c|}
\hline
Approximate/Compute & Infimum & Supremum \cite{DBLP:journals/ai/MadaniHC03} \\
\hline
Reachability & $\surd / ?$ & $\times / \times$
\\
\hline
Acceptance \cite{DBLP:journals/mst/BlondelC03} & $? / \times$
& $\times / \times$ \\
\hline
\end{tabular}
\caption{\label{tab:decidability}Existence of algorithms for PFA}
\end{table}
Note that the only $\surd$ in the table corresponds to the result in this
paper. The table also indicates two pending questions: whether there exists
an algorithm to effectively compute the infimum for reachability, and whether
the infimum for acceptance can be approximated.
The undecidability for the supremum probability has been used to prove that
quantitative model checking under partial
information~\cite{DBLP:conf/formats/GiroD07,DBLP:conf/sbmf/Giro09} is
undecidable for properties involving the supremum. The setting of these papers
is more general, as several entities might have different information about the
state of the system (in contrast, the problem we address in this paper concerns
only an environment that has no information about the state of the system).
However, we expect that the proof we presented sheds some light on whether this
more general problem is computable or not, in case we consider the infimum
instead of the supremum.
\bibliographystyle{elsarticle-num}
|
{
"timestamp": "2010-09-21T02:03:23",
"yymm": "1009",
"arxiv_id": "1009.3822",
"language": "en",
"url": "https://arxiv.org/abs/1009.3822"
}
|
\section{Introduction}
\label{sec:intro}
Motivated by the algorithmic problem of searching large graphs, we study the expansion properties of real-world networks and the extent to which these properties can be exploited to better understand and facilitate decentralized search. Complex networks of linked entities arise across diverse domains, from sociology (e.g. social networks) to biology (e.g. neural networks, protein interactions) to technological and information systems (e.g. P2P networks, the Web, power grids). Scientists from disparate fields, especially within the past decade, have attempted to characterize both the structure and function of these networked systems. The standard approach here has been to measure topological features of the graph representing the network and correlate them with functional or dynamic aspects of the network (e.g. evolution of the network, the behavior of processes that occur over the network). For instance, in their seminal paper on small-world networks, Watts and Strogatz \cite{Watts1998Collective} showed that many real-world networks simultaneously exhibit short average path lengths and relatively high degrees of clustering. They further showed that these features can facilitate spreading processes across a network (e.g. the spread of a virus) \cite{Watts1998Collective}. In this work, we study a feature that has received comparatively less attention in the study of real-world\footnote{There is a large body of work studying expansion in theoretical computer science and graph theory. However, much of this work focuses on 1) synthetic graphs that do not normally arise in the real-world such as $d$-regular graphs and 2) the \emph{minimum} (not maximum) expansion in these graphs \cite{Hoory2006Expander}.} networks: \emph{expansion}.
\subsection{Expansion}
Given a network $G=(V,E)$ where $V$ is the set of nodes and $E$ is the set of links among the nodes, the \emph{expansion} of a set of nodes $S \subset V$ is a function of the number of nodes in $V-S$ to which $S$ is connected (see Section \ref{sec:preliminaries} for more precise definitions). That is, if $N(S)$ is the set of nodes to which $S$ is connected, then the \emph{expansion} of $S$ is $\frac{|N(S)|}{|S|}$. Informally, an expander graph is a graph in which any subset of nodes has good expansion (i.e. has many neighbors) \cite{Hoory2006Expander}. For instance, a network is said to be a $\gamma$-expander if every $S \subset V$ has an expansion of \emph{at least} $\gamma$, where $|S| < \frac{|V|}{2}$ \cite{Hoory2006Expander}. Thus, the classical definition of an expander graph focuses on samples with the \emph{minimum} expansion (as $\gamma$ is a minimum if every sample $S$ has an expansion of at least $\gamma$). Expander graphs have been shown to have many applications from constructing error correcting codes to routing calls in telephone switching networks \cite{Hoory2006Expander}. Although these classical expander graphs are well-studied, there has been surprisingly less attention paid to studying the sample in a network with the \emph{maximum} expansion, and it is this in which we are most interested. Our overall aim in this work is to investigate the extent to which a vertex set with high expansion can be leveraged to both understand and facilitate efficient decentralized search in networks. We now describe the problem of decentralized search.
\subsection{Search}
A central algorithmic problem in the study of complex networks is how to efficiently search them in a decentralized manner. This scenario arises in many practical applications from querying peer-to-peer file sharing networks to focused Web crawling \cite{Kleinberg2006Complex}. Starting from some initial source node, we must locate, access, or route a message to some other target node in the network. Without full knowledge of the global network topology, we are unable to simply compute the shortest path or access the target node directly. Thus, we must hop from node to node until the target node is found. Decentralized search, then, is related to information diffusion or dissemination, as an efficient search will involve efficiently disseminating a query message to large portions of the network. In this work, we study the effect of network expansion on decentralized search. If, using only local information, nodes are visited in such a way that their overall distance is close to many other nodes (i.e. the set of visited nodes has high expansion), then the efficiency of search might be improved. Moreover, the magnitude of expansion in a network may shed light on the extent to which a network can be efficiently searched by \emph{any} search algorithm. These are precisely the questions we investigate here.
\subsection{Contributions and Summary of Findings}
\label{sec:intro.contributions}
We rigorously investigate decentralized search across a span of networks much wider and more diverse than previously studied in work on searching graphs. Our main contributions include the following:
\begin{itemize}
\item Borrowing from concepts in expander graphs, we introduce the concept of \emph{\esig s}, which concisely captures the overall expansion properties of a network. We find that, in many networks, relatively small samples of nodes can exhibit significantly high expansion. However, we also find that there are a few networks for which small samples with high expansion may not exist.
\item We propose an expansion-based, decentralized search strategy, which explicitly tries to locate these samples with high expansion in order to quickly discover the most nodes in a search. We evaluate a number of search algorithms such as degree-based searches, breadth-first searches, and random walks. We show that an expansion-based search strategy generally outperforms others.
\item We demonstrate that \emph{\esig s} correctly infer the extent to which a network can be efficiently searched (which we refer to as ``searchability''). Moreover, we show that it is the maximum expansion in a network, rather than the minimum expansion, that contributes most to searchability and information dissemination in a network. At the same time, we find that standard graph-theoretic measures, such as average path length, \emph{fail} to fully explain the extent to which a network is easily searched.
\end{itemize}
The last point is our most significant finding. Existing works have mostly studied the effect of \emph{minimum} expansion on the ease of dissemination in a network (e.g. \cite{Hoory2006Expander,Chierichetti2010Rumour}). Other works have focused on the effect of standard graph properties such as average path length on searchability and ease of dissemination (e.g. \cite{Hui2006Smallworld}). For the first time, we show that it is the \emph{maximum} expansion (rather than minimum expansion) that most affects efficient searchability. Our results, then, offer a more comprehensive picture of decentralized search and information diffusion in networks than has previously been appreciated.
\section{Background and Related Work}
\label{sec:relatedwork}
Interestingly, one of the first experiments on decentralized search in networks was the famous chain-letter study by the social psychologist Stanley Milgram \cite{Milgram1967Small}. In this experiment, participants were given the name, address, and occupation of an unknown target person and told to forward a chain letter to this person by passing the letter on to a single acquaintance meeting two main conditions: 1) the acquaintance must be someone with whom the individual knew on a first-name basis and 2) the acquaintance chosen should be the one perceived as closest to the target \cite{Milgram1967Small}. This study not only provided some evidence for short paths in social networks\footnote{As noted in \cite{Kleinfeld2002Could}, a number of issues exist in Milgram's results. For instance, many chain letters failed to ever reach the target. Nevertheless, the conclusion that short path lengths exist in social networks is generally accepted today and has been verified in many networked data \cite{Kleinberg2006Complex}.} (the median path length between sources and targets was $6$ among letters that reached their destination), but also showed that individuals were able to collectively discover these short paths \emph{without} full knowledge of the network \cite{Kleinberg2000SmallWorld}. Kleinberg \cite{Kleinberg2000SmallWorld} later modeled this problem algorithmically using a 2-dimensional grid with probabilistically-added long-range connections and shed light on the precise conditions that these short paths were discoverable using a decentralized search algorithm.
In both the works of Milgram \cite{Milgram1967Small} and Kleinberg \cite{Kleinberg2000SmallWorld}, although nodes had no knowledge of global network connectivity, there was, in fact, \emph{external} knowledge that aided searches. In the Milgram experiment, for instance, individuals were instructed to forward letters to the local acquaintance that was perceived as being closest to the target - as measured by geographic or occupational similarity in many cases. Thus, external knowledge of geographical distance and occupational similarity was employed as an aid in the search heuristic. In other words, there was knowledge (or, at least, an assumption) that individuals closer geographically or more similar occupationally are more likely to know each other. Liben-Nowell et al. \cite{LibenNowell2005Geographic}, in fact, showed some evidence for the geographical basis of online friendships in the LiveJournal social network. In Kleinberg's model also, it is assumed that message holders have access to external knowledge and know the local contacts of \emph{all} nodes in the network (i.e. nodes are aware of the Manhattan distances between all nodes in the underlying grid structure and use this information as a forwarding heuristic). Boguna et al. \cite{Boguna2008Navigability} have recently modeled this external information as a hidden metric space.
Unfortunately, in many real-world scenarios, such as unstructured peer-to-peer file sharing networks, these types of external information and similarity-based heuristics are unavailable as search aids, and the problem of decentralized search becomes even more challenging. Typical approaches here resort to variations on flooding the network (which can be unscalable), random walks (which may be less effective in finding information), or imposing structure on the network to improve searchability (which requires additional overhead) \cite{Tsoumakos2006Analysis}. For a review of decentralized search both in the contexts of complex networks and specifically P2P, one may refer to \cite{Kleinberg2006Complex,Mitra2009Technological,Tsoumakos2006Analysis}. Our focus in this work is to investigate efficient search on networks with \emph{arbitrary} structure in which similarity-based heuristics (e.g. geographic distance) are unavailable. For the first time, we investigate the relationship between \emph{expansion} and the extent to which a network is efficiently searchable. We further show that a search strategy based on expansion generally outperforms typical existing approaches such as random walks and flooding-based techniques. We begin a discussion of our work with some preliminaries.
\section{Preliminaries}
\label{sec:preliminaries}
\subsection{Notations and Definitions}
\label{sec:preliminaries.definitions}
We now briefly describe some notations and definitions used throughout this paper.
\begin{defn}
\label{defn:network}
$G=(V,\,E)$ is an undirected \emph{network} or \emph{graph} where $V$ is a set of vertices (or nodes) and $E \subseteq V \times V$ is a set of edges (or links between the nodes). We will use the terms \emph{network} and \emph{graph} interchangeably.
\end{defn}
\begin{defn}
\label{defn:sample}
A \emph{sample} $S$ is a subset of vertices, $S \subset V$.
\end{defn}
\begin{defn}
\label{defn:neighborhood}
$N(S)$ is the \emph{neighborhood} of $S$ if $N(S)=\{w \in V-S: \; \exists v \in S \; s.t. \; (v,\,w) \in E\}$. The \emph{neighborhood} may also be referred to as the \emph{frontier} of a sample $S$.
\end{defn}
\begin{defn}
\label{defn:expansion}
The \emph{expansion} of a sample\footnote{The \emph{expansion} of an entire graph is typically taken to mean $\min_{S \subset V}\frac{|N(S)|}{|S|}$ \cite{Hoory2006Expander}.} $S$ is: $$\frac{|N(S)|}{|S|}$$
\end{defn}
\begin{defn}
\label{defn:expanderset}
The \emph{maximum expander set} of size $k$ is a sample $S$ of size $k$ with the maximal expansion: $$\argmax_{S:\,|S|=k} \frac{|N(S)|}{|S|}$$
\end{defn}
\begin{defn}
\label{defn:expansionquality}
The \emph{expansion quality} of a sample $S$ is the normalized\footnote{Alternatively, one can normalize \emph{expansion} as $\frac{|N(S) \cup S|}{|V|}$.} \emph{expansion}:
$$ \frac{|N(S)|}{|S|} \div \frac{|V-S|}{|S|} = \frac{|N(S)|}{|V-S|}$$.
\end{defn}
Notice that, given a sample $S$, the maximum possible expansion on \emph{any} network of $|V|$ nodes is: $\frac{|V-S|}{|S|}$.
The \emph{expansion quality} $\frac{|N(S)|}{|V-S|}$, then, captures the extent to which a sample achieves this maximum possible expansion. A score of $1$ indicates that the sample ``touches'' or is one hop away from every other node in the network.
\subsection{Datasets}
\label{sec:preliminaries.datasets}
We study expansion and search in a total of ten different networks: two random graph models, a neural network, a power grid, a co-authorship network, an email network, a citation network, a P2P file-sharing network, and two online social networks. It should be noted that not all of these networks may require efficient decentralized search (e.g. a co-authorship network, the neural network of a worm). Nevertheless, these datasets represent a rich set of diverse networks from different domains. This allows us to more comprehensively study network expansion and thoroughly assess the performance of decentralized search strategies in the face of varying network topologies. Table \ref{tab:datasets} shows characteristics of each network. We now describe each dataset.
\noindent
\textbf{Erdos-Renyi Model.} One of the first random graph models proposed was that of Erdos and Renyi \cite{Erdos1959Random}. The Erdos-Renyi $G(n,p)$ model produces a random graph of $n$ nodes with each of the ${n \choose 2}$ possible edges existing with probability $p$. Erdos-Renyi graphs exhibit the short average path lengths found in many real-world networks, but lack the high clustering and skewed (or heavy-tailed) degree distributions found in reality.
\noindent
\textbf{Barabasi-Albert Model.} The Barabasi-Albert model follows a more, realistic generative process than previous models: the preferential attachment model \cite{Barabasi1999Emergence}. A graph of $n$ nodes is grown in a sequential fashion. Each subsequent node of $m$ edges is preferentially attached to previously added nodes with high degree (where the ``degree'' of a node is the number of neighbors). Graphs generated by this model exhibit skewed, power law degree distributions and short average path lengths, but lack the high clustering found in real networks. (Skewed degree distributions are ones in which there are many nodes with low connectivity and a few nodes with high connectivity that act as hubs. A power law distribution is one such example.)
\noindent
\textbf{C. elegans Neural Network} is the neural network of the C. elegans worm \cite{Watts1998Collective}.
\noindent
\textbf{Power Grid.} This technological network represents the power grid of the western United States \cite{Watts1998Collective}.
\noindent
\textbf{CondMat.} This is a co-authorship network of scientists publishing in Arxiv Cond-Mat (i.e. the Condensed Matter Physics category) from the e-print archive, arxiv.org \cite{Leskovec2005Graphs}.
\noindent
\textbf{Enron Emails} is the network comprised of email communications among Enron employees \cite{Klimt2004Enron}.
\noindent
\textbf{HEPPh} is a citation network between papers in Arxiv HEP-Ph (high energy physics phenomenology) from the e-print archive, arxiv.org \cite{Leskovec2005Graphs,Gehrke2003Overview}.
\noindent
\textbf{Gnutella.} This network is an August 31st, 2002 snapshot of the Gnutella peer-to-peer file-sharing network. Nodes represent hosts and edges represent connections among the hosts \cite{Ripeanu2002Mapping}.
\noindent
\textbf{Epinions} is a trust-based online social network of the consumer review site, Epinions.com \cite{Richardson2003Trust}.
\noindent
\textbf{Slashdot} is an online social network of the technology news site, Slashdot.com \cite{Leskovec2008Statistical}.
\begin{table} [th]
\centering
\footnotesize
\begin{tabular}{l|c|c|c|c|c} \hline \hline
Random Graphs & N & D & PL & CC & AD\\ \hline
Erdos-Renyi & 10,000 & 0.0005 & 4.2 & 0.0005 & 6.0 \\
Barabasi-Albert & 10,000 & 0.0005 & 3.0 & 0.006 & 6.0 \\ \hline \hline
Real-World & N & D & PL & CC & AD \\ \hline
C. elegans & 297 & 0.05 & 2.5 & 0.3 & 14.5 \\
Power Grid & 4941 & 0.0005 & 19 & 0.11 & 2.7 \\
CondMat & 21,363 & 0.0004 & 5.4 & 0.70 & 8.5 \\
Enron & 33,696 & 0.0003 & 4.0 & 0.71 & 10.7 \\
HEPPh & 34,401 & 0.0007 & 4.3 & 0.30 & 24.5 \\
Gnutella & 62,561 & 0.00008 & 5.9 & 0.01 & 4.7 \\
Epinions & 75,877 & 0.0001 & 4.3 & 0.26 & 10.7 \\
Slashdot & 82,168 & 0.0001 & 4.1 & 0.10 & 12.2 \\ \hline \hline
\end{tabular}
\caption{Network Properties. \textbf{Key:} \emph {N= \# of nodes, D= density, PL = characteristic path length, CC = clustering coefficient, AD = average degree.}}
\label{tab:datasets}
\end{table}
\section{Expansion Signatures}
\label{sec:expansion}
In this section we introduce the concept of \emph{\esig s}, which concisely captures the expansion properties of a network at different size scales. Intuitively, the \emph{\esig} plots the maximum (and minimum) \emph{expansion qualities} of samples at increasing sample sizes. As discussed, we are mostly interested in samples with the \emph{maximum} expansion, but we include the \emph{minimum} expansion for completeness. As we will see later, \emph{\esig s} reveal a number of interesting aspects of networks. But first, we address how precisely to compute the \emph{\esig} of a network.
\subsection{Problem Formalization}
To construct the \emph{\esig}, we must seek out the sample $S$ of size $k$ with the maximal (and minimal) expansion for progressively increasing values of $k$. In Definition \ref{defn:expanderset}, we defined the \emph{maximum expander set} as the sample of size $k$ with the maximum expansion. We now formally define the problem of finding this sample:
\begin{defn}
(\emph{Maximum Expansion Problem}) Given a graph $G=(V,E)$ and a sample size $k < |V|$, the {\sc Maximum Expansion} problem (\textbf{MEP)} is to find a sample $S \subset V$ of size $k$ with the maximum expansion, $\frac{|N(S)|}{|S|}$. That is, find: $\argmax_{S:\,|S|=k} \frac{|N(S)|}{|S|}$.
\end{defn}
The hardness of various problems related to expansion is well-studied (e.g. \cite{Hoory2006Expander}). For instance, determining that a graph is a $\gamma$-expander (where every $S \subset V$ has expansion of at least $\gamma$ and $|S| < \frac{|V|}{2}$) is known to be co-NP-complete \cite{Hoory2006Expander}. The {\sc Dominating Set} problem \cite{1997Approximation}, known to be NP-hard, is to find the smallest sample $S$ such that $N(S) = V-S$. {\sc Maximum Expansion} is clearly a generalization of {\sc Dominating Set} and is, thus, also NP-hard. There also exist reductions to and from the {\sc Maximum Coverage} problem (defined in \cite{1997Approximation} and also below). Proposition \ref{prop:maxexpnphard}, for instance, shows NP-hardness by reduction from {\sc Maximum Coverage}.
\begin{prop}
\label{prop:maxexpnphard}
The {\sc Maximum Expansion} problem is NP-hard.
\end{prop}
\begin{proof}
We show a reduction from the {\sc Maximum Coverage} problem, which is known to be NP-hard \cite{1997Approximation}. In {\sc Maximum Coverage}, given a set $\mathcal{U}$ of $n$ elements, a collection $\mathcal{F}=\{C_{i} \mid i \in I\}$ of $|I|$ subsets of $\mathcal{U}$ where $\bigcup_{i}C_{i}=\mathcal{U}$, and an integer $k < |I|$, the goal is to find $k$ subsets of $\mathcal{F}$ such that their union has the maximum cardinality. To construct a graph $G$ for the {\sc Maximum Expansion} instance, for each $i\in I$ we create a node $i$. For each element $u$ in $\mathcal{U}$, we create a node $u$. Thus, $V=I \cup \mathcal{U}$ is the vertex set of $G$. To create the edge set $E$ of $G$, an edge $\{i,j\}$ is created for each pair $i,j \in I$ (forming a clique among nodes in $I$). In addition, for each $i \in I$ and $u \in C_i$, an edge $\{i,u\}$ is created (forming an independent set among nodes in $\mathcal{U}$).
If $C=\{C_{i} \mid i \in S\}$ is a feasible solution to the {\sc Maximum Coverage} instance for some subset $S \subset I$ where $|S|=|C|=k$, then $S$ is a sample of nodes in $G$ with maximum expansion where $|S| = k$. By construction, each set $C_i \in C$ (where $i \in S$) is represented by a node $i \in I$ from $G$ that is both connected to every other node in $I$ and connected to the nodes in $\mathcal{U}$ that represent elements {\em contained} by $C_i$. Thus, $S$ is a $k$-size sample with the largest neighborhood in $G$. Conversely, let $S \subset V$ be a sample in $G$ with the maximum expansion. Note that, if $S \cap \mathcal{U} \neq \emptyset$, then a new sample $S^\prime$, with $\frac{|N(S^\prime)|}{|S^\prime|} \geq \frac{|N(S)|}{|S|}$, can be constructed by replacing each node $v \in S \cap \mathcal{U}$ with one of $v$'s neighbors $w \in I$. Thus, $C=\{C_{i} \mid i \in S^\prime\}$ is a feasible solution to {\sc Maximum Coverage}, since each node in $S^\prime$ represents a set in $\mathcal{F}$.
\end{proof}
Given the hardness of expansion-related problems, one typically resorts to spectral analysis, as the spectrum of a graph can be computed in polynomial time \cite{Hoory2006Expander}. A key difference in our work, however, is that we are not only interested in the magnitude of expansion, but the \emph{identity} of the sample producing it. Moreover, we are most interested in the \emph{maximum} expansion (as opposed to the minimum expansion, which is normally the focus in theoretical work on expander graphs). Our ultimate objective is to access these high expansion nodes during the course of a decentralized search to understand and facilitate search performance.\footnote{For directed networks that are very weakly connected, a sample with high maximum expansion (based on out-degree) may exist, but the nodes in the sample itself may not be reachable from substantial portions of the network. Samples such as this may shed little light on searchability. One possible approach to address these scenarios is to compute \esig s using the \emph{expected} maximum expansion of \emph{connected} samples. In the present work, however, for simplicity and brevity, we treat all links as bidirectional (or undirected).} Spectral analysis may be less useful here. Thus, we approximate expansion using a simple greedy algorithm ({\sc \greedy}). At each iteration, we greedily select the node that maximizes (or minimizes) the expansion of the currently constructed sample, as shown in Algorithm \ref{alg:greedy}. We now show that this simple greedy algorithm yields a $(1-1/e)$-approximation guarantee for the {\sc Maximum Expansion} problem.
\begin{prop}
\label{prop:maxexpapx}
{\sc GreedyAPX} approximates {\sc Maximum Expansion} within a ratio of at least $1 - 1/e \approxeq 0.632$.
\end{prop}
\begin{proof}
The structure of this proof follows that of the well-known proof for the {\sc Maximum Coverage} greedy approximation (see \cite{Hochbaum1998Analysis,Feige1998Threshold}, for instance). Let $S_{opt}$ be the optimal sample of size $k$ and $N$ be the set of nodes covered by $S_{opt}$ (where ``covered'' is taken to mean $N(S_{opt}) \cup S_{opt}$). Let $N_{i}$ be the set of \emph{new} nodes covered by the $i^{th}$ iteration of {\sc GreedyAPX}. Since $N$ can be covered by a sample of size $k$, by the pigeonhole principle:
$$ |N_{i}| \geq \frac{|N| - \sum_{j=1}^{i-1} |N_{j}|}{k}$$
Then, $\sum_{j=1}^{i} |N_{j}| \geq |N| - |N|(1 - \frac{1}{k})^i$ and
$$\sum_{i=1}^{k}|N_{i}| \geq |N| - |N|(1 - \frac{1}{k})^k \geq |N|(1-\frac{1}{e}).$$
\end{proof}
\begin{algorithm}[tb]
\caption{{\sc \greedy}}
\label{alg:greedy}
\begin{algorithmic}[1]
\STATE {\bfseries Input:} \\\quad Graph $G=(V, E)$ \\ \quad $k$, the sample size.
\STATE $S = \emptyset$ \qquad \qquad \qquad // initialize sample to empty set
\STATE $v = \argmax_w |N(\{w\})|$ \quad
\STATE $S = S \cup \{v\}$
\WHILE{$|S| \leq k$}
\STATE Select new node $v \in V-S$ that \\ maximizes (or minimizes): \\ \qquad \qquad $|N(\{v\})-(N(S) \cup S)|$
\STATE $S = S \cup \{v\}$
\ENDWHILE
\end{algorithmic}
\end{algorithm}
It should be noted that, during preliminary testing, we also experimented with using simulated annealing for finding the sample with maximum expansion, but {\sc \greedy} was shown to be superior. We now use {\sc \greedy} to construct \emph{\esig s} for both synthetic random graphs and real-world networks. We discuss each separately.
\subsection{Signatures for Random Graphs}
We examine the \emph{\esig s} of two well-known random graph models: Erdos-Renyi (ER) graphs \cite{Erdos1959Random} and the Barabasi-Albert preferential attachment model (BA) \cite{Barabasi1999Emergence}. The ER and BA models produce graphs with very different degree distributions. Whereas the BA model produces graphs with highly skewed, heavy-tailed degree distributions that follow the power law \cite{Barabasi1999Emergence}, the ER model produces graphs following a Poisson degree distribution \cite{Erdos1959Random}. It is clear that the BA model exhibits a higher and more rapidly increasing maximum expansion (which is a result of the highly connected hubs in its skewed degree distribution). However, the ER model exhibits a relatively higher \emph{minimum} expansion. In fact, random $d$-regular graphs, where every node has the same degree $d$, also have ``good'' minimum expansion with high probability (where \emph{every} $S \subset V$ will have high expansion) \cite{Hoory2006Expander}. In Section \ref{sec:evaluation}, we will examine whether it is the maximum or minimum expansion that most affects searchability.
\begin{figure}[ht]
\centering
\subfloat[Erdos-Renyi] {\label{fig:esig.er}\includegraphics[width=0.2\textwidth]{figures/esig-er}} \vspace{.05cm}
\subfloat[Barabasi-Albert]{\label{fig:esig.ba}\includegraphics[width=0.2\textwidth]{figures/esig-ba}}
\caption{\emph{\esigcap s} for ER and BA models.}
\label{fig:esig.randomgraphs}
\end{figure}
\subsection{Signatures for Real-World Networks}
We now turn our attention to the \emph{\esig s} of real-world networks. We examine eight different networks from diverse domains. \emph{\esigcap s} for each network are shown in Figure \ref{fig:esig.realnetworks}. We can immediately see that different types of networks exhibit very different expansion properties. For instance, the size scale required to obtain a maximum \emph{expansion quality} of 1 in the Enron network is only $7\%$. For the power grid, it is $49\%$.
We also see that the \emph{minimum} expansion varies across networks. We identify two different causes for low minimum expansion: 1) there is \emph{extreme} sparsity in the number of edges (imagine a simple line graph) or 2) the network is relatively sparse while exhibiting a high degree of clustering (imagine small sets of densely linked nodes linked together by \emph{sparse} connections). Both cases result in a relatively low minimum expansion (as the neighborhood size ($|N(S)|$) will be small for most samples). In the former case, the maximum expansion will tend to also be low (e.g. Power Grid). In the latter case, we find the maximum expansion to be relatively higher (e.g. CondMat, Enron). As will be discussed later, we posit the sparse links between relatively dense clusters in networks result in these higher values for maximum expansion. Recall also that the minimum expansion is related to the classic definition of an expander graph: every $S$ has expansion at least $\gamma$ in a $\gamma$-expander \cite{Hoory2006Expander}. The co-authorship, email, and social networks, with higher clustering and consequent lower minimum expansion, do \emph{not}, then, appear to be classic expander graphs. In Section \ref{sec:evaluation}, we will see whether or not this low minimum expansion affects searchability and information dissemination.
\begin{figure}[ht]
\centering
\subfloat[C. elegans] {\label{fig:esig.celegans}\includegraphics[width=0.2\textwidth]{figures/esig-celegans}} \vspace{.05cm}
\subfloat[Power Grid]{\label{fig:esig.power}\includegraphics[width=0.2\textwidth]{figures/esig-power}} \\ \vskip -0.01in
\subfloat[CondMat] {\label{fig:esig.condmat}\includegraphics[width=0.2\textwidth]{figures/esig-condmat}} \vspace{.05cm}
\subfloat[Enron] {\label{fig:esig.enron}\includegraphics[width=0.2\textwidth]{figures/esig-enron}} \\ \vskip -0.01in
\subfloat[HEPPh]{\label{fig:esig.hepph}\includegraphics[width=0.2\textwidth]{figures/esig-hepph}}\vspace{.05cm}
\subfloat[Gnutella]{\label{fig:esig.gnutella}\includegraphics[width=0.2\textwidth]{figures/esig-gnutella}} \\ \vskip -0.01in
\subfloat[Epinions]{\label{fig:esig.epinions}\includegraphics[width=0.2\textwidth]{figures/esig-epinions}}\vspace{.05cm}
\subfloat[Slashdot]{\label{fig:esig.slashdot}\includegraphics[width=0.2\textwidth]{figures/esig-slashdot}}
\caption{\emph{\esigcap s} for different networks.}
\label{fig:esig.realnetworks}
\end{figure}
\section{Searching Networks}
\label{sec:searching}
We now address the problem of decentralized search in networks. In a typical realistic scenario, starting from some initial node, we must locate some other node in the network \emph{without} full knowledge of global network topology. Thus, we are unable to simply compute the shortest path, and we must hop from node to node until the destination node is found. The running example application we employ is querying unstructured peer-to-peer file-sharing networks, where a search is comprised of sending a query message from node to node. The destination node in question, for instance, might host a particular file of interest to the querier. How can we locate this destination node? Flooding the network with the query (where a node receiving a query forwards it to all neighbors) is provably unscalable and impractical \cite{Adamic2001Search,Schmid2007Structuring}. In fact, when the music file-sharing service Napster became unavailable due to a court injunction in 2001, the Gnutella network (which employed a flooding-based search protocol at the time) crashed due to the large influx of former Napster users \cite{Schmid2007Structuring}.
In Section \ref{sec:expansion}, through \emph{\esig s}, we have seen that it is often a relatively small set of nodes that is connected to a large portion of the network. If one were able to easily locate this set of high expansion nodes, then the efficiency of search might vastly be improved (as this would quickly take us within one hop of many other nodes). But, how can these nodes be accessed during the course of a decentralized search? Our aforementioned greedy $(1-1/e)$-approximation algorithm to find high expansion nodes, shown in Algorithm \ref{alg:greedy}, assumes we have access to the network in its entirety (in which case decentralized search would not even be needed). As mentioned, we are interested in cases where there is \emph{limited} knowledge of global network connectivity. Therefore, we adapt the greedy $(1-1/e)$-approximation algorithm from Section \ref{sec:expansion} into a greedy search heuristic - one that does \emph{not} require full knowledge of network topology. We refer to this search heuristic as an \emph{expansion search}. We compare the \emph{expansion search} to several popular search strategies in complex networks. These include a \emph{degree search} \cite{Adamic2001Search}, a breadth-first search (BFS) \cite{Jin2007Novel,Jiang2008LightFlood,Yang2002Improving}, and a random walk \cite{Adamic2001Search,Li2006Searching}. We note that, in a BFS-based search strategy, there are \emph{multiple} copies of the search query traversing the network. In contrast, for the remaining three search strategies, there is a \emph{single} copy of the search query. For all search methods, unvisited nodes are always preferentially selected over previously visited nodes at each step in the search. We now describe each search strategy in detail.
~\\
\noindent
\textbf{Expansion Search (XS).} In an \emph{expansion search}, the next node in the search is selected so as to maximize the expansion. Let $S$ be the set of nodes visited thus far, let $N(S)$ be the neighborhood of the visited nodes, and let $c$ be the current, most recently visited node (where $c \in S$). Then, in an \emph{expansion search}, the next hop is selected from among the unvisited neighbors of $c$ (i.e. $N(\{c\})-S$) as the node that maximizes the expansion. That is, we visit node $v$ where $$v = \argmax_{v \in N(\{c\})-S} |N(\{v\}) - (N(S) \cup S)|$$ The key difference, then, is that the next hop is selected from the neighborhood of the current node $c$ (i.e. $N(\{c\})$), rather than all of $V-S$ (as is the case in the greedy approximation algorithm described in Algorithm \ref{alg:greedy}).
~\\
\noindent
\textbf{Degree Search (DS).} The degree-based search was proposed by Adamic et al. \cite{Adamic2001Search}. At each step in the search, the search query is forwarded to the unvisited neighbor with the highest degree (i.e. largest number of neighbors). That is, the next hop selected is node $v$ where $$v = \argmax_{v \in N(\{c\})-S} |N(\{v\})|.$$ Adamic et al. \cite{Adamic2001Search} analytically and empirically showed that, for power-law networks, if nodes with highest degree are preferentially selected during the search and visited first, substantial portions of the network can be covered and explored.
~\\
\noindent
\textbf{Breadth-First Search (BFS).} One type of search strategy used most often in practice is a breadth-first search \cite{Mitra2009Technological,Tsoumakos2006Analysis}. In its simplest form, this involves flooding the network, where each node sends a copy of the query to each and every one of its neighbors. These flooding and broadcast methods find targets quickly. But, as we have already mentioned, they are highly unscalable due to the tremendous overhead incurred from redundant forwards (as each node forwards the query regardless of whether its neighbors have already received it). As a result, a number of variations on flooding have been proposed to reduce this overhead (e.g. \cite{Jin2007Novel,Jiang2008LightFlood,Yang2002Improving}). In this work, we evaluate a hypothetical BFS strategy in which there are \emph{no} redundant messages. In other words, each message holder forwards a copy of the query only to those neighbors who have not yet received it, and all copies of the query terminate as soon as at least one copy of the query is successful and reaches its destination. Note that this avoidance of redundant forwards and immediate termination are somewhat unrealistic for BFS or flooding strategies. Unlike the other search methods we evaluate, there are \emph{multiple} copies of the query traversing the network in a BFS-based strategy. And, with no information transfer between the various copies of the query, it is difficult to determine which neighbors have already seen the query or when one of the copies reaches the intended target. Nevertheless, our implementation of pure BFS allows us to test the \emph{true} power of flooding-based strategies. If this strategy, with its unrealistic and unfair advantage, still cannot match the performance of other search strategies, then BFS-based methods may not hold as much promise as previously thought, and their utility for exploring networks (e.g. P2P, focused web crawling) should be possibly re-assessed.
~\\
\noindent
\textbf{Random Walk (RW).} The final search strategy we evaluate is the random walk \cite{Adamic2001Search,Li2006Searching} in which the current node forwards the query to exactly one randomly selected neighbor. We employ a \emph{self-avoiding} random walk \cite{Adamic2001Search} where the next hop is selected randomly from among the neighbors who have not yet been visited in the search. Note that, as opposed to BFS-based strategies, self-avoidance to eliminate redundant forwards is realistic here because there is a \emph{single} copy of the query traversing the network, within which a list of previously visited nodes can be stored. (The same is true for self-avoidance in the \emph{expansion search} and the \emph{degree search}.)
~\\
We conclude this section with two final remarks. First, for both the \emph{expansion search} and \emph{degree search}, each node must know both its neighbors \emph{and} its neighbors' neighbors. This is required so that the \emph{expansion search} and \emph{degree search} can compute expansion and degree (respectively). This, as it happens, is a modest and satisfiable requirement for many application domains. For instance, in a P2P network, nodes must communicate with their neighbors when joining or leaving the network anyway and neighbor lists can be exchanged during this communication. In fact, several existing search protocols exchange information with nodes at distances of even greater than two hops \cite{Adamic2001Search,Tsoumakos2006Analysis}. Even in a social network, one typically is aware of friends of friends.
Second, as mentioned previously, unvisited nodes are always preferentially chosen over visited nodes in all four search strategies. But, at some points during the search, it may be the case that all the neighbors of a given node are already visited. There are several approaches to dealing with these situations. The next step might be chosen uniformly at random from among the visited neighbors, for instance. For the \emph{expansion search} and the \emph{degree search}, another approach is to select the ``best'' unvisited node from among the neighborhood of all previously visited nodes (i.e. if $S$ is the set of visited nodes, select a node $v \in N(S)$ with the highest degree or best expansion). Note that, if using this approach, the partial topology of the network, learned during the course of the search, must be stored so that a path to the best next hop may be traversed. During preliminary testing, we did not find a significant performance difference between the two. Therefore, we only consider the former approach: when all neighbors of a current node are visited, the next hop is selected uniformly at random from among these visited neighbors.
\section{Experimental Evaluation}
\label{sec:evaluation}
\subsection{Experimental Setup}
\label{sec:evaluation.setup}
We evaluate each search strategy on each network and track performance over time. Each node is assumed to know its neighbors and passes received messages to them based on one of the four search strategies. The search ends when the message is passed to a neighbor of the target, at which time the message-holder can pass the message directly to its destination\footnote{In the context of P2P, we assume each node knows the \emph{identity} of its neighbors' neighbors, but not necessarily the \emph{files} stored by its neighbors' neighbors.}. We track the cumulative nodes discovered\footnote{For the Experimental Evaluation section, we employ the normalized cumulative nodes discovered ($\frac{|N(S) \cup S|}{|V|}$) as the evaluation measure rather than the \emph{expansion quality} ($\frac{|N(S)|}{|V-S|}$).} at each step of a search, which is comprised of both the nodes visited and the neighbors of nodes visited. We define a ``step'' in the search as a single hop taken by a single query message. If there are multiple copies of the message (as in the case of a BFS or flooding strategy), then the number of steps is defined as the total number of hops taken by all copies of the message in the system. Note that this setup is somewhat of a worst case scenario, as we are assuming there is but a single node in the entire network capable of satisfying a given search query. In the context of a P2P network, for instance, we are assuming that there is a single file residing on a single node in the whole network that must be located. As a result, actual performance in real applications, where multiple nodes can satisfy a search query, will be much higher. The extent will be domain-specific and depend on the extent of object (or file) replication in the network. This setup, then, allows us to evaluate the performance of each search strategy \emph{independent} of the effects of extraneous factors such as replication.
\subsection{Experimental Results}
\label{sec:evaluation.results}
\subsubsection{On the Performance of Search Strategies}
\label{sec:evaluation.results.performance}
We first examine the relative performance of each search strategy on each network. Table \ref{tab:searchresults} shows the number of steps required to discover $20\%$, $35\%$, and $50\%$ of the nodes in the network. As can be seen, the \emph{expansion search} (XS) exhibits the best overall performance. We also find a clear performance difference between the conventional search strategies (BFS and RW) and the less conventional approaches (XS and DS). We discuss each separately.
~\\
\textbf{XS and DS Performance}\\
Overall, we find the XS and DS strategies to exhibit the best general performance with the XS approach faring better. On most of the networks, the XS strategy either exceeds or ties the performance of other approaches. This leads us to a natural question: what causes performance differences between XS and DS? On networks in which XS and DS perform similarly, high degree nodes will tend to link to different sets of nodes (in which case high degree nodes and high expansion nodes will tend to be one and the same and will discover a similar amount of nodes). On the other hand, for networks where XS exceeds the performance of DS, we posit that these nodes may be more likely to have similar neighbors, in which case a high degree node may, in fact, have \emph{low expansion} if it links to neighbors already seen during the search. In these cases, the XS strategy will discover more nodes. Overall, despite the modestly better performance of the XS method, we find the DS strategy performs exceedingly well, which indicates that the former case may be more common in real-world networks. That is, on real-world networks, a \emph{degree search} may do well in finding high expansion nodes without explicitly looking.
The one network on which neither XS nor DS performs the best is the power grid. The power grid seems to be the least well-connected network evaluated (with mean degree of only $2.7$). In fact, it has such low connectivity that only a systematic BFS does best in exploring the network.
~\\
\textbf{BFS and RW Performance}\\
Conventional approaches to searching networks such as P2P systems include those based on random walks (RW) and breadth-first search (BFS) \cite{Mitra2009Technological}. As mentioned, flooding strategies, based on BFS, are used most often in real applications, as they tend to find answers quickly. BFS is also pervasively used in web crawling and graph sampling. It is striking, then, that our idealized version of BFS, one that avoids redundant communications and immediately terminates upon success, still cannot outperform other approaches (on all but the power grid). In general, we find that the BFS and RW approaches exhibit a relatively lower \emph{expansion quality} as compared to XS, fail to explore the network as well as other strategies in the same number of message forwards, and, consequently, discover less nodes.
~\\
\textbf{Comparison to \greedy}\\
Given the relatively better performance of the XS strategy as compared to other methods, we now examine the extent to which it matches the performance of {\sc \greedy} (our best known approximation of the maximum expansion in a network). Figure \ref{fig:xpl} shows the cumulative nodes discovered by {\sc \greedy} and XS for the first $1000$ steps. Interestingly, the XS strategy, which hops from node to node and performs the search \emph{without} complete access to the network in its entirety, often comes close to the performance of {\sc \greedy} (which \emph{does} have random access to the entire network). Once again, the most salient exception is the power grid, which seems to be the least searchable network evaluated. We discuss network searchability next.
\begin{figure}[htb]
\centering
\subfloat[Erdos-Renyi] {\label{fig:xplr.er}\includegraphics[width=0.2\textwidth]{figures/xplr-er}} \vspace{.05cm}
\subfloat[Barabasi-Albert]{\label{fig:xplr.ba}\includegraphics[width=0.2\textwidth]{figures/xplr-ba}} \\ \vskip -0.01in
\subfloat[C. elegans] {\label{fig:xpor.celegans}\includegraphics[width=0.2\textwidth]{figures/xplr-celegans}} \vspace{.05cm}
\subfloat[Power Grid]{\label{fig:xpor.power}\includegraphics[width=0.2\textwidth]{figures/xplr-power}} \\ \vskip -0.01in
\subfloat[CondMat] {\label{fig:xplr.condmat}\includegraphics[width=0.2\textwidth]{figures/xplr-condmat}} \vspace{.05cm}
\subfloat[Enron] {\label{fig:xplr.enron}\includegraphics[width=0.2\textwidth]{figures/xplr-enron}}\\ \vskip -0.01in
\subfloat[HEPPh]{\label{fig:xplr.hepph}\includegraphics[width=0.2\textwidth]{figures/xplr-hepph}} \vspace{.05cm}
\subfloat[Gnutella]{\label{fig:xplr.gnutella31}\includegraphics[width=0.2\textwidth]{figures/xplr-gnutella31}}\\ \vskip -0.01in
\subfloat[Epinions]{\label{fig:xplr.epinions}\includegraphics[width=0.2\textwidth]{figures/xplr-epinions}}\vspace{.05cm}
\subfloat[Slashdot]{\label{fig:xplr.slashdot}\includegraphics[width=0.2\textwidth]{figures/xplr-slashdot}}
\caption{[\textbf{Best viewed in color.}] Comparison of {\sc \greedy} and XS for first 1000 steps of a search. In most cases (save for the power grid), XS strategy closely matches {\sc \greedy} (our best approximation for the maximum expansion).}.
\label{fig:xpl}
\vskip -0.15in
\end{figure}
\subsubsection{On the Searchability of Networks}
\label{sec:evaluation.results.searchability}
~\\
\textbf{Expansion and Searchability}
\begin{sloppypar}
From Figures \ref{fig:esig.randomgraphs} and \ref{fig:esig.realnetworks} and Table \ref{tab:searchresults}, we can see that the magnitude of maximum expansion (as approximated by {\sc \greedy}), corresponds remarkably well to the extent to which each network is searchable. On any given network, when the maximum expansion is low, \emph{all} search strategies perform significantly worse. On the other hand, when the maximum expansion is high, \emph{all} search strategies fare relatively better. The \emph{\esig s}, then, correctly infer the ease of search and information dissemination in a network.
\end{sloppypar}
It is also striking to find that it is the \emph{maximum} expansion (rather than the \emph{minimum} expansion) most responsible for the level of searchability in a network. The classic definition of an expander graph is based on \emph{minimum expansion}. Recall that a graph is a $\gamma$-expander if $|N(S)| \geq \gamma |S|$ for each $S \subset V$ where $|S| \leq \frac{|V|}{2}$ \cite{Hoory2006Expander}. In the literature, expander graphs and minimum expansion are often connected to the ease of dissemination in network (e.g. \cite{Chierichetti2010Rumour,Barret2007Fighting}). For instance, \cite{Barret2007Fighting} has claimed social networks to be expander graphs as a means to explain the ease of diffusion across them. In contrast, our work shows that social networks are \emph{not} classic expander graphs and have a low \emph{minimum} expansion due to clustering. Moreover, we find that it is the \emph{maximum} expansion, not the minimum expansion, that is related to efficient searchability in social networks and other graphs.
\begin{figure*}[htb]
\centering
\subfloat[Expansion Search (XS)] {\label{fig:trace.xs}\includegraphics[width=0.3\textwidth]{figures/netscience-xp}} ~~~~~~~~~~
\subfloat[Random Walk (RW)]{\label{fig:trace.rw}\includegraphics[width=0.3\textwidth]{figures/netscience-rw}}
\caption{Numbers on each plot show the trace of the first 25 steps in a search by an \emph{expansion search} (XS) and a self-avoiding random walk (RW). Both searches were started from the same initial source node. The XS strategy explores a wider portion of the network and more clusters in the same number of steps}.
\label{fig:trace}
\vskip -0.15in
\end{figure*}
~\\
\textbf{Structural Properties and Possible Explanations}
\\
It is both surprising and ironic that the Gnutella network, which exists for the very purpose of search, turns out to be one of the \emph{least} searchable networks we evaluated. The only other network exhibiting \emph{less} searchability is the power grid. At the other end of the spectrum, the C. elegans and Enron networks appear to be the \emph{most} searchable. As shown in Table \ref{tab:searchresults}, for both Enron and C. elegans, all four search strategies are able to discover half the network in a very small number of steps. What causes a network to exhibit high maximum expansion and good searchability?
One of the more obvious explanations is that denser, more well-connected networks tend to be more searchable than extremely sparse networks. Nodes have larger neighborhoods in denser networks and are, therefore, easier to explore. This is true for the same reason a clique is intuitively more searchable than a long sequence or chain of nodes each connected by a single edge. For instance, the ``unsearchable'' power grid has a density of $0.0005$ and mean degree of only $2.7$ whereas the C. elegans network has a density of $0.05$ and mean degree of $14.5$. The Gnutella network, like the power grid, is also relatively more sparse than other networks of equivalent size.
Density, however, fails to explain the whole story. Consider the ER and BA graphs. Both were constructed to have similar densities but exhibit different \emph{\esig s} (see Figure \ref{fig:esig.randomgraphs}) and, correspondingly, different degrees of searchability (see Table \ref{tab:searchresults}). By virtue of its skewed degree distribution, the BA model seems to exhibit better searchability than that of the ER model, and the effect of degree distributions on search and dissemination is well-known (e.g. \cite{Barabasi1999Emergence,Adamic2001Search}). By visiting well-connected hubs, one can quickly cover significant portions of a network. But, once again, degree distributions fall short in adequately explaining the ease of search. Many of the networks considered exhibit skewed, heavy-tailed degree distributions (e.g. Enron, Epinions), but, nonetheless, exhibit different levels of searchability. Surprisingly, in stark contrast to previously held beliefs (e.g. \cite{Hui2006Smallworld}), even average path length fails to fully explain searchability. A number of networks have very similar average path lengths (see Table \ref{tab:datasets}), but very different levels of searchability (see Table \ref{tab:searchresults}).
Unlike density and degree distributions, the effect of clustering on searchability and dissemination is less studied and more nebulous. As mentioned in Section \ref{sec:evaluation.results.performance}, based on our results, we reason that clustering can also facilitate searchability. Real-world networks often exhibit what is known as \emph{community structure} \cite{Wasserman2005Models,Girvan2002Community}. Intuitively, a community in a network is a cluster of nodes more densely connected to each other than other nodes and exhibit higher clustering coefficients than one would expect at random \cite{Wasserman2005Models,Girvan2002Community}. By this intuitive definition, nodes in the same community will be expected to share more neighbors than nodes in different communities (by virtue of the dense connections within clusters and lower conductance). As a result, if one were to visit a small number of nodes from many different communities, the expansion (and, therefore, conductance) of these visited nodes would be high and many nodes would be discovered in the search. By searching based on expansion, more communities (and, consequently, larger portions of the network) are explored\footnote{This relationship between the maximum expansion and community structure has been demonstrated in \cite{Maiya2010Sampling}.}, and this can be demonstrated. Consider the network theory co-authorship network \cite{Newman2006Finding}, a small, sparse network considered by many to exhibit some degree of community structure. Figure \ref{fig:trace} shows a typical path taken by both an \emph{expansion search} (XS) and a random walk (RW) on this network. The XS strategy, by attempting to maximize expansion, jumps across the boundaries between different clusters more easily and is able to explore larger portions of the network. In this way, clustering and community structure, like high density and skewed degree distributions, can facilitate searchability in a network. However, the only common thread and unifying theme that fully and consistently explains searchability across different networks is the singular concept of \emph{expansion}.
\begin{table*}[htb]
\centering
\begin{tabular}{l|cccc|cccc|cccc} \hline \hline
{\bf ~} & \multicolumn{4}{c|}{\bf 20\%} & \multicolumn{4}{c|}{\bf 35\%} & \multicolumn{4}{c}{\bf 50\%} \\
& XS & DS & RW & BFS & XS & DS & RW & BFS & XS & DS & RW & BFS \\ \hline
& ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ & ~ \\
ER & \textbf{218} & 224 & 366 & 386 & \textbf{417} & 443 & 711 & 738 & \textbf{662} & 720 & 1141 & 1188 \\
BA & \textbf{18} & \textbf{18} & 154 & 91 & 59 & \textbf{56} & 288 & 285 & 149 & \textbf{145} & 537 & 393 \\
C. eleg. & \textbf{2} & \textbf{2} & \textbf{2} & 3 & \textbf{2} & \textbf{2} & 4 & 7 & \textbf{3} & \textbf{3} & 9 & 8 \\
Power & 1394 & 1450 & 1271 & \textbf{649} & 2220 & 2370 & 2794 & \textbf{1332} & 8051 & 6151 & 5148 & \textbf{2091} \\
CondMat & \textbf{72} & 93 & 474 & 413 & \textbf{208} & 317 & 1064 & 1071 & \textbf{495} & 827 & 2248 & 2336 \\
Enron & \textbf{9} & 10 & 125 & 266 & \textbf{20} & 22 & 342 & 801 & \textbf{49} & 58 & 559& 1941 \\
HEPPh & \textbf{26} & 37 & 204 & 446 & \textbf{56} & 80 & 469 & 1366 & \textbf{132} & 250 & 825 & 2205 \\
Gnutella & \textbf{659} & 720 & 1788 & 1730 & \textbf{1577} & 1836 & 3897 & 3829 & \textbf{2930} & 3615 & 7191 & 6875 \\
Epinions & \textbf{34} & 48 & 281 & 590 & \textbf{189} & 344 & 1059 & 2679 & \textbf{752} & 1213& 3029 & 5948 \\
Slashdot & \textbf{32} & 43 & 241 & 338 & \textbf{163} & 239 & 859 & 1612 & \textbf{492} & 725 & 1997 & 4210 \\ \hline \hline
\end{tabular}
\caption{Number of steps to discover 20\%, 35\%, and 50\% of the network. The best (i.e. lowest) value is in highlighted for each dataset. Overall, XS performs best. The variance for XS and DS was significantly small and standard error is omitted for ease of illustration. (Standard error for RW/BFS was larger, but not so large that either became a candidate for the best or even second-best performer.)}
\label{tab:searchresults}
\end{table*}
\section{Conclusions}
We have introduced the concept of \emph{\esig s} and have used them to study the effect of expansion on decentralized search in networks. We have shown that it is the magnitude of maximum expansion (rather than minimum expansion) that corresponds to the extent to which a network is efficiently searchable. Moreover, we have shown that traditional graph properties such as average path length and skewed degree distributions fail, by themselves, to fully explain the level of searchability in a network. Finally, we have shown that a search strategy based on maximizing expansion covers the network far better than some typical approaches to decentralized search. For future work, we plan to further investigate the interplay between expansion and various graph-theoretic properties and their effect on dissemination.
\balance
\bibliographystyle{abbrv}
|
{
"timestamp": "2011-09-05T02:00:39",
"yymm": "1009",
"arxiv_id": "1009.4383",
"language": "en",
"url": "https://arxiv.org/abs/1009.4383"
}
|
\section{Introduction}
There is much interest in studying the degrees of freedom (DoF) -- and thereby exploring the potential for interference alignment -- in wireless networks in the absence of instantaneous channel state information at the transmitters (CSIT) \cite{Caire_Shamai_ITrans, Jafar_scalar, Jafar_mobile, Lapidoth_Shamai_Wigger_BC, Weingarten_Shamai_Kramer, Gou_Jafar_Wang, Maddah_Compound, Huang_Jafar_Shamai_Vishwanath, Varanasi_noCSIT, Guo_noCSIT, Guo_isotropic, Jafar_corr, Wang_Gou_Jafar, Maddah_Tse}. On the one hand, there are pessmistic results that include \cite{Caire_Shamai_ITrans, Jafar_scalar, Jafar_mobile, Lapidoth_Shamai_Wigger_BC, Huang_Jafar_Shamai_Vishwanath, Varanasi_noCSIT, Guo_noCSIT, Guo_isotropic} where the DoF are found to collapse due to the inability of the transmitters to resolve spatial dimensions. On the other hand, there are more recent optimistic results \cite{Weingarten_Shamai_Kramer, Gou_Jafar_Wang, Maddah_Compound, Jafar_corr, Wang_Gou_Jafar, Maddah_Tse} where the feasibility of interference alignment is demonstrated under various models of channel uncertainty at the transmitter(s). Closely related to this work are the papers on blind interference alignment \cite{Jafar_corr, Wang_Gou_Jafar} and especially the recent work on interference alignment with delayed CSIT \cite{Maddah_Tse}.
Reference \cite{Jafar_corr} assumes block fading channels where the coherence blocks are staggered due to difference in coherence times between users. The channels stay constant within a coherence block and are assumed to change instantly across coherence block boundaries. The transmitter(s) have no knowledge of instantaneous channel coefficient values but are assumed to be aware of the coherence block structure of all users. The surprising finding of \cite{Jafar_corr} is the feasibility of alignment based on just the knowledge of the differences in the channel coherence structure across users. \cite{Wang_Gou_Jafar} applies the same principles under a different model where some of the nodes are equipped with reconfigurable antennas capable of switching their own channel states at predetermined time instants, thus allowing further flexibility in determining the channel coherence structure. With this added flexibility \cite{Wang_Gou_Jafar} show that X networks, even with no knowledge of channel coefficient values, do not lose DoF relative to the perfect CSIT setting. The key insight from these blind alignment schemes was that the commonly used i.i.d. fading model is not sufficient for studying the capacity limits of wireless networks even in the absence of CSIT -- because the knowledge of even relatively long term channel statistics can be exploited to achieve interference alignment.
More recently, reference \cite{Maddah_Tse} has introduced the delayed CSIT model, that will also be the main focus of this paper. The delayed CSIT model assumes i.i.d. fading channel conditions, with no knowledge of current channel state at the transmitter. However, perfect knowledge of channel states is available to the transmitter with some delay. The surprising finding of \cite{Maddah_Tse}, in the context of the vector broadcast (BC) channel, is that not only is CSIT helpful even when it is outdated, but also that it can have a very significant impact as it is capable of increasing the DoF. The delayed CSIT model studied in \cite{Maddah_Tse} is particularly relevant in practice where invariably a delay is involved in any feedback from the receivers to the transmitters. Several recent works point out that channel state information (CSI) can be estimated in principle with sufficient accuracy (estimation error scaling as O$(SNR^{-1})$) to enable the DoF results as SNR becomes large \cite{Jindal, Caire_Jindal_Shamai}. The main obstacle, from a practical perspective, has been the perceived necessity of delivering this CSI to the transmitter before it becomes outdated. The delayed CSIT model therefore opens a practically meaningful direction to explore the benefits of interference alignment. However, it is one of many possible forms that (delayed) feedback can take in a wireless network.
The terminology for three closely related delayed feedback models is delineated below through a simple example, for ease of reference in the sequel.
\subsubsection*{Delayed Feedback Models}
Consider a simple Gaussian channel:
\begin{eqnarray}
Y=HX+N
\end{eqnarray}
where $X, Y, N, H$ are the transmitted symbol, the channel output symbol, the additive noise and the i.i.d. time-varying channel, respectively. Perfect channel state information at the receiver (CSIR) is modeled by the assumption that in addition to the channel output symbol $Y$, the receiver also receives the instantaneous channel state $H$ over each channel use. By \emph{delayed} feedback, what is meant is that the information being made available to the transmitter through the feedback channel is based only on past observations at the receivers and, in particular, is independent of the current channel state. Three different settings may be considered.
\begin{enumerate}
\item {\bf Delayed CSIT:} This is the setting where the feedback provides the transmitters only the values of the past channel states $H$ but not the output signals.
\item {\bf Delayed Output Feedback:} This is the setting where the feedback provides the transmitters only the past received signals $Y$ seen by the receivers, but not the channel states explicitly.
\item{\bf Delayed Shannon Feedback:} This is the setting where the feedback provides the transmitters both the past received signals $Y$ as well as the past channel states $H$.
\end{enumerate}
The definitions extend naturally to multiuser settings, although the amount of feedback, and the possible associations between transmitters and receivers on the feedback channel may give rise to many special cases. Clearly, delayed Shannon feedback is the strongest delayed feedback setting, i.e., it can be weakened to obtain either delayed CSIT or the delayed output feedback model by discarding some of the fed back information. Between delayed CSIT and delayed output feedback, neither is a weakened form of the other because in general the output signals $Y$ (even while discounting noise) cannot be inferred from the knowledge of channel states $H$ (e.g., when there is more than one transmitter), and the channel states $H$ cannot be deduced in general from the channel outputs $Y$ even if these are noiselessly made available to the transmitter (e.g., when there are more transmit antennas than receive antennas). In this work, we will be concerned primarily with the delayed CSIT model, and to a lesser extent, with delayed output feedback. But first, we start with a broader review of the similarities and differences between the interference alignment schemes used for blind interference alignment and those used for the Delayed Feedback model.
\section{Similarities between Blind Interference Alignment and Interference Alignment with Delayed CSIT}
While the channel models studied in \cite{Jafar_corr,Wang_Gou_Jafar} and \cite{Maddah_Tse} are quite different, there are some remarkable \emph{essential} similarities in the achievable schemes used in both works that are further expanded in the present work. We start by pointing out the similar aspects of \cite{Jafar_corr, Wang_Gou_Jafar, Maddah_Tse} through a few examples before proceeding to the main contribution of this work.
\subsection{Vector BC with no instantaneous CSIT}
\begin{figure}[!h]
\centering
\includegraphics[width=2.5in]{MISOBC}
\caption{Vector BC with No Instantaneous CSIT}
\label{fig:X}
\end{figure}
Consider the broadcast channel (BC) with two single-antenna users, where the transmitter is equipped with two antennas. It is well known that with perfect channel knowledge this channel has $2$ DoF, which can be achieved quite simply by zero forcing. However, the DoF are unknown with partial/limited CSIT for most cases and it is believed that no more than 1 DoF may be achievable in general. An outer bound of $\frac{4}{3}$ DoF has been shown to be applicable to a wide variety of limited CSIT scenarios \cite{Lapidoth_Shamai_Wigger_BC,Weingarten_Shamai_Kramer, Maddah_Tse}. In particular, \cite{Maddah_Tse} explicitly shows the $\frac{4}{3}$ outer bound for delayed CSIT. It is easily verified that the outer bound applies under the stronger setting of delayed Shannon feedback as well. Further the same outer bound applies to the staggered block fading model. Recently, reference \cite{Jafar_corr} shows that $\frac{4}{3}$ DoF are achievable under a staggered block fading model where the two users have staggered coherent fading blocks. Reference \cite{Maddah_Tse} shows the achievability of $\frac{4}{3}$ DoF under the assumption that only delayed CSIT is available. In both cases the achievable scheme is described as follows:
\begin{itemize}
\item Use the channel three times to send two information symbols to each user.
\item In one time slot, two symbols for user 1 are sent simultaneously from the two transmit antennas, providing user 1 one linear combination of the two desired symbols. One more linear combination would be needed to resolve the desired symbols at user 1. User 2 simultaneously also obtains a linear combination of these undesired symbols, but does not need to resolve them.
\item In another time slot, two symbols for user 2 are sent simultaneously from the two transmit antennas, thus similarly providing user 2 one linear combination of his desired symbols, and providing user 1 a linear combination of these undesired symbols.
\item In the final transmission, both users are simultaneously provided another linear combination of their respective desired symbols. The key to alleviate interference in this time slot is that the linear combination of undesired symbols received by a user in the third time slot is \emph{the same linear combination} that he already received earlier, which allows the receiver to cancel the interference and then recover his desired symbols.
\end{itemize}
The key to both schemes is the third time slot. Because of different channel models, the manner in which the third time slot transmission is accomplished is different in \cite{Jafar_corr} and \cite{Maddah_Tse}. In \cite{Jafar_corr} the staggered coherence times ensure that each user receives the final transmission over a channel state that is identical to his channel state when he received interference in a prior time slot, but different from his channel state where he previously received his desired symbols. The channel state determines the linear combination and thus, the desired symbols are seen twice with different linear combinations while the interfering symbols are seen twice in the same linear combination, allowing interference cancellation. In \cite{Maddah_Tse} the same effect is accomplished by using delayed CSIT feedback. In the third time slot -- because the transmitter now knows the channel states from the previous two time slots -- the transmitter is able to send (from only one antenna) a superposition of the linear combinations of the undesired symbols seen previously by the two users. Since these linear combinations were received before, undesired information is easily cancelled, leaving only the linear combination of desired symbols that provides the second equation in order to solve for the two desired variables. Since each user is able to resolve his 2 desired symbols over a total of 3 time slots, the DoF of $\frac{4}{3}$ are achieved.
\subsection{Vector BC with instantaneous CSIT for User 1}
\begin{figure}[!h]
\centering
\includegraphics[width=2.5in]{bc3staggered}
\caption{Vector BC with Instantaneous CSIT for User 1}
\label{fig:X}
\end{figure}Consider the same vector BC as before, with the difference that now we assume the channel state of User 1 is instantaneously available to the transmitter. The DoF outer bound in this case is $\frac{3}{2}$ \cite{Weingarten_Shamai_Kramer} and is also applicable to a broad class of channel uncertainty models for User 2. In particular, it can be shown to be applicable to the delayed Shannon feedback model, by providing all the information available to Receiver 2 also to Receiver 1, thus making it a physically degraded broadcast channel for which it is known that feedback does not increase capacity \cite{ElGamal_FB}. Then, proceeding without feedback, the outer bound arguments for the compound setting in \cite{Weingarten_Shamai_Kramer} can be extended to this setting, producing the same DoF outer bound of $\frac{3}{2}$. Note that an outer bound for delayed Shannon feedback is also an outer bound for delayed CSIT. Similar arguments (without feedback) are applicable to show the $\frac{3}{2}$ DoF outer bound for blind interference alignment (staggered block fading) model as well. Further, the achievability of $\frac{3}{2}$ DoF can be shown under both settings of staggered block fading and delayed CSIT in essentially the same manner, as described below.
\begin{itemize}
\item Use the channel twice to send two information symbols to User 1 and one information symbol to User 2.
\item User 2's information symbol is sent along a beamforming vector orthogonal to User 1's known channel vector, so User 1 sees no interference due to User 2.
\item User 1's two symbols are sent twice in a manner that the same linear combination of the two undesired symbols is experienced by User 2 in both timeslots. This allows User 2 to cancel interference from one of the two time slots to recover his one desired symbol. However, User 1 sees two different linear combinations of his desired symbols and no interference, so he is able to recover both symbols.
\end{itemize}
Thus, 1 DoF is achieved by User 1 and 0.5 DoF by User 2, for a total of 1.5 DoF which is also the outer bound.
The key here is to transmit the same linear combination of undesired symbols for User 2 while User 1 sees two different linear combinations. \cite{Jafar_corr} does this by assuming that the channel stays constant for User 2 while it changes for User 1. The delayed CSIT setting can also accomplish the same effect because once the transmitter learns the channel states, it knows the linear combination of undesired symbols already seen by User 2 and re-sends the same linear combination from one antenna. Since this linear combination has not yet been seen by User 1, it gives him the second equation he needs, while for User 2 it is just a repetition of the previously seen interference which can be cancelled. Interestingly, in both cases (staggered block fading or delayed CSIT) the channel knowledge of User 1 (i.e., the user with the known channel) is only needed by the transmitter for one of the two channel uses (because User 2's symbol can be transmitted only once \cite{Jafar_corr}).
\subsubsection{The (1,2,3,4) Two User MIMO Interference Channel}
\begin{figure}[!h]
\centering
\includegraphics[width=3in]{MIMOIC}
\caption{(1,2,3,4) MIMO IC}
\label{fig:MIMOIC}
\end{figure}
Shown in Fig. \ref{fig:MIMOIC}, the (1,2,3,4) MIMO IC is a two user interference channel where User 1 has 1 transmit and 2 receive antennas, while User 2 has 3 transmit and 4 receive antennas. This particular channel configuration was first highlighted in \cite{Huang_Jafar_Shamai_Vishwanath} as an example where interference alignment was a possibility. Specifically, the question is, what is the maximum DoF possible for User 2 if User 1 simultaneously achieves his maximum value of 1 DoF? With no interference alignment the result would be only 1 DoF for user 2, but with interference alignment it may be possible to achieve up to 1.5 DoF for User 2. While recent work in \cite{Guo_noCSIT} showed that under i.i.d. isotropic fading, User 2 cannot achieve more than 1 DoF, thus precluding the possibility of interference alignment, the question remains open for non-iid, non-isotropic models. In particular, \cite{Jafar_corr} shows that under a staggered coherence block fading model, User 2 may indeed achieve 1.5 DoF at the same time that User 1 achieves his maximum DoF =1. Here we briefly summarize how this result is shown in \cite{Jafar_corr} and how the same scheme can be translated to the delayed CSIT setting (although it remains to be verified whether the same outer bound applies to delayed CSIT setting for the (1,2,3,4) MIMO IC), once again highlighting the similarity of the two.
\begin{itemize}
\item Operate the channel over two time slots, sending a new symbol in each time slot from Transmitter 1 and sending three symbols from Transmitter 2, repeated over two slots.
\item The key is to send the three symbols from Transmitter 2 in such a manner that User 1, at his two receive antennas, sees the same two linear combinations of undesired symbols twice. Thus, in each time slot he is able to free up one dimension by cancelling the corresponding interference using the same linear combination received in the other time slot, and recover his desired symbol.
\end{itemize}
In order to make sure that the same two linear combinations of undesired symbols are seen on the two receive dimensions twice by Receiver 1, reference\cite{Jafar_corr} exploits the assumption that the channel between Transmitter 2 and Receiver 1 stays constant over two time slots. However, in the delayed CSIT setting, after the first time slot Transmitter 2 knows the CSIT from the first time slot and therefore also knows the two different linear combinations of the 3 transmitted symbols observed at the two receive antennas of Receiver 1. In time slot 2 then Transmitter 2 sends the two linear combinations, each from a different transmit antenna. Since only $2$ antennas are used in the second time slot, the $2\times 2$channel can be inverted by Receiver 1 to essentially experience the same two linear combinations of undesired symbols on the two resulting receive antennas. Thus, one receive antenna can be cleared of interference in each time slot by using the corresponding linear combination of undesired symbols observed in the other time slot, allowing the desired symbol to be resolved in each time slot.
\section{Differences Between Staggered Fading and Delayed CSIT Settings}
In spite of the strong similarities between the staggered fading model \cite{Jafar_corr} and the delayed CSIT \cite{Maddah_Tse} model, the two settings have very marked differences in general. The difference becomes evident as soon as the question of alignment of signals from \emph{different} transmitters comes to light. Note that in all examples described above the signals being aligned are from the same transmitter. However, when we go to more distributed settings, e.g., the X or $K$ user interference channels the difference between the two settings is quite stark. With \emph{suitably} staggered coherent times/antenna switching, it is shown in \cite{Jafar_corr, Wang_Gou_Jafar} that neither the $X$ network with $M$ transmitters and $N$ receivers, nor the $K$ user interference channel lose \emph{any} DoF relative to the setting where perfect, global and instantaneous CSIT is available. The $M\times N$ user $X$ network can still achieve $\frac{MN}{M+N-1}$ DoF, and the $K$ user interference channel can still achieve $\frac{K}{2}$ DoF even with no instantaneous (or delayed) CSIT besides the knowledge of the channel coherence structure/antenna switching pattern. However, in the delayed CSIT setting, (albeit only with i.i.d fading) the outer bounds clearly show a loss in DoF. Thus, evidently there is no complete theoretical equivalence between the two settings.
The significance of distributed transmitters brings us to the issue central to the present work. We know that in the delayed CSIT setting, DoF are lost because the DoF outer bound for the vector BC (say with $K$ transmit antennas, and $K$ single antenna users) loses to the $\frac{K}{2}$ DoF that are achievable in the $K$ user interference channel both in the compound setting (with finite number of generic states \cite{Gou_Jafar_Wang}) as well as with suitable staggered coherence block model. However, a very interesting question remains open -- Does transmitter cooperation improve DoF even with channel uncertainty at transmitters?
The main difference between the $X$ channel and the vector BC is that in the vector BC the transmitters are allowed to cooperate as multiple antennas of one transmit node. With full CSIT it is well known that the vector BC has larger DoF than the X channel, so the cooperation between transmitters increases DoF. However, with limited CSIT, e.g. in the compound channel setting \cite{Weingarten_Shamai_Kramer, Gou_Jafar_Wang, Maddah_Compound}, somewhat surprisingly, several recent results have shown that the DoF are the same with and without transmitter cooperation, i.e., whether it is the X channel or the corresponding vector BC. Indeed, with the staggered fading model of \cite{Jafar_corr}, the same approach works for the X channel as does for the vector BC.
The delayed CSIT setting gives us another framework within which one can hope to gain additional insights into the role of transmitter cooperation in determining the available DoF under channel uncertainty. Consider the vector BC with 2 antennas at the transmitter and 2 single-antenna users, for which we know DoF = $\frac{4}{3}$ with delayed CSIT. However, does the same scheme work for the $X$ channel? More generally, what DoF can we achieve on the $X$ channel with delayed CSIT? These are the questions that motivate the current work.
We start with two interesting observations.
\begin{itemize}
\item {\bf The approach used for vector BC in \cite{Maddah_Tse} is not directly applicable to the X channel.} This is because in the vector BC, after the transmitter acquires delayed CSIT, it is able to reconstruct the linear combinations of undesired symbols previously seen by the receivers. However, in the X channel, even after the distributed transmitters learn the channel states, they cannot re-send the same linear combination of undesired symbols. For example, consider the first time slot where both transmitters send a symbol each intended for User 1, which provides User 2 a particular linear combination of the two undesired symbols. Future repetitions of this particular linear combination carry no interference cost to User 2, because he is able to cancel the interference. However, in the X channel, one cannot repeat this linear combination, because if each transmitter repeats its own symbol meant for User 1, the linear combination seen by User 2 also depends on the \emph{current} channel state which cannot be compensated for by the transmitters who have, at this point, no knowledge of it. This is why in the broadcast setting the repetitions of a particular linear combination take place from a single transmit antenna as a scalar value. For example, if the transmitter wants User 2 to see the linear combination $2u_1+u_2$, it cannot transmit $u_1$ and $u_2$ from two different antennas, which would leave the resulting linear combination in the hands of nature. Instead, in the vector BC with delayed CSIT, the transmitter sends $2u_1+u_2$ from the same antenna, so that even when nature scales the transmitted scalar value, the receiver will see $h(2u_1+u_2)$ from which $h$ can be scaled off. The BC can do this because both $u_1, u_2$ symbols are available to e.g. antenna 1. In the $X$ channel on the other hand, transmitter 1 may only know $u_1$ and transmitter $2$ only $u_2$, thus forcing the two symbols to be sent from different antennas, and since the channels change every time slot (or before the CSIT becomes available), it is not possible to repeat the linear combination seen by User 2\footnote{Note that if instead of delayed CSIT, we have delayed output feedback, then this problem does not arise.}.
\item {\bf Aligning interfering symbols from the same transmitter does not produce DoF benefits on the X channel.} The next natural thought is this -- since, as discussed above, it seems we cannot repeat the alignment of symbols from two different transmitters without knowledge of current channel state, can we instead achieve interference alignment between two undesired symbols coming from the \emph{same} transmitter? This is a subtle but important point. As we argue here, in the X channel, alignment of symbols (with linear beamforming schemes) from the same transmitter is not beneficial for DoF. The reason is that we are considering the X channel where all nodes have equal number of antennas. Suppose we have two symbols that originate at the same transmitter (say Transmitter 1) and are aligned at a particular receiver (say Receiver 1). Since this alignment does not depend on the other transmitter, let us ignore all signals received from the other transmitter and focus only on the aligned signals received from Transmitter 1. Even with symbol extensions, barring degenerate conditions, what it means is that Receiver 1 can invert the channel to Transmitter 1, and thus observe the transmitted symbols (within the noise distortion level, which is inconsequential for DoF arguments). Thus, if two undesired symbols from the same transmitter are seen aligned at a particular receiver, it must be because they are aligned at the transmitter itself. Now, if these symbols are aligned at the transmitter itself, then they cannot be resolved further downstream at any other receiver, and in particular, at the receiver that wants these two symbols and therefore must not see them aligned together. Thus the benefits of alignment are lost from a DoF perspective if the aligned symbols come from the same transmitter on the X channel.
\end{itemize}
The two observations listed above, seem to leave very little hope of achieving interference alignment on the X channel. If we cannot repeat a linear combination from two different transmitters without current channel knowledge, and as we argue above, it does not help to align interference from the same transmitter, then there are few alternatives left and it seems the DoF of the X channel will be limited to 1. However, as we show in this paper, this is not the case. The key to the positive result is that while the vector BC approach does not directly extend to the X channel, an interesting new approach -- that we call \emph{retrospective interference alignment} -- is able to achieve interference alignment on the X channel even with delayed CSIT.
We proceed to the main result next.
\section{The X Channel}
The X channel consists of two transmitters, two receivers, and four independent messages, one from each transmitter to each receiver. We assume that the channels vary in an i.i.d. fashion according to some continuous distribution with support that is bounded away from zero and infinity. The receivers are assumed to have perfect knowledge of all channel states. The transmitters do not know the current channel state, but they do have access to all channel states up to the previous channel use. This model is referred to as the \emph{delayed CSIT} model. Our goal is to explore the DoF that can be achieved by the X channel in this setting. The definitions of achievable data rates, capacity, power constraints, DoF are all used in the standard sense as in e.g. \cite{Maddah_Tse} and will not be repeated here. We also assume that the reader is familiar with DoF analysis when working with linear beamforming schemes, and in particular the requirements for interference alignment. For these and other standard issues such as -- why we ignore noise in this analysis, what are the conditions for desired signals to be recovered in the presence of interference, a literature survey of earlier works on interference alignment and DoF such as \cite{Zheng_Tse, Jafar_Shamai, MMK, Cadambe_Jafar_int} is recommended.
\begin{theorem}
The 2 user X channel with delayed CSIT, can achieve DoF = $\frac{8}{7}$ almost surely.
\end{theorem}
\begin{figure}[!h]
\centering
\includegraphics[width=3in]{Xchannel}
\caption{X Channel}
\label{fig:X}
\end{figure}
\noindent\hspace{2em}{\it Proof: }
We wish to show that $\frac{8}{7}$ DoF are achievable on the X channel with delayed CSIT. To this end, consider a 7 symbol extension of the channel, i.e., we will consider 7 channel uses over which the precoding vectors will be designed. Each of the 4 messages $W^{[kj]}$ will be assigned two precoding vectors $\overline{\bf V}^{[kj]}=\left[\overline{\bf V}^{[kj]}_{1} ~~\overline{\bf V}^{[kj]}_{2}\right]$, where $\overline{\bf V}^{[kj]}_{i}$ is the $i^{th}$ column of the $7\times 2$ matrix $\overline{\bf V}^{[kj]}$, $k=1,2$, and will carry the scalar coded information symbol $u^{[kj]}_i$. Thus, over the $7$ symbol block, the received signal at Receiver $k$, $k=1,2$, can be expressed as:
\begin{eqnarray}
\overline{\bf Y}^{[k]}&=&\overline{\bf H}^{[k1]}\left(\overline{\bf V}^{[11]}_{ 1}u^{[11]}_1+\overline{\bf V}^{[11]}_{ 2}u^{[11]}_2+\overline{\bf V}^{[21]}_{ 1}u^{[21]}_1+\overline{\bf V}^{[21]}_{ 2}u^{[21]}_2\right)\nonumber\\
&&~~~~+\overline{\bf H}^{[k2]}\left(\overline{\bf V}^{[12]}_{ 1}u^{[12]}_1+\overline{\bf V}^{[12]}_{ 2}u^{[12]}_2+\overline{\bf V}^{[22]}_{ 1}u^{[22]}_1+\overline{\bf V}^{[22]}_{ 2}u^{[22]}_2\right)+\overline{\bf Z}^{[k]}\\
&=& \overline{\bf H}^{[k1]}\left[ \overline{\bf V}^{[11]}~~~\overline{\bf V}^{[21]}\right]{\bf U}^{[\star 1]}+\overline{\bf H}^{[k2]}\left[ \overline{\bf V}^{[12]}~~~\overline{\bf V}^{[22]}\right]{\bf U}^{[\star 2]}+\overline{\bf Z}^{[k]}
\end{eqnarray}
Here $\overline{\bf Y}^{[k]}, \overline{\bf Z}^{[k]}$ are $7\times 1$ vectors, $\overline{\bf H}^{[kj]}$ are $7\times 7$ diagonal matrices and $u^{[kj]}_i$ are scalar symbols. ${\bf U}^{[\star i]}=\left[u^{[1i]}_1, u^{[1i]}_2, u^{[2i]}_1, u^{[2i]}_2\right]^T$ is the $4\times 1$ vector of information symbols originating at transmitter $i$.
The key task is to design the precoding vectors $\overline{\bf V}^{[kj]}_{ i}$ to achieve interference alignment. The elements of the precoding vector can depend on the channel state only retrospectively, i.e., at time $n$, $ n= 1,2, \cdots, 7$, the $n^{th}$ element of a precoding vector can depend on the past channel states $\overline{\bf H}^{[\star\star]}(1), \cdots, \overline{\bf H}^{[\star\star]}(n-1)$ but not the present or future channel states.
\subsubsection*{First 3 Channel Uses}
For the first $3$ channel uses each transmitter simply sends random linear combinations (the coefficients are generated randomly offline and shared between all transmitters and receivers before the beginning of communication) of the 4 symbols originating at that transmitter. A different linear combination is sent each time. This gives each receiver $3$ equations in $8$ variables. Because the combining coefficients are chosen randomly and independently of the channel realizations these equations contain no particular structure and may be regarded as \emph{generic} linear equations. Ignoring noise, the received signals at Receiver 1 at this stage can be expressed as:
\begin{allowdisplaybreaks}
\begin{eqnarray}
\left[\begin{array}{c}\overline{\bf Y}^{[1]}(1)\\\overline{\bf Y}^{[1]}(2)\\\overline{\bf Y}^{[1]}(3)\end{array}\right]&=&
\left[\begin{array}{cccc}
\overline{\bf H}^{[11]}(1)\overline{\bf V}^{[11]}_1(1) & \overline{\bf H}^{[11]}(1)\overline{\bf V}^{[11]}_2(1) &\overline{\bf H}^{[12]}(1)\overline{\bf V}^{[12]}_1(1)&\overline{\bf H}^{[12]}(1)\overline{\bf V}^{[12]}_2(1)\\
\overline{\bf H}^{[11]}(2)\overline{\bf V}^{[11]}_1(2) & \overline{\bf H}^{[11]}(2)\overline{\bf V}^{[11]}_2(2) &\overline{\bf H}^{[12]}(2)\overline{\bf V}^{[12]}_1(2)&\overline{\bf H}^{[12]}(2)\overline{\bf V}^{[12]}_2(2)\\
\overline{\bf H}^{[11]}(3)\overline{\bf V}^{[11]}_1(3) & \overline{\bf H}^{[11]}(3)\overline{\bf V}^{[11]}_2(3) &\overline{\bf H}^{[12]}(3)\overline{\bf V}^{[12]}_1(3)&\overline{\bf H}^{[12]}(3)\overline{\bf V}^{[12]}_2(3)
\end{array}\right]
\left[\begin{array}{c}
u^{[11]}_1\\ u^{[11]}_2\\ u^{[12]}_1\\ u^{[12]}_2
\end{array}\right]\nonumber\\
&&+\left[\begin{array}{cccc}
\overline{\bf H}^{[11]}(1)\overline{\bf V}^{[21]}_1(1) & \overline{\bf H}^{[11]}(1)\overline{\bf V}^{[21]}_2(1) &\overline{\bf H}^{[12]}(1)\overline{\bf V}^{[22]}_1(1)&\overline{\bf H}^{[12]}(1)\overline{\bf V}^{[22]}_2(1)\\
\overline{\bf H}^{[11]}(2)\overline{\bf V}^{[21]}_1(2) & \overline{\bf H}^{[11]}(2)\overline{\bf V}^{[21]}_2(2) &\overline{\bf H}^{[12]}(2)\overline{\bf V}^{[22]}_1(2)&\overline{\bf H}^{[12]}(2)\overline{\bf V}^{[22]}_2(2)\\
\overline{\bf H}^{[11]}(3)\overline{\bf V}^{[21]}_1(3) & \overline{\bf H}^{[11]}(3)\overline{\bf V}^{[21]}_2(3) &\overline{\bf H}^{[12]}(3)\overline{\bf V}^{[22]}_1(3)&\overline{\bf H}^{[12]}(3)\overline{\bf V}^{[22]}_2(3)
\end{array}\right]
\left[\begin{array}{c}
u^{[21]}_1\\ u^{[21]}_2\\ u^{[22]}_1\\ u^{[22]}_2
\end{array}\right]\nonumber\\
\Rightarrow {\bf Y}^{[1]}&=& \left[{\bf H}^{[11]}{\bf V}^{[11]}_1 ~~~ {\bf H}^{[11]}{\bf V}^{[11]}_2 ~~~ {\bf H}^{[12]}{\bf V}^{[12]}_1 ~~~{\bf H}^{[12]}{\bf V}^{[12]}_2\right]\left[u^{[11]}_1, u^{[11]}_2, u^{[12]}_1, u^{[12]}_2\right]^T\nonumber\\
&&+\left[{\bf H}^{[11]}{\bf V}^{[21]}_1 ~~~ {\bf H}^{[11]}{\bf V}^{[21]}_2 ~~~ {\bf H}^{[12]}{\bf V}^{[22]}_1 ~~~{\bf H}^{[12]}{\bf V}^{[22]}_2\right]\left[u^{[21]}_1, u^{[21]}_2, u^{[22]}_1, u^{[22]}_2\right]^T
\end{eqnarray}
\end{allowdisplaybreaks}
Note that the channel and precoding vectors without the overbar notation refer to the values over only the first 3 channel uses. The received signal for Receiver 2, is also defined similarly.
\begin{eqnarray}
\Rightarrow {\bf Y}^{[2]}&=& \left[{\bf H}^{[21]}{\bf V}^{[11]}_1 ~~~ {\bf H}^{[21]}{\bf V}^{[11]}_2 ~~~ {\bf H}^{[22]}{\bf V}^{[12]}_1 ~~~{\bf H}^{[22]}{\bf V}^{[12]}_2\right]\left[u^{[11]}_1, u^{[11]}_2, u^{[12]}_1, u^{[12]}_2\right]^T\nonumber\\
&&+\left[{\bf H}^{[21]}{\bf V}^{[21]}_1 ~~~ {\bf H}^{[21]}{\bf V}^{[21]}_2 ~~~ {\bf H}^{[22]}{\bf V}^{[22]}_1 ~~~{\bf H}^{[22]}{\bf V}^{[22]}_2\right]\left[u^{[21]}_1, u^{[21]}_2, u^{[22]}_1, u^{[22]}_2\right]^T
\end{eqnarray}
\subsubsection*{Last 4 Channel Uses}
During the last 4 channel uses we will operate over a new effective set of variables. Using the terminology of \cite{Maddah_Tse} we can call these the second layer variables. The goal will be to allow each receiver to resolve all layer-2 variables. Since we have only $4$ channel uses left, and each channel use will provide only one equation to each receiver, we will choose $4$ layer-2 variables.
While everything so far is consistent with the approaches of \cite{Jafar_corr, Maddah_Tse}, at this point our approach goes into a new direction. As mentioned earlier, because the transmitters are distributed, they do not have access to any of the equations available so far to the receivers -- because each equation contains symbols from both transmitters. Thus, unlike the broadcast setting studied in \cite{Maddah_Tse} where the transmit antennas are co-located, we cannot construct layer-2 variables out of the equations already available to the receivers. Instead, we will construct layer-2 variables out of the symbols available at each transmitter. Let us define our new variables:
\begin{eqnarray}
s^{[1]}_1&=&u^{[11]}_1 - \gamma^{[1]}_1 u^{[11]}_2\\
s^{[2]}_1&=&u^{[12]}_1- \gamma^{[2]}_1 u^{[12]}_2\\
s^{[1]}_2&=&u^{[21]}_1- \gamma^{[1]}_2 u^{[21]}_2\\
s^{[2]}_2&=&u^{[22]}_1- \gamma^{[2]}_2 u^{[22]}_2
\end{eqnarray}
where $\gamma^{[k]}_i$ are constants whose values will be specified soon after we arrive at the rationale for choosing these values. Consistent with our causality and delayed CSIT constraint, the constants will depend on the channel values only from phase-I, i.e., from the first 3 channel uses. The most important observation here is that the variables $s^{[k]}_1, s^{[k]}_2$ are available to Transmitter $k$.
Once the layer-2 variables are defined, the operation over the last 4 channel uses is very simple. Each transmitter sends a different linear combination of its two layer-2 variables over each channel use. Each receiver sees 4 different linear combinations of the 4 layer-2 variables (2 from each transmitter) over the 4 channel uses, and is therefore able to resolve each variable (again, ignoring noise -- for DoF arguments).
{\it Remark:} Note that no CSIT, not even delayed CSIT, is needed for the last 4 channel uses. This is an important observation that could significantly reduce the overhead of feeding back the delayed CSIT to the transmitters. Evidently, no more than $\frac{3}{7}$ of the channel states need to be fed back with the proposed scheme.
\subsubsection*{Retrospective Interference Alignment}
As mentioned above, the novelty of \emph{retrospective interference alignment} lies in the construction of layer-2 variables. In particular, we will choose the values of the constants $\gamma^{[k]}_i$ to align interference over the \emph{first 3 channel uses} -- i.e., acting retrospectively. Also note that the definitions of layer-2 variables seem to suggest at first that the variables from the same transmitter are being aligned into the layer-2 variables. This is not the correct interpretation, for the simple reason that alignment of variables from the same transmitter through linear schemes cannot provide DoF benefits on the SISO X channel, as argued earlier. As we show below, the actual alignment still happens between information variables coming from different transmitters over the first 3 channel uses.
From phase-2 we know that both users are able to solve for the layer-2 variables. Now, let us consider the $3$ equations accumulated at each receiver over the first 3 channel uses in 8 variables. Let us substitute for four of these variables in terms of the solved values of the layer-2 variables. Specifically, we make the following substitutions:
\begin{eqnarray}
u^{[11]}_1 &\longrightarrow& s^{[1]}_1+\gamma^{[1]}_1 u^{[11]}_2\\
u^{[12]}_1 &\longrightarrow& s^{[2]}_1+\gamma^{[2]}_1 u^{[12]}_2\\
u^{[21]}_1 &\longrightarrow& s^{[1]}_2+\gamma^{[1]}_2 u^{[21]}_2\\
u^{[22]}_1 &\longrightarrow& s^{[2]}_2+\gamma^{[2]}_2 u^{[22]}_2
\end{eqnarray}
Note that after these substitutions there are only four unknown variables left (since the $s^{[k]}_i$ are already known from Phase-2) --- $u^{[11]}_2, u^{[12]}_2, u^{[21]}_2, u^{[22]}_2$. Since we have four variables and only three equations we will need interference alignment. Out of the 4 remaining unknown variables each receiver only desires 2. As usual on the X channel, the 2 undesired variables will be aligned into one dimension, leaving the remaining two dimensions available to recover the two desired variables. This alignment will be enabled precisely by the choice of the $\gamma^{[k]}_i$ in the layer-2 variable definitions -- thus accomplishing retrospective interference alignment.
Following the substitutions, consider Receiver 1, where we have three equations in the remaining 4 variables (Note that the following quantities -- without overbar notation -- refer to only the first three channel uses).
\begin{eqnarray}
{\bf Y}^{[1]}-{\bf H}^{[11]}{\bf V}^{[11]}_1s^{[1]}_1-{\bf H}^{[12]}{\bf V}^{[12]}_1s^{[2]}_1-{\bf H}^{[11]}{\bf V}^{[21]}_1s^{[1]}_2-{\bf H}^{[12]}{\bf V}^{[22]}_1s^{[2]}_2=\nonumber\\
{\bf H}^{[11]}{\bf V}^{[11]}_1\gamma^{[1]}_1u^{[11]}_2+ {\bf H}^{[11]}{\bf V}^{[11]}_2 u^{[11]}_2+{\bf H}^{[12]}{\bf V}^{[12]}_1\gamma^{[2]}_1u^{[12]}_2+{\bf H}^{[12]}{\bf V}^{[12]}_2u^{[12]}_2\nonumber\\
{\bf H}^{[11]}{\bf V}^{[21]}_1\gamma^{[1]}_2u^{[21]}_2+ {\bf H}^{[11]}{\bf V}^{[21]}_2 u^{[21]}_2+{\bf H}^{[12]}{\bf V}^{[22]}_1\gamma^{[2]}_2u^{[22]}_2+{\bf H}^{[12]}{\bf V}^{[22]}_2u^{[22]}_2
\end{eqnarray}
\noindent The interfering symbols $u^{[21]}_2, u^{[22]}_2$ arrive along directions:
\begin{eqnarray}
u^{[21]}_2&\Rightarrow& {\bf H}^{[11]}{\bf V}^{[21]}_1\gamma^{[1]}_2+ {\bf H}^{[11]}{\bf V}^{[21]}_2 \\
u^{[22]}_2&\Rightarrow&{\bf H}^{[12]}{\bf V}^{[22]}_1\gamma^{[2]}_2+{\bf H}^{[12]}{\bf V}^{[22]}_2
\end{eqnarray}
The RHS of the above expressions are the $3\times 1$ vectors indicating the direction along which interference is seen by Receiver 1 from the two undesired symbols. For interference alignment we would like these directions to be co-linear.
\begin{eqnarray}
{\bf H}^{[11]}{\bf V}^{[21]}_1\gamma^{[1]}_2+ {\bf H}^{[11]}{\bf V}^{[21]}_2 &=&\beta\left({\bf H}^{[12]}{\bf V}^{[22]}_1\gamma^{[2]}_2+{\bf H}^{[12]}{\bf V}^{[22]}_2\right)
\end{eqnarray}
for some constant $\beta$. Equivalently
\begin{eqnarray}
\left[{\bf H}^{[11]}{\bf V}^{[21]}_1 ~~~ {\bf H}^{[11]}{\bf V}^{[21]}_2 ~~~ {\bf H}^{[12]}{\bf V}^{[22]}_1~~~{\bf H}^{[12]}{\bf V}^{[22]}_2\right]_{3\times 4}
\left[\begin{array}{c}
\gamma^{[1]}_2\\
1\\
-\beta\gamma^{[2]}_2\\
-\beta
\end{array}
\right]={\bf 0}_{3\times 1}
\end{eqnarray}
Since the matrix on the left is generic and of size $3\times 4$ it has a unique (upto scaling) null vector. The choice of the values of $\gamma^{[2]}_1, \gamma^{[2]}_2, \beta$ is made precisely to force the vector on the right to be this null vector, thus aligning interference.
Similarly, at Receiver 2, we have three equations in four information symbols:
\begin{eqnarray}
{\bf Y}^{[2]}-{\bf H}^{[21]}{\bf V}^{[11]}_1s^{[1]}_1-{\bf H}^{[22]}{\bf V}^{[12]}_1s^{[2]}_1-{\bf H}^{[21]}{\bf V}^{[21]}_1s^{[1]}_2-{\bf H}^{[22]}{\bf V}^{[22]}_1s^{[2]}_2=\nonumber\\
{\bf H}^{[21]}{\bf V}^{[11]}_1\gamma^{[1]}_1u^{[11]}_2+ {\bf H}^{[21]}{\bf V}^{[11]}_2 u^{[11]}_2+{\bf H}^{[22]}{\bf V}^{[12]}_1\gamma^{[2]}_1u^{[12]}_2+{\bf H}^{[22]}{\bf V}^{[12]}_2u^{[12]}_2\nonumber\\
{\bf H}^{[21]}{\bf V}^{[21]}_1\gamma^{[1]}_2u^{[21]}_2+ {\bf H}^{[21]}{\bf V}^{[21]}_2 u^{[21]}_2+{\bf H}^{[22]}{\bf V}^{[22]}_1\gamma^{[2]}_2u^{[22]}_2+{\bf H}^{[22]}{\bf V}^{[22]}_2u^{[22]}_2
\end{eqnarray}
and the interfering symbols $u^{[11]}_2, u^{[12]}_2$ arrive along the directions:
\begin{eqnarray}
u^{[11]}_2&\Rightarrow& {\bf H}^{[21]}{\bf V}^{[11]}_1\gamma^{[1]}_1+{\bf H}^{[21]}{\bf V}^{[11]}_2\\
u^{[12]}_2&\Rightarrow&{\bf H}^{[22]}{\bf V}^{[12]}_1\gamma^{[2]}_1+{\bf H}^{[22]}{\bf V}^{[12]}_2
\end{eqnarray}
Thus we would like
\begin{eqnarray}\label{eqn:null2}
\left[{\bf H}^{[21]}{\bf V}^{[11]}_1 ~~~ {\bf H}^{[21]}{\bf V}^{[11]}_2 ~~~ {\bf H}^{[22]}{\bf V}^{[12]}_1~~~{\bf H}^{[22]}{\bf V}^{[12]}_2\right]_{3\times 4}
\left[\begin{array}{c}
\gamma^{[1]}_1\\
1\\
-\delta\gamma^{[2]}_1\\
-\delta
\end{array}
\right]={\bf 0}_{3\times 1}
\end{eqnarray}
As before, we choose the values of $\gamma_1^{[1]}, \gamma^{[2]}_1, \delta$ to satisfy the equation above, and thereby achieve interference alignment at Receiver 2.
Lastly, we need to check that the desired symbols are not aligned either with each other or with the interference by this choice of the $\gamma^{[\star]}_\star$ constants.
Consider Receiver 1, where the desired symbols $u^{[11]}_2, u^{[12]}_2$ arrive along directions:
\begin{eqnarray}
u^{[11]}_2&\Rightarrow&{\bf H}^{[11]}{\bf V}^{[11]}_1\gamma^{[1]}_1+ {\bf H}^{[11]}{\bf V}^{[11]}_2\\
u^{[12]}_2&\Rightarrow&{\bf H}^{[12]}{\bf V}^{[12]}_1\gamma^{[2]}_1+{\bf H}^{[12]}{\bf V}^{[12]}_2
\end{eqnarray}
and the aligned interference arrives along the direction:
\begin{eqnarray}
{\bf H}^{[11]}{\bf V}^{[21]}_1\gamma^{[1]}_2+ {\bf H}^{[11]}{\bf V}^{[21]}_2
\end{eqnarray}
Thus, for desired symbols to be resolvable from the interference, we need the following $3\times 3$ matrix to have full rank:
\begin{eqnarray}
M_1&=&\left[{\bf H}^{[11]}{\bf V}^{[11]}_1\gamma^{[1]}_1+ {\bf H}^{[11]}{\bf V}^{[11]}_2 ~~~ {\bf H}^{[12]}{\bf V}^{[12]}_1\gamma^{[2]}_1+{\bf H}^{[12]}{\bf V}^{[12]}_2~~~~{\bf H}^{[11]}{\bf V}^{[21]}_1\gamma^{[1]}_2+ {\bf H}^{[11]}{\bf V}^{[21]}_2 \right]
\end{eqnarray}
Similarly we can define the $3\times 3$ matrix $M_2$ that also needs to be full rank for Receiver 2 to be able to obtain his desired symbols. Both conditions can be stated together in the following form:
\begin{eqnarray}
\det(M_1)\det(M_2)&\neq &0
\end{eqnarray}
However, both $\det(M_1)$ and $\det(M_2)$ correspond to polynomials in ${\bf V}^{[**]}, {\bf H}^{[**]}$ (the $\gamma$ can be evaluated in terms of ${\bf V}, {\bf H}$), and therefore, so does the product $\det(M_1)\det(M_2)$. Note that ${\bf V}, {\bf H}$ are picked independently over complex numbers. Therefore the polynomial $\det(M_1)\det(M_2)$ is either identically the zero polynomial, or it is non-zero almost surely for all realizations of ${\bf V}, {\bf H}$. To prove that it is non-zero almost surely, it suffices to show that it is not the zero polynomial, which in turn is established if there exists any non-zero evaluation of $\det(M_1)\det(M_2)$. Indeed this is easily verified by a numerical example. Since such examples are easy to find (almost all choices work fine) we will omit the explicit construction.
\section{Interference Channel with 3 Users}
Besides the $X$ channel \cite{MMK,Jafar_Shamai} and the MISO BC \cite{Weingarten_Shamai_Kramer}, the interference channel with more than 2 users is one of the earliest settings where interference alignment was first introduced \cite{Cadambe_Jafar_int}, and as such it is natural to ask if interference alignment is possible in this setting with only delayed CSIT? In this section we will study the interference channel with $3$ users. With full CSIT it is known that the $3$ user interference channel has $\frac{3}{2}$ DoF, which is higher than the $2$ user X channel's $\frac{4}{3}$ DoF. However, with delayed CSIT, because the transmitters are even more distributed in the $3$ user interference channel, it is not clear if it will continue to have a DoF advantage over the X channel. In this paper, we will show the achievability of only $\frac{9}{8}$ DoF for the $3$ user interference channel, which is less than the $\frac{8}{7}$ DoF that we are able to achieve for the $X$ channel with delayed CSIT. This is interesting because it shows that delayed CSIT is beneficial even in the $3$ user interference channel. It is also interesting because it raises the question -- whether the 3 user IC in fact pays a greater price in DoF than the X channel for delayed CSIT. The question remains wide open because the optimality of the schemes presented here is neither established nor conjectured.
The 3 user interference channel, shown in Fig. \ref{fig:3userIC}, consists of transmitters 1,2,3, who wish to communicate independent messages $W^{[1]}, W^{[2]},W^{[3]}$ to receivers 1,2,3 respectively. The assumptions regarding the delayed CSIT model are identical to the preceding sections, and the notation specific to the interference channel will become clear in the technical description of the proof.
\begin{figure}[!t]
\centering
\includegraphics[width=3.5in]{3userIC}
\caption{Interference Channel with 3 Users}
\label{fig:3userIC}
\end{figure}
The following theorem presents an achievability result for the DoF of the 3 user interference channel with delayed CSIT.
\begin{theorem}
The $3$ user interference channel with delayed CSIT, can achieve $\frac{9}{8}$ DoF almost surely.
\end{theorem}
\noindent\hspace{2em}{\it Proof: } In order to show the achievability of $\frac{9}{8}$ DoF, we will consider a $8$ symbol extension of the channel. Each user will send $3$ information symbols over these $8$ channel uses. At each receiver, in addition to the $3$ desired symbols, there are $6$ interfering symbols. Since the total number of dimensions is only $8$, one of the $6$ interfering symbols must align within the vector space spanned by the remaining $5$, to leave $3$ signal dimensions free of interference where the desired signals can be projected. Since we are again dealing with delayed CSIT and distributed transmitters, we will again use the retrospective interference alignment scheme. However, in this section, for the sake of providing a richer understanding of the scheme, we will arrive at it in an alternative fashion.
{\bf Phase I: }
As stated earlier, we wish that the 6 interfering symbols should span no more than $5$ dimensions. Since we have no instantaneous channel knowledge, let us start by sending random linear combinations of the symbols from each transmitter. Since interference is allowed to fill up $5$ dimensions, we can send $5$ random linear combinations of the symbols from each transmitter over the first $5$ channel uses without exceeding the quota of $5$ dimensions that are allowed to be spanned by interference. This is the end of Phase I. No special effort has been made to align anything so far, and we have exhausted the number of dimensions allowed for interference at each receiver.
At this point, consider the signal seen by Receiver 1 (ignoring noise as usual).
\begin{eqnarray*}
{\bf Y}^{[1]}&=&{\bf H}^{[11]}\left[{\bf V}^{[1]}_1 ~~{\bf V}^{[1]}_2~~{\bf V}^{[1]}_3\right]\left[\begin{array}{c}u^{[1]}_1\\u^{[1]}_2\\u^{[1]}_3\end{array}\right]+{\bf H}^{[12]}\left[{\bf V}^{[2]}_1 ~~{\bf V}^{[2]}_2~~{\bf V}^{[2]}_3\right]\left[\begin{array}{c}u^{[2]}_1\\u^{[2]}_2\\u^{[2]}_3\end{array}\right]+{\bf H}^{[13]}\left[{\bf V}^{[3]}_1 ~~{\bf V}^{[3]}_2~~{\bf V}^{[3]}_3\right]\left[\begin{array}{c}u^{[3]}_1\\u^{[3]}_2\\u^{[3]}_3\end{array}\right]
\end{eqnarray*}
Here ${\bf V}^{[k]}_i$ are the $5\times 1$ precoding vectors, ${\bf Y}^{[k]}$ is the $5\times 1$ vector of received signals so far, ${\bf H}^{[kj]}$ is the $5\times 5$ diagonal channel matrix representing the i.i.d. variations of the channel coefficient from Transmitter $j$ to Receiver $k$.
Consider the interference carrying vectors ${\bf V}^{[2]}_i, {\bf V}^{[3]}_i, i=1,2,3$, over the first $5$ channel uses. Since these six vectors are only in a five dimensional space, we can identify the $6\times 1$ null vector $\vec\alpha^{[1]}=[\alpha^{[1]}_1,\alpha^{[1]}_2,\cdots,\alpha^{[1]}_6]$ such that:
\begin{eqnarray}
\left[{\bf H}^{[12]}{\bf V}^{[2]}_1~~~{\bf H}^{[12]}{\bf V}^{[2]}_2~~~{\bf H}^{[12]}{\bf V}^{[2]}_3~~~{\bf H}^{[13]}{\bf V}^{[3]}_1~~~{\bf H}^{[13]}{\bf V}^{[3]}_2~~~{\bf H}^{[13]}{\bf V}^{[3]}_3\right]\vec\alpha^{[1]}={\bf 0}_{5\times 1}
\end{eqnarray}
Similarly, considering Receivers 2, and 3 respectively, we define null vectors $\vec{\alpha}^{[2]}, \vec\alpha^{[3]}$ so that
\begin{eqnarray}
\left[{\bf H}^{[21]}{\bf V}^{[1]}_1~~~{\bf H}^{[21]}{\bf V}^{[1]}_2~~~{\bf H}^{[21]}{\bf V}^{[1]}_3~~~{\bf H}^{[23]}{\bf V}^{[3]}_1~~~{\bf H}^{[23]}{\bf V}^{[3]}_2~~~{\bf H}^{[23]}{\bf V}^{[3]}_3\right]\vec\alpha^{[2]}={\bf 0}_{5\times 1}\\
\left[{\bf H}^{[31]}{\bf V}^{[1]}_1~~~{\bf H}^{[31]}{\bf V}^{[1]}_2~~~{\bf H}^{[31]}{\bf V}^{[1]}_3~~~{\bf H}^{[32]}{\bf V}^{[2]}_1~~~{\bf H}^{[32]}{\bf V}^{[2]}_2~~~{\bf H}^{[32]}{\bf V}^{[2]}_3\right]\vec\alpha^{[3]}={\bf 0}_{5\times 1}
\end{eqnarray}
{\bf Phase 2:} Phase 2 consists of the remaining $3$ channel uses. It is here that retrospective alignment will be used, based on the knowledge of the channel states from Phase I. No knowledge of channel states, not even delayed CSIT, will be used of the Phase-2 channels. Consider the $n^{th}$, $n=6,7,8$, transmission, i.e., any of the three transmissions of Phase 2. Suppose the transmitters choose to send the linear combinations:
\begin{eqnarray}
\mbox{Transmitter 1:}&\rightarrow& {\bf V}^{[1]}_1(n)u^{[1]}_1+{\bf V}^{[1]}_2(n)u^{[1]}_2+{\bf V}^{[1]}_3(n)u^{[1]}_3\\
\mbox{Transmitter 2:}&\rightarrow& {\bf V}^{[2]}_1(n)u^{[2]}_1+{\bf V}^{[2]}_2(n)u^{[2]}_2+{\bf V}^{[2]}_3(n)u^{[2]}_3\\
\mbox{Transmitter 3:}&\rightarrow& {\bf V}^{[3]}_1(n)u^{[3]}_1+{\bf V}^{[3]}_2(n)u^{[3]}_2+{\bf V}^{[3]}_3(n)u^{[3]}_3
\end{eqnarray}
The linear precoding coefficients ${\bf V}^{[k]}_i(n)$ can be chosen by the transmitters based on the delayed CSIT, i.e., the knowledge of the channel coefficients from Phase-I.
An important observation here is the following. In order to keep the interference contained in a 5 dimensional space at each receiver, the Phase-2 precoding coefficients must follow the same linear relationship as established in Phase-I. Mathematically, at Receiver 1:
\begin{eqnarray*}
\left[{\bf H}^{[12]}(n){\bf V}^{[2]}_1(n)~~~{\bf H}^{[12]}(n){\bf V}^{[2]}_2(n)~~~{\bf H}^{[12]}(n){\bf V}^{[2]}_3(n)~~~{\bf H}^{[13]}(n){\bf V}^{[3]}_1(n)~~~{\bf H}^{[13]}(n){\bf V}^{[3]}_2(n)~~~{\bf H}^{[13]}(n){\bf V}^{[3]}_3(n)\right]\vec\alpha^{[1]}=0
\end{eqnarray*}
and similarly at Receivers 2, 3:
\begin{eqnarray*}
\left[{\bf H}^{[21]}(n){\bf V}^{[1]}_1(n)~~~{\bf H}^{[21]}(n){\bf V}^{[1]}_2(n)~~~{\bf H}^{[21]}(n){\bf V}^{[1]}_3(n)~~~{\bf H}^{[23]}(n){\bf V}^{[3]}_1(n)~~~{\bf H}^{[23]}(n){\bf V}^{[3]}_2(n)~~~{\bf H}^{[23]}(n){\bf V}^{[3]}_3(n)\right]\vec\alpha^{[2]}=0\\
\left[{\bf H}^{[31]}(n){\bf V}^{[1]}_1(n)~~~{\bf H}^{[31]}(n){\bf V}^{[1]}_2(n)~~~{\bf H}^{[31]}(n){\bf V}^{[1]}_3(n)~~~{\bf H}^{[32]}(n){\bf V}^{[2]}_1(n)~~~{\bf H}^{[32]}(n){\bf V}^{[2]}_2(n)~~~{\bf H}^{[32]}(n){\bf V}^{[2]}_3(n)\right]\vec\alpha^{[3]}=0
\end{eqnarray*}
Since the current channel states ${\bf H}^{[\star\star]}(n)$ are not known to the transmitters, the only way to guarantee the above equations for all current channel realizations is to choose ${\bf V}^{[k]}_i$ so that
\begin{eqnarray}
{\bf V}^{[1]}_1(n)\alpha^{[2]}_1+{\bf V}^{[1]}_2(n)\alpha^{[2]}_2+{\bf V}^{[1]}_3(n)\alpha^{[2]}_3=0\label{eq:v11}\label{eq:alignfirst}\\
{\bf V}^{[1]}_1(n)\alpha^{[3]}_1+{\bf V}^{[1]}_2(n)\alpha^{[3]}_2+{\bf V}^{[1]}_3(n)\alpha^{[3]}_3=0\label{eq:v12}\\
{\bf V}^{[2]}_1(n)\alpha^{[1]}_1+{\bf V}^{[2]}_2(n)\alpha^{[1]}_2+{\bf V}^{[2]}_3(n)\alpha^{[1]}_3=0\\
{\bf V}^{[2]}_1(n)\alpha^{[3]}_4+{\bf V}^{[2]}_2(n)\alpha^{[3]}_5+{\bf V}^{[2]}_3(n)\alpha^{[3]}_6=0\\
{\bf V}^{[3]}_1(n)\alpha^{[1]}_4+{\bf V}^{[3]}_2(n)\alpha^{[1]}_5+{\bf V}^{[3]}_3(n)\alpha^{[1]}_6=0\\
{\bf V}^{[3]}_1(n)\alpha^{[2]}_4+{\bf V}^{[3]}_2(n)\alpha^{[2]}_5+{\bf V}^{[3]}_3(n)\alpha^{[2]}_6=0\label{eq:alignlast}
\end{eqnarray}
Now consider the precoding coefficients ${\bf V}^{[1]}_i(n), i=1,2,3$ that are to be chosen by Transmitter 1. Based on equations (\ref{eq:v11}), (\ref{eq:v12}), we can express ${\bf V}^{[1]}_2(n)$ and ${\bf V}^{[1]}_3(n)$ as linear functions of ${\bf V}^{[1]}_1(n)$, say
\begin{eqnarray}
{\bf V}^{[1]}_2(n)&=&\frac{\left|\begin{array}{rr}-\alpha^{[2]}_1&\alpha^{[2]}_3\\-\alpha^{[3]}_1&\alpha^{[3]}_3\end{array}\right|}{\left|\begin{array}{rr}\alpha^{[2]}_2&\alpha^{[2]}_3\\\alpha^{[3]}_2&\alpha^{[3]}_3\end{array}\right|}{\bf V}^{[1]}_1(n)\\
{\bf V}^{[1]}_3(n)&=&\frac{\left|\begin{array}{rr}\alpha^{[2]}_2&-\alpha^{[2]}_1\\\alpha^{[3]}_2&-\alpha^{[3]}_1\end{array}\right|}{\left|\begin{array}{rr}\alpha^{[2]}_2&\alpha^{[2]}_3\\\alpha^{[3]}_2&\alpha^{[3]}_3\end{array}\right|}{\bf V}^{[1]}_1(n)
\end{eqnarray}
Thus, Transmitter 1 is forced to send:
\begin{eqnarray}
{\bf X}^{[1]}(n)&=& {\bf V}^{[1]}_1(n)u^{[1]}_1+{\bf V}^{[1]}_2(n)u^{[1]}_2+{\bf V}^{[1]}_3(n)u^{[1]}_3\\
&=&c\underbrace{\left((\alpha^{[2]}_2\alpha^{[3]}_3-\alpha^{[2]}_3\alpha^{[3]}_2)u^{[1]}_1+(\alpha^{[2]}_3\alpha^{[3]}_1-\alpha^{[3]}_3\alpha^{[2]}_1)u^{[1]}_2+(\alpha^{[2]}_1\alpha^{[3]}_2-\alpha^{[2]}_2\alpha^{[3]}_1)u^{[1]}_3\right)}_{s^{[1]}}\nonumber
\end{eqnarray}
where $c$ is any constant, and without loss of generality we can set it to unity. The new information variable
\begin{eqnarray}
s^{[1]}=(\alpha^{[2]}_2\alpha^{[3]}_3-\alpha^{[2]}_3\alpha^{[3]}_2)u^{[1]}_1+(\alpha^{[2]}_3\alpha^{[3]}_1-\alpha^{[3]}_3\alpha^{[2]}_1)u^{[1]}_2+(\alpha^{[2]}_1\alpha^{[3]}_2-\alpha^{[2]}_2\alpha^{[3]}_1)u^{[1]}_3
\end{eqnarray}
is precisely our Phase-2 variable, available only to Transmitter 1, and composed of variables only available to Transmitter 1. Thus, in Phase-2, even though the transmitter has three information symbols to send, it can only send scaled versions of the same effective scalar symbol $s^{[1]}$ in order to keep the interference aligned within $5$ dimensions at each receiver. Similarly, we can define the effective variables $s^{[2]}$ and $s^{[3]}$ to be sent by transmitters $2$ and $3$ over Phase 2.
\begin{eqnarray}
s^{[2]}=(\alpha^{[1]}_2\alpha^{[3]}_6-\alpha^{[1]}_3\alpha^{[3]}_5)u^{[2]}_1+(\alpha^{[3]}_4\alpha^{[1]}_3-\alpha^{[3]}_6\alpha^{[1]}_1)u^{[2]}_2+(\alpha^{[1]}_1\alpha^{[3]}_5-\alpha^{[1]}_2\alpha^{[3]}_4)u^{[2]}_3\\
s^{[3]}=(\alpha^{[1]}_5\alpha^{[2]}_6-\alpha^{[1]}_6\alpha^{[2]}_5)u^{[3]}_1+(\alpha^{[1]}_6\alpha^{[2]}_4-\alpha^{[2]}_6\alpha^{[1]}_4)u^{[3]}_2+(\alpha^{[1]}_4\alpha^{[2]}_5-\alpha^{[2]}_4\alpha^{[1]}_5)u^{[3]}_3
\end{eqnarray}
Since there are only $3$ Phase-2 symbols, and there are $3$ channel uses, the operation over Phase-2 can be simply interpreted as each transmitter repeating its own effective information symbol, so that the channel variations provide each receiver with a different linear combination of the $3$ effective Phase-2 symbols each time, so that at the end of Phase-2, each receiver is able to decode all three symbols $s^{[1]}, s^{[2]}, s^{[3]}$.
Thus, we have completely determined the resulting precoding vectors sent over the $8$ symbols. Putting everything together, the transmitted symbols from e.g., Transmitter 1 are shown below:
\begin{eqnarray}
\left[\begin{array}{c}
{\bf X}^{[1]}(1)\\
{\bf X}^{[1]}(2)\\
{\bf X}^{[1]}(3)\\
{\bf X}^{[1]}(4)\\
{\bf X}^{[1]}(5)\\
{\bf X}^{[1]}(6)\\
{\bf X}^{[1]}(7)\\
{\bf X}^{[1]}(8)
\end{array}\right]&=&
\left[\begin{array}{ccc}
{\bf V}^{[1]}_1(1)&{\bf V}^{[1]}_2(1)&{\bf V}^{[1]}_3(1)\\
{\bf V}^{[1]}_1(2)&{\bf V}^{[1]}_2(2)&{\bf V}^{[1]}_3(2)\\
{\bf V}^{[1]}_1(3)&{\bf V}^{[1]}_2(3)&{\bf V}^{[1]}_3(3)\\
{\bf V}^{[1]}_1(4)&{\bf V}^{[1]}_2(4)&{\bf V}^{[1]}_3(4)\\
{\bf V}^{[1]}_1(5)&{\bf V}^{[1]}_2(5)&{\bf V}^{[1]}_3(5)\\
\alpha^{[2]}_2\alpha^{[3]}_3-\alpha^{[2]}_3\alpha^{[3]}_2&
\alpha^{[2]}_3\alpha^{[3]}_1-\alpha^{[3]}_3\alpha^{[2]}_1&
\alpha^{[2]}_1\alpha^{[3]}_2-\alpha^{[2]}_2\alpha^{[3]}_1\\
\alpha^{[2]}_2\alpha^{[3]}_3-\alpha^{[2]}_3\alpha^{[3]}_2&
\alpha^{[2]}_3\alpha^{[3]}_1-\alpha^{[3]}_3\alpha^{[2]}_1&
\alpha^{[2]}_1\alpha^{[3]}_2-\alpha^{[2]}_2\alpha^{[3]}_1\\
\alpha^{[2]}_2\alpha^{[3]}_3-\alpha^{[2]}_3\alpha^{[3]}_2&
\alpha^{[2]}_3\alpha^{[3]}_1-\alpha^{[3]}_3\alpha^{[2]}_1&
\alpha^{[2]}_1\alpha^{[3]}_2-\alpha^{[2]}_2\alpha^{[3]}_1
\end{array}\right]\left[\begin{array}{c}u^{[1]}_1\\u^{[1]}_2\\u^{[1]}_3\end{array}\right]
\end{eqnarray}
The first 5 channel uses correspond to Phase 1. All these precoding coefficients ${\bf V}^{[1]}_i(n)$ are chosen independently, randomly, before the beginning of communication and with no knowledge of CSIT. The last three channel uses correspond to Phase 2, and can be easily seen to be repetitions of the Phase-2 variable $s^{[1]}$. The transmitted symbols for all other transmitters can be described similarly. Interference alignment is accomplished because this choice of precoding vectors satisfies equations (\ref{eq:alignfirst})-(\ref{eq:alignlast}).
Keeping the interference aligned within $5$ dimensions at each receiver, allows the receiver to null out the $5$ interference dimensions and recover the $3$ desired symbols from the remaining $3$ dimensions from the overall $8$ dimensional vector space. Once again, while the construction above guarantees that interference is restricted within $5$ dimensions, one must also show that the desired signal vectors are not aligned within the interference or aligned among themselves. This is proven as before, by constructing the $8\times 8$ matrix consisting of 3 desired signal vectors and 5 interference vectors that span the interference space received at each receiver, and showing that the determinant of this matrix, which is equivalent to a polynomial in Phase 1 variables, is not identically a zero polynomial. While we have established this through numerical evaluations, the details of an explicit numerical example are omitted here (because almost all examples work).
\section{Delayed Output Feedback}
So far we assumed that the only feedback available to the transmitters consists of delayed CSIT. Another commonly studied model for feedback is channel \emph{output} feedback (without explicitly providing the CSI). In this section we study the X channel and the $3$ user interference channel with delayed \emph{output} feedback, i.e., the channel output is available to the transmitters only after the channel state associated with the observed output is no longer current. While delayed CSIT created difficulties because of the transmitters' inability to reconstruct the previously received linear combinations of undesired received symbols because of the distributed nature of the information, delayed output feedback automatically provides the transmitters with information that has been previously observed at one of the receivers. Retransmitting this information provides the transmitters an opportunity to provide new observations to the receivers who desire this information, while allowing the receivers who have already observed this interference to cancel it entirely. In this sense, delayed output feedback allows a direct extension of the alignment techniques explored in \cite{Maddah_Tse}.
\subsection{X Channel with Delayed Output Feedback}
\begin{theorem}
The X channel with delayed output feedback can achieve $\frac{4}{3}$ DoF almost surely.
\end{theorem}
\noindent\hspace{2em}{\it Proof: }
In light of the earlier discussion on the vector BC, it is easily seen that the X channel, with only delayed channel output feedback can easily achieve the $\frac{4}{3}$ DoF outer bound. Since the achievable scheme is essentially the same as the schemes studied \cite{Jafar_corr} and \cite{Maddah_Tse}, we only present a brief description.
\begin{itemize}
\item Send two symbols to each receiver over three time slots to achieve $\frac{4}{3}$ DoF.
\item In the first time slot, each transmitter sends its own symbol intended for Receiver 1. Receiver 1 observes a linear combination (along with noise) of the two desired symbols, while Receiver 2 sees a linear combination of undesired symbols (and noise).
\item The second time slot is similar to the first time slot, except the information symbols transmitted are for User 2.
\item In the third time slot, the transmitters send a superposition of the previous undesired outputs. This is possible due to delayed output feedback as long as each undesired output signal is available to one of the transmitters.
\end{itemize}
The third time slot provides each receiver with a second linear combination of desired symbols while the interfering undesired information is cancelled because it has been received previously. Note that we are assuming that each receiver has \emph{global} channel knowledge, i.e., it knows not only the channels associated with itself but also the other receivers' channels as well. Further we are once again ignoring noise in this discussion because, as stated earlier, for such linear beamforming schemes, noise does not affect the DoF.
\subsection{3 User Interference Channel}
The following theorem presents an achievability result for the 3 user interference channel.
\begin{theorem}
The 3 user interference channel with delayed output feedback available from each receiver to only its corresponding transmitter can achieve $\frac{6}{5}$ DoF almost surely.
\end{theorem}
{\it Remark:} While it is remarkable that the achievability results presented in this work for both the X channel and the 3 user interference channel under delayed output feedback correspond to higher DoF than with delayed CSIT, it should be noted that these are only achievability results and in the absence of outer bounds it is not possible to make categorical comparisons between the two settings.
\noindent\hspace{2em}{\it Proof: }
In order to achieve $\frac{6}{5}$ DoF we will operate over a $5$ channel-use block. Each user will communicate two coded information symbols over these $5$ channel uses using linear schemes that can be simply seen as swapping output symbols to help resolve desired signals \cite{Maddah_Tse}. A summary of the transmission scheme is described below.
\begin{enumerate}
\item Over the first time slot, Transmitter 1 sends its first information symbol $u^{[1]}_1$ and Transmitter 2 simultaneously sends its first information symbol $u^{[2]}_1$. Ignoring noise, the received signals are described below:
\begin{eqnarray}
\mbox{Receiver 1: }Y^{[1]}(1)&=&H^{[11]}(1)u^{[1]}_1+H^{[12]}(1)u^{[2]}_1\\
\mbox{Receiver 2: }Y^{[2]}(1)&=&H^{[21]}(1)u^{[1]}_1+H^{[22]}(1)u^{[2]}_1\\
\mbox{Receiver 3: }Y^{[3]}(1)&=&H^{[31]}(1)u^{[1]}_1+H^{[32]}(1)u^{[2]}_1
\end{eqnarray}
\item Over the second time slot, Transmitter 1 sends its second information symbol $u^{[1]}_2$ and Transmitter 3 simultaneously sends its first information symbol $u^{[3]}_1$. Ignoring noise, the received signals are described below:
\begin{eqnarray}
\mbox{Receiver 1: }Y^{[1]}(2)&=&H^{[11]}(2)u^{[1]}_1+H^{[13]}(2)u^{[3]}_1\\
\mbox{Receiver 2: }Y^{[2]}(2)&=&H^{[21]}(2)u^{[1]}_1+H^{[23]}(2)u^{[3]}_1\\
\mbox{Receiver 3: }Y^{[3]}(2)&=&H^{[31]}(2)u^{[1]}_1+H^{[33]}(2)u^{[3]}_1
\end{eqnarray}
\item Over the third time slot, Transmitter 2 sends its second information symbol $u^{[2]}_2$ and Transmitter 3 simultaneously sends its second information symbol $u^{[3]}_2$. Ignoring noise, the received signals are described below:
\begin{eqnarray}
\mbox{Receiver 1: }Y^{[1]}(3)&=&H^{[11]}(3)u^{[2]}_2+H^{[12]}(3)u^{[3]}_2\\
\mbox{Receiver 2: }Y^{[2]}(3)&=&H^{[21]}(3)u^{[2]}_2+H^{[22]}(3)u^{[3]}_2\\
\mbox{Receiver 3: }Y^{[3]}(3)&=&H^{[31]}(3)u^{[2]}_2+H^{[32]}(3)u^{[3]}_2
\end{eqnarray}
\item Over the fourth time slot, Transmitter 3 retransmits $Y^{[3]}(1)$ and Transmitter 2 retransmits $Y^{[2]}(2)$.
\item Over the fifth time slot, $Y^{[3]}(1)$ and $Y^{[1]}(3)$ are retransmitted from Transmitters $3$ and $1$ respectively.
\end{enumerate}
Next we explain how every receiver has enough information to recover its two desired symbols.
{\bf Receiver 1:} Consider Receiver 1. From the linear combination of $Y^{[3]}(1)$ and $Y^{[1]}(3)$ received over the fifth symbol, this receiver is able to remove $Y^{[1]}(3)$ which it has previously received, to obtain $Y^{[3]}(1)$. Combining $Y^{[1]}(1)$ and $Y^{[3]}(1)$ the receiver has enough information to resolve the first received symbol $u^{[1]}_1$.
Further, from the linear combination of $Y^{[3]}(1), Y^{[2]}(2)$ received over the fourth symbol, the receiver removes $Y^{[3]}(1)$ to obtain a clean $Y^{[2]}(2)$. Combining $Y^{[2]}(2)$ with $Y^{[1]}(2)$, the receiver is able to resolve the second desired symbol $u^{[1]}_2$.
{\bf Receiver 2:} Consider Receiver 2. From the linear combination of $Y^{[3]}(1), Y^{[2]}(2)$ received over the fourth symbol, this receiver is able to remove $Y^{[2]}(2)$ which it has previously received, to obtain $Y^{[3]}(1)$. Combining $Y^{[2]}(1)$ and $Y^{[3]}(1)$ the receiver has enough information to resolve the first desired symbol $u^{[2]}_1$.
Further, from the linear combination of $Y^{[3]}(1)$ and $Y^{[1]}(3)$ received over the fifth channel use, the receiver removes $Y^{[3]}(1)$ to obtain a clean $Y^{[1]}(3)$. Combining $Y^{[1]}(3)$ with $Y^{[2]}(3)$, the receiver is able to resolve the second desired symbol $u^{[2]}_2$.
{\bf Receiver 3:} Consider Receiver 3. From the linear combination of $Y^{[3]}(1)$ and $Y^{[2]}(2)$ received over the fourth symbol, this receiver is able to remove $Y^{[3]}(1)$ which it has previously received, to obtain $Y^{[2]}(2)$. Combining $Y^{[2]}(2)$ and $Y^{[3]}(2)$ the receiver has enough information to resolve the first received symbol $u^{[3]}_1$.
Further, from the linear combination of $Y^{[1]}(3), Y^{[3]}(1)$ received over the fifth symbol, the receiver removes $Y^{[3]}(1)$ to obtain a clean $Y^{[1]}(3)$. Combining $Y^{[1]}(3)$ with $Y^{[3]}(3)$, the receiver is able to resolve the second desired symbol $u^{[3]}_2$.
Thus, all symbols are resolved and $\frac{6}{5}$ DoF are achieved on the 3 user interference channel.
\section{Conclusion}
We explored similarities, differences, and the apparent difficulties in achieving interference alignment with channel uncertainty at the transmitters based on recent works that assume two different channel uncertainty models -- staggered block fading and delayed CSIT. While there are many shared aspects that allow the schemes to be translated from one setting to another for many cases, overall the two settings are indeed fundamentally different and face different challenges. In particular, the delayed CSIT setting appears to be more sensitive to whether the transmitters are co-located or distributed, unlike previous results where both for compound channels and suitably staggered block fading models the two were found to be equivalent from a DoF perspective. While the $2$ user MISO BC with delayed CSIT easily achieves the outer bound of $\frac{4}{3}$, it is not known if the same DoF can be achieved on the X channel, i.e., without cooperation between transmitters. We were able to show that delayed CSIT is still useful in the X channel from a DoF perspective, as one can achieve $\frac{8}{7}$ DoF. The achievability was shown here using an interesting new scheme that we call retrospective interference alignment. While the scheme operates in two phases, and with two layers of variables as the scheme proposed in \cite{Maddah_Tse}, the novelty of retrospective alignment appears in the construction of auxiliary (layer - 2 in the terminology of \cite{Maddah_Tse}) variables that aid in the alignment of the previously transmitted information symbols based on only the information symbols available to each transmitter. The same scheme was used to prove the achievability of 9/8 DoF for the 3 user interference channel with delayed CSIT. We also found that the X channel and the 3 user interference channel can achieve $4/3$ and $6/5$ DoF respectively when delayed output feedback is available to the transmitters. It is remarkable that with perfect and instantaneous CSIT, output feedback does not increase DoF for X networks or interference channels \cite{Cadambe_Jafar_XFB}.
Another issue that is of both theoretical and practical interest is the feedback rate. The delayed feedback models discussed in this work could be seen as essentially noiseless, infinite capacity feedback links. It is also clear that the benefits of feedback will be retained if the accuracy of feedback information is scaled appropriately with SNR \cite{Jindal, Caire_Jindal_Shamai}. An interesting question to make further progress in this direction would be to explore how a delayed feedback link whose capacity is itself limited in DoF becomes a bottleneck on the forward channel capacity. In conclusion, the results reported here only scratch the surface and much more remains to be done in order to understand the true potential for interference alignment in delayed feedback settings.
|
{
"timestamp": "2010-09-21T02:01:11",
"yymm": "1009",
"arxiv_id": "1009.3593",
"language": "en",
"url": "https://arxiv.org/abs/1009.3593"
}
|
\section{Introduction} \label{s1}
Berezin-Toeplitz operators are important in geometric quantization and
the properties of their kernels turn out to be
deeply related to various problems in K{\"a}hler geometry
(see e.g.\ \cite{Fine08,Fine10}). In this paper,
we will study the precise asymptotic expansion of these kernels.
We refer the reader to the book \cite{MM07}
for a comprehensive study of the Bergman kernel,
Berezin-Toeplitz quantization and its applications.
See also the survey \cite{Ma10}.
The setting of Berezin-Toeplitz quantization
on K{\"a}hler manifolds is the following.
Let $(X,\omega, J)$ be a compact K{\"a}hler manifold of
$\dim_{\field{C}}X=n$ with K{\"a}hler form $\omega$ and complex structure $J$.
Let $(L, h^L)$ be a holomorphic Hermitian line bundle on $X$,
and let $(E, h^E)$ be a holomorphic Hermitian vector bundle on $X$.
Let $\nabla ^L, \nabla^E$ be the holomorphic Hermitian
connections on $(L,h^L)$, $(E, h^E)$ with curvatures
$R^L=(\nabla^L)^2$, $R^E=(\nabla^E)^2$, respectively.
We assume that $(L,h^L,\nabla^L)$ is a prequantum line bundle, i.e.,
\begin{align} \label{toe2.1}
\omega= \frac{\sqrt{-1}}{2 \pi} R^L.
\end{align}
Let $g^{TX}(\cdot,\cdot):= \omega(\cdot,J\cdot)$ be the Riemannian
metric on $TX$
induced by $\omega$ and $J$. The Riemannian volume
form $dv_X$ of $(X, g^{TX})$ has the form $dv_X= \omega^n/n!$.
The $L^2$--Hermitian product on the space
$\mathscr{C}^\infty(X,L^p\otimes E)$
of smooth sections of $L^p\otimes E$ on $X$,
with $L^{p}:=L^{\otimes p}$, is given by
\begin{equation}\label{toe2.2}
\langle s_1,s_2\rangle=\int_X\langle s_1,s_2\rangle(x)\,dv_X(x)\,.
\end{equation}
We denote the corresponding norm by $\norm{\cdot}_{L^2}$ and
by $L^2(X,L^p\otimes E)$ the completion of
$\mathscr{C}^\infty(X,L^p\otimes E)$ with respect to this norm.
Given a continuous smoothing linear operator
$K:L^2(X,L^p\otimes E)\longrightarrow L^2(X,L^p\otimes E)$,
the Schwartz kernel theorem \cite[Th.\,B.2.7]{MM07}
guarantees the existence of an integral kernel with respect to $dv_X$,
denoted by $K(x,x')\in(L^p\otimes E)_x\otimes(L^p\otimes E)^*_{x'}$,
for $x,x'\in X$, i.e.,
\begin{equation}\label{toe2.21}
(KS)(x)=\int_X K(x,x')S(x')\,dv_X(x')\,,\quad S\in L^2(X,L^p\otimes E)\,.
\end{equation}
Consider now the space $H^0(X,L^p\otimes E)$ of holomorphic sections
of $L^p\otimes E$ on $X$ and let
$P_p:L^2(X,L^p\otimes E)\to H^0(X,L^p\otimes E)$
be the orthogonal (Bergman) projection.
Its kernel $P_{p}(x,x')$ with respect to $dv_X(x')$ is smooth;
it is called the {\em Bergman kernel}.
The {\em Berezin-Toeplitz quantization}
of a section $f\in \mathscr{C}^\infty(X, \End(E))$
is the {\em Berezin-Toeplitz operator} $\{T_{f,p}\}_{p\in \field{N}}$
which is a sequence of linear operators $T_{f,\,p}$ defined by
\begin{equation}\label{toe2.4}
T_{f,\,p}:L^2(X,L^p\otimes E)\longrightarrow L^2(X,L^p\otimes E)\,,
\quad T_{f,\,p}=P_p\,f\,P_p\,.
\end{equation}
The kernel $T_{f,\,p}(x,x')$ of $T_{f,\,p}$ with respect to
$dv_X(x')$ is also smooth. Since $\End(L)=\field{C}$,
we have $T_{f,\,p}(x,x)\in \End(E)_{x}$ for $x\in X$.
We introduce now the relevant geometric objects used in
Theorems \ref{toet4.1}, \ref{toet4.6} and \ref{toet4.5}.
Let $T^{(1,0)}X$ be the holomorphic tangent bundle on $X$,
and $T^{*(1,0)}X$ its dual bundle.
Let $\nabla^{TX}$ be the Levi-Civita connection on $(X, g^{TX})$. We denote by
$R^{TX}=(\nabla^{TX})^2$ the curvature, by $\ric$ the Ricci curvature and
by $\br$ the scalar curvature of $\nabla^{TX}$ (cf.\,\eqref{alm01.1}).
We still denote by $\nabla ^{E}$ the connection on
$\End (E)$ induced by $\nabla ^E$.
Consider
the (positive) Laplacian $\Delta$
acting on the functions on $(X, g^{TX})$ and the
Bochner Laplacian $\Delta^{E}$
on $\mathscr{C}^{\infty}(X, E)$ and on $\mathscr{C}^{\infty}(X, \End(E))$.
Let $\{e_k\}$ be a (local) orthonormal frame of $(TX, g^{TX})$. Then
\begin{align} \label{toe2.3}
\Delta^{E} = - \sum_k (\nabla^{E}_{e_{k}}\nabla^{E}_{e_{k}}
- \nabla^{E}_{\nabla^{TX}_{e_{k}}e_{k}}).
\end{align}
Let $\Omega^{q,\,r}(X, \End(E))$ be the space of $(q,r)$-forms on $X$
with values in $\End(E)$, and let
\begin{align} \label{abk2.5}
\nabla^{1,0}:\Omega^{q,*}(X, \End(E))\to \Omega^{q+1,*}(X,\End(E))
\end{align}
be the $(1,0)$-component of the connection $\nabla^E$.
Let $(\nabla^{E})^*$, $\nabla^{1,0*}, \overline{\partial}^{E*}$
be the adjoints of $\nabla^{E}$, $\nabla^{1,0}, \overline{\partial}^{E}$,
respectively.
Let $D^{1,0}, D^{0,1}$ be the $(1,0)$ and $(0,1)$ components of the
connection
$\nabla^{T^{*}X}: \mathscr{C}^\infty(X,T^* X)\to \mathscr{C}^\infty(X,T^* X\otimes T^*X)$
induced by $\nabla^{TX}$.
In the following, we denote by
\[
\langle\cdot\,,\cdot \rangle_{\omega}:
\Omega^{*,\,*}(X,\End(E))\times\Omega^{*,\,*}(X,\End(E))
\to\mathscr{C}^\infty(X,\End(E))
\]
the $\field{C}$-bilinear pairing
$\langle\alpha\otimes f,\beta\otimes g \rangle_{\omega}
=\langle\alpha,\beta\rangle f\cdot g$,
for forms $\alpha,\beta\in\Omega^{*,\,*}(X)$ and sections
$f,g\in\mathscr{C}^\infty(X,\End(E))$ (cf. \eqref{abk4.4},
\eqref{toe4.31}, \eqref{toe4.32}). Put
\begin{align} \label{bk2.5}
\begin{split}
&R^E_{\Lambda}
=\left \langle R^E, \omega\right \rangle_{\omega}\, .
\end{split}\end{align}
Let $\ric_\omega= \ric(J\cdot,\cdot)$
be the $(1,1)$-form associated to $\ric$.
Set
\[
|\ric_\omega|^2 = \sum_{i<j}\ric_\omega(e_{i},e_{j})^{2}\,,
\quad |R^{TX}|^2 = \sum_{i<j}\sum_{k<l} \langle
R^{TX}(e_{i},e_{j})e_{k},e_{l}\rangle ^{2},
\]
and let
\begin{equation}\label{abk2.6}\begin{split}
\bb_{2\field{C}}= & - \frac{\Delta \br}{48}+ \frac{1}{96}|R^{TX}|^2
- \frac{1}{24} |\ric_\omega|^2 + \frac{1}{128} \br^2,\\
\bb_{2E}=&\frac{\sqrt{-1}}{32} \Big(2 \br R^E_{\Lambda}
- 4 \langle\ric_\omega, R^ E\rangle_{\omega} + \Delta^E R^E_{\Lambda}\Big) \\
&- \frac{1}{8} (R^E_{\Lambda})^2 +
\frac{1}{8}\langle R^ E, R^ E\rangle_{\omega}
+ \frac{3}{16} \overline{\partial}^{E*}\nabla^{1,0*} R^ E,\\
\bb_{1} =& \frac{\br}{8\pi}
+ \frac{\sqrt{-1}}{2\pi}R^E_{\Lambda},\quad \,
\bb_{2} = \frac{1}{\pi^2}(\bb_{2\field{C}} + \bb_{2E}).
\end{split}\end{equation}
We use now the notation from \eqref{lm01.4}.
By our convention (cf.\;\eqref{lm01.3}), we have at $x_{0}\in X$,
\[\big\langle\alpha_{\ell\overline{m}}\, dz_\ell\wedge d\overline{z}_m,\beta_{k\overline{q}}\,
dz_k\wedge d\overline{z}_q\big\rangle
= -4 \alpha_{\ell\overline{m}}\,\beta_{m\overline{\ell}}\;\;,\quad
\big\langle \alpha_{\overline{m}\,\overline{q}}\, d\overline{z}_m\otimes d\overline{z}_q,
\beta_{k\ell}\, dz_k\otimes dz_\ell\big\rangle
= 4 \alpha_{\overline{m}\,\overline{q}}\, \beta_{mq}
\]
(note that $|dz_q|^2=2$).
Then by Lemma \ref{lmt1.6}, \eqref{abk4.3} and \eqref{bk4.2a},
we have at $x_{0}\in X$,\footnote{
The Bianchi identity reads $[\nabla^E, R^E]=0$. Take the derivative of
$[\nabla^E, R^E]( \widetilde{\tfrac{\partial}{\partial z_i}},
\widetilde{\tfrac{\partial}{\partial z_k}},
\widetilde{\tfrac{\partial}{\partial \overline{z}_k}})=0,$
and use \eqref{lm01.5}, \eqref{lm01.27}, \eqref{bk2.85}
to obtain $[\widetilde{\tfrac{\partial}{\partial z_i}},
\widetilde{\tfrac{\partial}{\partial \overline{z}_k}}]=
-\frac{1}{2}R^{TX}_{x_{0}}(\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_k}) \mathcal{R} + \mathscr{O}(|Z|^{2})$,
$[\widetilde{\tfrac{\partial}{\partial z_i}},
\widetilde{\tfrac{\partial}{\partial z_k}}]=\mathscr{O}(|Z|^{2})$.
We conclude that
$$\quad R^E_{m\overline{k}\,;\,k\overline{m}}
= R^E_{k\overline{k}\,;\,m\overline{m}}\,.$$
The term $\frac{1}{4}\left(-R^E_{k\overline{k}\,;\,m\overline{m}}
+3 R^E_{m\overline{k}\,;\,k\overline{m}}\right)$ in (\ref{bk2.6})
can be thus replaced by
$\frac{1}{2}R^E_{k\overline{k}\,;\,m\overline{m}}$\,. Equivalently,
by using (\ref{bk4.3a}),
one can replace
$+\frac{\sqrt{-1}}{32}\Delta^E R^E_{\Lambda}
+ \frac{3}{16} \overline{\partial}^{E*}\nabla^{1,0*} R^ E$
in (\ref{abk2.6}) by
$- \frac{\sqrt{-1}}{16}\Delta^E R^E_{\Lambda}.$}
\begin{equation}\label{bk2.6}\begin{split}
&\ric_{\ell\overline{k}}=2R_{\ell\overline{k}q\overline{q}}=2R_{\ell\overline{q}q\overline{k}}\,,
\quad \br=8R_{\ell\overline{\ell}q\overline{q}},\quad
\ric_\omega= \sqrt{-1} \ric_{\ell\overline{k}}\, dz_\ell\wedge d\overline{z}_k\,, \\
&\sqrt{-1}R^E_{\Lambda}= 2 R^E_{k\overline{k}}, \quad
\bb_{1} = \frac{1}{\pi}\left( R_{k\overline{k} m\overline{m}}+ R^E_{m\overline{m}}\right)\,,\\
&\bb_{2\field{C}}= - \frac{\Delta \br}{48}
+ \frac{1}{6} R_{k\overline{\ell} m\overline{q}} R_{\ell\overline{k}q\overline{m}}
- \frac{2}{3} R_{\ell\overline{\ell} m\overline{q}} R_{k\overline{k}q\overline{m}}
+ \frac{1}{2} R_{\ell\overline{\ell}q\overline{q}} R_{k\overline{k}m\overline{m}} \,,\\
&\bb_{2E}= R^E_{q\overline{q}} R_{k\overline{k}m\overline{m}}
- R^E_{m\overline{q}} R_{k\overline{k}q\overline{m}}
+ \frac{1}{2}\left(R^E_{q\overline{q}} R^E_{m\overline{m}}
- R^E_{m\overline{q}} R^E_{q\overline{m}}\right)\\
&\hspace{15mm}+\frac{1}{4}\left(-R^E_{k\overline{k}\,;\,m\overline{m}}
+3 R^E_{m\overline{k}\,;\,k\overline{m}}\right)\,.
\end{split}\end{equation}
We say that a sequence $\Theta_{p}\in\mathscr{C}^\infty(X,\End(E))$
has an asymptotic expansion of the form
\begin{equation}\label{bk4.2o}
\Theta_{p}(x)= \sum_{r=0}^{\infty} \boldsym{A}_{r}(x) p^{n-r}
+\mathscr{O}(p^{-\infty})\,,
\quad \boldsym{A}_{r}\in\mathscr{C}^\infty(X,\End(E))\,,
\end{equation}
if for any $k,l\in \field{N}$, there exists
$C_{k,l}>0$ such that for any $p\in \field{N}^*$, we have
\begin{equation}\label{bk4.23}
\Big |\Theta_{p}(x)- \sum_{r=0}^{k} \boldsym{A}_{r}(x) p^{n-r} \Big |_{\mathscr{C}^l(X)}
\leqslant C_{k,l} \,p^{n-k-1},
\end{equation}
where $|\cdot|_{\mathscr{C}^{l}(X)}$ is the $\mathscr{C}^{l}$-norm on $X$.
\begin{thm} \label{toet4.1}
For any $f\in\mathscr{C}^\infty(X,\End(E))$, we have
\begin{equation}\label{bk4.2}
T_{f,\,p}(x,x)= \sum_{r=0}^{\infty} \bb_{r,f}(x) p^{n-r}+\mathscr{O}(p^{-\infty})\,,
\quad \bb_{r,f}\in\mathscr{C}^\infty(X,\End(E))\,.
\end{equation}
Moreover,
\begin{align}\label{bk4.3}
\bb_{0,f}=&f, \quad \bb_{1,f} = \frac{\br}{8\pi} f
+ \frac{\sqrt{-1}}{4\pi}
\left(R^E_{\Lambda} f + fR^E_{\Lambda} \right)
- \frac{1}{4\pi} \Delta^{E} f.
\end{align}
If $f\in \mathscr{C}^\infty(X)$, then
\begin{equation}\label{abk4.4}
\begin{split}
\pi^2 \bb_{2,f} & = \pi^2 \bb_{2} f
+ \frac{1}{32}\Delta^2 f
- \frac{1}{32} \br \Delta f
- \frac{\sqrt{-1}}{8} \big\langle \ric_\omega,
\partial\overline{\partial}f\big\rangle
+ \frac{\sqrt{-1}}{24} \big\langle df,
\nabla^E R^E_{\Lambda}\big\rangle_{\omega}\\
&+ \frac{1}{24} \big\langle \partial f, \nabla^{1,0*} R^E\big\rangle_{\omega}
- \frac{1}{24} \big\langle \overline{\partial} f,
\overline{\partial}^{E*} R^E\big\rangle_{\omega}
- \frac{\sqrt{-1}}{8} (\Delta f)R^E_{\Lambda}
+ \frac{1}{4}\big\langle \partial \overline{\partial} f, R^E\big\rangle_{\omega}.
\end{split}
\end{equation}
\end{thm}
\begin{thm} \label{toet4.6}
For any $f,g\in\mathscr{C}^\infty(X,\End(E))$, the kernel of
the composition $T_{f,\,p}\circ T_{g,\,p}$
has an asymptotic expansion on the diagonal
\begin{equation}\label{toe4.30}
(T_{f,\,p}\,\circ T_{g,\,p})(x,x)
=\sum_{r=0}^{\infty} \bb_{r,\,f,\,g}(x) p^{n-r}+\mathscr{O}(p^{-\infty})\,,
\quad \bb_{r,\,f,\,g}\in \mathscr{C}^\infty(X,\End(E))\,,
\end{equation}
in the sense of \eqref{bk4.23}.
\comment{that for any $k,l\in \field{N}$, there exists
$C_{k,l}>0$ such that the following estimate
\begin{equation}\label{toe4.301}
\Big | (T_{f,\,p}\, \cdot T_{g,\,p})(x,x)
- \sum_{r=0}^{k} \bb_{r,f,g}(x) p^{n-r} \Big |_{\mathscr{C}^l(X)}
\leqslant C_{k,l} \, p^{n-k-1}
\end{equation}
holds for any $p\in \field{N}$.}
Moreover, $\bb_{0,\,f,\,g}=fg$ and
\begin{equation}\label{toe4.31} \begin{split}
\bb_{1,\,f,\,g}= &\frac{1}{8\pi} \br fg
+ \frac{\sqrt{-1}}{4\pi}\big(R^E_{\Lambda} fg + fgR^E_{\Lambda} \big)\\
&-\frac{1}{4\pi}\big(f\Delta^{E} g + (\Delta^{E} f)g\big)
+ \frac{1}{2\pi}\big\langle\, \overline{\partial}^{E} f , \nabla^{1,0} g
\big\rangle_{\omega}\; .
\end{split}\end{equation}
If $f,g\in \mathscr{C}^\infty(X)$, then
\begin{equation}\label{toe4.32}
\begin{split}
\bb_{2,\,f,\,g} &= f\, \bb_{2,g} + g \,\bb_{2,f} - fg \,\bb_{2}
+ \frac{1}{\pi^2}\Big\{
-\frac{1}{8}\big\langle \,\overline{\partial} f, \partial \Delta g\big\rangle
- \frac{1}{8}\big\langle \,\overline{\partial} \Delta f, \partial g\big\rangle\\
&+ \frac{1}{2}\big\langle \,\overline{\partial} f, \partial g\big\rangle \pi \bb_{1}
- \frac{1}{4}\big\langle \,\overline{\partial} f\wedge\partial g, R^E\big\rangle_{\omega}
+ \frac{1}{16} \Delta f \cdot\Delta g
+ \frac{1}{8} \big\langle D^{0,1}\overline{\partial} f,
D^{1,0}\partial g\big\rangle\Big\}.
\end{split}\end{equation}
\end{thm}
The existence of the expansions \eqref{bk4.2} and \eqref{toe4.30}
and the formulas for
the leading terms hold in fact in the symplectic setting
and are consequences of \cite[Lemma\,4.6 and (4.79)]{MM08b}
or \cite[Lemma\,7.2.4 and (7.4.6)]{MM07} (cf.\ Lemma \ref{toet2.3}).
The novel point of Theorems \ref{toet4.1}, \ref{toet4.6} is the calculation of
the coefficients $\bb_{1,f}$, $\bb_{2,f}$,
$\bb_{1,f,g}$ and $\bb_{2,f,g}$\,.
Note that the precise formula $\bb_{1,f}$ for a function $f\in \mathscr{C}^\infty(X)$
was already given in \cite[Problem\,7.2]{MM07}.
In Theorem \ref{toet4.1a}, we find a general formula
of $ \bb_{2,f}$ for any $f\in \mathscr{C}^\infty(X,\End(E))$.
If $f=1$, then $T_{f,\,p}=P_{p}$, and
the existence of the expansion (\ref{bk4.2}) and the form of the leading term
was proved by \cite{T90}, \cite{Catlin99}, \cite{Ze98}.
The terms $\bb_{1}$, $\bb_{2}$ were computed by
Lu \cite{Lu00} (for $E=\field{C}$, the trivial line bundle with trivial
metric), X.~Wang \cite{Wang05},
L.~Wang \cite{Wangl03}, in various degree of generality.
The method of these authors is to construct appropriate peak sections
as in \cite{T90}, using
H{\"o}rmander's $L^2$ $\overline\partial$-method.
In \cite[\S 5.1]{DLM04a}, Dai-Liu-Ma computed $\bb_{1}$
by using the heat kernel, and in \cite[\S 2]{MM08a}, \cite[\S 2]{MM06}
(cf.\ also \cite[\S 4.1.8, \S 8.3.4]{MM07}), we
computed $\bb_{1}$ in the symplectic case.
The expansion of the Bergman kernel $P_{p}(x,x)$ on the diagonal,
for $E=\field{C}$, was rederived
by Douglas and Klevtsov \cite{DouKle10} by using path integral
and perturbation theory. They give physics interpretations
in terms of supersymmetric quantum mechanics,
the quantum Hall effect and black holes (cf.\;also \cite{DouKle08}).
An interesting consequence of Theorem \ref{toet4.6}
is the following precise computation of the expansion of the composition
of two Berezin-Toeplitz operators.
\begin{thm} \label{toet4.5}
Let $f,g\in\mathscr{C}^\infty(X,\End(E))$. The product of the Toeplitz operators
$T_{f,\,p}$ and $T_{g,p}$ is a Toeplitz operator,
more precisely, it admits the asymptotic expansion
\begin{equation}\label{toe4.2}
T_{f,\,p}\circ T_{g,p}=\sum^\infty_{r=0}p^{-r}T_{C_r(f,g),p}
+\mathcal{O}(p^{-\infty}),
\end{equation}
where $C_r$ are bidifferential operators, in the sense
that for any $k\geqslant0$, there exists
$c_k>0$ with
\begin{equation}\label{toe4.1}
\Big\|T_{f,\,p}\circ T_{g,\,p}- \sum_{l=0}^k p^{-l}T_{C_r(f,g),p}\Big\|
\leqslant c_k\,p^{-k-1},
\end{equation}
where $\norm{\cdot}$ denotes the operator norm on the space of
bounded operators.
We have
\begin{equation}\label{toe4.3} \begin{split}
C_0(f,g)=&fg, \\
C_1(f,g)=&- \frac{1}{2\pi}
\langle \nabla^{1,0} f, \overline{\partial}^{E} g\rangle_{\omega}\in
\mathscr{C}^{\infty}(X,\End(E)),\\
C_2(f,g)=& \, \bb_{2,f,g} - \bb_{2,fg}- \bb_{1,C_1(f,g)}.
\end{split}\end{equation}
If $f,g\in \mathscr{C}^\infty(X)$, then
\begin{align}\label{toe4.3a} \begin{split}
C_2(f,g)= &
\frac{1}{8\pi^2 } \langle D^{1,0}\partial f, D^{0,1}\overline{\partial} g\rangle
+ \frac{\sqrt{-1}}{4\pi^2 } \langle \ric_\omega,
\partial f \wedge\overline{\partial} g\rangle
-\frac{1}{4\pi^2} \langle\partial f\wedge \overline{\partial} g, R^E\rangle_{\omega}\,.
\end{split}\end{align}
\end{thm}
The existence of the expansion \eqref{toe4.2} is
a special case of \cite[Th.\;1.1]{MM08b}
(cf.\;also \cite[Th.\;7.4.1,\;8.1.10]{MM07}),
where we found a symplectic version in which the Toeplitz operators
\eqref{toe2.4} are constructed by using the projection to
the kernel of the Dirac operator.
Note that the precise values of $\bb_{2}$ are not used to derive
\eqref{toe4.3a} (cf.\ Section \ref{toes4.3}).
The existence of the expansion \eqref{toe4.2}
for $E=\field{C}$
was first established by
Bordemann, Meinrenken and Schlichenmaier \cite{BMS94},
Schlichenmaier \cite{Schlich:00} (cf.\ also \cite{KS01})
using the theory of Toeplitz structures
by Boutet de Monvel and Guillemin \cite{BouGu81}.
Charles \cite{Charles03} calculated $C_1(f,g)$ for $E=\field{C}$.
The asymptotic expansion \eqref{toe4.2} with a twisting bundle $E$
was derived by Hawkins \cite[Lemma 4.1]{Haw00} up to order one
(i.e., \eqref{toe4.1} for $k=0$).
Also, there is related work of Engli\v{s} \cite{Englis00,Englis02} dealing with
expressing asymptotic expansions of Bergman kernel and coefficients of
the Berezin-Toeplitz expansion \eqref{toe4.2} in terms of the metric.
Engli\v{s} \cite[Cor.\,15]{Englis02} computed $C_{1}(f,g)$ and $C_{2}(f,g)$
for a smoothly bounded pseudoconvex domain
$X=\{z\in\field{C}^{n}:\varphi(z)>0\}$, where
$\varphi$ is a defining function such that $-\log\varphi$
is strictly plurisubharmonic, and for the trivial line bundle $L=\field{C}$ over $X$,
equipped with the nontrivial metric $h^{L}=\varphi$ of positive curvature.
Note that we work throughout the paper with a non-trivial twisting bundle $E$.
Moreover, we have shown in \cite[\S5-6]{MM08b} (cf.\ also \cite[\S 7.5]{MM07})
that Berezin-Toeplitz quantization
holds for complete K{\"a}hler manifolds and orbifolds endowed
with a prequantum line bundle.
The calculations of the coefficients in the present paper
being local in nature, they hold also for the above cases.
For some applications of the results of this paper
to K\"ahler geometry see the paper \cite{Fine10} by Fine.
We close the introduction with some remarks about the Berezin-Toeplitz
star-product. Following the ground-breaking work of Berezin \cite{Berez:74},
one can define a star-product by using Toeplitz operators.
Note that formal star-products are known to exist on symplectic manifolds by
\cite{DeLe83, Fedo:96}.
The Berezin-Toeplitz star-product gives
a very concrete and geometric realization of such product.
For general symplectic manifolds this was realized in \cite{MM07,MM08b}
by using Toeplitz operators obtained by projecting on the kernel of
the Dirac operator.
Consider now a compact K\"ahler manifold $(X,\omega)$ and
a prequantum line bundle $L$. For every $f,g\in\mathscr{C}^{\infty}(X)$
one defines the Berezin-Toeplitz star-product (cf.\ \cite{KS01,Schlich:00}
and \cite{MM07,MM08b} for the symplectic case) by
\begin{equation}\label{toe4.4}
f*g:=\sum_{k=0}^\infty C_k(f,g) \hbar^{k}\in\mathscr{C}^\infty(X)[[\hbar]].
\end{equation}
This star-product is associative. Moreover, for $f,g\in\mathscr{C}^{\infty}(X)$
we have (cf.\ \cite[(7.4.3)]{MM07}, \cite[(4.89)]{MM08b})
\begin{equation}\label{toe4.5}
C_{0}(f,g)=fg=C_{0}(g,f)\,,\quad C_1(f,g)-C_1(g,f)=\sqrt{-1}\{f,g\}\,,
\end{equation}
where $\{f,g\}$ is the Poisson bracket associated to $2\pi\omega$. Therefore
\begin{equation}\label{toe4.4a}
\big[T_{f,\,p}\,,T_{g,\,p}\big]=\tfrac{\sqrt{-1}}{\, p}\,T_{\{f,g\},\,p}
+\mathcal{O}\big(p^{-2}\big)\,,\quad p\to\infty.
\end{equation}
Consider a twisting holomorphic Hermitian vector bundle $E$
and $f,g\in\mathscr{C}^\infty(X,\End(E))$ as in Theorem \ref{toet4.5}.
This corresponds to matrix-valued Berezin-Toeplitz quantization,
which models a quantum system with $r=\rank E$ degrees of freedom.
By \eqref{toe4.2}, this Berezin-Toeplitz quantization has
the expected semi-classical behaviour. Moreover,
by \cite[Th.\,7.4.2]{MM07}, \cite[Th.\,4.19]{MM08b} we have
\begin{equation}\label{toe4.17}
\lim_{p\to\infty}\norm{T_{f,\,p}}={\norm f}_\infty
:=\sup_{0\neq u\in E_x, x\in X} |f(x)(u)|_{h^{E}}/ |u|_{h^{E}}\,.
\end{equation}
\begin{cor}
Let $f,g\in\mathscr{C}^\infty(X,\End(E))$. Set
\begin{equation}\label{toe4.4c}
f*g:=\sum_{k=0}^\infty C_k(f,g) \hbar^{k}\in\mathscr{C}^\infty(X,\End(E))[[\hbar]].
\end{equation}
where $C_{r}(f,g)$ are determined by \eqref{toe4.2}. Then \eqref{toe4.4c} defines an associative star-product on $\mathscr{C}^\infty(X,\End(E))$. Set moreover
\begin{equation}\label{toe4.5a}
\{\!\{f,g\}\!\}:
\frac{1}{2\pi\sqrt{-1}}
\big(\langle \nabla^{1,0} g, \overline{\partial}^{E} f\rangle_{\omega}-\langle \nabla^{1,0} f, \overline{\partial}^{E} g\rangle_{\omega}\big)\,.
\end{equation}
If $fg=gf$ on $X$ we have
\begin{equation}\label{toe4.4b}
\big[T_{f,\,p}\,,T_{g,\,p}\big]=\tfrac{\sqrt{-1}}{\, p}\,T_{\{\!\{f,g\}\!\},\,p}
+\mathcal{O}\big(p^{-2}\big)\,,\quad p\to\infty.
\end{equation}
\end{cor}
The associativity of the star-product \eqref{toe4.4c} follows immediately
from the associativity rule for the composition of Toeplitz operators,
$(T_{f,\,p}\circ T_{g,\,p})\circ T_{k,\,p}=
T_{f,\,p}\circ (T_{g,\,p}\circ T_{k,\,p})$ for any
$f,g,k\in\mathscr{C}^\infty(X,\End(E))$, and from the asymptotic expansion
\eqref{toe4.2} applied to both sides of the latter equality.
Due to the fact that $\{\!\{f,g\}\!\}=\{f,g\}$ if $E$ is trivial and
comparing \eqref{toe4.4a} to \eqref{toe4.4b}, one can regard
$\{\!\{f,g\}\!\}$ defined in \eqref{toe4.5a}
as a non-commutative Poisson bracket
\begin{rem} \label{toet4.25}
Throughout the paper we suppose that
$g^{TX}(u,v)=\omega(u,Jv)$.
The results presented so far still hold for a general non-K\"ahler
$J$-invariant Riemannian
metric $g^{TX}$. To explain this point we follow \cite[\S 4.1.9]{MM07}.
Let us denote the metric associated to $\omega$ by
$g^{TX}_{\omega}:=\omega(\cdot,J\cdot)$.
We identify the $2$-form $R^L$ with the Hermitian matrix
$\dot{R}^L \in \End(T^{(1,0)}X)$ via $g^{TX}$.
Then the Riemannian volume form of $g^{TX}_{\omega}$
is given by $dv_{X,\,\omega} =(2\pi)^{-n}{\det}(\dot{R}^L) dv_{X}$
(where $dv_{X}$ is the Riemannian volume form of $g^{TX}$).
Moreover, $h^E_\omega:={\det} (\frac{\dot{R}^L}{2\pi})^{-1} h^E$
defines a metric on $E$.
We add a subscript $\omega$ to indicate the objects associated to $g^{TX}_{\omega}$, $h^L$ and $h^E_{\omega}$.
Hence $\left\langle\,\cdot,\cdot \right \rangle_\omega$ denotes
the $L^2$ Hermitian product on
$\mathscr{C}^\infty (X, L^p\otimes E)$
induced by $g^{TX}_\omega$, $h^L$, $h^E_\omega$.
This product is equivalent to the product
$\left\langle\,\cdot,\cdot \right \rangle$ induced by $g^{TX}$, $h^L$, $h^E$.
Moreover, $H^{0}(X, L^p\otimes E)$ does not depend on the Riemannian metric
on $X$ or on the Hermitian metrics on $L$, $E$. Therefore, the orthogonal
projections from
$(\mathscr{C}^\infty (X, L^p\otimes E), \left\langle\,\cdot,\cdot \right \rangle_\omega)$ and
$(\mathscr{C}^\infty (X, L^p\otimes E), \left\langle\,\cdot,\cdot \right \rangle)$
onto $H^{0}(X, L^p\otimes E)$ are the same. Hence $P_p=P_{p,\,\omega}$
and therefore $T_{f,\,p}=T_{f,\,p,\,\omega}$ as operators.
However, their kernels are different.
If
$T_{f,\,p,\,\omega}(x,x')$, ($x,x'\in X$), denotes the smooth
kernels of
$T_{f,\,p,\,\omega}$ with respect to $dv_{X,\omega}(x')$,
we have
\begin{equation}\label{bk2.95}
\begin{split}
&T_{f,\,p}(x,x')=(2\pi)^{-n} {\det}(\dot{R}^L)(x') T_{f,\,p,\,\omega}(x,x')\,.
\end{split}
\end{equation}
For the kernel $T_{f,\,p,\,\omega}(x,x')$, we can apply Theorem \ref{toet4.1}
since $g^{TX}_\omega(\cdot, \cdot)= \omega(\cdot, J\cdot)$
is a K\"ahler metric on $TX$.
We obtain in this way the expansion of $T_{f,\,p}(x,x)$ for
a non-K\"ahler metric $g^{TX}$ on $X$.
By \eqref{bk2.95}, the coefficients of these
expansions (\ref{bk4.2}), (\ref{toe4.2}) satisfy
\begin{equation}\label{bk2.951}
\begin{split}
&\bb_{r,\,f}=(2\pi)^{-n} {\det}(\dot{R}^L)\bb_{r,\, f,\,\omega}\,,\\
&C_r(f,g)=C_{r,\,\omega}(f,g)\,.
\end{split}
\end{equation}
\end{rem}
This paper is organized as follows. In Section~\ref{toes1}, we recall
the formal calculus on $\field{C}^n$ for the model operator $\mathscr{L}$, which is the
main ingredient of our approach. In Section~\ref{toes2}, we review the
asymptotic expansion of the kernel of Berezin-Toeplitz operators and
explain the strategy of our computation.
In Section \ref{toes3}, we obtain explicitly the first terms of the
Taylor expansion of our rescaled operator $\mathscr{L}_t$\,.
In Section \ref{bks3}, we study in detail the contribution of
$\mathcal{O}_2,\mathcal{O}_4$ to the term $\mathscr{F}_4$ from \eqref{bk2.32}.
In Section \ref{toes4}, by applying the formal calculi on $\field{C}^n$ and
the results from Section \ref{bks3}, we establish Theorems
\ref{toet4.1}, \ref{toet4.6} and \ref{toet4.5}. We also verify that
our calculations are compatible with Riemann-Roch-Hirzebruch Theorem.
In Section \ref{toes5}, we estimate the $\mathscr{C}^m$-norm of
Donaldson's $Q$-operator, thus continuing
\cite{LM07,LM09}.
We shall use the following notations. For
$\alpha=(\alpha_1, \ldots, \alpha_n)\in \field{N}^{m}$,
$Z\in \field{C}^{m}$, we set $|\alpha|:=\sum_{j=1}^{m}
\alpha_j$ and $Z^\alpha:= Z_1^{\alpha_1}\cdots Z_{m}^{\alpha_{m}}$.
Moreover, when an index variable appears twice in a single term,
it means that we are summing over all its possible values.
\smallskip
{\small\emph{
\textbf{Acknowledgments.} We would like to thank Joel Fine for motivating
and helpful discussions which led to the writing of this paper.
We are grateful to the referee for pointing out some interesting references.
X.~M. thanks Institut Universitaire de France for support.
G.~M. was partially supported by SFB/TR 12 and
Fondation Sciences Math{\'e}matiques de Paris.
We were also supported by the DAAD Procope Program.
}}
\section{Kernel calculus on $\field{C}^n$}\label{toes1
In this section we recall
the formal calculus on $\field{C}^n$ for the model operator $\mathscr{L}$
introduced in \cite[\S 2]{MM08b}, \cite[\S 7.1]{MM07}
(with $a_{j}=2\pi$ therein),
and we derive the properties of the calculus
of the kernels $(F\mathscr{P})(Z,Z^\prime)$, where $F\in\field{C}[Z,Z^\prime]$ and
$\mathscr{P}(Z,Z^\prime)$ is the kernel of the projection on the null space
of the model operator $\mathscr{L}$.
This calculus is the main ingredient of our approach.
Let us consider the canonical coordinates $(Z_1,\dotsc,Z_{2n})$
on the real vector space $\field{R}^{2n}$. On the complex vector space $\field{C}^n$
we consider the complex coordinates $(z_1,\dotsc,z_n)$.
The two sets of coordinates are linked by the relation
$z_j=Z_{2j-1}+\sqrt{-1} Z_{2j}$, $j=1,\dotsc,n$.
We consider the $L^2$-norm
$\norm{\,\cdot\,}_{L^2}=\big(\int_{\field{R}^{2n}}\abs{{\,\cdot\,}}^2\,dZ\big)^{1/2}$
on $\field{R}^{2n}$, where $dZ=dZ_1\cdots dZ_{2n}$
is the standard Euclidean volume form.
We define the differential operators:
\begin{equation}\label{toe1.1}
\begin{split}
&b_i=-2{\frac{\partial}{\partial z
_i}}+\pi \overline{z}_i\,,\quad
b^{+}_i=2{\frac{\partial}{\partial\overline{z}_i}}+\pi
z_i\,,\quad b=(b_1,\ldots,b_n)\,,\quad
\mathscr{L}=\sum_i b_i\, b^{+}_i\,,
\end{split}
\end{equation}
which extend to closed densely defined operators on
$(L^2(\field{R}^{2n}), \norm{\,\cdot\,}_{L^2})$. As such,
$b^{+}_i$ is the adjoint of $b_i$
and $\mathscr{L}$ defines as a densely defined self-adjoint operator on
$(L^2(\field{R}^{2n}),\norm{\,\cdot\,}_{L^2})$.
\noindent
The following result was established in \cite[Th.\,1.15]{MM08a}
(cf.\,also \cite[Th.\,4.1.20]{MM07}).
\begin{thm}\label{bkt2.17}
The spectrum of $\mathscr{L}$ on $L^2(\field{R}^{2n})$ is given by
\begin{equation}\label{bk2.68}
{\spec}(\mathscr{L})=\Big\{ 4\pi |\alpha| \,:\, \alpha\in\field{N} ^n\Big\}\,.
\end{equation}
Each $\lambda\in\spec(\mathscr{L})$ is an eigenvalue of infinite multiplicity and
an orthogonal basis of the corresponding eigenspace is given by
\begin{equation}\label{bk2.69}
B_\lambda=\Big\{b^{\alpha}\big(z^{\beta} e^{-\pi\sum_i |z_i|^2/2}\big):
\text{$\alpha\in\field{N}^n$ with $4\pi|\alpha|=\lambda$, $\beta\in\field{N}^n$}\Big\}
\end{equation}
and $\bigcup\{B_\lambda:\lambda\in\spec(\mathscr{L})\}$
forms a complete orthogonal basis of $L^2(\field{R}^{2n})$.
In particular, an orthonormal basis of
$\Ker (\mathscr{L})$ is
\begin{equation}\label{bk2.70}
\Big\{\varphi_\beta(z)=\big(\tfrac{\pi ^{|\beta|}}{\beta!}\big)^{1/2}z^\beta
e^{-\pi\sum_i |z_i|^2/2}\,:\beta\in\field{N}^n\Big\}\,.
\end{equation}
\end{thm}
\noindent
Let $\mathscr{P}(Z,Z')$ denote the kernel of the orthogonal projection
$\mathscr{P}:L^2 (\field{R}^{2n})\longrightarrow\Ker(\mathscr{L})$ with respect to $dZ'$.
Let $\mathscr{P}^\bot = \Id -\mathscr{P}$.
We call $\mathscr{P}(\cdot,\cdot)$ the Bergman kernel of $\mathscr{L}$.
Obviously
$\mathscr{P}(Z,Z')=\sum_{\beta}\varphi_\beta(z)\,\overline{\varphi_\beta(z')}$ so
we infer from \eqref{bk2.70} that
\begin{equation}\label{toe1.3}
\mathscr{P}(Z,Z') = \exp\big(-\tfrac{\pi}{2}\textstyle\sum_{i=1}^n
\big(|z_i|^2+|z^{\prime}_i|^2 -2z_i\overline{z}_i'\big)\big)\,.
\end{equation}
In the calculations involving the kernel $\mathscr{P}(\cdot,\cdot)$,
we prefer however to use the orthogonal decomposition of $L^2(\field{R}^{2n})$
given in Theorem \ref{bkt2.17} and the fact that $\mathscr{P}$ is
an orthogonal projection, rather than
integrating against the expression \eqref{toe1.3} of $\mathscr{P}(\cdot,\cdot)$.
This point of view helps simplify a lot the computations
and understand better the operations.
As an example, if $\varphi(Z)=b^{\alpha}\big(z^{\beta}
e^{-\pi\sum_i |z_i|^2/2}\big)$
with $\alpha,\beta\in \field{N}^n$, then Theorem \ref{bkt2.17}
implies immediately that
\begin{align}\label{toe1.4}
(\mathscr{P} \varphi)(Z)= \left \{ \begin{array}{ll}
\displaystyle{z^\beta e^{-\pi\sum_i |z_i|^2/2}\,, }
& \mbox{if} \,\, |\alpha|=0, \\
0\,,& \mbox{if} \,\, |\alpha|>0.\end{array}\right.
\end{align}
\noindent
The following commutation relations are very useful in the computations.
Namely, for any polynomial $g(z,\overline{z})$ in $z$ and $\overline{z}$, we have
\begin{align}\label{bk2.66}
\begin{split}
&[b_i,b^{+}_j]=b_i b^{+}_j-b^{+}_j b_i =-4\pi \delta_{i\,j},\\
&[b_i,b_j]=[b^{+}_i,b^{+}_j]=0\, ,\\
& [g(z,\overline{z}),b_j]= 2 \frac{\partial}{\partial z_j}g(z,\overline{z}), \\
& [g(z,\overline{z}),b_j^+]
= - 2\frac{\partial}{\partial \overline{z}_j}g(z,\overline{z})\,.
\end{split}\end{align}
\noindent
For a polynomial $F$ in $Z,Z^\prime$, we denote by $F\mathscr{P}$
the operator on $L^2(\field{R}^{2n})$ defined by the kernel
$F(Z,Z^\prime)\mathscr{P}(Z,Z^\prime)$ and the volume form $dZ$
according to \eqref{toe2.21}.
The following very useful Lemma \cite[Lemma 7.1.1]{MM07}
describes the calculus of the kernels
$(F\mathscr{P})(Z,Z^\prime):=F(Z,Z^\prime)\mathscr{P}(Z,Z^\prime)$.
\begin{lemma}\label{toet1.1}
For any
$F,G\in\field{C}[Z, Z^{\prime}]$ there exists a polynomial
$\mathscr{K}[F,G]\in\field{C}[Z, Z^{\prime}]$
with degree $\deg\mathscr{K}[F,G]$ of the same parity as
$\deg F+\deg G$, such that
\begin{equation}\label{toe1.6}
((F\mathscr{P}) \circ (G\mathscr{P}))(Z, Z^{\prime}) =
\mathscr{K}[F,G](Z, Z^{\prime}) \mathscr{P}( Z, Z^{\prime}).
\end{equation}
\end{lemma}
\section{Expansion of the kernel of
Berezin-Toeplitz operators}\label{toes2}
In this section, we review some results from \cite{MM08a}, \cite{MM08b}
(cf.\ also \cite[\S\,7.2]{MM07}). We explain then how to compute
the coefficients of various expansions considered in this paper.
We keep the notations and assumptions from the Introduction.
\comment{
Let $(X,\omega, J)$ be a compact K{\"a}hler manifold of
$\dim_{\field{C}}X=n$ with complex structure $J$,
and let $(L, h^L)$ be a holomorphic Hermitian line
bundle on $X$. Let $\nabla ^L$ be the holomorphic Hermitian
connection on $(L,h^L)$ with curvature $R^L$. We assume that
\begin{align} \label{toe2.1}
\frac{\sqrt{-1}}{2 \pi} R^L=\omega.
\end{align}
Let $(E, h^E)$ be a holomorphic Hermitian vector bundle on $X$. Let
$\nabla^E$ be the holomorphic Hermitian connection on $(E, h^E)$
with curvature $R^E$.
Let $\nabla ^{\End (E)}$ be the connection on
$\End (E)$ induced by $\nabla ^E$.
Let $g^{TX}(\cdot,\cdot):= \omega(\cdot,J\cdot)$ be
the Riemannian metric on $TX$ induced by $\omega, J$.
Let $dv_X$ be the Riemannian volume
form of $(TX, g^{TX})$, then $dv_X= \omega^n/n!$.
Let $\Delta$ be the (positive) Laplace operator on $(X, g^{TX})$
acting on the functions on $X$.
The $L^2$--Hermitian product on the space
$\mathscr{C}^\infty(X,L^p\otimes E)$
of smooth sections of $L^p\otimes E$ is given by
\begin{equation}\label{toe2.2}
\langle s_1,s_2\rangle=\int_X\langle s_1(x),s_2(x)\rangle\,dv_X(x)\,.
\end{equation}
We denote the corresponding norm with $\norm{\cdot}_{L^2}$ and
with $L^2(X,L^p\otimes E)$ the completion of
$\mathscr{C}^\infty(X,L^p\otimes E)$ with respect to this norm.
Let $\nabla^{TX}$ be the Levi-Civita connection on $(X, g^{TX})$.
}
\noindent
\textbf{\emph{Kodaira-Laplace operator.\/}}
Let $\partial ^{L^{p}\otimes E,*}$ be the adjoint of the Dolbeault
operator $\partial ^{L^{p}\otimes E}$.
Let $\Box_{p}= \partial ^{L^{p}\otimes E,*}\partial ^{L^{p}\otimes E}$
be the restriction of the Kodaira Laplacian to
$\mathscr{C}^{\infty}(X, L^p\otimes E)$.
Let $\Delta^{L^{p}\otimes E}$ be the Bochner Laplacian
on $\mathscr{C}^{\infty}(X, L^p\otimes E)$ associated to $\nabla^{L}$,
$\nabla^{E}$, $g^{TX}$, defined as in \eqref{toe2.3}.
\comment{Let
\begin{align} \label{toe2.3}
\Delta^{L^{p}\otimes E}
= - \nabla^{L^{p}\otimes E}_{e_{i}}\nabla^{L^{p}\otimes E}_{e_{i}}
- \nabla^{L^{p}\otimes E}_{\nabla^{TX}_{e_{i}}e_{i}}
\end{align}
be the Bochner Laplacian
on $\mathscr{C}^{\infty}(X, L^p\otimes E)$. }
Then we have (cf. \cite[Remark\,1.4.8]{MM07})
\begin{align}\label{toe2.5}
2 \, \Box_{p} = \Delta^{L^{p}\otimes E}
- \tfrac{\sqrt{-1}}{2}R^{E}(e_{i}, Je_i) - 2n p.
\end{align}
Moreover, by Hodge theory (cf.\ \cite[Th.\,1.4.1]{MM07}) we have
\begin{align}\label{toe2.6}
\Ker (\Box_{p}) =H^0(X,L^p\otimes E).
\end{align}
This identification is important since the computations
are performed by rescaling $\Box_{p}$
and expanding the rescaled operator.
\comment{
Let $P_{p}(x,x')$ $(x,x'\in X)$ be the smooth kernel of
the orthogonal projection $P_{p}$ from
$(\mathscr{C} ^\infty(X, L^p\otimes E),\langle\cdot\,,\cdot \rangle )$
onto $H^0(X,L^p\otimes E)$,
the space of the holomorphic sections of $L^p\otimes E$ on $X$,
with respect to $dv_X$. For $f\in \mathscr{C}^\infty(X, \End(E))$,
its Berezin-Toeplitz quantization $T_{f,\,p}$ is defined by
\begin{equation}\label{toe2.4}
T_{f,\,p}:L^2(X,L^p\otimes E)\longrightarrow L^2(X,L^p\otimes E)\,,
\quad T_{f,\,p}=P_p\,f\,P_p\,.
\end{equation}
Let $T_{f,\,p}(x,x')$ $(x,x'\in X)$ be the smooth kernel of
$T_{f,\,p}$ with respect to $dv_X$.
}
\noindent
\textbf{\emph{Normal coordinates.\/}}
Let $a^X$ be the injectivity radius of $(X, g^{TX})$.
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 exponential map $ T_x X\ni Z \to \exp^X_x(Z)\in X$ is a
diffeomorphism from $B^{T_xX} (0,\varepsilon)$ onto $B^{X} (x,\varepsilon)$ for
$\varepsilon\leqslant a^X$. {}From now on, we identify $B^{T_xX}(0,\varepsilon)$
with $B^{X}(x,\varepsilon)$ via the exponential map for $\varepsilon \leqslant a^X$.
Throughout what follows, $\varepsilon$ runs in the
fixed interval $]0, a^X/4[$.
\noindent
\textbf{\emph{Basic trivialization.\/}}
We fix $x_0\in X$.
For $Z\in B^{T_{x_0}X}(0,\varepsilon)$ we identify
$(L_Z, h^L_Z)$, $(E_Z, h^E_Z)$ and $(L^p\otimes E)_Z$
to $(L_{x_0},h^L_{x_0})$, $(E_{x_0},h^E_{x_0})$ 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
$\gamma_Z :[0,1]\ni u \to \exp^X_{x_0} (uZ)$.
This is the basic trivialization we use in this paper.
Using this trivialization we identify $f\in \mathscr{C}^\infty(X,\End(E))$ to a family
$\{f_{x_0}\}_{x_0\in X}$ where $f_{x_0}$ is the function $f$ in
normal coordinates near $x_0$, i.e.,
$f_{x_0}:B^{T_{x_0}X}(0,\varepsilon)\to\End(E_{x_0})$,
$f_{x_0}(Z)=f\circ\exp^X_{x_0}(Z)$.
In general, for functions in the normal coordinates,
we will add a subscript $x_0$ to indicate the base point $x_0\in X$.
Similarly,
$P_p(x,x')$ induces in terms of the basic trivialization a smooth section
$(Z,Z')\mapsto P_{p,\,x_0}(Z,Z')$
of $\pi ^* \End(E)$ over $\{(Z,Z')\in TX\times_{X} TX:|Z|,|Z'|<\varepsilon\}$,
which depends smoothly on $x_0$. Here we identify a section
$S\in \mathscr{C}^\infty \big(TX\times_{X}TX,\pi ^* \End (E)\big)$
with the family $(S_x)_{x\in X}$, where
$S_x=S|_{\pi^{-1}(x)}$.
\comment{
Then under our identification,
we can view $P_{p,x_0}(Z,Z')$ as a smooth section of $\pi^*(\End( E))$ on
$TX\times_{X} TX$ (which is defined for $|Z|,|Z^\prime|\leq \varepsilon$)
by identifying a section $S\in \mathscr{C}^\infty (TX\times_{X}TX,\pi ^* \End (E))$
with the family $(S_x)_{x\in X}$, where
$S_x=S|_{\pi^{-1}(x)}$, $\End(E)=\field{C}$.
}
\noindent
\textbf{\emph{Coordinates on $T_{x_0}X$.\/}}
Let us choose
an orthonormal basis $\{ w_i\}_{i=1}^n$ of $T^{(1,0)}_{x_0} X$.
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\, $
form an orthonormal basis of $T_{x_0}X$.
We use coordinates on $T_{x_0}X\simeq\field{R}^{2n}$
given by the identification
\begin{equation}\label{n11}
\field{R}^{2n}\ni (Z_1,\ldots, Z_{2n}) \longmapsto \sum_i
Z_i e_i\in T_{x_0}X.
\end{equation}
In what follows we also use complex coordinates $z=(z_1,\ldots,z_n)$
on $\field{C}^n\simeq\field{R}^{2n}$.
\noindent
\textbf{\emph{Volume form on $T_{x_0}X$.\/}}
If $dv_{TX}$ is the Riemannian volume form
on $(T_{x_0}X, g^{T_{x_0}X})$, there exists
a smooth positive function $\kappa_{x_0}:T_{x_0}X\to\field{R}$,
$Z\mapsto\kappa_{x_0}(Z)$ defined by
\begin{equation} \label{atoe2.7}
dv_X(Z)= \kappa_{x_0}(Z) dv_{TX}(Z),\quad \kappa_{x_0}(0)=1,
\end{equation}
where the subscript $x_0$ of
$\kappa_{x_0}(Z)$ indicates the base point $x_0\in X$.
\noindent
\textbf{\emph{Sequences of operators.\/}}
Let $\Theta_p: L^2(X,L^p\otimes E)\longrightarrow L^2(X,L^p\otimes E)$
be a sequence of continuous linear operators with smooth kernel
$\Theta_p(\cdot,\cdot)$ with respect to $dv_X$ (e.g.\,$\Theta_p=T_{f,p}$).
Let $\pi : TX\times_{X} TX \to X$ be the natural projection from the
fiberwise product of $TX$ on $X$.
In terms of our basic trivialization, $\Theta_p(x,y)$ induces
a family of smooth sections
$Z,Z'\mapsto \Theta_{p,\,x_0}(Z,Z^\prime)$
of $\pi^*\End(E)$ over $\{(Z,Z')\in TX\times_{X} TX:|Z|,|Z'|<\varepsilon\}$,
which depends smoothly on $x_0$.
We denote by $\abs{\Theta_{p,\,x_0}(Z,Z^\prime)}_{\mathscr{C}^l(X)}$
the $\mathscr{C}^{l}$ norm with respect to the parameter $x_0\in X$. We say that
$\Theta_{p,\,x_0}(Z,Z^\prime)=\mathcal{O}(p^{-\infty})$ if
for any $l,m\in \field{N}$, there exists $C_{l,m}>0$ such that
$\abs{\Theta_{p,\,x_0}(Z,Z^\prime)}_{\mathscr{C}^{m}(X)}\leqslant C_{l,m}\, p^{-l}$.
\begin{notation}\label{noe2.7}
Recall that $\mathscr{P}_{x_0}=\mathscr{P}$ was defined in \eqref{toe1.3}.
Fix $k\in\field{N}$ and $\varepsilon^\prime\in\,]0,a^X[$\,.
Let $\{Q_{r,\,x_0}\}_{0\leqslant r\leqslant k,x_0\in X}$ be a family
of polynomials $Q_{r,\,x_0}\in \End(E)_{x_0}[Z,Z^{\prime}]$
in $Z,Z^ \prime$, which is smooth with respect to
the parameter $x_0\in X$. We say that
\begin{equation} \label{toe2.7}
p^{-n} \Theta_{p,x_0}(Z,Z^\prime)\cong \sum_{r=0}^k
(Q_{r,\,x_0} \mathscr{P}_{x_0})(\sqrt{p}Z,\sqrt{p}Z^{\prime})p^{-r/2}
+\mathcal{O}(p^{-(k+1)/2})\,,
\end{equation}
on $\{(Z,Z^\prime)\in TX\times_X TX:\abs{Z},\abs{Z^{\prime}}<\varepsilon^\prime\}$
if there exist $C_0>0$ and a decomposition
\begin{equation} \label{toe2.71}
\begin{split}
p^{-n} \Theta_{p,x_0}(Z,Z^\prime)\kappa^{1/2}_{x_0}(Z)\kappa^{1/2}_{x_0}(Z')-\sum_{r=0}^k
(Q_{r,\,x_0} \mathscr{P}_{x_0})(\sqrt{p}Z,\sqrt{p}Z^{\prime})p^{-r/2}\\
=\Psi_{p,k,x_0}(Z,Z^\prime)+\mathcal{O}(p^{-\infty})\,,
\end{split}
\end{equation}
where $\Psi_{p,k,x_0}$ satisfies the following estimate on
$\{(Z,Z^\prime)\in TX\times_X TX:\abs{Z},\abs{Z^{\prime}}<\varepsilon^\prime\}$:
for every $l\in\field{N}$ there exist $C_{k,\,l}>0$, $M>0$ such that for all
$p\in\field{N}^{*}$
\begin{equation}
|\Psi_{p,k,x_0}(Z,Z^\prime)|_{\mathscr{C}^l(X)}\leqslant \,C_{k,\,l}\,p^{-(k+1)/2}
(1+\sqrt{p}\,|Z|+\sqrt{p}\,|Z^{\prime}|)^M \,
e^{-C_0\,\sqrt{ p}\,|Z-Z^{\prime}|}\,.
\end{equation}
\end{notation}
\noindent
\textbf{\emph{The sequence $P_p$\,.\/}}
By \cite[Proposition\,4.1]{DLM04a} we know that the Bergman kernel
decays very fast outside the diagonal of $X\times X$.
Namely, for any $l,m\in \field{N}$, $\varepsilon>0$, there exists $C_{l,m,\varepsilon}>0$
such that for all $p\geqslant 1$ we have
\begin{align}\label{1n13}
&|P_{p}(x,x')|_{\mathscr{C}^m}
\leqslant C_{l,m,\varepsilon}\, p^{-l} \quad \text{on $\{(x,x')
\in X\times X: d(x,x') \geqslant \varepsilon\}$} \,.
\end{align}
Here the $\mathscr{C}^m$-norm is induced by $\nabla ^L$, $\nabla ^E$,
$\nabla ^{TX}$ and $h^L, h^E, g^{TX}$.
By \cite[Th.\,\,4.18$^\prime$]{DLM04a},
there exist polynomials $J_{r,\,x_{0}}(Z,Z')\in \End(E)_{x_0}$
in $Z,Z'$ with the same parity as $r$, such that
for any $k\in \field{N}$, $\varepsilon\in]0, a^X/4[$\,,
we have
\begin{equation} \label{toe2.9}
p^{-n} P_{p,\,x_0}(Z,Z^\prime)\cong \sum_{r=0}^k
(J_{r,\,x_0} \mathscr{P}_{x_0})(\sqrt{p}Z,\sqrt{p}Z^{\prime})p^{-\frac{r}{2}}
+\mathcal{O}(p^{-\frac{k+1}{2}})\,,
\end{equation}
on the set $\{(Z,Z^\prime)\in TX\times_X TX:\abs{Z},\abs{Z^{\prime}}<2\varepsilon\}$,
in the sense of Notation \ref{noe2.7}.
\noindent
\textbf{\emph{The sequence $T_{f,\,p}$\,.\/}}
From (\ref{toe2.9}), we get the following result
(cf.\,\cite[Lemma\,4.6]{MM08b}, \cite[Lemma\,7.2.4]{MM07}).
\begin{lemma} \label{toet2.3}
Let $f\in\mathscr{C}^\infty(X,\End(E))$.
There exists a family $\{Q_{r,\,x_0}(f)\}_{r\in\field{N},\,x_0\in X}$,
depending smoothly on the parameter $x_0\in X$, where
$Q_{r,\,x_0}(f)\in\End(E)_{x_0}[Z,Z^{\prime}]$
are polynomials with the same parity as $r$ and such that for every $k\in \field{N}$,
$\varepsilon\in]0, a^X/4[$\,,
\begin{equation} \label{toe2.13}
p^{-n}T_{f,\,p,\,x_0}(Z,Z^{\prime})
\cong \sum^k_{r=0}(Q_{r,\,x_0}(f)\mathscr{P}_{x_0})(\sqrt{p}Z,\sqrt{p}Z^{\prime})
p^{-r/2} + \mathcal{O}(p^{-(k+1)/2})\,,
\end{equation}
on the set $\{(Z,Z^\prime)\in TX\times_X TX:\abs{Z},\abs{Z^{\prime}}<2\varepsilon\}$,
in the sense of Notation \ref{noe2.7}. Moreover,
$Q_{r,\,x_0}(f)$ are expressed by
\begin{equation} \label{toe2.14}
Q_{r,\,x_0}(f) = \sum_{r_1+r_2+|\alpha|=r}
\mathscr{K}\Big[J_{r_1,\,x_0}\;,\;
\frac{\partial ^\alpha f_{\,x_0}}{\partial Z^\alpha}(0)
\frac{Z^\alpha}{\alpha !} J_{r_2,\,x_0}\Big]\,.
\end{equation}
Especially,
\begin{align} \label{toe2.15}
Q_{0,\,x_0}(f)= f(x_0) .
\end{align}
\end{lemma}
Our goal is of course to compute the coefficients $Q_{r,\,x_0}(f)$.
For this we need $J_{r,\,x_{0}}$, which are obtained by computing
the operators $\mathscr{F}_{r,\,x_{0}}$ defined by the smooth kernels
\begin{align}\label{bk2.24}
\mathscr{F}_{r,\,x_{0}}(Z,Z')= J_{r,\,x_{0}}(Z,Z')\mathscr{P}(Z,Z')
\end{align}
with respect to $dZ'$. Our strategy (already used in \cite{MM07,MM08a})
is to rescale the Kodaira-Laplace operator, take the Taylor expansion
of the rescaled operator and apply resolvent analysis.
In the remaining of this section we outline the main steps
and continue the calculation in Section \ref{toes3}.
\noindent
\textbf{\emph{Rescaling $\Box_p$ and Taylor expansion.\/}}
For $s \in \mathscr{C}^{\infty}(\field{R}^{2n}, E_{x_0})$, $Z\in \field{R}^{2n}$,
$|Z|\leq 2\varepsilon$, and
for $t=\frac{1}{\sqrt{p}}$, set
\begin{align}\label{bk2.21}
\begin{split}
&(S_{t} s ) (Z) :=s (Z/t), \\
& \nabla'_{t}:= S_t^{-1}
t \, \nabla ^{L^p\otimes E} S_t,\\
& \nabla_{t}:= S_t^{-1}
t \, \kappa^{1/2}\nabla ^{L^p\otimes E} \kappa^{-1/2} S_t
= \kappa^{1/2}(tZ) \nabla'_{t}\, \kappa^{-1/2}(tZ), \\
& \mathscr{L}_{t}:= S_t^{-1} \kappa^{1/2}\, t^2 (2\, \Box_{p})\kappa^{-1/2} S_t.
\end{split}\end{align}
Then by \cite[Th.\,4.1.7]{MM07}, there exist
second order differential operators $\mathcal{O}_{r}$
such that we have an asymptotic expansion in $t$ when $t\to 0$,
\begin{align}\label{bk2.22}
\mathscr{L}_{t} = \mathscr{L}_{0} + \sum_{r=1}^{m} t^r \mathcal{O}_{r} + \mathscr{O}(t^{m+1}).
\end{align}
\noindent
From \cite[Th.\,\,4.1.21,\,4.1.25]{MM07}
(cf. also Theorem \ref{bkt2.21}), we obtain
\begin{align}\label{bk2.30}
\mathscr{L}_0=& \sum_j b_jb^+_j=\mathscr{L}, \qquad \mathcal{O}_1=0.
\end{align}
\noindent
\textbf{\emph{Resolvent analysis.\/}}
We define by recurrence
$f_r(\lambda)\in \End(L^2(\field{R}^{2n}, E_{x_0}))$ by
\begin{align}\label{bk2.23}
f_{0}(\lambda) = (\lambda -\mathscr{L}_0)^{-1},
\quad f_r(\lambda)=(\lambda -\mathscr{L}_0)^{-1}
\sum_{j=1}^{r} \mathcal{O}_j f_{r-j}(\lambda).
\end{align}
Let $\delta$ be the counterclockwise
oriented circle in $\field{C}$ of center $0$ and radius $\pi/2$.
Then by \cite[(1.110)]{MM08a} (cf. also \cite[(4.1.91)]{MM07})
\begin{equation}\label{bk2.77}
\mathscr{F}_{r,\,x_{0}}= \frac{1}{2\pi \sqrt{-1}} \int_{\delta} f_r (\lambda)d \lambda.
\end{equation}
Since the spectrum of $\mathscr{L}$ is well understood we can calculate
the coefficients $\mathscr{F}_{r,\,x_{0}}$. Recall than $\mathscr{P}^\bot= \Id - \mathscr{P}$.
From Theorem \ref{bkt2.17}, \eqref{bk2.30} and \eqref{bk2.77}, we get
\begin{align}\label{bk2.31}
\begin{split}
\mathscr{F}_{0,\,x_{0}}=& \mathscr{P}, \quad \mathscr{F}_{1,\,x_{0}}= 0,\\
\mathscr{F}_{2,\,x_{0}}=&- \mathscr{L}^{-1} \mathscr{P}^\bot\mathcal{O}_2 \mathscr{P}
- \mathscr{P} \mathcal{O}_2\mathscr{L}^{-1}\mathscr{P}^\bot,\\
\mathscr{F}_{3,\,x_{0}}=&- \mathscr{L}^{-1} \mathscr{P}^\bot\mathcal{O}_3\mathscr{P}
- \mathscr{P} \mathcal{O}_3\mathscr{L}^{-1}\mathscr{P}^\bot,
\end{split}\end{align}
and
\begin{align}\nonumber
\mathscr{F}_{4,\,x_{0}}&= \frac{1}{2 \pi \sqrt{-1}} \int_{\delta}
\Big[(\lambda -\mathscr{L})^{-1} \mathscr{P}^\bot (\mathcal{O}_2 f_2 + \mathcal{O}_4 f_0)(\lambda)
+ \frac{1}{\lambda} \mathscr{P} (\mathcal{O}_2 f_2+ \mathcal{O}_4 f_0)(\lambda)\Big]d\lambda \\
\label{bk2.32}
\begin{split}&=\mathscr{L}^{-1}\mathscr{P}^\bot \mathcal{O}_{2}\mathscr{L}^{-1}\mathscr{P}^\bot \mathcal{O}_{2} \mathscr{P}
- \mathscr{L}^{-1}\mathscr{P}^\bot \mathcal{O}_{4} \mathscr{P}\\
&\quad+ \mathscr{P}\mathcal{O}_{2}\mathscr{L}^{-1}\mathscr{P}^\bot \mathcal{O}_{2}\mathscr{L}^{-1}\mathscr{P}^\bot
- \mathscr{P} \mathcal{O}_{4}\mathscr{L}^{-1}\mathscr{P}^\bot \\
&\quad+ \mathscr{L}^{-1}\mathscr{P}^\bot \mathcal{O}_{2} \mathscr{P} \mathcal{O}_{2} \mathscr{L}^{-1}\mathscr{P}^\bot
- \mathscr{P} \mathcal{O}_{2} \mathscr{L}^{-2} \mathscr{P}^\bot \mathcal{O}_{2} \mathscr{P}\\
&\quad- \mathscr{P}\mathcal{O}_{2} \mathscr{P} \mathcal{O}_{2} \mathscr{L}^{-2}\mathscr{P}^\bot
- \mathscr{P}^\bot \mathscr{L}^{-2}\mathcal{O}_{2} \mathscr{P} \mathcal{O}_{2} \mathscr{P}.
\end{split}\end{align}
In particular, the first two identities of \eqref{bk2.31} imply
\begin{align}\label{bk2.33}
J_{0,\,x_{0}}= 1, \quad J_{1,\,x_{0}}=0.
\end{align}
\begin{rem} \label{toet2.7}
$\mathscr{L}_t$ is a formally self-adjoint elliptic operator on
$\mathscr{C}^\infty(\field{R}^{2n}, E_{x_0})$ with respect to the norm
$\|\cdot\|_{L^2}$
induced by $h^{E_{x_0}}$, $dZ$. Thus $\mathscr{L}_0$
and $\mathcal{O}_r$ are also formally self-adjoint with respect to
$\|\cdot\|_{L^2}$.
Therefore the third and fourth terms in \eqref{bk2.32} are the adjoints
of the first and second terms, respectively.
In Lemma \ref{bkt3.1}, we will show that $\mathscr{P}\mathcal{O}_2\mathscr{P}=0$, hence the
last two terms in \eqref{bk2.32} vanish.
Set
\begin{align}\label{bk2.34}
\mathscr{F}_{41}= \mathscr{L}^{-1}\mathscr{P}^\bot \mathcal{O}_{2}\mathscr{L}^{-1}\mathscr{P}^\bot \mathcal{O}_{2} \mathscr{P}
- \mathscr{L}^{-1}\mathscr{P}^\bot \mathcal{O}_{4} \mathscr{P}.
\end{align}
\end{rem}
\section{Taylor expansion of the rescaled operator $\mathscr{L}_t$}\label{toes3}
In this section we compute
the operators $\mathscr{L}_{0}$ and $\mathcal{O}_{i}$ (for $1\leqslant i \leqslant 4$)
from \eqref{bk2.22} (see Theorem \ref{bkt2.21}), which will
be used in Sections \ref{bks3}, \ref{toes4} for the evaluation of the
coefficients of the expansion of the kernels of the Berezin-Toeplitz
operators.
We denote by $\left\langle\cdot,\cdot\right\rangle$ the
$\field{C}$-bilinear form on $TX\otimes_\field{R} \field{C}$ induced by $g^{TX}$.
Let $R^{TX}$ be the curvature of the Levi-Civita connection $\nabla^{TX}$.
Let $\ric$ and $\br$ be the Ricci and scalar curvature of $\nabla^{TX}$.
Then we have the following well know facts:
for $U,V, W,Y$ vector fields on $X$,
\begin{align}\label{alm01.0}
\begin{split}
&R^{TX}(U,V)W+ R^{TX}(V,W)U + R^{TX}(W,U)V=0,\\
&\langle R^{TX}(U,V)W,Y\rangle= \langle R^{TX}(W,Y)U,V\rangle
= - \langle R^{TX}(V,U)W,Y\rangle.
\end{split}
\end{align}
Now we work on $T_{x_0}X\simeq \field{R}^{2n}$ as in \eqref{n11}.
Recall that we have trivialized $L, E$.
Let $\nabla_U$ denote the ordinary differentiation
operator on $T_{x_0}X$ in the direction $U$.
We adopt the convention that all tensors will be evaluated at the base
point $x_0\in X$ and most of time, we will omit the subscript $x_0$.
For $W\in T_{x_0}X$, $Z\in\field{R}^{2n}$,
let $\widetilde{W}(Z)$ be the parallel transport of $W$
with respect to $\nabla^{TX}$ along the curve $[0,1]\ni u\to uZ$.
Because the complex structure $J$ is parallel with respect to
$\nabla^{TX}$, we know that
\begin{align}\label{lm01.1}
J_Z \widetilde{W}(Z) = \widetilde{J_{x_0}W}(Z).
\end{align}
Recall that $\{e_i\}$ is a fixed orthonormal basis of
$(T_{x_0}X, g^{TX})$. Then for $U,V\in T_{x_0}X$,
\begin{align}\label{alm01.1}
\ric_{x_0}(U,V)=- \langle R^{TX}(U,e_j)V,e_j\rangle_{x_0},\quad
\br_{x_0}= - \langle R^{TX}(e_i,e_j)e_i,e_j\rangle_{x_0}.
\end{align}
We define
\begin{align*}
&R^{TX}_{\, ;\, \bullet}\in (T^*X\otimes \Lambda^2(T^*X) \otimes
\End(TX))_{x_0}\,,\\
&R^{TX}_{\, ; \,(\bullet,\bullet)}\in
((T^*X)^{\otimes 2}\otimes \Lambda^2(T^*X) \otimes \End(TX))_{x_0}\,,\\
&\ric_{\, ; \,\bullet}\in (T^*X\otimes (T^*X)^{\otimes 2})_{x_0}\,,\\
&R^E_{\, ;\, \bullet}\in (T^*X\otimes\Lambda^2(T^*X)\otimes \End(E))_{x_0}\,,\\
&R^{E}_{\, ; \,(\bullet,\bullet)}\in ((T^*X)^{\otimes 2}\otimes
\Lambda^2(T^*X) \otimes \End(E))_{x_0}\,,
\end{align*}
by
\begin{align} \label{lm01.2}
\begin{split}
&\left \langle R^{TX}_{\, ;\, e_{k}}(e_{m}, e_j) e_{q}, e_i\right\rangle
= \Big(\nabla_{e_k}\left \langle R^{TX} (\widetilde{e}_{m}, \widetilde{e}_j) \widetilde{e}_{q},
\widetilde{e}_i\right\rangle\Big)_{x_0},\\
&\left \langle R^{TX}_{\, ; (e_k, \,e_\ell)}
(e_{m}, e_j) e_{q}, e_i\right\rangle
= \Big(\nabla_{e_\ell}\nabla_{e_k}\left \langle
R^{TX} (\widetilde{e}_{m}, \widetilde{e}_j) \widetilde{e}_{q}, \widetilde{e}_i\right\rangle\Big)_{x_0},\\
&\ric_{\, ; \,e_{k}}(e_i,e_j)= (\nabla_{e_k} \ric(\widetilde{e}_i,\widetilde{e}_j))_{x_0},\\
&R^E_{\, ; \,e_{k}}(e_i,e_j)= (\nabla_{e_k} R^E(\widetilde{e}_i,\widetilde{e}_j))_{x_0},\\
&R^E_{\, ;\, (e_{k},\, e_{\ell})}(e_i,e_j)
= (\nabla_{e_\ell}\nabla_{e_k} R^E(\widetilde{e}_i,\widetilde{e}_j))_{x_0}.
\end{split}
\end{align}
We will also use the complex coordinates $z=(z_1,\ldots,z_n)$. Note that
\begin{equation}\label{lm01.3}
e_{2j-1}=\frac{\partial}{\partial Z_{2j-1}}
=\frac{\partial}{\partial z_j}+ \frac{\partial}{\partial \overline{z}_j},
\quad e_{2j}=\frac{\partial}{\partial Z_{2j}}
=\sqrt{-1}\Big(\frac{\partial}{\partial z_j}
- \frac{\partial}{\partial \overline{z}_j}\Big),
\quad \left|\frac{\partial}{\partial z_j}\right|^2=\frac{1}{2}.
\end{equation}
Set
\begin{equation} \label{lm01.4} \begin{split}
&R_{k\overline{m}\ell\overline{q}} =\left \langle R^{TX} \Big(\frac{\partial}{\partial z_k},
\frac{\partial}{\partial \overline{z}_m}\Big) \frac{\partial}{\partial z_\ell},
\frac{\partial}{\partial \overline{z}_q}\right\rangle_{x_0},\quad
R^E_{k\overline{\ell}}= R^E_{x_0}\Big(\frac{\partial}{\partial z_k},
\frac{\partial}{\partial \overline{z}_\ell}\Big),\\
&\ric_{k\overline{\ell}}= \ric_{x_0}\Big(\frac{\partial}{\partial z_k},
\frac{\partial}{\partial \overline{z}_\ell}\Big)\,,\\
&R_{k\overline{m}\ell\overline{q};\,s} =\left \langle R^{TX}_{\, ;
\frac{\partial}{\partial z_{s}}}\Big(\frac{\partial}{\partial z_k},
\frac{\partial}{\partial\overline{z}_m}\Big) \frac{\partial}{\partial z_\ell},
\frac{\partial}{\partial\overline{z}_q}\right\rangle\,,\quad
R^E_{k\overline{q}; \,s}=
R^E_{\, ; \frac{\partial}{\partial z_{s}}}\Big(\frac{\partial}{\partial z_k},
\frac{\partial}{\partial \overline{z}_q}\Big),
\end{split}\end{equation}
and in the same way, we define $R_{k\overline{m}\ell\overline{q};\, \overline{s}}$\,,
$R_{k\overline{m}\ell\overline{q};\, t\overline{s}}$\,,
$\ric_{k\overline{q};\, \overline{s}}$\,, $R^E_{k\overline{q};\, \overline{s}}$,
$R^E_{k\overline{q};\, t\overline{s}}$\,.
Since $R^{TX}$ is a $(1,1)$-form
and $\nabla^E$ is the Chern connection on $(E,h^E)$, we deduce from
\eqref{alm01.0}--\eqref{alm01.1} the following.
\begin{lemma}\label{lmt1.6}
\noindent
\\[2pt]
{\rm(1)} $R_{k\overline{m}\ell\overline{q}} =R_{\ell\overline{m}k\overline{q}}=
R_{k\overline{q}\ell\overline{m}}= R_{\ell\overline{q}k\overline{m}}$\,,
$\br= 8\, R_{m\overline{q}q\overline{m}}$\,, $(R^E_{k\overline{q}})^*
=R^E_{q\overline{k}}$\,, \\[2pt]
{\rm(2)} $R_{k\overline{m}\ell\overline{q}\,;\, \overline{s}} =R_{\ell\overline{m}k\overline{q}\,;\, \overline{s}}=
R_{k\overline{q}\ell\overline{m}\,; \,\overline{s}}= R_{\ell\overline{q}k\overline{m}\,; \,\overline{s}}$\,,
\\[2pt]
{\rm(3)} $\ric$ is a symmetric $(1,1)$-tensor and
$\ric_{m\overline{q}}=2 \, R_{m\overline{k}k\overline{q}}$\,,
$\ric_{m\overline{q}\,; \,\overline{s}}=2 \, R_{m\overline{k}k\overline{q}\,;\, \overline{s}}$\,,
\\[2pt]
{\rm(4)} $R^{TX}_{\, ;\, e_k}$, $R^{TX}_{\, ;\, (e_k,\,e_\ell)}$
are
$(1,1)$-forms with values in $\End(T_{x_0}X)$ which commute with
$J_{x_0}$\,,
\\[2pt]
{\rm(5)}
$R^{E}_{\, ; \,e_k}$\,, $R^{E}_{\, ; \,(e_k,\,e_\ell)}\in \End(E_{x_0})$.
\end{lemma}
Let $\text{div}(\ric)$ be the divergence of $\ric$.
By \cite[\S 2.3.4, Prop. 6]{Petersen06},
\begin{align}\label{alm01.4}
d\br = 2\, \text{div} (\ric) = 2 \,
(\nabla^{T^{*}X}_{\widetilde{e}_{m}}\ric)(\widetilde{e}_{m}, \cdot)\,.
\end{align}
Lemma \ref{lmt1.6} and \eqref{alm01.4} entail
\begin{align}\label{alm01.5}\begin{split}
&R_{\ell\overline{\ell}m\overline{m}\,;\, \overline{k}}
= R_{\ell\overline{\ell}m\overline{k}\,;\, \overline{m}}\,,\quad
R_{\ell\overline{\ell}m\overline{m}\,;\, k}= R_{\ell\overline{\ell}k\overline{m}\,;\, m},\\
-(&\Delta\br)_{\,x_{0}} =2 e_qe_m(\ric(\widetilde{e}_q,\widetilde{e}_m))_{\,x_{0}}
= 32 R_{k\overline{m} q\overline{q}\,;\,m\overline{k}}
= 32 R_{m\overline{m} q\overline{q}\,;\, k\overline{k}}\;\;.
\end{split}\end{align}
Set
\begin{equation}\label{bk2.62}
\mathcal{R}:= \sum_i Z_i e_i =Z, \qquad
\nabla_{0,\scriptstyle\bullet}:= \nabla_{\scriptstyle\bullet}
+ \frac{1}{2} R^L_{x_0}(\mathcal{R}, \,{\scriptstyle\bullet}\,).
\end{equation}
Thus $\mathcal{R}$ is the radial vector field on
$\field{R}^{2n}$. We also introduce the vector fields
$z= \sum_{i} z_{i} \tfrac{\partial}{\partial z_i}$ and
$\overline{z}= \sum_{i} \overline{z}_{i} \tfrac{\partial}{\partial \overline{z}_i}$.
By \cite[Prop. 1.2.2]{MM07}, \eqref{lm01.1}, we have
\begin{align} \label{lm01.5}
\mathcal{R}= \sum_i Z_i \widetilde{e}_i,\quad
z= \sum_{i} z_{i} \widetilde{\tfrac{\partial}{\partial z_i}},\quad
\overline{z}= \sum_{i} \overline{z}_{i} \widetilde{\tfrac{\partial}{\partial \overline{z}_i}}.
\end{align}
By \eqref{toe2.1} and \eqref{toe1.1}, we get
\begin{equation}\label{bk2.64}
b_i=-2\nabla_{0,\tfrac{\partial}{\partial z_i}},\quad
b^{+}_i=2\nabla_{0,\tfrac{\partial}{\partial \overline{z}_i}},\quad
R^L_{x_0} = -2\pi\sqrt{-1}\left \langle J\,\scriptstyle\bullet\,,\,
\scriptstyle\bullet\, \right \rangle_{x_0}.
\end{equation}
Let $A_{1Z}, A_{2Z}\in (T^*X)^{\otimes 2}$ be polynomials in $Z$ with
values symmetric tensors, defined by
\begin{align} \label{lm01.7} \begin{split}
&A_{1Z}(e_i,e_j)=\left\langle R^{TX}_{\, ;(Z,Z)} (\mathcal{R},e_i) \mathcal{R},
e_j\right\rangle_{x_0},\\
&A_{2Z}(e_i,e_j)=\left \langle R^{TX}_{x_0} (\mathcal{R},e_i) \mathcal{R},
R^{TX}_{x_0} (\mathcal{R},e_j) \mathcal{R}\right \rangle_{x_0}.
\end{split}\end{align}
\comment{
Then by (\ref{alm01.0}), $R^{TX}$ is a $(1,1)$-form
and $\mathcal{R}= z + \overline{z}$ as vector fields, we get
\begin{align} \label{lm01.8} \begin{split}
A_{1Z}(e_j,e_j)=& 4A_{1Z}(\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial \overline{z}_j})= 4 \left\langle R^{TX}_{\, ;(Z,Z)}
(z,\tfrac{\partial}{\partial \overline{z}_j})\overline{z} ,
\tfrac{\partial}{\partial z_j}\right\rangle_{x_0},\\
A_{2Z}(e_j,e_j)=& 4A_{2Z}(\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial \overline{z}_j})=4 \left \langle R^{TX}_{x_0}
(\overline{z}, \tfrac{\partial}{\partial z_j}) \mathcal{R},
R^{TX}_{x_0} (z,\tfrac{\partial}{\partial \overline{z}_j}) \mathcal{R}\right \rangle_{x_0}.
\end{split}\end{align}
}
Recall that the operator $\mathscr{L}$ was defined in \eqref{toe1.1}. Set
\begin{align} \label{lm01.9a}
\begin{split}
\mathcal{O}^{\,\prime}_2=&\frac{1}{3} \left \langle R^{TX}_{x_0} (\mathcal{R},e_i)
\mathcal{R}, e_j\right \rangle
\nabla_{0,e_i} \nabla_{0,e_j}
- 2 R^E_{k\overline{k}} \\
&\hspace{10mm} + \left(\left \langle
\frac{\pi}{3} R^{TX}_{x_0} (z,\overline{z}) \mathcal{R}, e_j\right \rangle
+ \frac{2}{3} \ric_{x_0} (\mathcal{R},e_j) -R^E_{x_0} (\mathcal{R}, e_j)\right)
\nabla_{0,e_j},
\end{split}\end{align}
and
\begin{align} \label{lm01.9}
\begin{split}
\mathcal{O}_{41}= & \frac{1}{20}\Big( A_{1Z}
- \frac{4}{3}A_{2Z}\Big) (e_{i}, e_{j})
\nabla_{0,e_i}\nabla_{0,e_j},\\
\mathcal{O}_{42}=&\Big[\mathscr{L}, -\Big(\frac{1}{80}A_{1Z}
- \frac{1}{360}A_{2Z}\Big)(e_{j}, e_{j})
- \frac{1}{288} \ric(\mathcal{R},\mathcal{R})^{2} \Big]
+ \frac{\mathscr{L}}{144} \ric(\mathcal{R},\mathcal{R})^{2},\\
\mathcal{O}_{43}=&-\frac{1}{144} \ric(\mathcal{R},\mathcal{R}) \mathscr{L} \ric(\mathcal{R},\mathcal{R}),\\
\mathcal{O}_{44}=&\Big \{ \frac{\pi}{30} A_{1Z}(\overline{z}, e_{i})
- \frac{\pi}{10} A_{2Z}(\overline{z}, e_{i})
+ \tfrac{\partial}{\partial Z_j} \Big(\frac{1}{20}A_{1Z}
+ \frac{2}{45}A_{2Z} \Big)(e_{i}, e_{j}) \\
&
- \tfrac{\partial}{\partial Z_i} \Big(\frac{1}{40}A_{1Z}
+ \frac{1}{45}A_{2Z} \Big)(e_{j}, e_{j})\Big\}
\nabla_{0,e_i},
\end{split}\end{align}
and
\begin{align} \label{lm01.9b}
\begin{split}
\mathcal{O}_{45}=&\Big \{\frac{2}{9}\left \langle R^{TX}_{x_0} (\mathcal{R},e_k) \mathcal{R},
R^{TX}_{x_0} (\mathcal{R},e_k)e_\ell\right\rangle_{x_0}
- \frac{1}{9} \left \langle R^{TX}_{x_0} (\mathcal{R},e_\ell) \mathcal{R},
e_k\right\rangle_{x_0} \ric(\mathcal{R},e_k)\\
&+\frac{1}{4}
\left \langle R^{TX}_{x_0} (\mathcal{R},e_\ell) \mathcal{R}, e_m\right \rangle_{x_0}
R^E_{x_0} (\mathcal{R},e_m)
- \frac{1}{4} R^E_{\, ;(Z,Z)} (\mathcal{R}, e_\ell)\Big\}
\nabla_{0,e_\ell},\\
\mathcal{O}_{46}=&-\frac{\pi^2}{36} A_{2Z}(\overline{z}, \overline{z})
+ \frac{\pi}{30} \left \langle
R^{TX}_{\, ;(Z,e_\ell)} (z,\overline{z})\mathcal{R}, e_\ell\right \rangle_{x_0}\\
&\hspace{-3mm}
- \frac{\pi}{20}\left \langle R^{TX}_{x_0} (z,\overline{z})\mathcal{R},
e_m \right \rangle_{x_0} \ric(\mathcal{R},e_m)
+ \frac{4}{9} \ric_{k\overline{m}}\ric_{m\overline{\ell}}z_{k}\overline{z}_{\ell}
- \frac{4}{9} R_{k\overline{\ell}m\overline{q}}\ric_{\ell\overline{m}}z_{k}\overline{z}_{q}\\
&\hspace{-3mm}+ \frac{1}{6}\left\langle
\pi R^{TX}_{x_0}(z,\overline{z})\mathcal{R} , e_m\right\rangle
R^E_{x_0} (\mathcal{R},e_m)
+ \frac{1}{8} \ric(\mathcal{R},e_m)R^E_{x_0} (\mathcal{R},e_m)\\
&\hspace{-3mm}+ \frac{1}{2} (R^E_{k\overline{m}}R^E_{m\overline{\ell}}
+ R^E_{m\overline{\ell}}R^E_{k\overline{m}}) z_{k}\overline{z}_{\ell}
- \frac{1}{4}R^E_{\, ;(Z,e_\ell)} (\mathcal{R}, e_\ell)
- R^E_{\, ;(Z,Z)}
(\tfrac{\partial}{\partial z_\ell},\tfrac{\partial}{\partial \overline{z}_\ell}).
\end{split}\end{align}
The following result extends \cite[Th.\,4.1.25]{MM07} where
$\mathscr{L}_0, \mathcal{O}_1,\mathcal{O}_2$ were computed.
\begin{thm} \label{bkt2.21} The following identities hold for the operators
$\mathcal{O}_r$ introduced in \eqref{bk2.22}\,\rm{:}
\begin{subequations}
\begin{align}
\mathscr{L}_0=& \sum_j b_jb^+_j=\mathscr{L}
=-\sum_{i} \nabla_{0,e_{i}}\nabla_{0,e_{i}}- 2\pi n, \qquad \mathcal{O}_1=0,
\label{lm01.101}\\
\mathcal{O}_2=& \mathcal{O}^{\,\prime}_2 - \frac{1}{3} \ric_{x_0}(\mathcal{R},e_j)\nabla_{0,e_j}
- \frac{\br_{x_0}}{6},\label{lm01.102}
\end{align}
\end{subequations}
and
\begin{subequations}
\begin{align}\label{lm01.11}
\mathcal{O}_3=& \frac{1}{6}\left \langle R^{TX}_{\, ; Z} (\mathcal{R},e_i) \mathcal{R}, e_j\right
\rangle_{x_0} \nabla_{0,e_i} \nabla_{0,e_j}\\
&+ \Big[\frac{2\pi}{15}\left \langle
R^{TX}_{\, ; Z} (z,\overline{z})\mathcal{R}, e_i\right\rangle_{x_0}
+\frac{1}{6} \ric_{\, ; Z} (\mathcal{R},e_i)\nonumber\\
&\hspace{10mm}+ \frac{1}{6}
\left\langle R^{TX}_{\, ; e_j} (\mathcal{R}, e_j) \mathcal{R}, e_i\right\rangle_{x_0}
- \frac{2}{3} R^E_{\, ; Z} (\mathcal{R}, e_i)\Big]\nabla_{0,e_i} \nonumber\\
&+ \frac{\pi}{15}\left \langle
R^{TX}_{\, ; e_{j}} (z,\overline{z})\mathcal{R}, e_j\right \rangle_{x_0}
-\frac{1}{6} \ric_{\, ; e_{i}} (\mathcal{R}, e_i)\nonumber\\
& -\frac{1}{12} \ric_{\, ; Z} (e_i, e_i)
-\frac{1}{3} R^E_{\, ; e_{i}} (\mathcal{R}, e_i)- \frac{\sqrt{-1}}{2}
R^E_{\, ; Z} (e_i, Je_i),\nonumber\\
\mathcal{O}_4=&\mathcal{O}_{41}+ \mathcal{O}_{42}+\mathcal{O}_{43}+\mathcal{O}_{44}
+\mathcal{O}_{45}+\mathcal{O}_{46}.\label{lm01.12}
\end{align}
\end{subequations}
\end{thm}
\begin{proof}
Recall that $\widetilde{e}_i (Z)$ is the parallel
transport of ${e}_i$ with respect to $\nabla^{TX}$
along the curve $[0,1]\ni u \to uZ$.
Let $\widetilde{\theta} (Z) = (\theta _j^i (Z))_{i,j=1}^{2n}$
be the $2 n \times 2n$-matrix such that
\begin{eqnarray}\label{lm01.29}
e_i = \sum_j \theta ^j_i(Z) \widetilde{e}_j (Z), \quad
\widetilde{e}_j (Z)= (\widetilde{\theta} (Z)^ {-1})_j^k e_k.
\end{eqnarray}
Taking into account the Taylor expansion
of $\theta ^i_j$ at $0$ we have (cf.\,\cite[(1.2.27)]{MM07})
\begin{align}\label{lm01.35}
\sum_{|\alpha| \geqslant 1} ( |\alpha|^2 + |\alpha|)
(\partial ^\alpha\theta ^i_j)(0) \frac{Z^\alpha}{\alpha !}=
\left \langle R^{TX} ( \mathcal{R},e_j) \mathcal{R}, \widetilde{e}_i\right \rangle_Z .
\end{align}
From this equation, we obtain first that
\begin{eqnarray}\label{lm01.27}
e_j(Z) = \widetilde{e}_j(Z) + \frac{1}{6}
\left \langle R^{TX}_{x_0} (\mathcal{R},e_j) \mathcal{R}, e_k\right \rangle_{x_0}
\widetilde{e}_k(Z) + \mathscr{O}(|Z|^3).
\end{eqnarray}
From \eqref{lm01.2}, \eqref{lm01.5}, \eqref{lm01.7},
\eqref{lm01.35} and \eqref{lm01.27}, we get further
\begin{equation}\label{lm01.36}
\begin{split}
\theta ^i_j = \delta_{ij}
& + \frac{1}{6}
\left \langle R^{TX}_{x_0} (\mathcal{R},e_i) \mathcal{R}, e_j\right \rangle_{x_0}
+ \frac{1}{12}
\left \langle R^{TX}_{\, ; Z} (\mathcal{R},e_i) \mathcal{R}, e_j\right
\rangle_{x_0}\\
& + \frac{1}{20} \Big(\frac{1}{2}A_{1Z}(e_i,e_j)
+ \frac{1}{6}A_{2Z}(e_i,e_j)
\Big) + \mathscr{O}(|Z|^5)\,.
\end{split}
\end{equation}
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 $(g_{ij}(Z))$.
Then by \eqref{lm01.7}, \eqref{lm01.29} and \eqref{lm01.36}, we have
\begin{equation}\label{lm01.37}
\begin{split}
g_{ij}(Z)= \theta^k_i(Z) \theta^k_j(Z)
= \delta_{ij} &+ \frac{1}{3}
\left \langle R^{TX}_{x_0} (\mathcal{R},e_i) \mathcal{R}, e_j\right \rangle_{x_0}
+ \frac{1}{6}\left \langle R^{TX}_{\, ; Z} (\mathcal{R},e_i) \mathcal{R}, e_j\right
\rangle_{x_0}\\
& + \frac{1}{20} A_{1Z}(e_i,e_j)
+ \frac{2}{45}A_{2Z}(e_i,e_j)
+ \mathscr{O} (|Z|^5).
\end{split}
\end{equation}
In view of the expansion $(1+a)^{-1}= 1-a +a^2+\ldots$\,, we obtain
\begin{equation}\label{lm01.38}
\begin{split}
g^{ij}(Z)
= \delta_{ij} & - \frac{1}{3}
\left \langle R^{TX}_{x_0} (\mathcal{R},e_i) \mathcal{R}, e_j\right \rangle_{x_0}
- \frac{1}{6}\left \langle R^{TX}_{\, ; Z} (\mathcal{R},e_i) \mathcal{R}, e_j\right
\rangle_{x_0}\\
&- \frac{1}{20} A_{1Z}(e_i,e_j)
+ \frac{1}{15}A_{2Z}(e_i,e_j)
+ \mathscr{O} (|Z|^5).
\end{split}
\end{equation}
If $\Gamma _{ij}^\ell$ are the Christoffel symbols
of $\nabla ^{TX}$ with respect to the frame $\{e_i\}$, then
$(\nabla ^{TX}_{e_i}e_j)(Z)$ $= \Gamma _{ij}^\ell (Z) e_\ell$.
By
the explicit formula for $\nabla^{TX}$, we get
(cf.\,\cite[(4.1.102)]{MM07}) with $\partial_j:=\frac{\partial}{\partial Z_j}$
\begin{equation}\label{bk2.83}
\begin{split}
\Gamma _{ij}^\ell (Z)& = \frac{1}{2} g^{\ell k} (\partial_i g_{jk}
+ \partial_j g_{ik}-\partial_k g_{ij})(Z)\\
&= \frac{1}{3}\Big [ \left \langle R^{TX}_{x_0}
(\mathcal{R}, e_j) e_i, e_\ell\right \rangle _{x_0}
+ \left \langle R^{TX}_{x_0} (\mathcal{R}, e_i) e_j, e_\ell\right \rangle_{x_0}\Big ]
+ \mathscr{O}(|Z|^2)\,.
\end{split}
\end{equation}
For $j$ fixed, $\Gamma _{jj}^\ell (Z)=\frac{1}{2} g^{\ell k} ( 2\partial_j g_{jk}
-\partial_k g_{jj})(Z)$, thus by \eqref{lm01.37} and \eqref{lm01.38},
\begin{equation}\label{abk2.83}
\begin{split}
\Gamma&_{jj}^\ell (Z) = \frac{2}{3}
\left \langle R^{TX}_{x_0} (\mathcal{R}, e_j) e_j, e_\ell\right \rangle _{x_0}\\
&+ \frac{1}{12} \Big[
4 \left \langle R^{TX}_{\, ; Z} (\mathcal{R}, e_j) e_j, e_\ell\right
\rangle_{x_0}
+ 2\left \langle R^{TX}_{\, ; e_j} (\mathcal{R}, e_j) \mathcal{R}, e_\ell\right
\rangle_{x_0}
+ \left \langle R^{TX}_{\, ; e_\ell} (\mathcal{R}, e_j) e_j,\mathcal{R} \right
\rangle_{x_0} \Big ] \\
&- \frac{2}{9} \left \langle R^{TX}_{x_0} (\mathcal{R},e_\ell) \mathcal{R}, R^{TX}_{x_0}
(\mathcal{R},e_j) e_j\right\rangle_{x_0}
+ \frac{\partial}{\partial Z_j} \Big(
\frac{1}{20}A_{1Z}
+ \frac{2}{45}A_{2Z}\Big)(e_j,e_\ell) \\
&- \frac{1}{2} \frac{\partial}{\partial Z_l}\Big (
\frac{1}{20}A_{1Z}
+ \frac{2}{45}A_{2Z}\Big)(e_j,e_j)
+ \mathscr{O}(|Z|^4).
\end{split}
\end{equation}
Note that
\[
\det(\delta_{ij}+ a_{ij})= 1 + \sum_{i} a_{ii} +
\sum_{i<j}(a_{ii}a_{jj}- a_{ij}a_{ji}) + \ldots
\]
and
\[(1+a)^{1/4}= 1+ \frac{1}{4} a - \frac{3}{32} a^{2} + \ldots
\]
By \eqref{alm01.1} and \eqref{lm01.37}, we get
\begin{equation}\label{bk2.82}
\begin{split}
\kappa(&Z)^{1/2}= |\det (g_{ij}(Z))|^{1/4} = 1 + \frac{1}{12}
\left \langle R^{TX}_{x_0} (\mathcal{R},e_j) \mathcal{R}, e_j\right \rangle_{x_0}\\
&- \frac{1}{24} \ric_{\, ; Z}(\mathcal{R},\mathcal{R})
+ \frac{1}{80} A_{1Z}(e_j,e_j) + \frac{1}{90} A_{2Z}(e_j,e_j)\\
&+ \frac{1}{36} \sum_{i<j} \Big( \left \langle R^{TX}_{x_0} (\mathcal{R},e_i)
\mathcal{R}, e_i\right \rangle_{x_0} \left \langle R^{TX}_{x_0} (\mathcal{R},e_j)
\mathcal{R}, e_j\right \rangle_{x_0}- \left \langle R^{TX}_{x_0} (\mathcal{R},e_i) \mathcal{R},
e_j\right \rangle_{x_0} ^2 \Big)\\
&- \frac{1}{96} \Big(\sum_j\left \langle R^{TX}_{x_0} (\mathcal{R},e_j) \mathcal{R},
e_j\right \rangle_{x_0}\Big) ^2+ \mathscr{O} (|Z|^5)\\
=&\, 1-\frac{1}{12} \ric(\mathcal{R},\mathcal{R}) -\frac{1}{24} \ric_{\, ; Z}(\mathcal{R},\mathcal{R})
+ \Big(\frac{1}{80} A_{1Z}
- \frac{1}{360} A_{2Z}\Big)(e_j,e_j)\\
&+ \frac{1}{288} \ric(\mathcal{R},\mathcal{R})^2 + \mathscr{O} (|Z|^5).
\end{split}
\end{equation}
Thus
\begin{equation}\label{abk2.82}
\begin{split}
\kappa(Z)^{-1/2}
= 1&+\frac{1}{12} \ric(\mathcal{R},\mathcal{R}) +\frac{1}{24} \ric_{\, ; Z}(\mathcal{R},\mathcal{R})
- \Big(\frac{1}{80} A_{1Z}
- \frac{1}{360} A_{2Z}\Big)(e_j,e_j) \\
&+ \Big(\frac{1}{144} -\frac{1}{288}\Big) \ric(\mathcal{R},\mathcal{R})^2 + \mathscr{O} (|Z|^5)\,.
\end{split}
\end{equation}
Observe that $J$ is parallel with respect to $\nabla ^{TX}$,
thus $\left \langle J\widetilde{e}_i,\widetilde{e}_j \right \rangle_{Z}=\left
\langle Je_i,e_j \right \rangle_{x_0}$.
From (\ref{bk2.62}), (\ref{lm01.5}),
(\ref{lm01.29}) and (\ref{lm01.36}), we get
\begin{equation}\label{bk2.84}
\begin{split}
\frac{\sqrt{-1}}{2 \pi} R^L_Z(\mathcal{R},e_\ell)=&\theta ^j_\ell(Z)
\left \langle J\widetilde{e}_i,\widetilde{e}_j \right \rangle_{Z} Z_i
=\theta ^j_\ell(Z)
\left \langle J\mathcal{R},e_j \right \rangle_{x_0} \\
= &\left \langle J\mathcal{R}, e_\ell \right \rangle_{x_0}
+\frac{1}{6}\left \langle R^{TX}_{x_0}(\mathcal{R}, J\mathcal{R})\mathcal{R},
e_\ell\right \rangle_{x_0}\\
+&\frac{1}{12}\left \langle R^{TX}_{\, ; Z}
(\mathcal{R}, J\mathcal{R})\mathcal{R}, e_\ell\right \rangle_{x_0}
+ \frac{1}{40}\left \langle R^{TX}_{\, ;(Z,Z)} (\mathcal{R},
J\mathcal{R})\mathcal{R}, e_\ell\right \rangle_{x_0}\\
+& \frac{1}{120}\left \langle R^{TX}_{x_0} (\mathcal{R}, J\mathcal{R})\mathcal{R},
R^{TX}_{x_0} (\mathcal{R}, e_\ell)\mathcal{R}\right \rangle_{x_0}
+ \mathscr{O}(|Z|^6).
\end{split}
\end{equation}
Let $\Gamma ^\bullet= \Gamma ^E, \Gamma ^L$ and
$R^\bullet= R^E, R^L$, respectively.
By \cite[Lemma 1.2.4]{MM07}, the Taylor coefficients of
$\Gamma ^\bullet (e_\ell) (Z)$ at $x_0$ up to order $r$ are only
determined by those of $R^\bullet$ up to order $r-1$, and
\begin{align}\label{0c39}
\sum_{|\alpha|=r} (\partial^\alpha
\Gamma ^\bullet ) _{x_0} (e_\ell) \frac{Z^\alpha}{\alpha !}
=\frac{1}{r+1} \sum_{|\alpha|=r-1}
(\partial^\alpha R^\bullet ) _{x_0}(\mathcal{R}, e_\ell)
\frac{Z^\alpha}{\alpha !}.
\end{align}
Thus by (\ref{bk2.84}), (\ref{0c39}) and since $R^{TX}$ is a $(1,1)$-form,
we obtain
\[
R^{TX} (\mathcal{R},J\mathcal{R})= -2\sqrt{-1}R^{TX}(z,\overline{z})\,,\quad
\left \langle R^{TX}_{\, ;(Z,Z)} (\mathcal{R}, J\mathcal{R})\mathcal{R},
e_i\right \rangle_{x_0}= -2\sqrt{-1} A_{1Z}(\overline{z}, e_i)
\] and
\begin{equation}\label{0c40}
\begin{split}
t^{-1} \Gamma ^L &(e_i)(tZ)
= -\pi \sqrt{-1} \left \langle J\mathcal{R}, e_i \right \rangle_{x_0}
-t^2\frac{\pi}{6}\left \langle R^{TX}_{x_0} (z,\overline{z})\mathcal{R},
e_i\right \rangle_{x_0}\\
&-t^3\frac{\pi}{15}\left \langle R^{TX}_{\, ; Z} (z,\overline{z})\mathcal{R},
e_i\right \rangle_{x_0}
-t^4 \frac{\pi}{60}A_{1Z}(\overline{z}, e_i)
- t^4 \frac{\pi}{180}A_{2Z}(\overline{z}, e_i)
+ \mathscr{O}(t^5).
\end{split}
\end{equation}
By (\ref{bk2.21}), (\ref{lm01.2}), (\ref{lm01.27}), (\ref{0c39})
and (\ref{0c40}), for $t=\frac{1}{\sqrt{p}}$, we get
\begin{align}\label{bk2.85}\begin{split}
& \Gamma ^E (e_i)(Z) = \frac{1}{2} R^E_{x_0} (\mathcal{R},e_i)
+ \frac{1}{3} R^E_{\, ; Z} (\mathcal{R},e_i) \\
&\hspace*{10mm}+\frac{1}{8}
\Big( R^E_{\, ; (Z,Z)} (\mathcal{R},e_i)
+ \frac{1}{3}\left \langle R^{TX} (\mathcal{R},e_i)\mathcal{R},
e_k\right \rangle_{x_0} R^E_{x_0} (\mathcal{R},e_k) \Big) + \mathscr{O}(|Z|^4),\\
&\nabla'_{t, e_i} =\nabla_{e_i}
+ \frac{1}{t} \Gamma ^L (e_i)(tZ) + t \Gamma ^E (e_i)(tZ) \\
&\hspace*{10mm}
= \nabla_{0,e_i} -\frac{t^2}{6} \left\langle \pi R^{TX}_{x_0}(z,\overline{z})\mathcal{R}
, e_i\right\rangle_{x_0}
+ \frac{t^2}{2} R^E_{x_0} (\mathcal{R},e_i) + \mathscr{O}(t^3).
\end{split}\end{align}
\comment{
we get
\begin{align}\label{bk2.85}
&\nabla_{t, e_i} = \kappa(tZ)^{1/2} \Big [\nabla_{e_i}
+ \Big(\frac{1}{t} \Gamma ^L (e_i) + t \Gamma ^E (e_i)
\Big)(tZ) \Big]\kappa(tZ)^{-1/2}\\
&\hspace*{3mm}
= \nabla_{0,e_i} -\frac{t^2}{6} \left\langle \pi R^{TX}_{x_0}(z,\overline{z})\mathcal{R}
- R^{TX}_{x_0} (\mathcal{R}, e_k)e_k , e_i\right\rangle
+ \frac{t^2}{2} R^E_{x_0} (\mathcal{R},e_i) + \mathscr{O}(t^3). \nonumber
\end{align}
}
By \eqref{toe2.5} and \eqref{bk2.21},
we get,
\begin{equation}\label{bk2.87}
\begin{split}
\mathscr{L}_t= -\kappa(tZ)^{1/2} g^{ij} (tZ) \Big [ \nabla'_{t, e_i}\nabla'_{t, e_j}
- t \Gamma _{ij}^l(t\,{\scriptscriptstyle\bullet}) \nabla'_{t, e_l} \Big ](Z)
\kappa(tZ)^{-1/2} \\
- t^2\tfrac{\sqrt{-1}}{2} R^E (\widetilde{e}_i, J \widetilde{e}_i) (tZ) -2 \pi n.
\end{split}
\end{equation}
We will derive now \eqref{lm01.101} and \eqref {lm01.102} (they
were already obtained in \cite[Th.\,4.1.25]{MM07}).
By using the Taylor expansion of the expressions from \eqref{bk2.87}
(see \eqref{lm01.38}, \eqref{bk2.83}, \eqref{bk2.82}, \eqref{abk2.82},
\eqref{bk2.85}) we obtain immediately the formulas for
$\mathscr{L}_0$ and $\mathcal{O}_1$ given in \eqref{lm01.101}.
In order to compute $\mathcal{O}_2$, observe first that by \eqref{alm01.0}
and the fact that $R^{TX}$ is a (1,1)-form with values in
$\End(TX)$, we get
\begin{multline}\label{abk2.87}
\nabla_{e_j}\left \langle R^{TX}_{x_0} (z,\overline{z}) \mathcal{R},
e_j\right \rangle
= 2\left(\tfrac{\partial}{\partial\overline{z}_j}
\left \langle R^{TX}_{x_0} (z,\overline{z})\overline{z},
\tfrac{\partial}{\partial z_j}\right \rangle
+ \tfrac{\partial}{\partial z_j}\left \langle R^{TX}_{x_0} (z,\overline{z})z,
\tfrac{\partial}{\partial\overline{z}_j}\right \rangle\right) =0.
\end{multline}
Thus from \eqref{lm01.9a}, \eqref{lm01.38}, \eqref{bk2.83}, \eqref{bk2.82},
\eqref{bk2.85}--\eqref{abk2.87}, we have
\begin{align}\label{0c41}
\mathcal{O}_{2}= \mathcal{O}^{\,\prime}_{2} + \Big[\mathscr{L}_{0}, \frac{1}{12} \ric(\mathcal{R},\mathcal{R})\Big].
\end{align}
By the formula of $\mathscr{L}_0$ (see \eqref{lm01.101}) and since
\[
[\mathscr{L}_{0}, \ric(\mathcal{R},\mathcal{R})]= -4 \ric(\mathcal{R},e_{j})\nabla_{0,e_j}
-2 \ric(e_{j},e_{j})\,,
\]
we get from \eqref{0c41} the formula for $\mathcal{O}_{2}$ given in \eqref{lm01.102}.
{}From \eqref{bk2.87}, we have also
\begin{equation}\label{0c43}
\begin{split}
\mathcal{O}_3\;\;= &\;\;\;\frac{1}{6}\left \langle R^{TX}_{\, ; Z}
(\mathcal{R},e_i) \mathcal{R}, e_j\right
\rangle_{x_0} \nabla_{0,e_i} \nabla_{0,e_j}\\
&- \Big[-\frac{2\pi}{15}\left \langle
R^{TX}_{\, ; Z} (z,\overline{z})\mathcal{R}, e_i\right\rangle_{x_0}
+ \frac{2}{3} R^E_{\, ; Z} (\mathcal{R}, e_i)\Big ]\nabla_{0,e_i} \\
&-\frac{\partial}{\partial Z_i}\Big[ -\frac{\pi}{15}\left \langle
R^{TX}_{\, ; Z} (z,\overline{z})\mathcal{R}, e_i\right \rangle_{x_0}
+ \frac{1}{3} R^E_{\, ; Z} (\mathcal{R}, e_i)\Big]\\
&+ \frac{1}{12} \Big[ 4 \ric_{\, ; Z}(\mathcal{R},e_l)
+ 2 \left \langle R^{TX}_{\, ; e_j} (\mathcal{R}, e_j) \mathcal{R}, e_l\right
\rangle_{x_0}
+ \ric_{\, ; e_l} (\mathcal{R},\mathcal{R})\Big ]\nabla_{0,e_l} \\
&+ \Big[\mathscr{L}_0, \frac{1}{24} \ric_{\, ; Z}(\mathcal{R},\mathcal{R})\Big]
- \frac{\sqrt{-1}}{2} R^E_{\, ; Z} (e_i, Je_i).
\end{split}
\end{equation}
In \eqref{0c43}, the first (resp. second and third, resp. fourth,
resp. fifth) term is the contribution of the coefficient
of $t^3$ in $g^{ij}(tZ)$ (resp. $\nabla'_{t,e_i}$,
resp. $t\Gamma^l_{ii}(t\cdot)$, resp. $\kappa^{1/2}(tZ)$).
By the same argument in (\ref{abk2.87})
and the formula of $\mathscr{L}_0$ given in \eqref{lm01.101}, we get
\begin{align}\label{0c44}\begin{split}
\frac{\partial}{\partial Z_i}
\left \langle R^{TX}_{\, ; Z} (z,\overline{z}) \mathcal{R}, e_i\right \rangle_{x_0}
=& \left \langle R^{TX}_{\, ; e_{j}} (z,\overline{z}) \mathcal{R},
e_j\right \rangle_{x_0},\\
[\mathscr{L}_0, \ric_{\, ; Z}(\mathcal{R},\mathcal{R})] =& -2 ( \ric_{\, ; e_i}(\mathcal{R},\mathcal{R})
+2 \ric_{\, ; Z}(\mathcal{R},e_i))\nabla_{0,e_i} \\
&- 4 \ric_{\, ; e_i}(\mathcal{R},e_i)-2 \ric_{\, ; Z}(e_i,e_i).
\end{split}\end{align}
From \eqref{0c43} and \eqref{0c44} we get the formula for $\mathcal{O}_{3}$
asserted in \eqref{lm01.11}. Moreover,
\begin{equation}\label{0c45}
\begin{split}
\mathcal{O}_4\;\;= &\;\;\;\mathcal{O}_{42}
+\Big[\mathcal{O}^{\,\prime}_2, \frac{1}{12} \ric(\mathcal{R},\mathcal{R})\Big] +\mathcal{O}_{43}
+ \mathcal{O}_{41}\\
&+\frac{1}{3}
\left \langle R^{TX}_{x_0} (\mathcal{R},e_i) \mathcal{R}, e_j\right \rangle_{x_0}
\Big\{ \Big(-\frac{1}{3} \left\langle \pi R^{TX}_{x_0}(z,\overline{z})\mathcal{R} ,
e_j\right\rangle_{x_0}
+ R^E_{x_0} (\mathcal{R},e_j)\Big)\nabla_{0,e_i} \\
&\hspace{20mm}-\frac{1}{6} \frac{\partial}{\partial Z_i}
\left\langle \pi R^{TX}_{x_0}(z,\overline{z})\mathcal{R} , e_j\right\rangle_{x_0}
- \frac{2}{3} \left \langle R^{TX}_{x_0} (\mathcal{R}, e_i) e_j,
e_l\right \rangle _{x_0} \nabla_{0,e_l}\Big\} \\
&- \Big(-\frac{1}{6} \left\langle \pi R^{TX}_{x_0}(z,\overline{z})\mathcal{R},
e_i\right\rangle_{x_0}
+ \frac{1}{2} R^E_{x_0} (\mathcal{R},e_i)
- \frac{2}{3} \left \langle R^{TX}_{x_0} (\mathcal{R}, e_j) e_j,
e_i\right \rangle _{x_0}\Big)\\
&\hspace{45mm}
\times \Big(-\frac{1}{6} \left\langle \pi R^{TX}_{x_0}(z,\overline{z})\mathcal{R}
, e_i\right\rangle_{x_0}
+ \frac{1}{2} R^E_{x_0} (\mathcal{R},e_i)\Big)\\
&-\Big \{ -\mathcal{O}_{44} + \Big[- \frac{\pi}{9} A_{2Z}(\overline{z},e_i)
+ \frac{1}{12}\left \langle R^{TX} (\mathcal{R},e_i)\mathcal{R},
e_m\right \rangle_{x_0} R^E_{x_0} (\mathcal{R},e_m)\\
&\hspace{20mm}+ \frac{2}{9} \left \langle R^{TX}_{x_0} (\mathcal{R},e_i) \mathcal{R},
e_k\right\rangle_{x_0} \ric(\mathcal{R},e_k)+ \frac{1}{4} R^E_{\, ;(Z,Z)} (\mathcal{R}, e_i)
\Big]\nabla_{0,e_i} \Big\}\\
&- \frac{\partial}{\partial Z_i}
\Big(-\frac{\pi}{60}A_{1Z}(\overline{z},e_i)
- \frac{\pi}{180}A_{2Z}(\overline{z},e_i)
+ \frac{1}{8} R^E_{\, ;(Z,Z)} (\mathcal{R}, e_i)\\
&\hspace{55mm}+ \frac{1}{24}\left \langle R^{TX}_{x_0} (\mathcal{R},e_i)\mathcal{R},
e_k\right \rangle_{x_0} R^E_{x_0} (\mathcal{R},e_k)\Big)\\
&-\frac{\sqrt{-1}}{4} R^E_{\, ;(Z,Z)} (e_i, J e_i)\,.
\end{split}
\end{equation
Here
\begin{itemize}
\item $\mathcal{O}_{42}$ is the contribution of
the coefficients of $t^4$ in $\kappa^{1/2}(tZ)$ and $\kappa^{-1/2}(tZ)$,
\item the second term is the contribution of
the coefficients of $t^2$ in
$\kappa ^{1/2}(tZ)$, $\kappa ^{-1/2}(tZ)$ and in
$-g^{ij}(tZ) (\nabla'_{t,e_i}\nabla'_{t,e_j}
- t \Gamma^\ell_{ij}(tZ)\nabla'_{t,e_\ell})$,
\item $\mathcal{O}_{43}$ is the contribution of
the coefficients of $t^2$ in
$\kappa ^{1/2}(tZ)$ and $\kappa ^{-1/2}(tZ)$,
\item $\mathcal{O}_{41}$ is the contribution of
the coefficients of $t^4$ in $g^{ij}(tZ)$,
\item the fifth term is the contribution of
the coefficients of $t^2$ in
$-g^{ij}(tZ)$ and in
$ (\nabla'_{t,e_i}\nabla'_{t,e_j}
- t \Gamma^\ell_{ij}(tZ)\nabla'_{t,e_\ell})$,
\item the sixth, seventh and eight terms are the contributions of
the coefficients of $t^4$ in $-(\nabla'_{t,e_i}\nabla'_{t,e_i}
- t \Gamma^\ell_{ii}(tZ)\nabla'_{t,e_\ell})$:
the sixth term is the contribution of the coefficients of $t^2$ in
$\nabla'_{t,e_i}$, $- t \Gamma^\ell_{ii}(tZ)$ and $t^2$ in $\nabla'_{t,e_i}$;
the seventh and eighth terms are the contributions
of the coefficients of $t^4$ in $\nabla'_{t,e_i}$ and $-t \Gamma^\ell_{ii}(tZ)$.
\end{itemize}
\noindent
Now by \eqref{lm01.9a},
\begin{equation}\label{0c46}
\begin{split}
\frac{1}{12}[\mathcal{O}^{\,\prime}_2, \ric(\mathcal{R},\mathcal{R})] =
\frac{1}{36}\left \langle R^{TX}_{x_0} (\mathcal{R},e_i) \mathcal{R}, e_j\right
\rangle_{x_0}
( 4 \ric(\mathcal{R},e_j) \nabla_{0,e_i} + 2 \ric(e_i,e_j))\\
+ \frac{1}{6} \left(\left \langle
\frac{\pi}{3} R^{TX}_{x_0} (z,\overline{z}) \mathcal{R}, e_j\right \rangle
+ \frac{2}{3} \ric (\mathcal{R},e_j) -R^E_{x_0} (\mathcal{R}, e_j)\right)\ric(\mathcal{R},e_j).
\end{split}
\end{equation}
and the same argument used to obtain \eqref{abk2.87} shows that
\begin{align} \label{0c47}
\begin{split}
&\left \langle R^{TX}_{x_0} (\mathcal{R},e_i) \mathcal{R}, e_j\right \rangle_{x_0}
\frac{\partial}{\partial Z_i}
\left\langle R^{TX}_{x_0}(z,\overline{z})\mathcal{R} , e_j\right\rangle
= \left \langle R^{TX}_{x_0} (\mathcal{R},e_i) \mathcal{R},
R^{TX}_{x_0}(z,\overline{z}) e_i\right\rangle,\\
& \frac{\partial}{\partial Z_i}(A_{1Z}(\overline{z},e_i))
= \frac{\partial}{\partial Z_i}\left \langle
R^{TX}_{\, ;(Z,Z)} (z,\overline{z})\mathcal{R}, e_i\right \rangle_{x_0}
= 2 \left \langle R^{TX}_{\, ;(Z,e_i)} (z,\overline{z})\mathcal{R}, e_i\right \rangle_{x_0},\\
&\frac{\partial}{\partial Z_i}(A_{2Z}(\overline{z},e_i))
= \frac{\partial}{\partial Z_i}\left \langle R^{TX}_{x_0} (z,\overline{z})\mathcal{R},
R^{TX}_{x_0} (\mathcal{R}, e_i)\mathcal{R}\right \rangle_{x_0}\\
&\hspace{10mm}= \left \langle R^{TX}_{x_0} (z,\overline{z})e_i,
R^{TX}_{x_0} (\mathcal{R}, e_i)\mathcal{R}\right \rangle_{x_0}
+ \left \langle R^{TX}_{x_0} (z,\overline{z})\mathcal{R}, e_j \right \rangle_{x_0}
\ric(\mathcal{R},e_j) .
\end{split}\end{align}
Finally, by \eqref{alm01.0} and since $R^{TX}$ is a $(1,1)$-form, we obtain
\begin{align}\label{0c50}\begin{split}
& \left \langle R^{TX}_{x_0} (z,\overline{z})e_\ell,
R^{TX}_{x_0} (\mathcal{R}, e_\ell)\mathcal{R}\right \rangle
= \left \langle R^{TX}_{x_0} (z,\overline{z})e_\ell, e_m \right \rangle
\left \langle R^{TX}_{x_0} (\mathcal{R}, e_\ell)\mathcal{R}, e_m\right \rangle =0,\\
&\left \langle R^{TX}_{x_0} (\mathcal{R},e_\ell) \mathcal{R}, e_m\right
\rangle_{x_0} \ric(e_\ell,e_m)
= -8 R_{k\overline{\ell}m\overline{q}} \ric_{\ell\overline{m}} \, z_{k} \overline{z}_{q}.
\end{split}\end{align}
Thus
\begin{equation}\label{0c49}
\begin{split}
\mathcal{O}_4=&\sum_{\alpha=1}^{5}\mathcal{O}_{4\alpha}
-\frac{\pi^2}{36} \left\langle R^{TX}_{x_0}(z,\overline{z})\mathcal{R},
R^{TX}_{x_0}(z,\overline{z})\mathcal{R}\right\rangle - \frac{1}{4}R^E_{x_0} (\mathcal{R},e_j)^2 \\
&+ \frac{1}{6}\left\langle \pi R^{TX}_{x_0}(z,\overline{z})\mathcal{R} , e_j\right\rangle
R^E_{x_0} (\mathcal{R},e_j)
+ \frac{\pi}{30} \left \langle
R^{TX}_{\, ;(Z,e_i)} (z,\overline{z})\mathcal{R}, e_i\right \rangle_{x_0}\\
&-\frac{\pi}{20} \left \langle R^{TX}_{x_0} (z,\overline{z})\mathcal{R},
e_j \right \rangle_{x_0} \ric(\mathcal{R},e_j)
+ \frac{1}{9} \ric(\mathcal{R},e_j)\ric(\mathcal{R},e_j)\\
&+ \frac{1}{18}\left \langle R^{TX}_{x_0} (\mathcal{R},e_i) \mathcal{R}, e_j\right
\rangle_{x_0} \ric(e_i,e_j)
+
\frac{1}{8} \ric(\mathcal{R},e_j)R^E_{x_0} (\mathcal{R},e_j)\\
& - \frac{1}{4} R^E_{\, ;(Z,e_i)} (\mathcal{R}, e_i)
- R^E_{\, ;(Z,Z)}
(\tfrac{\partial}{\partial z_i},\tfrac{\partial}{\partial \overline{z}_i}).
\end{split}
\end{equation}
\noindent
{}Putting together Lemma \ref{lmt1.6}, \eqref{lm01.7}, \eqref{lm01.9}, \eqref{0c50},
\eqref{0c49} and the fact that $R^{TX}$ is a $(1,1)$-form,
we infer \eqref{lm01.12}.
The proof of Theorem \ref{bkt2.21} is completed.
\end{proof}
\section{Evaluation of $\mathscr{F}_4$ from \eqref{bk2.32}}\label{bks3}
We calculate in this section an explicit formula for the operator $\mathscr{F}_4$,
defined by \eqref{bk2.24} and appearing in the Bergman kernel expansion
\eqref{toe2.9}. This
is necessary in Section \ref{toes4} in order to evaluate the
expansion of the kernel of the Berezin-Toeplitz operators.
We use formula \eqref{bk2.32} to achieve our aim.
Recall that explicit formulas for the operators $\mathscr{L}$, $\mathcal{O}_2$, $\mathcal{O}_4$
appearing in \eqref{bk2.32} were given in Theorem~\ref{bkt2.21}.
This section is organized as follows.
In Section \ref{bks3.2}, we determine the terms in \eqref{bk2.32} which
involve $\mathcal{O}_{2}$. In
Section \ref{bks3.3}, we calculate the terms in \eqref{bk2.32} which involve
$\mathcal{O}_{4}$.
In Section \ref{bks3.4}, we
obtain the formula for $\mathscr{F}_4(0,0)$ (cf.\;Theorem \ref{bkt3.0}).
We adopt the convention that all tensors will be evaluated at the base
point $x_0\in X$ and most of time, we will omit the subscript $x_0$.
\subsection{Contribution of
$\mathcal{O}_{2}$ to $\mathscr{F}_4(0,0)$}\label{bks3.2}
\begin{lemma} \label{bkt3.1} The following identities hold:
\begin{subequations}
\begin{align}
&\mathscr{P} \mathcal{O}_2 \mathscr{P}=0,\label{bk3.0a}\\
&(\mathscr{L}^{-1} \mathcal{O}_2 \mathscr{P}\mathcal{O}_2\mathscr{L}^{-1})(0,0)
=\frac{1}{4\pi^2} \Big(\sum_{km}R_{m\overline{m}k\overline{k}}
+ \sum_k R^E_{k\overline{k}}\Big)^2,\label{bk3.0b}\\
& \label{bk3.0c}(\mathscr{P} \mathcal{O}_2\mathscr{L}^{-2} \mathcal{O}_2 \mathscr{P})(0,0)\\
&\hspace{10mm} =
\frac{1}{36\pi^2} R_{m\overline{k}q\overline{\ell}} R_{k\overline{m}\ell\overline{q}}
+ \frac{1}{4\pi^2} \Big(\frac{4}{3}R_{q\overline{m}m\overline{\ell}}
+ R^E_{q\overline{\ell}}\Big)
\Big(\frac{4}{3} R_{\ell\overline{k}k\overline{q}} + R^E_{\ell\overline{q}}\Big).\nonumber
\end{align}
\end{subequations}
\end{lemma}
\begin{proof}
Note that by \eqref{toe1.1} and (\ref{toe1.3}),
\begin{equation} \label{bk3.1}
(b^+_i\mathscr{P})(Z,Z^{\prime}) =0\,,\quad \,
(b_i\mathscr{P})(Z,Z^{\prime})=2\pi (\overline{z}_i-\overline{z}^{\,\prime}_i)\mathscr{P}(Z,Z^{\prime}).
\end{equation}
For $\phi\in T^{*}X$, by (\ref{lm01.3}),
(\ref{bk2.64}) and (\ref{bk3.1}), we have
\begin{align}\label{bk3.2}\begin{split}
&\phi(e_{i}) e_{i}= 2 \phi(\tfrac{\partial}{\partial z_j})
\tfrac{\partial}{\partial\overline{z}_j}
+ 2 \phi(\tfrac{\partial}{\partial\overline{z}_j})
\tfrac{\partial}{\partial z_j},\quad
\phi(e_{i}) \nabla_{0, e_{i}}= \phi(\tfrac{\partial}{\partial z_j})
b^{+}_{j}
- \phi(\tfrac{\partial}{\partial\overline{z}_j}) b_{j},\\
&\phi(e_{i}) \nabla_{0, e_{i}}\mathscr{P}(Z,0)
= -2\pi \phi(\overline{z}) \mathscr{P}(Z,0) .
\end{split}\end{align}
\noindent
By Lemma \ref{lmt1.6}, \eqref{alm01.0}, \eqref{lm01.4}, \eqref{lm01.102},
\eqref{bk3.2} and the fact that $R^{TX}$ is a $(1,1)$-form, we get
\begin{equation} \label{bk3.3}
\begin{split}
\mathcal{O}_2 & = \frac{1}{3} R_{k\overline{m}\ell\overline{q}} z_{k} z_{\ell} b_{m} b_{q}
+ \frac{1}{3} R_{k\overline{q}\ell\overline{m}} z_{k} \overline{z}_{m} (b_{q} b^+_{\ell}+
b^+_{\ell}b_{q} )
+ \frac{1}{3} R_{k\overline{m}\ell\overline{q}} \overline{z}_{m} \overline{z}_{q} b^+_{k} b^+_{\ell}
- \frac{4}{3} R_{m\overline{m}q\overline{q}}\\
&\quad+ \Big( \frac{2}{3}R_{k\overline{m}\ell\overline{k}} \overline{z}_{m}
- \frac{\pi}{3} R_{k\overline{m}\ell\overline{q}} z_k \overline{z}_{m} \overline{z}_{q}\Big) b^+_l
- \Big( \frac{2}{3}R_{\ell\overline{k}k\overline{q}} z_\ell
+ \frac{\pi}{3} R_{k\overline{m}\ell\overline{q}} z_k \overline{z}_{m} z_{\ell}\Big) b_q\\
&\quad- 2 R^E_{q\overline{q}}
- R^E(\overline{z},\tfrac{\partial}{\partial z_\ell}) b^+_\ell
+ R^E(z,\tfrac{\partial}{\partial\overline{z}_q}) b_q\,.
\end{split}
\end{equation}
By Lemma \ref{lmt1.6}, \eqref{bk2.66} and \eqref{bk3.3}, we get
$ [R_{k\overline{m}\ell\overline{q}} z_{k} z_{\ell}, b_{m} b_{q}]
= 8 b_q R_{k\overline{k}\ell\overline{q}} z_{\ell} + 8 R_{m\overline{m}q\overline{q}}$, and
\begin{equation} \label{bk3.4}
\begin{split}
\mathcal{O}_2 & = \frac{1}{3} b_{m} b_{q} R_{k\overline{m}\ell\overline{q}} z_{k} z_{\ell}
+ b_{q} \Big(- \frac{\pi}{3} R_{k\overline{m}\ell\overline{q}} z_k z_{\ell}\overline{z}_{m}
+ 2R_{\ell\overline{k}k\overline{q}} z_\ell+R^E_{\ell\overline{q}}z_\ell\Big)\\
&\quad+ \Big( \frac{2 b_q}{3} R_{k\overline{m}\ell\overline{q}} z_{k} \overline{z}_{m}
- \frac{\pi}{3} R_{k\overline{m}\ell\overline{q}} z_k \overline{z}_{m}\overline{z}_{q}
+ 2R_{k\overline{k}\ell\overline{m}} \overline{z}_{m}
+R^E_{\ell\overline{m}}\overline{z}_m \Big) b^+_{\ell}\\
&\quad+ \frac{1}{3} R_{k\overline{m}\ell\overline{q}} \overline{z}_{m} \overline{z}_{q}
b^+_{k} b^+_{\ell}.
\end{split}
\end{equation}
Thus Lemma \ref{lmt1.6}, \eqref{bk2.66}, \eqref{bk3.1} and \eqref{bk3.4} yield
\begin{equation} \label{bk3.5}
\begin{split}
\mathcal{O}_2\mathscr{P} &= \Big\{\frac{1}{3} b_{m} b_{q} R_{k\overline{m}\ell\overline{q}} z_{k} z_{\ell}
+ b_{q} \Big(- \frac{\pi}{3} R_{k\overline{m}\ell\overline{q}} z_k z_{\ell}
(\frac{b_m}{2\pi} +\overline{z}_{m}')
+ 2R_{\ell\overline{k}k\overline{q}} z_\ell +R^E_{\ell\overline{q}}z_\ell\Big)\Big\}\mathscr{P}\\
&= \Big\{\frac{1}{6} b_{m} b_{q} R_{k\overline{m}\ell\overline{q}} z_{k} z_{\ell}
+ \frac{4}{3}b_{q} R_{\ell\overline{k}k\overline{q}} z_\ell
- \frac{\pi}{3}b_{q} R_{k\overline{m}\ell\overline{q}} z_kz_\ell \overline{z}_{m}'
+ b_q R^E_{\ell\overline{q}}z_\ell\Big\}\mathscr{P} .
\end{split}
\end{equation}
Now, \eqref{bk3.0a} follows from Theorem \ref{bkt2.17}, \eqref{toe1.4} and \eqref{bk3.5}. These imply also
\begin{equation} \label{bk3.6}
\begin{split}
&\mathscr{L}^{-1} \mathcal{O}_2 \mathscr{P} =
\Big\{\frac{b_{m} b_{q}}{48\pi} R_{k\overline{m}\ell\overline{q}}\, z_{k}\, z_{l}
+ \frac{b_{q}}{3\pi} R_{\ell\overline{k}k\overline{q}}\, z_\ell
- \frac{b_{q}}{12} R_{k\overline{m}\ell\overline{q}}\, z_k\,z_\ell\, \overline{z}_{m}^{\,\prime}
+ \frac{b_q}{4\pi} R^E_{\ell\overline{q}}\,z_\ell\Big\}\mathscr{P} .
\end{split}
\end{equation}
Due to \eqref{bk2.66} and \eqref{bk3.6} we have
\begin{equation} \label{bk3.7}
\begin{split}
&(\mathscr{L}^{-1} \mathcal{O}_2 \mathscr{P})(Z,0) =
\Big\{\frac{b_{m} b_{q}}{48\pi} R_{k\overline{m}\ell\overline{q}} z_{k} z_{\ell}
+ \frac{b_q}{4\pi}\Big( \frac{4}{3} R_{\ell\overline{k}k\overline{q}}
+ R^E_{\ell\overline{q}}\Big)z_\ell\Big\}\mathscr{P}(Z,0) ,\\
&(\mathscr{L}^{-1} \mathcal{O}_2 \mathscr{P})(0,Z) =
- \frac{1}{2\pi} (R_{m\overline{m}q\overline{q}} + R^E_{q\overline{q}})\mathscr{P}(0,Z).
\end{split}
\end{equation}
Since $\mathcal{O}_2$, $\mathscr{L}$ are symmetric (as explained in Remark
\ref{toet2.7}) and $(R^E_{\ell\overline{q}})^{*} = R^E_{q\overline{\ell}}$\,, we get
by \eqref{toe1.1}, \eqref{lm01.4} and \eqref{bk3.7},
\begin{equation} \label{bk3.8}
\begin{split}
&(\mathscr{P} \mathcal{O}_2 \mathscr{L}^{-1})(0,Z) = ((\mathscr{L}^{-1} \mathcal{O}_2 \mathscr{P})(Z,0))^* \\
&\hspace{32mm}
= \Big\{\mathscr{P} \Big[ R_{u\overline{v}q\overline{s}} \overline{z}_{v} \overline{z}_{s}
\frac{b^+_{u} b^+_{q}}{48\pi}
+ \Big(\frac{4}{3} R_{q\overline{v}v\overline{s}} + R^E_{q\overline{s}}\Big)
\overline{z}_{s}\frac{b^+_{q}}{4\pi}\Big]\Big\}(0,Z) ,\\
&(\mathscr{P} \mathcal{O}_2 \mathscr{L}^{-1})(Z,0) = ( (\mathscr{L}^{-1} \mathcal{O}_2 \mathscr{P})(0,Z))^*
=- \frac{1}{2\pi} (R_{k\overline{k}q\overline{q}} + R^E_{q\overline{q}})\mathscr{P}(Z,0).
\end{split}
\end{equation}
Note that $\mathscr{P}(0,0)=1$ by \eqref{toe1.3}.
From \eqref{toe1.4}, \eqref{bk2.66}, \eqref{bk3.1}, \eqref{bk3.7} and \eqref{bk3.8},
we get \eqref{bk3.0b}, and
\begin{equation} \label{bk3.9}
\begin{split}
(\mathscr{P} \mathcal{O}_2\mathscr{L}^{-2} \mathcal{O}_2 \mathscr{P})(0,0)
= \mathscr{P} \Big\{&\frac{32\pi^2}{(48\pi)^2}
R_{m\overline{s}q\overline{t}} \overline{z}_{s} \overline{z}_{t}
R_{k\overline{m}\ell\overline{q}} z_{k} z_{\ell} \\
&+ \frac{1}{4\pi} \Big(\frac{4}{3}R_{q\overline{s}s\overline{t}}
+ R^E_{q\overline{t}}\Big) \overline{z}_{t}
\Big(\frac{4}{3} R_{\ell\overline{k}k\overline{q}} +
R^E_{\ell\overline{q}}\Big)z_\ell\Big\}\mathscr{P}(0,0) .
\end{split}
\end{equation}
Let $\phi\in\field{C}[b,z]$ be a polynomial in
$b,z$. By \eqref{bk2.66}, \eqref{bk3.1}, we have
\begin{align}\label{bk3.10}\begin{split}
&(\overline{z}_k \phi(b,z) \mathscr{P})(Z,0) =
\phi(b,z)\frac{b_k}{2\pi} \mathscr{P}(Z,0)
= \Big(\frac{b_k}{2\pi}\phi
+ \frac{1}{\pi}\frac{\partial \phi}{\partial z_k}\Big)\mathscr{P}(Z,0),\\
&(\overline{z}_k\overline{z}_l \phi(b,z) \mathscr{P})(Z,0)=(\phi(b,z)\overline{z}_k\overline{z}_l \mathscr{P})(Z,0) \\
&\hspace{30mm} = \Big(\frac{b_k b_l}{4\pi^2}\phi
+\frac{b_k}{2\pi^2}\frac{\partial \phi}{\partial z_l}
+\frac{b_l}{2\pi^2}\frac{\partial \phi}{\partial z_k}
+ \frac{1}{\pi^2}\frac{\partial^2 \phi}{\partial z_k\partial z_l}
\Big)\mathscr{P}(Z,0).
\end{split}\end{align}
\noindent
Let $F(Z)$ be a homogeneous degree $2$ polynomial in $Z$.
By \eqref{bk3.1},
\begin{align}\label{bk3.15}
\left(F(Z)\mathscr{P}\right)(Z,0) = \Big(\frac{1}{2}
\frac{\partial ^2 F}{\partial z_i\partial z_j} z_iz_j +
\frac{\partial ^2 F}{\partial z_i\partial \overline{z}_j} z_i
\frac{b_j}{2\pi} + \frac{1}{2}\frac{\partial ^2 F}{\partial
\overline{z}_i\partial \overline{z}_j}
\frac{b_ib_j}{4\pi^2} \Big) \mathscr{P}(Z,0).
\end{align}
Thus from \eqref{toe1.4}, \eqref{bk3.10} and \eqref{bk3.15},
we have
\begin{align}\label{abk3.8}\begin{split}
&(\mathscr{P} F \mathscr{P})(Z,0)=
\Big(\sum_{|\alpha|=2}\frac{\partial^2 F}{\partial {z}^\alpha}
\frac{z^\alpha}{\alpha !}
+\frac{1}{\pi} \frac{\partial ^2 F}{\partial z_{i} \partial \overline{z}_{i}}
\Big) \mathscr{P}(Z,0),\\
&(\mathscr{P} F \overline{z}_i\overline{z}_j \mathscr{P})(0,0)= \frac{1}{\pi^2}
\frac{\partial ^2 F}{\partial z_{i} \partial z_{j}}.
\end{split}\end{align}
By \eqref{bk3.9} and \eqref{abk3.8}, we get \eqref{bk3.0c}.
The proof of Lemma \ref{bkt3.1} is completed.
\end{proof}
\begin{lemma}\label{bkt3.2} The following identity holds
\begin{equation}\label{bk3.12}
\begin{split}
\pi^2 (\mathscr{L}^{-1}\mathscr{P}^{\bot}\mathcal{O}_2\mathscr{L}^{-1} \mathcal{O}_2 \mathscr{P})(0,0)=
- \frac{25}{2^3\cdot 3^3} R_{m\overline{k}q\overline{\ell}}R_{k\overline{m}\ell\overline{q}}
- \frac{47}{54} R_{k\overline{k}q\overline{\ell}} R_{m\overline{m}\ell\overline{q}} \\
+ \frac{1}{8} R_{k\overline{k}\ell\overline{\ell}} R_{m\overline{m}q\overline{q}}
+ \frac{1}{4}R^E_{\ell\overline{\ell}} R_{m\overline{m}q\overline{q}}
- \frac{7}{6} R^E_{q\overline{\ell}} R_{m\overline{m}\ell\overline{q}}
+ \frac{1}{8} (R^E_{\ell\overline{\ell}} R^E_{q\overline{q}}
-3 R^E_{q\overline{\ell}} R^E_{\ell\overline{q}}).
\end{split}
\end{equation}
\end{lemma}
\begin{proof}
Set
\begin{align}\label{bk3.13}\begin{split}
\bI_1 =& \Big\{\frac{1}{3} b_{k} b_{\ell} R_{s\overline{k}t\overline{\ell}} z_{s} z_{t}
+ b_{\ell} \Big[- \frac{\pi}{3} R_{s\overline{k}t\overline{\ell}}
z_{s} z_{t} \overline{z}_{k}
+ \big(2R_{t\overline{s}s\overline{\ell}} + R^E_{t\overline{\ell}}) z_{t}\big) \Big]\Big\}
\\
&\times \Big[\frac{b_{m} b_{q}}{48\pi} R_{i\overline{m}j\overline{q}} z_{i} z_{j}
+ \frac{b_{q}}{4\pi} \big( \frac{4}{3}R_{j\overline{i}i\overline{q}} +
R^E_{j\overline{q}}\big) z_j\Big],\\
\bI_2=& \Big[ \frac{2 b_{\ell}}{3} R_{s\overline{k}q\overline{\ell}} z_{s} \overline{z}_{k}
- \frac{\pi}{3} R_{s\overline{k}q\overline{\ell}} z_{s}\overline{z}_{k}\overline{z}_{\ell}
+ \big(2R_{s\overline{s}q\overline{k}} + R^E_{q\overline{k}}\big) \overline{z}_{k} \Big]\\
&\times \Big(\frac{b_{m}}{6} R_{i\overline{m}j\overline{q}} z_{i} z_{j}
+ \frac{4}{3} R_{j\overline{i}i\overline{q}} z_j + R^E_{j\overline{q}}z_j \Big).
\end{split}\end{align}
Then by Lemma \ref{lmt1.6}, \eqref{bk2.66}, \eqref{bk3.1}, \eqref{bk3.4}
and \eqref{bk3.7}, we get as in \eqref{bk3.9} that
\begin{align} \label{bk3.14}
(\mathcal{O}_2\mathscr{L}^{-1} \mathcal{O}_2 \mathscr{P})(Z,0)= \Big\{\bI_1 +
\bI_2
+ \frac{2\pi}{9} R_{m\overline{s}q\overline{t}}\,\overline{z}_{s}\overline{z}_{t}
R_{k\overline{m}\ell\overline{q}} z_{k} z_{\ell}\Big\} \mathscr{P} (Z,0).
\end{align}
Let $h(z)=\sum_i h_i z_i $, $h'(z)=\sum_i h'_i z_i$
with $h_i, h'_i\in\field{C}$, and
let $F(Z)$ be a homogeneous degree $2$ polynomial in $Z$.
\comment{
By \eqref{bk3.1},
\begin{align}\label{bk3.15}
\left(F(Z)\mathscr{P}\right)(Z,0) = \Big(\frac{1}{2}
\frac{\partial ^2 F}{\partial z_i\partial z_j} z_iz_j +
\frac{\partial ^2 F}{\partial z_i\partial \overline{z}_j} z_i
\frac{b_j}{2\pi} + \frac{1}{2}\frac{\partial ^2 F}{\partial
\overline{z}_i\partial \overline{z}_j}
\frac{b_ib_j}{4\pi^2} \Big) \mathscr{P}(Z,0).
\end{align}
}By Theorem \ref{bkt2.17}, \eqref{toe1.4}, \eqref{bk2.66},
\eqref{bk3.1}, \eqref{bk3.10} and \eqref{bk3.15}, we have
\begin{subequations}
\begin{align}
&(\mathscr{P}^{\bot} F\mathscr{P})(0,0)
=-\frac{1}{\pi} \frac{\partial ^2 F}{\partial z_i\partial \overline{z}_i}\;,
\label{bk3.16a}\\
&\left(\mathscr{L}^{-1} \mathscr{P}^{\bot} h b_i \mathscr{P}\right)(0,0)=
\left(\mathscr{L}^{-1} b_ih \mathscr{P}\right)(0,0)
=-\frac{1}{2\pi} h_i\;,\label{bk3.16c}\\
&\left(\mathscr{L}^{-1} \mathscr{P}^{\bot} F \mathscr{P}\right)(0,0)= -\frac{1}{4\pi ^2}
\frac{\partial ^2 F}{\partial z_i\partial \overline{z}_i}\;,\label{bk3.16d}\\
&\left(\mathscr{L}^{-1} b_j F b_i \mathscr{P}\right)(0,0)=
-\left(\mathscr{L}^{-1} b_i b_j F \mathscr{P}\right)(0,0)= -\frac{1}{2\pi}
\frac{\partial ^2 F}{\partial z_i\partial z_j}\;,\label{bk3.16e}\\
&\left(\mathscr{L}^{-1} \mathscr{P}^{\bot}F b_i b_j \mathscr{P}\right)(0,0)
= \mathscr{L}^{-1} \mathscr{P}^{\bot}\Big( b_jF b_i
+ 2 \frac{\partial F}{\partial z_j} b_i\Big) \mathscr{P}(0,0)
= -\frac{3}{2\pi} \frac{\partial ^2 F}{\partial z_i\partial z_j}\;,
\label{bk3.16f}\\
&\left(\mathscr{L}^{-1} \mathscr{P}^{\bot}F \overline{z}_{i} \overline{z}_{j}\mathscr{P}\right)(0,0)
=\mathscr{L}^{-1} \mathscr{P}^{\bot}F\,\frac{b_{i} b_{j}}{4\pi^2}\mathscr{P}(0,0)
= -\frac{3}{8\pi^3} \frac{\partial ^2 F}{\partial z_i\partial z_j}\;,
\label{bk3.16g}\\
&\left(\mathscr{L}^{-1} b_k F \overline{z}_{l} \mathscr{P}\right)(0,0)
=\mathscr{L}^{-1} b_k F \frac{b_{l}}{2\pi}\mathscr{P}(0,0)
= -\frac{1}{4\pi^{2}} \frac{\partial ^2 F}{\partial z_k\partial z_l}\;.
\label{bk3.16h}
\end{align}
\end{subequations}
In \eqref{bk3.16d} and \eqref{bk3.16e}, we have used
$F b_i= b_i F + 2 \frac{\partial F}{\partial z_i}$.
Observe that \eqref{toe1.3}, \eqref{bk2.66} imply that for every
homogeneous degree $k$ polynomial $G$ in $Z$, and every
$\alpha\in\field{N}^n$, we have
\begin{align}\label{abk3.16}
(b^\alpha G\mathscr{P})(0,0) = \begin{cases} 0\,, &\quad \text{ if $|\alpha|\neq k$},\\
(-2)^k \displaystyle\frac{\partial^\alpha G}{\partial
z^\alpha}\,,&\quad \text{ if $|\alpha|=k$}\;.
\end{cases}
\end{align}
By Theorem \ref{bkt2.17}, \eqref{toe1.4} and \eqref{bk2.66}, we also have
\begin{subequations}
\begin{align}
&\left(\mathscr{L}^{-1} b_{i} h b_{j} h' \mathscr{P}\right)(0,0)
= \left(\frac{b_j b_{i}}{8\pi} h h' + \frac{b_{i}}{2\pi} h_{j}
h'\right) \mathscr{P}(0,0) = \frac{1}{2\pi} ( h_{i}h'_{j}- h_{j}h'_{i}),\label{bk3.17a} \\
&\left(\mathscr{L}^{-1} \mathscr{P}^{\bot} h b_i h' b_j
\mathscr{P}\right)(0,0)
= -\frac{1}{2\pi}h_j h'_i -\frac{3}{2\pi}h_i h'_j.\label{bk3.17b}
\end{align}
\end{subequations}
where we use in the last equation that $ h b_i h' b_j= b_i h h' b_j + 2
h_i h' b_j$ and further \eqref{bk3.16c}, \eqref{bk3.16e}.
Let $\phi(z)= \phi_{ij} z_iz_j,\psi=\psi_{ij}
z_iz_j$ be degree $2$ polynomials in $z$ with symmetric matrices
$(\phi_{ij})$, $(\psi_{ij})$. Then
by Theorem \ref{bkt2.17}, (\ref{bk2.66}), (\ref{bk3.10}), (\ref{bk3.16e}),
(\ref{bk3.16f}) and (\ref{bk3.17a}), we obtain
\begin{subequations}
\begin{align}
\left(\mathscr{L}^{-1} b_{\ell }b_{k} \phi b_{j}h \mathscr{P}\right) (0,0)
&= \left(\frac{b_{\ell }b_{k} b_{j}}{12\pi} \phi h +
\frac{b_{\ell }b_{k}}{4\pi} \frac{\partial \phi}{\partial z_{j}}
h\right) \mathscr{P} (0,0)\label{bk3.18a}\\
& = \frac{-2}{3\pi}\frac{\partial^{3} (\phi h)}{\partial z_{\ell }\partial
z_{k}\partial z_{j}} + \frac{2}{\pi}(\phi_{\ell j}h_{k}
+\phi_{kj}h_{\ell }),\nonumber\\
\left(\mathscr{L}^{-1} b_{k} \phi b_{\ell } b_{j}h \mathscr{P}\right) (0,0)
& = \mathscr{L}^{-1} \left(b_{\ell }b_{k} \phi + 2 b_{k}
\frac{\partial \phi}{\partial z_{\ell }} \right)
b_{j}h \mathscr{P} (0,0)\label{bk3.18b}\\
& = \frac{-2}{3\pi}\frac{\partial^{3} (\phi h)}{\partial z_{\ell }\partial
z_{k}\partial z_{j}} + \frac{2}{\pi}(\phi_{kj}h_{\ell }
+ \phi_{\ell k}h_{j}),\nonumber\\
- \pi \left(\mathscr{L}^{-1} b_{k} \phi \overline{z}_{\ell } b_{j}h \mathscr{P}\right) (0,0)
&= - \frac{1}{2} \mathscr{L}^{-1}b_{k} \phi b_{j} \left(b_{\ell } h
+ 2 h_{\ell } \right)\mathscr{P} (0,0)\label{bk3.18c}\\
&= \frac{1}{3\pi}\frac{\partial^{3} (\phi h)}{\partial z_{\ell }\partial
z_{k}\partial z_{j}} - \frac{1}{\pi}\phi_{\ell k}h_{j},\nonumber
\end{align}
\end{subequations}
and
\begin{subequations}
\begin{align}
\left( \mathscr{L}^{-1}b_{k} h b_{i}b_{j}\psi \mathscr{P}\right) (0,0)
& = \left(\frac{b_{k}b_{i} b_{j}}{12\pi} h \psi+
\frac{b_{k}}{4\pi}(b_{i}h_{j}+ b_{j} h_{i})
\psi\right) \mathscr{P} (0,0)\label{bk3.19a}\\
& = \frac{-2}{3\pi}\frac{\partial^{3} (h\psi)}
{\partial z_{k}\partial z_{i}\partial z_{j}}
+ \frac{2}{\pi}(\psi_{ki}h_{j} + \psi_{kj}h_{i}),\nonumber\\
(\mathscr{L}^{-1} b_{k} h \overline{z}_{\ell } b_{j} \psi\mathscr{P}) (0,0)
&= \mathscr{L}^{-1}b_{k} h b_{j} \left( \frac{b_{\ell }}{2\pi}\psi
+ \frac{1}{\pi}\frac{\partial \psi}{\partial z_{\ell }} \right)\mathscr{P} (0,0)
\label{bk3.19b}\\
& = \frac{-1}{3\pi^{2}}
\frac{\partial^{3} (h\psi)}{\partial z_{\ell }\partial z_{k}\partial z_{j}}
+ \frac{1}{\pi^{2}}(\psi_{kj}h_{\ell }+ \psi_{\ell j}h_{k}),\nonumber\\
- \pi \left(\mathscr{L}^{-1} h \overline{z}_{k}\overline{z}_{\ell } b_{j}\psi \mathscr{P}\right) (0,0)
&= \frac{1}{6\pi^{2}}\frac{\partial^{3} (h \psi)}{\partial z_{\ell }\partial
z_{k}\partial z_{j}} + \frac{3}{2\pi^{2}}\psi _{\ell k}h_{j},\label{bk3.19c}
\end{align}
\end{subequations}
In fact, by \eqref{bk3.1}, \eqref{bk3.10} implies
\[
\pi (h \overline{z}_{k}\overline{z}_{\ell } b_{j}\psi \mathscr{P}) (Z,0)
= \frac{1}{2}\Big(b_{k} h b_{j} \psi \overline{z}_{\ell }
+ \Big(2 h_{k}b_{j} \psi + 2 h b_j \frac{\partial \psi}{\partial
z_{k}}\Big) \frac{b_{\ell }}{2 \pi} \Big)\mathscr{P}(Z,0)\,,
\]
so from \eqref{bk3.16e}, \eqref{bk3.17b} and \eqref{bk3.19b}
we get \eqref{bk3.19c}.
Set $R_{\overline{m}\,\overline{q}}(z)= R_{k\overline{m}l\overline{q}}\, z_kz_l$.
By \eqref{bk2.66} and Lemma \ref{lmt1.6},
\[
R_{\overline{s}\overline{k}}(z) b_m b_qR_{\overline{m}\,\overline{q}}(z)
= \big(b_m b_q R_{\overline{s}\overline{k}}(z) + 8 b_m R_{q\overline{s}\ell\overline{k}}\, z_{\ell}
+ 8 R_{m\overline{s}q\overline{k}} \big) R_{\overline{m}\,\overline{q}}(z)\,,
\]
thus Theorem \ref{bkt2.17}, \eqref{bk3.10},
\eqref{bk3.18b} and \eqref{bk3.19a} show that
\begin{equation}\label{bk3.21}
\begin{split}
& \left( \mathscr{L}^{-1}b_{s}b_{k} R_{\overline{s}\overline{k}}
b_{m}b_{q}R_{\overline{m}\,\overline{q}} \mathscr{P}\right) (0,0)\\
& \hspace{5mm}= b_{s}b_{k} \Big(\frac{b_m b_q}{16\pi} R_{\overline{s}\overline{k}}
+ \frac{2 b_m}{3\pi} R_{q\overline{s}\ell \overline{k}} z_{\ell }
+ \frac{1}{\pi} R_{m\overline{s}q\overline{k}} \Big) R_{\overline{m}\,\overline{q}}\mathscr{P}(0,0)\\
& \hspace{5mm}
= \frac{1}{\pi}\frac{\partial^{4} (R_{\overline{s}\overline{k}}R_{\overline{m}\,\overline{q}})}
{\partial z_{s}\partial z_{k}\partial z_{m}\partial z_{q}}
- \frac{16}{3\pi}\frac{\partial^{3} (R_{q\overline{s}\ell \overline{k}}z_{\ell }
R_{\overline{m}\,\overline{q}})}
{\partial z_{s}\partial z_{k}\partial z_{m}}
+ \frac{8}{\pi}R_{m\overline{s}\,q\overline{k}} R_{s\overline{m}k\overline{q}}\, ,\\
& - \left( \pi \mathscr{L}^{-1}b_{k} \overline{z}_{s} R_{\overline{s}\overline{k}}
b_{m}b_{q}R_{\overline{m}\,\overline{q}} \mathscr{P}\right) (0,0)\\
& \hspace{5mm}
= - \mathscr{L}^{-1}b_{k} \Big( \frac{1}{2}b_{s} R_{\overline{s}\overline{k}}
b_{m}b_{q}R_{\overline{m}\,\overline{q}}
+ \frac{\partial R_{\overline{s}\overline{k}}}{\partial z_{s}} b_m
b_qR_{\overline{m}\,\overline{q}}
+ R_{\overline{s}\overline{k}}b_{m}b_{q}\frac{\partial
R_{\overline{m}\,\overline{q}}}{\partial z_{s}} \Big) \mathscr{P}(0,0)\\
& \hspace{5mm}
= \frac{1}{6\pi}\frac{\partial^{4} (R_{\overline{s}\overline{k}}R_{\overline{m}\,\overline{q}})}
{\partial z_{s}\partial z_{k}\partial z_{m}\partial z_{q}}
+ \frac{8}{3\pi}\frac{\partial^{3} (R_{q\overline{s}\ell \overline{k}}z_{\ell }
R_{\overline{m}\,\overline{q}})}
{\partial z_{s}\partial z_{k}\partial z_{m}}
- \frac{4}{\pi}R_{m\overline{s}q\overline{k}} R_{s\overline{m}k\overline{q}}
- \frac{16}{\pi}R_{s\overline{s}q\overline{k}} R_{m\overline{m}k\overline{q}}\, .
\end{split}
\end{equation}
Due to \eqref{bk3.13}, \eqref{bk3.16a}--\eqref{bk3.21}, we get
\begin{multline}\label{bk3.22}
\pi^2 (\mathscr{L}^{-1}\bI_1\mathscr{P})(0,0)=
\frac{1}{144} \Big[\frac{7}{6}
\frac{\partial^{4} (R_{\overline{s}\overline{t}}R_{\overline{m}\overline{q}})}
{\partial z_{s}\partial z_{t}\partial z_{m}\partial z_{q}}
- \frac{8}{3}\frac{\partial^{3} (R_{q\overline{s}\ell \overline{t}}z_{v}
R_{\overline{m}\overline{q}})}
{\partial z_{s}\partial z_{t}\partial z_{m}}
+ 4 R_{m\overline{s}q\overline{t}} R_{s\overline{m}t\overline{q}}\\
- 16 R_{s\overline{s}q\overline{t}} R_{m\overline{m}t\overline{q}}
- 2 \frac{\partial^{3} ( ( 2R_{v\overline{u}u\overline{t}} + R^E_{v\overline{t}})z_{v}
R_{\overline{m}\overline{q}})}{\partial z_{t}\partial z_{m}\partial z_{q}}
+ 12 (2R_{q\overline{u}u\overline{t}} + R^E_{q\overline{t}}) R_{t\overline{m}m\overline{q}}\Big]\\
+ \frac{1}{12} \Big[ - \frac{1}{3}
\frac{\partial^{3} ( R_{\overline{s}\overline{t}}(\frac{4}{3}R_{\ell\overline{k}k\overline{q}}
+ R^E_{\ell\overline{q}})z_{\ell} )}
{\partial z_{s}\partial z_{t}\partial z_{q}}
+ 4 R_{s\overline{s}q\overline{t}}\big(\frac{4}{3}R_{t\overline{k}k\overline{q}}
+ R^E_{t\overline{q}}\big) - R_{s\overline{s}t\overline{t}}\big(\frac{4}{3}R_{q\overline{k}k\overline{q}}
+ R^E_{q\overline{q}}\big)\Big]\\
+ \frac{1}{8} \Big[\big(2R_{t\overline{u}u\overline{t}}
+ R^E_{t\overline{t}}\big)\big(\frac{4}{3}R_{q\overline{k}k\overline{q}}
+ R^E_{q\overline{q}}\big)-\big(2R_{q\overline{u}u\overline{t}} + R^E_{q\overline{t}}\big)
\big(\frac{4}{3}R_{t\overline{k}k\overline{q}} + R^E_{t\overline{q}}\big)\Big].
\end{multline}
But from Lemma \ref{lmt1.6}, we have
\begin{equation}\label{bk3.23}
\begin{split}
& \frac{\partial^{4} (R_{\overline{s}\overline{t}}R_{\overline{m}\overline{q}})}
{\partial z_{s}\partial z_{t}\partial z_{m}\partial z_{q}}
= 4 R_{m\overline{s}q\overline{t}} R_{s\overline{m}t\overline{q}}
+ 4R_{s\overline{s}t\overline{t}} R_{m\overline{m}q\overline{q}}
+ 16 R_{s\overline{s}m\overline{t}} R_{q\overline{m}t\overline{q}},\\
&\frac{\partial^{3} (R_{s\overline{s}v\overline{t}} z_{v} R_{\overline{m}\overline{q}})}
{\partial z_{t}\partial z_{m}\partial z_{q}}
= 2R_{s\overline{s}t\overline{t}} R_{m\overline{m}q\overline{q}}
+ 4 R_{s\overline{s}q\overline{t}} R_{m\overline{m}t\overline{q}},\\
&\frac{\partial^{3} (R_{m\overline{s}v\overline{t}}z_{v} R_{\overline{m}\overline{q}})}
{\partial z_{s}\partial z_{t}\partial z_{q}}
= 2 R_{m\overline{s}q\overline{t}} R_{s\overline{m}t\overline{q}}
+ 4 R_{m\overline{s}s\overline{t}} R_{t\overline{m}q\overline{q}}.
\end{split}
\end{equation}
Plugging \eqref{bk3.23} in \eqref{bk3.22} we see that
the coefficient of $R_{s\overline{s}q\overline{t}} R_{m\overline{m}t\overline{q}}$ in
the term $\frac{1}{144}[\cdots]$ of \eqref{bk3.22} is
$\frac{1}{144}( \frac{56}{3}- \frac{32}{3}-16 -16+24)=0$ and
\begin{equation}\label{bk3.24}
\begin{split}
\pi^2 &(\mathscr{L}^{-1}\bI_1\mathscr{P})(0,0)= \frac{1}{144}
\frac{10}{3} R_{m\overline{s}q\overline{t}} R_{s\overline{m}t\overline{q}}
- \frac{1}{27} R_{s\overline{s}q\overline{t}} R_{m\overline{m}t\overline{q}}\\
&+ \Big[\frac{1}{144}\cdot\frac{-10}{3}+ \frac{1}{12}\cdot\frac{-20}{9}
+\frac{1}{3}\Big ] R_{s\overline{s}t\overline{t}} R_{m\overline{m}q\overline{q}}
+ \Big[\frac{-4}{144}+ \frac{1}{12}\cdot\frac{-5}{3
+ \frac{5}{12} \Big]
R^E_{t\overline{t}} R_{m\overline{m}q\overline{q}} \\
&+ \Big [\frac{4}{144} + \frac{1}{12} \cdot\frac{8}{3}
-\frac{10}{3\cdot 8} \Big ] R^E_{q\overline{t}} R_{m\overline{m}t\overline{q}}
+ \frac{1}{8} (R^E_{t\overline{t}} R^E_{q\overline{q}}
- R^E_{q\overline{t}} R^E_{t\overline{q}}) \\
= &\, \frac{5}{2^3\cdot 3^3} R_{m\overline{s}q\overline{t}} R_{s\overline{m}t\overline{q}}
- \frac{1}{27} R_{s\overline{s}q\overline{t}} R_{m\overline{m}t\overline{q}}
+ \frac{1}{8} R_{s\overline{s}t\overline{t}} R_{m\overline{m}q\overline{q}} \\
&+ \frac{1}{4} R^E_{t\overline{t}} R_{m\overline{m}q\overline{q}}
- \frac{1}{6} R^E_{q\overline{t}} R_{m\overline{m}t\overline{q}}
+ \frac{1}{8} (R^E_{t\overline{t}} R^E_{q\overline{q}}
- R^E_{q\overline{t}} R^E_{t\overline{q}}).
\end{split}
\end{equation}
From \eqref{bk3.13}, \eqref{bk3.16d}, \eqref{bk3.16g},
\eqref{bk3.16h}, \eqref{bk3.19b}, \eqref{bk3.19c} and \eqref{bk3.23}, we get
\begin{equation}\label{bk3.25}
\begin{split}
\pi^2 (\mathscr{L}^{-1} \bI_2 & \mathscr{P})(0,0)=
\frac{1}{18} \Big[ -\frac{1}{2}
\frac{\partial^{3} (R_{u\overline{s}q\overline{t}}z_{u}
R_{\overline{m}\overline{q}})}
{\partial z_{s}\partial z_{t}\partial z_{m}}
+ 4 R_{s\overline{s}q\overline{t}} R_{t\overline{m}m\overline{q}}\\
&+ \frac{3}{2} R_{m\overline{s}q\overline{t}} R_{s\overline{m}t\overline{q}}
+ 3 (2 R_{u\overline{u}q\overline{s}}
+ R^E_{q\overline{s}}) (-\frac{2}{4}) R_{m\overline{m}s\overline{q}} \Big] \\
&+ \Big[
-\frac{1}{12}R_{t\overline{s}q\overline{t}}
- \frac{1}{4}( 2 R_{u\overline{u}q\overline{s}}
+ R^E_{q\overline{s}}) \Big] \big(\frac{4}{3}R_{s\overline{k}k\overline{q}}
+ R^E_{s\overline{q}}\big) \\
=&\frac{1}{36} R_{m\overline{s}q\overline{t}}
R_{s\overline{m}t\overline{q}}
- \frac{5}{6} R_{s\overline{s}q\overline{t}} R_{m\overline{m}t\overline{q}}
- R^E_{s\overline{q}} R_{t\overline{s}q\overline{t}}
- \frac{1}{4} R^E_{q\overline{t}} R^E_{t\overline{q}}.
\end{split}
\end{equation}
By \eqref{bk3.16g}, we get
\begin{align}\label{bk3.26}
\Big(\mathscr{L}^{-1}\frac{2\pi}{9} R_{m\overline{s}q\overline{t}}\overline{z}_{s}\overline{z}_{t}
R_{k\overline{m}\ell\overline{q}} z_{k} z_{\ell} \mathscr{P}\Big) (0,0)
= - \frac{1}{6\pi^2} R_{m\overline{s}q\overline{t}}R_{s\overline{m}t\overline{q}}.
\end{align}
{}Relations \eqref{bk3.14}, \eqref{bk3.24}, \eqref{bk3.25} and \eqref{bk3.26}
imply the desired formula \eqref{bk3.12}.
\end{proof}
\subsection{Contribution of
$\mathcal{O}_{4}$ to $\mathscr{F}_4(0,0)$}\label{bks3.3
We will use the following remark repeatedly in our computation.
\begin{rem} \label{bkt3.5}
Let $\Phi$ be a polynomial in $b^+,z, b, \overline{z}$. Due to
\eqref{bk2.66} and \eqref{bk3.1}, the value of the kernels of
$\mathscr{P} \Phi \mathscr{P}$, $\mathscr{P}^\bot \Phi \mathscr{P}$,
$\mathscr{L}^{-1}\mathscr{P}^\bot \Phi \mathscr{P}$ at $(0,0)$ consists of the terms of $\Phi$ whose
total degree in $b$ and $\overline{z}$ is the same as the total degree in
$b^+$ and $z$.
\end{rem}
\begin{lemma}\label{bkt3.4} We have the following identity\,\rm{:}
\begin{align}\label{lm3.31}
\begin{split}
-\pi^2 (\mathscr{L}^{-1} \mathcal{O}_4 \mathscr{P})(0,0)= &\,-\frac{\Delta \br}{96}
+ \frac{23}{108}
R_{m\overline{s} q\overline{t}} R_{s\overline{m}t\overline{q}}
+ \frac{41}{54}
R_{s\overline{s} q\overline{t}} R_{m\overline{m}t\overline{q}}\\
&+ R_{m\overline{m}q\overline{k}} R^E_{k\overline{q}}
+ \frac{1}{8}(-R^E_{m\overline{m}; q\overline{q}}+3 R^E_{q\overline{m}; m\overline{q}})
+ \frac{1}{4} R^E_{k\overline{q}} R^E_{q\overline{k}}\,.
\end{split}
\end{align}
\end{lemma}
\begin{proof} By (\ref{bk2.66}), \eqref{lm01.9}, (\ref{bk3.1}) and
(\ref{bk3.2}), as in (\ref{bk3.3}), we have
\begin{align}\label{lm3.33}
\begin{split}
-&(\mathcal{O}_{41} \mathscr{P})(Z,0)= - \Big\{ \frac{1}{20}\Big(A_{1Z}
- \frac{4}{3}A_{2Z}\Big) (\tfrac{\partial}{\partial \overline{z}_i},
\tfrac{\partial}{\partial \overline{z}_j}) b_i b_j \\
&\hspace{45mm}- \frac{1}{20}\Big(A_{1Z}
- \frac{4}{3}A_{2Z}\Big) (\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_j})b^+_i b_j\Big\} \mathscr{P}(Z,0)\\
&= \Big[- \frac{\pi^2}{5}\Big(A_{1Z}
- \frac{4}{3}A_{2Z}\Big) (\overline{z},\overline{z})
+ \frac{\pi}{5}\Big(A_{1Z}
- \frac{4}{3}A_{2Z}\Big) (\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial \overline{z}_j})\Big] \mathscr{P}(Z,0).
\end{split}
\end{align}
and
\begin{align} \label{lm3.34}
-\mathscr{L}^{-1} \mathcal{O}_{42} \mathscr{P}= & \mathscr{P}^\bot\Big[4 \Big(\frac{1}{80}A_{1Z}
- \frac{1}{360}A_{2Z}\Big)(\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial \overline{z}_j})
- \frac{1}{72} \ric(z, \overline{z})^{2} \Big]\mathscr{P}.
\end{align}
From \eqref{lm01.7}, \eqref{lm01.9}, \eqref{bk3.1} and
\eqref{bk3.2}, and since $R^{TX}$ is $(1,1)$-form, we have
\begin{align} \label{lm3.35}
\begin{split}
- (\mathcal{O}_{44} \mathscr{P})(Z,&0)
= 2\pi \Big \{ \frac{\pi}{30}A_{1Z}(\overline{z}, \overline{z})
+ \Big(\frac{\partial}{\partial Z_j} \Big(\frac{1}{20}A_{1Z}
+ \frac{2}{45}A_{2Z} \Big)\Big)( \overline{z}, e_j) \\
&- \frac{\pi}{10}A_{2Z}(\overline{z}, \overline{z})- \overline{z}_i
\frac{\partial}{\partial \overline{z}_i} \Big(\frac{1}{10}A_{1Z}
+ \frac{4}{45}A_{2Z} \Big)(\tfrac{\partial}{\partial z_j},
\tfrac{\partial}{\partial \overline{z}_j})\Big\}\mathscr{P}(Z,0),\\
- (\mathcal{O}_{45} \mathscr{P})(Z,&0) = 2\pi \Big \{
\frac{2}{9}\left \langle R^{TX} (\mathcal{R},e_k) z, R^{TX}_{x_0}
(\mathcal{R},e_k) \overline{z}\right\rangle\\
& - \frac{1}{9} \left \langle R^{TX} (z, \overline{z}) \mathcal{R},
e_k\right\rangle \ric(\mathcal{R},e_k)\\
&+\frac{1}{4}
\left \langle R^{TX} (z, \overline{z}) \mathcal{R}, e_j\right \rangle
R^E (\mathcal{R},e_j)
- \frac{1}{4}R^E_{\, ;(Z,Z)} (z,
\overline{z})\Big\}\mathscr{P}(Z,0).
\end{split}\end{align}
Let $\psi_{ijk}$ be degree $3$ polynomials in $z$
which are symmetric in $i,j$. By \eqref{bk2.66}, \eqref{bk3.1}, we get
\begin{align} \label{lm3.37}
\begin{split}
(&\psi_{ijk} \overline{z}_i \overline{z}_j \overline{z}_k\mathscr{P})(Z,0)
= \frac{1}{8\pi^3} ( \psi_{ijk} b_i b_j b_k \mathscr{P})(Z,0)
= \frac{1}{8\pi^3} \Big\{ b_i b_j b_k \psi_{ijk} \\
& + 2 b_i b_j \frac{\partial\psi_{ijk}}{\partial z_k}
+ 4 b_i b_k \frac{\partial\psi_{ijk}}{\partial z_j}
+ 8 b_i \frac{\partial^2\psi_{ijk}}{\partial z_j\partial z_k}
+ 4 b_k \frac{\partial^2\psi_{ijk}}{\partial z_i\partial z_j}
+ 8 \frac{\partial^3\psi_{ijk}}
{\partial z_i\partial z_j\partial z_k}\Big\}\mathscr{P}(Z,0).
\end{split}
\end{align}
Thus by Theorem \ref{bkt2.17}, (\ref{toe1.1}) and (\ref{lm3.37}), we obtain
\begin{align} \label{lm3.38}
\begin{split}
\pi^2 (\mathscr{L}^{-1} & \mathscr{P}^\bot \psi_{ijk} \overline{z}_i \overline{z}_j \overline{z}_k
\mathscr{P})(0,0)
= \frac{1}{8\pi^2} \Big\{ \frac{b_i b_j b_k}{12} \psi_{ijk}
+ \frac{1}{4} b_i b_j \frac{\partial\psi_{ijk}}{\partial z_k} \\
&+ \frac{1}{2} b_i b_k \frac{\partial\psi_{ijk}}{\partial z_j}
+ 2 b_i \frac{\partial^2\psi_{ijk}}{\partial z_j\partial z_k}
+ b_k \frac{\partial^2\psi_{ijk}}{\partial z_i\partial z_j}\Big\}\mathscr{P}(0,0)
= - \frac{11}{24\pi^2}\frac{\partial^3\psi_{ijk}}
{\partial z_i\partial z_j\partial z_k} \;.
\end{split}
\end{align}
Since $|\tfrac{\partial}{\partial z_j} |^2 = \frac{1}{2}$\,,
Lemma \ref{lmt1.6}, \eqref{lm01.4}, \eqref{lm01.7} and
the fact that $R^{TX}$ is a $(1,1)$-form entail
\begin{align} \label{lm3.39}
\begin{split}
&A_{1Z}(\overline{z},\overline{z})=\left\langle R^{TX}_{\, ; (Z,Z)}
(z,\overline{z}) z,\overline{z}\right\rangle,\\
&A_{2Z}(\overline{z},\overline{z})=2 \left \langle R^{TX} (z,\overline{z}) z,
R^{TX} (z,\overline{z}) \overline{z}\right \rangle
= -4 R_{u\overline{s}v\overline{t}} R_{k\overline{m} t\overline{q}}
z_{u} \overline{z}_{s} z_{v} z_{k}\overline{z}_{m}\overline{z}_{q},\\
&A_{1Z}(\tfrac{\partial}{\partial z_q},\tfrac{\partial}{\partial\overline{z}_q})
=\left\langle R^{TX}_{\, ;(Z,Z)} (\overline{z},
\tfrac{\partial}{\partial z_q}) z,
\tfrac{\partial}{\partial\overline{z}_q}\right\rangle,\\
&A_{2Z}(\tfrac{\partial}{\partial z_q},\tfrac{\partial}{\partial\overline{z}_q})
= \left \langle R^{TX} (\overline{z},
\tfrac{\partial}{\partial z_q}) \mathcal{R},
R^{TX} ( z,\tfrac{\partial}{\partial\overline{z}_q})\mathcal{R}\right \rangle\\
&\hspace{23mm} = 2(R_{q\overline{s}k\overline{t}} R_{\ell\overline{q} t\overline{u}}
+ R_{q\overline{s}t \overline{u}} R_{\ell\overline{q} k\overline{t}} )
\overline{z}_{s}\overline{z}_{u} z_{k} z_{\ell}.
\end{split}\end{align}
By Remark \ref{bkt3.5}, we can replace $A_{1Z}(\overline{z},\overline{z})$
by $2R_{k\overline{m} \ell\overline{q}; s\overline{t}} z_{k} \overline{z}_{m} z_{\ell}
\overline{z}_{q} z_{s}\overline{z}_{t}$ in our computation.
We deduce from \eqref{lm3.38} and \eqref{lm3.39} that
\begin{align} \label{lm3.40}
\begin{split}
\pi^2 (\mathscr{L}^{-1}\mathscr{P}^\bot A_{1Z}(\overline{z},\overline{z}) \mathscr{P})(0,0)
=& - \frac{11}{6\pi^2} (R_{m\overline{m} q\overline{q}; t\overline{t}}
+ 2 R_{t\overline{m} q\overline{q}; m\overline{t}}),\\
\pi^2 (\mathscr{L}^{-1}\mathscr{P}^\bot A_{2Z}(\overline{z},\overline{z}) \mathscr{P})(0,0)
=& \frac{11}{3\pi^2} ( R_{m\overline{s} q\overline{t}} R_{s\overline{m} t\overline{q}}
+ 2 R_{s\overline{s} q\overline{t}} R_{m\overline{m} t\overline{q}}).
\end{split}\end{align}
Let $F_{ij}$
be homogeneous degree $2$ polynomials in $Z$. Then by
\eqref{toe1.1}, \eqref{bk3.10} and \eqref{bk3.16g}, we obtain
\begin{subequations}
\begin{align}
&(\mathscr{P}^\bot F_{ij}\overline{z}_i \overline{z}_j \mathscr{P})(0,0)
= - \frac{1}{\pi^2}\frac{\partial^2 F_{ij}}{\partial z_i \partial z_j}\,,
\label{lm3.36a}\\
&(\mathscr{L}^{-1}\mathscr{P}^\bot \overline{z}_k \frac{\partial (F_{ij}\overline{z}_i \overline{z}_j)}
{\partial \overline{z}_k}\mathscr{P})(0,0)
= - \frac{3}{4\pi^3}\frac{\partial^2 F_{ij}}{\partial z_i \partial z_j}\,.
\label{lm3.36d}
\end{align}\end{subequations}
(Note that by Remark \ref{bkt3.5} the contributions of
$\overline{z}_k \frac{\partial (F_{ij}\overline{z}_i \overline{z}_j)}{\partial \overline{z}_k}$
and $2 F_{ij}\overline{z}_i \overline{z}_j $
to \eqref{lm3.36d} are the same, so \eqref{lm3.36d}
follows from \eqref{bk3.16g}.)
By Remark \ref{bkt3.5} and \eqref{lm3.39}, only the term
$-2 R_{q\overline{m} k\overline{q}; \ell\overline{t}}\overline{z}_{m}z_{k}\overline{z}_{t}z_{\ell}$
from $A_{1Z}(\tfrac{\partial}{\partial z_q},
\tfrac{\partial}{\partial\overline{z}_q})$
has a nontrivial contribution in our computation at $(0,0)$,
and from \eqref{bk3.16g}, \eqref{lm3.36a} and \eqref{lm3.36d}, we get
\begin{align} \label{lm3.45}
\begin{split}
&\left(\mathscr{P}^{\bot}A_{1Z}(\tfrac{\partial}{\partial z_q},
\tfrac{\partial}{\partial \overline{z}_q})\mathscr{P}\right) (0,0)
=\frac{2}{\pi ^2} (R_{q\overline{m} m\overline{q}; t\overline{t}}
+ R_{q\overline{m} t\overline{q}; m\overline{t}}),\\
&\frac{1}{2} \left(\mathscr{L}^{-1}\mathscr{P}^{\bot} \overline{z}_m
\frac{\partial}{\partial\overline{z}_m} A_{1Z}(\tfrac{\partial}{\partial z_q},
\tfrac{\partial}{\partial \overline{z}_q})\mathscr{P}\right) (0,0) =
\left(\mathscr{L}^{-1} \mathscr{P}^{\bot}A_{1Z}(\tfrac{\partial}{\partial z_q},
\tfrac{\partial}{\partial \overline{z}_q})\mathscr{P}\right) (0,0)\\
&\hspace{20mm}= \frac{3}{4\pi ^3} (R_{q\overline{m} m\overline{q}; t\overline{t}}
+ R_{q\overline{m} t\overline{q}; m\overline{t}}),\\
&\left(\mathscr{P}^{\bot}A_{2Z}(\tfrac{\partial}{\partial z_q},
\tfrac{\partial}{\partial \overline{z}_q})\mathscr{P}\right) (0,0)
= - \frac{2}{\pi ^2} (R_{q\overline{s}s\overline{t}} R_{u\overline{q} t\overline{u}}
+ 3 R_{q\overline{s}u\overline{t}} R_{s\overline{q} t\overline{u}} ),\\
& \frac{1}{2} \left(\mathscr{L}^{-1} \mathscr{P}^{\bot}\overline{z}_m
\frac{\partial}{\partial\overline{z}_m }
A_{2Z}(\tfrac{\partial}{\partial z_q},
\tfrac{\partial}{\partial \overline{z}_q})\mathscr{P}\right) (0,0)
=\left(\mathscr{L}^{-1} \mathscr{P}^{\bot}A_{2Z}(\tfrac{\partial}{\partial z_q},
\tfrac{\partial}{\partial \overline{z}_q})\mathscr{P}\right) (0,0)\\
&\hspace{20mm}
= - \frac{3}{4\pi ^3} (R_{q\overline{s}s\overline{t}} R_{u\overline{q} t\overline{u}}
+ 3 R_{q\overline{s}u\overline{t}} R_{s\overline{q} t\overline{u}} ).
\end{split}\end{align}
\comment{
From (\ref{lm3.36d}) and (\ref{lm3.39}), we get
\begin{align} \label{lm3.46}
\begin{split}
\left(\mathscr{L}^{-1}\mathscr{P}^{\bot} \overline{z}_i \tfrac{\partial}{\partial\overline{z}_j}
A_{1Z}(\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_j})\mathscr{P}\right) &(0,0) =
\frac{3}{4\pi ^2} (R_{j\overline{j} i\overline{i}; l\overline{l}} + R_{j\overline{j}
l\overline{i}; i\overline{l}} + R_{j\overline{l} i\overline{i}; l\overline{j}}+ R_{j\overline{l}
l\overline{i}; \overline{j}} )\\
&= \frac{3}{4\pi ^2} (R_{j\overline{j} i\overline{i}; l\overline{l}} + 3 R_{j\overline{j}
l\overline{i}; i\overline{l}}),\\
\left(\mathscr{L}^{-1} \mathscr{P}^{\bot}\overline{z}_i \tfrac{\partial}{\partial\overline{z}_j}
A_{2Z}(\tfrac{\partial}{\partial z_i},
\tfrac{\partial}{\partial \overline{z}_j})\mathscr{P}\right) &(0,0)
= - \frac{3}{4\pi ^2} ( R_{j\overline{l}i\overline{j'}} R_{l\overline{i} j'\overline{j}}
+ 3 R_{j\overline{j}i\overline{j'}} R_{l\overline{i} j'\overline{l}} + 4
R_{j\overline{l}j'\overline{j}} R_{i\overline{i} l\overline{j'}} )\\
&= - \frac{3}{4\pi ^2} ( R_{j\overline{l}i\overline{j'}} R_{l\overline{i} j'\overline{j}}
+ 7 R_{j\overline{j}i\overline{j'}} R_{l\overline{i} j'\overline{l}} ).
\end{split}\end{align}
}
Remark \ref{bkt3.5}, \eqref{bk3.2}, \eqref{bk3.16d}
and \eqref{bk3.16g} yield
\begin{align} \label{lm3.47}
\begin{split}
&\left(\mathscr{L}^{-1}\mathscr{P}^{\bot} \left \langle R^{TX}_{\, ;(Z,e_m)} (z,\overline{z})\mathcal{R},
e_m\right \rangle_{x_0}\mathscr{P}\right) (0,0) \\
&\hspace{10mm} = 2 \left(\mathscr{L}^{-1}\mathscr{P}^{\bot} (R_{k\overline{q} \ell\overline{m}; m\overline{s}}
- R_{k\overline{q} m\overline{s}\,;\, \ell\overline{m}})z_{k} \overline{z}_{q}
z_{\ell}\overline{z}_{s}\mathscr{P}\right) (0,0)=0,\\
& \left(\mathscr{L}^{-1}\mathscr{P}^{\bot} \left \langle R^{TX}_{x_0} (z, \overline{z}) \mathcal{R},
e_k\right\rangle_{x_0} \ric(\mathcal{R},e_k)\mathscr{P}\right) (0,0) \\
&\hspace{10mm}= 2 \left(\mathscr{L}^{-1}\mathscr{P}^{\bot}
(-R_{\ell\overline{s}k\overline{q}} \ric_{m\overline{k}}
+ R_{\ell\overline{s}m\overline{k}} \ric_{k\overline{q}}) z_{\ell} \overline{z}_{s}
\overline{z}_{q}z_{m}\mathscr{P}\right) (0,0)=0,\\
&\left(\mathscr{L}^{-1}\mathscr{P}^{\bot} \ric(\mathcal{R},e_q)R^E_{x_0} (\mathcal{R},e_q) \mathscr{P}\right)
(0,0) \\
&\hspace{10mm}=2 \left(\mathscr{L}^{-1}\mathscr{P}^{\bot} ( - \ric_{k\overline{q}} R^E_{q\overline{s}}
+ \ric_{q\overline{s}} R^E_{k\overline{q}}) z_{k} \overline{z}_{s} \mathscr{P}\right) (0,0) = 0,\\
&\left(\mathscr{L}^{-1}\mathscr{P}^{\bot} R^E_{\, ;(Z,e_s)} (\mathcal{R}, e_s)
\mathscr{P}\right) (0,0) \\
&\hspace{10mm}
= 2\left(\mathscr{L}^{-1}\mathscr{P}^{\bot} (R^E_{k\overline{s}; s\overline{q}}
- R^E_{s\overline{q}; k\overline{s}})z_{k} \overline{z}_{q}\mathscr{P}\right) (0,0) =0.
\end{split}\end{align}
By \eqref{0c47}, \eqref{0c50}, (\ref{bk3.2}) and \eqref{lm3.47},
we know that for $\alpha=1,2$,
\begin{align} \label{lm3.48}\begin{split}
&\Big(\mathscr{L}^{-1}\mathscr{P}^{\bot} \frac{\partial}{\partial Z_q}
\Big(A_{\alpha Z}( \overline{z}, e_q) \Big)\mathscr{P}\Big) (0,0)=0,\\
&\Big(\mathscr{L}^{-1}\mathscr{P}^{\bot} \Big(\frac{\partial}{\partial Z_q}
A_{\alpha Z}\Big)( \overline{z}, e_q) \mathscr{P}\Big) (0,0)
= -2 \Big(\mathscr{L}^{-1}\mathscr{P}^{\bot}
A_{\alpha Z}(\tfrac{\partial}{\partial \overline{z}_q},
\tfrac{\partial}{\partial z_q} ) \mathscr{P}\Big) (0,0).
\end{split}\end{align}
By \eqref{lm01.3} and \eqref{bk3.16g}, we have
\begin{align} \label{lm3.49}
\begin{split}
&\left(\mathscr{L}^{-1}\mathscr{P}^{\bot} \left \langle R^{TX}_{x_0} (\mathcal{R},e_k) z, R^{TX}_{x_0}
(\mathcal{R},e_k) \overline{z}\right\rangle_{x_0}\mathscr{P}\right) (0,0) \\
&\hspace{5mm}= 4 \left(\mathscr{L}^{-1}\mathscr{P}^{\bot}
(R_{k\overline{s} \ell \overline{u}} R_{m\overline{k}u\overline{q}}
+ R_{m\overline{k} \ell \overline{u}}R_{k\overline{s} u\overline{q}}) \overline{z}_{s} z_{\ell }
z_{m} \overline{z}_{q}\mathscr{P}\right) (0,0) \\
&\hspace{5mm}= - \frac{3}{2\pi ^3} (R_{k\overline{s}s\overline{u}} R_{q\overline{k}u\overline{q}}
+ 3 R_{s\overline{k}q\overline{u}} R_{k\overline{s}u\overline{q}}),\\
& \left(\mathscr{L}^{-1}\mathscr{P}^{\bot} \left\langle \pi
R^{TX}_{x_0}(z,\overline{z})\mathcal{R} , e_k\right\rangle
R^E_{x_0} (\mathcal{R},e_k) \mathscr{P}\right) (0,0) \\
&\hspace{5mm}= 2\pi \left(\mathscr{L}^{-1}\mathscr{P}^{\bot}
(-R_{\ell \overline{s}k\overline{q}} R^{E}_{m\overline{k}}
- R_{\ell \overline{s}m\overline{k}} R^{E}_{k\overline{q}}) z_{\ell } \overline{z}_{s}
\overline{z}_{q}z_{m}\mathscr{P}\right) (0,0)
= \frac{3}{\pi ^2}R_{s\overline{s}q\overline{k}} R^{E}_{k\overline{q}},\\
&\left(\mathscr{L}^{-1}\mathscr{P}^{\bot}R^E_{\, ;(Z,Z)} (z,
\overline{z})\mathscr{P}\right) (0,0)
=- \frac{3}{4\pi ^3} (R^E_{s\overline{s}\,;\, q\overline{q}}+ R^E_{q\overline{s}\,;\, s\overline{q}}) .
\end{split}\end{align}
Note that by Lemma \ref{lmt1.6}, $\ric(\mathcal{R},\mathcal{R})= 2
\ric_{k\overline{q}}z_{k}\overline{z}_{q}$, and by (\ref{toet1.1}), (\ref{bk2.66})
and (\ref{bk3.1}),
\begin{align}\label{alm3.49}
\mathscr{L} \ric(\mathcal{R},\mathcal{R}) \mathscr{P} = 2 b_{m} b^+_{m} \ric_{k\overline{q}}z_{k}\overline{z}_{q} \mathscr{P}
= 4 b_{m}\ric_{k\overline{m}}\,z_{k}\mathscr{P}.
\end{align}
Thus by \eqref{lm01.9}, \eqref{bk3.17b} and \eqref{alm3.49}, we obtain
\begin{align} \label{alm3.50}
\begin{split}
-(\mathscr{L}^{-1} \mathscr{P}^\bot \mathcal{O}_{43} \mathscr{P})(0,0)
&= \frac{1}{18} (\mathscr{L}^{-1} \mathscr{P}^\bot \ric_{l\overline{q}}z_{l} b_{m}
\ric_{k\overline{m}}z_{k}\frac{b_{q}}{2\pi} \mathscr{P})(0,0)\\
& = - \frac{1}{72\pi ^2} (\ric_{m\overline{m}}\ric_{q\overline{q}}
+ 3 \ric_{m\overline{q}}\ric_{q\overline{m}}).
\end{split}\end{align}
From \eqref{lm3.33}--\eqref{lm3.48} and \eqref{alm3.50}, we get
\begin{align} \label{lm3.50}
\begin{split}
-\pi^2 (\mathscr{L}^{-1}&(\mathcal{O}_{41}+ \mathcal{O}_{42}+ \mathcal{O}_{43}+ \mathcal{O}_{44}) \mathscr{P})(0,0)
= -\frac{2}{15} \cdot \frac{-11}{6}(R_{\ell \overline{\ell } q\overline{q}\,;\, u\overline{u}}
+ 2 R_{u\overline{\ell } q\overline{q}\,;\, \ell \overline{u}})\\
&+ \frac{1}{15} \cdot \frac{11}{3}
( R_{\ell \overline{s} q\overline{u}} R_{s\overline{\ell} u\overline{q}}
+ 2 R_{s\overline{s} q\overline{u}} R_{\ell\overline{\ell} u\overline{q}}) \\
&+ \Big( - \frac{2}{5}\cdot \frac{3}{4}
+ \frac{1}{10}\Big) (R_{q\overline{m} m\overline{q}; u\overline{u}}
+ R_{q\overline{m} u\overline{q}; m\overline{u}})\\
& - \Big( \frac{-12}{15}\cdot \frac{3}{4}
- \frac{2}{90}\Big) (R_{q\overline{s}s\overline{u}} R_{v\overline{q} u\overline{v}}
+ 3 R_{q\overline{s}v\overline{u}} R_{s\overline{q} u\overline{v}} )\\
&+ \frac{1}{72} ( \ric_{\ell\overline{\ell}}\ric_{q\overline{q}}
+\ric_{\ell\overline{q}}\ric_{q\overline{\ell}})
- \frac{1}{72} ( \ric_{\ell\overline{\ell}}\ric_{q\overline{q}}
+ 3 \ric_{\ell\overline{q}}\ric_{q\overline{\ell}})\\
=& \frac{2}{45} R_{\ell\overline{\ell} q\overline{q}\,;\, u\overline{u}}
+\frac{13}{45} R_{u\overline{\ell} q\overline{q}\,;\, \ell\overline{u}}
+ \frac{19}{9} R_{\ell\overline{s} q\overline{u}} R_{s\overline{\ell}u\overline{q}}
+ R_{s\overline{s} q\overline{u}} R_{\ell\overline{\ell}u\overline{q}}\;.
\end{split}\end{align}
Moreover, \eqref{lm3.35}, \eqref{lm3.47} and \eqref{lm3.49} yield
\begin{align} \label{lm3.51}
\begin{split}
-\pi^2 (\mathscr{L}^{-1}\mathcal{O}_{45}\mathscr{P})(0,0)
= &-\frac{2}{3} (R_{k\overline{\ell}\ell\overline{u}} R_{q\overline{k}u\overline{q}}
+ 3 R_{\ell\overline{k}q\overline{u}} R_{k\overline{\ell}u\overline{q}})\\
&+ \frac{3}{2} R_{\ell\overline{\ell}q\overline{k}} R^{E}_{k\overline{q}}
+ \frac{3}{8}(R^E_{\ell\overline{\ell}; q\overline{q}}+ R^E_{q\overline{\ell}\,;\, \ell\overline{q}}) .
\end{split}\end{align}
Further, \eqref{lm01.9b}, \eqref{bk3.16d}, \eqref{lm3.40},
\eqref{lm3.47} and \eqref{lm3.49} imply
\begin{align} \label{lm3.52}
\begin{split}
&-\pi^2 (\mathscr{L}^{-1}\mathcal{O}_{46}\mathscr{P})(0,0)=\frac{11}{108}
( R_{\ell\overline{s} q\overline{u}} R_{s\overline{\ell} u\overline{q}}
+ 2 R_{s\overline{s} q\overline{u}} R_{\ell\overline{\ell} u\overline{q}})
+ \frac{1}{9} \ric_{k\overline{q}} \ric_{q\overline{k}}\\
&\hspace{45mm}- \frac{1}{9} R_{k\overline{\ell}q\overline{k}} \ric_{\ell\overline{q}}
- \frac{1}{2} R_{\ell\overline{\ell}q\overline{k}} R^E_{k\overline{q}}
+\frac{1}{4} R^E_{k\overline{q}} R^E_{q\overline{k}}
- \frac{1}{2} R^E_{\ell\overline{\ell}; q\overline{q}}\\
&\hspace{3mm}= \frac{11}{108} R_{\ell\overline{s} q\overline{u}} R_{s\overline{\ell} u\overline{q}}
+ \frac{23}{54}R_{s\overline{s} q\overline{u}} R_{\ell\overline{\ell}
u\overline{q}}
- \frac{1}{2} R_{\ell\overline{\ell}q\overline{k}} R^E_{k\overline{q}}
+ \frac{1}{4} R^E_{k\overline{q}} R^E_{q\overline{k}}
-\frac{1}{2} R^E_{\ell\overline{\ell}\,;\, q\overline{q}}.
\end{split}\end{align}
\comment{
By \cite[\S 2.3.4, Prop. 6]{Petersen06},
\begin{align}\label{lm3.56}
d\br = 2div (\ric),
\end{align}
thus by Lemma \ref{lmt1.6} and \eqref{lm3.56}, we have
\begin{align}\label{lm3.57}
-\Delta\br =2 e_je_i(\ric(e_j,e_i)) = 32 R_{j'\overline{i} j\overline{j};
i\overline{j'}} ,
\quad -\Delta\br =32 R_{i\overline{i} j\overline{j}; j'\overline{j'}}.
\end{align}
}
By \eqref{alm01.5}, \eqref{lm01.12}, \eqref{lm3.50}--\,\eqref{lm3.52},
we get \eqref{lm3.31}.
The proof of Lemma \ref{bkt3.4} is completed.
\end{proof}
\subsection{Evaluation of $\mathscr{F}_4(0,0)$}\label{bks3.4}
\comment{
Set
\begin{align}\label{bk2.6}\begin{split}
&\bb_{2\field{C}}=
\frac{4}{45} R_{i\overline{i} j\overline{j}; j'\overline{j'}}
+\frac{26}{45} R_{j'\overline{i} j\overline{j}; i\overline{j'}}
+ \frac{1}{6}
R_{i\overline{i'} j\overline{j'}} R_{i'\overline{i}j'\overline{j}}
- \frac{2}{3}
R_{i'\overline{i'} j\overline{j'}} R_{i\overline{i}j'\overline{j}}
+ \frac{1}{2} R_{i'\overline{i'}j'\overline{j'}} R_{i\overline{i}j\overline{j}} ,\\
&\bb_{2E}= R^E_{j'\overline{j'}} R_{i\overline{i}j\overline{j}}
- R^E_{j\overline{j'}} R_{i\overline{i}j'\overline{j}}
+ \frac{1}{2} (R^E_{j'\overline{j'}} R^E_{j\overline{j}}
- R^E_{j\overline{j'}} R^E_{j'\overline{j}})
+\frac{1}{4}(-R^E_{i\overline{i}; j\overline{j}}+3 R^E_{j\overline{i}; i\overline{j}}),\\
&
\bb_{2} = \frac{1}{\pi^2}(\bb_{2\field{C}} + \bb_{2E}).
\end{split}\end{align}
}
\begin{thm}\label{bkt3.0} The following identity holds\rm{:}
\begin{align}\label{bk2.8}
\mathscr{F}_{4,\,x_{0}}(0,0)= \bb_{2} .
\end{align}
\end{thm}
\begin{proof}
By Lemmas \ref{bkt3.2}, \ref{bkt3.4} and (\ref{bk2.34}), we have
\begin{multline}\label{lm3.54}
\pi^2 \mathscr{F}_{41}(0,0) =- \frac{\Delta \br}{96}
+ \frac{7}{72}
R_{m\overline{s} q\overline{u}} R_{s\overline{m}u\overline{q}}
- \frac{1}{9}
R_{s\overline{s} q\overline{u}} R_{m\overline{m}u\overline{q}}
+ \frac{1}{8} R_{s\overline{s}u\overline{u}} R_{m\overline{m}q\overline{q}} \\
+ \frac{1}{4}R^E_{u\overline{u}} R_{m\overline{m}q\overline{q}}
- \frac{1}{6} R^E_{q\overline{u}} R_{m\overline{m}u\overline{q}}
+ \frac{1}{8} (R^E_{u\overline{u}} R^E_{q\overline{q}}
- R^E_{q\overline{u}} R^E_{u\overline{q}}
-R^E_{m\overline{m}; q\overline{q}}+3 R^E_{q\overline{m}\,;\, m\overline{q}})\,.
\end{multline}
Remark \ref{toet2.7}, Lemmas \ref{lmt1.6}, \ref{bkt3.1},
(\ref{bk2.6}), \eqref{bk2.32},
\eqref{lm3.54} and formula $\mathscr{P}(0,0)=1$ entail
\begin{equation}\label{lm3.55}
\begin{split}
J_{4,x_0}(0,0) = &\,\mathscr{F}_{41}(0,0)+ \mathscr{F}_{41}(0,0)^*
+\frac{1}{4\pi^2} \Big[\sum_{mq}R_{m\overline{m}q\overline{q}}
+ \sum_q R^E_{q\overline{q}}\Big]^2\\
&- \frac{1}{36\pi^2} R_{m\overline{k}q\overline{\ell}} R_{k\overline{m}\ell\overline{q}}
- \frac{1}{4\pi^2} \Big(\frac{4}{3}R_{q\overline{v}v\overline{\ell}} + R^E_{q\overline{\ell}}\Big)
\Big(\frac{4}{3} R_{\ell\overline{k}k\overline{q}} + R^E_{\ell\overline{q}}\Big)
=\bb_{2}\,.
\end{split}
\end{equation}
The proof of Theorem \ref{bkt3.0} is completed.
\end{proof}
\section{The first coefficients of the asymptotic
expansion}\label{toes4}
The lay-out of this section is as follows. In Section \ref{toes4.1}, we
explain the expansion of the kernel of Berezin-Toeplitz operators
and verify its compatibility with Riemann-Roch-Hirzebruch Theorem.
In Section \ref{toes4.2}, we establish Theorem \ref{toet4.1}. The
results from Sections \ref{bks3.2}, \ref{bks3.3} play an important role here.
In Section \ref{toes4.3}, we prove Theorems \ref{toet4.6}, \ref{toet4.5},
i.e., the expansion of the composition of two Berezin-Toeplitz operators.
We use the notations and assumptions from Introduction and Section \ref{toes3}.
\subsection{Expansion of the kernel of Berezin-Toeplitz operators}
\label{toes4.1}
For $U\in TX$, we have (cf. \eqref{lm01.1})
\begin{align} \label{abk4.1}
\nabla^{T^{*}X}_U\widetilde{dz_j} = 2 \langle \nabla^{TX}_U
\widetilde{\tfrac{\partial}{\partial \overline{z}_j}},
\widetilde{\tfrac{\partial}{\partial z_m}}\rangle \widetilde{dz_m},\quad
\nabla^{T^{*}X}_U\widetilde{d\overline{z}_j} = 2 \langle \nabla^{TX}_U
\widetilde{\tfrac{\partial}{\partial z_j}},
\widetilde{\tfrac{\partial}{\partial \overline{z}_m}}\rangle \widetilde{d\overline{z}_m}.
\end{align}
For
$\sigma=\sum_{kq} \sigma_{k\overline{q}}\widetilde{dz_k}\wedge\widetilde{d\overline{z}_q}
\in \Omega^{1,1}(X,\End(E))$,
by \cite[Lemma 1.4.4]{MM07}, \eqref{abk2.5} and \eqref{abk4.1}, we get
\begin{align} \label{abk4.2}
\begin{split}
\nabla^{1,0*} \sigma =&
- \Big(2 \nabla^E_{\widetilde{\tfrac{\partial}{\partial \overline{z}_m}}} \sigma_{m\overline{q}}
+ 4 \sigma_{k\overline{q}}
\Big\langle \nabla^{TX}_{\widetilde{\tfrac{\partial}{\partial \overline{z}_m}}}
\widetilde{\tfrac{\partial}{\partial \overline{z}_k}},
\widetilde{\tfrac{\partial}{\partial z_m}}\Big\rangle
+ 4 \sigma_{m\overline{l}}
\Big\langle \nabla^{TX}_{\widetilde{\tfrac{\partial}{\partial \overline{z}_m}}}
\widetilde{\tfrac{\partial}{\partial z_l}},
\widetilde{\tfrac{\partial}{\partial \overline{z}_q}}\Big\rangle \Big) \widetilde{d\overline{z}_q},\\
\overline{\partial}^{E*} \sigma = &
\Big(2 \nabla^E_{\widetilde{\tfrac{\partial}{\partial z_m}}} \sigma_{k\overline{m}}
+ 4 \sigma_{k\overline{q}}
\Big\langle \nabla^{TX}_{\widetilde{\tfrac{\partial}{\partial z_m}}}
\widetilde{\tfrac{\partial}{\partial z_q}},
\widetilde{\tfrac{\partial}{\partial \overline{z}_m}}\Big\rangle
+ 4 \sigma_{l\overline{m}}
\Big\langle \nabla^{TX}_{\widetilde{\tfrac{\partial}{\partial \overline{z}_m}}}
\widetilde{\tfrac{\partial}{\partial \overline{z}_l}},
\widetilde{\tfrac{\partial}{\partial z_k}}\Big\rangle \Big) \widetilde{d z_k}.
\end{split}\end{align}
We evaluate now \eqref{abk4.2} at the point $x_0$
(identified to $0\in\field{R}^{2n}$). By using \eqref{0c39} applied for $r=0,1$
associated with the vector bundles $E$, $T^{(1,0)}X$, we get
\begin{align} \label{abk4.3}
\begin{split}
&(\nabla^{1,0*} \sigma)_{x_0} =
-2 \frac{\partial\sigma_{m\overline{q}}}{\partial \overline{z}_m}(0)\,d\overline{z}_q,
\quad (\overline{\partial}^{E*} \sigma)_{x_0}
= 2 \frac{\partial \sigma_{k\overline{m}}}{\partial z_m}(0)\, d z_k,\, \, \\
&(\overline{\partial}^{E*}\nabla^{1,0*} \sigma)_{x_0}
=4 \frac{\partial \sigma_{k\overline{m}} }{\partial z_m\partial \overline{z}_k}(0)
+ 2 \left[R^E_{m\overline{k}}\,, \sigma_{k\overline{m}}(0)\right].
\end{split}\end{align}
Note that by
\eqref{bk2.6}, \eqref{lm01.4} and \eqref{abk4.3}, we have at $x_0$,
\begin{align} \label{bk4.2a}
\begin{split}
&\omega= \frac{\sqrt{-1}}{2} dz_q\wedge d\overline{z}_q,\quad
\tr[R^{T^{(1,0)}X}]= \ric_{k\overline{q}} dz_k \wedge d\overline{z}_q
= -\sqrt{-1} \ric_\omega,\\
& R^E=R^E_{k\overline{q}} dz_k\wedge d\overline{z}_q,\quad
\nabla^{1,0*} R^E= -2 R^E_{m\overline{q}; \overline{m}} d\overline{z}_q,\quad
\overline{\partial}^{E*} R^E= 2 R^E_{k\overline{q};q} dz_k.
\end{split}\end{align}
For $f\in\mathscr{C}^\infty(X,\End(E))$, recall that $T_{f,\,p}(x,x')$
is the smooth kernel of the Berezin-Toeplitz operator $T_{f,\,p}$
defined according to \eqref{toe2.4}.
Then by \eqref{0c39}, at $x_0$,
\begin{align} \label{bk4.3a}
\frac{\partial ^2 f_{\,x_0}}{\partial z_{q}
\partial \overline{z}_{\ell}} (0) =
(\nabla^E_{\widetilde{\tfrac{\partial}{\partial z_q}}}
\nabla^E_{\widetilde{\tfrac{\partial}{\partial \overline{z}_\ell}}} f)(x_0)
- \frac{1}{2} [R^E_{q\overline{\ell}}, f(x_0)],\quad
\Delta^E f = -4 \frac{\partial ^2 f_{\,x_0}}{\partial z_{q}
\partial \overline{z}_{q}} (0).
\end{align}
In view of Lemma \ref{lmt1.6}, \eqref{bk2.5} and \eqref{bk4.3a},
we introduce the following coefficients:
\begin{align}\label{bk4.1}\begin{split}
\bb_{\field{C} f}:= & R_{m\overline{m} q\overline{q}} \frac{\partial ^2 f_{\,x_0}}{\partial z_{k}
\partial \overline{z}_{k}} (0)
- R_{\ell\overline{k}k\overline{q}}\frac{\partial ^2 f_{\,x_0}}{\partial z_{q}
\partial \overline{z}_{\ell}} (0)\\
=&- \frac{\br}{32} \Delta^E f - \frac{\sqrt{-1}}{8}
\big\langle \ric_\omega, \nabla^{1,0}\overline{\partial}^Ef
-\frac{1}{2} [R^E, f] \big\rangle_{\omega}\,,
\end{split}\end{align}
and
\begin{align}\label{bk4.1a}\begin{split}
\bb_{E f1}:=& \frac{\partial f_{\,x_0}}{\partial z_{u}}(0)
\Big( \frac{1}{6} R^E_{k\overline{k} \,;\, \overline{u}}
-\frac{5}{12} R^E_{q\overline{u}\,;\,\overline{q}}\Big)
+ \frac{1}{4} R^E_{m\overline{u}\,;\, \overline{m}}
\frac{\partial f_{\,x_0}}{\partial z_{u}}(0) \\
&+ \Big( \frac{1}{6} R^E_{k\overline{k}\,;\, u}
- \frac{5}{12} R^E_{u\overline{q}\,;\,q}\Big)
\frac{\partial f_{\,x_0}}{\partial \overline{z}_{u}}(0)
+ \frac{1}{4}\frac{\partial f_{\,x_0}}{\partial \overline{z}_{u}}(0)
R^E_{u\overline{m} \,;\, m}\\
=& \frac{1}{48} \Big\langle \nabla^{1,0} f, 2\sqrt{-1}\,
\overline{\partial}^E R^E_{\Lambda}
+ 5 \nabla^{1,0*} R^E\Big\rangle_{\omega}
- \frac{1}{16} \langle \nabla^{1,0*} R^E, \nabla^{1,0} f \rangle_{\omega}\\
&+ \frac{1}{16} \langle \overline{\partial}^E f, \overline{\partial}^{E*}R^E \rangle_{\omega}
+ \frac{1}{48}\Big\langle 2\sqrt{-1}\nabla^{1,0}R^E_{\Lambda}
- 5 \overline{\partial}^{E*}R^E, \overline{\partial}^E f\Big\rangle_{\omega},\\
\bb_{E f2}:=&\frac{1}{2} \frac{\partial ^2 f_{\,x_0}}{\partial z_{k}
\partial \overline{z}_{k}} (0) R^E_{q\overline{q}}
- \frac{1}{2} \frac{\partial ^2 f_{\,x_0}}{\partial z_{q}
\partial \overline{z}_{\ell}} (0) R^E_{\ell\overline{q}}
+ \frac{1}{2} R^E_{q\overline{q}}
\frac{\partial ^2 f_{\,x_0}}{\partial z_{k}\partial \overline{z}_{k}} (0)
- \frac{1}{2}R^E_{\ell\overline{q}}
\frac{\partial ^2 f_{\,x_0}}{\partial z_{q}\partial \overline{z}_{\ell}} (0)\\
= &- \frac{\sqrt{-1}}{16} \Big[R^E_{\Lambda} \Delta^E f
+ (\Delta^E f)R^E_{\Lambda} \Big]
+\frac{1}{8} \Big\langle \nabla^{1,0} \overline{\partial}^E f
-\frac{1}{2} [R^E, f], R^E\Big\rangle_{\omega}\\
&+\frac{1}{8} \Big \langle R^E, \nabla^{1,0} \overline{\partial}^E f
-\frac{1}{2} [R^E, f]\Big\rangle_{\omega}.
\end{split}\end{align}
The following result implies Theorem \ref{toet4.1}.
\begin{thm} \label{toet4.1a}
Let $f\in\mathscr{C}^\infty(X,\End(E))$.
There exist smooth sections
$\bb_{r,f}(x)\in \End(E)_x$ such that \eqref{bk4.2} and \eqref{bk4.3}
hold and
\begin{equation}\label{bk4.4}
\begin{split}
\pi^2 \bb_{2,f} = \bb_{2\field{C}} f(x_0)
&+ \frac{1}{2} \Big(\bb_{2E}+ \frac{1}{16}(R^E_{\Lambda})^2\Big) f(x_0)
+ \frac{1}{2} f(x_0) \Big(\bb_{2E}+ \frac{1}{16}(R^E_{\Lambda})^2\Big)\\
&- \frac{1}{16}R^E_{\Lambda} f(x_0)R^E_{\Lambda}
+ \frac{1}{32}(\Delta^{E})^2 f + \bb_{\field{C} f}
+\bb_{Ef1}+\bb_{Ef2}.
\end{split}
\end{equation}
\end{thm}
Before giving the proof, we verify that Theorem \ref{toet4.1} is
compatible with the
Riemann-Roch-Hirzebruch Theorem. Note that by \eqref{toe2.1},
the first Chern class $c_1(L)$ of $L$ is represented by $\omega$.
By the Kodaira vanishing Theorem and the
Riemann-Roch-Hirzebruch Theorem, we have for $p$ large enough:
\begin{equation}\label{bk2.10}
\begin{split}
\dim H^0 (&X, L^p\otimes E)= \int_X \td(T^{(1,0)}X)\ch( E) \, e^{p\, \omega}\\
=&\,{\rm rk} (E) \int_X \frac{\omega^n}{n!} p^n
+ \int_X \Big(c_1(E) + \frac{{\rm rk} (E)}{2} c_1(X)\Big)
\frac{\omega^{n-1} p^{n-1}}{(n-1)!} \\
&+ \int_X \Big({\rm rk} (E)\{\td(T^{(1,0)}X)\}^{(4)}
+ \frac{1}{2} c_1(X)c_1(E)
+\{\ch( E)\}^{(4)}\Big)
\frac{\omega^{n-2} p^{n-2}}{(n-2)!}\\
&+ \mathscr{O}(p^{n-3})\,.
\end{split}
\end{equation}
As usual, $\ch(\cdot), c_1(\cdot), \td(\cdot)$ are the Chern character,
the first Chern class and the Todd class of the corresponding complex
vector bundles, $\{\cdot\}^{(4)}$ is the degree $4$-part of the
corresponding differential forms.
Note that \[\frac{x}{1-e^{-x}}= 1+ \frac{x}{2} + \frac{x^2}{12}
+\ldots\,,\] thus $\{\td(T^{(1,0)}X)\}^{(4)}= \frac{1}{12} (c_1(X)^2 + c_2(X))$.
Let $R^{T^{(1,0)}X}$ be the curvature of the Chern connection
on $T^{(1,0)}X$ which is the restriction of the Levi-Civita connection
in our case.
Then by \eqref{bk4.2a},
we have the following identities at the cohomology level:
\begin{align} \label{bk2.11}
\begin{split}
&\{\ch( E)\}^{(4)} = -\frac{1}{8\pi^2} \tr[(R^E)^2],
\qquad c_1(X)= \frac{1}{2\pi} \ric_\omega,\\
&\{\td(T^{(1,0)}X)\}^{(4)}
= \frac{1}{32\pi^2} (\ric_\omega)^2
+ \frac{1}{96\pi^2} \tr\Big[(R^{T^{(1,0)}X})^2 \Big].
\end{split}\end{align}
By Lemma \ref{lmt1.6} and \eqref{bk4.2a}, we have
\begin{align} \label{bk2.14}
\begin{split}
& \frac{1}{32}\left\langle (\ric_\omega)^2, \omega^2/2\right\rangle
= \frac{1}{2} (R_{u\overline{u}v\overline{v}}
R_{k\overline{k}q\overline{q}}- R_{u\overline{u} q\overline{v}} R_{k\overline{k}v\overline{q}}),\\
&\frac{1}{96}\left\langle \tr\Big[(R^{T^{(1,0)}X})^2 \Big],
\omega^2/2\right\rangle
= \frac{1}{6}(R_{k\overline{u} q\overline{v}} R_{u\overline{k}v\overline{q}}
- R_{u\overline{u} q\overline{v}} R_{k\overline{k}v\overline{q}}),\\
& \langle \{\ch( E)\}^{(4)}, \omega^2/2\rangle
= \frac{1}{2 \pi^2} \tr \Big[R^E_{k\overline{k}} R^E_{q\overline{q}}
- R^E_{k\overline{q}} R^E_{q\overline{k}}\Big],\\
& \left\langle \frac{1}{2} c_1(X) c_1(E), \omega^2/2\right\rangle = \frac{1}{2\pi^2}
\tr (R^E_{k\overline{k}} \ric_{q\overline{q}} - \ric_{q\overline{k}} R^E_{k\overline{q}}).
\end{split}\end{align}
\comment{
Let $\overline{\partial}^*$, $\partial^*$ be the adjoints of the operators
$\overline{\partial}$, $\partial$.
Let $\nabla^{TX}$ be the connection on
$\Lambda(T^{*}X)$ induced by the Levi-Civita connection $\nabla^{TX}$.
Finally, for $\alpha$ a $(1,1)$-form, we have
\begin{align} \label{bk2.15}
(\overline{\partial}^* \partial^* \alpha)_{x_0}
= 4 \Big(i_{\widetilde{\frac{\partial}{\partial \overline{z}_{q}}}}\,
\nabla^{TX}_{\widetilde{\frac{\partial}{\partial z_{q}}}}\,
i_{\widetilde{\frac{\partial}{\partial z_{i}}}}\,
\nabla^{TX}_{\widetilde{\frac{\partial}{\partial \overline{z}_{i}}}}\alpha\Big)_{x_0}
=4 \frac{\partial}{\partial z_{q}}
\frac{\partial}{\partial \overline{z}_{i}}
\Big(\alpha\Big(\widetilde{\tfrac{\partial}{\partial z_{i}}},
\widetilde{\tfrac{\partial}{\partial \overline{z}_{q}}}\Big)\Big)_{x_0},
\end{align}
by using \eqref{0c39} for $r=1$ and the bundle $T^{(1,0)}X$.
}
We set now $f=1$ in Theorem \ref{toet4.1}, take the pointwise trace
of the expansion \eqref{bk4.2} relative to $E$ and then integrate
the result over $X$ with respect to the volume form $\omega^{n}/n!$\,.
Taking into account \eqref{bk2.6} and \eqref{bk2.11}, \eqref{bk2.14},
we recover the expansion
up to $\mathscr{O}(p^{n-3})$ given in \eqref{bk2.10} for the Hilbert polynomial.
Thus the value of $\bb_{2}$ obtained in Theorem \ref{toet4.1}
is compatible with the Riemann-Roch-Hirzebruch Theorem.
\subsection{Proof of Theorem \ref{toet4.1a}}\label{toes4.2}
The first part of Theorem \ref{toet4.1a} follows from Lemma \ref{toet2.3}.
Moreover, by \eqref{toe2.13}, we have for any $r\in \field{N}$,
\begin{align}\label{bk4.5}
\bb_{r,f}(x_0) = Q_{2r,x_0}(f) (0,0).
\end{align}
Thus by \eqref{toe2.15}, the formula $\bb_{0,f}=f$,
and by \eqref{toe2.14} and \eqref{bk2.33}, we get
\begin{equation} \label{bk4.6}
Q_{2,\,x_0}(f) =
\mathscr{K}\big[1, f(x_0)J_{2,x_0} \big]
+ \mathscr{K}\big[J_{2,x_0}, f(x_0)\big]
+ \sum_{|\alpha|=2}
\mathscr{K}\Big[1\;,\;
\frac{\partial ^\alpha f_{\,x_0}}{\partial Z^\alpha}(0)
\frac{Z^\alpha}{\alpha !} \Big]\,.
\end{equation}
Further, \eqref{toe1.6}, \eqref{bk2.24} and \eqref{bk2.31},
entail
\begin{align}\label{bk4.7}
\begin{split}
&\mathscr{K}\big[1, f(x_0)J_{2,x_0} \big]\mathscr{P}
= - f(x_0)\mathscr{P} \mathcal{O}_2 \mathscr{L}^{-1}\mathscr{P}^{\bot},\\
&\mathscr{K}\big[J_{2,x_0}, f(x_0) \big]\mathscr{P}
= - ( \mathscr{L}^{-1}\mathscr{P}^{\bot}\mathcal{O}_2\mathscr{P} ) f(x_0).
\end{split}\end{align}
From \eqref{toe1.6} and \eqref{abk3.8}, we deduce
\begin{align}\label{bk4.8}\begin{split}
&\sum_{|\alpha|=2}
\mathscr{K}\Big[1\;,\;
\frac{\partial ^\alpha f_{\,x_0}}{\partial Z^\alpha}(0)
\frac{Z^\alpha}{\alpha !} \Big]\mathscr{P}(Z,0)
=\bigg(\sum_{|\alpha|=2}\frac{\partial^\alpha f_{x_0}}{\partial {z}^\alpha}(0)
\frac{z^\alpha}{\alpha !}
+ \frac{1}{\pi} \frac{\partial ^2 f_{x_0}}
{\partial z_{i} \partial \overline{z}_{i}}(0)\!\bigg)\mathscr{P}(Z,0).
\end{split}\end{align}
Lemma \ref{lmt1.6}, \eqref{bk3.7}, \eqref{bk3.8}, \eqref{bk4.3a}
and \eqref{bk4.5}--\eqref{bk4.8} yield the formula for $\bb_{1,f}$
from \eqref{bk4.2}.
It remains to compute $\bb_{2,f}$ for
a self-adjoint section $f\in \mathscr{C}^\infty(X,\End(E))$ in order to complete
the proof of Theorem \ref{toet4.1a}.
Set
\begin{align}\label{abk4.9}
\mathscr{K}_{2f}= \sum_{|\alpha|=2}
\mathscr{K}\Big[1, \frac{\partial ^\alpha f_{\,x_0}}{\partial Z^\alpha}(0)
\frac{Z^\alpha}{\alpha !}J_{2,x_0} \Big].
\end{align}
By \eqref{toe2.14} and \eqref{bk2.33}, we get
\begin{equation} \label{bk4.9}
\begin{split}
Q&_{4,\,x_0}(f) = \mathscr{K}\Big[1, f(x_0)J_{4,x_0} \Big]
+\mathscr{K}\Big[J_{2,x_0}, f(x_0)J_{2,x_0} \Big]
+ \mathscr{K}\Big[J_{4,x_0}, f(x_0)\Big]\\
&+ \sum_{|\alpha|=2} \mathscr{K}\Big[J_{2,x_0}\;,\;
\frac{\partial ^\alpha f_{\,x_0}}{\partial Z^\alpha}(0)
\frac{Z^\alpha}{\alpha !} \Big]+ \mathscr{K}_{2f}\\
&+ \mathscr{K}\Big[1, \frac{\partial f_{\,x_0}}{\partial Z_i}(0) Z_i J_{3,x_0} \Big]
+ \mathscr{K}\Big[J_{3,x_0}, \frac{\partial f_{\,x_0}}{\partial Z_i}(0) Z_i \Big]
+\sum_{|\alpha|=4}
\mathscr{K}\Big[1, \frac{\partial ^\alpha f_{\,x_0}}{\partial Z^\alpha}(0)
\tfrac{Z^\alpha}{\alpha !}\Big] \,.
\end{split}
\end{equation}
Since $\mathscr{L}_0$ and $\mathcal{O}_r$ are formally self-adjoint,
\eqref{bk2.23} and \eqref{bk2.77} show that $(\mathscr{F}_{r,x_0})^* =\mathscr{F}_{r,x_0}$.
Hence, in the right hand side of \eqref{bk4.9},
the first, fourth and sixth terms
are adjoints of the third, fifth and seventh terms, respectively.
When we take $f=1$ in \eqref{bk4.9}, we get
\begin{align} \label{bk4.10}
J_{4,x_0}= \mathscr{K}\big[1, J_{4,x_0} \big]
+\mathscr{K}\big[J_{2,x_0}, J_{2,x_0} \big]
+ \mathscr{K}\big[J_{4,x_0}, 1\big],
\end{align}
which is also a direct consequence of \eqref{bk2.31},
\eqref{bk2.32} and \eqref{bk3.0a}, as by \eqref{toe1.6},
\begin{align}\label{bk4.11}
\begin{split}
&\mathscr{K}\big[1, J_{4,x_0} \big] \mathscr{P}=
\mathscr{P}\mathcal{O}_{2}\mathscr{L}^{-1}\mathscr{P}^\bot \mathcal{O}_{2}\mathscr{L}^{-1}\mathscr{P}^\bot
- \mathscr{P} \mathcal{O}_{4}\mathscr{L}^{-1}\mathscr{P}^\bot
- \mathscr{P} \mathcal{O}_{2} \mathscr{L}^{-2} \mathcal{O}_{2} \mathscr{P},\\
&\mathscr{K}\big[J_{4,x_0}, 1\big]\mathscr{P}= \mathscr{L}^{-1}\mathscr{P}^\bot
\mathcal{O}_{2}\mathscr{L}^{-1}\mathscr{P}^\bot \mathcal{O}_{2} \mathscr{P}- \mathscr{L}^{-1}\mathscr{P}^\bot \mathcal{O}_{4} \mathscr{P}-
\mathscr{P} \mathcal{O}_{2} \mathscr{L}^{-2} \mathcal{O}_{2} \mathscr{P},\\
&\mathscr{K}\big[J_{2,x_0}, J_{2,x_0} \big]\mathscr{P}
= \mathscr{L}^{-1}\mathscr{P}^\bot \mathcal{O}_{2} \mathscr{P} \mathcal{O}_{2} \mathscr{L}^{-1}\mathscr{P}^\bot
+ \mathscr{P} \mathcal{O}_{2} \mathscr{L}^{-2} \mathcal{O}_{2} \mathscr{P}.
\end{split}\end{align}
Set
\begin{align}\label{abk4.10}\begin{split}
&K_{41}:= -\frac{\Delta \br}{96}
+ \frac{5}{72} R_{m\overline{u} q\overline{v}} R_{u\overline{m}v\overline{q}}
- \frac{5}{9} R_{u\overline{u} q\overline{v}} R_{m\overline{m}v\overline{q}}
+ \frac{1}{8} R_{u\overline{u}v\overline{v}} R_{m\overline{m}q\overline{q}} ,\\
&K_{42}:= \frac{1}{4}R^E_{v\overline{v}} R_{m\overline{m}q\overline{q}}
- \frac{5}{6} R^E_{q\overline{v}} R_{m\overline{m}v\overline{q}}
+ \frac{1}{8} \Big(R^E_{v\overline{v}} R^E_{q\overline{q}}
- 3 R^E_{q\overline{v}} R^E_{v\overline{q}}
-R^E_{m\overline{m}; q\overline{q}}+3 R^E_{q\overline{m}; m\overline{q}}\Big),\\
&K_{2f}:= \frac{1}{4} (R_{m\overline{m}q\overline{q}}
+ R^E_{q\overline{q}}) f(x_{0})(R_{u\overline{u}v\overline{v}} + R^E_{v\overline{v}})
+\frac{1}{36} R_{m\overline{k}q\overline{\ell}} R_{k\overline{m}\ell\overline{q}} f(x_{0}) \\
&\hspace{15mm}
+ \frac{1}{4} \Big(\frac{4}{3}R_{q\overline{s}s\overline{\ell}} + R^E_{q\overline{\ell}}\Big)
f(x_{0}) \Big(\frac{4}{3} R_{\ell\overline{k}k\overline{q}} + R^E_{\ell\overline{q}}\Big).
\end{split}\end{align}
By \eqref{bk2.34}, (\ref{bk3.0c}), (\ref{lm3.54}) and (\ref{bk4.11}), we have
\begin{align}\label{abk4.11}
&\mathscr{K}\big[J_{4,x_0},1 \big] (0,0)= \frac{1}{\pi^2} (K_{41}+ K_{42}).
\end{align}
By \eqref{toe1.6} and (\ref{bk2.31}), we see as in (\ref{bk4.11}) that
\begin{equation}\label{abk4.12}
\begin{split}
\mathscr{K}\big[J_{2,x_0}, f(x_{0})J_{2,x_0} \big] \mathscr{P}
= \mathscr{L}^{-1} \mathscr{P}^\bot\mathcal{O}_2 f(x_{0})\mathscr{P} \mathcal{O}_2\mathscr{L}^{-1}\mathscr{P}^\bot\\
+ \mathscr{P} \mathcal{O}_2\mathscr{L}^{-1} f(x_{0})\mathscr{L}^{-1} \mathcal{O}_2 \mathscr{P}.
\end{split}
\end{equation}
Thus by (\ref{bk3.7}), (\ref{bk3.8}) and (\ref{abk4.12}),
as in Lemma \ref{bkt3.1}, we get
\begin{align}\label{abk4.13}
\mathscr{K}\big[J_{2,x_0}, f(x_{0})J_{2,x_0} \big] (0,0)
=\frac{1}{\pi^2} K_{2f}.
\end{align}
We next compute the fifth term in \eqref{bk4.9}.
From \eqref{bk2.31} and (\ref{abk4.9}), we get
\begin{align}\label{bk4.12}
\mathscr{K}_{2f}= \mathscr{P}\sum_{|\alpha|=2} \frac{\partial^\alpha f_{\,x_0}}
{\partial Z^\alpha}(0)
\frac{Z^\alpha}{\alpha !} \Big(-\mathscr{L}^{-1} \mathcal{O}_2 \mathscr{P}
-\mathscr{P} \mathcal{O}_2\mathscr{L}^{-1}\mathscr{P}^\bot\Big)\,.
\end{align}
For a degree $2$ polynomial $F(Z)$ we have
by Remark \ref{bkt3.5}, \eqref{toe1.4}, \eqref{bk2.66}, \eqref{bk3.1}
and \eqref{bk3.7}
\begin{equation}\label{bk4.13}
\begin{split}
- (& \mathscr{P} F \mathscr{L}^{-1} \mathcal{O}_2 \mathscr{P}) (0,0)\\
&=- \Big(\mathscr{P} \frac{\partial ^2 F}{\partial z_{u}
\partial \overline{z}_{v}} z_{u} \overline{z}_{v}
\Big\{\frac{b_{m} b_{q}}{48\pi} R_{k\overline{m}l\overline{q}} z_{k} z_{l}
+ \frac{b_q}{4\pi}\Big( \frac{4}{3} R_{l\overline{k}k\overline{q}}
+ R^E_{l\overline{q}}\Big)z_l\Big\}
\mathscr{P}\Big) (0,0)\\
&= - \Big(\mathscr{P} \frac{\partial ^2 F}{\partial z_{q}
\partial \overline{z}_{v}} \frac{ \overline{z}_{v}}{2\pi}
\Big( \frac{4}{3} R_{l\overline{k}k\overline{q}}+ R^E_{l\overline{q}}\Big)z_l
\mathscr{P}\Big) (0,0)
= - \frac{1}{2\pi^2} \frac{\partial ^2 F}{\partial z_{q}
\partial \overline{z}_{l}}\Big( \frac{4}{3} R_{l\overline{k}k\overline{q}}
+ R^E_{l\overline{q}}\Big),
\end{split}
\end{equation}
where we have used \eqref{toe1.4}, \eqref{bk2.66}
and \eqref{abk3.8} in the last two equalities.
By \eqref{bk3.8}, \eqref{abk3.8}, \eqref{bk4.12} and \eqref{bk4.13},
we get
\begin{align}\label{bk4.14}
\mathscr{K}_{2f}(0,0)= \frac{1}{2\pi^2} \frac{\partial ^2 f_{\,x_0}}{\partial z_{k}
\partial \overline{z}_{k}} (0) (R_{m\overline{m}q\overline{q}} + R^E_{q\overline{q}})
- \frac{1}{2\pi^2} \frac{\partial ^2 f_{\,x_0}}{\partial z_{q}
\partial \overline{z}_{l}} (0)
\Big( \frac{4}{3} R_{l\overline{k}k\overline{q}}+ R^E_{l\overline{q}}\Big).
\end{align}
By Lemma \ref{lmt1.6}, (\ref{bk4.1}), (\ref{bk4.1a})
and (\ref{bk4.14}), we get
\begin{align}\label{bk4.15}
\mathscr{K}_{2f}(0,0) + \mathscr{K}_{2f}(0,0)^*
=\frac{1}{\pi^2} (\bb_{\field{C} f}+ \bb_{Ef2})
- \frac{1}{3\pi^2} R_{l\overline{k}k\overline{q}}
\frac{\partial ^2 f_{\,x_0}}{\partial z_{q}\partial \overline{z}_{l}} (0) .
\end{align}
We compute now the last term in \eqref{bk4.9}.
By Remark \ref{bkt3.5}, \eqref{toe1.6} and \eqref{abk3.8}, we have
\begin{align}\label{bk4.16}
\sum_{|\alpha|=4}
\mathscr{K}\Big[1, \frac{\partial ^\alpha f_{\,x_0}}{\partial Z^\alpha}(0)
\frac{Z^\alpha}{\alpha !}\Big](0,0)
= \frac{1}{2\pi^2}\frac{\partial ^4
f_{\,x_0} }{\partial z_i \partial z_q\partial\overline{z}_i \partial\overline{z}_q}(0).
\end{align}
We next turn to the computation of the sixth term in \eqref{bk4.9}. Set
\begin{equation}\label{bk4.17}
\begin{split}
K_{3f}: &= \Big(\frac{1}{6}R_{k\overline{k}m\overline{m}; \overline{u}}
- \frac{1}{3} R_{k\overline{k}m\overline{u}; \overline{m}} \Big)
\frac{\partial f_{\,x_0}}{\partial z_{u}}(0)
+ \Big(\frac{1}{6}R_{k\overline{k}m\overline{m}; u}- \frac{1}{3} R_{k\overline{k}u\overline{m}; m}
\Big)\frac{\partial f_{\,x_0}}{\partial \overline{z}_{u}}(0) \\
&+ \frac{\partial f_{\,x_0}}{\partial z_{u}}(0)
\Big( \frac{1}{6} R^E_{k\overline{k} ; \overline{u}}
-\frac{1}{2} R^E_{q\overline{u};\overline{q}}\Big)
+ \frac{1}{3} R^E_{m\overline{u}; \overline{m}}
\frac{\partial f_{\,x_0}}{\partial z_{u}}(0) \\
&+ \Big( \frac{1}{6} R^E_{k\overline{k}; u}
- \frac{1}{2} R^E_{u\overline{q};q}\Big)
\frac{\partial f_{\,x_0}}{\partial \overline{z}_{u}}(0)\
+ \frac{1}{3}\frac{\partial f_{\,x_0}}{\partial \overline{z}_{u}}(0)
R^E_{u\overline{m} ; m}.
\end{split}
\end{equation}
\begin{lemma} \label{bkt4.3} The following identity holds
with $K_{3f}$ defined in \eqref{bk4.17}:
\begin{align}\label{bk4.18}
\mathscr{K}\Big[ J_{3,\,x_0}\,, \frac{\partial f_{\,x_0}}{\partial Z_{u}}(0) Z_{u}
\Big](0,0)
+ \mathscr{K}\Big[1, \frac{\partial f_{\,x_0}}{\partial Z_{u}}(0) Z_{u}
J_{3,\,x_0}\Big](0,0)
=\frac{1}{\pi^2} K_{3f}.
\end{align}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{bkt4.3}] Set
\begin{subequations}
\begin{align} \label{bk4.19}
B_1(b,Z):=&\Big\{ \frac{1}{6} R_{k\overline{s}\ell\overline{q}; \overline{m}} z_{k} z_{\ell}
b_s b_q + \frac{2\pi}{3} R_{q\overline{s}k\overline{q}; \overline{m}} \overline{z}_{s} z_{k}\\
&- \Big[\frac{2\pi}{15} R_{k\overline{q}\ell\overline{s}; \overline{m}}
z_{\ell} \overline{z}_q +\frac{1}{3} R_{\ell\overline{\ell}k\overline{s} ; \overline{m}}
- \frac{1}{3} R_{k\overline{s}q\overline{m}; \overline{q}}
- \frac{2}{3} R^E_{k\overline{s}; \overline{m}} \Big] z_{k} b_s \nonumber\\
&- \frac{2\pi}{15} R_{k\overline{s}q\overline{m}; \overline{q}} z_{k} \overline{z}_s
-\frac{2}{3} R_{\ell\overline{\ell}q\overline{m};\overline{q}}
-\frac{2}{3} R_{\ell\overline{\ell}q\overline{q}; \overline{m}}
+ \frac{2}{3} R^E_{q\overline{m};\overline{q}}
- 2R^E_{q\overline{q} ; \overline{m}} \Big\} \overline{z}_m ,\nonumber\\
\label{bk4.21}
\mathcal{B}_1(Z): =&\frac{2\pi^2}{5} R_{k\overline{s}\ell\overline{q}; \overline{m}} z_{k} z_{\ell}
\overline{z}_s \overline{z}_q \overline{z}_m
+ \frac{8\pi}{15} R_{k\overline{s}q\overline{m}; \overline{q}} z_{k} \overline{z}_s
\overline{z}_m\\
&+ \frac{4\pi}{3} R^E_{k\overline{s} ; \overline{m}} z_k \overline{z}_{s}\overline{z}_m
-\frac{2}{3} \Big[ R_{\ell\overline{\ell}q\overline{m};\overline{q}}
+ R_{\ell\overline{\ell}q\overline{q}; \overline{m}}
- R^E_{q\overline{m};\overline{q}}
+ 3 R^E_{q\overline{q} ; \overline{m}} \Big] \overline{z}_m. \nonumber
\end{align}
\end{subequations}
Then by Lemma \ref{lmt1.6}, (\ref{bk3.1}), we have
\begin{equation}\label{bk4.22}
\begin{split}
B_1(b,Z) & \mathscr{P}(Z,0)=\Big\{
\frac{4\pi^2}{6} R_{k\overline{s}\ell\overline{q}; \overline{m}} z_{k} z_{\ell}\overline{z}_s \overline{z}_q
+ \frac{2\pi}{3} R_{q\overline{s}k\overline{q}; \overline{m}} \overline{z}_{s}z_{k}\\
&- \Big[\frac{2\pi}{15} R_{k\overline{q}\ell\overline{s}; \overline{m}}
z_{\ell} \overline{z}_q +\frac{1}{3} R_{\ell\overline{\ell}k\overline{s} ; \overline{m}}
- \frac{1}{3} R_{k\overline{s}q\overline{m}; \overline{q}}
- \frac{2}{3} R^E_{k\overline{s} ; \overline{m}}\Big] z_k \cdot 2\pi\overline{z}_s \\
&- \frac{2 \pi}{15} R_{k\overline{s}q\overline{m}; \overline{q}} z_{k} \overline{z}_s
- \frac{2}{3} R_{\ell\overline{\ell}q\overline{m};\overline{q}}
-\frac{2}{3} R_{\ell\overline{\ell}q\overline{q}; \overline{m}}
+ \frac{2}{3} R^E_{q\overline{m};\overline{q}}
- 2R^E_{q\overline{q} ; \overline{m}} \Big\} \overline{z}_{m}\mathscr{P}(Z,0)\\
&= \mathcal{B}_1(Z)\mathscr{P}(Z,0).
\end{split}
\end{equation}
Observe that the commutation relations \eqref{bk2.66} imply that
\begin{equation*}
\begin{split}
R_{k\overline{s}\ell\overline{q}; \overline{m}} z_{k} z_{\ell} b_{s}b_{q}b_{m}
= b_{s}b_{q}b_{m}R_{k\overline{s}\ell\overline{q}; \overline{m}} z_{k} z_{\ell}
&+ b_{s}b_{m} ( 8 R_{q\overline{s}k\overline{q}; \overline{m}} + 4R_{k\overline{s}q\overline{m};
\overline{q}} )z_{k} \\
&+ b_{m}(8 R_{q\overline{s}s\overline{q}; \overline{m}} + 16R_{s\overline{s}q\overline{m};
\overline{q}} )\,.
\end{split}
\end{equation*}
By (\ref{bk2.66}), (\ref{bk3.1}) and (\ref{bk4.21}), we have
\begin{equation}\label{bk4.24}
\begin{split}
\mathcal{B}_1(Z)\mathscr{P}(Z,0)
=\frac{1}{\pi} \Big\{ \Big(\frac{1}{20}
R_{k\overline{s}l\overline{q}; \overline{m}} z_{k} z_{\ell} b_q
+ \frac{2}{15} R_{k\overline{s}q\overline{m}; \overline{q}} z_{k}
+ \frac{1}{3} R^E_{k\overline{s} ; \overline{m}} z_k \Big) b_{s}b_m\\
- \frac{1}{3} \Big[ R_{\ell\overline{\ell}q\overline{m};\overline{q}}
+ R_{\ell\overline{\ell}q\overline{q}; \overline{m}}
- R^E_{q\overline{m};\overline{q}}
+ 3 R^E_{q\overline{q} ; \overline{m}} \Big] b_m\Big\}\mathscr{P}(Z,0)\\
=\frac{1}{\pi} \Big\{ \frac{b_s b_q b_m}{20}
R_{k\overline{s}\ell\overline{q}\,;\,\overline{m}} z_{k} z_{\ell}
+ b_{s}b_m\Big(\frac{2}{5} R_{q\overline{s}k\overline{q}\,;\,\overline{m}}
+ \frac{1}{3}R_{k\overline{s}q\overline{m}; \overline{q}}
+ \frac{1}{3} R^E_{k\overline{s} ; \overline{m}} \Big) z_{k} \\
+ b_m\Big( \frac{1}{15}
R_{s\overline{s}q\overline{q}; \overline{m}}+
R_{s\overline{s}q\overline{m}; \overline{q}}
- \frac{1}{3} R^E_{s\overline{s} ; \overline{m}} + R^E_{q\overline{m};\overline{q}}
\Big) \Big\}\mathscr{P}(Z,0).
\end{split}
\end{equation}
Note that by Theorem \ref{bkt2.17} and \eqref{bk3.1}, we have
$(\mathscr{P}^\bot z_{u} \mathscr{P})(Z,0)=(\mathscr{P} \overline{z}_{u} \mathscr{P})(Z,0)=0$.
Taking into account that $\mathscr{P}(0,0)=1$ and
relations \eqref{toe1.6}, \eqref{bk2.24} and \eqref{bk2.31}, we get
\begin{equation}\label{bk4.25}
\begin{split}
\mathscr{K}\Big[ J_{3,x_0}, \frac{\partial f_{\,x_0}}{\partial Z_{u}}(0) Z_{u}
\Big](0,0)
= -( \mathscr{L}^{-1} \mathscr{P}^\bot\mathcal{O}_3\mathscr{P} \frac{\partial f_{\,x_0}}{\partial z_{u}}(0)
z_{u}\mathscr{P} )(0,0) \\
-( \mathscr{P} \mathcal{O}_3\mathscr{L}^{-1}\mathscr{P}^\bot
\tfrac{\partial f_{\,x_0}}{\partial \overline{z}_{u}}(0) \overline{z}_{u} \mathscr{P} )(0,0) .
\end{split}
\end{equation}
By Remark \ref{toet2.7}, $(\mathscr{P} \mathcal{O}_3\mathscr{L}^{-1} \overline{z}_{u} \mathscr{P} )(0,0)$ is the
adjoint of $(\mathscr{P} z_{u}\mathscr{L}^{-1}\mathscr{P}^\bot\mathcal{O}_3\mathscr{P} )(0,0)$,
thus we will compute only the latter.
By Lemma \ref{lmt1.6}, \eqref{bk2.66}, \eqref{lm01.11} and \eqref{bk3.2},
we get as in \eqref{bk3.3},
\begin{equation}\label{bk4.26}
\begin{split}
\mathcal{O}_3=&\, \frac{1}{6} R_{k\overline{s}\ell\overline{q}; Z} z_{k} z_{\ell}
b_s b_q + \frac{2\pi}{3} R_{q\overline{s}k\overline{q}; Z} \overline{z}_{s}z_{k}
+ c_s(b,b^+, Z) b^+_s\\
&- \Big(\frac{2\pi}{15} R_{\ell\overline{q}k\overline{s}; Z} z_{\ell} \overline{z}_q
+\frac{1}{3} R_{\ell\overline{\ell}k\overline{s} ; Z}
+\frac{1}{3} \big(R_{\ell\overline{q}k\overline{s}; q} z_{\ell}
-R_{q\overline{m}k\overline{s}; \overline{q}} \overline{z}_{m}\big)
- \frac{2}{3} R^E_{k\overline{s} ; Z} \Big) z_{k} b_s \\
&+ \frac{2\pi}{15} (R_{k\overline{s}\ell\overline{q}; q}z_\ell
-R_{k\overline{s}q\overline{m}\,;\, \overline{q}} \overline{z}_{m} ) z_{k} \overline{z}_s
-\frac{2}{3} \big( R_{\ell\overline{\ell}k\overline{q} ;q} z_k
+ R_{\ell\overline{\ell}q\overline{m};\overline{q}} \overline{z}_m\big)\\
& -\frac{2}{3} R_{\ell\overline{\ell}q\overline{q}; Z}
-\frac{2}{3} \big( R^E_{k\overline{q} ;q} z_k -R^E_{q\overline{m};\overline{q}}\overline{z}_m \big)
- 2R^E_{q\overline{q} ; Z},
\end{split}
\end{equation}
and $c_s(b,b^+, Z)$ are polynomials in $b, b^+, Z$, whose precise
formula will not be used,
and $R_{k\overline{q}\ell\overline{s}; Z} $, $R^E_{k\overline{s} ; Z}$ are defined by
replacing $\frac{\partial}{\partial z_{s}}$ by $Z$ in \eqref{lm01.4}.
From Lemma \ref{lmt1.6}, \eqref{bk2.66}, \eqref{bk4.19}
and \eqref{bk4.26} we deduce that
the only term in $\mathcal{O}_3$ not containing
$b^+$ and
having total degree in $b,\overline{z}$ bigger than its degree in $z$, is $B_1(b,Z)$.
Now, Theorem \ref{bkt2.17}, Remark \ref{bkt3.5}, \eqref{toe1.4},
\eqref{bk2.66}, \eqref{bk4.22}, \eqref{bk4.24} and \eqref{bk4.26} imply
\begin{equation}\label{bk4.27}
\begin{split}
- (&\mathscr{P} z_{u} \mathscr{L}^{-1} \mathscr{P}^\bot\mathcal{O}_3\mathscr{P} )(0,0)
= - (\mathscr{P} z_{u}\mathscr{L}^{-1} \mathscr{P}^\bot B_{1}(b,Z)\mathscr{P} )(0,0)\\
&=- \Big(\mathscr{P} z_{u} \frac{b_m}{4 \pi^2}\Big(\frac{1}{15}
R_{s\overline{s}q\overline{q}; \overline{m}}+ R_{s\overline{s}q\overline{m}; \overline{q}}
- \frac{1}{3} R^E_{s\overline{s} ; \overline{m}}
+ R^E_{q\overline{m};\overline{q}}\Big) \mathscr{P} \Big)(0,0)\\
&= - \frac{1}{2 \pi^2}\Big(\frac{1}{15}
R_{s\overline{s}q\overline{q}; \overline{u}}+ R_{s\overline{s}q\overline{u}; \overline{q}}
- \frac{1}{3} R^E_{s\overline{s} ; \overline{u}} + R^E_{q\overline{u};\overline{q}}\Big).
\end{split}
\end{equation}
By Remark \ref{bkt3.5}, (\ref{toe1.4}), (\ref{bk2.66}) and (\ref{bk4.26}),
we also have
\begin{equation}\label{bk4.28}
\begin{split}
- ( \mathscr{L}^{-1} \mathscr{P}^\bot\mathcal{O}_3\mathscr{P} z_{u} \mathscr{P} )(0,0)
= - ( \mathscr{L}^{-1} \mathscr{P}^\bot B_{1}(b,Z) z_{u}\mathscr{P} )(0,0)\\
= - \Big\{\mathscr{L}^{-1} \mathscr{P}^\bot \big[z_{u}B_{1}(b,Z)
-2 \tfrac{\partial}{\partial b_{u}}B_{1}(b,Z)\big]\mathscr{P} \Big\}(0,0) .
\end{split}
\end{equation}
Lemma \ref{lmt1.6}, (\ref{bk3.16d}), (\ref{bk3.16g}),
(\ref{lm3.38}), (\ref{bk4.21}) and (\ref{bk4.22}) yield
\begin{equation}\label{bk4.29}
\begin{split}
- &(\mathscr{L}^{-1} \mathscr{P}^\bot z_{u}B_{1}(b,Z) \mathscr{P} )(0,0)
= - (\mathscr{L}^{-1} \mathscr{P}^\bot z_{u}\mathcal{B}_{1}(Z) \mathscr{P} )(0,0) \\
&= \frac{11}{24\pi^2} \cdot
\frac{2}{5} \big(4 R_{s\overline{s}m\overline{u}; \overline{m}} +2 R_{s\overline{s}q\overline{q}; \overline{u}}\big)
+\frac{3}{8\pi^2} \cdot \Big(\frac{16}{15} R_{s\overline{s}q\overline{u}; \overline{q}}
+\frac{4}{3} R^E_{s\overline{s} ; \overline{u}}
+\frac{4}{3} R^E_{m\overline{u} ; \overline{m}} \Big)\\
&\hspace{4cm}- \frac{1}{6 \pi^2}\Big( R_{\ell\overline{\ell}q\overline{u};\overline{q}}
+ R_{\ell\overline{\ell}q\overline{q}; \overline{u}} - R^E_{q\overline{u};\overline{q}}
+ 3 R^E_{q\overline{q} ; \overline{u}} \Big) \\
&= \frac{1}{\pi^2} \Big(
\frac{29}{30} R_{s\overline{s}m\overline{u}; \overline{m}}
+ \frac{1}{5} R_{s\overline{s}q\overline{q}; \overline{u}}
+\frac{2}{3} R^E_{m\overline{u} ; \overline{m}}\Big)\,.
\end{split}
\end{equation}
Moreover, by (\ref{bk4.19}),
\begin{equation}\label{bk4.30}
\begin{split}
\frac{\partial}{\partial
b_{u}}B_{1}(b,Z) = &\,\frac{1}{3} R_{k\overline{s}\ell\overline{u}; \overline{m}}
z_{k} z_{\ell} b_{s} \overline{z}_{m}\\
&- \Big(\frac{2\pi}{15} R_{k\overline{q}\ell\overline{u}; \overline{m}}
z_{\ell} \overline{z}_q +\frac{1}{3} R_{\ell\overline{\ell}k\overline{u} ; \overline{m}}
- \frac{1}{3} R_{k\overline{u}q\overline{m}; \overline{q}}
- \frac{2}{3} R^E_{k\overline{u} ; \overline{m}} \Big)
z_{k} \overline{z}_{m} .
\end{split}
\end{equation}
Lemma \ref{lmt1.6}, (\ref{bk3.1}), (\ref{bk3.16d}),
(\ref{bk3.16g}) and (\ref{bk4.30}) yield
\begin{equation}\label{bk4.31}
\begin{split}
( &\mathscr{L}^{-1} \mathscr{P}^\bot (\tfrac{\partial}{\partial b_{u}}B_{1}(b,Z))\mathscr{P})(0,0) \\
&= -\frac{3}{8\pi^2} \cdot
\frac{8}{15} \cdot 2
R_{s\overline{s}m\overline{u}; \overline{m}}
+ \frac{1}{4\pi^2} \Big(\frac{1}{3} R_{m\overline{q}q\overline{u} ; \overline{m}}
- \frac{1}{3} R_{m\overline{u}q\overline{m}; \overline{q}}
- \frac{2}{3} R^E_{m\overline{u} ; \overline{m}} \Big) \\
&= -\frac{2}{5\pi^2} R_{s\overline{s}m\overline{u}; \overline{m}}
- \frac{1}{6\pi^2} R^E_{m\overline{u} ; \overline{m}}.
\end{split}
\end{equation}
Formulas \eqref{bk4.28}--\eqref{bk4.31} entail altogether
\begin{equation}\label{bk4.32}
-\pi^2 ( \mathscr{L}^{-1} \mathscr{P}^\bot\mathcal{O}_3\mathscr{P} z_{u} \mathscr{P} )(0,0)
=\frac{1}{6}R_{s\overline{s}m\overline{u}; \overline{m}}
+ \frac{1}{5} R_{s\overline{s}q\overline{q}; \overline{u}}
+\frac{1}{3}R^E_{m\overline{u} ; \overline{m}} .
\end{equation}
Combining \eqref{bk4.25}, \eqref{bk4.27} with \eqref{bk4.32}, we get
\begin{equation}\label{bk4.33}
\begin{split}
\pi^2 \mathscr{K}\Big[ J_{3,x_0}, \frac{\partial f_{\,x_0}}{\partial Z_{u}}(0) Z_{u}
\Big](0,0)= \Big(\frac{1}{6}R_{s\overline{s}m\overline{u}; \overline{m}}
+ \frac{1}{5} R_{s\overline{s}q\overline{q}; \overline{u}}
+\frac{1}{3}R^E_{m\overline{u} ; \overline{m}} \Big)
\frac{\partial f_{\,x_0}}{\partial z_{u}}(0) \\
+ \Big(-\frac{1}{30} R_{s\overline{s}q\overline{q}; u}
- \frac{1}{2} R_{s\overline{s}u\overline{q}; q}
+ \frac{1}{6} R^E_{s\overline{s}; u} -\frac{1}{2}R^E_{u\overline{q}; q} \Big)
\frac{\partial f_{\,x_0}}{\partial \overline{z}_{u}}(0).
\end{split}
\end{equation}
Since $\mathscr{K}\Big[1,\frac{\partial f_{\,x_0}}{\partial Z_{u}}(0) Z_{u}
J_{3,x_0} \Big]$ is the adjoint of
$\mathscr{K}\Big[ J_{3,x_0}, \tfrac{\partial f_{\,x_0}}{\partial Z_{u}}(0) Z_{u}
\Big]$, Lemma \ref{lmt1.6} yield
\begin{equation}\label{bk4.34}
\begin{split}
\pi^2 \mathscr{K}\Big[1, \frac{\partial f_{\,x_0}}{\partial Z_{u}}(0) Z_{u}
J_{3,x_0} \Big](0,0)
=\frac{\partial f_{\,x_0}}{\partial \overline{z}_{u}}(0)
\Big(\frac{1}{6}R_{s\overline{s}u\overline{m}; m}
+ \frac{1}{5} R_{s\overline{s}q\overline{q}; u}
+\frac{1}{3}R^E_{u\overline{m}; m} \Big)\\
+\frac{\partial f_{\,x_0}}{\partial z_{u}}(0)
\Big(-\frac{1}{30} R_{s\overline{s}q\overline{q}; \overline{u}}
- \frac{1}{2} R_{s\overline{s}q\overline{u}; \overline{q}}
+ \frac{1}{6} R^E_{s\overline{s}; \overline{u}} -\frac{1}{2}R^E_{q\overline{u};\overline{q}} \Big).
\end{split}
\end{equation}
Finally, \eqref{bk4.33} and \eqref{bk4.34} deliver \eqref{bk4.18}.
The proof of Lemma \ref{bkt4.3} is completed.
\end{proof}
We continue with the proof of Theorem \ref{toet4.1a}. We'll write now
the formulas in terms of connections.
For $f\in \mathscr{C}^{\infty}(X,\End(E))$, we obtain
by \eqref{toe2.5} (as in \eqref{bk2.87}) the following formula
in normal coordinates:
\begin{align}\label{bk4.41}
(\Delta^{E} f)(Z) = -g^{ij}(\nabla^{E}_{e_{i}}\nabla^{E}_{e_{j}}
- \Gamma^{l}_{ij}\nabla^{E}_{e_{l}}) f, \quad
\nabla^{E} f = df + [\Gamma^{E}(\cdot), f].
\end{align}
By (\ref{bk2.85}),
\begin{equation}\label{bk4.42}
\begin{split}
\nabla^{E}_{e_{i}}\nabla^{E}_{e_{i}}
&= \frac{\partial^{2}}{\partial Z_{i}^{2}}
+ R^E(\mathcal{R},e_{i}) \frac{\partial}{\partial Z_{i}}
+ \frac{1}{4} R^E (\mathcal{R},e_{i}) R^E (\mathcal{R},e_i)
+ \frac{2}{3} R^E_{\, ; Z} (\mathcal{R},e_i) \frac{\partial}{\partial
Z_{i}}\\
&+ \frac{1}{3} R^E_{\, ; e_{i}} (\mathcal{R},e_i)
+\frac{1}{8} \Big(2 R^E_{\, ; (Z,e_{i})} (\mathcal{R},e_i)
+ \frac{1}{3}\left \langle R^{TX} (\mathcal{R},e_i)e_{i},
e_k\right \rangle R^E (\mathcal{R},e_k) \Big)\\& + \mathscr{O}(|Z|^3).
\end{split}
\end{equation}
Note that by \eqref{lm01.2},
$\dfrac{\partial^{2}}{\partial Z_{m}^{2}}R^E_{\, ; (Z,e_{i})} (\mathcal{R},e_i)
= 2 R^E_{\, ; (e_{m},e_{i})} (e_{m},e_i) =0$.
Thus from (\ref{alm01.0}), (\ref{bk3.2}), (\ref{bk4.42}),
and taking into account that $R^E(e_{m}, e_{i})$ is anti-symmetric in
$m,i$, and $\ric(e_{m}, e_{i})$ is symmetric in $m,i$, we infer
\begin{multline}\label{bk4.43}
\Big( \frac{\partial^{2}}{\partial Z_{m}^{2}}
\nabla^{E}_{e_{i}}\nabla^{E}_{e_{i}} f\Big)(0)
= \frac{\partial^{4} f_{x_0}}{\partial Z_{m}^{2}\partial Z_{i}^{2}}(0)
+ 2 \bigg[R^E(e_{m}, e_{i}), \frac{\partial^{2}f_{x_0}}{\partial Z_{i}\partial
Z_{m}}(0)\bigg] \\
+ \frac{1}{2} \Bigl[R^E(e_{m},e_{i}), \Bigl[R^E(e_{m},e_{i}),
f(x_{0})\Bigr] \Bigr]
+ \frac{2}{3} \bigg[R^E_{\, ; e_{m}} (e_{m},e_i) , \frac{\partial
f_{x_0}}{\partial Z_{i}}(0)\bigg]\\
= 16\frac{\partial ^4
f_{x_0} }{\partial z_i \partial z_q\partial\overline{z}_i \partial\overline{z}_q}(0)
- 4 \Big[R^{E}_{k\overline{\ell}}, \Big[R^{E}_{\ell\overline{k}}, f(x_{0})\Big]\Big]
+ \frac{8}{3} \bigg[R^E_{m\overline{q} ; \overline{m}} , \frac{\partial
f_{x_0}}{\partial z_{q}}(0)\bigg]
-\frac{8}{3} \bigg[R^E_{q\overline{m} ; m} , \frac{\partial
f_{x_0}}{\partial \overline{z}_{q}}(0)\bigg] .
\end{multline}
By \eqref{alm01.1}, \eqref{abk2.83}, \eqref{bk2.85} and \eqref{bk4.41},
\begin{equation}\label{bk4.44}
\begin{split}
- \frac{\partial^{2}}{\partial Z_{m}^{2}}&
(\Gamma^{\ell}_{ii}\nabla^{E}_{e_{\ell}} f)(0)\\
&= - \frac{4}{3} \ric(e_{m}, e_{\ell})
\Big( \frac{\partial^{2}f_{x_0}}{\partial Z_{\ell}\partial
Z_{m}}(0) + \frac{1}{2} \left[R^E(e_{m},e_{\ell}),
f(x_{0})\right]\Big)\\
&\quad - \frac{1}{6}\Big(4\ric_{\, ; e_{m}}(e_{m}, e_{\ell})
-2 \ric_{\, ; e_{q}}(e_{q}, e_{\ell})
+ \ric_{\, ; e_{\ell}}(e_{m}, e_{m}) \Big) \frac{\partial f_{x_0}}{\partial
Z_{\ell}}(0)\\
&= - \frac{32}{3} \ric_{m\overline{\ell}}\frac{\partial ^2
f_{x_0} }{\partial z_{\ell} \partial\overline{z}_{m}}(0)\\
&\quad - \frac{4}{3} \Big( \big(\ric_{m\overline{\ell}; \overline{m}}
+ \ric_{m\overline{m}; \overline{\ell}}\big) \frac{\partial
f_{x_0} }{\partial z_\ell}(0) + \big(\ric_{\ell\overline{m}; m} + \ric_{m\overline{m}; \ell}\big)
\frac{\partial f_{x_0} }{\partial\overline{z}_\ell}(0)\Big)\,.
\end{split}
\end{equation}
Formulas \eqref{alm01.1}, \eqref{lm01.38}, \eqref{bk2.85}, \eqref{bk4.3a}
and \eqref{bk4.41}--\eqref{bk4.44} yield
\begin{equation}\label{bk4.45}
\begin{split}
((\Delta^E)^2 f&)(x_{0}) =
- \Big( \frac{\partial^{2}}{\partial Z_{m}^{2}} \Delta^E f\Big)(0) \\
= &\,\frac{2}{3} \ric(e_{i}, e_{q}) (
\nabla^{E}_{e_{i}}\nabla^{E}_{e_{q}}f)(0)
+ \Big( \frac{\partial^{2}}{\partial Z_{m}^{2}}
\nabla^{E}_{e_{i}}\nabla^{E}_{e_{i}} f\Big)(0)
- \frac{\partial^{2}}{\partial Z_{m}^{2}}
(\Gamma^{l}_{ii}\nabla^{E}_{e_{l}} f)(0)\\
= &\,16\frac{\partial ^4
f_{x_0} }{\partial z_i \partial z_q\partial\overline{z}_i \partial\overline{z}_q}(0)
- \frac{16}{3} \ric_{m\overline{\ell}}\frac{\partial ^2
f_{x_0} }{\partial z_{\ell} \partial\overline{z}_{m}}(0)
- 4 \left[R^{E}_{k\overline{\ell}}\,, [R^{E}_{\ell\overline{k}}\,, f(x_{0})]\right]\\
& + \frac{8}{3} \bigg[R^E_{m\overline{q} ; \overline{m}} \,, \frac{\partial
f_{x_0}}{\partial z_{q}}(0)\bigg]
-\frac{8}{3} \bigg[R^E_{q\overline{m} ; m} \,, \frac{\partial
f_{x_0}}{\partial \overline{z}_{q}}(0)\bigg]\\
& - \frac{4}{3} \Big( \big(\ric_{m\overline{\ell}; \overline{m}}
+ \ric_{m\overline{m}; \overline{\ell}}\big) \frac{\partial
f_{x_0} }{\partial z_\ell}(0) + \big(\ric_{\ell\overline{m}; m} + \ric_{m\overline{m}; \ell}\big)
\frac{\partial f_{x_0} }{\partial\overline{z}_\ell}(0)\Big)\,.
\end{split}
\end{equation}
By Lemma \ref{bkt4.3}, the discussion after \eqref{bk4.9},
formulas \eqref{bk4.9}, \eqref{abk4.11}, \eqref{abk4.13},
\eqref{bk4.15} and \eqref{bk4.16}, we have
\begin{equation}\label{bk4.46}
\begin{split}
\pi^2 Q_{4,\,x_0}(f) (0,0)= &\, 2 K_{41} f(x_{0})
+ K_{42} f(x_{0}) +f(x_{0}) K_{42}+K_{2f}+ K_{3f}\\
&+\bb_{\field{C} f}+ \bb_{Ef2}
- \frac{1}{3} R_{\ell\overline{k}k\overline{q}}
\frac{\partial ^2 f_{\,x_0}}{\partial z_{q}\partial \overline{z}_{\ell}} (0)
+ \frac{1}{2}\frac{\partial ^4
f_{\,x_0} }{\partial z_i \partial z_q\partial\overline{z}_i
\partial\overline{z}_q}(0).
\end{split}
\end{equation}
Note that
$\left[R^{E}_{k\overline{\ell}}\,, [R^{E}_{\ell\overline{k}}\,, f(x_{0})]\right]
= R^{E}_{k\overline{\ell}}R^{E}_{\ell\overline{k}}f(x_{0})
- 2 R^{E}_{k\overline{\ell}}f(x_{0}) R^{E}_{\ell\overline{k}}
+ f(x_{0}) R^{E}_{k\overline{\ell}}R^{E}_{\ell\overline{k}}$\,,
so by \eqref{alm01.5}, \eqref{bk4.5}, (\ref{bk4.45}) and \eqref{bk4.46},
we get \eqref{bk4.4}.
The proof of Theorem \ref{toet4.1a} is completed.
\subsection{Composition of Berezin-Toeplitz operators: proofs of Theorems
\ref{toet4.6}, \ref{toet4.5}}\label{toes4.3}
\begin{proof}[\textbf{Proof of Theorem \ref{toet4.6}}]
By Lemma \ref{toet2.3}, we deduce as in the proof of
Lemma \ref{toet2.3}, that for $Z,Z^\prime \in T_{x_0}X$,
$\abs{Z},\abs{Z^{\prime}}<\varepsilon/4$, we have
(cf.\,\cite[(4.79)]{MM08b}, \cite[(7.4.6)]{MM07})
\begin{align} \label{toe4.6}
p^{-n}(T_{f,\,p}\circ T_{g,\,p})_{x_0}(Z,Z^\prime)\cong
\sum^k_{r=0}(Q_{r,\,x_0}(f,g)\mathscr{P}_{x_0})(\sqrt{p}\,Z,\sqrt{p}\,Z^{\prime})
p^{-\frac{r}{2}} + \mathcal{O}(p^{-\frac{k+1}{2}}),
\end{align}
where
\begin{align} \label{toe4.7}
Q_{r,\,x_0}(f,g)= \sum_{r_1+r_2=r} \mathscr{K}[Q_{r_1,\,x_0}(f),
Q_{r_2,\,x_0}(g)]\in\End(E)_{\,x_{0}}[Z,Z^{\prime}],
\end{align}
is a polynomial in $Z,Z^{\prime}$ with the same parity as $r$.
The existence of the expansion \eqref{toe4.30} and the expression of
$\bb_{0,\,f,\,g}$ follow from
\eqref{toe2.15}, \eqref{toe4.6} and \eqref{toe4.7}; we get also
\begin{align}\label{toe4.33}
\bb_{r,\,f,\,g}(x_{0}) = Q_{2r,\,x_{0}}(f,g)(0,0).
\end{align}
By \eqref{toe4.7},
\begin{align}\label{toe4.10}
Q_{2,\,x_{0}}(f,g) = \mathscr{K}[f(x_{0}), Q_{2,\,x_{0}}(g)]
+ \mathscr{K}[Q_{1,\,x_{0}}(f), Q_{1,\,x_{0}}(g)]
+ \mathscr{K}[Q_{2,\,x_{0}}(f), g(x_{0})].
\end{align}
Formulas (\ref{toe1.6}), (\ref{bk4.6})--(\ref{bk4.7}) yield
\begin{align}\label{toe4.11}
\begin{split}
\mathscr{K}[Q_{2,\,x_{0}}(f), g(x_{0})] \mathscr{P}
&= (Q_{2,\,x_{0}}(f)\mathscr{P})\mathscr{P} g(x_{0}) \\
&= \Big( \mathscr{P} \sum_{|\alpha|=2}\frac{\partial ^\alpha f_{x_{0}}}
{\partial Z^\alpha}(0) \frac{Z^\alpha}{\alpha !} \mathscr{P}
- \mathscr{L}^{-1}\mathcal{O}_2\mathscr{P} f(x_{0}) \Big) g(x_{0}),\\
\mathscr{K}[f(x_{0}), Q_{2,\,x_{0}}(g)] \mathscr{P}
&= f(x_{0}) \mathscr{P} (Q_{2,\,x_{0}}(g)\mathscr{P})\\
&= f(x_{0}) \Big( \mathscr{P} \sum_{|\alpha|=2}
\frac{\partial ^\alpha g_{x_{0}}}{\partial Z^\alpha}(0)
\frac{Z^\alpha}{\alpha !} \mathscr{P} - g(x_{0}) \mathscr{P}
\mathcal{O}_2\mathscr{L}^{-1}\mathscr{P}^\bot\Big).
\end{split}
\end{align}
Using \eqref{bk2.6}, \eqref{bk3.7}, \eqref{bk3.8}, \eqref{abk3.8},
\eqref{bk4.3a} and \eqref{toe4.11}, we obtain
\begin{align}\label{toe4.12} \begin{split}
\mathscr{K}[Q_{2,\,x_{0}}(f), g(x_{0})] (0,0)
=& \left(- \frac{1}{4\pi} (\Delta^{E}f)(x_{0})
+ \frac{1}{2} (\bb_{1} f)(x_{0})\right)g(x_{0}),\\
\mathscr{K}[f(x_{0}), Q_{2,\,x_{0}}(g)] (0,0)
=&f(x_{0}) \left( - \frac{1}{4\pi} (\Delta^{E} g)(x_{0})
+ \frac{1}{2} (g\bb_{1})(x_{0})\right).
\end{split}\end{align}
By (\ref{toe1.6}), (\ref{toe2.14}) and (\ref{bk2.33}),
we have (cf.\,\cite[(7.4.14)]{MM07}),
\begin{align}\label{toe4.13}
Q_{1,\,x_{0}}(f) (Z,Z^{\prime})
= \mathscr{K}\Big[1, \frac{\partial f_x}{\partial Z_q}(0) Z_q\Big](Z,Z^{\prime})
= \frac{\partial f_{x_{0}}}{\partial z_i}(0)
z_i+\frac{\partial f_{x_{0}}}{\partial \overline{z}_i}(0)
\overline{z}^{\,\prime}_{i}.
\end{align}
Thus from (\ref{abk3.8}), (\ref{toe4.12}),
we get as in \cite[(7.4.15)]{MM07} (cf.\,also \cite[(7.1.11)]{MM07}),
\begin{align}\label{toe4.16}
\mathscr{K}[Q_{1,\,x_{0}}(f), Q_{1,\,x_{0}}(g)] (0,0)=
\sum_{i=1}^n \frac{1}{\pi}
\frac{\partial f_{x_{0}}}{\partial \overline{z}_i}(0)\,
\frac{\partial g_{x_{0}}}{\partial z_i}(0).
\end{align}
Now, \eqref{toe4.33}, \eqref{toe4.10}, \eqref{toe4.12} and \eqref{toe4.16}
imply the formula for $\bb_{1,\,f,\,g}$ given in (\ref{toe4.31}).
We prove next \eqref{toe4.32}. It suffices to consider
$f,g\in \mathscr{C}^\infty(X,\field{R})$, which we henceforth assume.
By \eqref{toe2.9}, we have $Q_{r,\,x_0}(1)=J_{r,\,x_0}$.
Hence, taking $f=1$ in \eqref{toe4.7}, we get
\begin{equation}\label{toe4.36}
Q_{4,\,x}(g)= \mathscr{K}[1, Q_{4,\,x}(g)]
+ \mathscr{K}[J_{2,\,x}, Q_{2,\,x}(g)]
+ \mathscr{K}[J_{3,\,x}, Q_{1,\,x}(g)]+ g(x) \mathscr{K}[J_{4,\,x}, 1].
\end{equation}
Taking $g=1$ in \eqref{toe4.7} yields
\begin{equation}\label{toe4.37}
Q_{4,\,x}(f)= f(x) \mathscr{K}[1 , J_{4,\,x}]
+ \mathscr{K}[Q_{1,\,x}(f), J_{3,\,x}]\\
+ \mathscr{K}[Q_{2,\,x}(f), J_{2,\,x}]
+ \mathscr{K}[Q_{4,\,x}(f), 1].
\end{equation}
By (\ref{toe2.15}) and (\ref{toe4.7}), we get
\begin{equation}\label{toe4.34}
\begin{split}
Q_{4,\,x_0}&(f,g)= \mathscr{K}[ f(x_0), Q_{4,\,x_0}(g)]
+ \mathscr{K}[Q_{1,\,x_0}(f), Q_{3,\,x_0}(g)]\\
&+ \mathscr{K}[Q_{2,\,x_0}(f), Q_{2,\,x_0}(g)]
+ \mathscr{K}[Q_{3,\,x_0}(f), Q_{1,\,x_0}(g)]+ \mathscr{K}[Q_{4,\,x_0}(f), g(x_0)].
\end{split}
\end{equation}
Set
\begin{align}\label{toe4.39}\begin{split}
&\widetilde{Q}_{3,\,x_0}(g) = \mathscr{K}\Big[1, \frac{\partial g_{x_{0}}}
{\partial Z_{q}}(0) Z_{q} J_{2,\,x_0}\Big]
+\mathscr{K}\Big[ J_{2,\,x_0}, \frac{\partial g_{x_{0}}}
{\partial Z_{q}}(0) Z_{q} \Big]\\
&\hspace{20mm}+ \mathscr{K}\Big[1, \sum_{|\alpha|=3}\frac{\partial ^\alpha g_{x_{0}}}
{\partial Z^\alpha}(0) \frac{Z^\alpha}{\alpha !}\Big],\\
&\bI_{4,f,g}= \mathscr{K}\bigg[\mathscr{K}\Big[1, \sum_{|\alpha|=2}
\frac{\partial ^\alpha g_{x_{0}}}
{\partial Z^\alpha}(0) \frac{Z^\alpha}{\alpha !}\Big],\,
\mathscr{K}\Big[1, \sum_{|\alpha|=2}\frac{\partial ^\alpha g_{x_{0}}}
{\partial Z^\alpha}(0) \frac{Z^\alpha}{\alpha !}\Big]\bigg].
\end{split}\end{align}
Note that by (\ref{bk2.24}) and (\ref{bk2.31}),
we have
\begin{align}\label{toe4.38}
J_{3,\,x_0} = \mathscr{K}[1, J_{3,\,x_0}]+ \mathscr{K}[J_{3,\,x_0}, 1].
\end{align}
and \eqref{toe2.14}, \eqref{bk2.33} together with \eqref{toe4.38} imply
\begin{align}\label{toe4.40}
Q_{3,\,x_0}(g) = g(x_{0}) J_{3,\,x_0} + \widetilde{Q}_{3,\,x_0}(g) .
\end{align}
Since $g\in\mathscr{C}^{\infty}(X,\field{R})$, \eqref{bk4.6} entails
\begin{align}\label{toe4.41}
Q_{2,\,x_0}(g) = g(x_{0}) J_{2,\,x_0}
+ \mathscr{K} \Big[1, \sum_{|\alpha|=2}\frac{\partial ^\alpha g_{x_{0}}}
{\partial Z^\alpha}(0) \frac{Z^\alpha}{\alpha !} \Big].
\end{align}
By Remark \ref{bkt3.5}, \eqref{bk4.10}, \eqref{toe4.36}--\eqref{toe4.41}, we get
\begin{equation}\label{toe4.42}
\begin{split}
Q_{4,\,x_0}(f,g)= &\,f(x_0) Q_{4,\,x_0}(g) + g(x_0)Q_{4,\,x_0}(f)
- f(x_0) g(x_0) J_{4,\,x_0}\\
&+ \mathscr{K}[Q_{1,\,x_0}(f), \widetilde{Q}_{3,\,x_0}(g)]
+ \mathscr{K}[ \widetilde{Q}_{3,\,x_0}(f), Q_{1,\,x_0}(g)]
+ \bI_{4,f,g}.
\end{split}
\end{equation}
By (\ref{abk3.8}) and (\ref{bk4.8}), we get
\begin{equation}\label{toe4.43}
\begin{split}
(\bI_{4,f,g}&\mathscr{P})(0,0)\\
& =
\Big\{ \mathscr{P} \sum_{|\beta|=2}\frac{\partial ^\beta f_{x_{0}}}
{\partial Z^\beta}(0) \frac{Z^\beta}{\beta !}
\Big(\sum_{|\alpha|=2}\frac{\partial^\alpha g_{x_0}}{\partial {z}^\alpha} (0)
\frac{z^\alpha}{\alpha !}
+ \frac{1}{\pi} \frac{\partial ^2 g_{x_0}}{\partial z_{i} \partial \overline{z}_{i}} (0)
\Big) \mathscr{P}\Big\}(0,0)\\
&= \frac{1}{\pi^2} \Big( \frac{1}{2}
\frac{\partial ^2 f_{x_0}}{\partial \overline{z}_{i} \partial \overline{z}_{q}} (0)
\frac{\partial ^2 g_{x_0}}{\partial z_{i} \partial z_{q}} (0)
+ \frac{\partial ^2 f_{x_0}}{\partial z_{q} \partial \overline{z}_{q}} (0)
\frac{\partial ^2 g_{x_0}}{\partial z_{i} \partial \overline{z}_{i}}
(0)\Big).
\end{split}
\end{equation}
By Remark \ref{toet2.7},
$Q_{1,\,x_0}(f)$, $Q_{3,\,x_0}(g)$ are self-adjoint
for $f,g \in \mathscr{C}^{\infty}(X,\field{R})$, thus by (\ref{toe4.40}),
\begin{align}\label{toe4.44}
\mathscr{K}[ \widetilde{Q}_{3,\,x_0}(f), Q_{1,\,x_0}(g)]
= \mathscr{K}[ Q_{1,\,x_0}(g), \widetilde{Q}_{3,\,x_0}(f)]^{*}.
\end{align}
Thus we only need to compute the fourth term in \eqref{toe4.42}.
An examination of \eqref{bk3.3} shows that in each term of the sum
giving $\mathcal{O}_2$, the total degree in
$b^+,z$ equals the total degree in $b,\overline{z}$.
Hence Remark \ref{bkt3.5}, \eqref{toe1.6}, \eqref{bk2.31},
\eqref{bk3.7}, \eqref{bk3.8}, \eqref{abk3.8}, \eqref{bk4.13} and \eqref{toe4.13} yield
\begin{equation}\label{toe4.48}
\begin{split}
\mathscr{K}\Big[Q_{1,\,x_0}(f)&, \mathscr{K}\Big[1, \frac{\partial g_{x_{0}}}
{\partial Z_{q}}(0) Z_{q} J_{2,\,x_0}\Big]\Big] \mathscr{P}(0,0)\\
&= \Big(\mathscr{P} \frac{\partial f_{x_0}}{\partial \overline{z}_{u}} (0)\overline{z}_{u}
\frac{\partial g_{x_0}}{\partial z_{v}} (0) z_{v}
\mathscr{F}_{2,\,x_{0}}\Big)(0,0)\\
&= \frac{1}{2\pi^{2}} \frac{\partial f_{x_0}}{\partial \overline{z}_{u}} (0)
\frac{\partial g_{x_0}}{\partial z_{v}}(0)
\Big[ \delta_{uv} (R_{s\overline{s}q\overline{q}} + R^E_{q\overline{q}})
-\frac{4}{3} R_{u\overline{k}k\overline{v}}- R^E_{u\overline{v}} \Big].
\end{split}
\end{equation}
Using \eqref{bk2.31}, and the formula
$(\mathscr{P}^{\bot}z_{i}\mathscr{P})(Z,0)=(\mathscr{P}\overline{z}_{i}\mathscr{P})(Z,0)=0$ (cf. \eqref{toe1.4}),
we get
\begin{equation}\label{toe4.49}
\begin{split}
&( \mathscr{K}[ J_{2,\,x_0}, \frac{\partial g_{x_{0}}}
{\partial Z_{q}}(0) Z_{q} ] \mathscr{P})(Z,0) =
(\mathscr{F}_{2,\,x_{0}}\frac{\partial g_{x_{0}}}
{\partial Z_{q}}(0) Z_{q} \mathscr{P})(Z,0) \\
&=(-\mathscr{L}^{-1} \mathcal{O}_2 \mathscr{P} \frac{\partial g_{x_{0}}}
{\partial z_{q}}(0) z_{q} \mathscr{P}
-\mathscr{P} \mathcal{O}_2 \mathscr{L}^{-1}\mathscr{P}^{\bot} \frac{\partial g_{x_{0}}}
{\partial \overline{z}_{q}}(0) \overline{z}_{q} \mathscr{P})(Z,0 ).
\end{split}
\end{equation}
By Remark \ref{bkt3.5}, \eqref{bk2.66},
\eqref{toe1.6}, \eqref{toe4.13} and
\eqref{toe4.49}, we obtain as in \eqref{toe4.48}
\begin{equation}\label{toe4.50}
\begin{split}
&\mathscr{K}\bigg[Q_{1,\,x_0}(f), \mathscr{K}\Big[ J_{2,\,x_0}, \frac{\partial g_{x_{0}}}
{\partial Z_{q}}(0) Z_{q} \Big] \bigg]\mathscr{P} (0,0)\\
&= \bigg(\mathscr{P} \frac{\partial f_{x_{0}}}{\partial \overline{z}_{u}}(0) \overline{z}_{u}
\mathscr{K}\Big[ J_{2,\,x_0}, \frac{\partial g_{x_{0}}}
{\partial Z_{q}}(0) Z_{q} \Big] \mathscr{P}\bigg) (0,0)\\
&=- \Big(\mathscr{P} \frac{\partial f_{x_{0}}}{\partial \overline{z}_{u}}(0)
\overline{z}_{u} \mathscr{L}^{-1} \mathcal{O}_2 \mathscr{P} \frac{\partial g_{x_{0}}}
{\partial z_{q}}(0) z_{q} \mathscr{P}\Big) (0,0)=0,
\end{split}
\end{equation}
since by \eqref{toe1.4} and \eqref{bk3.10} we have
$(\mathscr{P} \overline{z}_{u} b^{\alpha}z^{\beta}\mathscr{P})(Z,0)=0$
for $|\alpha|\geqslant1$.
Note that for homogeneous degree $3$ polynomials $H$ in $Z$
the analogue of formula \eqref{bk3.15} holds for $(H\mathscr{P})(Z,0)$.
Using this analogue together with
\eqref{toe1.4} and \eqref{bk3.10} we obtain
\begin{align}\label{abk4.8b}
&(\mathscr{P} H \mathscr{P})(Z,0)=
\Big(\sum_{|\alpha|=3}\frac{\partial^3 H}{\partial {z}^\alpha}
\frac{z^\alpha}{\alpha !}
+\frac{z_q}{\pi} \frac{\partial ^3 H}
{\partial z_{q}\partial z_{i} \partial \overline{z}_{i}}
\Big) \mathscr{P}(Z,0).
\end{align}
Finally, \eqref{toe1.6}, \eqref{abk3.8},
\eqref{toe4.13}, \eqref{abk4.8b} and the equality $\mathscr{P}(0,0)=1$ imply
\begin{align}\label{toe4.51}
\mathscr{K}\Big[Q_{1,\,x_0}(f),
\mathscr{K}\Big[1, \sum_{|\alpha|=3}\frac{\partial ^\alpha g_{x_{0}}}
{\partial Z^\alpha}(0) \frac{Z^\alpha}{\alpha !}\Big]\Big](0,0)
= \frac{1}{\pi^2} \frac{\partial f_{x_{0}}}
{\partial \overline{z}_{u}}(0) \frac{\partial^3 g_{x_{0}}}
{\partial z_{u}\partial z_{i}\partial \overline{z}_{i}}(0) .
\end{align}
By Lemma \ref{lmt1.6}, \eqref{lm01.38},
\eqref{abk2.83} and \eqref{bk4.41} for $E=\field{C}$, we get
\begin{equation}\label{toe4.53}
\frac{\partial}{\partial z_{q}} (\Delta g) (0)
= - 4 \frac{\partial^3 g_{x_{0}}}
{\partial z_{q}\partial z_{i}\partial \overline{z}_{i}}(0)
+ \frac{4}{3} \ric_{q\overline{\ell}} \frac{\partial g_{x_{0}}}{\partial z_{\ell}}(0).
\end{equation}
Lemma \ref{lmt1.6}, \eqref{toe4.39} and
\eqref{toe4.48}--\eqref{toe4.53} entail
\begin{equation}\label{toe4.54}
\begin{split}
&\pi^2\mathscr{K}[Q_{1,\,x_0}(f), \widetilde{Q}_{3,\,x_0}(g)](0,0)\\
&= \frac{\partial f_{x_{0}}}
{\partial \overline{z}_{u}}(0) \frac{\partial^3 g_{x_{0}}}
{\partial z_{u}\partial z_{i}\partial \overline{z}_{i}}(0)
+\frac{1}{2} \frac{\partial f_{x_0}}{\partial \overline{z}_{u}} (0)
\frac{\partial g_{x_0}}{\partial z_{v}}(0)
\Big[ \delta_{uv} (R_{s\overline{s}q\overline{q}} + R^E_{q\overline{q}})
- \frac{4}{3} R_{u\overline{k}k\overline{v}}- R^E_{u\overline{v}} \Big]\\
&= - \frac{1}{8} \langle \overline{\partial} f, \partial \Delta g\rangle
+ \frac{1}{4} \langle \overline{\partial} f, \partial g\rangle
( R_{s\overline{s}q\overline{q}} + R^E_{q\overline{q}})
- \frac{1}{8} \langle \overline{\partial} f\wedge \partial g, R^E\rangle_{\omega}.
\end{split}
\end{equation}
From \eqref{toe4.44} and \eqref{toe4.54}, we have
\begin{equation}\label{toe4.55}
\begin{split}
\pi^2\mathscr{K}[ & \widetilde{Q}_{3,\,x_0}(f), Q_{1,\,x_0}(g)](0,0)\\
&= - \frac{1}{8}\langle \overline{\partial} \Delta f, \partial g\rangle
+ \frac{1}{4} \langle \overline{\partial} f, \partial g\rangle
( R_{s\overline{s}q\overline{q}} + R^E_{q\overline{q}})
- \frac{1}{8} \langle \overline{\partial} f\wedge \partial g, R^E\rangle_{\omega}.
\end{split}
\end{equation}
By \eqref{bk2.6}, \eqref{bk4.3a}, \eqref{toe4.42},
\eqref{toe4.43}, \eqref{toe4.54} and \eqref{toe4.55},
we get (\ref{toe4.32}). The proof of Theorem \ref{toet4.6} is completed.
\end{proof}
\begin{proof}[\textbf{Proof of Theorem \ref{toet4.5}}]
The existence of the expansion \eqref{toe4.2} and formula
$C_0(f,g)=fg$ were established in \cite[Th.\,1.1]{MM08b} (cf.\,also \cite[Th.\,7.4.1]{MM07})
in general symplectic settings.
By \eqref{toe4.2}, \eqref{toe2.13}, \eqref{bk4.5},
\eqref{toe4.6} and \eqref{toe4.33}, we obtain (cf.\;also \cite[(7.4.9)]{MM07}),
\begin{equation}\label{toe4.8}
C_1(f,g)= (Q_{2,x}(f,g)- Q_{2,x}(fg))(0,0) = \bb_{1,\,f,\,g}-\bb_{1,\,fg} .
\end{equation}
Hence \eqref{toe2.3}, \eqref{bk4.3}, \eqref{toe4.31} and \eqref{toe4.8}
yield the formula for $C_1(f,g)$ given in \eqref{toe4.3}.
Moreover, \eqref{toe4.2}, \eqref{bk4.5} and \eqref{toe4.33} imply
the formula for $C_2(f,g)$ from \eqref{toe4.3}.
We will prove \eqref{toe4.3a} now.
Let $\{e_i\}$ be an orthonormal frame of $(TX, g^ {TX})$,
and $\{w_i\}$ be an orthonormal frame of
$T^{(1,0)}X$.
Let $\square= \overline{\partial}^*\overline{\partial}+ \overline{\partial}\,\overline{\partial}^*$
be the Kodaira Laplacian on $\Lambda(T^*X)\otimes _{\field{R}}\field{C}$,
and let $\Delta$ be the Bochner Laplacian on
$\Lambda(T^*X)\otimes _{\field{R}}\field{C}$
associated with the connection $\nabla^{\Lambda(T^{*}X)}$
on $\Lambda(T^*X)$
induced by $\nabla^{TX}$ (cf.\,\eqref{toe2.3}).
Let $R^{\Lambda(T^{*(1,0)}X)}$
be the curvature of the holomorphic Hermitian connection
on $\Lambda(T^{*(1,0)}X)$.
By Lichnerowicz formula \cite[Remark 1.4.8]{MM07},
and \eqref{bk4.2a}, we have
\begin{align}\label{toe4.61}\begin{split}
&R^{\Lambda(T^{*(1,0)}X)}=
- \langle R^{TX}w_l,\overline{w}_k\rangle w^l\wedge i_{w_k},\\
&2\square =\Delta - R^{\Lambda(T^{*(1,0)}X)}(w_q,\overline{w}_q)
+ ( 2 R^{\Lambda(T^{*(1,0)}X)} + \ric) (w_l,\overline{w}_k)
\overline{w}^k \wedge i_{\overline{w}_l}.
\end{split}\end{align}
Since $(X,\omega,J)$ is K{\"a}hler, $\square$ commutes
with the operators $\partial,\overline{\partial},d$ (cf.\,\cite[Cor.\,1.4.13]{MM07}),
and \eqref{toe4.61} shows that $2\square f=\Delta f$
for any $f\in\mathscr{C}^\infty(X)$. From Lemma \ref{lmt1.6},
\eqref{bk4.2a} and \eqref{toe4.61}, we have for any $f\in\mathscr{C}^\infty(X)$:
\begin{align}\label{toe4.62}\begin{split}
\Delta \partial f =& \,\partial \Delta f - \ric(\cdot, \overline{w}_k) \, w_k(f),\\
\Delta \overline{\partial} f=& \, \overline{\partial} \Delta f
- \ric(\cdot, w_k) \, \overline{w}_k(f),\\
\Delta d f=& \, d \Delta f
- \ric(\cdot, e_j) \, e_j(f).
\end{split}\end{align}
Thus \eqref{toe2.3}, \eqref{toe4.62} yield for any $f,g\in \mathscr{C}^\infty(X)$:
\begin{align}\label{toe4.63}\begin{split}
&\Delta(fg)= g \Delta f + f\Delta g -2 \langle d f,dg\rangle,\\
&\Delta\langle \partial f,\overline{\partial}g\rangle
=\langle \Delta\partial f,\overline{\partial}g\rangle
+\langle \partial f,\Delta\overline{\partial}g\rangle
-2\langle \nabla^{T^{*}X}\partial f,\nabla^{T^{*}X}\overline{\partial}g\rangle\\
&= \langle \partial \Delta f,\overline{\partial}g\rangle
+\langle \partial f,\overline{\partial}\Delta g\rangle
-2\langle \nabla^{T^{*}X}\partial f,\nabla^{T^{*}X}\overline{\partial}g\rangle
-2 \ric(w_m,\overline{w}_q)\, \overline{w}_m(g) w_q(f).
\end{split}\end{align}
Using \eqref{toe2.3}, \eqref{toe4.62} and \eqref{toe4.63}, we infer
\begin{align}\label{toe4.64}\begin{split}
\Delta^2 (fg)=& f\Delta^2g + g\Delta^2 f + 2 (\Delta f)\Delta g
- 4 \langle d\Delta f,d g\rangle
- 4 \langle d f,d\Delta g\rangle\\
&+ 4 \langle \nabla^{T^{*}X}d f,\nabla^{T^{*}X}d g\rangle
+ 4 \ric(e_i, e_j) \, e_i(g) e_j(f).
\end{split}\end{align}
We examine now closely the expression of $\pi^2 C_2(f,g)$ given
by \eqref{toe4.3}. Using
\eqref{abk4.4}, \eqref{toe4.32}, \eqref{toe4.3},
we see that the term of differential order $0$ in $f,g$
from the expression of $\pi^2 C_2(f,g)$ is zero, and
the term of total differential order $2$ in $f,g$,
disregarding the term involving $R^E$, in $\pi^2 C_2(f,g)$ is
\begin{align}\label{toe4.67}\begin{split}
C_{22}= \frac{\sqrt{-1}}{8} \left\langle \ric_\omega,
- f\partial\overline{\partial}g -g \partial\overline{\partial}f
+\partial\overline{\partial}(fg)\right\rangle.
\end{split}\end{align}
The term of total differential order $4$ in $f,g$
in the expression of $\pi^2 C_2(f,g)$ is
\begin{align}\label{toe4.65}\begin{split}
C_{24}=& \frac{1}{32}f \Delta^2 g + \frac{1}{32}g \Delta^2 f
-\frac{1}{8}\langle \overline{\partial} f, \partial \Delta g\rangle
- \frac{1}{8}\langle \overline{\partial} \Delta f, \partial g\rangle
+ \frac{1}{16} (\Delta f) \Delta g\\
&+ \frac{1}{8} \langle D^{0,1}\overline{\partial} f, D^{1,0}\partial g\rangle
- \frac{1}{32} \Delta^2 (fg)
- \frac{1}{8} \Delta \langle\partial f,\overline{\partial} g\rangle.
\end{split}\end{align}
By \eqref{toe4.63}, \eqref{toe4.64}, \eqref{toe4.65} and by the formula
$\langle D^{1,0}\overline{\partial} f, D^{0,1}\partial g\rangle
=\langle D^{0,1}\partial f, D^{1,0}\overline{\partial} g\rangle$, we get
\begin{align}\label{toe4.66}\begin{split}
C_{24}=\frac{1}{8} \langle D^{1,0}\partial f, D^{0,1}\overline{\partial} g\rangle
+\frac{\sqrt{-1}}{8} \left\langle \ric_\omega,
\partial f\wedge \overline{\partial}g
-\partial g \wedge\overline{\partial}f \right\rangle.
\end{split}\end{align}
Finally, by inspecting \eqref{abk4.4}, \eqref{toe4.32}, \eqref{toe4.3},
we see that the term involving $R^E$ in the expression of
$\pi^{2} C_2(f,g)$ is
\begin{equation}\label{toe4.68}
\begin{split}
\frac{\sqrt{-1}}{8} \Big(-f\Delta g - g \Delta f +
\Delta(fg)\Big)R^E_{\Lambda}
+ \frac{1}{4}\langle g\partial \overline{\partial} f +
f\partial\overline{\partial}g - \partial\overline{\partial}(fg), R^E\rangle_{\omega}\\
+ \frac{\sqrt{-1}}{4}\langle \overline{\partial} f,
\partial g\rangle R^E_{\Lambda}
- \frac{1}{4} \langle \overline{\partial} f\wedge \partial g,
R^E\rangle_{\omega}
+\frac{\sqrt{-1}}{4} \langle \partial f,\overline{\partial} g\rangle R^E_{\Lambda}.
\end{split}
\end{equation}
Combining \eqref{toe4.67}, \eqref{toe4.66} and \eqref{toe4.68},
we get \eqref{toe4.3a}.
The proof of Theorem \ref{toet4.5} is completed.
\end{proof}
\section{Donaldson's $Q$-operator}\label{toes5}
In this section
we study the asymptotics of the sequence of operators introduced by
Donaldson \cite{D09}. We suppose henceforth that $E=\field{C}$.
Set ${\rm Vol}(X,dv_X):= \int_X dv_X$. Following \cite[\S 4]{D09}, set
\begin{align}\label{1n1}
K_p(x,x'):= |P_{p}(x,x')|^2_{h^{L^p}_x\otimes h^{L^{p*}}_{x'}}, \quad
R_p:= (\dim H^0(X, L^p))/ {\rm Vol}(X, dv_X).
\end{align}
Let $K_p$, $Q_{K_p}$ be the integral operators associated to $K_p$,
defined for $f\in \mathscr{C}^\infty (X)$ by
\begin{equation}\label{1n2}
(K_p f)(x):= \int_X K_p(x,y) f(y)dv_X(y),
\quad Q_{K_p} (f) = \frac{1}{R_p} K_p f.
\end{equation}
Recall that, just as the Bergman kernel appears when comparing
a K{\"a}hler metric $\omega$ to its algebraic approximations $\omega_p$
(i.e.\ pull-backs of the Fubini-Study metrics by the Kodaira embeddings),
the operators $Q_{K_p}$ appear when one
relates infinitesimal deformations of the metric $\omega$
to the corresponding deformations of the approximations $\omega_p$\,.
The asymptotics of the operator $K_p$ were obtained
in \cite[Th.\,26]{LM09}\footnote{Note that in the present context
\cite[Th.\,26]{LM09} should be modified as follows:
$\lim_{p\to \infty}Q_{K_p}f=\frac{{\rm Vol}(X, \nu)}{{\rm Vol}(X,
dv_X)}\,\eta f$ in $\mathscr{C}^{m}(X)$, or
\begin{align*}
\begin{split}
& \left| Q_{K_p}f- \frac{{\rm Vol}(X, \nu)}{{\rm Vol}(X, dv_X)}\,\eta f
\right|_{\mathscr{C}^m(X)}
\leqslant C p^{-1/2} \left|f\right|_{\mathscr{C}^{m+1}(X)}
\text{ or } Cp^{-1} \left|f\right|_{\mathscr{C}^{m+2}(X)},
\end{split}
\end{align*}
since the right hand side of the second equation of \cite[(33)]{LM09}
and \cite[(34)]{LM09} should read
as convergence in $\mathscr{C}^{m}(X)$ without the speed, or
$C p^{-1/2} \left|f\right|_{\mathscr{C}^{m+1}(X)}$
or $Cp^{-1} \left|f\right|_{\mathscr{C}^{m+2}(X)}$.}
and used in \cite{Fine08}. The following result refines \cite[Th.\,26]{LM09}
and is applied in the recent paper \cite{Fine10}.
\begin{thm} \label{nt1} For every $m\in \field{N}$,
there exists $C>0$ such that for any
$f\in \mathscr{C}^\infty (X)$, $p\in \field{N}^{*}$,
\begin{align}\label{1n4}
\begin{split}
& \left|\frac{1}{p^n} K_p f - f + \frac{1}{8\pi p} (-\br f + 2\Delta f)
\right|_{\mathscr{C}^m(X)}
\leqslant C p^{-3/2} \left|f\right|_{\mathscr{C}^{m+3}(X)}
\text{ or } C p^{-2} \left|f\right|_{\mathscr{C}^{m+4}(X)}.
\end{split}
\end{align}
\end{thm}
\begin{proof}
By \eqref{toe2.9} with $Z=0$, \eqref{bk2.24}, \eqref{bk2.31} and \eqref{bk2.33}, we get
\begin{equation}\label{n50}
\begin{split}
\left |\Big(\frac{1}{p^{2n}} K_{p,x_0}(0,Z')\kappa_{x_0}(Z')
- \Big(1+ \sum_{r=2}^k p^{-r/2} J^\prime_{r} (0,\sqrt{p} Z')\Big)
e ^{-\pi p |Z^\prime|^2} \Big)\right|_{\mathscr{C}^m(X)}\\
\leqslant C p^{-(k+1)/2}
(1+|\sqrt{p} Z'|)^N
\exp (- C_0 \sqrt{p} |Z'|)+ \mathscr{O}(p^{-\infty}),
\end{split}
\end{equation}
with
\begin{align}\label{n51}
J'_2(0,Z') =(J_2 + \overline{J_2})(0,Z') .
\end{align}
Now we have the analogue of \cite[(32)]{LM09},
\begin{equation}\label{n52}
\begin{split}
\left|p^{-n} K_{p} f
- p^n \int_{|Z^\prime|\leqslant \varepsilon}
\Big(1+ \sum_{r=2}^k p^{-r/2} J^\prime_{r} (0,\sqrt{p} Z')\Big)
e^{-\pi p |Z^\prime|^2}f_{x_0}(Z^\prime)
dv_{T_{x_0}X} (Z^\prime)\right|_{\mathscr{C}^m(X)}\\
\leqslant C p^{-(k+1)/2} \left|f\right|_{\mathscr{C}^m(X)}.
\end{split}
\end{equation}
But as in the proof of \cite[Th\,.2.29\,(2)]{BeGeVe}, we get
\begin{align}\label{n53}
\begin{split}
&\left| p^n \int_{|Z^\prime|\leqslant \varepsilon} J^\prime_{r} (0,\sqrt{p} Z')
e^{-\pi p |Z^\prime|^2}f_{x_0}(Z^\prime)
dv_{T_{x_0}X}(Z^\prime)\right|_{\mathscr{C}^m(X)}
\leqslant C \left|f\right|_{\mathscr{C}^m(X)}.
\end{split}\end{align}
Moreover
\begin{equation}\label{n54}
\begin{split}
\left| p^n \int_{|Z^\prime|\leqslant \varepsilon}
e^{-\pi p |Z^\prime|^2}f_{x_0}(Z^\prime)
dv_{T_{x_0}X}(Z^\prime)- f(x_{0})
+ \frac{1}{4\pi p}(\Delta f) (x_{0}) \right|_{\mathscr{C}^m(X)}\\
\leqslant C p^{-3/2} \left|f\right|_{\mathscr{C}^{m+3}(X)}
\text{ or } C p^{-2} \left|f\right|_{\mathscr{C}^{m+4}(X)}.
\end{split}
\end{equation}
Finally,
by \eqref{bk3.8} (cf.\,also \cite[(4.1.110)]{MM07}), we have
\begin{equation}\label{n55}
\begin{split}
\int_{Z'\in \field{C}^n} \overline{J_2}(0,Z') |\mathscr{P}|^2(0, Z') dZ'
= \int_{Z'\in \field{C}^n} \mathscr{P} (0, Z')J_2(Z', 0) \mathscr{P} (Z', 0)dZ'\\
= (\mathscr{P} J_2 \mathscr{P})(0,0)
= - (\mathscr{P} \mathcal{O}_2\mathscr{L}^{-1}\mathscr{P}^\bot)(0,0)
= \frac{1}{16\pi} \br.
\end{split}
\end{equation}
Thus
\begin{equation}\label{n56}
\begin{split}
\left| p^n \int_{|Z^\prime|\leqslant \varepsilon} J'_2(0,\sqrt{p} Z')
e^{-\pi p |Z^\prime|^2}f_{x_0}(Z^\prime)dv_{T_{x_0}X}(Z^\prime)
- \big(\frac{ \br}{8\pi} f\big) (x_{0}) \right|_{\mathscr{C}^m(X)}\\
\leqslant C p^{-1/2} \left|f\right|_{\mathscr{C}^{m+1}(X)}
\text{ or } C p^{-1} \left|f\right|_{\mathscr{C}^{m+2}(X)}.
\end{split}
\end{equation}
The proof of Theorem \ref{nt1} is completed.
\end{proof}
|
{
"timestamp": "2012-03-14T01:05:07",
"yymm": "1009",
"arxiv_id": "1009.4405",
"language": "en",
"url": "https://arxiv.org/abs/1009.4405"
}
|
\section{Introduction}
Isometries of spacetimes are generated by Killing vector fields, and one can associate a
conserved quantity to each of these. A spacetime may also possess so-called hidden
symmetries generated by higher-rank tensor fields. Following Carter's celebrated
work \cite{Car68} on the Kerr black hole, it was realized \cite{WP70} that his new integral
of motion is quadratic in momenta and generated by a rank-2 Killing tensor of the Kerr spacetime.
Similar results have now been obtained for many classes of black holes and in various
dimensions. We refer to the recent reviews \cite{ER0801,FK0802} for details and references.
Here we consider the family of extremal Kerr-NUT-AdS-dS black holes and our main focus is on the
near-horizon geometries. Such geometries have attracted a lot of attention lately due to their role
in the recently proposed Kerr/CFT correspondence \cite{GHSS0809}.
Extensive lists of references on this topic can be found in \cite{Ras1005}.
First, we review the Kerr-NUT-AdS-dS black hole in the metric form discussed in \cite{CLP0604}.
This spacetime admits two linearly independent Killing vectors and possesses a hidden
symmetry generated by a rank-2 Killing tensor. It is also recalled \cite{KF07} that this Killing
tensor follows from a Killing-Yano potential. We then determine the metric of the near-horizon
geometry of the extremal Kerr-NUT-AdS-dS black hole
and verify that it satisfies Einstein's field equations with the same cosmological constant as for the
black hole itself. The isometry group, on the other hand, is enhanced as the
near-horizon geometry admits four linearly independent Killing vectors.
In the limiting procedure used to obtain the near-horizon geometry, the Killing-Yano
potential diverges. However, a gauge freedom in the definition of the Killing-Yano potential
allows us to construct a well-defined potential for the near-horizon geometry.
We finally demonstrate that the corresponding rank-2 Killing tensor is reducible as it can be
expressed in terms of the Casimir operators formed by the four Killing vectors of the near-horizon
geometry.
\section{Kerr-NUT-AdS-dS black holes}
\subsection{Geometry}
Adopting the unit convention where $G=c=1$, a general Kerr-NUT-AdS-dS black hole of mass $M$ and with
rotation parameter $a$ is described by the metric \cite{CLP0604}
\be
d\sh^2=-\frac{\D_r}{\rh^2+y^2}\big(d\thh+y^2d\psi\big)^2+\frac{\D_y}{\rh^2+y^2}\big(d\thh-\rh^2d\psi\big)^2
+\frac{\rh^2+y^2}{\D_r}d\rh^2+\frac{\rh^2+y^2}{\D_y}dy^2
\label{ds2alg}
\ee
where
\be
\D_r=(\rh^2+a^2)\big(1+\frac{\rh^2}{\ell^2}\big)-2M\rh,\qquad
\D_y=(a^2-y^2)\big(1-\frac{y^2}{\ell^2}\big)+2Ly
\label{Dy}
\ee
Here $L$ is the NUT parameter (see also \cite{GP0702}) and the determinant $\hat{g}$ of the metric
only depends on $\rh$ and $y$ as
\be
\sqrt{-\hat{g}}=\rh^2+y^2
\ee
This spacetime satisfies Einstein's field equations
\be
\Gh_{\mu\nu}+\frac{3}{\ell^2}\gh_{\mu\nu}=0
\ee
and is AdS (dS) for positive (negative) renormalized cosmological constant $\ell^{-2}$.
It reduces to Kerr for $L=\ell^{-2}=0$.
The horizons of the black hole are located at the positive zeros of $\D_r$, and the value of $\rh$ at the outer
horizon is denoted by $r_+$.
The isometry group $U(1)\times U(1)$ of (\ref{ds2alg}) is generated by the Killing vector fields
\be
\{\pab_{\psi}\}\cup\{\pab_{\thh}\}
\ee
\subsection{Hidden symmetry}
As observed in \cite{KF07} (see also the reviews \cite{FK0802}),
the metric (\ref{ds2alg}) admits the so-called Killing-Yano potential (1-form)
\be
\bbh=\frac{y^2-\rh^2}{2}\db\thh-\frac{\rh^2y^2}{2}\db\psi
\label{bhat}
\ee
This implies the existence of the principal conformal Killing-Yano tensor \cite{Kas68}
\be
\hbh=\db\bbh
\label{hbh}
\ee
and its Hodge dual, the Killing-Yano tensor \cite{Yano52}
\be
\fbh=\ast\hbh=\ast\db\bbh
\ee
{}From this, one can construct a symmetric Killing tensor by contraction
\be
\hat{K}_{\mu\nu}=\hat{f}_{\mu\lambda}\hat{f}_{\nu}^{\phantom{\nu}\lambda}
\label{Kh}
\ee
This tensor $\Kbh$ is responsible for the hidden symmetry associated with the conserved quantity
$\hat{K}^{\mu\nu}\hat{p}_\mu\hat{p}_\nu$ quadratic in the momenta $\hat{p}_\mu$.
Considering the construction of $\Kbh$ above, it appears natural to refer to the entire quadruplet $(\bbh,\hbh,\fbh,\Kbh)$
as the generator of the hidden symmetry.
It is straightforward to verify that
\be
\Kbh=\Qbh+\rh^2\gbh
\ee
where $\gbh$ is the metric tensor while the components of $\Qbh$ in the ordered basis $\{\thh,\rh,\psi,y\}$
are given by
\be
[\hat{Q}_{\mu\nu}]=\left(\!\!\!\begin{array}{cccc} \D_r&0&\D_r y^2&0\\ 0&\displaystyle{-\frac{(\rh^2+y^2)^2}{\D_r}}&0&0\\
\D_r y^2&0&\D_r y^4&0\\ 0&0&0&0 \end{array} \!\!\right)
\ee
With raised indices, the components are
\be
[\hat{Q}^{\mu\nu}]=\frac{1}{\D_r}\left(\!\!\begin{array}{cccc} \rh^4&0&\rh^2&0\\ 0&-\D_r^2&0&0\\
\rh^2&0&1&0\\ 0&0&0&0 \end{array} \!\!\right)
\ee
\section{Extremal Kerr-NUT-AdS-dS black holes}
When the inner and outer horizons of the black hole (\ref{ds2alg}) coalesce, the black hole is said to be extremal.
This happens when the otherwise single pole $r_+$ of $\D_r$ is a double pole in which case
\be
\D_r(r_+)=\D'_r(r_+)=0
\label{DhDh}
\ee
We denote the value of $\rh$ at this single horizon by $\rb$.
The conditions (\ref{DhDh}) can be used to express $M$ and $a^2$ at extremality in terms of $\rb$
\be
M=\frac{\rb\big(1+\frac{\rb^2}{\ell^2}\big)^2}{1-\frac{\rb^2}{\ell^2}},
\qquad
a^2=\frac{\rb^2\big(1+\frac{3\rb^2}{\ell^2}\big)}{1-\frac{\rb^2}{\ell^2}}
\ee
\subsection{Near-horizon geometry}
To describe the near-horizon geometry of an extremal Kerr-NUT-AdS-dS black hole,
we introduce (in the spirit of \cite{BH9905}) the new coordinates $t,r,\phi$
\be
t=\frac{(\rb^2+a^2)\eps\thh}{\rb^2\rb_0},\qquad
r=\frac{\rh-\rb}{\eps\rb_0},\qquad
\phi=\rb^2\psi-\thh
\label{subs}
\ee
where
\be
\rb_0^2=\frac{(\rb^2+a^2)\big(1-\frac{\rb^2}{\ell^2}\big)}{1+\frac{6\rb^2}{\ell^2}
-\frac{3\rb^4}{\ell^4}}
\label{rrho}
\ee
The transformation of the radial coordinate facilitates zooming in on the near-horizon region while
the accompanying transformations ensure that the line element (\ref{ds2alg}) is well-defined in the
limit $\eps\to0$. The near-horizon geometry is then obtained by taking the limit $\eps\to0$ in which case
the metric becomes\footnote{Using different coordinates, a similar limit and near-horizon geometry is discussed in
\cite{Ghe0902}, although Einstein's field equations (\ref{Gg}) are not addressed there.}
\be
d\bar{s}^2=\frac{\rb_0^2(\rb^2+y^2)}{\rb^2+a^2}\big(-r^2dt^2+\frac{1}{r^2}dr^2\big)
+\frac{\D_y}{\rb^2+y^2}\big(d\phi+\kb rdt\big)^2+\frac{\rb^2+y^2}{\D_y}dy^2
\label{ds2Near}
\ee
where
\be
\kb=\frac{2\rb\rb_0^2}{\rb^2+a^2}
\label{kb}
\ee
It is straightforward, albeit tedious, to verify explicitly that this metric satisfies Einstein's field
equations
\be
G_{\mu\nu}+\frac{3}{\ell^2}g_{\mu\nu}=0
\label{Gg}
\ee
We find that the determinant $g$ of the metric (\ref{ds2Near}) only depends on the coordinate $y$ as
\be
\sqrt{-g}=\frac{\rb_0^2(\rb^2+y^2)}{\rb^2+a^2}
\label{g}
\ee
and it is noted that the metric describes the near-horizon geometry of the Kerr black hole when $N=\ell^{-2}=0$.
The exact isometry group of the spacetime (\ref{ds2Near})
is generated by the Killing vectors
\be
\big\{\Pb=\pab_\phi\big\}\cup\big\{\Tb=\pab_t,\ \Dbb=t\pab_t-r\pab_r,\ \Fb=\big(t^2+\frac{1}{r^2}\big)\pab_t
-2tr\pab_r-\frac{2\kb}{r}\pab_\phi\big\}
\label{iso}
\ee
It follows that the isometry group $U(1)\times U(1)$ of the Kerr-NUT-AdS-dS black hole is enhanced to
$U(1)\times SL(2)$ in the near-horizon geometry.
\subsection{Hidden symmetry}
After the substitution (\ref{subs}), the Killing-Yano potential $\bbh$ (\ref{bhat}) {\em diverges} in
the limit $\eps\to0$. This could indicate that the near-horizon geometry does not possess
a hidden symmetry admitting a
Killing-Yano potential, at least not one carried over from the original Kerr-NUT-AdS-dS
spacetime. This is not the case, though, as the divergence problem can be resolved
and a well-defined Killing-Yano potential can be constructed.
To demonstrate this, we use that a Killing-Yano potential is defined only up to a {\em constant}
1-form. This corresponds to gauge invariance of the corresponding Killing-Yano and Killing
tensors.
We thus find that
\be
\bbh+\tfrac{1}{2}\big(\alpha\db\thh+\beta\db\psi\big)
\ee
with $\alpha$ and $\beta$ constant, remains well-defined in the limit $\eps\to0$ provided
\be
-\rb^4+\alpha \rb^2+\beta=0
\ee
The 1-form thereby obtained is given by
\be
\bb=-\frac{\kb(\rb^2+y^2)r}{2}\db t-\frac{y^2}{2}\db\phi
\ee
and we have verified explicitly that the tensor $\Kb$ constructed as in (\ref{hbh})-(\ref{Kh}) by
\be
\hb=\db\bb,\qquad
\fb=\ast\hb=\ast\db\bb,\qquad
K_{\mu\nu}=f_{\mu\lambda}f_{\nu}^{\phantom{\nu}\lambda}
\ee
is indeed a Killing tensor.
The 1-form $\bb$ is therefore the sought-after Killing-Yano potential for the Killing tensor $\Kb$.
We conclude that the quadruplet $(\bb,\hb,\fb,\Kb)$ generates a hidden symmetry of the near-horizon
geometry (\ref{ds2Near}) of a Kerr-NUT-AdS-dS black hole.
It is observed that the Killing tensor $\Kb$ decomposes as
\be
\Kb=\Qb+\rb^2\gb
\label{KQg}
\ee
where $\gb$ is the metric tensor of the near-horizon geometry (\ref{ds2Near}) while the components of $\Qb$ are
given by
\be
[Q_{\mu\nu}]=-\sqrt{-g}\left(\!\!\begin{array}{cccc}
-r^2&0&0&0\\
0&\displaystyle{\frac{1}{r^2}}&0&0\\
0&0&0&0\\
0&0&0&0 \end{array} \!\!\right)
\ee
Aside from the $y$-dependent factor $\sqrt{-g}$, this is recognized as the metric tensor of $AdS_2$ in the
coordinates $t,r$. With raised indices, the components of $\Qb$ only depend on the radial coordinate and are given by
\be
[Q^{\mu\nu}]=\left(\!\!\begin{array}{cccc}
\displaystyle{\frac{\rb^2+a^2}{\rb_0^2 r^2}}&0&\displaystyle{-\frac{2\rb}{r}}&0\\
0&\displaystyle{-\frac{(\rb^2+a^2)r^2}{\rb_0^2}}&0&0\\
\displaystyle{-\frac{2\rb}{r}}&0&\displaystyle{\frac{4\rb^2\rb_0^2}{\rb^2+a^2}}&0\\
0&0&0&0 \end{array} \!\!\right)
\ee
Since the metric tensor is multiplied by a {\em constant} in the decomposition (\ref{KQg}), the `essential' part of the hidden
symmetry is generated by $\Qb$ implying that the metric-tensor term can be ignored when
analyzing the hidden symmetry generated by $\Kb$.
We find that this hidden symmetry is {\em reducible} in the sense that $Q^{\mu\nu}$ can be expressed in terms of
the Killing vectors (\ref{iso}) by
\be
Q^{\mu\nu}=\gamma
\big(\Tb\otimes\Fb+\Fb\otimes\Tb-2\Dbb\otimes\Dbb\big)^{\mu\nu}
-\frac{2}{\gamma\rb^2}(\Pb\otimes\Pb)^{\mu\nu}
\label{Qgamma}
\ee
where
\be
\gamma=\frac{\rb^2(1+\rb^2/\ell^2)}{\rb_0^2(1-\rb^2/\ell^2)}
\ee
The Killing tensor $\Qb$ is thus a linear combination of the quadratic Casimir operators formed out of the two sets of
Killing vectors (\ref{iso}).
The various constants appearing in this decomposition can be absorbed by rescaling the Killing vectors.
\subsubsection*{Note added}
\vskip.1cm
\noindent
During the completion of this work, the paper \cite{Gal1009} appeared.
It has some overlap with the present work as it also discusses the hidden symmetry of the
near-horizon geometry of the extremal Kerr black hole. The reducibility of the corresponding Killing tensor is
discussed using Poisson brackets in the context of particle dynamics near the extremal Kerr throat.
The paper \cite{Gal1009}
does not, however, consider the near-horizon geometry of the general Kerr-NUT-AdS-dS black hole,
and it does not address the Killing-Yano potential underlying the hidden symmetry.
On the other hand, it proposes an ${\cal N}=2$ supersymmetric extension of the particle dynamics
and it may be of interest to generalize this to the Kerr-NUT-AdS-dS scenario considered here.
\subsection*{Acknowledgments}
\vskip.1cm
\noindent
This work is supported by the Australian Research Council.
|
{
"timestamp": "2010-09-27T02:01:13",
"yymm": "1009",
"arxiv_id": "1009.4388",
"language": "en",
"url": "https://arxiv.org/abs/1009.4388"
}
|
\section{Introduction}
The application of the renormalization group (RG) and boson representation methods to one-dimensional (1D) models of interacting electrons have provided over the last four decades considerable insight into the nature of correlations in low dimensional systems \cite{Emery79,Solyom79,Voit95,Giamarchi04}. This has been largely achieved by treating the models in the continuum field-theory limit, corresponding to the so-called weak coupling 1D electron gas (EG) model. There are notable exceptions, however, where the 1D-EG model clearly fails to reveal the nature of correlations at long distance. These situations are likely to occur in models for which lattice effects, albeit related to irrelevant terms in the RG sense, do affect the asymptotic behavior of electronic correlations and then the nature of the ground state.
A well documented case is encountered in the extended Hubbard model at half-filling, which is defined on a lattice in terms of intersite hopping and the on-site and nearest-neighbor sites couplings $U$ and $V$. On numerical side, exact diagonalization\cite{Nakamura99,Nakamura00}, quantum Monte Carlo \cite{Sandvik04} and density-matrix renormalization group analysis \cite{Zhang04} have established the incursion of a bond order-wave (BOW) state over a finite region of the phase diagram surrounding the line $U=2V>0$, a result at variance with the spin density-wave (SDW) to charge density-wave (CDW) transition found in the theory of the 1D EG\cite{Emery79,Voit92,Japaridze99}.
Using perturbation theory arguments, Tsuchiizu and Furasaki\cite{Tsuchiizu02} showed how high-energy or short-distance degrees of freedom can modify the initial conditions of an effective low energy continuum theory and favor the occurrence of a BOW phase that enfolds the $U=2V>0$ line in weak coupling. The influence of the lattice on the nature of the ordered phase in this region of the phase diagram has been investigated by Tam {\it et al.,}\cite{Tam06} using the functional RG method. The scaling transformation of interactions, which in this framework gather both their marginal and momentum dependent parts, was obtained for a tight-binding electron spectrum in a finite momentum space. The CDW/SDW degeneracy that takes place at $U=2V$ in the continuum limit, is thus lifted and a BOW state stabilized over a portion of the phase diagram that grows in size with increasing $U$, consistently with numerical calculations at weak coupling. The functional RG method, however, tells us not as much about the structure of leading irrelevant terms and how these modify scaling and the nature of ordered states of the continuum theory.
In this work we address this issue from a different perspective that generalizes the weak coupling momentum shell Kadanoff-Wilson (K-W) RG method to lattice models \cite{Bourbon91,Bourbon03}. The proposed approach exceeds the limitations of the continuum approximation and takes into account the tight-binding structure of the spectrum and its impact on the scaling transformation of both local and momentum dependent interactions of the extended Hubbard model \cite{Dumoulin96b}. The latter couplings, though irrelevant, are found to affect the flow of the former interactions. A modification of certain portions of the phase diagram follows; in particular, the BOW phase is found to insert in a finite region near the $U=2V>0$ line, in agreement with the results of numerical calculations.
In Sec.~II, we introduce the model and set out the basic steps of the momentum shell RG transformation for the partition function. In Sec.~III, the RG flow equations for the coupling constants and the most singular response functions are analyzed at the one-loop level and different $U$ and $V$. The phase diagram is mapped out in weak coupling.
We conclude in Sec. IV.
\section{The extended Hubbard model and the renormalization group formulation}
\subsection{The model}
We consider the extended Hubbard Hamiltonian for a one-dimensional lattice,
\begin{equation}
\label{ExHR}
\begin{split}
H= & -t\sum_{i,\sigma}(c^\dagger_{i+1,\sigma}c_{i,\sigma}+ c^\dagger_{i,\sigma}c_{i+1,\sigma})\cr
&+ U\sum_{i}n_{i,\uparrow}n_{i,\downarrow} + V\sum_{i}n_in_{i+1},
\end{split}
\end{equation}
where $t$ is is the hopping integral, $n_{i,\sigma}= c^{\dagger}_{i,\sigma} c_{i,\sigma}$ is the occupation number on site $i$ for the spin orientation $\sigma=\uparrow,\downarrow$, and $n_i= n_{i,\uparrow}+n_{i,\downarrow}$. In Fourier space the Hamiltonian can be written in the form
\begin{widetext}
\begin{equation}
\label{ExHF}
\begin{split}
H= \sum_{p,k,\sigma} \epsilon_k c^\dagger_{p,k,\sigma}c_{p,k,\sigma} \
& + {1\over L}\sum_{\{k,q,\sigma\}} \left(g_1+ 2\bar{g}_1\sin^2 {q\over 2} \right) c^\dagger_{+,k_1+q +2k_F,\sigma_1}c^\dagger_{-,k_2-q-2k_F,\sigma_2}c_{+,k_2,\sigma_2}c_{-,k_1,\sigma_1}\cr
& + {1\over L}\sum_{\{k,q,\sigma\}} \left(g_2+ 2\bar{g}_2\sin^2 {q\over 2} \right) c^\dagger_{+,k_1+q,\sigma_1}c^\dagger_{-,k_2-q,\sigma_2}c_{-,k_2,\sigma_2}c_{+,k_1,\sigma_1} \cr
& + {1\over 2L}\sum_{\{k,q,\sigma\}} \left(g_3+ 2\bar{g}_3\sin^2 {q\over 2} \right) \big(c^\dagger_{+,k_1+q+2k_F,\sigma_1}c^\dagger_{+,k_2-q -2k_F +G,\sigma_2}c_{-,k_2,\sigma_2}c_{-,k_1,\sigma_1} + \text{H.c.}\big)\cr
&+ {1\over 2L}\sum_{\{k,q,\sigma\}} \left(g_4+ 2\bar{g}_4\sin^2 {q \over 2} \right) c^\dagger_{+,k_1+q,\sigma_1}c^\dagger_{+,k_2-q ,\sigma_2}c_{+,k_2,\sigma_2}c_{+,k_1,\sigma_1} \cr
& + {1\over 2L}\sum_{\{k,q,\sigma\}} \left(g_4+ 2\bar{g}_4\sin^2 {q \over 2} \right) c^\dagger_{-,k_1+q,\sigma_1}c^\dagger_{-,k_2-q ,\sigma_2}c_{-,k_2,\sigma_2}c_{-,k_1,\sigma_1},
\end{split}
\end{equation}
\end{widetext}
where $\epsilon_k = -2t\cos k$ is the tight-binding spectrum, and $v= 2t$ is the bare Fermi velocity; the Fermi points are $k_F=\pm {\pi\over 2}$ at half-filling (here the lattice constant has been set to unity, and $\hbar= 1 = k_B)$. By analogy with the `g-ology' description of interactions, we have proceeded to the splitting of the $U$ and $V$ interaction terms into couplings for right ($p=+$, $k>0$) and left ($p=-$, $k<0$) moving electrons. We thus obtain momentum independent (local) as well as momentum dependent (non local) couplings, denoted in (\ref{ExHF}) by $g_{i=1\ldots4}$ and $\bar{g}_{1\ldots4}$, respectively. The pairs of couplings for backscattering ($g_1,\bar{g}_1$) and Umklapp ($g_3,\bar{g}_3$) have the bare amplitudes $g_{1,3}= U-2V, \bar{g}_{1,3}= 2V$, whereas $ g_{2,4}= U +2V$ and $\bar{g}_{2,4}=-2V$ stand for the amplitudes for the forward scattering between opposite ($g_2,\bar{g}_2$) and parallel ($g_4,\bar{g}_4$) $k$ electrons.
The information about the lattice in (\ref{ExHF}) is present by the use of the tight binding spectrum $\epsilon_k$ for $k\in [-\pi,\pi]$ in the Brillouin zone and in the momentum dependent couplings $\bar{g}_i\sin^2 q/2$. In the continuum limit, the latter amplitudes vanish when evaluated at zero momentum transfer, while the spectrum $\epsilon_k\to \epsilon_p(k) \approx v(pk-k_F)$ is taken as linear around each Fermi points. One thus recovers the standard electron gas formulation of the extended Hubbard model \cite{Emery79,Solyom79,Voit95,Giamarchi04}.
\subsection{The renormalization group transformation}
We write the partition function $Z=\text{Tr}\, e^{-\beta H} $ as a functional integral \begin{equation}
\label{Z}
Z= \int\!\!\int\mathfrak{D}\psi^*\mathfrak{D}\psi\, e^{S[\psi^*,\psi]},
\end{equation}
over anticommuting Grassmann fields $\psi^{(*)}$. The action $S[\psi^*,\psi] =S_0[\psi^*,\psi] + S_{I}[\psi^*,\psi]$ consists of a free and an interacting parts. In the Fourier-Matsubara space, the former part $S_0[\psi^*,\psi]$ reads
\begin{equation}
\label{S0}
S_0[\psi^*,\psi]= \sum_{p,\bar{k},\sigma} [G^{0}_p(\bar{k})]^{-1}\psi^*_{p,\sigma}(\bar{k})\psi_{p,\sigma}(\bar{k}),
\end{equation}
where
\begin{equation}
\label{G0}
G^{0}_p(\bar{k}) = [i\omega_n - \epsilon_k]^{-1},
\end{equation}
is the free electron propagator. Here $ \bar{k} = (k,\omega_n)$ and $\omega_n=(2n+1)\pi T$ is the fermion Mastubara frequency. The interacting part is given by
\begin{widetext}
\begin{equation}
\begin{split}
S_I[\psi^*,\psi] = & -{T\over L} \sum_{\{\bar{k},\bar{q},\sigma\}} \left(g_1+ 2\bar{g}_1\sin^2 {q\over 2} \right) \psi^*_{+,\sigma_1}( \bar{k} _1 + \bar{q}_0 + \bar{q} )\psi^*_{-,\sigma_2}( \bar{k} _2 - \bar{q}_0 - \bar{q} )\psi_{+,\sigma_2}( \bar{k} _2 )\psi_{-,\sigma_1}( \bar{k} _1) \cr
&-{T\over L} \sum_{\{\bar{k},\bar{q},\sigma\}} \left(g_2+ 2\bar{g}_2\sin^2 {q\over 2} \right) \psi^*_{+,\sigma_1}( \bar{k} _1 + \bar{q} )\psi^*_{-,\sigma_2}( \bar{k} _2 - \bar{q} )\psi_{-,\sigma_2}( \bar{k} _2 )\psi_{+,\sigma_1}( \bar{k} _1)\cr
& - {T\over 2 L} \sum_{\{\bar{k},\bar{q},\sigma\}} \left(g_3+ 2\bar{g}_3\sin^2 {q\over 2} \right) \big(\psi^*_{+,\sigma_1}( \bar{k} _1 + \bar{q}_0 + \bar{q} )\psi^*_{+,\sigma_2}( \bar{k} _2 - \bar{q}_0 - \bar{q} + \bar{G} )\psi_{-,\sigma_2}( \bar{k} _2 )\psi_{-,\sigma_1}( \bar{k} _1) + \text{c.c.}\big)\cr
& - {T\over 2 L} \sum_{\{\bar{k},\bar{q},\sigma\}} \left(g_4+ 2\bar{g}_4\sin^2 {q\over 2} \right) \psi^*_{+,\sigma_1}( \bar{k} _1 + \bar{q} )\psi^*_{+,\sigma_2}( \bar{k} _2 - \bar{q} )\psi_{+,\sigma_2}( \bar{k} _2 )\psi_{+,\sigma_1}( \bar{k} _1) \cr
&- {T\over 2 L} \sum_{\{\bar{k},\bar{q},\sigma\}} \left(g_4+ 2\bar{g}_4\sin^2 {q\over 2} \right) \psi^*_{-,\sigma_1}( \bar{k} _1 + \bar{q} )\psi^*_{-,\sigma_2}( \bar{k} _2 - \bar{q} )\psi_{-,\sigma_2}( \bar{k} _2 )\psi_{-,\sigma_1}( \bar{k} _1),
\end{split}
\end{equation}
\end{widetext}
where $ \bar{q} =(q,\omega_m)$, $\omega_m=2\pi mT$, $ \bar{q}_0 =(2k_F,0)$; here $ \bar{G} =(4k_F,0)$ is a reciprocal lattice vector that enters in the definition of Umklapp scattering at half-filling.
The momentum shell K-W RG transformation is based upon the recursive application of the two following steps for the partition function. In the first step, a partial trace of $Z$ over outer shell electronic degrees of freedom denoted by $\bar{\psi}_{p,\sigma}(k,\omega_n)$, is carried out at all $\omega_n$ and spin $\sigma$. The outer momentum shell is defined by the intervals of momentum
\begin{align}
\label{shell}
k \in \ & [0,k_F-k_0/s[ \ \cup \ ]k_F+k_0/s,\pi], \ \ p=+ \cr
\in \ & ]\!-k_F+k_0/s,0] \ \cup \ [-\pi,-k_F-k_0/s[, \ \ p=-. \cr
\end{align}
above and below the Fermi level for each branch $p$. Here $k_0=\pi/2$ is a cutoff wave vector defined with respect to the Fermi points (\hbox{$\pm k_F\pm k_0=\pm \pi$}), and $s=e^{d\ell}>1$ is the momentum scaling factor for $d\ell\ll 1$. The second step consists in the rescaling of the momentum distance from the Fermi points we call $\delta k$; this gives $k'= \pm k_F + s \delta k $, which restores the initial cutoff $k_0=\pi/2$ of the lattice model.
The two recursive steps of the RG transformation can be expressed as
\begin{align}
\label{RG}
& Z =\! \Big[\int\!\!\int_<\mathfrak{D}\psi^*\mathfrak{D}\,\psi e^{S[\psi^*,\psi]_\ell} \!\! \int\!\!\int \mathfrak{D}\bar{\psi}^*\mathfrak{D} \bar{\psi} \,e^{ S_0[\bar{\psi}^*,\bar{\psi}] }\cr
&\hskip 4 truecm \ \ \times e^{ \sum_{i=1}^4 S_{I,i}[\bar{\psi}^*,\bar{\psi},\psi^*,\psi]}\Big]_{\psi\to\zeta^{1\over 2}_s \psi'} \cr
&\propto\!\! \Big[\!\ \! \int\!\!\int_<\! \mathfrak{D}\psi^*\mathfrak{D}\psi \ e^{S[\psi^*,\psi]_\ell + \langle S_{I,2}\rangle_{\bar{0},c} +\frac{1}{2} \langle (S_{I,2})^2\rangle_{\bar{0},c} +\ldots }\Big]_{\psi\to\zeta^{1\over 2}_s \psi'},\cr
\end{align}
where $S_{I,i}$ is the interacting part of the action with $i=1,\ldots,4$, $\bar{\psi}$ fields in the outer momentum shell. The outer shell statistical averages $\langle ....\rangle_{\bar{0},c}$ over the variables $\bar{\psi}^{(*)} $ are performed with respect to $S_0[\bar{\psi}^*,\bar{\psi}]$. These averages correspond to the sum of all connected diagrams with even number of external fields $\psi^*,\psi$ pertaining to the inner momentum shell ($<$) degrees of freedom, which are kept fixed in the partial trace operation. At the one-loop level the partial trace and rescaling lead to the recursion relations
\begin{align}
\label{RecurS0}
\Big[S_0[\psi^*,\psi]_{\ell + d\ell} = &\ S_0[\psi^*,\psi]_\ell + \langle S_{I,2}\rangle_{\bar{0},c} + \ldots \Big]_{\psi\to\zeta^{1\over 2}_s \psi'}, \\
\label{RecurSI}
\Big[S_I[\psi^*,\psi]_{\ell + d\ell} = &\ S_I[\psi^*,\psi]_\ell + \frac{1}{2} \langle S^2_{I,2}\rangle_{\bar{0},c} + \ldots \Big]_{\psi\to\zeta^{1\over 2}_s \psi'},
\end{align}
for the free and interacting parts of the action.
Following the momentum rescaling, the inner shell fields $\psi$ are rescaled by the factor $\zeta_s^{1/2}$, which can be derived from a dimensional analysis of the parameters that define the bare action $S_0$. Thus assuming that the rescaling of the tight-binding spectrum is of the form $\epsilon'_{k'}\equiv \zeta_s \epsilon_k $, by taking $k'= \pm k_F + s\delta k$, one gets
in the limit $d\ell \to0$
\begin{equation}
\zeta_s \to \zeta_s(\delta k)= s^{\delta k\cot \delta k},
\end{equation}
which can be expressed in the form $s^y$ compatible with an iterative transformation in renormalization group. At variance with the usual case, however, the scaling dimension $y$ is here $k$ dependent. Thus at either the edge or the bottom of the band where the group velocity vanishes, $\zeta_s(\pm k_0) \to s^0$ and $\epsilon_k$ is dimensionless. It is only when the Fermi points is approached in the limit $\delta k\to 0$, that $\zeta_s(\delta k) \to s^1$ and the result of the continuum limit for a linear spectrum is recovered\cite{Bourbon91}. This also indicates that repetition of rescaling turns down the curvature of the band, which continuously evolves toward a linear shape. Since $\omega_n$ or the temperature $T$ enters on the same footing as $\epsilon_k$ in the inverse propagator $[G^0]^{-1}$, the temperature then transforms according to $T'=\zeta_s(\delta k)T$. Now referring to the form of $S_0$ in (\ref{S0}), this yields the transformation assumed above for the field, namely $\psi^{(*)\prime}=\zeta^{-1/2}_s(\delta k) \psi^{(*)}$.
When applied to the interacting part $S_I$ of the the action, the above relations for the field and temperature, combined to the shrinking of the number sites $L'=L/s$ under rescaling, will impose the following $k$-dependent transformations of interactions
\begin{align}
\label{recura}
g'_i & = \big(g_i +{\cal O}(g^2) \big) s^{-1+ \delta k \cot \delta k}, \\
\bar{g}'_i& = \big( \bar{g}_i + {\cal O}(g\bar{g}) \big) s^{-1-\delta k \csc \delta k}.
\label{recurb}
\end{align}
It ensues that for $\delta k\to \pm k_0$, we have $g_i'\to g_i s^{-1}$, and the local couplings are then irrelevant instead of being marginal variables near the bottom or the edge of the band. At the approach of the Fermi level, when $\delta k\to 0$, we have $g_i'=g_i$ and the dimensionless or marginal character of the local interactions of the electron gas model is retrieved\cite{Solyom79,Bourbon91}. In the same way, the non local terms transform according to $ \bar{g}'_i = \bar{g}_i s^{-1-\pi/2}$ at the boundaries or the bottom of the band and are therefore strongly irrelevant. In the limit $\delta k \to 0$ near the Fermi points, $ \bar{g}'_i = \bar{g}_i s^{-2}$, which corresponds to the usual negative bare scaling dimension of nearest-neighbor couplings of the continuum theory\cite{Haldane82,Giamarchi04}.
\subsection{The Fermi velocity and coupling constant flow equations}
We now proceed to the partial trace operation that defines the first step of the renormalization group transformation (\ref{RG}). At the one-loop level, this amounts to evaluate the outer shell statistical averages $\langle S_{I,2}\rangle_{\bar{0},c}$ and $\langle S^2_{I,2}\rangle_{\bar{0},c}$ of the recursion relations (\ref{RecurS0}) and (\ref{RecurSI}). The former contribution $\langle S_{I,2}\rangle_{\bar{0},c}$ is composed of Hartree and Fock self-energy corrections. In these, enter $k$ independent or constant terms that correct the chemical potential, a quantity that can be simply redefined to keep the filling of the band constant. These terms can be safely ignored. The presence of non local interactions give rise to momentum dependent Fock terms, which at the step $\ell$ of the iterative RG procedure read
\begin{align}
\label{HF}
\langle S_{I,2}\rangle_{\bar{0},c} = {T(\ell)\over L(\ell)}\sum_{\bar{k}}\mathop{\sum \kern-1.4em -\kern 0.5em}_{\bar{k}'}& [\, \bar{g}_1(\ell)G_{-p}^0(\bar{k}')
- \bar{g}_4(\ell)G_{p}^0(\bar{k}')] \cr
&\times \cos(k_F + \delta k) \psi^*_p(\bar{k})\psi_p(\bar{k}),
\end{align}
where the slashed summation contains an integration over $k'$ in the outer momentum shell interval (\ref{shell}) at a given $p$. The Fock terms contribute to the renormalization of the spectrum, that is the Fermi velocity. Carrying the $\bar{k}'$ summation, one gets the flow equation for the velocity,
\begin{equation}
\label{velocity}
{d_\ell \ln v(\ell)} = {\pi \over 4} \big(\tilde{\bar{g}}_4(\ell) -\tilde{\bar{g}}_1(\ell)\big) \tanh[ v(\ell) \sin \delta k_\ell/2T],
\end{equation}
where $\delta k_\ell = k_0e^{-\ell}$ and the couplings $\tilde{\bar{g}}\equiv \bar{g}/\pi v(\ell)$ are henceforth taken as normalized by the scale dependent Fermi velocity $v(\ell)$.
The recursion relations (\ref{recura}) for the local normalized couplings $\tilde{g} (\equiv{g}/\pi v(\ell))$ are obtained from the outer shell contractions $\langle S^2_{I,2}\rangle_{\bar{0},c}$ in the logarithmically singular Cooper (electron-electron) and Peierls ($2k_F$ electron-hole) channels. Their insertion in (\ref{recura}), leads after rescaling to the recursion relations
\begin{align}
\label{}
\tilde{g}_1' = &\ [\, \tilde{g}_1+ ( -\tilde{g}_1^2 +\tilde{g}_3\tilde{\bar{g}}_3 ) I_P + \tilde{\bar{g}}_1(\tilde{g}_2 +\tilde{\bar{g}}_2) I_C\, ]s^{-f_g} \\
\tilde{g}_2' = &\ [ \, \tilde{g}_2+ ( \tilde{g}_1+ \tilde{\bar{g}}_1)^2 I_C + ( \tilde{g}_3 + \tilde{\bar{g}}_3)^2 I_P\,] s^{-f_g} \\
\tilde{g}_3' = & \ [ \, \tilde{g}_3+ ( \tilde{g}_2+ \tilde{\bar{g}}_2) (2 \tilde{g}_3+ \tilde{\bar{g}}_3)I_P \cr
& \hskip 1 truecm - ( \tilde{g}_1+ \tilde{\bar{g}}_1)( \tilde{g}_3- \tilde{\bar{g}}_3)I_P\, ] s^{-f_g} \\
\tilde{g}_4' = & \ \tilde{g}_4 s^{-f_g},
\end{align}
where $f_g= 1-\delta k_\ell \cot \delta k_\ell + d_\ell \ln v(\ell) $, which contains the rescaling exponent of (\ref{recura}), and the correction due to the normalization from the scale dependent Fermi velocity. We note that the one-loop level, there is no logarithmic correction to the forward scattering amplitude $g_4$. The outer shell Cooper and Peierls loops evaluated at zero external variables are respectively given by
\begin{align}
\label{ }
I_C= &- 2\pi v(\ell) {T(\ell) \over L(\ell)}\mathop{\sum \kern-1.4em -\kern 0.5em}_{k>0} \sum_{\omega_n} G^0_+(\bar{k}+\bar{q}_C)G^0_-(-\bar{k})\cr
= &- \pi v(\ell) {1\over L(\ell)} \mathop{\sum \kern-1.4em -\kern 0.5em}_{k>0} {\tanh [\epsilon(k)/2T(\ell)]\over \epsilon({k})}\cr
= & - {\pi \over 2} \tanh [\epsilon(\ell)/2T] \ d\ell
\end{align}
at $\bar{q}_C=0$, where $\epsilon(\ell) = v(\ell)\sin\delta k_\ell$ and
\begin{align}
\label{ }
I_P(\ell) = &- 2\pi v(\ell) {T(\ell) \over L(\ell)}\mathop{\sum \kern-1.4em -\kern 0.5em}_k \sum_{\omega_n} G^0_+(\bar{k}+\bar{q}_P)G^0_-(k)\cr
= & -I_C
\end{align}
at $\bar{q}_P= (2k_F,0)$. It is worth stressing that neglecting the dependence of $I_{P,C}$ on external variables does not generate new momentum dependent interactions whose number is kept fixed along the RG flow.
The flow equations for the local interactions then become
\begin{align}
\label{flowg1}
d_\ell \tilde{g}_1 = & - f_g \tilde{g}_1 + f_1[-\tilde{g}_1^2 -\tilde{\bar{g}}_1(\tilde{g}_2+\tilde{\bar{g}}_2) + \tilde{g}_3\tilde{\bar{g}}_3\, ],\\
\label{flowg2}
d_\ell \tilde{g}_2 = & - f_g \tilde{g}_2 + {1\over 2} f_1[ \,(\tilde{g}_3+\tilde{\bar{g}}_3)^2 - (\tilde{g}_1+\tilde{\bar{g}}_1)^2], \\
\label{flowg3}
d_\ell \tilde{g}_3 = & - f_g \tilde{g}_3 + f_1 [ \, (\tilde{g}_2+ \tilde{\bar{g}}_2)(2\tilde{g}_3+\tilde{\bar{g}}_3)\cr
& \hskip 2.6 truecm - (\tilde{g}_1+\tilde{\bar{g}}_1)(\tilde{g}_3-\tilde{\bar{g}}_3)], \\
\label{flowg4}
d_\ell \tilde{g}_4 = & - f_g \tilde{g}_4,
\end{align}
where $f_1= {\pi\over 2} \tanh [\epsilon(\ell)/2T ]$. These equations differs from the usual scaling equations of the 1D-EG model in two respects. First, the rescaling for a tight-binding spectrum and velocity renormalization introduce linear terms; second, there are additional corrections coming to the coupling to momentum dependent interactions. These latter corrections are by far the most likely to influence the flow of local couplings if not the nature of the ground state as we will see.
As for the non local irrelevant interactions, the corrections due to loop contractions are small and will be neglected in weak coupling. From the rescaling transformation (\ref{recurb}) and the normalization of the couplings by $\pi v(\ell)$, we get
\begin{align}
\label{gbar}
d_\ell \tilde{\bar{g}}_{i} = &- \big(1+\delta k_\ell \csc \delta k_\ell + d_\ell \ln v(\ell)\big) \tilde{\bar{g}}_{i}
\end{align}
for $i=1,\ldots 4$. In the zero temperature limit, the solution of Eqs. (\ref{velocity}) and (\ref{gbar}) yields the following expressions
\begin{equation}
\label{velocityb}
v(\ell) = v\left(1- {V\over \pi t}\ln[ 2\cos^2(\delta k_\ell/2)]\right),
\end{equation}
for the Fermi velocity and
\begin{equation}
\label{gbarb}
{\bar{g}}_{i}(\ell) = {\bar{g}}_{i} {v\over v(\ell)} e^{-\ell}\tan (\delta k_\ell/2),
\end{equation}
for the non-local couplings. The Fermi velocity is thus renormalized downward due to the presence of the $V$ term; it reaches the value $v^*=v(1-{V\over\pi t}\ln2)$ in the limit of large $\ell$.
\subsection{Response Functions}
To determine the nature of long-range correlations in the ground state, we consider the most singular response functions or susceptibilities, which are denoted $\chi_\mu$.
These latter are obtained by adding to the $\ell=0$ action an additional term $S_h$\cite{Bourbon91}, which consists of source fields $h_\mu$ linearly coupled to the composite fields $O_\mu^*$,
\begin{equation}
\label{ }
S_h[\psi^*,\psi] = \sum_{\mu,\bar{q}} [\, h_\mu(\bar{q}) z_\mu O_\mu^*(\bar{q}) + {\rm c.c}\,],
\end{equation}
where $z_\mu$ is a pair vertex renormalization factor ($z_\mu=1$ at $\ell=0$). In what follows we shall examine the site spin density-wave ($\mu=$ SDW), bond spin density-wave ($\mu=$ BSDW), site charge density-wave ($\mu=$ CDW) and the bond order-wave ($\mu=$ BOW) susceptibilities of the Peierls channel; the singlet (SS) and triplet (TS) superconducting susceptibilities of the Cooper channel. These are defined with the aid of the following expressions for the composite pair fields,
\begin{align}
\label{ }
O_{{\rm SDW/BSDW}}(\bar{q}) = & {1\over2} \big(O^*_{x,y,z}(-\bar{q}) \pm O_{x,y,z}(\bar{q}) \big)
\end{align}
and
\begin{align}
\label{ }
O_{{\rm CDW/BOW}}(\bar{q}) = & {1\over2} \big(O^*_{0}(-\bar{q}) \pm O_{0}(\bar{q})\big)
\end{align}
in the Peierls channel, where
$$
O_{\mu}(\bar{q}) = \sqrt{T\over L} \sum_{\bar{k}} \psi(\bar{k}-\bar{q})^*_{-,\alpha}\sigma_\mu^{\alpha\beta} \psi_{+,\beta}(\bar{k}),
$$
and
\begin{align}
\label{ }
O_{{\rm SS}}(\bar{q}) = & \sqrt{T\over L} \sum_{\bar{k}} \alpha \psi(-\bar{k}+\bar{q})^*_{-,-\alpha} \psi_{+,\alpha}(\bar{k}), \\
O_{{\rm TS}_{\mu}}(\bar{q}) = & \sqrt{T\over L} \sum_{\bar{k}} \alpha \psi(-\bar{k}+\bar{q})^*_{-,-\alpha}\sigma_\mu^{\alpha\beta} \psi_{+,\beta}(\bar{k})
\end{align}
in the Cooper channel. Here $\sigma_{\mu=x,y,z} (\sigma_0)$ are the Pauli (identity) matrices.
The renormalization group transformation (\ref{RG}) at the one-loop level, will modify $S_h$ according to
\begin{align}
\label{h}
S_h[O^*,O]_{\ell +d\ell} =\! \Big[ S_h[O^*,O]_{\ell} &+ \langle S_h S_{I,2} \rangle_{\bar{0},c} + \ldots \Big]_{O^{(*)} \to s^0 O'^{(*)}}\cr
& + {1\over 2} \langle S^2_h \rangle_{\bar{0},c} + \ldots,
\end{align}
where the pair fields, having zero canonical dimension, remain unchanged under rescaling. The last term is a constant $\propto d\ell z_\mu^2 h_\mu^*h_\mu$ that adds at each iteration and yields the expression of the susceptibility
\begin{equation}
\label{ }
\pi v\chi_\mu(\bar{q}^0_\mu)= {\pi\over 2} \int_\ell {v\over v(\ell)}z_\mu^2 \tanh [\epsilon(\ell)/2T] d\ell,
\end{equation}
which is defined positive and evaluated in the static limit at $\bar{q}^0_\mu =(2k_F,0)$ and $(0,0)$ for the Peierls and Cooper channels, respectively. From the one-loop outer shell corrections to the linear coupling, which read
$$
\langle S_h S_{I,2} \rangle_{\bar{0},c} ={\pi\over 2} \tanh [\epsilon(\ell)/2T]d\ell \sum_{\mu, \bar{q}} [\, h_\mu(\bar{q}) \tilde{g}_\mu z_\mu O_\mu^*(\bar{q}) + {\rm c.c}\,],
$$
one gets the one-loop equation for the pair vertex part $z_\mu$ at $\bar{q}_\mu^0$,
\begin{equation}
\label{zmu}
{d_\ell \ln z_\mu} = \tilde{g}_\mu {\pi\over 2} \tanh [\epsilon(\ell)/2T].
\end{equation}
For the density-wave type susceptibilities, the normalized couplings $\tilde{g}_\mu$ are given by the combinations
\begin{align}
\label{gmuP}
\tilde{g}_{\rm CDW/BOW} & = -2\tilde{g}_1+ \tilde{g}_2 + \tilde{\bar{g}}_2 \mp \tilde{g}_3 \pm \tilde{\bar{g}}_3, \\
\tilde{g}_{\rm SDW/BSDW} & = \tilde{g}_2\pm \tilde{g}_3 + \tilde{\bar{g}}_2 \pm \tilde{\bar{g}}_3.
\end{align}
The corresponding expressions for the superconducting susceptibilities are
\begin{align}
\label{gmuC}
\tilde{g}_{\rm SS/TS} & = \mp \tilde{g}_1-\tilde{g}_2 \mp \tilde{\bar{g}}_1 - \tilde{\bar{g}}_2.
\end{align}
A positive value for $\tilde{g}_\mu$ at $\ell \to \infty$ signals a singularity in $z_\mu$ and then in $\chi_\mu$ in that limit.
\section{Results}
\begin{figure}
\includegraphics[width=7.0cm]{phaseUV.eps} \includegraphics[width=7.0cm]{phaseUVzoom.eps}
\caption{ a) The phase diagram of the 1D extended Hubbard model. The bold (thin) lines refer to the boundaries between the primary (secondary) phases indicated in bold (regular) characters. The dashed lines correspond to the boundaries of the phase diagram of the electron gas model in the continuum limit; b) zoom in the neighborhood of the $U=2V$ (dashed) line in the repulsive sector. \label{Dphases}
%
}
\end{figure}
The solution of the flow equations for the pair vertices (\ref{zmu}) and the couplings (\ref{flowg1}-\ref{gbar}) in the $T\to 0$ limit leads to the determination of the most singular susceptibilities. These in turn serve to the determination of the dominant and subdominant phases of the model in the ground state. This is summarized in the one-loop phase diagram of Fig.~\ref{Dphases}, as a function of weak $U$ and $V$. The results are compared with those obtained in the continuum limit \cite{Emery79,Giamarchi04}.
\subsection{Repulsive U }
We commence by looking at the first quadrant of the phase diagram, in the region surrounding the $U=2V>0$ line. At the point A below the separatrix in Fig.~\ref{Dphases}-b, where $U>2V$, the $\tilde{g}_2$ and $\tilde{g}_3$ couplings scale to strong repulsive values and become singular at a finite $\ell_\rho$, a singularity at one-loop level that is indicative of a (Mott) gap in the charge sector compatible with the initial conditions satisfying the inequality $\tilde{g}_1-2\tilde{g}_2 < \tilde{g}_3$. The repulsive $\tilde{g}_1$ coupling is marginally irrelevant and attributed to gapless spin degrees of freedom. The SDW response then develops a singularity similar to the one of BOW at large $\ell\sim\ell_\rho$, as shown by the behavior of $z_{\rm SDW}$ and $z_{\rm BOW}$ in the inset of Fig.~\ref{A}-b.
\begin{figure}
\includegraphics[width=7.0cm]{R1g.eps}
\par
\includegraphics[width=7.0cm]{R1Ki.eps}
\caption{ a) Flow of the coupling constants $\tilde{g}_{1,2,3}$ at the point A (1, 0.4) of the phase diagram in Fig.~\ref{Dphases}. b) The density-wave susceptibilities {\it vs} $\ell$; inset: the flow of the pair vertices $d_\ell \ln z_\mu$ for $\mu=$ SDW, BOW and SDW. \label{A}}
\end{figure}
From the same Figure, however, the amplitude of the SDW susceptibility is larger, and SDW (BOW) is then taken as the dominant (subdominant) phase in the ground state. These one-loop results indicate that in this region irrelevant non local couplings introduce no qualitative changes with respect to known results of the continuum theory\cite{Kimura75,Emery79,Solyom79}.
\begin{figure}
\includegraphics[width=7.0cm]{R2g.eps} \includegraphics[width=7.0cm]{R2Ki.eps}
\caption{ a) Flow of the coupling constants $\tilde{g}_{1,2,3}$ at point B : (1, 0.495) of the phase diagram in Fig.~\ref{Dphases}. b) The density-wave susceptibilities {\it vs} $\ell$; inset: the difference between the BOW and SDW flows of the pair vertices showing the dominance of the BOW phase. \label{B}}
\end{figure}
If we now move up to the point B in the phase diagram of Fig.~\ref{Dphases}-b, close but below the $U=2V$ line, a qualitative change with respect to the results of the continuum limit emerges. While $\tilde{g}_2$ and $\tilde{g}_3$ still scale to strong repulsive coupling, signaling the formation of a charge gap at a finite $\ell_\rho$ (Fig.~\ref{B}-a), the backscattering amplitude $\tilde{g}_1$ no longer scales toward zero, but extends across the $\tilde{g}_1=0$ line to then level off at a small non universal negative value (inset of Fig.~\ref{B}-a). According to the expressions in (\ref{gmuP}), this change of sign of $\tilde{g}_1$ yields $\tilde{g}_{\rm BOW} > \tilde{g}_{\rm SDW}$, indicating that the strongest singularity now occurs for the BOW response (inset of Fig.~\ref{B}-b). The BOW phase then becomes the dominant phase, whereas SDW closely follows as the secondary phase. The change of sign of $\tilde{g}_1$ takes its origin in the presence of non local couplings in the flow equations (\ref{flowg1}-\ref{flowg3}). Although irrelevant, these interactions push the renormalization of $\tilde{g}_1$ ($\tilde{g}_3$) downward (upward) through their coupling to local variables.
\begin{figure}
\includegraphics[width=7.0cm]{RCg.eps} \includegraphics[width=7.0cm]{RCKi.eps}
\caption{ a) Flow of the $\tilde{g}_{1,2,3}$ couplings at C (1, 0.55) in the phase diagram of Fig.~\ref{Dphases}. b) The susceptibilities {\it vs} $\ell$; \label{C}}
\end{figure}
The dominance of the BOW phase becomes more pronounced as one moves up across the line $U=2V$ (point C of Fig.~\ref{Dphases}-b). In this region, the initial local couplings $\tilde{g}_1$ and $\tilde{g}_3$ are negative, but the latter interaction is still pushed to strong repulsive sector by non-local couplings (Fig.~\ref{C}-a). The BOW susceptibility then develops the strongest singularity with the largest amplitude (Fig.~\ref{C}-b). These features of the flow and the predominance of BOW order keep on up the BOW-CDW boundary passing just below point D in Fig.~\ref{Dphases}-b. At that point, strong attractive coupling in $\tilde{g}_1$ and $\tilde{g}_2$ is occurring while $\tilde{g}_3$ remains small and attractive (Fig.~\ref{D}-a), implying the formation of a gap in the spin sector instead of the charge. In these conditions, we have $\tilde{g}_{\rm CDW}$ $>$ $\tilde{g}_{\rm BOW}$, which marks the onset of a dominant CDW phase. The BOW order is subdominant and SDW correlations are non longer singular and are strongly reduced by the presence of a spin gap.
It is worth noting that the emergence of a spin gap regime on the BOW-CDW frontier is compatible with the results of quantum Monte Carlo simulations\cite{Sandvik04}, which find the onset of a Luther-Emery liquid with a spin gap at the boundary.
\begin{figure}
\includegraphics[width=7.0cm]{R3g.eps} \includegraphics[width=7.0cm]{R3Ki.eps}
\caption{ a) Flow of the $\tilde{g}_{1,2,3}$ couplings at D (1, 0.609) in the phase diagram of Fig.~\ref{Dphases}. b) The susceptibilities {\it vs} $\ell$; inset: the flow of the pair vertices $d_\ell \ln z_\mu$ for $\mu=$ CDW, BOW, and SDW \label{D}}
\end{figure}
The same analysis carried out as a function of $U$ allows for the delimitation of a small but finite fan-shape region of the weak coupling phase diagram of Fig.~\ref{Dphases} where the BOW order intervenes as the ground state around the $U=2V$ line. This well known result of numerical calculations \cite{Nakamura99,Nakamura00,Sandvik04} and functional RG\cite{Tam06} contrasts with the direct SDW to CDW transition predicted for the 1D-EG model\cite{Emery79}.
\begin{figure}
\includegraphics[width=7.5cm]{R4g.eps}\par\hskip 0.3 truecm \includegraphics[width=7.0cm]{R4Ki.eps}
\caption{ a) Flow of the $\tilde{g}_{1,2,3}$ couplings at E (1, 0.65) in the phase diagram of Fig.~\ref{Dphases}. b) The susceptibilities {\it vs} $\ell$. \label{E}}
\end{figure}
We proceed on the analysis of the repulsive $U$ region by looking at the point E, that is above the intermediate BOW region. In this domain, $\tilde{g}_2$ and $\tilde{g}_3$ scale to strong repulsive and attractive couplings, respectively, while $\tilde{g}_1$ is non universal and weakly attractive (Fig.~\ref{E}-a), contrary to what is found for the electron gas model\cite{Kimura75,Emery79,Solyom79}. The CDW singularity is stronger and accompanied by a weaker singularity in the BSDW response (Fig.~\ref{E}-b). The BSDW replaces BOW as the subdominant phase over a finite domain of the phase diagram at $V>0$ (Fig.~\ref{Dphases}-a).
\begin{figure}
\includegraphics[width=7.0cm]{R5g.eps} \includegraphics[width=7.0cm]{R5Ki.eps}
\caption{ a) Flow of the $\tilde{g}_{1,2,3}$ couplings at F (1, -0.55) in the phase diagram of Fig.~\ref{Dphases}. b) The susceptibilities {\it vs} $\ell$. \label{F}}
\end{figure}
We turn to the point F located in the $V<0$ region below, but near the $U=-2V$ line, where qualitative changes with respect to the continuum limit are also found. In the framework of the 1D-EG model, the region below the $U=-2V$ line is characterized by the conditions $\tilde{g}_1 >0$ and $ \tilde{g}_1-2\tilde{g}_2 >\tilde{g}_3$, respectively for gapless spin and charge degrees of freedom with dominant TS and subdominant SS phases\cite{Emery79,Solyom79}. In the presence of non local couplings, however, while $\tilde{g}_1$ is marginally irrelevant, both $\tilde{g}_2$ and $\tilde{g}_3$ scale to strong repulsive coupling signaling that the charge degrees of freedom are still gapped (Fig.~\ref{F}-a). Therefore the SDW phase remains dominant contrary to the 1D-EG prediction of a gapless TS phase \cite{Emery79,Solyom79,Kimura75} (Fig.~\ref{F}-b); the SDW incursion below the $U=-2V$ line expands in size as $U$ increases as shown in Fig.~\ref{Dphases}-a. It is worth mentioning that the resulting inward bending of the TS-SDW boundary line which becomes more pronounced with increasing $U$ is consistent with the numerical results of Nakamura \cite{Nakamura00}.
Finally, as one moves sufficiently downward along the $V$ axis, one reaches a region where $\tilde{g}_1$ and $\tilde{g}_3$ behave the way marginally irrelevant variables do (Fig~\ref{G}-a), as shown for instance at the point G of the phase diagram of Fig.~\ref{Dphases}-a. One then essentially recovers the behavior of the 1D-EG model with a dominant (subdominant) power law singularity $\chi_{\rm TS(SS)} \propto \exp(\gamma_{\rm TS(SS)}\ell)$ for TS (SS) response at large $\ell$ (Fig~\ref{G}-b) with $\gamma_{\rm TS} \gtrsim \gamma_{\rm SS}>0$.
\begin{figure}
\includegraphics[width=7.0cm]{R6g.eps} \includegraphics[width=7.0cm]{R6Ki.eps}
\caption{ a) Flow of the $\tilde{g}_{1,2,3}$ couplings at G (1, -1) in the phase diagram of Fig.~\ref{Dphases}. b) The susceptibilities {\it vs} $\ell$. \label{G}}
\end{figure}
\subsection{Attractive U }
\begin{figure}
\includegraphics[width=7.0cm]{R7g.eps} \includegraphics[width=7.0cm]{R7Ki.eps}
\caption{ a) Flow of the $\tilde{g}_{1,2,3}$ couplings at H (-1, -0.46) in the phase diagram of Fig.~\ref{Dphases}. b) The susceptibilities (logarithmic scale) {\it vs} $\ell$ . \label{H} }
\end{figure}
We now consider the region of negative $U$ near the $U= 2V$ line. In this region, we encounter an alteration of the 1D-EG phase diagram boundary that is similar to the one discussed in the last paragraph at $U=- 2V>0$. At H in Fig.~\ref{Dphases}-b, a portion of the phase diagram with dominant (subdominant) TS (SS) gapless phase is lost, this time to the benefit of a SS phase with a spin gap. Strong attractive coupling in the spin sector is induced by non local couplings that push downward the renormalization of $\tilde{g}_1$ (Fig.~\ref{H}-a). As for Umklapp scattering, it stays weakly attractive indicating that the charge sector is gapless. The SS-TS boundary is then distorted inward compared to the straight line 1D-EG prediction, which is in fair agreement with the results of exact diagonalisation by Nakamura.\cite{Nakamura00} The SS phase expands from the bent boundary with the TS phase up to the $V=0$ symmetry line for the transition to CDW (Fig.~\ref{Dphases}-a). We exemplify the SS region by the point H of the phase diagram (Fig.~\ref{Dphases}-a), where the $\tilde{g}_1$ and $\tilde{g}_2$ scale to strong attractive coupling for the formation of a spin gap at $ \ell_\sigma$ (Fig.~\ref{H}-a). The SS response is the only singular response of the system and the whole region has no subdominant phase (Fig.~\ref{H}-b).
\begin{figure}
\includegraphics[width=7.0cm]{R8g.eps} \includegraphics[width=7.0cm]{R8Ki.eps}
\caption{ a) Flow of the $\tilde{g}_{1,2,3}$ couplings at I (-1, 0.6) in the phase diagram of Fig.~\ref{Dphases}. b) The susceptibilities {\it vs} $\ell$. \label{I} }
\end{figure}
We end the tour of the phase diagram with the second quadrant above the $V=0$ SS-CDW frontier at the point I. There, the rapid flow to strong attractive coupling for $\tilde{g}_1$ marks the onset of a spin gap at relatively small $\ell_\sigma$ (Fig.~\ref{I}-a). The strong attraction for $\tilde{g}_1$ prevails over the Umklapp term, though also marginally relevant. The singularity of the CDW response is thus by far prevalent, being followed by a much weaker BOW susceptibility, whose subdominance is less guaranteed since it occurs in the strong coupling domain where the perturbative RG becomes less reliable.
\section{Conclusion}
In this work we have proposed a generalization of the momentum shell renormalization group transformation that is applicable to 1D lattice models of interacting electrons. The approach has been put to the test for the determination of the phase diagram of the extended Hubbard model in weak coupling. The method discloses the influence of a finite number of dangerous irrelevant couplings on the scaling of marginal interaction terms of the model. Modification of scaling gives rise in some regions of the phase diagram to unexpected phases from the standpoint of the theory in the continuum limit. Among the results obtained, let us mention the incursion of BOW order in a finite portion of the repulsive $U\simeq 2V$ sector of the phase diagram, which agrees with previous results of numerical and functional RG methods. The approach is also able to capture the deformation of boundaries between Luttinger liquid and gapped phases in the phase diagram of the model as found previously
by exact diagonalisation.
These findings are encouraging for applications to other weak coupling 1D or quasi-1D interacting electron models in which lattice details can play an important role in the properties of correlations at long distance.
\acknowledgments
C. B thanks the National Science and Engineering Research Council of Canada (NSERC), the R\'eseau Qu\'ebcois des Mat\'eriaux de Pointe (RQMP) and the {\it Quantum materials} program of Canadian Institute of Advanced Research (CIFAR) for financial support.
|
{
"timestamp": "2010-09-22T02:03:14",
"yymm": "1009",
"arxiv_id": "1009.4181",
"language": "en",
"url": "https://arxiv.org/abs/1009.4181"
}
|
\section{Introduction}
We continue our program devoted to Laplacian operators on quantum spaces with the study of such operators on the quantum (standard) Podle\'s sphere $S^{2}_{q}$ and their coupling with gauge connections on the quantum principal $\U(1)$-fibration $\Asq\hookrightarrow\ASU$. While in \cite{lareza} one worked with a left $3D$ covariant differential calculus on $\SU$ and its restriction to the (unique) $2D$ left covariant differential calculus on the sphere $S^{2}_{q}$, in the present paper we use the somewhat more complicate $4D_{+}$ bicovariant calculus on $\SU$ introduced in \cite{wor89} and its restriction to a $3D$ left covariant calculus on the sphere $S^{2}_{q}$.
Laplacian operators on all Podle\'s spheres, related to the $4D_{+}$ bicovariant calculus on $\SU$ were already studied in \cite{poddc}. Our contribution to Laplacian operators comes from the use of Hodge $\star$-operators on both the manifold of $\SU$ and $S^{2}_{q}$ that we introduce by improving and diversifying on existing definitions.
We then move on to line bundles on the standard sphere $S^{2}_{q}$ and to a class of operators on such bundles that are `gauged' with the use of a suitable class of connections on the principal bundle $\Asq\hookrightarrow\ASU$ and of the corresponding covariant derivative on (module of sections of) the line bundles. These gauged Laplacians are completely diagonalized and are split in terms of a Laplacian operator on the total space $\SU$ of the bundle minus vertical operators, paralleling what happens on a classical principal bundle (see e.g. \cite[Prop.~5.6]{bgv}) and on the Hopf fibration of the sphere $S^{2}_{q}$ with calculi coming from the left covariant one on $\SU$ as shown in \cite{lareza,ale09}
In \S\ref{s:qsb} we describe all we need of the principal fibration $\Asq\hookrightarrow\ASU$ and associated line bundles over $S^{2}_{q}$. We also give a systematic description of the differential calculi we are interested in, the
$4D_{+}$ bicovariant calculus on $\SU$ and its restriction to a $3D$ left covariant calculus on the sphere $S^{2}_{q}$.
A thoughtful construction of Hodge $\star$-dualities on $\SU$ are in \S\ref{s:Hop} while the ones on $S^{2}_{q}$ are in
\S\ref{s:hl3}. These are used in \S\ref{se:L} for the definition of Laplacian operators. A digression on connections on the principal bundle and of covariant derivatives on the line bundles is in \S\ref{s:ccd} and the following
\S\ref{GLoLB} is devoted to the corresponding gauged Laplacian operators of modules of sections if the line bundles. To make the paper relatively self-contained it concludes with two appendices, \S\ref{ass:a1} giving general facts on differential calculi on Hopf algebras and \S\ref{ap:qpb} concerning with general facts on quantum principal bundles endowed with connections.
We like to mention that examples of Hodge operators on the exterior algebras of the quantum homogeneous $q$-Minkowski and $q$-Euclidean spaces -- satisfying a covariance requirement with respect to the action of the quantum groups $\mathrm{SO}_{q}(3,1)$ and $\mathrm{SO}_{q}(4)$ -- have been given in \cite{um94, majqe} using the formalism of braided geometry and with a construction of a $q$-epsilon tensor. On the exterior algebra over the quantum planes $\IR_{q}^{N}$ a Hodge operator has been studied in \cite{gf}.
\subsection*{Conventions and notations} When writing about connections and covariant derivatives we shall pay attention in keeping the two notions distinct: a connection will be a projection on a principal bundle while a covariant derivative will be an operator on section, both of concepts fulfilling suitable properties. The `$q$-number' is defined as:
\begin{equation}
[x] = [x]_q := \frac{q^x - q^{-x}}{q - q^{-1}} ,
\label{eq:q-integer}
\end{equation}
for $q \neq 1$ and any $x \in \IR$. For a coproduct
$\Delta$ we use the conventional Sweedler notation
$\Delta(x)=x_{(1)}\otimes x_{(2)}$ (with implicit summation) with iterations. The
convention is iterated to $(\id\otimes\Delta)\circ\Delta(x) = (\Delta\otimes\id)\circ\Delta(x) = x_{(1)}\otimes x_{(2)}\otimes x_{(3)}$, and so on.
\subsection*{Acknowledgments}
We are grateful to S. Albeverio, L.S. Cirio and I. Heckenberger for comments and suggestions.
AZ thanks P. Lucignano for his help with Maple.
GL was partially supported by the Italian Project `Cofin08--Noncommutative Geometry, Quantum Groups and Applications'. AZ gratefully acknowledges the support of the Max-Planck-Institut f\"ur Mathematik in Bonn, the Hausdorff Zentrum f\"ur Mathematik der Universit\"at Bonn, the Stiftelsen Blanceflor Boncompagni-Ludovisi f\"odd Bildt (Stockholm), the I.H.E.S. (Bures sur Yvette, Paris).
\section{Prelude: calculi and line bundles on quantum spheres}\label{s:qsb}
We introduce the manifolds of the quantum group $\SU$ and its quantum homogeneous space $S^{2}_{q}$ -- the standard Podle\'s sphere. The corresponding inclusion $\Asq\hookrightarrow\ASU$ of the corresponding coordinate algebras is a (topological) quantum principal bundle. Following App.~\ref{ass:a1} we then equip $\ASU$ with a 4-dimensional bicovariant calculus, whose restriction gives a 3-dimensional left covariant calculus on $\Asq$.
\subsection{Spheres and bundles}
The polynomial algebra $\ASU$ of the quantum group
$\SU$ is the unital $*$-algebra generated by elements $a$ and $c$, with relations
\begin{align}
\label{derel}
& ac=qca\quad ac^*=qc^*a\quad cc^*=c^*c , \nonumber \\
& a^*a+c^*c=aa^*+q^{2}cc^*=1 .
\end{align}
For the sake of the present paper, the deformation parameter
$q\in\IR$ can be restricted to the interval $0<q<1$ without loss of
generality. In the limit $q \to1$ one recovers the commutative coordinate algebra on the group manifold $\mathrm{SU(2)}$.
If we use the matrix
$$ U =
\left(
\begin{array}{cc} a & -qc^* \\ c & a^*
\end{array}\right) ,
$$
whose being unitary is equivalent to relations \eqref{derel}, the Hopf algebra structure for $\ASU$ is given by coproduct,
antipode and counit:
$$
\Delta\, U = U \otimes U , \qquad S(U) = U^* , \qquad \eps(U) = 1 ,
$$
that is $\Delta(a)= a \otimes a - q c^* \otimes c$, and $\Delta(c)= c \otimes a + a^* \otimes c$;
$S(a)=a^*$ and $S(c)=-qc$; $\eps(a)=1$ and $\eps(c)=0$ and their $*$-conjugated relations.
\medskip
The quantum universal envelopping algebra $\su$ is the unital Hopf $*$-algebra
generated as an algebra by four elements $K^{\pm 1},E,F$ with $K K^{-1}=1$ and subject to
relations:
\begin{equation}
K^{\pm}E=q^{\pm}EK^{\pm}, \qquad
K^{\pm}F=q^{\mp}FK^{\pm}, \qquad
[E,F] =\frac{K^{2}-K^{-2}}{q-q^{-1}} .
\label{relsu}
\end{equation}
The $*$-structure is
$K^*=K, \, E^*=F $,
and the Hopf algebra structure is provided by coproduct
$$\Delta(K^{\pm}) =K^{\pm}\otimes K^{\pm}, \quad
\Delta(E) =E\otimes K+K^{-1}\otimes E, \quad
\Delta(F)
=F\otimes K+K^{-1}\otimes F,$$
while the antipode is
$S(K) =K^{-1}, \,
S(E) =-qE, \,
S(F) =-q^{-1}F$
and the counit reads
$\varepsilon(K)=1, \,\varepsilon(E)=\varepsilon(F)=0$. The quadratic element
\begin{equation}
C_{q}=\frac{qK^2-2+q^{-1}K^{-2}}{(q-q^{-1})^2}+FE-\tfrac{1}{4}
\label{cas}
\end{equation}
is a quantum Casimir operator that generates the centre of $\su$.
\medskip
The Hopf $*$-algebras $\su$ and $\ASU$ are dually paired. The $*$-compatible bilinear mapping $\hs{~}{~}:\su\times\ASU\to\IC$ is on the generators given by
\begin{align}
&\langle K^{\pm},a\rangle=q^{\mp 1/2}, \qquad \langle K^{\pm},a^*\rangle=q^{\mp 1/2}, \nonumber\\
&\langle E,c\rangle=1, \qquad \langle F,c^*\rangle=-q^{-1}, \label{ndp}
\end{align}
with all other couples of generators pairing to zero. This pairing is proved \cite{KS97} to be non-degenerate. The algebra $\su$ is recovered as a $*$-Hopf subalgebra in the dual algebra $\ASU^o$, the largest Hopf $*$-subalgebra contained in the vector space dual $\ASU^{\prime}$.
There are \cite{wor87} $*$-compatible canonical commuting actions of $\su$ on $\ASU$:
$$
h \lt x := \co{x}{1} \,\hs{h}{\co{x}{2}}, \qquad
x {\triangleleft} h := \hs{h}{\co{x}{1}}\, \co{x}{2}.
$$
On powers of generators one computes, for $s\in\,\IN$, that
\begin{equation}
\label{lact}
\begin{array}{lll}
K^{\pm}\triangleright a^{s} =q^{\mp\frac{s}{2}}a^{s} &
F\triangleright a^{s} =0 &
E\triangleright a^{s} =-q^{(3-s)/2} [s] a^{s-1} c^{*} \\
K^{\pm}\triangleright a^{* s} =q^{\pm\frac{s}{2}}a^{* s} &
F\triangleright
a^{*s} =q^{(1-s)/2} [s] c a^{* s-1} &
E\triangleright a^{* s} =0 \\
K^{\pm}\triangleright c^{s} =q^{\mp\frac{s}{2}}c^{s} &
F\triangleright c^{s} =0 &
E\triangleright c^{s} =q^{(1-s)/2} [s] c^{s-1} a^* \\
K^{\pm}\triangleright c^{* s} =q^{\pm\frac{s}{2}}c^{* s} &
F\triangleright c^{*s} =-q^{-(1+s)/2} [s] a c^{*s-1} &
E\triangleright c^{* s} =0;
\end{array}
\end{equation}
and:
\begin{equation}
\label{ract}
\begin{array}{lll}
a^{s}\triangleleft K^{\pm} =q^{\mp\frac{s}{2}}a^{s} &
a^{s}\triangleleft F =q^{(s-1)/2} [s] c a^{s-1} &
a^{s}\triangleleft E =0 \\
a^{* s}\triangleleft K^{\pm} =q^{\pm\frac{s}{2}}a^{* s} &
a^{* s}\triangleleft F =0 &
a^{*s}\triangleleft E =-q^{(3-s)/2} [s] c^{*}a^{*s-1} \\
c^{s}\triangleleft K^{\pm} =q^{\pm\frac{s}{2}}c^{s} &
c^{s}\triangleleft F =0 &
c^{s}\triangleleft E =q^{(s-1)/2} [s] c^{s-1} a \\
c^{* s}\triangleleft K^{\pm} =q^{\mp\frac{s}{2}}c^{* s} &
c^{* s}\triangleleft F =-q^{(s-3)/2} [s] a^{*}c^{*s-1} &
c^{* s}\triangleleft E =0.
\end{array}
\end{equation}
\medskip
Consider the algebra $\mathcal{A}(\U(1)):=\IC[z,z^*] \big/ \!\!<zz^* -1>$. The map
\begin{equation} \label{qprp}
\pi: \ASU \, \to\,\mathcal{A}(\U(1)) , \qquad
\pi\,\left(
\begin{array}{cc} a & -qc^* \\ c & a^*
\end{array}\right):=
\left(
\begin{array}{cc} z & 0 \\ 0 & z^*
\end{array}\right)
\end{equation}
is a surjective Hopf $*$-algebra homomorphism. As a consequence, $\U(1)$
is a quantum subgroup of $\SU$ with right coaction:
\begin{equation}
\delta_{R}:= (\id\otimes\pi) \circ \Delta \, : \, \ASU \,\to\,\ASU \otimes
\mathcal{A}(\U(1)) . \label{cancoa}
\end{equation}
The coinvariant elements for this
coaction,
elements $b\in\ASU$ for which $\delta_{R}(b)=b\otimes 1$,
form the algebra of the standard Podle\'s sphere $\Asq\hookrightarrow\ASU$. This inclusion gives a topological quantum principal bundle, following the formulation reviewed in appendix~\ref{ap:qpb}.
\medskip
The above right $\U(1)$ coaction on $\SU$ is dual to the left action of the element $K$, and allows one \cite{maetal} to give a decomposition
\begin{equation}\label{dcmp}
\ASU=\oplus_{n\in\IZ} \mathcal{L}_{n}
\end{equation}
in terms of $\Asq$-bimodules defined by
\begin{equation}
\label{libu}
\mathcal{L}_{n} := \{x \in \ASU ~:~ K \lt x = q^{n/2} x\quad\Leftrightarrow\quad\delta_{R}(x)=x\otimes z^{-n}\},
\end{equation}
with $\Asq = \mathcal{L}_{0}$. It is easy to see that $\mathcal{L}_{n}^{*} = \mathcal{L}_{-n}$ and $\mathcal{L}_{n}\mathcal{L}_{m} = \mathcal{L}_{n+m}$, with
\begin{equation}
E \lt \mathcal{L}_{n} \subset \mathcal{L}_{n+2}, \qquad
F \lt \mathcal{L}_{n} \subset \mathcal{L}_{n-2}, \qquad
\mathcal{L}_{n} {\triangleleft} u \subset \mathcal{L}_{n},
\label{rellb}
\end{equation}
for any $u\in \su$. The bimodules $\mathcal{L}_{n}$ will be described at length later on when we endow them with connections.
Here we only mention that the bimodules $\mathcal{L}_{n}$ have a vector space decomposition (cf. e.g. \cite{maj95}):
\begin{equation}
\label{decoln}
\mathcal{L}_{n}:=\bigoplus_{J=\tfrac{|n|}{2}, \tfrac{|n|}{2} +1,
\tfrac{|n|}{2} +2, \cdots}V_{J}^{\left(n\right)},
\end{equation}
where $V_{J}^{\left(n\right)}$ is the spin
$J$ (with $J\in{\tfrac{1}{2}}\IN$) irreducible $*$-re\-pre\-sen\-ta\-tion spaces for the right action of $\su$, and basis elements
\begin{equation}
\phi_{n,J,l}= (c^{J-n/2} a^{*J+n/2}){\triangleleft} E^l
\label{bsphi}
\end{equation}
with $n\in\IZ,\,J=\tfrac{|n|}{2}+\IN, \,l=0,\ldots,2J$.
\subsection{The 4D exterior algebra over the quantum group $\SU$}\label{se:4dc}
We present here the exterior algebra over the so called $4D_{+}$ bicovariant calculus on $\SU$, which was introduced as a first order differential calculus in \cite{wor89}, and described in details in \cite{sta}.
The ideal $\mathcal{Q}_{\SU}\subset\ker\varepsilon_{\SU}$ corresponding to the $4D_+$ calculus is generated by the nine elements $\{c^{2}; \, c(a^{*}-a); \, q^{2}a^{*2}-(1+q^{2})(aa^{*}-cc^{*})+a^{2}; \, c^{*}(a^{*}-a); \, c^{*2}; \, [q^{2}a+a^{*}-q^{-1}(1+q^{4})]c; \,
[q^{2}a+a^{*}-q^{-1}(1+q^{4})](a^{*}-a); \, [q^{2}a+a^{*}-q^{-1}(1+q^{4})]c^{*}; \, [q^{2}a+a^{*}-q^{-1}(1+q^{4})][q^{2}a+a^{*}-(1+q^{2})]\}$. One has $\mathrm{Ad}(\mathcal{Q}_{\SU})\subset\mathcal{Q}_{\SU}\otimes\ASU$ and $\dim(\ker\varepsilon_{\SU}/{\mathcal{Q}_{\SU}})=4$. The quantum tangent space turns out to be a four dimensional $\mathcal{X}_{\mathcal{Q}}\subset\su$. A choice for a basis is given by the elements
\begin{align}
& L_{-}=q^{\frac{1}{2}}FK^{-1},\qquad L_{z}=\frac{K^{-2}-1}{q-q^{-1}},\qquad L_{+}=q^{-\frac{1}{2}}EK^{-1}; \nonumber \\
& L_{0}=\frac{q(K^{2}-1)+q^{-1}(K^{-2}-1)}{(q-q^{-1})^{2}}\,+FE=
\frac{q(K^{-2}-1)+q^{-1}(K^{2}-1)}{(q-q^{-1})^{2}}\,+EF,
\label{Lq}
\end{align}
from the last commutation rule in \eqref{relsu}. The vector $L_{0}$ belongs to the centre of $\su$: it differs from the quantum Casimir \eqref{cas} by a constant term,
\begin{equation}\label{casbis}
C_{q}=L_{0}+\left(\frac{q^{\frac{1}{2}}-q^{-\frac{1}{2}}}{q-q^{-1}}\right)^{2}-\tfrac{1}{4}=L_{0}+[\tfrac{1}{2}]^{2}-\tfrac{1}{4}.
\end{equation}
The coproducts of the basis \eqref{cpuh} give $\Delta L_{b}=1\otimes L_{b}+\sum_a L_{a}\otimes f_{ab}$: once chosen the ordering $(-,z,+,0)$, such a tensor product can be represented as a row by column matrix product where
\begin{equation}
f_{ab}=\left(\begin{array}{cccc} 1 & 0 & 0 & q^{-\frac{1}{2}}KE \\
(q-q^{-1})q^{\frac{1}{2}} FK^{-1} & K^{-2} & (q-q^{-1})q^{-\frac{1}{2}} EK^{-1} & (q-q^{-1})[FE+q^{-1}\,\frac{K^{-2}-K^{2}}{(q-q^{-1})^{2}}] \\ 0 & 0 & 1 & q^{-\frac{1}{2}} FK \\ 0 & 0 & 0 & K^{2}
\end{array}
\right).
\label{fab}
\end{equation}
The differential ${\rm d}:\ASU\mapsto\Omega^{1}(\SU)$ is written for any $x\in \ASU$ as
\begin{equation}
{\rm d} x=\sum_a (L_{a}\lt x)\omega_{a}=\sum_a \omega_{a}(R_{a}\lt x
\label{d4}
\end{equation}
on the dual basis of left invariant forms $\omega_{a}\in\Omega^{1}(\SU)$ with $\Delta_{L}^{(1)}(\omega_{a})=1\otimes\omega_{a}$. Here $R_{a}:=-S^{-1}(L_{a})$ and explicitly:
\begin{equation}
\label{Rder}
R_{\pm}=L_{\pm}K^2, \quad\quad R_{z}=L_{z}K^2, \quad\quad R_{0}=-L_{0}.
\end{equation}
On the generators of the algebra the differential acts as:
\begin{align}
&{\rm d} a=(q-q^{-1})^{-1}(q-1)a\omega_{z}-qc^{*}\omega_{+}+\lambda_{1} a\omega_{0}, \nonumber \\
&{\rm d} a^{*}=c\omega_{-}+(q-q^{-1})^{-1}(q^{-1}-1)a^{*}\omega_{z}+\lambda_{1} a^{*}\omega_{0}, \nonumber \\
&{\rm d} c=(q-q^{-1})^{-1}(q-1)c\omega_{z}+a^{*}\omega_{+}+\lambda_{1} c\omega_{0}, \nonumber \\
&{\rm d} c^{\star}=-q^{-1}a\omega_{-}+(q-q^{-1})^{-1}(q^{-1}-1)c^{*}\omega_{z}+\lambda_{1} c^{\star}\omega_{0},
\label{dcf}
\end{align}
with $\lambda_1= [{\tfrac{1}{2}}][\frac{3}{2}]$.
These relations can be inverted, giving
\begin{align}
&\omega_{-}=c^{*}{\rm d} a^{*}-qa^{*}{\rm d} c^{*}, \qquad \omega_{+}=a{\rm d} c-qc{\rm d} a, \nonumber \\
&\omega_{z}=a^{*}{\rm d} a+c^{*}{\rm d} c-(a{\rm d} a^{*}+q^{2}c{\rm d} c^{*}),\nonumber \\
&\omega_{0}=(1+q)^{-1}\lambda^{-1}_{1}[a^{*}{\rm d} a+c^{*}{\rm d} c+q(a{\rm d} a^{*}+q^{2}c{\rm d} c^{*})].
\label{om4}
\end{align}
It is then easy to see that for $q\to 1$ one has $\omega_{0}\to 0$. This differential calculus reduces in the classical limit to the natural three-dimensional bicovariant calculus on $\mathrm{SU(2)}$.
This first order differential $4D_{+}$ calculus is a $*$-calculus: the $*$-structure on $\ASU$ is extended to an antilinear $*$-structure on $\Omega^{1}(\SU)$, such that $({\rm d} x)^*={\rm d} (x^*)$ for any $x\in \ASU$. For the basis of left invariant 1-forms is just
\begin{equation}
\omega_{-}^{*}=-\omega_{+},\qquad\omega_{z}^{*}=-\omega_{z},\qquad\omega_{0}^{*}=-\omega_{0}.
\label{ss}
\end{equation}
From \eqref{bi-struct} one works out the bimodule structure of the calculus, obtaining:
\begin{equation}
\begin{array}{lll}
\omega_{-}a=a\omega_{-}-qc^{*}\omega_{0},\qquad &\omega_{+}a=a\omega_{+},\qquad & \omega_{0}a=q^{-1}a\omega_{0}, \\
\omega_{-}a^{*}=a^{*}\omega_{-}, \qquad &\omega_{+}a^{*}=a^{*}\omega_{+}+c\omega_{0}, \qquad &\omega_{0}a^{*}=qa^{*}\omega_{0}, \\
\omega_{-}c=c\omega_{-}+a^{*}\omega_{0}, \qquad &\omega_{+}c=c\omega_{+}, \qquad & \omega_{0}c=q^{-1}c\omega_{0}, \\
\omega_{-}c^{*}=c^{*}\omega_{-}, \qquad &\omega_{+}c^{*}=c^{*}\omega_{+}-q^{-1}a\omega_{0}, \qquad & \omega_{0}c^{*}=qc^{*}\omega_{0};
\end{array}
\label{biuno}
\end{equation}
as well as:
\begin{align}
&\omega_{z}a=qa\omega_{z}-q(q-q^{-1})c^{*}\omega_{+}+qa\omega_{0}, \nonumber \\
&\omega_{z}a^{*}=(q-q^{-1})c\omega_{-}+q^{-1}a^{*}\omega_{z}-q^{-1}a^{*}\omega_{0}, \nonumber \\
& \omega_{z}c=qc\omega_{z}+(q-q^{-1})a^{*}\omega_{+}+qc\omega_{0}, \nonumber \\
& \omega_{z}c^{*}=-q^{-1}(q-q^{-1})a\omega_{-}+q^{-1}c^{*}\omega_{z}-q^{-1}c^{*}\omega_{0}.
\label{bidue}
\end{align}
The $\ASU$-bicovariant bimodule $\Omega^{2}(\SU)$ of exterior 2-forms is defined by the projection given in \eqref{wedk}, with $\mathcal{S}_{\mathcal{Q}}^{(2)}=\ker\,\mathfrak{A}^{(2)}=\ker\,(1-\sigma)\subset\Omega^{1}(\SU)^{\otimes2}$. This necessitates computing the braiding as in \eqref{sigco}, a preliminary step being the computation as in \eqref{ri-co-form} of the right coaction on the left invariant basis forms, $\Delta_{R}^{(1)}(\omega_{a})=\omega_{b}\otimes J_{ba}$. For the calculus at hand:
\begin{equation}
J_{ba}=\left(\begin{array}{cccc}
a^{*2} & (1+q^{2})a^{*}c & -qc^{2} & (1-q^{2})a^{*}c \\
-qa^{*}c^{*} & aa^{*}-cc^{*} & -ac & (q^{2}-1)cc^{*} \\
-qc^{*2} & (q+q^{-1})ac^{*} & a^{2} & (q^{-1}-q)ac^{*} \\
0 & 0 & 0 & 1
\end{array}
\right).
\label{Jba}
\end{equation}
The braiding map $\sigma:\Omega^{1}(\SU)^{\otimes2}\to\Omega^{1}(\SU)^{\otimes2}$ is then worked out \cite{cla10} to be:
\begin{align}
&\sigma(\omega_{-}\otimes\omega_{-})=\omega_{-}\otimes\omega_{-}, \qquad \sigma(\omega_{+}\otimes\omega_{+})=\omega_{+}\otimes\omega_{+}, \qquad \sigma(\omega_{0}\otimes\omega_{0})=\omega_{0}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{z}\otimes\omega_{z})=\omega_{z}\otimes\omega_{z}+(q^{2}-q^{-2})(\omega_{z}\otimes\omega_{0}+\omega_{-}\otimes\omega_{+}-\omega_{+}\otimes\omega_{-}), \nonumber \\
&\sigma(\omega_{-}\otimes\omega_{+})=\omega_{+}\otimes\omega_{-}-\omega_{z}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{+}\otimes\omega_{-})=\omega_{-}\otimes\omega_{+}+\omega_{z}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{-}\otimes\omega_{z})=\omega_{z}\otimes\omega_{-}+(1+q^{2})\omega_{-}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{z}\otimes\omega_{-})=(1-q^{-2})\omega_{z}\otimes\omega_{-}+q^{-2}\omega_{-}\otimes\omega_{z}-(1+q^{-2})\omega_{-}\otimes\omega_{0},\nonumber \\
&\sigma(\omega_{-}\otimes\omega_{0})=\omega_{0}\otimes\omega_{-}+(1-q^{2})\omega_{-}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{0}\otimes\omega_{-})=q^{2}\omega_{-}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{z}\otimes\omega_{+})=q^{2}\omega_{+}\otimes\omega_{z}+(1-q^{2})\omega_{z}\otimes\omega_{+}+(1+q^{2})\omega_{+}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{+}\otimes\omega_{z})=\omega_{z}\otimes\omega_{+}-(1+q^{-2})\omega_{+}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{z}\otimes\omega_{0})=\omega_{0}\otimes\omega_{z}+(q-q^{-1})^{2}(\omega_{+}\otimes\omega_{-}-\omega_{-}\otimes\omega_{+})-(q-q^{-1})^{2}\omega_{z}\otimes\omega_{0}, \nonumber \\
& \sigma(\omega_{0}\otimes\omega_{z})=\omega_{z}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{+}\otimes\omega_{0})=\omega_{0}\otimes\omega_{+}+(1-q^{-2})\omega_{+}\otimes\omega_{0},\nonumber \\
&\sigma(\omega_{0}\otimes\omega_{+})=q^{-2}\omega_{+}\otimes\omega_{0}.
\label{sig}
\end{align}
By defining $\theta\wedge\theta^{\prime}=(1-\sigma)(\theta\otimes\theta^{\prime})$, the $q$-wedge product on 1-forms is:
\begin{align}
&\omega_{-}\wedge\omega_{-}=\omega_{+}\wedge\omega_{+}=\omega_{0}\wedge\omega_{0}=0, \nonumber \\
&\omega_{z}\wedge\omega_{z}-(q^{2}-q^{-2})\omega_{+}\wedge\omega_{-}=0, \nonumber \\
& \omega_{z}\wedge\omega_{\pm}+q^{\pm 2}\omega_{
\pm}\wedge\omega_{z}=0, \nonumber \\
&\omega_{\pm}\wedge\omega_{0}+\omega_{0}\wedge\omega_{\pm}=0,\nonumber \\
&\omega_{+}\wedge\omega_{-}+\omega_{-}\wedge\omega_{+}=0, \nonumber \\
&\omega_{z}\wedge\omega_{0}+\omega_{0}\wedge\omega_{z}-(q-q^{-1})^{2}\omega_{-}\wedge\omega_{+}=0 .
\label{2fw}
\end{align}
These relations show that $\dim\Omega^{2}(\SU)=6$. The exterior derivative on basis 1-forms result into:
\begin{align}
&{\rm d}\omega_{\pm}= \mp\, q^{\pm 1}\omega_{-}\wedge\omega_{z}, \nonumber \\
&{\rm d}\omega_{z}=(q+q^{-1})\omega_{+}\wedge\omega_{-}, \nonumber \\
&{\rm d}\omega_{0}=\frac{(q-q^{-1})^{2}(q+q^{-1}-1)}{(q-q^{-2})(1+q)}\,\omega_{-}\wedge\omega_{+}.
\label{d2f}
\end{align}
The antisymmetriser operator $\mathfrak{A}^{(2)}:\Omega^{2}(\SU)\to\Omega^{2}(\SU)$
has an interesting spectral decomposition, which will be used later on to introduce Hodge operators.
On the basis
\begin{equation}
\label{p12}
\begin{array}{lll}
\varphi_{0}=\omega_{-}\wedge\omega_{0},& & \varphi_{z}=\omega_{-}\wedge\omega_{0}+(1-q^{-2})\omega_{-}\wedge\omega_{z} \\
\psi_{0}=\omega_{+}\wedge\omega_{0}, & &\psi_{z}=\omega_{+}\wedge\omega_{0}-(1-q^2)\omega_{+}\wedge\omega_{z} \\
\psi_{\pm}=\omega_{0}\wedge\omega_{z}+(1-q^{\pm 2})\omega_{-}\wedge\omega_{+}, & &
\end{array}
\end{equation}
which is such that $\varphi_{0}^*=\psi_{0}$, $\varphi_{z}^*=\psi_{z}$ and $\psi_{-}^*=\psi_{+}$, it holds that
\begin{equation}
\label{p14}
\begin{array}{lll}
\mathfrak{A}^{(2)}(\varphi_{0})=(1+q^2)\varphi_{0}, &
\mathfrak{A}^{(2)}(\psi_{z})=(1+q^2)\psi_{z}, &
\mathfrak{A}^{(2)}(\psi_{+})=(1+q^2)\psi_{+} \\
\mathfrak{A}^{(2)}(\varphi_{z})=(1+q^{-2})\varphi_{z}, &
\mathfrak{A}^{(2)}(\psi_{0})=(1+q^{-2})\psi_{0}, &
\mathfrak{A}^{(2)}(\psi_{-})=(1+q^{-2})\psi_{-}.
\end{array}
\end{equation}
For later use we shall use the labelling $\xi_{(\pm)}\in\,\mathcal{E}_{(\pm)}$ with
\begin{equation}\label{p12bis}
\mathcal{E}_{(+)}=\{\varphi_{0}, \psi_{z}, \psi_{+}\}, \quad
\mathrm{and} \qquad \mathcal{E}_{(-)}=\{\varphi_{z},\psi_{0}, \psi_{-}\} .
\end{equation}
By proceeding further, the $\ASU$-bimodule $\Omega^{3}(\SU)$ is found to be 4-dimensional with left invariant basis elements:
\begin{equation}
\label{p13}
\begin{array}{lll}
\chi_{-}=\omega_{+}\wedge\omega_{0}\wedge\omega_{z},& &
\chi_{+}=\omega_{-}\wedge\omega_{0}\wedge\omega_{z} \\
\chi_{0}=\omega_{-}\wedge\omega_{+}\wedge\omega_{z}, & &
\chi_{z}=\omega_{-}\wedge\omega_{+}\wedge\omega_{0},
\end{array}
\end{equation}
with $\chi_{-}^*=-q^{-2}\chi_{+}$, $\chi_{0}^*=\chi_{0}$ and $\chi_{z}^*=\chi_{z}$.
These exterior forms are closed,
\begin{equation}
{\rm d}\chi_{a}=0, \label{d3f}
\end{equation}
and in addition satisfy
\begin{equation}
\mathfrak{A}^{(3)}(\chi_{a})=2(1+q^2+q^{-2})\chi_{a}
\label{p15-}
\end{equation}
for $a=-,+,z,0$, thus providing the spectral decomposition for the antisymmetriser operator $\mathfrak{A}^{(3)}:\Omega^3(\SU)\to\Omega^3(\SU)$:
The $\ASU$-bimodule $\Omega^{4}(\SU)$ of top forms ($\Omega^{k}(\SU)=\emptyset$ for $k>4$) is 1 dimensional. Its left invariant basis element $\mu=\omega_{-}\wedge\omega_{+}\wedge\omega_{z}\wedge\omega_{0}$ is central, i.e. $x\,\mu=\mu\,x$ for any $x\in\,\ASU$ and its eigenvalue for the action of the antisymmetriser is
\begin{equation}
\label{p15}
\mathfrak{A}^{(4)}(\mu)=2(q^4+2q^2+6+2q^{-2}+q^{-4})\mu.
\end{equation}
\subsection{The exterior algebra over the quantum sphere $S^{2}_{q}$}\label{se:cals2}
The restriction of the $4D_+$ bicovariant calculus endows the sphere $S^{2}_{q}$ with a left covariant 3-dimensional calculus \cite{ap94,poddc}. The exterior algebra $\Omega(S^{2}_{q})$ over such a calculus can be characterised in terms of some of the bimodules $\mathcal{L}_{n}$ introduced in \S\ref{s:qsb} .
Given $f\in \Asq\simeq\mathcal{L}_{0}$, the exterior derivative ${\rm d}:\Asq\mapsto\Omega^{1}(S^{2}_{q})$ from \eqref{d4} acquires the form:
\begin{equation}
{\rm d} f=(L_{-}\lt f)\omega_{-}+(L_{+}\lt f)\omega_{+}+(L_{0}\lt f)\omega_{0}.
\label{d3d}
\end{equation}
Notice that the basis 1-forms $\{\omega_{a}, a=-,+,0\}$ are graded commutative (cf. \eqref{2fw}). Furthermore, relation \eqref{rellb} shows that $(L_{\pm}\lt f)\in \mathcal{L}_{\pm2}$ and that $(L_{0}\lt f)\in \mathcal{L}_{0}$, while the $\ASU$-bimodule structure of $\Omega^{1}(\SU)$ described by the coproduct \eqref{fab} of the quantum derivations $L_{a}$ gives:
\begin{equation}
\begin{array}{ll}
\phi\,\omega_{-}=\omega_{-}\,\phi-q^{-1}\omega_{0}(L_{+}\lt \phi), & \qquad\qquad \omega_{-}\phi=\phi\,\omega_{-}+q(L_{+}K^{2}\lt \phi)\omega_{0}, \\
\phi^{\prime}\omega_{+}=\omega_{+}\phi^{\prime}-q\omega_{0}(L_{-}\lt\phi^{\prime}), & \qquad\qquad \omega_{+}\phi^{\prime}=\phi^{\prime}\omega_{+}+q^{-1}(L_{-}K^{2}\lt\phi^{\prime})\omega_{0}, \\
\phi^{\prime\prime}\omega_{0}=\omega_{0}(K^{-2}\lt\phi^{\prime\prime}), & \qquad\qquad \omega_{0}\phi^{\prime\prime}=(K^{2}\lt\phi^{\prime\prime})\omega_{0}.
\end{array}
\label{bis2}
\end{equation}
These identities are valid for any $\phi,\phi^{\prime},\phi^{\prime\prime}\in \ASU$.
They allow one to prove by explicit calculations the following identities:
\begin{align}
&\phi\in \mathcal{L}_{-2}:\qquad{\rm d}(\phi\,\omega_{-})=(L_{+}\lt \phi)\omega_{+}\wedge\omega_{-}+(L_{0}\lt \phi)\omega_{0}\wedge\omega_{-}, \nonumber \\
&\phi^{\prime}\in \mathcal{L}_{2}:\qquad {\rm d}(\phi^{\prime}\omega_{+})=(L_{-}\lt\phi^{\prime})\omega_{-}\wedge\omega_{+}+(L_{0}\lt\phi^{\prime})\omega_{0}\wedge\omega_{+}, \nonumber \\
&\phi^{\prime\prime}\in\mathcal{L}_{0}:\qquad{\rm d}(\phi^{\prime\prime}\omega_{0})=(L_{-}\lt\phi^{\prime\prime})\omega_{-}\wedge\omega_{0}+(L_{+}\lt\phi^{\prime\prime})\omega_{-}\wedge\omega_{0}+\phi^{\prime\prime}{\rm d}\omega_{0},
\label{deom}
\end{align}
and
\begin{align}
&\phi\in \mathcal{L}_{-2}:\qquad{\rm d}(\phi\,\omega_{-}\wedge\omega_{0})=(L_{+}\lt \phi)\omega_{+}\wedge\omega_{-}\wedge\omega_{0}, \nonumber \\
&\phi^{\prime}\in \mathcal{L}_{2}:\qquad {\rm d}(\phi^{\prime}\omega_{0}\wedge\omega_{+})=(L_{-}\lt\phi^{\prime})\omega_{-}\wedge\omega_{0}\wedge\omega_{+} , \nonumber \\
&\phi^{\prime\prime}\in\mathcal{L}_{0}:\qquad{\rm d}(\phi^{\prime\prime}\omega_{-}\wedge\omega_{+})=(L_{0}\lt\phi^{\prime\prime})\omega_{0}\wedge\omega_{-}\wedge\omega_{+}.
\label{deom2}
\end{align}
Together with the anti-symmetry properties \eqref{2fw} of the wedge product in $\Omega(\SU)$, these identities give:
\begin{prop}
\label{lea}
The exterior algebra $\Omega(S^{2}_{q})$ obtained as a restriction of $\Omega(\SU)$ associated to $4D_{+}$ calculus on $\SU$ can be written in terms of $\Asq$-bimodule isomorphisms:
\begin{align}
\Omega^{1}(S^{2}_{q})&\simeq\mathcal{L}_{-2}\,\omega_{-}\oplus\mathcal{L}_{2}\,\omega_{+}\oplus\mathcal{L}_{0}\,\omega_{0
\nonumber \\
\Omega^{2}(S^{2}_{q})&\simeq\mathcal{L}_{-2}\,(\omega_{-}\wedge\omega_{0})\oplus\mathcal{L}_{0}\,(\omega_{-}\wedge\omega_{+})\oplus\mathcal{L}_{2}\,(\omega_{0}\wedge\omega_{+})
\nonumber\\
\Omega^{3}(S^{2}_{q})&\simeq\mathcal{L}_{0}\,\omega_{-}\wedge\omega_{+}\wedge\omega_{0
\nonumber
\end{align}
\end{prop}
The basis element $\omega_{-}\wedge\omega_{+}\wedge\omega_{0}$ commutes with all elements in $\mathcal{L}_{0}\simeq\Asq$.
Such a calculus is 3 dimensional, since from \eqref{d3f} one has ${\rm d}(\phi^{\prime\prime}\omega_{-}\wedge\omega_{+}\wedge\omega_{0})=0$, for any $\phi^{\prime\prime}\in \Asq$, and from \eqref{2fw} one has that $\Omega^{1}(S^{2}_{q})\wedge(\omega_{-}\wedge\omega_{+}\wedge\omega_{0})=0$.
From \eqref{d4} and \eqref{Rder} the differential can also be written as
\begin{equation}
{\rm d} f=\omega_{-}(R_{-}\lt f)+\omega_{+}(R_{+}\lt f)+\omega_{0}(R_{0}\lt f),
\label{d3di}
\end{equation}
and it is easy to check the following relations, analogues of the previous \eqref{deom}, \eqref{deom2}:
\begin{align}
&\phi\in \mathcal{L}_{-2}:\qquad{\rm d}(\omega_{-}\,\phi)=-\omega_{-}\wedge\omega_{+}\,(R_{+}\lt \phi)-\omega_{-}\wedge\omega_{0}\,(R_{0}\lt \phi), \nonumber \\
&\phi^{\prime}\in \mathcal{L}_{2}:\qquad {\rm d}(\omega_{+}\,\phi^{\prime})=
-\omega_{+}\wedge\omega_{-}\,(R_{-}\lt \phi^{\prime})-\omega_{+}\wedge\omega_{0}\,(R_{0}\lt \phi^{\prime}), \nonumber \\
&\phi^{\prime\prime}\in \mathcal{L}_{0}:\qquad {\rm d}(\omega_{0}\,\phi^{\prime\prime})=
{\rm d}\omega_{0}\wedge\phi^{\prime\prime}-\omega_{0}\wedge\omega_{-}\,(R_{-}\lt \phi^{\prime\prime})-\omega_{0}\wedge\omega_{+}\,(R_{+}\lt \phi^{\prime\prime});
\label{deomi}
\end{align}
and
\begin{align}
&\phi\in \mathcal{L}_{-2}:\qquad{\rm d}(\omega_{-}\wedge\omega_{0}\,\phi)=\omega_{-}\wedge\omega_{0}\wedge\omega_{+}\,(R_{+}\lt\phi), \nonumber \\
&\phi^{\prime}\in \mathcal{L}_{2}:\qquad {\rm d}(\omega_{0}\wedge\omega_{+}\,\phi^{\prime})=\omega_{0}\wedge\omega_{+}\wedge\omega_{-}\,(R_{-}\lt\phi^{\prime}) , \nonumber \\
&\phi^{\prime\prime}\in\mathcal{L}_{0}:\qquad{\rm d}(\omega_{-}\wedge\omega_{+}\,\phi^{\prime\prime})=\omega_{-}\wedge\omega_{+}\wedge\omega_{0}\,(R_{0}\lt\phi^{\prime\prime}).
\label{deom2i}
\end{align}
\section{Hodge structures on $\Omega(\SU)$}
\label{s:Hop}
As described in \S\ref{se:4dc}, it holds for the bicovariant forms of the $4D_{+}$ first order bicovariant calculus that the spaces $\Omega^{k}(\SU)$ of forms are free $\ASU$-bimodules with $\dim\,\Omega^{k}(\SU)=\dim\,\Omega^{4-k}(\SU)$, and $\dim\Omega^4(\SU)=1$. Our strategy to introduce Hodge operators on
$\Omega(\SU)$ in \S\ref{sub:H-1} uses first suitable contraction maps in order to define Hodge operators on the vector spaces $\Omega^{k}_{inv}(\SU)$ of left invariant $k$-forms; we extend them next to the whole $\Omega^k(\SU)$ by requiring (one side) linearity over $\ASU$. This follows an alternative although equivalent
approach to Hodge operators on classical group manifold that we describe first in \S\ref{sse:Hcla}.
A somewhat complementary approach to the one of \S\ref{sub:H-1}, more suitable when restricting to the sphere $S^{2}_{q}$, is then given in \S\ref{sub:H}.
\subsection{Hodge operators on classical group manifolds}
\label{sse:Hcla}
Let $G$ be an $N$-dimensional compact connected Lie group given as a real form of a complex connected
Lie group. The algebra $\mathcal{A}(G)=Fun(G)$ of complex valued coordinate functions on $G$ is a $*$-algebra, whose $*$-structure can be extended to the whole tensor algebra. A metric on the group $G$ is a non degenerate tensor $g:\mathfrak{X}(G)\otimes\mathfrak{X}(G)\to\mathcal{A}(G)$ which is symmetric -- i.e. $g(X,Y)=g(Y,X)$, with $X,Y\in\,\mathfrak{X}(G)$ -- and real -- i.e. $g^*(X,Y)=g(Y^*,X^*)$ --. Any metric has a normal form: there exists a basis $\{\theta^{a}, a=1, \dots N\}$ of the $\mathcal{A}(G)$-bimodule $\Omega^{1}(G)$ of 1-forms which is real, $\theta^{a*}=\theta^a$, such that
\begin{equation}
\label{gmf}
g=\sum_{a,b=1}^{N}\eta_{ab}\,\theta^{a}\otimes\theta^b
\end{equation}
with $\eta_{ab}=\pm1\cdot\delta_{ab}$.
Given the volume $N$-form $\mu=\mu^*:=\theta^{1}\wedge\ldots\wedge\theta^{N}$, the corresponding Hodge operator $\star:\Omega^{k}(G)\to\Omega^{N-k}(G)$ is the $\mathcal{A}(G)$-linear operator whose action on the above basis is
\begin{align}
&\star(1)=\mu, \nonumber \\
\label{Hdui}
&\star(\theta^{a_{1}}\wedge\ldots\wedge\theta^{a_{k}})=\frac{1}{(N-k)!} \sum_{b_j} \epsilon^{a_{1}\ldots a_{k}}_{\qquad b_{1}\ldots b_{N-k}}\theta^{b_{1}}\wedge\ldots\wedge b^{N-k},
\end{align}
with $\epsilon^{a_{1}\ldots a_{k}}_{\qquad b_{1}\ldots b_{N-k}}:=\sum_{s_k} \eta^{a_{1}s_{1}}\ldots \eta^{a_{k}s_{k}}\epsilon_{s_{1}\ldots s_{k} b_{1}\ldots b_{N-k}}$ from the Levi-Civita tensor and the usual expression for the inverse metric tensor
$g^{-1}=\sum_{a,b=1}^{N} \eta^{ab} L_{a}\otimes L_{b}$
with $\sum_{b} \eta^{ab}\eta_{bc}=\delta^{a}_{c}$ on the dual vector field basis such that $\theta^{b}(L_{a})=\delta^{b}_{a}$. The Hodge operator \eqref{Hdui} satisfies the identity:
\begin{equation}
\label{quhs}
\star^{2}(\xi)=sgn(g)(-1)^{k(N-k)}\xi
\end{equation}
on any $\xi\in\,\Omega^k(G)$. Here $sgn(g)=\det(\eta_{ab})$ is the signature of the metric.
Hodge operators can indeed be equivalently introduced in terms of contraction maps. By this we mean an
$\mathcal{A}(G)$-sesquilinear map $\Gamma:\Omega^1(G)\times\Omega^1(G)\to\mathcal{A}(G)$ such that
$\Gamma(f\,\phi,\eta)=f^*\Gamma(\phi,\eta)$ while $\Gamma(\phi,\eta\,f )=\Gamma(\phi,\eta) f$ for
$f\in\,\mathcal{A}(G)$. Such a map can be uniquely extended to a consistent map $\Gamma:\Omega^k(G)\times\Omega^{k+k^\prime}(G)\to\Omega^{k^{\prime}}(G)$. We postpone showing this to the later \S\ref{sub:H-1}
where we prove a similar statement for the bicovariant calculus on $\SU$.
Having a contraction map, define the tensor $\tilde{g}:\Omega^1(G)\times\Omega^1(G)\to\mathcal{A}(G)$:
\begin{equation}
\tilde{g}(\phi, \eta):=\Gamma(\phi^{*},\eta).
\label{p7}
\end{equation}
Next, with a volume form $\mu$, such that $\mu^*=\mu$, define the operator $L:\Omega^{k}(G)\to\Omega^{N-k}(G)$ as
\begin{equation}
\label{p3}
L(\xi):=\frac{1}{k!} \Gamma^*(\xi,\mu)
\end{equation}
on $\xi\in\,\Omega^k(G)$, having used the notation $\Gamma^*(\cdot,\cdot)=(\Gamma(\cdot,\cdot))^*$. A second $\mathcal{A}(G)$-sesquilinear map $\{~,~\}:\Omega^{k}(G)\times\Omega^{k}(G)\to\mathcal{A}(G)$ can be implicitly introduced by the relation
\begin{equation}
\label{p4}
\{\xi,\xi^{\prime}\}\mu:=\xi^*\wedge L(\xi^{\prime}).
\end{equation}
For any pair of $k$-forms $\xi,\xi^{\prime}$ it is straightforward to recover that
\begin{equation}
\label{pi5}
\{\xi,\xi^{\prime}\}=\frac{1}{k!} \Gamma^*(\xi^{\prime},\xi) .
\end{equation}
The operator \eqref{p3} is not in general an Hodge operator: one has for example $L(1)=\mu$ as well as $L(\mu)= \det (\Gamma^*(\mu,\mu))$ which is not necessarily $\pm 1$.
To recover the standard formulation for a Hodge operator, one has to impose two constraints:
\begin{enumerate}[(a)]
\item An hermitianity condition. The sesquilinear map $\Gamma$ is said hermitian provided it satisfies:
\begin{equation}
\label{p10}
\{\phi,\eta\}=\Gamma(\phi,\eta),
\end{equation}
for any couple of 1-forms $\phi$ and $\eta$.
\end{enumerate}
From \eqref{pi5} and \eqref{p4} it holds that $\{\phi,\eta\}=\Gamma^*(\eta,\phi)$.Then
\begin{equation}
\label{pi10}
\{\phi,\eta\}=\Gamma(\phi,\eta)\quad\Leftrightarrow\quad\Gamma(\phi,\eta)=\Gamma^{*}(\eta,\phi).
\end{equation}
If the sesquilinear form $\Gamma$ is hermitian, one can prove that the expression \eqref{pi5} becomes
\begin{equation}
\label{pii10}
\{\xi,\xi^{\prime}\}=\frac{1}{k!}\Gamma(\xi,\xi^{\prime}).
\end{equation}
\begin{enumerate}[(b)]
\item A reality condition, namely a compatibility of the operator $L$ with the $*$-conjugation:
\begin{equation}
\label{p11}
L(\phi^{*})=(L(\phi))^*
\end{equation}
on 1-forms.
\end{enumerate}
If these two constraints are fullfilled, the tensor $\tilde{g}$ in \eqref{p7} is symmetric and real: it is (the inverse of) a metric tensor on the group manifold $G$. The operator $L$ turns out to be the standard Hodge operator corresponding to the metric given by $\tilde{g}$, and satisfies the identities:
\begin{align}
\label{p8}
L^2(\xi)=(-1)^{k(N-k)}sgn(\Gamma)\xi , \qquad
\{\xi,\xi^{\prime}\}= sgn(\Gamma)\{L(\xi),L(\xi^{\prime})\}
\end{align}
with
$$sgn(\Gamma):=(\det(\Gamma(\phi^{a},\phi^{b}))|\det(\Gamma(\phi^{a},\phi^{b})|^{-1}=sgn(\tilde{g}).
$$
Moreover, the operator $L$ turns out to be real, that is it commute with the hermitian conjugation $*$, on the whole exterior algebra $\Omega(G)$.
The above procedure could be inverted somehow. That is, given an hermitian contraction map $\Gamma$ as in \eqref{p10}, define the operator $L$ by \eqref{p3}. The corresponding tensor $\tilde{g}$ turns out to be real, but non necessarily symmetric. Imposing $L$ to satisfy one of the two conditions in \eqref{p8} -- they are proven to be equivalent -- makes the tensor $\tilde{g}$ symmetric, that is the inverse of a metric tensor, whose Hodge operator is $L$.
\subsection{Hodge operators on $\Omega(\SU)$}\label{sub:H-1}
In this section we shall describe how the classical geometry analysis of the previous section can be used
to introduce an Hodge operator on both the exterior algebras $\Omega(\SU)$ and $\Omega(S^{2}_{q})$ built out of the $4D$-bicovariant calculus \`a la Woronowicz on $\ASU$.
A somewhat different formulation of contraction maps was also used
in \cite{hec99,hec03} for a family of Hodge operators on the exterior algebras of bicovariant differential calculi over quantum groups.
We shall then start with a contraction map $\Gamma:\Omega^1_{inv}(\SU)\times\Omega^1_{inv}(\SU)\to\IC$
required to satisfy $\Gamma(\lambda\,\omega,\eta) = \lambda^* \Gamma(\omega, \eta)$ and
$\Gamma(\omega,\eta\, \lambda) = \Gamma(\omega, \eta) \lambda$, for $\lambda\in\IC$.
The natural extension to $\Gamma:\Omega^{\otimes k}(G)\times\Omega^{\otimes k+k^{\prime}}(G)\to\Omega^{\otimes k^{\prime}}(G)$ given by
\begin{equation}
\label{p2i}
\Gamma(\phi^{a_{1}}\otimes\ldots\otimes\phi^{a_k},\phi^{b_1}\otimes\ldots\otimes\phi^{b_{k+k^{\prime}}}):=\left(\Pi_{j=1}^{k}\,\Gamma(\phi^{a_{j}},\phi^{ b_{j}})\right)\,\phi^{b_{k+1}}\otimes\cdots\otimes\phi^{b_{k+k^{\prime}}} ,
\end{equation}
with the assumption that $\Gamma(1,\phi)=\phi$ for any $\phi\in\,\Omega(\SU)$, can be consistently extended to give a contraction map
$\Gamma:\Omega^{ k}(G)\times\Omega^{ k+k^{\prime}}(G)\to\Omega^{ k^{\prime}}(G)$, via
\begin{equation}
\label{c0}
\Gamma(\phi^{a_{1}}\wedge\ldots\wedge\phi^{a_k},\phi^{b_1}\wedge\ldots\wedge\phi^{b_{k+k^{\prime}}})
:=\Gamma(\mathfrak{A}^{(k)}(\phi^{a_{1}}\otimes\ldots\otimes\phi^{a_k}), \mathfrak{A}^{(k+k^{\prime})}(\phi^{b_1}\otimes\ldots\otimes\phi^{b_{k+k^{\prime}}})) .
\end{equation}
This comes from the $k$-th order anti-symmetriser $\mathfrak{A}^{(k)}$, constructed from the braiding of the calculus, and used to define the exterior product of forms,
\begin{equation}
\label{1c}
\phi^{a_{1}}\wedge\ldots\wedge\phi^{a_{k}}:=\mathfrak{A}^{(k)}(\phi^{a_{1}}\otimes\cdots\otimes\phi^{a_{k}});
\end{equation}
the key identity for the consistency of \eqref{c0} is
\begin{equation}
\label{2c}
\mathfrak{A}^{(k+k^{\prime})}(\phi^{a_{1}}\otimes\cdots\otimes\phi^{a_{k+k^{\prime}}})=(\mathfrak{A}^{(k)}\otimes\mathfrak{A}^{(k^{\prime})})(\sum_{\sigma_j\in\mathit{S}(k,k^{\prime})}\sigma_j(\phi^{a_{1}}\otimes\cdots\otimes\phi^{a_{k+k^{\prime}}})),
\end{equation}
where $\mathit{S}(k,k^{\prime})$ is the collection of the $(k,k^{\prime})$-shuffles, permutations $\sigma_{j}$ of $\{1,\ldots,k+k^{\prime}\}$ such that $\sigma_{j}(1)<\cdots<\sigma_{j}(k)$ and $\sigma_{j}(k+1)<\ldots<\sigma_{j}(k+k^{\prime})$. The identity \eqref{2c} is valid for any bicovariant calculus \`a la Woronowicz on a quantum group. It allows to show that any $(k+k)^{\prime}$-form can be written as the tensor product of a $k$-form times a $k^{\prime}$-form.
To proceed further, we use a slightly more general volume form by taking $\mu=\mu^*= \mathrm{i} \, m\,\omega_{-}\wedge\omega_{+}\wedge\omega_{0}\wedge\omega_{z}$, with $m\in\,\IR$. Then we define an operator
$$
\star:\Omega^{k}_{inv}(\SU)\to\Omega^{4-k}_{inv}(\SU),
$$
in degree zero and one by
\begin{equation}\label{p17}
\star(1):= \Gamma^*(1,\mu) =\mu \qquad \mathrm{and} \qquad
\star(\omega_{a}):= \Gamma^*(\omega_{a},\mu) .
\end{equation}
For $\Omega^{k}_{inv}(\SU)$ with $k\geq2$ we use the diagonal bases of the antisymmetriser, that is
\begin{equation}
\mathfrak{A}^{(k)}(\xi)=\lambda_{\xi}\xi ,
\label{eixi}
\end{equation}
with coefficients in \eqref{p14}, \eqref{p15-} and \eqref{p15} respectively. On these basis we define
\begin{equation}
\star(\xi):=\frac{1}{\lambda_{\xi}}\, \Gamma^*(\xi,\mu) .
\label{p17bis}
\end{equation}
Here and in the following we denote $(\Gamma(~,~))^*=\Gamma^*(~,~)$.
The definition \eqref{p17bis} is a natural generalisation of the classical \eqref{p3}: the classical factor $k!$ -- the spectrum of the antisymmetriser operator on $k$-forms in the classical case, where the braiding is the flip operator -- is replaced by the spectrum of the quantum antisymmetriser. Also, the presence of the *-conjugate comes from consistency and in order to have non trivial solutions.
Before we proceed it is useful to re-express the volume forms in terms of the diagonal bases of the ant-symmetrietrizer operators. Some little algebra shows that
\begin{align}\label{p16}
&\mu=\mathrm{i} m \{-\omega_{-}\otimes\chi_{+}^*+\omega_{+}\otimes\chi_{-}^*+\omega_{0}\otimes\chi_{0}^*-\omega_{z}\otimes\chi_{z}^*\} \nonumber \\
&\mu=\mathrm{i} m \{-\chi_{z}\otimes\omega_{z}^*+\chi_{-}\otimes\omega_{+}^* -\chi_{+}\otimes\omega_{-}^* +\chi_{0}\otimes\omega_{0}^*\},
\end{align}
and
\begin{multline}\label{p16bis}
\mu=\frac{\mathrm{i} m}{q^2-1}\Big\{\frac{1}{1+q^2}(q^4\psi_{-}\otimes\psi_{+}^*-\psi_{+}\otimes\psi_{-}^*) \\
+(q^4\varphi_{z}\otimes\varphi_{0}^*-\varphi_{0}\otimes\varphi_{z}^*+q^2\psi_{0}\otimes\psi_{z}^*-q^{-2}\psi_{z}\otimes\psi_{0}^*) \Big\} .
\end{multline}
A little more algebra shows in turn that on 1-forms
\begin{equation}\label{p19-1}
\star(\omega_{a})=\mathrm{i} m \big\{ \Gamma^*(\omega_{a},\omega_{-}) \chi_{+}
- \Gamma^*(\omega_{a},\omega_{+}) \chi_{-}
- \Gamma^*(\omega_{a},\omega_{0}) \chi_{0}
+ \Gamma^*(\omega_{a},\omega_{z}) \chi_{z}
\big\} ;
\end{equation}
and that using the bases \eqref{p12bis}, on 2-forms
\begin{multline*}
\star(\xi_{(+)})=\frac{\mathrm{i} m}{1-q^4} \Big\{
\frac{1}{1+q^2} \left(q^4\Gamma^*(\xi_{(+)},\psi_{-})\psi_{+}-\Gamma^*(\xi_{(+)},\psi_{+})\psi_{-} \right)+(q^4\Gamma^*(\xi_{(+)},\varphi_{z})\varphi_{0} \\
-\Gamma^*(\xi_{(+)},\varphi_{0})\varphi_{z}+q^2\Gamma^*(\xi_{(+)},\psi_{0})\psi_{z}-q^{-2}\Gamma^*(\xi_{(+)},\psi_{z})\psi_{0}) \Big\}
\end{multline*}
\begin{multline}\label{p19-2}
\star(\xi_{(-)})=\frac{\mathrm{i} m}{q^2-q^{-2}} \Big\{
\frac{1}{1+q^2} \left( q^4\Gamma^*(\xi_{(-)},\psi_{-})\psi_{+}-\Gamma^*(\xi_{(-)},\psi_{+})\psi_{-} \right)+(q^4\Gamma^*(\xi_{(-)},\varphi_{z})\varphi_{0} \\
-\Gamma^*(\xi_{(-)},\varphi_{0})\varphi_{z}+q^2\Gamma^*(\xi_{(-)},\psi_{0})\psi_{z}-q^{-2}\Gamma^*(\xi_{(-)},\psi_{z})\psi_{0})
\Big\}.
\end{multline}
As for 3-forms one finds
\begin{multline}\label{p19-3}
\star(\chi_{a})=-\frac{\mathrm{i} m}{2(1+q^2+q^{-2})} \Big\{-\Gamma^*(\chi_{a},\chi_{+})\omega_{-} \\ +\Gamma^*(\chi_{a},\chi_{-})\omega_{+}+\Gamma^*(\chi_{a},\chi_{0})\omega_{0}-\Gamma^*(\chi_{a},\chi_{z})\omega_{z} \Big\}, \end{multline}
and finally for the top form
\begin{equation}\label{p19-4}
\star(\mu)=\frac{1}{2(q^4+2q^2+6+2q^{-2}+q^{-4})} \Gamma^*(\mu,\mu).
\end{equation}
As in \eqref{p4} we define the sesquilinear map $\{~,~\}:\Omega^{k}_{inv}(\SU)\times\Omega^{k}_{inv}(\SU)\to\IC$ by
\begin{equation}
\{\xi,\xi^{\prime}\}\mu:=\xi^*\wedge\star(\xi^{\prime}).
\label{p4q}
\end{equation}
Then, mimicking the analogous construction of \S\ref{sse:Hcla} we impose both an hermitianity and a reality condition on the contraction map.
\begin{enumerate}[(a)]
\item A contraction map is hermitian provided it satisfies:
\begin{equation}
\label{pi21}
\{\omega_{a},\omega_{b}\}=\Gamma(\omega_{a},\omega_{b}), \qquad \mathrm{for} \qquad a,b=-,+,z,0 .
\end{equation}
\end{enumerate}
Given contraction maps fullfilling such an hermitianity constraint, from the first line in \eqref{p19-1} one has
that $\Gamma(\omega_{a},\omega_{b}) = \Gamma^*(\omega_{b},\omega_{a})$. i.e. $\Gamma_{ab}=\Gamma^{*}_{ba}$. With such a condition it is moreover possible to prove, that for with $k=2,3,4$,
\begin{equation}
\label{p21}
\{\xi,\xi^{\prime}\}=\frac{\lambda_{\xi^*}}{\lambda_{\xi}\lambda_{\xi^{\prime}}}\,\Gamma(\xi,\xi^{\prime}).
\end{equation}
on any $\xi,\xi^{\prime}\in\,\Omega^{k}_{inv}(\SU)$ of a diagonal basis of the antisymmetrizer as in \eqref{eixi}.
The above expression is the counterpart of \eqref{pii10} for a braiding which is not just the flip operator.
\begin{enumerate}[(b)]
\item
An hermitian contraction map is real provided one has
\begin{equation}
\label{pi22}
\lambda_{\xi^*}(\star\xi^*)=(\lambda_{\xi}(\star\xi))^*.
\end{equation}
\end{enumerate}
again on a diagonal basis of $\mathfrak{A}^{(k)}(\xi)$.
This expression generalises the classical one \eqref{p11}. Notice that it is set on any $\Omega^{k}_{inv}(\SU)$, and not only on 1-forms as in the classical case.
The requirement that the contraction be hermitian and real results in a series of constraints. Firstly,
the action on $\Omega^1_{inv}(\SU)$ of the corresponding operator $\star$ as defined in \eqref{p17} is worked out to be given b
\begin{equation}
\label{piii22}
\star\left(\begin{array}{c} \omega_{-} \\ \omega_{+} \\ \omega_{0} \\ \omega_{z} \end{array}\right)=\mathrm{i} m \left(\begin{array}{cccc} 0 & \alpha & 0 & 0 \\ -q^2\alpha & 0 & 0 & 0 \\ 0 & 0 & -\nu & \epsilon \\ 0 & 0 & -\epsilon & \gamma \end{array} \right)\left(\begin{array}{c}\chi_{-} \\ \chi_{+} \\ \chi_{0} \\ \chi_{z} \end{array}\right).
\end{equation}
The only non zero terms of the contraction $\Gamma$ are given by
\begin{align}
\Gamma_{--}=q^{-2}\Gamma_{++}=\alpha, \qquad \Gamma_{0z}=\Gamma_{z0}=\epsilon, \
\qquad \Gamma_{00}=\nu, \qquad
\Gamma_{zz}=\gamma,
\label{piv22}
\end{align}
with parameters that are real and satisfy in addition the conditions:
\begin{align}
&2\nu+(q^2-q^{-2})\epsilon=0, \nonumber \\
&2(\epsilon^2-\gamma\nu)+(q-q^{-1})^2(2q^2\alpha^2+\epsilon^2)=0.
\label{pv22}
\end{align}
On $\Omega^{2}_{inv}(\SU)$ the action of such operator is block off-diagonal,
\begin{align}
\label{pvi22}
&\star\left(\begin{array}{c}\varphi_{0} \\ \psi_{z} \\ \psi_{+} \end{array}\right)\,=
\frac{\mathrm{i} m}{q^4-1}\left(\begin{array}{ccc} \Gamma(\varphi_{0},\varphi_{0}) & 0 & 0 \\
0 & q^4\Gamma(\varphi_{z},\varphi_{z}) & 0 \\ 0 & 0 & \frac{\Gamma(\psi_{+},\psi_{+})}{1+q^{2}}
\end{array}
\right)\left(\begin{array}{c} \varphi_{z} \\ \psi_{0} \\ \psi_{-} \end{array}\right),\nonumber
\\
~\nonumber \\
&\star\left(\begin{array}{c} \varphi_{z} \\ \psi_{0} \\ \psi_{-} \end{array}\right)\,=
\frac{\mathrm{i} m}{1-q^4}\left(\begin{array}{ccc} q^6\Gamma(\varphi_{z},\varphi_{z}) & 0 & 0 \\
0 & q^2\Gamma(\varphi_{0},\varphi_{0}) & 0 \\ 0 & 0 & \frac{q^4\Gamma(\psi_{+},\psi_{+})}{1+q^{2}}
\end{array}
\right)\left(\begin{array}{c}\varphi_{0} \\ \psi_{z} \\ \psi_{+} \end{array}\right),
\end{align}
\noindent
while on $\Omega^{3}_{inv}(\SU)$ is
\begin{multline}
\label{pvii22}
\star\left(\begin{array}{c} \chi_{-} \\ \chi_{+} \\ \chi_{0} \\ \chi_{z} \end{array}\right)= \\ =
\frac{\mathrm{i} m}{2(1+q^2+q^{-2})}\left(\begin{array}{cccc}0 & -\Gamma(\chi_{-},\chi_{-}) & 0 & 0 \\ q^2\Gamma(\chi_{-},\chi_{-}) & 0 & 0 & 0 \\ 0 & 0 & -\Gamma(\chi_{0},\chi_{0}) & \Gamma(\chi_{z},\chi_{0}) \\ 0 & 0 & -\Gamma(\chi_{0},\chi_{z}) & \Gamma(\chi_{z},\chi_{z}) \end{array}\right)
\left(\begin{array}{c} \omega_{-} \\ \omega_{+} \\ \omega_{0} \\ \omega_{z} \end{array}\right).
\end{multline}
\bigskip
It turns out that the square of the operator $\star$ is not necessarily diagonal.
An explicit computation shows moreover that when $q\neq1$, given the constraints \eqref{pv22} there is no choice for the contraction $\Gamma$, nor for the value of the scale parameter $m\in\,\IR$ in the volume form such that the spectrum of the operator $\star^2$ is constant on any vector space $\Omega^k_{inv}(\SU)$.
This means that the operator $\star$ does not satisfy the classical expressions in \eqref{p8}.
We choose a particular value for the parameter $m$ defining
\begin{align}
\det\Gamma:=\frac{1}{\lambda_{\mu}}\Gamma(\omega_{-}\wedge\omega_{+}\wedge\omega_{0}\wedge\omega_{z},\omega_{-}\wedge\omega_{+}\wedge\omega_{0}\wedge\omega_{z}), \qquad
sgn(\Gamma):=\frac{\det\Gamma}{|\det\Gamma|}
\label{pviii22}
\end{align}
and imposing
\begin{equation}
\label{pix22}
\star^2(1)=sgn(\Gamma),
\end{equation}
which is clearly equivalent to the constraint
\begin{equation}
\label{px22}
m^2=|\det\Gamma|^{-1}.
\end{equation}
From now on we fix the orientation, and set $sgn(\Gamma)=1$.
We finally extend the operator $\star$ to the whole exterior algebra. This can be defined in two ways, i.e. we define Hodge operators $\star^{L},\star^{R}:\Omega^{k}(\SU)\to\Omega^{4-k}(\SU)$ by:
\begin{align}
\star^{L}(x\,\omega):=x\star(\omega), \qquad\qquad\qquad
\star^{R}(\omega\,x):=(\star\, \omega) x ,
\label{Hlr}
\end{align}
with $x\in\,\ASU$ and $\omega\in\,\Omega_{inv}(\SU)$. Both operators will find their use later on.
\subsection{Hodge operators on $\Omega(\SU)$ -- a complementary approach} \label{sub:H}
The procedure used in the previous section can not be ipso facto extended to introduce an Hodge operator on $\Omega(S^{2}_{q})$: it is well-known that $\Omega^{k}(S^{2}_{q})$ are not free $\Asq$-bimodules (as also evident from the description in \S\ref{se:cals2}) and the tensor product $\Omega^{\otimes2}(S^{2}_{q})$ has no braiding.
In order to construct a suitable Hodge operator on the quantum sphere, we shall export to this quantum homogeneous space the construction of \cite{kmt}, originally conceived on the exterior algebra over a quantum group.
The strategy largely coincides with the one described in \cite{ale09} and presents similarities to that used
in \cite{dal09} where a Hodge operator has been introduced on a quantum projective plane.
We start by briefly recalling the formulation from \cite{kmt}.
Consider a $*$-Hopf algebra $\mathcal{H}$ and the exterior algebra $\Omega(\mathcal{H})$ over an $N$-dimensional left covariant first order calculus
$(\Omega^{1}(\mathcal{H}),{\rm d})$, with $\dim \Omega^{N-k}(\mathcal{H})=\dim \Omega^{k}(\mathcal{H})$ and $\dim \Omega^{N}(\mathcal{H})=1$. Suppose in addition that $\mathcal{H}$ has an Haar state $h:\mathcal{H}\to\IC$, i.e. a unital functional, which is
invariant, i.e. $(\id\otimes h)\Delta x=(h\otimes \id)\Delta x=h(x)1$ for any $x\in \mathcal{H}$, and positive, i.e. $h(x^{*}x)\geq0$ for all $x\in \mathcal{H}$. An Haar state so defined is unique and automatically faithful: $h(x^{*}x)=0$ implies $x=0$.
Upon fixing an inner product on a left invariant basis of forms, the state $h$ is then used to endow
the whole exterior algebra with a left and a right inner product, when requiring left or right invariance,
\begin{align}
&\hs{x\,\omega}{x^{\prime}\,\omega^{\prime}}^{L}:=h(x^{*}x^{\prime})\hs{\omega}{\omega^{\prime}} ,
\nonumber \\
&\hs{\omega\,x}{\omega^{\prime}\,x^{\prime}}^{R}:=h(x^*x^{\prime})\hs{\omega}{\omega^{\prime}}
\label{inpo}
\end{align}
for any $x,x^{\prime}\in \mathcal{H}$ and $\omega,\omega^{\prime}$ in $\Omega_{inv}(\mathcal{H})$.
The spaces $\Omega^{k}(\mathcal{H})$ are taken to be pairwise orthogonal (this is stated by saying that
the inner product is graded).
The differential calculus is said to be non-degenerate if, whenever $\eta\in \Omega^{k}(\mathcal{H})$ and $\eta^{\prime}\wedge\eta=0$ for any $\eta^{\prime}\in \Omega^{N-k}(\mathcal{H})$, then necessarily $\eta=0$.
Choose in $\Omega^{N}(\mathcal{H})$ a left invariant hermitian basis element $\mu=\mu^{*}$, referred to as the volume form of the calculus. For the sake of the present paper, we assume that the differential calculus has a volume form such that $\mu\,x=x\,\mu$ for any $x\in\,\mathcal{H}$ (this condition is satisfied by the $4D_{+}$ bicovariant calculus on $\SU$ that we are considering). Then one defines an `integral'
\begin{align*}
&\int _{\mu} \, : \, \Omega(\mathcal{H})\to\IC, \qquad\qquad
\int _{\mu} x\, \mu =h(x) , \qquad \mathrm{for} \quad x\in \mathcal{H} ,
\end{align*}
and $\int_{\mu} \eta=0$ for any $k$-form
$\eta$ with $k<N$. For a non-degenerate calculus the functional $\int_{\mu}$ is left-faithful: if $\eta \in \Omega^{k}(\mathcal{H})$ is such that $\int_{\mu}\eta^{\prime}\wedge\eta=0$ for all $\eta^{\prime}\in \Omega^{N-k}(\mathcal{H})$, then $\eta=0$.
The central result is \cite{kmt}:
\begin{prop}
\label{Lge}
Consider a left covariant, non-degenerate differential calculus on $*$-Hopf algebra, whose corresponding exterior algebra is
such that $\dim\, \Omega^{N-k}(\mathcal{H})=\dim\, \Omega^{k}(\mathcal{H})$ and $\dim\, \Omega^{N}(\mathcal{H})=1$, with a left-invariant volume form $\mu=\mu^*$ satisfying $x\,\mu=\mu\,x$ for any $x\in\,\mathcal{H}$. If $\Omega(\mathcal{H})$ is endowed with inner products and integrals as before, there exists a unique left $\mathcal{H}$-linear bijective operator
$L:\Omega^{k}(\mathcal{H})\to\Omega^{N-k}(\mathcal{H})$ for $k=0,\ldots,N$ (resp. a unique right $\mathcal{H}$-linear bijective operator $R$) such that
\begin{equation}\label{Lop}
\int_{\mu}\eta^{*}\wedge L(\eta^{\prime})=\hs{\eta}{\eta^{\prime}}^L, \qquad\qquad
\int_{\mu}\eta^{*}\wedge R(\eta^{\prime})=\hs{\eta}{\eta^{\prime}}^R
\end{equation}
for any $\eta,\eta^{\prime}\in \Omega^{k}(\mathcal{H})$.
\end{prop}
We mention that there is no $R$ operator in \cite{kmt}. It is just to prove its right $\mathcal{H}$-linearity that one needs the condition $x\mu=\mu x$ for the volume form $\mu$ with $x\in\,\mathcal{H}$.
We are now ready to make contact with the previous \S\ref{sub:H-1}. The $4D_{+}$ differential calculus on
$\SU$ is easily seen to be non degenerate. On the other hand, the Haar state functional $h$ is given by (cf. \cite{KS97}):
\begin{equation}
h(1)=1;\qquad h((cc^{*})^{k})=(\sum_{j=0}^{k}\,q^{2j})^{-1}=\frac{1}{1+q^{2}+\ldots+q^{2k}},
\label{Has}
\end{equation}
with $k\in \IN$, all other generators mapping to zero.
Now, use the sesquilinear map \eqref{p4q} for an inner product
$\hs{\omega}{\omega^{\prime}}:=\{\omega,\omega^{\prime}\}$ on generators of $\Omega_{inv}(\SU)$ and extend it to a left invariant and a right invariant ones to the whole of $\Omega_{inv}(\SU)$ as in \eqref{inpo} using the state $h$. The uniqueness of the operators $L$ and $R$ from Proposition~\ref{Lge} then implies that the extended left and right inner products are related to the left and right Hodge operators \eqref{Hlr} by
\begin{equation}
\label{Holr}
\int_{\mu}\eta^{*}\wedge (\star^{L}\,\eta^{\prime})=\hs{\eta}{\eta^{\prime}}^L, \qquad\qquad
\int_{\mu}\eta^{*}\wedge (\star^{R}\,\eta^{\prime})=\hs{\eta}{\eta^{\prime}}^R
\end{equation}
for any $\eta,\eta^{\prime}\in \Omega^{k}(\mathcal{H})$.
\section{Hodge structures on $\Omega(S^{2}_{q})$} \label{s:hl3}
From the previous section,
the procedure to introduce Hodge operators on the quantum sphere appears outlined. Inner products on $\Omega(\SU)$ naturally induce inner products on $\Omega(S^{2}_{q})$, and we shall explore the use of relations like the \eqref{Holr} above to define a class of Hodge operators.
The exterior algebra $\Omega(S^{2}_{q})$ over the quantum sphere $S^{2}_{q}$ is described in \S\ref{se:cals2}. In particular, we recall its description in terms of the $\Asq$-bimodules $\mathcal{L}_{n}$ given in \eqref{libu}:
\begin{align}
&\Omega^{0}(S^{2}_{q}) \simeq \Asq \simeq \mathcal{L}_{0} \, \nonumber \\
&\Omega^{1}(S^{2}_{q}) \simeq \mathcal{L}_{-2}\,\omega_{-}\oplus\mathcal{L}_{2}\,\omega_{+}\oplus\mathcal{L}_{0}\,\omega_{0}\,\simeq \omega_{-}\,\mathcal{L}_{-2}\oplus\omega_{+}\,\mathcal{L}_{2}\oplus \omega_{0}\,\mathcal{L}_{0} \nonumber \\
&\Omega^{2}(S^{2}_{q}) \simeq \mathcal{L}_{-2}\,(\omega_{-}\wedge\omega_{0})\oplus\mathcal{L}_{0}\,(\omega_{-}\wedge\omega_{+})\oplus\mathcal{L}_{2}\,(\omega_{0}\wedge\omega_{+})\nonumber \\
&\qquad\qquad\qquad\qquad \simeq
(\omega_{-}\wedge\omega_{0})\,\mathcal{L}_{-2}\oplus(\omega_{-}\wedge\omega_{+})\,\mathcal{L}_{0}\oplus(\omega_{0}\wedge\omega_{+})\,\mathcal{L}_{2} \, \nonumber \\
&\Omega^{3}(S^{2}_{q}) \simeq \mathcal{L}_{0}\,\omega_{-}\wedge\omega_{+}\wedge\omega_{0} \label{isoc2} \,\simeq\,\omega_{-}\wedge\omega_{+}\wedge\omega_{0}\,\mathcal{L}_{0} \, .
\end{align}
In the rest of this section, to be consistent with the notation introduced in \S\ref{se:cals2}, we shall consider elements $\phi,\psi\in\,\mathcal{L}_{-2}$, elements $\phi^{\prime},\psi^{\prime}\in\,\mathcal{L}_{2}$ and elements $\phi^{\prime\prime},\psi^{\prime\prime}\in\,\mathcal{L}_{0}$.
\begin{lemm}
The above left covariant 3D calculus on $S^{2}_{q}$ is non-degenerate.
\label{nddc}
\begin{proof}
Given $\theta\in \Omega^{k}(S^{2}_{q})$ the condition of non degeneracy, namely $\theta^{\prime}\wedge\theta=0$ for any $\theta^{\prime}\in \Omega^{3-k}(S^{2}_{q})$ only if $\theta=0$, is trivially satisfied for $k=0,3$.
From \eqref{isoc2} take the 1-form
$\theta=\phi\,\omega_{-}$ and a 2-form $\theta^{\prime}=\psi\,\omega_{-}\wedge\omega_{0}+\psi^{\prime}\omega_{+}\wedge\omega_{0}+\psi^{\prime\prime}\omega_{-}\wedge\omega_{+}$. Using the commutation properties \eqref{bis2} between 1-forms and elements in $\ASU$, one has $\theta^{\prime}\wedge\theta=\{\psi^{\prime}(K^{2}\lt\phi)-\psi^{\prime\prime}(q^{-\frac{1}{2}} KE\lt\phi)\}\omega_{-}\wedge\omega_{+}\wedge\omega_{0}$, so that the equation $\theta^{\prime}\wedge\theta=0$ for any $\theta^{\prime}\in \Omega^{2}(S^{2}_{q})$ is equivalent to the condition $\{\psi^{\prime}(K^{2}\lt\phi)-\psi^{\prime\prime}(q^{-\frac{1}{2}} KE\lt\phi)\}=0$ for any $\theta^{\prime}\in \Omega^{2}(S^{2}_{q})$; taking $\theta^{\prime}=\psi^{\prime}\omega_{+}\wedge\omega_{0}$, one shows that this condition is satisfied only if $\phi=0$. A similar conclusion is reached with a 1-form $\theta=\phi^{\prime}\omega_{+}$, and with a 1-form $\theta=\phi^{\prime\prime}\omega_{0}$.
Consider then a 2-form $\theta=\phi\,\omega_{-}\wedge\omega_{0}$, and a 1-form $\theta^{\prime}=\psi\omega_{-}+\psi^{\prime}\omega_{+}+\psi^{\prime\prime}\omega_{0}$. Their product is $\theta^{\prime}\wedge\theta=(\psi^{\prime}\phi)\omega_{+}\wedge\omega_{-}\wedge\omega_{0}$, so that the condition $\theta^{\prime}\wedge\theta=0$, for all $\theta^{\prime}\in \Omega^{1}(S^{2}_{q})$ is equivalent to the condition $\psi^{\prime}\phi=0$ for any $\psi^{\prime}$; this condition is obviously satisfied only by $\phi=0$. It is clear that a similar analysis can be performed for any 2-form $\theta\in \Omega^{2}(S^{2}_{q})$.
\end{proof}
\end{lemm}
The Haar state $h$ of $\ASU$ given in \eqref{Has} yields a faithful and invariant state when restricted to
$\Asq$. As a volume form we take $\check{\mu}=\check{m}\,\omega_{-}\wedge\omega_{+}\wedge\omega_{0}=\check{\mu}^*$ with $\check{m}\in \IR$. It commute with algebra element, $f\,\check{\mu}=\check{\mu}\,f$ for $f\in\,\Asq$, so the integral on the exterior algebra $\Omega(S^{2}_{q})$ can be defined by
\begin{align}
&\int_{\check{\mu}}\theta=0,\qquad &\mathrm{on}\quad \theta\in \Omega^{k}(S^{2}_{q}),\,\mathrm{for}\,k=0,1,2 \, ,
\nonumber \\
&\int_{\check{\mu}}f\,\check{\mu}=h(f) ,\qquad &\mathrm{on}\quad f\,\check{\mu} \in \Omega^{3}(S^{2}_{q}) \, .
\label{ints2}
\end{align}
\begin{lemm}
\label{lef}
The integral $\int_{\check{\mu}}:\Omega(S^{2}_{q})\to\IC$ defined by \eqref{ints2} is left-faithful.
\begin{proof}
The proof of the left-faithfulness of the integral can be easily established from a direct analysis, using the faithfulness of the Haar state $h$. \end{proof}
\end{lemm}
The restriction to $\Omega(S^{2}_{q})$ of the left and right $\ASU$-linear graded inner products on $\Omega(\SU)$ in \eqref{Holr} gives left and right
$\Asq$-linear
graded inner products on $\Omega(S^{2}_{q})$:
\begin{equation}
\hs{\theta}{\theta^{\prime}}_{S^{2}_{q}}^{L}:=
\hs{\theta}{\theta^{\prime}}^{L}; \qquad\qquad\qquad
\hs{\theta}{\theta^{\prime}}_{S^{2}_{q}}^{R}:=
\hs{\theta}{\theta^{\prime}}^{R}
\label{prosca}
\end{equation}
with $\theta,\theta^{\prime}\in\,\Omega(S^{2}_{q})$. The analogue result to relation \eqref{Holr} is given in the following
\begin{prop}
\label{LH}
On the exterior algebra on the sphere $S^{2}_{q}$ endowed with the above graded left (resp. right) inner product, there exists a unique invertible left
$\Asq$-linear
Hodge operator $\check{L}:\Omega^{k}(S^{2}_{q})\to\Omega^{3-k}(S^{2}_{q})$, (resp. a unique invertible right $\Asq$-linear Hodge operator $\check{R}$) for $k=0,1,2,3$, satisfying
\begin{equation}\label{Les}
\int_{\check{\mu}}\theta^{*}\wedge \check{L}(\theta^{\prime})=\hs{\theta}{\theta^{\prime}}^{L}_{S^{2}_{q}} \, , \qquad\qquad\qquad
\int_{\check{\mu}}\theta^{*}\wedge \check{R}(\theta^{\prime})=\hs{\theta}{\theta^{\prime}}^{R}_{S^{2}_{q}}
\end{equation}
for any $\theta,\theta^{\prime}\in \Omega^{k}(S^{2}_{q})$.
They can be written in terms of the sesquilinear map \eqref{p4q} as:
\begin{equation}
\begin{array}{lcl}
\check{L}(1)=\check{\mu} \, , & \quad\quad\quad & \check{L}(\check{\mu})= \{\check{\mu},\check{\mu}\} \\
\check{L}(\phi\,\omega_{-})=\check{m}\alpha\,\phi\,\omega_{-}\wedge\omega_{0} \, , &\quad\quad\quad
& \check{L}(\phi\,\omega_{-}\wedge\omega_{0})=\check{m}\,\{\omega_-\wedge\omega_{0},\omega_-\wedge\omega_0\}\phi\,\omega_{-} \, ,\\
\check{L}(\phi^{\prime}\omega_{+})=\check{m}\,q^2\alpha\,\phi^{\prime}\omega_{0}\wedge\omega_{+} \, , &\quad\quad\quad & \check{L}(\phi^{\prime}\omega_{0}\wedge\omega_{+})=\check{m}\,\{\omega_{+}\wedge\omega_{0},\omega_{+}\wedge\omega_{0}\}\,\phi^{\prime}\omega_{+} \, , \\
\check{L}(\omega_{0})=-\check{m}\nu\,\omega_{-}\wedge\omega_{+} \, , &\quad\quad\quad& \check{L}(\omega_{-}\wedge\omega_{+})=-\check{m}\,\{\omega_{-}\wedge\omega_{+},\omega_{-}\wedge\omega_{+}\}\,\omega_{0} \,
\end{array}
\label{cLo}
\end{equation}
and
\begin{equation}
\begin{array}{lcl}
\check{R}(1)=\check{\mu} \, , & \quad\quad\quad & \check{R}(\check{\mu})= \{\check{\mu},\check{\mu}\} \\
\check{R}(\omega_{-}\,\phi)=\check{m}q^2\alpha\,\omega_{-}\wedge\omega_{0}\,\phi \, , &\quad\quad\quad
& \check{R}(\omega_{-}\wedge\omega_{0}\,\phi)=\check{m}\,q^2\{\omega_-\wedge\omega_{0},\omega_-\wedge\omega_0\}\,\omega_{-} \,\phi ,\\
\check{R}(\omega_{+}\,\phi^{\prime})=\check{m}\,\alpha\,\omega_{0}\wedge\omega_{+} \,\phi^{\prime} , &\quad\quad\quad & \check{R}(\omega_{0}\wedge\omega_{+}\,\phi^{\prime})=\check{m}\,q^{-2}\{\omega_{+}\wedge\omega_{0},\omega_{+}\wedge\omega_{0}\}\,\omega_{+} \,\phi^{\prime} , \\
\check{R}(\omega_{0})=-\check{m}\nu\,\omega_{-}\wedge\omega_{+} \, , &\quad\quad\quad& \check{R}(\omega_{-}\wedge\omega_{+})=-\check{m}\,\{\omega_{-}\wedge\omega_{+},\omega_{-}\wedge\omega_{+}\}\,\omega_{0} \, .
\end{array}
\label{cRo}
\end{equation}
\begin{proof}
For the rather technical proof we refer to \cite{ale09}, where the same strategy has been adopted for the analysis of a Hodge operator on a two dimensional exterior algebra on $S^{2}_{q}$. Here we only observe that the uniqueness follows from the result in Lemma~\ref{lef}. Given two operators $\check{L},\check{L}^{\prime}:\Omega^{k}(S^{2}_{q})\to\Omega^{3-k}(S^{2}_{q})$ satisfying \eqref{Les} (or equivalently $\check{R},\check{R}^{\prime}$), their difference must satisfy the relation $\int_{\check{\mu}}\theta^{\prime*}\wedge (\check{L}(\theta)-\check{L}^{\prime}(\theta))=0$ for any $\theta,\theta^{\prime}\in \Omega^{k}(S^{2}_{q})$. The left-faithfulness of the integral allows one then eventually to get $\check{L}(\theta)=\check{L}^{\prime}(\theta)$.
\end{proof}
\end{prop}
From \eqref{p13} and \eqref{p15-} it is $\check{\mu}=\check{m}\,\chi_{z}$, so we define
\begin{align}
\det\check{\Gamma}:=\frac{\Gamma(\chi_z,\chi_z)}{2(1+q^2+q^{-2})} ,
\qquad sgn(\check{\Gamma}):=\frac{\det(\check{\Gamma})}{|\det\check{\Gamma}|}
\label{pxiv22}
\end{align}
and set
$$
\check{m}^2\det\check{\Gamma}:=sgn(\check{\Gamma})
$$ as a definition for the scale factor $\check{m}\in\,\IR$. This choice clearly gives $\check{L}^2(1)=\check{R}^2(1)=sgn(\check{\Gamma})$. Again we fix the `orientation' so that $sgn(\check{\Gamma})=1$.
We conclude by noticing that the Hodge operators \eqref{cLo} and \eqref{cRo}
are diagonal, but still there is no choice for the parameters \eqref{piv22} and \eqref{pv22} of a real and hermitian contraction map such that a relation like \eqref{p8} is satisfied.
\section{Laplacian operators}
\label{se:L}
Given the Hodge structures constructed in the previous sections, the corresponding Laplacian operators on the quantum group $\SU$,
\begin{align*}
&\Box^L_{\SU}:\ASU\to\ASU , \qquad\qquad \Box_{\SU}(x) := - \star^L{\rm d}\star^L{\rm d} x , \\
&\Box^R_{\SU}:\ASU\to\ASU , \qquad\qquad \Box_{\SU}(x) := - \star^R{\rm d}\star^R{\rm d} x
\end{align*}
can be readily written in terms of the basic derivations \eqref{Lq} and \eqref{Rder} for the first order differential calculus as
\begin{align}
\label{elp4}
\Box^L_{\SU} x =
\left\{\alpha \left( L_{+}L_{-}+q^2 L_{-}L_{+}\right) + \nu\,L_{0}L_{0}+\gamma\,L_{z}L_{z}+2\epsilon L_{0}L_{z}\right\}\lt x,
\end{align}
and
\begin{align}
\Box^R_{\SU} x =
\left\{\alpha \left( q^2 R_{+}R_{-}+R_{-}R_{+} \right) + \nu\,R_{0}R_{0}+\gamma\,R_{z}R_{z}+2\epsilon R_{0}R_{z}\right\}\lt x,
\label{erp4}
\end{align}
with parameters given in \eqref{piv22}.
From the decomposition \eqref{dcmp} and the action \eqref{rellb} it is immediate to see that such Laplacians restrict to operators $:\mathcal{L}_{n}\to\mathcal{L}_{n}$. In order to diagonalise them, we recall the decomposition \eqref{decoln}.
The action of each term of the Laplacians on the basis elements $\{\phi_{n,J,l} \}$ in \eqref{bsphi} can be explicitly computed by \eqref{lact}, giving:
\begin{align}
&L_{-}L_{+}\lt\,\phi_{n,J,l}=q^{-1-n} \, [J-{\tfrac{1}{2}} n][J+1- {\tfrac{1}{2}} n ]\,\phi_{n,J,l} \, , \nonumber \\
&L_{+}L_{-}\lt\,\phi_{n,J,l}=q^{1-n} \, [J+{\tfrac{1}{2}} n][J+1-{\tfrac{1}{2}} n]\,\phi_{n,J,l} \, , \nonumber \\
&L_{z}\lt\,\phi_{n,J,l}=-q^{-\frac{1}{2} n } \, [{\tfrac{1}{2}} n]\,\phi_{n,J,l} \, , \nonumber \\
&L_{0}\lt\,\phi_{n,J,l}=([J+{\tfrac{1}{2}}]^{2}-[{\tfrac{1}{2}}]^2)\,\phi_{n,J,l}=[J][J+1]\,\phi_{n,J,l} \, .
\label{eopep}
\end{align}
Here for the labels one has $n\in\IN$ with $J=\tfrac{|n|}{2}+\IZ$ and $l=0,\ldots,2J$.
The Laplacians on the quantum sphere are, with $f\in\,\Asq$:
\begin{align}
\Box_{S^{2}_{q}}^L f:=-\check{L}{\rm d}\check{L}{\rm d} f=
\left\{\alpha\,L_{+}L_{-}+q^2\alpha\,L_{-}L_{+}+\nu\,L_{0}L_{0}\right\}\lt f,
\label{sfl}
\end{align}
and
\begin{align}
\Box_{S^{2}_{q}}^R f:=-\check{R}{\rm d}\check{R}{\rm d} f=
\left\{q^2 \alpha\,R_{+}R_{-}+\alpha\,R_{-}R_{+}+\nu\,R_{0}R_{0}\right\}\lt f.
\label{sfr}
\end{align}
They are both the restriction to $S^{2}_{q}$ of the Laplacian on $\SU$, the left and right one respectively.
Their actions can be written in terms of the action of the Casimir element $C_{q}$ of $\su$, immediately giving their spectra. In fact they coincide on $S^{2}_{q}$.
\begin{align}
\Box_{S^{2}_{q}}^{L,R} &= 2q\alpha(C_{q}+\tfrac{1}{4}-[\tfrac{1}{2}]^2)
+\nu(C_{q}+\tfrac{1}{4}-[\tfrac{1}{2}]^2)^2 , \nonumber \\
&= 2q\alpha \,L_{0} +\nu \, L^2_{0} \qquad \mathrm{on} \quad \Asq .
\end{align}
Using \eqref{eopep}, their spectra are readily found:
\begin{equation}
\Box_{S^{2}_{q}}^{L,R} (\phi_{0,J,l}) = \left( 2q\alpha[J][J+1]+\nu[J]^2[J+1]^2 \right) \phi_{0,J,l} ,
\label{specL}
\end{equation}
with $J\in\,\IN, l=0,\ldots,2J$.
We end this section by comparing these spectra to the spectrum of $D^2$, the square of the Dirac operator on $S^{2}_{q}$ studied in \cite{gs10}. Some straightforward computation leads to:
\begin{align}\label{cosp}
\text{spec}( \,\Box_{S^{2}_{q}}^{L,R})=\text{spec}(D^2-[\tfrac{1}{2}]^2)\quad\quad\Leftrightarrow\quad\quad2q\alpha=1, \quad \nu=q^{-2}(q-q^{-1})^4 .
\end{align}
\section{A digression: connections the Hopf fibration over the quantum sphere} \label{s:ccd}
A monopole connection for the quantum fibration $\Asq\hookrightarrow\ASU$ on the standard Podle\'s sphere -- with a left-covariant 3d calculus on $\SU$ and the (corresponding restriction to a) 2d left-covariant calculus on $S^{2}_{q}$ -- was explicitly described in \cite{bm93}.
A slightly different, but to large extent equivalent \cite{durcomm} formulation of this and of a fibration constructed on the same topological data $\Asq\hookrightarrow\ASU$, but with $\SU$ equipped with a bicovariant 4D calculus inducing on
$S^{2}_{q}$ a left-covariant 3d calculus, are presented in \cite{durII}.
The general problem of finding the conditions between the differential calculi on a base space algebra and on a `structure' group, in a way giving a principal bundle structure with compatible calculi and a consistent definition of connections on it has been deeply studied \cite{bm98,ps97,dur98,haj97}. The slightly different perspective of this digression is to follow the path reviewed in appendix~\ref{ap:qpb}, namely to recall from \cite{gs10} the formulation of a Hopf bundle on the standard Podle\'s sphere starting from the 4D bicovariant calculus \`a la Woronowicz on the total space $\SU$, in order to fully describe the set of its connections. The first step in this analysis consists in describing how the differential calculus on $\SU$ naturally induces a 1 dimensional bicovariant calculus on the structure group $\U(1)$, and in which sense these two calculi are compatible.
\subsection{A 1D bicovariant calculus on $\U(1)$}
\label{se:cu1}
The Hopf projection \eqref{qprp} allows one to define an ideal $\mathcal{Q}_{\U(1)}\subset\ker\varepsilon_{\U(1)}$ as the projection
$\mathcal{Q}_{\U(1)}=\pi(\mathcal{Q}_{\SU})$. Then $\mathcal{Q}_{\U(1)}$ is generated by the three elements
\begin{align*}
\xi_{1}&=(z^{2}-1)+q^{2}(z^{-2}-1), \\
\xi_{2}&=(q^{2}z+z^{-1}-(q^{3}+q^{-1}))(q^{2}z+z^{-1}-(1+q^{2})), \\
\xi_{3}&=(q^{2}z+z^{-1}-(q^{-1}+q^{3}))(z^{-1}-z),
\end{align*}
and, since $\mathrm{Ad}(\mathcal{Q}_{\U(1)})\subset\mathcal{Q}_{\U(1)}\otimes\mathcal{A}(\U(1))$, it corresponds to a bicovariant differential calculus on $\U(1)$. The identity
$$
-q(1+q^{4})^{-1}(1+q^{2}+q^{3}+q^{5})^{-1}\{(q^{6}-1)\xi_{3}+(1+q^{4})\xi_{2}-q^{2}(1+q^{2})\xi_{1}\}=(z-1)+q(z^{-1}-1)
$$
shows that $\xi=(z-1)+q(z^{-1}-1)$ is in $\mathcal{Q}_{\U(1)}$. By induction one also sees that
\begin{align}
&j>0:\qquad z^{j}(z-1)=\xi(\sum_{n=0}^{j-1}\,q^{n}z^{j-n})+q^{j}(z-1), \nonumber \\
&j<0:\qquad z^{-\abs{j}}(z-1)=-\xi(\sum_{n=1}^{\abs{j}-1}\,q^{-n}z^{n-\abs{j}})+q^{-\abs{j}}(z-1).
\label{idzj}
\end{align}
From these relations it is immediate to prove (as in \cite{gs10}) that there is a complex vector space isomorphism
$\ker\varepsilon_{\U(1)}/\mathcal{Q}_{\U(1)} \simeq\IC$. The differential calculus induced by $\mathcal{Q}_{\U(1)}$ is 1-dimensional, and the projection $\pi_{\mathcal{Q}_{\U(1)}}:\ker\varepsilon_{\U(1)}\to\ker\varepsilon_{\U(1)}/\mathcal{Q}_{\U(1)}$ can be written as
\begin{equation}
\label{lam4}
\pi_{\mathcal{Q}_{\U(1)}}:\quad z^{j}(z-1)\to q^{j}[z-1],
\end{equation}
on the vector space basis $\varphi(j)=z^{j}(z-1)$ in $\ker\varepsilon_{\U(1)}$, with notation $[z-1]\,\in\,\ker\varepsilon_{\U(1)}/\mathcal{Q}_{\U(1)}$. The projection \eqref{lam4} will be used later on to define connection 1-forms on the fibration.
As a basis element for the quantum tangent space $\mathcal{X}_{\mathcal{Q}_{\U(1)}}$ we take
\begin{equation}
X=L_z=\frac{K^{-2}-1}{q-q^{-1}}.
\label{X1}
\end{equation}
The $*$-Hopf algebras $\mathcal{A}(\U(1))$ and $\cu(1)\simeq\{K,K^{-1}\}$ are dually paired via the pairing, induced by the one in \eqref{ndp} between $\ASU$ and $\su$, with
\begin{equation}
\hs{K^{\pm}}{z}=q^{\mp\frac{1}{2}},\qquad\hs{K^{\pm}}{z^{-1}}=q^{\pm\frac{1}{2}},
\label{bp1}
\end{equation}
on the generators. Thus, the exterior derivative ${\rm d}:\mathcal{A}(\U(1))\to\Omega^{1}(\U(1))$ can be written,
for any $u\in\,\mathcal{A}(\U(1))$,
as $ {\rm d} u=(X\lt u)\ \theta $ on the left invariant basis 1-form $\theta\sim[z-1]$. On the generators of the coordinate algebra one has
\begin{equation}
{\rm d} z=\frac{q-1}{q-q^{-1}}\,z \ \theta,
\qquad {\rm d} z^{-1}=\frac{q^{-1}-1}{q-q^{-1}}\,z^{-1} \ \theta,
\label{dz1}
\end{equation}
so to have $\theta=(q-1)(q-q^{-1})^{-1}z^{-1}{\rm d} z$. From the coproduct $\Delta X=1\otimes X+X\otimes K^{-2}$ the
$\mathcal{A}(\U(1))$-bimodule structure in $\Omega^{1}(\U(1))$ is
$$
\theta \ z^{\pm}=q^{\pm}z^{\pm}\ \theta.
$$
\subsection{Connections on the principal bundle}\label{conHf}
The compatibility -- as described in App.~\ref{ap:qpb} and expressed by the exactness of the sequence \eqref{des} -- of the differential calculus $\U(1)$ presented above with the 4D differential calculus on $\SU$ presented in \S\ref{se:4dc}, has been proved in \cite{gs10}. As a consequence, collecting the various terms, the data
$$
\left(\ASU, \Asq, \mathcal{A}(\U(1)); \mathcal{N}_{\SU}=r^{-1}(\SU\otimes\mathcal{Q}_{\SU}), \mathcal{Q}_{\U(1)}\right)
$$
is a quantum principal bundle with the described calculi.
In order to obtain connections on this bundle, that is maps \eqref{si} splitting the sequence \eqref{des}, we need to compute the action of the map
$$\sim_{\mathcal{N}_{\SU}}:\Omega^{1}(\SU)\to\ASU\otimes(\ker\varepsilon_{\U(1)}/\mathcal{Q}_{\U(1)})$$ defined via the diagram \eqref{qdia}.
Since it is left $\ASU$-linear, we take as representative universal 1-forms corresponding to the left invariant 1-forms \eqref{om4} in $\Omega^1(\SU)$:
\begin{align*}
&\pi^{-1}_{\mathcal{N}_{\SU}}(\omega_{+})=(a\delta c-qc\delta a) \\
&\pi^{-1}_{\mathcal{N}_{\SU}}(\omega_{-})= (c^* \delta a^*- qa^*\delta c^*) \\
&\pi^{-1}_{\mathcal{N}_{\SU}}(\omega_{0})=\{a^{*}\delta a+c^{*}\delta c+ q(a\delta a^{*}+q^{2}c\delta c^{*})\}/ (q+1)\lambda_{1} \\%\in[\pi_{\mathcal{Q}_{\SU}}]^{-1}(\omega_{0}),
&\pi^{-1}_{\mathcal{N}_{\SU}}(\omega_{z})=a^{*}\delta a+c^*\delta c - (a\delta a^*+q^2 c\delta c^*) .
\end{align*}
On them the action of the canonical map \eqref{chimap} is found to be:
\begin{align*}
&\chi(a\delta c-qc\delta a)=(ac-qca)\otimes(z-1)=0 \\
&\chi(c^* \delta a^*- qa^*\delta c^*) = (c^*a^* - qa^*c^*)\otimes(z^{*}-1)=0 \\
&\chi\left((1+q)^{-1}\lambda_{1}^{-1}\{a^{*}\delta a+c^{*}\delta c+ q(a\delta a^{*}+q^{2}c\delta c^{*})\}\right)=1\otimes\{(z-1)+q(z^{-1}-1)\}=1\otimes\xi \\
&\chi(a^{*}\delta a+c^*\delta c - (a\delta a^*+q^2 c\delta c^*))=1\otimes(z-z^{-1})
\end{align*}
with $\xi\in\mathcal{Q}_{\SU}$ introduced in \S\ref{se:cu1}. From the isomorphism \eqref{lam4} one finally has:
\begin{align}
&\sim_{\mathcal{N}_{\SU}}(\omega_{\pm})= \sim_{\mathcal{N}_{\SU}}(\omega_{0})=0 \nonumber \\
&\sim_{\mathcal{N}_{\SU}}(\omega_{z})=1\otimes(1+q^{-1})[z-1].
\label{omzv}
\end{align}
{}From these one recovers $\Omega^{1}_{\mathrm{hor}}(\SU)=\ker\sim_{\mathcal{N}_{\SU}}$ with, using \eqref{bis2},
\begin{equation}
\label{le:hf}
\ker\sim_{\mathcal{N}_{\SU}}\simeq\ASU\{\omega_{\pm},\omega_{0}\}\simeq\{\omega_{\pm},\omega_{0}\}\ASU.
\end{equation}
\begin{rema} \label{horv}
\textup{
From \eqref{omzv}, for the generator $X=L_z$ in \eqref{X1} one gets that
$$
\widetilde{X}(\omega_z) = \langle{X},{\sim_{\mathcal{N}_{\SU}}(\omega_{z})}\rangle
=1 ,
$$
which identifies $L_{z}\in\,\mathcal{X}_{\mathcal{Q}}$ as a vertical vector for the fibration. In turn it is used to extend the notion of horizontality to higher order forms in $\Omega(\SU)$. One defines \cite{KS97} a contraction operator $i_{L_{z}}:\Omega^{k}(\SU)\to\Omega^{k-1}(\SU)$, giving $i_{L_{z}}(\omega_{\pm})=i_{L_{z}}(\omega_{0})=0$, and $i_{L_{z}}(\omega_{z})=1$ on 1-forms, so that $\ker i_{L_{z}}\simeq\Omega^{1}_{\mathrm{hor}}(\SU)$. Then one define
\begin{equation}
\Omega^{k}_{\mathrm{hor}}(\SU):=\left.\ker\right|_{\Omega^{k}(\SU)}i_{L_{z}},
\label{khor}
\end{equation}
that is the kernel of the contraction map when restricted to the bimodule of $k$-forms. It is then easy to show
that $\Omega^{k}_{\mathrm{hor}}(\SU)\simeq\ASU\Omega^{k}(S^{2}_{q})\ASU$.
}
\end{rema}
Given the explicit expression \eqref{omzv} for the canonical map compatible with the differential calculi we are using, and the $\mathcal{A}(\U(1))$-coaction
\begin{equation}
\delta_{R}^{(1)}\omega_{z}=\omega_{z}\otimes 1, \qquad
\delta_{R}^{(1)}\omega_{0}=\omega_{0}\otimes 1, \qquad
\delta_{R}^{(1)}\omega_{\pm}=\omega_{\pm}\otimes z^{\pm2},
\label{rifo}
\end{equation}
using the vector space basis $\varphi(j)$ in $\ker\varepsilon_{\U(1)}$ of \S\ref{se:cu1},
a connection \eqref{si} is given by
\begin{equation}
\tilde{\sigma}(\phi\otimes[\varphi(j)])=q^{-2j}(1+q^{-1})^{-1}\phi(\omega_{z}+\mathrm{a})
\label{si3}
\end{equation}
for any $\phi\in\,\ASU$ and any element $\mathrm{a}\in\,\Omega^1(S^{2}_{q})$. The projection $\Pi$ on vertical forms, associated to this connection turns out to be
\begin{align}
&\Pi(\omega_{\pm})=0=\Pi(\omega_{0}),\nonumber \\
&\Pi(\omega_{z})=\tilde{\sigma}(\sim_{\mathcal{N}_{\SU}}(\omega_{z}))=\tilde{\sigma}(1\otimes[\varphi(0)])=\omega_{z}+\mathrm{a},
\label{Pi3}
\end{align}
while the corresponding connection 1-form $\omega:\mathcal{A}(\U(1))\to\Omega^{1}(\SU)$ is given by
\begin{equation}
\omega(z^{n})=\tilde{\sigma}(1\otimes[z^{n}-1])
=q^{n/2}[\tfrac{n}{2}] (\omega_{z}+\mathrm{a}).
\label{ome3}
\end{equation}
Connections corresponding to $\mathrm{a}=s\omega_{0}$ with $s\in\,\IR$ were already considered in \cite{durII}.
The vertical projector \eqref{Pi3} allows one to define a covariant derivative
$$
\mathfrak{D}:\ASU\to\Omega^{1}_{\mathrm{hor}}(\SU),
$$
given (as usual) as the horizontal projection of the exterior derivative:
\begin{equation}
\label{mfD}
\mathfrak{D}\phi:=(1-\Pi){\rm d}\phi.
\end{equation}
Covariance here clearly refers to the right coaction of the structure group $\U(1)$ of the bundle, since it is that $\delta_{R}\phi=\phi\otimes z^{-n} \,\Leftrightarrow\,\delta_{R}^{(1)}(\mathfrak{D}\phi)=(\mathfrak{D}\phi)\otimes z^{-n}$. From \eqref{ome} the action of this operator can be written as
\begin{equation}
\label{mfDo}
\mathfrak{D}\phi={\rm d}\phi-\phi\wedge\omega(z^{-n})
\end{equation}
for any $\phi\in\,\mathcal{L}_{n}$. From the bimodule structure \eqref{bis2} it is easy to check that all the above connections are \emph{strong} connections in the sense of \cite{hajsc}.
\section{Gauged Laplacians on line bundles} \label{GLoLB}
Each $\Asq$-bimodule $\mathcal{L}_{n}$ defined in \eqref{libu} is a bimodule of co-equivariant elements in $\ASU$ for the right $\U(1)$-coaction \eqref{cancoa}, and as such can be thought of as a module of `sections of a line bundle' over the quantum sphere $S^{2}_{q}$. Without requiring any compatibility with additional structures, any $\mathcal{L}_{n}$ can be realized both as a projective right or left $\Asq$-module (of rank 1 and winding number $-n$). One of such structures is that of a connection
on the quantum principal bundle $\Asq\hookrightarrow\ASU$. By transporting the covariant derivative \eqref{mfD} on the principal bundle to a derivative on sections forces to break the symmetry between the left or the right $\Asq$-module realization of $\mathcal{L}_{n}$.
With the choice in
\S\ref{s:qsb} for the principal bundle, we need an isomorphism $\mathcal{L}_{n}\simeq\mathcal{F}_n$ with $\mathcal{F}_{n}$ a projective left $\Asq$-module \cite{HM98}. Given this identification, in \S\ref{se:lbss} we shall describe the complete equivalence between covariant derivatives on $\mathcal{F}_{n}$ (associated to the $3{\rm d}$ left covariant differential calculus over
$S^{2}_{q}$) and connections (as described in \S\ref{s:ccd}) on the principal bundle $\Asq\hookrightarrow\ASU$, corresponding to compatible $4D_{+}$ bicovariant calculus over $\SU$ and $3{\rm d}$ left covariant calculus over $S^{2}_{q}$. We shall then move to a family of gauged Laplacian operators on $\mathcal{F}_{n}$, obtained by coupling the Laplacian operator over the quantum sphere with a set of suitable gauge potentials. We
finally show that among them there is one whose action extends to $\mathcal{L}_n$ the action of the Laplacian \eqref{sfr} on $\mathcal{L}_0\simeqS^{2}_{q}$. As we noticed in \S\ref{se:L}, the action of the (right) Laplacian \eqref{sfr} on $S^{2}_{q}$ is given by the restriction of the action \eqref{erp4} of the (right) Laplacian $\Box^R_{\SU}$. Here we obtain that the action of such gauged Laplacian can be written in terms of the ungauged (right) Laplacian on $\SU$, in parallel to what happens on a classical principal bundle (see e.g. \cite[Prop.~5.6]{bgv}) and on the Hopf fibration of the sphere $S^{2}_{q}$ with calculi coming from the left covariant one on $\SU$ as shown in \cite{lareza,ale09}.
\subsection{Line bundles as projective left $\Asq$-modules}\label{se:lbss}
With $n\in\,\IZ$, we consider the projective left
$\Asq$-module $\mathcal{F}_{n}=(\Asq)^{\mid n\mid+1}\mathfrak{p}^{\left(n\right)}$, with projections \cite{BM98,HM98} (cf. also \cite{lareza})
\begin{equation}
\mathfrak{p}^{\left(n\right)}=\ket{\Psi^{\left(n\right)}}\bra{\Psi^{\left(n\right)}}
\label{dP},
\end{equation}
written in terms of elements
$\ket{\Psi^{\left(n\right)}}\in \ASU^{\mid n\mid+1}$ and their duals $\bra{\Psi^{\left(n\right)}}$ as follows. One has:
\begin{align}
&n\leq0:\qquad\ket{\Psi^{\left(n\right)}}_{\mu}=\sqrt{\alpha_{n,\mu}} ~ c^{\mid n\mid-\mu}a^{\mu}\,\in \mathcal{L}_{n}, \nonumber \\
&\mathrm{where}\qquad\alpha_{n,\mid n\mid}=1;\qquad\alpha_{n,\mu}=\prod\nolimits_{j=0}^{\mid n\mid-\mu-1}\left(\frac{1-q^{2\left(\mid n\mid-j\right)}}
{1-q^{2\left(j+1\right)}}\right), \quad \mu = 0, \ldots, \mid n\mid-1
\label{ketpi}
\end{align}
\begin{align}
&n\geq 0:\qquad\ket{\Psi^{\left(n\right)}}_{\mu}=\sqrt{\beta_{n,\mu}} ~ c^{* \mu}a^{* n-\mu}\,\in \mathcal{L}_{n}, \nonumber\\
&\mathrm{where} \qquad \beta_{n,0}=1;\qquad
\beta_{n,\mu}=q^{2\mu}\prod\nolimits_{j=0}^{\mu-1}\left(\frac{1-q^{-2\left(n-j\right)}}{1-q^{-2\left(j+1\right)}}\right),
\quad \mu = 1, \ldots, n
\label{ketni} .
\end{align}
The coefficients are chosen so that $\hs{\Psi^{\left(n\right)}}{\Psi^{\left(n\right)}}=1$, as a consequence $(\mathfrak{p}^{\left(n\right)})^{2}=\mathfrak{p}^{\left(n\right)}$. Also by construction it holds that $(\mathfrak{p}^{\left(n\right)})^{\dagger}=\mathfrak{p}^{\left(n\right)}$.
The isomorphism $\mathcal{L}_n \simeq \mathcal{F}_{n}=(\Asq)^{\mid n\mid+1}\mathfrak{p}^{\left(n\right)}$ is realized as follows.
Given any element in the free module $(\Asq)^{\mid n\mid +1}$ as $\bra{f}=(f_0, f_1,\dots, f_{\mid n\mid})$ with $f_{\mu} \in \Asq$, the definition
\begin{align*} \phi_f :=\hs{f}{ \Psi^{(n)} }
&= \sum\nolimits_{\mu=0}^{n} \sqrt{\alpha_{n,\mu}}\,c^{\mid n\mid -\mu}a^{\mu} f_{\mu}
\qquad \qquad \mathrm{for} \quad n \leq 0 , \\
&= \sum\nolimits_{\mu=0}^{\mid n\mid} \sqrt{\beta_{n,\mu}}\,c^{*\mu}a^{*n-\mu} f_{\mu}
\qquad \qquad \mathrm{for} \quad n \geq 0,
\end{align*}
gives the left $\Asq$-modules isomorphism:
\begin{equation}
\mathcal{L}_{n} ~\xrightarrow{~\simeq~}~ \mathcal{F}_{n}, \quad \phi_f \mapsto \bra{\sigma_f} = \phi_{f} \bra{\Psi^{(n)} }
= \bra{f}\mathfrak{p}^{\left(n\right)} ,
\label{iso1n}
\end{equation}
with inverse
$$
\mathcal{F}_{n} ~\xrightarrow{~\simeq~}~ \mathcal{L}_{n}, \quad \bra{\sigma_f} = \bra{f}\mathfrak{p}^{\left(n\right)} \mapsto \phi_f =\hs{f}{ \Psi^{(n)} }.
$$
\bigskip
Given the exterior algebra $(\Omega(S^{2}_{q}),{\rm d})$ on the quantum sphere we are considering,
a covariant derivative on the left $\Asq$-modules $\mathcal{F}_n$ is a $\IC$-linear map
\begin{equation}
\nabla:\Omega^k(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}\,\to\,\Omega^{k+1}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}
\label{cdev}
\end{equation}
that satisfies the \emph{left} Leibniz rule
$$
\nabla(\xi\otimes_{\Asq}\bra{\sigma})=({\rm d}\xi)\otimes_{\Asq}\bra{\sigma}+(-1)^m \xi\otimes_{\Asq} \nabla\bra{\sigma}
$$
for any $\xi\,\in\,\Omega^{m}(S^{2}_{q})$ and $\bra{\sigma}\,\in\,\Omega^{k}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}$.
The curvature associated to a covariant derivative is
$\nabla^{2}:\mathcal{F}_{n}\to\Omega^{2}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}$, that is $\nabla^{2}(\xi\,\bra{\sigma})=\xi\,\nabla^2(\bra{\sigma})=\xi\,F_{\nabla}(\bra{\sigma})$ with the last equality defining the curvature 2-form $F_{\nabla}\in\,\textup{End}_{\Asq}\Omega^{2}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}$.
Any covariant derivative -- an element in $C(\mathcal{F}_{n})$ -- and its curvature
can be written as
\begin{align}
\nabla\bra{\sigma}&=({\rm d}\bra{\sigma})\mathfrak{p}^{\left(n\right)}\,+(-1)^k\,\bra{\sigma}\mathrm{A^{(n)}},
\label{cocug}
\\
\nabla^{2}\bra{\sigma}&=\bra{\sigma}\{-{\rm d}\mathfrak{p}^{\left(n\right)}\wedge{\rm d}\mathfrak{p}^{\left(n\right)}\,+\,{\rm d}\mathrm{A^{(n)}}\,-\,\mathrm{A^{(n)}}\wedge\mathrm{A^{(n)}}\}\mathfrak{p}^{\left(n\right)} .
\label{curvg}
\end{align}
with $\bra{\sigma}\in\,\Omega^{k}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}$.
For the `gauge potential' $\mathrm{A^{(n)}}$ one has
\begin{equation}
\mathrm{A^{(n)}}=\mathfrak{p}^{\left(n\right)} \mathrm{A^{(n)}}=\mathrm{A^{(n)}}\mathfrak{p}^{\left(n\right)}=\ket{\Psi^{\left(n\right)}}\mathrm{a}^{(n)}\bra{\Psi^{\left(n\right)}}\,\in\,\mathrm{Hom}_{\Asq}(\mathcal{F}_n, \Omega^{1}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}),
\label{apA}
\end{equation}
with $\mathrm{a}^{(n)}\in\,\Omega^{1}(S^{2}_{q})$. The monopole (Grassmann) connection corresponds to $\mathrm{a}^{(n)}=0$.
In analogy with the identification \eqref{iso1n}, the covariant derivative $\nabla$ naturally induces an operator
$D:\mathcal{L}_{n}\to\mathcal{L}_{n}\otimes_{\Asq}\Omega^{1}(S^{2}_{q})$ that can be written as
\begin{equation}
D\phi:=(\nabla\bra{\sigma_\phi})\ket{\tilde{\Psi}^{\left(n\right)}}={\rm d}\phi-\phi\wedge\{\hs{\Psi^{\left(n\right)}}{{\rm d}\Psi^{\left(n\right)}}-\mathrm{a}^{(n)}\}.
\label{domeg}
\end{equation}
We refer to the 1-form
\begin{equation}
\label{coAp}
\Omega^{1}(\SU)\,\ni\,\varpi^{(n)}=\left(\hs{\Psi^{\left(n\right)}}{{\rm d}\Psi^{\left(n\right)}}-\mathrm{a}^{(n)}\right)
\end{equation}
as the connection 1-form of the gauge potential. It allows one to express the curvature as
\begin{equation}
F_{\nabla}=-\ket{\Psi^{\left(n\right)}}\left( {\rm d}\varpi^{(n)}+\varpi^{(n)}\wedge\varpi^{(n)} \right)\bra{\Psi^{\left(n\right)}}
\label{Fp}
\end{equation}
where $({\rm d}\varpi^{(n)}+\varpi^{(n)}\wedge\varpi^{(n)})\in\,\Omega^{2}(S^{2}_{q})$.
The covariant derivatives defined above on the left modules $\mathcal{F}_n$ fit in the general theory of connections on the quantum Hopf bundle as described in the \S\ref{conHf}: any covariant vertical projector, as in \eqref{Pi3}, induces a gauge potential $\mathrm{A^{(n)}}$ as in \eqref{apA}. The notion \eqref{coAp} of connection 1-form of a given gauge potential in $C(\mathcal{F}_{n})$ matches the notion \eqref{ome3} of connection 1-form $\omega:\mathcal{A}(\U(1))\to\Omega^{1}(\SU)$ on the Hopf bundle. From the $\Asq$-bimodule isomorphism $\mathcal{L}_{n}\otimes_{\Asq}\Omega^{1}(S^{2}_{q})\,\simeq\,\Omega^{1}_{\mathrm{hor}}(\SU)$ (see Remark~\ref{horv}), the matching amounts to equate the actions of the covariant derivative operators \eqref{domeg} and \eqref{mfD},
\begin{equation}
\forall\,\phi\in\,\mathcal{L}_{n}: \quad D\phi=\mathfrak{D}\phi\qquad\Leftrightarrow\qquad\varpi^{(n)}=\omega(z^{-n}).
\label{omom}
\end{equation}
From formula \eqref{ome3}, this correspondence can be written as
\begin{equation}
\mathrm{a}^{(n)}=\lambda_{n}\omega_{0}-\xi_{-n}\mathrm{a},
\label{figi}
\end{equation}
where the coefficients refer to the eigenvalue equations:
\begin{align}
&L_{z}\lt\ket{\Psi^{\left(n\right)}}:=\xi_{n}\ket{\Psi^{\left(n\right)}} \qquad\Rightarrow\qquad\xi_{n}=-q^{-\tfrac{n}{2}}\,[\frac{n}{2}]\nonumber \\
&L_{0}\lt\ket{\Psi^{\left(n\right)}}:=\lambda_{n}\ket{\Psi^{\left(n\right)}}\qquad\Rightarrow\qquad\lambda_{n}=[\frac{\mid n\mid}{2}] \label{lamni}
[\frac{\mid n\mid}{2}+1].
\end{align}
Finally, the equivalence \eqref{omom} allows one to introduce a covariant derivative
$$ D:\Omega^{k}_{\mathrm{hor}}(\SU)\to\Omega^{k+1}_{\mathrm{hor}}(\SU),
$$ thus extending to horizontal forms on the total space of the quantum Hopf bundle the covariant derivative operator on $\ASU$ as given in \eqref{mfD}. This follows the formulation described in \cite{hajsc}, since any connection on the principal bundle is strong. Upon defining
$$
\mathcal{L}_{n}^{(k)}:=\{\phi\in\,\Omega^{k}_{\mathrm{hor}}(\SU)\,:\,\delta_{R}^{(k)}\phi=\phi\otimes z^{-n}\},
$$
where $\delta_{R}^{(k)}$ is the natural right $\U(1)$-coaction on $\Omega^{k}(\SU)$, one obtains:
\begin{equation}
D\phi={\rm d}\phi-(-1)^{k}\phi\wedge\omega(z^{-n}).
\label{esD}
\end{equation}
A further extension to the whole exterior algebra $\Omega(\SU)$ is proposed in \cite{durII}: a generalisation of the analysis in \cite[\S9]{ale09} shows how this extension is far from being unique.
We restrict our analysis again to covariant derivatives $\nabla_{s}\ket{\sigma}$ in \eqref{cocug}
whose gauge potential and corresponding connection 1-form
are of the form:
\begin{align}
\mathrm{A}_{s}^{(n)}=s\ket{\Psi^{\left(n\right)}}\omega_{0}\bra{\Psi^{\left(n\right)}}, \qquad\qquad \varpi_{s}^{(n)}=\xi_{n}\omega_{z}\,+\,(\lambda_{n}-s)\omega_{0} ,
\label{osn}
\end{align}
for $s\in\IR$ and coefficients as in \eqref{lamni}, since they reduce in the classical limit to the monopole connection on line bundles associated to the classical Hopf bundle $\pi:S^3\to S^{2}$.
Relations \eqref{2fw} and \eqref{d2f} allow to compute the curvature 2-form \eqref{Fp} as
\begin{align*}
&{\rm d}\varpi_{s}^{(n)}=\left( (q+q^{-1})\xi_{n}+(s-\lambda_{n})\rho \right) \omega_+\wedge\omega_- \nonumber \\
&\varpi_{s}^{(n)}\wedge\varpi_{s}^{(n)}=(q-q^{-1})\xi_{n} \left( (q+q^{-1})\xi_{n}+(q-q^{-1})(s-\lambda_{n}) \right)
\omega_{+}\wedge\omega_{-}.
\end{align*}
\subsection{Gauged Laplacians}
In order to introduce an Hodge operator
\begin{equation}
\star^{\mathcal{R}}:\Omega^{k}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}\to\Omega^{3-k}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n},
\label{scR}
\end{equation}
we use the right $\Asq$-linear Hodge operator \eqref{cRo} on $\Omega(S^{2}_{q})$:
\begin{equation}
\star^{\mathcal{R}}(\xi\,\bra{\sigma}):=(\check{R}\xi)\bra{\sigma}
\label{scRe}
\end{equation}
so that a gauged Laplacian operator is defined as
$$
\Box^{\mathcal{R}}_{\nabla}:\mathcal{F}_{n}\to\mathcal{F}_{n},\qquad\qquad\qquad\Box^{\mathcal{R}}_{\nabla}\bra{\sigma}:=-\star^{\mathcal{R}}\nabla(\star^{\mathcal{R}}\nabla\bra{\sigma}) .
$$
Equivalently we have an operator on $\mathcal{L}_{n}\simeq\mathcal{F}_{n}$ via the left $\Asq$-modules isomorphism \eqref{iso1n}. With $\phi=\hs{\sigma}{\Psi^{\left(n\right)}}$, it holds that
\begin{equation}
\Box^{\mathcal{R}}_{\nabla}:\mathcal{L}_{n}\to\mathcal{L}_{n}, \qquad\qquad\qquad \Box^{\mathcal{R}}_{\nabla}\phi=(\Box^{\mathcal{R}}_{\nabla}\bra{\sigma})\ket{\Psi^{\left(n\right)}}.
\label{glp}
\end{equation}
With the family of connections \eqref{osn} and using the identities
\begin{align}
&\left(R_{\pm}\lt\,\bra{\sigma}\right)\ket{\Psi^{\left(n\right)}}=q^{-n}\,R_{\pm}\lt \phi ,\nonumber \\
&\left(R_{0}\lt\,\bra{\sigma}\right)\ket{\Psi^{\left(n\right)}}=q^{-n}\left(R_{0}-[\frac{\mid n\mid}{2}][1-\frac{\mid n\mid}{2}]\right)\lt\phi
\label{idL}
\end{align}
one readily computes:
\begin{equation}
\label{rfB}
\Box_{\nabla_{s}}^{\mathcal{R}}\phi=
q^{-2n}\left\{ \alpha\, ( q^2 R_{+}R_{-}+ R_{-}R_{+} ) +\nu\,(R_{0}+sq^{-n}-[\frac{\mid n\mid}{2}][1-\frac{\mid n\mid}{2}])^2\right\}\lt \phi.
\end{equation}
Finally, fixing the parameter to be
\begin{equation}
\label{sCr}
\tilde{s}^{\prime} (n)= q^{n} [\frac{\mid n\mid}{2}][1-\frac{\mid n\mid}{2}],
\end{equation}
the action of the gauged Laplacians extends, apart from a multiplicative factor depending on the label $n$, to elements in the line bundles $\mathcal{L}_{n}$ the action of the Laplacian operator \eqref{sfr} on the quantum sphere, that is,
\begin{equation}
q^{2n}\left(\Box^{\mathcal{R}}_{\nabla_{\tilde{s}^{\prime}}}\phi\right)
=\left\{ \alpha ( q^2 R_{+}R_{-}\,+\,R_{-}R_{+} )\,+\,\nu R^2_{0} \right\}\lt\phi.
\label{finr}
\end{equation}
As an operator on $\mathcal{L}_{n}$ we get
\begin{align}\label{zed}
\Box^{\mathcal{R}}_{\nabla_{s^{\prime}}} &= (2q \alpha\, L_0 K^{2}+\nu\,L_{0}^2) K^{-4} - q\alpha
\frac{(q+q^{-1})(K-K^{-1})^2}{(q-q^{-1})^2}\,K^{-2} \\
&=2q \alpha\, (C_{q} - [\tfrac{1}{2}]^{2}+\tfrac{1}{4} ) K^{-2}+\nu\,(C_{q} - [\tfrac{1}{2}]^{2}+\tfrac{1}{4} )^2) K^{-4} - q\alpha
\frac{(q+q^{-1})(K-K^{-1})^2}{(q-q^{-1})^2}\,K^{-2} \nonumber ,
\end{align}
having used the relation \eqref{casbis}. This relation is the counterpart of what happens on a classical principal bundle (see e.g. \cite[Prop.~5.6]{bgv}) and on the Hopf fibration of the sphere $S^{2}_{q}$ with calculi coming from the left covariant one on $\SU$ as shown in \cite{lareza,ale09}.
\section{Introduction}
We continue our program devoted to Laplacian operators on quantum spaces with the study of such operators on the quantum (standard) Podle\'s sphere $S^{2}_{q}$ and their coupling with gauge connections on the quantum principal $\U(1)$-fibration $\Asq\hookrightarrow\ASU$. While in \cite{lareza} one worked with a left $3D$ covariant differential calculus on $\SU$ and its restriction to the (unique) $2D$ left covariant differential calculus on the sphere $S^{2}_{q}$, in the present paper we use the somewhat more complicate $4D_{+}$ bicovariant calculus on $\SU$ introduced in \cite{wor89} and its restriction to a $3D$ left covariant calculus on the sphere $S^{2}_{q}$.
Laplacian operators on all Podle\'s spheres, related to the $4D_{+}$ bicovariant calculus on $\SU$ were already studied in \cite{poddc}. Our contribution to Laplacian operators comes from the use of Hodge $\star$-operators on both the manifold of $\SU$ and $S^{2}_{q}$ that we introduce by improving and diversifying on existing definitions.
We then move on to line bundles on the standard sphere $S^{2}_{q}$ and to a class of operators on such bundles that are `gauged' with the use of a suitable class of connections on the principal bundle $\Asq\hookrightarrow\ASU$ and of the corresponding covariant derivatives on (module of sections of) the line bundles. These gauged Laplacians are completely diagonalized and are split in terms of a Laplacian operator on the total space $\SU$ of the bundle minus vertical operators, paralleling what happens on a classical principal bundle (see e.g. \cite[Prop.~5.6]{bgv}) and on the Hopf fibration of the sphere $S^{2}_{q}$ with calculi coming from the left covariant one on $\SU$ as shown in \cite{lareza,ale09}
In \S\ref{s:qsb} we describe all we need of the principal fibration $\Asq\hookrightarrow\ASU$ and associated line bundles over $S^{2}_{q}$. We also give a systematic description of the differential calculi we are interested in, the
$4D_{+}$ bicovariant calculus on $\SU$ and its restriction to a $3D$ left covariant calculus on the sphere $S^{2}_{q}$.
A thoughtful construction of Hodge $\star$-dualities on $\SU$ is in \S\ref{s:Hop}, while that on $S^{2}_{q}$ is in
\S\ref{s:hl3}. These are used in \S\ref{se:L} for the definition of Laplacian operators. A digression on connections on the principal bundle and of covariant derivatives on the line bundles is in \S\ref{s:ccd} and the following
\S\ref{GLoLB} is devoted to the corresponding gauged Laplacian operators on modules of sections of the line bundles. To make the paper relatively self-contained it concludes with two appendices,
App.\ref{ass:a1} giving general facts on differential calculi on Hopf algebras and App.\ref{ap:qpb} concerning with general facts on quantum principal bundles endowed with connections.
We like to mention that besides the constructions in \cite{hec99} and \cite{kmt}, examples of Hodge operators on the exterior algebras of the quantum homogeneous $q$-Minkowski and $q$-Euclidean spaces -- satisfying a covariance requirement with respect to the action of the quantum groups $\mathrm{SO}_{q}(3,1)$ and $\mathrm{SO}_{q}(4)$ -- have been given in \cite{um94, majqe} using the formalism of braided geometry and with a construction of a $q$-epsilon tensor. On the exterior algebra over the quantum planes $\IR_{q}^{N}$ a Hodge operator has been studied in \cite{gf}.
\subsection*{Conventions and notations} When writing about connections and covariant derivatives we shall pay attention in keeping the two notions distinct: a connection will be a projection on a principal bundle while a covariant derivative will be an operator on section, both objects fulfilling suitable properties. For $q \neq 1$ the `$q$-number' is defined as
\begin{equation}
[x] = [x]_q := \frac{q^x - q^{-x}}{q - q^{-1}} ,
\label{eq:q-integer}
\end{equation}
for any $x \in \IR$. For a coproduct
$\Delta$ we use the Sweedler notation
$\Delta(x)=x_{(1)}\otimes x_{(2)}$, with implicit summation. This is iterated to $(\id\otimes\Delta)\circ\Delta(x) = (\Delta\otimes\id)\circ\Delta(x) = x_{(1)}\otimes x_{(2)}\otimes x_{(3)}$, and so on.
\subsection*{Acknowledgments}
We are grateful to S. Albeverio, L.S. Cirio and I. Heckenberger for comments and suggestions.
AZ thanks P. Lucignano for his help with Maple.
GL was partially supported by the Italian Project `Cofin08--Noncommutative Geometry, Quantum Groups and Applications'. AZ gratefully acknowledges the support of the Max-Planck-Institut f\"ur Mathematik in Bonn, the Hausdorff Zentrum f\"ur Mathematik der Universit\"at Bonn, the Stiftelsen Blanceflor Boncompagni-Ludovisi f\"odd Bildt (Stockholm), the I.H.E.S. (Bures sur Yvette, Paris).
\section{Prelude: calculi and line bundles on quantum spheres}\label{s:qsb}
We introduce the manifolds of the quantum group $\SU$ and its quantum homogeneous space $S^{2}_{q}$ -- the standard Podle\'s sphere. The corresponding inclusion $\Asq\hookrightarrow\ASU$ of the corresponding coordinate algebras is a (topological) quantum principal bundle. Following App.~\S\ref{ass:a1} we then equip $\ASU$ with a 4-dimensional bicovariant calculus, whose restriction gives a 3-dimensional left covariant calculus on $\Asq$.
\subsection{Spheres and bundles}
For the quantum group
$\SU$ its polynomial algebra $\ASU$ is the unital $*$-algebra generated by elements $a$ and $c$, with relations
\begin{align}
\label{derel}
& ac=qca\quad ac^*=qc^*a\quad cc^*=c^*c , \nonumber \\
& a^*a+c^*c=aa^*+q^{2}cc^*=1 .
\end{align}
In the limit $q \to1$ one recovers the commutative coordinate algebra on the group manifold
$\mathrm{SU(2)}$. The algebra $\ASU$ can be completed to a $C^*$-algebra in a usual way by considering all its admissible representations and the supremum (universal) norm on them \cite{wor87}.
For the sake of the present paper this is not necessary since we are interested in Laplacian operators on $\SU$ (and on its homogeneous space, the quantum sphere) and their spectra. Thus we only exhibit a vector space basis for $\ASU$ in \eqref{bsphi} below, giving an analogue of the classical Wigner $D$-functions for the $\mathrm{SU}(2)$ group, i.e. matrix elements of its unitary irreducible (co)-representations. Also, without loss of generality, the deformation parameter
$q\in\IR$ will be restricted to the interval $0<q<1$, the map $q \to q^{-1}$ giving isomorphic algebras.
If we use the matrix
$$ U =
\left(
\begin{array}{cc} a & -qc^* \\ c & a^*
\end{array}\right) ,
$$
whose being unitary is equivalent to relations \eqref{derel}, the Hopf algebra structure for $\ASU$ is given by coproduct,
antipode and counit:
$$
\Delta\, U = U \otimes U , \qquad S(U) = U^* , \qquad \eps(U) = 1 ,
$$
that is $\Delta(a)= a \otimes a - q c^* \otimes c$, and $\Delta(c)= c \otimes a + a^* \otimes c$;
$S(a)=a^*$ and $S(c)=-qc$; $\eps(a)=1$ and $\eps(c)=0$ and their $*$-conjugated relations.
\medskip
The quantum universal envelopping algebra $\su$ is the unital Hopf $*$-algebra
generated as an algebra by four elements $K^{\pm 1},E,F$ with $K K^{-1}=1=K^{-1}K$ and relations:
\begin{equation}
K^{\pm1}E=q^{\pm1}EK^{\pm1}, \qquad
K^{\pm1}F=q^{\mp1}FK^{\pm1}, \qquad
[E,F] =\frac{K^{2}-K^{-2}}{q-q^{-1}} .
\label{relsu}
\end{equation}
The $*$-structure is
$K^*=K, \, E^*=F $,
and the Hopf algebra structure is provided by coproduct
$$\Delta(K^{\pm1}) =K^{\pm1}\otimes K^{\pm1}, \quad
\Delta(E) =E\otimes K+K^{-1}\otimes E, \quad
\Delta(F)
=F\otimes K+K^{-1}\otimes F,$$
while the antipode is
$S(K) =K^{-1}, \,
S(E) =-qE, \,
S(F) =-q^{-1}F$
and the counit reads
$\varepsilon(K)=1, \,\varepsilon(E)=\varepsilon(F)=0$. The quadratic element
\begin{equation}
C_{q}=\frac{qK^2-2+q^{-1}K^{-2}}{(q-q^{-1})^2}+FE-\tfrac{1}{4}
\label{cas}
\end{equation}
is a quantum Casimir operator that generates the centre of $\su$.
\medskip
The Hopf $*$-algebras $\su$ and $\ASU$ are dually paired. The $*$-compatible bilinear mapping $\hs{~}{~}:\su\times\ASU\to\IC$ is on the generators given by
\begin{align}
&\langle K^{\pm1},a\rangle=q^{\mp 1/2}, \qquad \langle K^{\pm1},a^*\rangle=q^{\mp 1/2}, \nonumber\\
&\langle E,c\rangle=1, \qquad \langle F,c^*\rangle=-q^{-1}, \label{ndp}
\end{align}
with all other couples of generators pairing to zero. This pairing is proved \cite{KS97} to be non-degenerate. The algebra $\su$ is recovered as a $*$-Hopf subalgebra in the dual algebra $\ASU^o$, the largest Hopf $*$-subalgebra contained in the vector space dual $\ASU^{\prime}$.
There are \cite{wor87} $*$-compatible canonical commuting actions of $\su$ on $\ASU$:
$$
h \lt x := \co{x}{1} \,\hs{h}{\co{x}{2}}, \qquad
x {\triangleleft} h := \hs{h}{\co{x}{1}}\, \co{x}{2}.
$$
On powers of generators one computes, for $s\in\,\IN$, that
\begin{equation}
\label{lact}
\begin{array}{lll}
K^{\pm1}\triangleright a^{s} =q^{\mp\frac{s}{2}}a^{s} &
F\triangleright a^{s} =0 &
E\triangleright a^{s} =-q^{(3-s)/2} [s] a^{s-1} c^{*} \\
K^{\pm1}\triangleright a^{* s} =q^{\pm\frac{s}{2}}a^{* s} &
F\triangleright
a^{*s} =q^{(1-s)/2} [s] c a^{* s-1} &
E\triangleright a^{* s} =0 \\
K^{\pm1}\triangleright c^{s} =q^{\mp\frac{s}{2}}c^{s} &
F\triangleright c^{s} =0 &
E\triangleright c^{s} =q^{(1-s)/2} [s] c^{s-1} a^* \\
K^{\pm1}\triangleright c^{* s} =q^{\pm\frac{s}{2}}c^{* s} &
F\triangleright c^{*s} =-q^{-(1+s)/2} [s] a c^{*s-1} &
E\triangleright c^{* s} =0;
\end{array}
\end{equation}
and:
\begin{equation}
\label{ract}
\begin{array}{lll}
a^{s}\triangleleft K^{\pm1} =q^{\mp\frac{s}{2}}a^{s} &
a^{s}\triangleleft F =q^{(s-1)/2} [s] c a^{s-1} &
a^{s}\triangleleft E =0 \\
a^{* s}\triangleleft K^{\pm1} =q^{\pm\frac{s}{2}}a^{* s} &
a^{* s}\triangleleft F =0 &
a^{*s}\triangleleft E =-q^{(3-s)/2} [s] c^{*}a^{*s-1} \\
c^{s}\triangleleft K^{\pm1} =q^{\pm\frac{s}{2}}c^{s} &
c^{s}\triangleleft F =0 &
c^{s}\triangleleft E =q^{(s-1)/2} [s] c^{s-1} a \\
c^{* s}\triangleleft K^{\pm1} =q^{\mp\frac{s}{2}}c^{* s} &
c^{* s}\triangleleft F =-q^{(s-3)/2} [s] a^{*}c^{*s-1} &
c^{* s}\triangleleft E =0.
\end{array}
\end{equation}
\medskip
Consider the algebra $\mathcal{A}(\U(1)):=\IC[z,z^*] \big/ \!\!<zz^* -1>$. The map
\begin{equation} \label{qprp}
\pi: \ASU \, \to\,\mathcal{A}(\U(1)) , \qquad
\pi\,\left(
\begin{array}{cc} a & -qc^* \\ c & a^*
\end{array}\right):=
\left(
\begin{array}{cc} z & 0 \\ 0 & z^*
\end{array}\right)
\end{equation}
is a surjective Hopf $*$-algebra homomorphism. As a consequence, $\U(1)$
is a quantum subgroup of $\SU$ with right coaction:
\begin{equation}
\delta_{R}:= (\id\otimes\pi) \circ \Delta \, : \, \ASU \,\to\,\ASU \otimes
\mathcal{A}(\U(1)) . \label{cancoa}
\end{equation}
The coinvariant elements for this
coaction,
elements $b\in\ASU$ for which $\delta_{R}(b)=b\otimes 1$,
form the algebra of the standard Podle\'s sphere $\Asq\hookrightarrow\ASU$. This inclusion gives a topological quantum principal bundle, following the formulation reviewed in appendix~\S\ref{ap:qpb}.
\medskip
The above right $\U(1)$ coaction on $\SU$ is dual to the left action of the element $K$, and allows one \cite{maetal} to give a decomposition
\begin{equation}\label{dcmp}
\ASU=\oplus_{n\in\IZ} \mathcal{L}_{n}
\end{equation}
in terms of $\Asq$-bimodules defined by
\begin{equation}
\label{libu}
\mathcal{L}_{n} := \{x \in \ASU ~:~ K \lt x = q^{n/2} x\quad\Leftrightarrow\quad\delta_{R}(x)=x\otimes z^{-n}\},
\end{equation}
with $\Asq = \mathcal{L}_{0}$. It is easy to see (cf. \cite[Prop. 3.1]{mgw}) that $\mathcal{L}_{n}^{*} = \mathcal{L}_{-n}$ and $\mathcal{L}_{n}\mathcal{L}_{m} = \mathcal{L}_{n+m}$. Also
\begin{equation}
E \lt \mathcal{L}_{n} \subset \mathcal{L}_{n+2}, \qquad
F \lt \mathcal{L}_{n} \subset \mathcal{L}_{n-2}, \qquad
\mathcal{L}_{n} {\triangleleft} u \subset \mathcal{L}_{n},
\label{rellb}
\end{equation}
for any $u\in \su$. The bimodules $\mathcal{L}_{n}$ will be described at length later on when we endow them with connections.
Here we only mention that the bimodules $\mathcal{L}_{n}$ have a vector space decomposition (cf. e.g. \cite{maj95}):
\begin{equation}
\label{decoln}
\mathcal{L}_{n}:=\bigoplus_{J=\tfrac{|n|}{2}, \tfrac{|n|}{2} +1,
\tfrac{|n|}{2} +2, \cdots}V_{J}^{\left(n\right)},
\end{equation}
where $V_{J}^{\left(n\right)}$ is the spin
$J$ (with $J\in{\tfrac{1}{2}}\IN$) irreducible $*$-re\-pre\-sen\-ta\-tion spaces for the right action of $\su$, and basis elements
\begin{equation}
\phi_{n,J,l}= (c^{J-n/2} a^{*J+n/2}){\triangleleft} E^l
\label{bsphi}
\end{equation}
with $n\in\IZ,\,J=\tfrac{|n|}{2}+\IN, \,l=0,\ldots,2J$.
\subsection{The 4D exterior algebra over the quantum group $\SU$}\label{se:4dc}
Following the formulation reviewed in App.\ref{ass:a1}, we present here the exterior algebra over the so called $4D_{+}$ bicovariant calculus on $\SU$, which was introduced as a first order differential calculus in \cite{wor89}, and described in details in \cite{sta}.
One needs an ideal $\mathcal{Q}_{\SU}\subset\ker\varepsilon_{\SU}$. The one corresponding to the $4D_+$ calculus is generated by the nine elements $\{c^{2}; \, c(a^{*}-a); \, q^{2}a^{*2}-(1+q^{2})(aa^{*}-cc^{*})+a^{2}; \, c^{*}(a^{*}-a); \, c^{*2}; \, [q^{2}a+a^{*}-q^{-1}(1+q^{4})]c; \,
[q^{2}a+a^{*}-q^{-1}(1+q^{4})](a^{*}-a); \, [q^{2}a+a^{*}-q^{-1}(1+q^{4})]c^{*}; \, [q^{2}a+a^{*}-q^{-1}(1+q^{4})][q^{2}a+a^{*}-(1+q^{2})]\}$. One has $\mathrm{Ad}(\mathcal{Q}_{\SU})\subset\mathcal{Q}_{\SU}\otimes\ASU$ and $\dim(\ker\varepsilon_{\SU}/{\mathcal{Q}_{\SU}})=4$. The associated quantum tangent space as in \eqref{qTv} turns out to be a four dimensional $\mathcal{X}_{\mathcal{Q}}\subset\su$. A choice for a basis is given by the elements
\begin{align}
& L_{-}=q^{\frac{1}{2}}FK^{-1},\qquad L_{z}=\frac{K^{-2}-1}{q-q^{-1}},\qquad L_{+}=q^{-\frac{1}{2}}EK^{-1}; \nonumber \\
& L_{0}=\frac{q(K^{2}-1)+q^{-1}(K^{-2}-1)}{(q-q^{-1})^{2}}\,+FE=
\frac{q(K^{-2}-1)+q^{-1}(K^{2}-1)}{(q-q^{-1})^{2}}\,+EF,
\label{Lq}
\end{align}
from the last commutation rule in \eqref{relsu}. The vector $L_{0}$ belongs to the centre of $\su$: it differs from the quantum Casimir \eqref{cas} by a constant term,
\begin{equation}\label{casbis}
C_{q}=L_{0}+\left(\frac{q^{\frac{1}{2}}-q^{-\frac{1}{2}}}{q-q^{-1}}\right)^{2}-\tfrac{1}{4}=L_{0}+[\tfrac{1}{2}]^{2}-\tfrac{1}{4}.
\end{equation}
The coproducts of the basis \eqref{cpuh} give $\Delta L_{b}=1\otimes L_{b}+\sum_a L_{a}\otimes f_{ab}$: once chosen the ordering $(-,z,+,0)$, such a tensor product can be represented as a row by column matrix product where
\begin{equation}
f_{ab}=\left(\begin{array}{cccc} 1 & 0 & 0 & q^{-\frac{1}{2}}KE \\
(q-q^{-1})q^{\frac{1}{2}} FK^{-1} & K^{-2} & (q-q^{-1})q^{-\frac{1}{2}} EK^{-1} & (q-q^{-1})[FE+q^{-1}\,\frac{K^{-2}-K^{2}}{(q-q^{-1})^{2}}] \\ 0 & 0 & 1 & q^{-\frac{1}{2}} FK \\ 0 & 0 & 0 & K^{2}
\end{array}
\right).
\label{fab}
\end{equation}
The differential ${\rm d}:\ASU\mapsto\Omega^{1}(\SU)$ in \eqref{ded} is written for any $x\in \ASU$ as
\begin{equation}
{\rm d} x=\sum_a (L_{a}\lt x)\omega_{a}=\sum_a \omega_{a}(R_{a}\lt x)
\label{d4}
\end{equation}
on the dual basis of left invariant forms $\omega_{a}\in\Omega^{1}(\SU)$ with $\Delta_{L}^{(1)}(\omega_{a})=1\otimes\omega_{a}$. Here $R_{a}:=-S^{-1}(L_{a})$ and explicitly:
\begin{equation}
\label{Rder}
R_{\pm}=L_{\pm}K^2, \quad\quad R_{z}=L_{z}K^2, \quad\quad R_{0}=-L_{0}.
\end{equation}
On the generators of the algebra the differential acts as:
\begin{align}
&{\rm d} a=(q-q^{-1})^{-1}(q-1)a\omega_{z}-qc^{*}\omega_{+}+\lambda_{1} a\omega_{0}, \nonumber \\
&{\rm d} a^{*}=c\omega_{-}+(q-q^{-1})^{-1}(q^{-1}-1)a^{*}\omega_{z}+\lambda_{1} a^{*}\omega_{0}, \nonumber \\
&{\rm d} c=(q-q^{-1})^{-1}(q-1)c\omega_{z}+a^{*}\omega_{+}+\lambda_{1} c\omega_{0}, \nonumber \\
&{\rm d} c^{*}=-q^{-1}a\omega_{-}+(q-q^{-1})^{-1}(q^{-1}-1)c^{*}\omega_{z}+\lambda_{1} c^{*}\omega_{0},
\label{dcf}
\end{align}
with $\lambda_1= [{\tfrac{1}{2}}][\frac{3}{2}]$.
These relations can be inverted, giving
\begin{align}
&\omega_{-}=c^{*}{\rm d} a^{*}-qa^{*}{\rm d} c^{*}, \qquad \omega_{+}=a{\rm d} c-qc{\rm d} a, \nonumber \\
&\omega_{z}=a^{*}{\rm d} a+c^{*}{\rm d} c-(a{\rm d} a^{*}+q^{2}c{\rm d} c^{*}),\nonumber \\
&\omega_{0}=(1+q)^{-1}\lambda^{-1}_{1}[a^{*}{\rm d} a+c^{*}{\rm d} c+q(a{\rm d} a^{*}+q^{2}c{\rm d} c^{*})].
\label{om4}
\end{align}
It is then easy to see that for $q\to 1$ one has $\omega_{0}\to 0$. This differential calculus reduces in the classical limit to the standard three-dimensional bicovariant calculus on $\mathrm{SU(2)}$.
This first order differential $4D_{+}$ calculus is a $*$-calculus: the $*$-structure on $\ASU$ is extended to an antilinear $*$-structure on $\Omega^{1}(\SU)$, such that $({\rm d} x)^*={\rm d} (x^*)$ for any $x\in \ASU$. For the basis of left invariant 1-forms is just
\begin{equation}
\omega_{-}^{*}=-\omega_{+},\qquad\omega_{z}^{*}=-\omega_{z},\qquad\omega_{0}^{*}=-\omega_{0}.
\label{ss}
\end{equation}
From \eqref{bi-struct} one works out the bimodule structure of the calculus, obtaining:
\begin{equation}
\begin{array}{lll}
\omega_{-}a=a\omega_{-}-qc^{*}\omega_{0},\qquad &\omega_{+}a=a\omega_{+},\qquad & \omega_{0}a=q^{-1}a\omega_{0}, \\
\omega_{-}a^{*}=a^{*}\omega_{-}, \qquad &\omega_{+}a^{*}=a^{*}\omega_{+}+c\omega_{0}, \qquad &\omega_{0}a^{*}=qa^{*}\omega_{0}, \\
\omega_{-}c=c\omega_{-}+a^{*}\omega_{0}, \qquad &\omega_{+}c=c\omega_{+}, \qquad & \omega_{0}c=q^{-1}c\omega_{0}, \\
\omega_{-}c^{*}=c^{*}\omega_{-}, \qquad &\omega_{+}c^{*}=c^{*}\omega_{+}-q^{-1}a\omega_{0}, \qquad & \omega_{0}c^{*}=qc^{*}\omega_{0};
\end{array}
\label{biuno}
\end{equation}
as well as:
\begin{align}
&\omega_{z}a=qa\omega_{z}-q(q-q^{-1})c^{*}\omega_{+}+qa\omega_{0}, \nonumber \\
&\omega_{z}a^{*}=(q-q^{-1})c\omega_{-}+q^{-1}a^{*}\omega_{z}-q^{-1}a^{*}\omega_{0}, \nonumber \\
& \omega_{z}c=qc\omega_{z}+(q-q^{-1})a^{*}\omega_{+}+qc\omega_{0}, \nonumber \\
& \omega_{z}c^{*}=-q^{-1}(q-q^{-1})a\omega_{-}+q^{-1}c^{*}\omega_{z}-q^{-1}c^{*}\omega_{0}.
\label{bidue}
\end{align}
The $\ASU$-bicovariant bimodule $\Omega^{2}(\SU)$ of exterior 2-forms is defined by the projection given in \eqref{wedk}, with $\mathcal{S}_{\mathcal{Q}}^{(2)}=\ker\,\mathfrak{A}^{(2)}=\ker\,(1-\sigma)\subset\Omega^{1}(\SU)^{\otimes2}$. This necessitates computing the braiding as in \eqref{sigco}, a preliminary step being the computation as in \eqref{ri-co-form} of the right coaction on the left invariant basis forms, $\Delta_{R}^{(1)}(\omega_{a})=\sum_{b}\omega_{b}\otimes J_{ba}$. For the calculus at hand:
\begin{equation}
J_{ba}=\left(\begin{array}{cccc}
a^{*2} & (1+q^{2})a^{*}c & -qc^{2} & (1-q^{2})a^{*}c \\
-qa^{*}c^{*} & aa^{*}-cc^{*} & -ac & (q^{2}-1)cc^{*} \\
-qc^{*2} & (q+q^{-1})ac^{*} & a^{2} & (q^{-1}-q)ac^{*} \\
0 & 0 & 0 & 1
\end{array}
\right).
\label{Jba}
\end{equation}
The braiding map $\sigma:\Omega^{1}(\SU)^{\otimes2}\to\Omega^{1}(\SU)^{\otimes2}$ is then worked out \cite{cla10} to be:
\begin{align}
&\sigma(\omega_{-}\otimes\omega_{-})=\omega_{-}\otimes\omega_{-}, \qquad \sigma(\omega_{+}\otimes\omega_{+})=\omega_{+}\otimes\omega_{+}, \qquad \sigma(\omega_{0}\otimes\omega_{0})=\omega_{0}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{z}\otimes\omega_{z})=\omega_{z}\otimes\omega_{z}+(q^{2}-q^{-2})(\omega_{z}\otimes\omega_{0}+\omega_{-}\otimes\omega_{+}-\omega_{+}\otimes\omega_{-}), \nonumber \\
&\sigma(\omega_{-}\otimes\omega_{+})=\omega_{+}\otimes\omega_{-}-\omega_{z}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{+}\otimes\omega_{-})=\omega_{-}\otimes\omega_{+}+\omega_{z}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{-}\otimes\omega_{z})=\omega_{z}\otimes\omega_{-}+(1+q^{2})\omega_{-}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{z}\otimes\omega_{-})=(1-q^{-2})\omega_{z}\otimes\omega_{-}+q^{-2}\omega_{-}\otimes\omega_{z}-(1+q^{-2})\omega_{-}\otimes\omega_{0},\nonumber \\
&\sigma(\omega_{-}\otimes\omega_{0})=\omega_{0}\otimes\omega_{-}+(1-q^{2})\omega_{-}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{0}\otimes\omega_{-})=q^{2}\omega_{-}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{z}\otimes\omega_{+})=q^{2}\omega_{+}\otimes\omega_{z}+(1-q^{2})\omega_{z}\otimes\omega_{+}+(1+q^{2})\omega_{+}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{+}\otimes\omega_{z})=\omega_{z}\otimes\omega_{+}-(1+q^{-2})\omega_{+}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{z}\otimes\omega_{0})=\omega_{0}\otimes\omega_{z}+(q-q^{-1})^{2}(\omega_{+}\otimes\omega_{-}-\omega_{-}\otimes\omega_{+})-(q-q^{-1})^{2}\omega_{z}\otimes\omega_{0}, \nonumber \\
& \sigma(\omega_{0}\otimes\omega_{z})=\omega_{z}\otimes\omega_{0}, \nonumber \\
&\sigma(\omega_{+}\otimes\omega_{0})=\omega_{0}\otimes\omega_{+}+(1-q^{-2})\omega_{+}\otimes\omega_{0},\nonumber \\
&\sigma(\omega_{0}\otimes\omega_{+})=q^{-2}\omega_{+}\otimes\omega_{0}.
\label{sig}
\end{align}
Using the general construction of App.\ref{ass:a1}, the $q$-wedge product on 1-forms is defined as $\theta\wedge\theta^{\prime}\,=\,\mathfrak{A}^{(2)}(\theta\otimes\theta^{\prime})\,
=\,(1-\sigma) (\theta\otimes\theta^{\prime})\,\subset\,\mathrm{Range}\,\mathfrak{A}^{(2)}$. On generators:
\begin{align}
&\omega_{-}\wedge\omega_{-}=\omega_{+}\wedge\omega_{+}=\omega_{0}\wedge\omega_{0}=0, \nonumber \\
&\omega_{z}\wedge\omega_{z}-(q^{2}-q^{-2})\omega_{+}\wedge\omega_{-}=0, \nonumber \\
& \omega_{z}\wedge\omega_{\pm}+q^{\pm 2}\omega_{
\pm}\wedge\omega_{z}=0, \nonumber \\
&\omega_{\pm}\wedge\omega_{0}+\omega_{0}\wedge\omega_{\pm}=0,\nonumber \\
&\omega_{+}\wedge\omega_{-}+\omega_{-}\wedge\omega_{+}=0, \nonumber \\
&\omega_{z}\wedge\omega_{0}+\omega_{0}\wedge\omega_{z}-(q-q^{-1})^{2}\omega_{-}\wedge\omega_{+}=0 .
\label{2fw}
\end{align}
These relations show that $\dim\Omega^{2}(\SU)=6$. The exterior derivative on basis 1-forms results into:
\begin{align}
&{\rm d}\omega_{\pm}= \mp\, q^{\pm 1}\omega_{-}\wedge\omega_{z}, \nonumber \\
&{\rm d}\omega_{z}=(q+q^{-1})\omega_{+}\wedge\omega_{-}, \nonumber \\
&{\rm d}\omega_{0}=(q-q^{-1})\,\omega_{-}\wedge\omega_{+}.
\label{d2f}
\end{align}
The antisymmetriser operator $\mathfrak{A}^{(2)}:\Omega^{2}(\SU)\to\Omega^{2}(\SU)$
has a natural spectral decomposition. This is what we need later on to introduce Hodge operators. A more general analysis of the spectral properties of the antisymmetriser operators associated to a class of bicovariant differential calculi over $\mathrm{SL}_{q}(N)$ (for $N\geq2$) is in \cite{sch99}.
On the basis
\begin{equation}
\label{p12}
\begin{array}{lll}
\varphi_{0}=\omega_{-}\wedge\omega_{0},& & \varphi_{z}=\omega_{-}\wedge\omega_{0}+(1-q^{-2})\omega_{-}\wedge\omega_{z} \\
\psi_{0}=\omega_{+}\wedge\omega_{0}, & &\psi_{z}=\omega_{+}\wedge\omega_{0}-(1-q^2)\omega_{+}\wedge\omega_{z} \\
\psi_{\pm}=\omega_{0}\wedge\omega_{z}+(1-q^{\pm 2})\omega_{-}\wedge\omega_{+}, & &
\end{array}
\end{equation}
which is such that $\varphi_{0}^*=\psi_{0}$, $\varphi_{z}^*=\psi_{z}$ and $\psi_{-}^*=\psi_{+}$, it holds that
\begin{equation}
\label{p14}
\begin{array}{lll}
\mathfrak{A}^{(2)}(\varphi_{0})=(1+q^2)\varphi_{0}, &
\mathfrak{A}^{(2)}(\psi_{z})=(1+q^2)\psi_{z}, &
\mathfrak{A}^{(2)}(\psi_{+})=(1+q^2)\psi_{+} \\
\mathfrak{A}^{(2)}(\varphi_{z})=(1+q^{-2})\varphi_{z}, &
\mathfrak{A}^{(2)}(\psi_{0})=(1+q^{-2})\psi_{0}, &
\mathfrak{A}^{(2)}(\psi_{-})=(1+q^{-2})\psi_{-}.
\end{array}
\end{equation}
For later use we shall adopt the labelling $\xi_{(\pm)}\in\,\mathcal{E}_{(\pm)}$ with
\begin{equation}\label{p12bis}
\mathcal{E}_{(+)}=\{\varphi_{0}, \psi_{z}, \psi_{+}\}, \quad
\mathrm{and} \qquad \mathcal{E}_{(-)}=\{\varphi_{z},\psi_{0}, \psi_{-}\} .
\end{equation}
By proceeding further, the $\ASU$-bimodule $\Omega^{3}(\SU)$ is found to be 4-dimensional with left invariant basis elements:
\begin{equation}
\label{p13}
\begin{array}{lll}
\chi_{-}=\omega_{+}\wedge\omega_{0}\wedge\omega_{z},& &
\chi_{+}=\omega_{-}\wedge\omega_{0}\wedge\omega_{z} \\
\chi_{0}=\omega_{-}\wedge\omega_{+}\wedge\omega_{z}, & &
\chi_{z}=\omega_{-}\wedge\omega_{+}\wedge\omega_{0},
\end{array}
\end{equation}
with $\chi_{-}^*=-q^{-2}\chi_{+}$, $\chi_{0}^*=\chi_{0}$ and $\chi_{z}^*=\chi_{z}$.
These exterior forms are closed,
\begin{equation}
{\rm d}\chi_{a}=0, \label{d3f}
\end{equation}
and in addition satisfy
\begin{equation}
\mathfrak{A}^{(3)}(\chi_{a})=2(1+q^2+q^{-2})\chi_{a}
\label{p15-}
\end{equation}
for $a=-,+,z,0$, thus providing the spectral decomposition for the antisymmetriser operator $\mathfrak{A}^{(3)}:\Omega^3(\SU)\to\Omega^3(\SU)$.
The $\ASU$-bimodule $\Omega^{4}(\SU)$ of top forms ($\Omega^{k}(\SU)=\emptyset$ for $k>4$) is 1 dimensional. Its left invariant basis element $\mu=\omega_{-}\wedge\omega_{+}\wedge\omega_{z}\wedge\omega_{0}$ is central, i.e. $x\,\mu=\mu\,x$ for any $x\in\,\ASU$ and its eigenvalue for the action of the antisymmetriser is
\begin{equation}
\label{p15}
\mathfrak{A}^{(4)}(\mu)=2(q^4+2q^2+6+2q^{-2}+q^{-4})\mu.
\end{equation}
\subsection{The exterior algebra over the quantum sphere $S^{2}_{q}$}\label{se:cals2}
The restriction of the first order $4D_+$ bicovariant calculus endows the sphere $S^{2}_{q}$ with a first order left covariant 3-dimensional calculus \cite{ap94,poddc}.
The exterior algebra $\Omega(S^{2}_{q})$ can be characterised in terms of some of the bimodules $\mathcal{L}_{n}$ introduced in \S\ref{s:qsb}.
Given $f\in \Asq\simeq\mathcal{L}_{0}$, the exterior derivative ${\rm d}:\Asq\mapsto\Omega^{1}(S^{2}_{q})$ from \eqref{d4} reduces to:
\begin{equation}
{\rm d} f=(L_{-}\lt f)\omega_{-}+(L_{+}\lt f)\omega_{+}+(L_{0}\lt f)\omega_{0}.
\label{d3d}
\end{equation}
Notice that the basis 1-forms $\{\omega_{a}, a=-,+,0\}$ are graded commutative (cf. \eqref{2fw}). Furthermore, relation \eqref{rellb} shows that $(L_{\pm}\lt f)\in \mathcal{L}_{\pm2}$ and that $(L_{0}\lt f)\in \mathcal{L}_{0}$, while the $\ASU$-bimodule structure of $\Omega^{1}(\SU)$ described by the coproduct \eqref{fab} of the quantum derivations $L_{a}$ gives:
\begin{equation}
\begin{array}{ll}
\phi\,\omega_{-}=\omega_{-}\,\phi-q^{-1}\omega_{0}(L_{+}\lt \phi), & \qquad\qquad \omega_{-}\phi=\phi\,\omega_{-}+q(L_{+}K^{2}\lt \phi)\omega_{0}, \\
\phi^{\prime}\omega_{+}=\omega_{+}\phi^{\prime}-q\omega_{0}(L_{-}\lt\phi^{\prime}), & \qquad\qquad \omega_{+}\phi^{\prime}=\phi^{\prime}\omega_{+}+q^{-1}(L_{-}K^{2}\lt\phi^{\prime})\omega_{0}, \\
\phi^{\prime\prime}\omega_{0}=\omega_{0}(K^{-2}\lt\phi^{\prime\prime}), & \qquad\qquad \omega_{0}\phi^{\prime\prime}=(K^{2}\lt\phi^{\prime\prime})\omega_{0}.
\end{array}
\label{bis2}
\end{equation}
These identities are valid for any $\phi,\phi^{\prime},\phi^{\prime\prime}\in \ASU$.
They allow to prove by explicit calculations the following identities:
\begin{align}
&\phi\in \mathcal{L}_{-2}:\qquad{\rm d}(\phi\,\omega_{-})=(L_{+}\lt \phi)\omega_{+}\wedge\omega_{-}+(L_{0}\lt \phi)\omega_{0}\wedge\omega_{-}, \nonumber \\
&\phi^{\prime}\in \mathcal{L}_{2}:\qquad {\rm d}(\phi^{\prime}\omega_{+})=(L_{-}\lt\phi^{\prime})\omega_{-}\wedge\omega_{+}+(L_{0}\lt\phi^{\prime})\omega_{0}\wedge\omega_{+}, \nonumber \\
&\phi^{\prime\prime}\in\mathcal{L}_{0}:\qquad{\rm d}(\phi^{\prime\prime}\omega_{0})=(L_{-}\lt\phi^{\prime\prime})\omega_{-}\wedge\omega_{0}+(L_{+}\lt\phi^{\prime\prime})\omega_{+}\wedge\omega_{0}+\phi^{\prime\prime}{\rm d}\omega_{0},
\label{deom}
\end{align}
and
\begin{align}
&\phi\in \mathcal{L}_{-2}:\qquad{\rm d}(\phi\,\omega_{-}\wedge\omega_{0})=(L_{+}\lt \phi)\omega_{+}\wedge\omega_{-}\wedge\omega_{0}, \nonumber \\
&\phi^{\prime}\in \mathcal{L}_{2}:\qquad {\rm d}(\phi^{\prime}\omega_{0}\wedge\omega_{+})=(L_{-}\lt\phi^{\prime})\omega_{-}\wedge\omega_{0}\wedge\omega_{+} , \nonumber \\
&\phi^{\prime\prime}\in\mathcal{L}_{0}:\qquad{\rm d}(\phi^{\prime\prime}\omega_{-}\wedge\omega_{+})=(L_{0}\lt\phi^{\prime\prime})\omega_{0}\wedge\omega_{-}\wedge\omega_{+}.
\label{deom2}
\end{align}
Together with the anti-symmetry properties \eqref{2fw} of the wedge product in $\Omega(\SU)$, these identities suggest that the following proposition holds.
\begin{prop}
\label{lea}
The exterior algebra $\Omega(S^{2}_{q})$ obtained as a restriction of $\Omega(\SU)$ associated to $4D_{+}$ calculus on $\SU$ can be written in terms of $\Asq$-bimodule isomorphisms:
\begin{align}
\Omega^{1}(S^{2}_{q})&\simeq\mathcal{L}_{-2}\,\omega_{-}\oplus\mathcal{L}_{2}\,\omega_{+}\oplus\mathcal{L}_{0}\,\omega_{0
\nonumber \\
\Omega^{2}(S^{2}_{q})&\simeq\mathcal{L}_{-2}\,(\omega_{-}\wedge\omega_{0})\oplus\mathcal{L}_{0}\,(\omega_{-}\wedge\omega_{+})\oplus\mathcal{L}_{2}\,(\omega_{0}\wedge\omega_{+})
\nonumber\\
\Omega^{3}(S^{2}_{q})&\simeq\mathcal{L}_{0}\,\omega_{-}\wedge\omega_{+}\wedge\omega_{0
\label{ntd}
\end{align}
\begin{proof}
The analysis above proves only the inclusion $\Omega^{1}(S^{2}_{q})\subset\mathcal{L}_{-2}\,\omega_{-}\oplus\mathcal{L}_{2}\,\omega_{+}\oplus\mathcal{L}_{0}\,\omega_{0}$ and the analogue ones for higher order forms. The proof of the inverse inclusion will be given at the end of \S\ref{s:ccd}, out of the compatibility of the calculi on the principal Hopf bundle.
\end{proof}
\end{prop}
The basis element $\omega_{-}\wedge\omega_{+}\wedge\omega_{0}$ commutes with all elements in $\mathcal{L}_{0}\simeq\Asq$.
Such a calculus is 3 dimensional, since from \eqref{d3f} one has ${\rm d}(\phi^{\prime\prime}\omega_{-}\wedge\omega_{+}\wedge\omega_{0})=0$, for any $\phi^{\prime\prime}\in \Asq$, and from \eqref{2fw} one has that $\Omega^{1}(S^{2}_{q})\wedge(\omega_{-}\wedge\omega_{+}\wedge\omega_{0})=0$.
From \eqref{d4} and \eqref{Rder} the differential can also be written as
\begin{equation}
{\rm d} f=\omega_{-}(R_{-}\lt f)+\omega_{+}(R_{+}\lt f)+\omega_{0}(R_{0}\lt f),
\label{d3di}
\end{equation}
and it is easy to check the following relations, analogues of the previous \eqref{deom}, \eqref{deom2}:
\begin{align}
&\phi\in \mathcal{L}_{-2}:\qquad{\rm d}(\omega_{-}\,\phi)=-\omega_{-}\wedge\omega_{+}\,(R_{+}\lt \phi)-\omega_{-}\wedge\omega_{0}\,(R_{0}\lt \phi), \nonumber \\
&\phi^{\prime}\in \mathcal{L}_{2}:\qquad {\rm d}(\omega_{+}\,\phi^{\prime})=
-\omega_{+}\wedge\omega_{-}\,(R_{-}\lt \phi^{\prime})-\omega_{+}\wedge\omega_{0}\,(R_{0}\lt \phi^{\prime}), \nonumber \\
&\phi^{\prime\prime}\in \mathcal{L}_{0}:\qquad {\rm d}(\omega_{0}\,\phi^{\prime\prime})=
{\rm d}\omega_{0}\wedge\phi^{\prime\prime}-\omega_{0}\wedge\omega_{-}\,(R_{-}\lt \phi^{\prime\prime})-\omega_{0}\wedge\omega_{+}\,(R_{+}\lt \phi^{\prime\prime});
\label{deomi}
\end{align}
and
\begin{align}
&\phi\in \mathcal{L}_{-2}:\qquad{\rm d}(\omega_{-}\wedge\omega_{0}\,\phi)=\omega_{-}\wedge\omega_{0}\wedge\omega_{+}\,(R_{+}\lt\phi), \nonumber \\
&\phi^{\prime}\in \mathcal{L}_{2}:\qquad {\rm d}(\omega_{0}\wedge\omega_{+}\,\phi^{\prime})=\omega_{0}\wedge\omega_{+}\wedge\omega_{-}\,(R_{-}\lt\phi^{\prime}) , \nonumber \\
&\phi^{\prime\prime}\in\mathcal{L}_{0}:\qquad{\rm d}(\omega_{-}\wedge\omega_{+}\,\phi^{\prime\prime})=\omega_{-}\wedge\omega_{+}\wedge\omega_{0}\,(R_{0}\lt\phi^{\prime\prime}).
\label{deom2i}
\end{align}
\section{Hodge operators on $\Omega(\SU)$}
\label{s:Hop}
As described in \S\ref{se:4dc}, it holds for the bicovariant forms of the $4D_{+}$ first order bicovariant calculus that the spaces $\Omega^{k}(\SU)$ of forms are free $\ASU$-bimodules with $\dim\,\Omega^{k}(\SU)=\dim\,\Omega^{4-k}(\SU)$, and $\dim\Omega^4(\SU)=1$. Our strategy to introduce Hodge operators on
$\Omega(\SU)$ in \S\ref{sub:H-1} uses first suitable contraction maps in order to define Hodge operators on the vector spaces $\Omega^{k}_{inv}(\SU)$ of left invariant $k$-forms; we extend them next to the whole $\Omega^k(\SU)$ by requiring (one side) linearity over $\ASU$. This follows an alternative although equivalent
approach to Hodge operators on classical group manifold that we describe first in \S\ref{sse:Hcla}.
A somewhat complementary approach to the one of \S\ref{sub:H-1}, more suitable when restricting to the sphere $S^{2}_{q}$, is then given in \S\ref{sub:H}.
\subsection{Hodge operators on classical group manifolds}
\label{sse:Hcla}
Let $G$ be an $N$-dimensional compact connected Lie group given as a real form of a complex connected
Lie group. The algebra $\mathcal{A}(G)=Fun(G)$ of complex valued coordinate functions on $G$ is a $*$-algebra, whose $*$-structure can be extended to the whole tensor algebra. A metric on the group $G$ is a non degenerate tensor $g:\mathfrak{X}(G)\otimes\mathfrak{X}(G)\to\mathcal{A}(G)$ which is symmetric -- i.e. $g(X,Y)=g(Y,X)$, with $X,Y\in\,\mathfrak{X}(G)$ -- and real -- i.e. $g^*(X,Y)=g(Y^*,X^*)$ --. Any metric has a normal form: there exists a basis $\{\theta^{a}, a=1, \dots N\}$ of the $\mathcal{A}(G)$-bimodule $\Omega^{1}(G)$ of 1-forms which is real, $\theta^{a*}=\theta^a$, such that
\begin{equation}
\label{gmf}
g=\sum_{a,b=1}^{N}\eta_{ab}\,\theta^{a}\otimes\theta^b
\end{equation}
with $\eta_{ab}=\pm1\cdot\delta_{ab}$.
Given the volume $N$-form $\mu=\mu^*:=\theta^{1}\wedge\ldots\wedge\theta^{N}$, the corresponding Hodge operator $\star:\Omega^{k}(G)\to\Omega^{N-k}(G)$ is the $\mathcal{A}(G)$-linear operator whose action on the above basis is
\begin{align}
&\star(1)=\mu, \nonumber \\
\label{Hdui}
&\star(\theta^{a_{1}}\wedge\ldots\wedge\theta^{a_{k}})=\frac{1}{(N-k)!} \sum_{b_j} \epsilon^{a_{1}\ldots a_{k}}_{\qquad b_{1}\ldots b_{N-k}}\theta^{b_{1}}\wedge\ldots\wedge \theta^{b_{N-k}},
\end{align}
with $\epsilon^{a_{1}\ldots a_{k}}_{\qquad b_{1}\ldots b_{N-k}}:=\sum_{s_1\ldots s_k} \eta^{a_{1}s_{1}}\ldots \eta^{a_{k}s_{k}}\epsilon_{s_{1}\ldots s_{k} b_{1}\ldots b_{N-k}}$ from the Levi-Civita tensor and the usual expression for the inverse metric tensor
$g^{-1}=\sum_{a,b=1}^{N} \eta^{ab} L_{a}\otimes L_{b}$
with $\sum_{b} \eta^{ab}\eta_{bc}=\delta^{a}_{c}$ on the dual vector field basis such that $\theta^{b}(L_{a})=\delta^{b}_{a}$. The Hodge operator \eqref{Hdui} satisfies the identity:
\begin{equation}
\label{quhs}
\star^{2}(\xi)=sgn(g)(-1)^{k(N-k)}\xi
\end{equation}
on any $\xi\in\,\Omega^k(G)$. Here $sgn(g)=\det(\eta_{ab})$ is the signature of the metric.
Hodge operators can indeed be equivalently introduced in terms of contraction maps. By this we mean an
$\mathcal{A}(G)$-sesquilinear map $\Gamma:\Omega^1(G)\times\Omega^1(G)\to\mathcal{A}(G)$ such that
$\Gamma(f\,\phi,\eta)=f^*\Gamma(\phi,\eta)$ while $\Gamma(\phi,\eta\,f )=\Gamma(\phi,\eta) f$ for
$f\in\,\mathcal{A}(G)$. Such a map can be uniquely extended to a consistent map $\Gamma:\Omega^k(G)\times\Omega^{k+k^\prime}(G)\to\Omega^{k^{\prime}}(G)$. We postpone showing this to the later \S\ref{sub:H-1}
where we prove a similar statement for the bicovariant calculus on $\SU$.
Having a contraction map, define the tensor $\tilde{g}:\Omega^1(G)\times\Omega^1(G)\to\mathcal{A}(G)$:
\begin{equation}
\tilde{g}(\phi, \eta):=\Gamma(\phi^{*},\eta).
\label{p7}
\end{equation}
Next, with a volume form $\mu$, such that $\mu^*=\mu$, define the operator $L:\Omega^{k}(G)\to\Omega^{N-k}(G)$ as
\begin{equation}
\label{p3}
L(\xi):=\frac{1}{k!} \Gamma^*(\xi,\mu)
\end{equation}
on $\xi\in\,\Omega^k(G)$, having used the notation $\Gamma^*(\cdot,\cdot)=(\Gamma(\cdot,\cdot))^*$. A second $\mathcal{A}(G)$-sesquilinear map $\{~,~\}:\Omega^{k}(G)\times\Omega^{k}(G)\to\mathcal{A}(G)$ can be implicitly introduced by the relation
\begin{equation}
\label{p4}
\{\xi,\xi^{\prime}\}\mu:=\xi^*\wedge L(\xi^{\prime}).
\end{equation}
For any pair of $k$-forms $\xi,\xi^{\prime}$ it is straightforward to recover that
\begin{equation}
\label{pi5}
\{\xi,\xi^{\prime}\}=\frac{1}{k!} \Gamma^*(\xi^{\prime},\xi) .
\end{equation}
The operator \eqref{p3} is not in general an Hodge operator: one has for example $L(1)=\mu$ as well as $L(\mu)= \det (\Gamma^*(\mu,\mu))$ which is not necessarily $\pm 1$.
To recover the standard formulation for a Hodge operator, one has to impose two constraints:
\begin{enumerate}[(a)]
\item An hermitianity condition. The sesquilinear map $\Gamma$ is said hermitian provided it satisfies:
\begin{equation}
\label{p10}
\{\phi,\eta\}=\Gamma(\phi,\eta),
\end{equation}
for any couple of 1-forms $\phi$ and $\eta$.
\end{enumerate}
From \eqref{pi5} and \eqref{p4} it holds that $\{\phi,\eta\}=\Gamma^*(\eta,\phi)$.Then
\begin{equation}
\label{pi10}
\{\phi,\eta\}=\Gamma(\phi,\eta)\quad\Leftrightarrow\quad\Gamma(\phi,\eta)=\Gamma^{*}(\eta,\phi).
\end{equation}
If the sesquilinear form $\Gamma$ is hermitian, one can prove that the expression \eqref{pi5} becomes
\begin{equation}
\label{pii10}
\{\xi,\xi^{\prime}\}=\frac{1}{k!}\Gamma(\xi,\xi^{\prime}).
\end{equation}
\begin{enumerate}[(b)]
\item A reality condition, namely a compatibility of the operator $L$ with the $*$-conjugation:
\begin{equation}
\label{p11}
L(\phi^{*})=(L(\phi))^*
\end{equation}
on 1-forms.
\end{enumerate}
If these two constraints are fullfilled, the tensor $\tilde{g}$ in \eqref{p7} is symmetric and real: it is (the inverse of) a metric tensor on the group manifold $G$. The operator $L$ turns out to be the standard Hodge operator corresponding to the metric given by $\tilde{g}$, and satisfies the identities:
\begin{align}
\label{p8}
L^2(\xi)=(-1)^{k(N-k)}sgn(\Gamma)\xi , \qquad
\{\xi,\xi^{\prime}\}= sgn(\Gamma)\{L(\xi),L(\xi^{\prime})\}
\end{align}
with
$$sgn(\Gamma):=(\det(\Gamma(\phi^{a},\phi^{b}))|\det(\Gamma(\phi^{a},\phi^{b})|^{-1}=sgn(\tilde{g}).
$$
Moreover, the operator $L$ turns out to be real, that is, it commutes with the hermitian conjugation $*$, on the whole exterior algebra $\Omega(G)$.
The above procedure could be somehow inverted. That is, given an hermitian contraction map $\Gamma$ as in \eqref{p10}, define the operator $L$ by \eqref{p3}. The corresponding tensor $\tilde{g}$ turns out to be real, but non necessarily symmetric. Imposing $L$ to satisfy one of the two conditions in \eqref{p8} -- they are proven to be equivalent -- makes the tensor $\tilde{g}$ symmetric, that is the inverse of a metric tensor, whose Hodge operator is $L$.
\subsection{Hodge operators on $\Omega(\SU)$}\label{sub:H-1}
In this section we shall describe how the classical geometry analysis of the previous section can be used
to introduce an Hodge operator on both the exterior algebras $\Omega(\SU)$ and $\Omega(S^{2}_{q})$ built out of the $4D$-bicovariant calculus \`a la Woronowicz on $\ASU$.
A somewhat different formulation of contraction maps was also used
in \cite{hec99,hec03} for a family of Hodge operators on the exterior algebras of bicovariant differential calculi over quantum groups.
We shall then start with a contraction map $\Gamma:\Omega^1_{inv}(\SU)\times\Omega^1_{inv}(\SU)\to\IC$,
required to satisfy $\Gamma(\lambda\,\omega,\omega^{\prime}) = \lambda^* \Gamma(\omega, \omega^{\prime})$ and
$\Gamma(\omega,\omega^{\prime}\, \lambda) = \Gamma(\omega, \omega^{\prime}) \lambda$, for $\lambda\in\IC$.
The natural extension to $\Gamma:\Omega_{inv}^{\otimes k}(\SU)\times\Omega_{inv}^{\otimes k+k^{\prime}}(SU)\to\Omega_{inv}^{\otimes k^{\prime}}(SU)$ given by
\begin{equation}
\label{p2i}
\Gamma(\omega_{a_{1}}\otimes\ldots\otimes\omega_{a_k},\omega_{b_1}\otimes\ldots\otimes\omega_{b_{k+k^{\prime}}}):=\left(\Pi_{j=1}^{k}\,\Gamma(\omega_{a_{j}},\omega_{ b_{j}})\right)\,\omega_{b_{k+1}}\otimes\cdots\otimes\omega_{b_{k+k^{\prime}}} ,
\end{equation}
with the assumption that $\Gamma(1,\omega)=\omega$ for any $\omega\in\,\Omega(\SU)$, can be used to define a consistent contraction map
$\Gamma:\Omega^{ k}(\SU)\times\Omega^{ k+k^{\prime}}(\SU)\to\Omega^{ k^{\prime}}(\SU)$, via
\begin{multline}
\label{c0}
\Gamma(\omega_{a_{1}}\wedge\ldots\wedge\omega_{a_k},\omega_{b_1}\wedge\ldots\wedge\omega_{b_{k+k^{\prime}}}) \\
:=\Gamma(\mathfrak{A}^{(k)}(\omega_{a_{1}}\otimes\ldots\otimes\omega_{a_k}), \mathfrak{A}^{(k+k^{\prime})}(\omega_{b_1}\otimes\ldots\otimes\omega_{b_{k+k^{\prime}}})) .
\end{multline}
This comes from the $k$-th order anti-symmetriser $\mathfrak{A}^{(k)}$, constructed from the braiding of the calculus, and used to define the exterior product of forms,
\begin{equation}
\label{1c}
\omega_{a_{1}}\wedge\ldots\wedge\omega_{a_{k}}:=\mathfrak{A}^{(k)}(\omega_{a_{1}}\otimes\cdots\otimes\omega_{a_{k}});
\end{equation}
the key identity for the consistency of \eqref{c0} is
\begin{equation}
\label{2c}
\mathfrak{A}^{(k+k^{\prime})}(\omega_{a_{1}}\otimes\cdots\otimes\omega_{a_{k+k^{\prime}}})=(\mathfrak{A}^{(k)}\otimes\mathfrak{A}^{(k^{\prime})})(\sum_{\sigma_j\in\mathit{S}(k,k^{\prime})}(-1)^{\pi_{\sigma_{j}}}\sigma_j(\omega_{a_{1}}\otimes\cdots\otimes\omega_{a_{k+k^{\prime}}})),
\end{equation}
where $\mathit{S}(k,k^{\prime})$ is the collection of the $(k,k^{\prime})$-shuffles, permutations $\sigma_{j}$ of $\{1,\ldots,k+k^{\prime}\}$ such that $\sigma_{j}(1)<\cdots<\sigma_{j}(k)$ and $\sigma_{j}(k+1)<\ldots<\sigma_{j}(k+k^{\prime})$, and $\pi_{\sigma_{j}}$ is the parity of $\sigma_{j}$. The identity \eqref{2c} is valid on the whole exterior algebra over any bicovariant calculus \`a la Woronowicz on a quantum group. It allows to show that any $(k+k^{\prime})$-form can be written as a linear combination of tensor products of $k$-forms times $k^{\prime}$-forms.
To proceed further, we use a slightly more general volume form by taking $\mu=\mu^*= \mathrm{i} \, m\,\omega_{-}\wedge\omega_{+}\wedge\omega_{0}\wedge\omega_{z}$, with $m\in\,\IR$. Then we define an operator
$$
\star:\Omega^{k}_{inv}(\SU)\to\Omega^{4-k}_{inv}(\SU),
$$
in degree zero and one by
\begin{equation}\label{p17}
\star(1):= \Gamma^*(1,\mu) =\mu \qquad \mathrm{and} \qquad
\star(\omega_{a}):= \Gamma^*(\omega_{a},\mu) .
\end{equation}
For $\Omega^{k}_{inv}(\SU)$ with $k\geq2$ we use the diagonal bases of the antisymmetriser, that is
\begin{equation}
\mathfrak{A}^{(k)}(\xi)=\lambda_{\xi}\xi ,
\label{eixi}
\end{equation}
with coefficients in \eqref{p14}, \eqref{p15-} and \eqref{p15} respectively. On these basis we define
\begin{equation}
\star(\xi):=\frac{1}{\lambda_{\xi}}\, \Gamma^*(\xi,\mu) .
\label{p17bis}
\end{equation}
Here and in the following we denote $(\Gamma(~,~))^*=\Gamma^*(~,~)$.
The definition \eqref{p17bis} is a natural generalisation of the classical \eqref{p3}: the classical factor $k!$ -- the spectrum of the antisymmetriser operator on $k$-forms in the classical case, where the braiding is the flip operator -- is replaced by the spectrum of the quantum antisymmetriser. Also, the presence of the *-conjugate comes from consistency and in order to have non trivial solutions.
Before we proceed, it is useful to re-express the volume forms in terms of the diagonal bases of the anti-symmetriser operators. Some little algebra shows that
\begin{align}\label{p16}
&\mu=\mathrm{i} m \{-\omega_{-}\otimes\chi_{+}^*+\omega_{+}\otimes\chi_{-}^*+\omega_{0}\otimes\chi_{0}^*-\omega_{z}\otimes\chi_{z}^*\} \nonumber \\
&\mu=\mathrm{i} m \{-\chi_{z}\otimes\omega_{z}^*+\chi_{-}\otimes\omega_{+}^* -\chi_{+}\otimes\omega_{-}^* +\chi_{0}\otimes\omega_{0}^*\},
\end{align}
and
\begin{multline}\label{p16bis}
\mu=\frac{\mathrm{i} m}{q^2-1}\Big\{\frac{1}{1+q^2}(q^4\psi_{-}\otimes\psi_{+}^*-\psi_{+}\otimes\psi_{-}^*) \\
+(q^4\varphi_{z}\otimes\varphi_{0}^*-\varphi_{0}\otimes\varphi_{z}^*+q^2\psi_{0}\otimes\psi_{z}^*-q^{-2}\psi_{z}\otimes\psi_{0}^*) \Big\} .
\end{multline}
A little more algebra shows in turn that on 1-forms
\begin{equation}\label{p19-1}
\star(\omega_{a})=\mathrm{i} m \big\{ \Gamma^*(\omega_{a},\omega_{-}) \chi_{+}
- \Gamma^*(\omega_{a},\omega_{+}) \chi_{-}
- \Gamma^*(\omega_{a},\omega_{0}) \chi_{0}
+ \Gamma^*(\omega_{a},\omega_{z}) \chi_{z}
\big\} ;
\end{equation}
and that using the bases \eqref{p12bis}, on 2-forms
\begin{multline}\label{p19-2}
\star(\xi_{(+)})=\frac{\mathrm{i} m}{1-q^4} \Big\{
\frac{1}{1+q^2} \left(q^4\Gamma^*(\xi_{(+)},\psi_{-})\psi_{+}-\Gamma^*(\xi_{(+)},\psi_{+})\psi_{-} \right)+(q^4\Gamma^*(\xi_{(+)},\varphi_{z})\varphi_{0} \\
-\Gamma^*(\xi_{(+)},\varphi_{0})\varphi_{z}+q^2\Gamma^*(\xi_{(+)},\psi_{0})\psi_{z}-q^{-2}\Gamma^*(\xi_{(+)},\psi_{z})\psi_{0}) \Big\}
\end{multline}
\begin{multline*}
\star(\xi_{(-)})=\frac{\mathrm{i} m}{q^{-2}-q^{2}} \Big\{
\frac{1}{1+q^2} \left( q^4\Gamma^*(\xi_{(-)},\psi_{-})\psi_{+}-\Gamma^*(\xi_{(-)},\psi_{+})\psi_{-} \right)+(q^4\Gamma^*(\xi_{(-)},\varphi_{z})\varphi_{0} \\
-\Gamma^*(\xi_{(-)},\varphi_{0})\varphi_{z}+q^2\Gamma^*(\xi_{(-)},\psi_{0})\psi_{z}-q^{-2}\Gamma^*(\xi_{(-)},\psi_{z})\psi_{0})
\Big\}.
\end{multline*}
As for 3-forms one finds
\begin{multline}\label{p19-3}
\star(\chi_{a})=-\frac{\mathrm{i} m}{2(1+q^2+q^{-2})} \Big\{-\Gamma^*(\chi_{a},\chi_{+})\omega_{-} \\ +\Gamma^*(\chi_{a},\chi_{-})\omega_{+}+\Gamma^*(\chi_{a},\chi_{0})\omega_{0}-\Gamma^*(\chi_{a},\chi_{z})\omega_{z} \Big\}, \end{multline}
and finally for the top form
\begin{equation}\label{p19-4}
\star(\mu)=\frac{1}{2(q^4+2q^2+6+2q^{-2}+q^{-4})} \Gamma^*(\mu,\mu).
\end{equation}
As in \eqref{p4} we define the sesquilinear map $\{~,~\}:\Omega^{k}_{inv}(\SU)\times\Omega^{k}_{inv}(\SU)\to\IC$ by
\begin{equation}
\{\xi,\xi^{\prime}\}\mu:=\xi^*\wedge\star(\xi^{\prime}).
\label{p4q}
\end{equation}
Then, mimicking the analogous construction of \S\ref{sse:Hcla} we impose both an hermitianity and a reality condition on the contraction map.
\begin{enumerate}[(a)]
\item A contraction map is hermitian provided it satisfies:
\begin{equation}
\label{pi21}
\{\omega_{a},\omega_{b}\}=\Gamma(\omega_{a},\omega_{b}), \qquad \mathrm{for} \qquad a,b=-,+,z,0 .
\end{equation}
\end{enumerate}
Given contraction maps fullfilling such an hermitianity constraint, from the first line in \eqref{p19-1} one has
that $\Gamma(\omega_{a},\omega_{b}) = \Gamma^*(\omega_{b},\omega_{a})$. i.e. $\Gamma_{ab}=\Gamma^{*}_{ba}$. With such a condition it is moreover possible to prove, that for with $k=2,3,4$,
\begin{equation}
\label{p21}
\{\xi,\xi^{\prime}\}=\frac{\lambda_{\xi^*}}{\lambda_{\xi}\lambda_{\xi^{\prime}}}\,\Gamma(\xi,\xi^{\prime}).
\end{equation}
on any $\xi,\xi^{\prime}\in\,\Omega^{k}_{inv}(\SU)$ of a diagonal basis of the antisymmetrizer as in \eqref{eixi}.
The above expression is the counterpart of \eqref{pii10} for a braiding which is not just the flip operator.
\begin{enumerate}[(b)]
\item
An hermitian contraction map is real provided one has
\begin{equation}
\label{pi22}
\lambda_{\xi^*}(\star\xi^*)=(\lambda_{\xi}(\star\xi))^*.
\end{equation}
\end{enumerate}
again on a diagonal basis of $\mathfrak{A}^{(k)}(\xi)$.
This expression generalises the classical one \eqref{p11}. Notice that it is set on any $\Omega^{k}_{inv}(\SU)$, and not only on 1-forms as in the classical case.
The requirement that the contraction be hermitian and real results in a series of constraints. Firstly,
the action on $\Omega^1_{inv}(\SU)$ of the corresponding operator $\star$ as defined in \eqref{p17} is worked out to be given by
\begin{equation}
\label{piii22}
\star\left(\begin{array}{c} \omega_{-} \\ \omega_{+} \\ \omega_{0} \\ \omega_{z} \end{array}\right)=\mathrm{i} m \left(\begin{array}{cccc} 0 & \alpha & 0 & 0 \\ -q^2\alpha & 0 & 0 & 0 \\ 0 & 0 & -\nu & \epsilon \\ 0 & 0 & -\epsilon & \gamma \end{array} \right)\left(\begin{array}{c}\chi_{-} \\ \chi_{+} \\ \chi_{0} \\ \chi_{z} \end{array}\right).
\end{equation}
The only non zero terms of the contraction $\Gamma$ are given by
\begin{align}
\Gamma_{--}=q^{-2}\Gamma_{++}=\alpha, \qquad \Gamma_{0z}=\Gamma_{z0}=\epsilon, \
\qquad \Gamma_{00}=\nu, \qquad
\Gamma_{zz}=\gamma,
\label{piv22}
\end{align}
with parameters that are real and satisfy in addition the conditions:
\begin{align}
&2\nu+(q^2-q^{-2})\epsilon=0, \nonumber \\
&2(\epsilon^2-\gamma\nu)+(q-q^{-1})^2(2q^2\alpha^2+\epsilon^2)=0.
\label{pv22}
\end{align}
On $\Omega^{2}_{inv}(\SU)$ the action of such operator is block off-diagonal,
\begin{align}
\label{pvi22}
&\star\left(\begin{array}{c}\varphi_{0} \\ \psi_{z} \\ \psi_{+} \end{array}\right)\,=
\frac{\mathrm{i} m}{q^4-1}\left(\begin{array}{ccc} \Gamma(\varphi_{0},\varphi_{0}) & 0 & 0 \\
0 & q^4\Gamma(\varphi_{z},\varphi_{z}) & 0 \\ 0 & 0 & \frac{\Gamma(\psi_{+},\psi_{+})}{1+q^{2}}
\end{array}
\right)\left(\begin{array}{c} \varphi_{z} \\ \psi_{0} \\ \psi_{-} \end{array}\right),\nonumber
\\
~\nonumber \\
&\star\left(\begin{array}{c} \varphi_{z} \\ \psi_{0} \\ \psi_{-} \end{array}\right)\,=
\frac{\mathrm{i} m}{1-q^4}\left(\begin{array}{ccc} q^6\Gamma(\varphi_{z},\varphi_{z}) & 0 & 0 \\
0 & q^2\Gamma(\varphi_{0},\varphi_{0}) & 0 \\ 0 & 0 & \frac{q^4\Gamma(\psi_{+},\psi_{+})}{1+q^{2}}
\end{array}
\right)\left(\begin{array}{c}\varphi_{0} \\ \psi_{z} \\ \psi_{+} \end{array}\right),
\end{align}
\noindent
while on $\Omega^{3}_{inv}(\SU)$ is
\begin{multline}
\label{pvii22}
\star\left(\begin{array}{c} \chi_{-} \\ \chi_{+} \\ \chi_{0} \\ \chi_{z} \end{array}\right)= \\ =
\frac{\mathrm{i} m}{2(1+q^2+q^{-2})}\left(\begin{array}{cccc}0 & -\Gamma(\chi_{-},\chi_{-}) & 0 & 0 \\ q^2\Gamma(\chi_{-},\chi_{-}) & 0 & 0 & 0 \\ 0 & 0 & -\Gamma(\chi_{0},\chi_{0}) & \Gamma(\chi_{z},\chi_{0}) \\ 0 & 0 & -\Gamma(\chi_{0},\chi_{z}) & \Gamma(\chi_{z},\chi_{z}) \end{array}\right)
\left(\begin{array}{c} \omega_{-} \\ \omega_{+} \\ \omega_{0} \\ \omega_{z} \end{array}\right).
\end{multline}
\bigskip
It turns out that the square of the operator $\star$ is not necessarily diagonal.
An explicit computation shows moreover that when $q\neq1$, given the constraints \eqref{pv22} there is no choice for the contraction $\Gamma$, nor for the value of the scale parameter $m\in\,\IR$ in the volume form such that the spectrum of the operator $\star^2$ is constant on any vector space $\Omega^k_{inv}(\SU)$.
This means that the operator $\star$ does not satisfy the classical expressions in \eqref{p8}.
We choose a particular value for the parameter $m$ defining
\begin{align}
\det\Gamma:=\frac{1}{\lambda_{\mu}}\Gamma(\omega_{-}\wedge\omega_{+}\wedge\omega_{0}\wedge\omega_{z},\omega_{-}\wedge\omega_{+}\wedge\omega_{0}\wedge\omega_{z}), \qquad
sgn(\Gamma):=\frac{\det\Gamma}{|\det\Gamma|}
\label{pviii22}
\end{align}
and imposing
\begin{equation}
\label{pix22}
\star^2(1)=sgn(\Gamma),
\end{equation}
which is clearly equivalent to the constraint
\begin{equation}
\label{px22}
m^2=|\det\Gamma|^{-1}.
\end{equation}
An explicit calculation shows that conditions \eqref{pi21} and \eqref{pi22} fix the quantum determinant \eqref{pviii22} to be positive, so that we have $sgn(\Gamma)=1$.
We finally extend the operator $\star$ to the whole exterior algebra. This can be defined in two ways, i.e. we define Hodge operators $\star^{L},\star^{R}:\Omega^{k}(\SU)\to\Omega^{4-k}(\SU)$ by:
\begin{align}
\star^{L}(x\,\omega):=x\star(\omega), \qquad\qquad\qquad
\star^{R}(\omega\,x):=(\star\, \omega) x ,
\label{Hlr}
\end{align}
with $x\in\,\ASU$ and $\omega\in\,\Omega_{inv}(\SU)$. Both operators will find their use later on.
\subsection{Hodge operators on $\Omega(\SU)$ -- a complementary approach} \label{sub:H}
The procedure used in the previous section cannot be extended \emph{ipso facto} to introduce an Hodge operator on the exterior algebra $\Omega(S^{2}_{q})$: although all $\Omega^{k}(S^{2}_{q})$ are free left $\Asq$-modules \cite{HK03}, the tensor product $\Omega^{\otimes2}(S^{2}_{q})$ has no braiding like the $\sigma$ above.
In order to construct a suitable Hodge operator on the quantum sphere, we shall export to this quantum homogeneous space the construction of \cite{kmt}, originally conceived on the exterior algebra over a quantum group.
The strategy largely coincides with the one described in \cite{ale09} and presents similarities to that used
in \cite{dal09} where a Hodge operator has been introduced on a quantum projective plane.
We start by briefly recalling the formulation from \cite{kmt}.
Consider a $*$-Hopf algebra $\mathcal{H}$ and the exterior algebra $\Omega(\mathcal{H})$ over an $N$-dimensional left covariant first order calculus
$(\Omega^{1}(\mathcal{H}),{\rm d})$, with $\dim \Omega^{N-k}(\mathcal{H})=\dim \Omega^{k}(\mathcal{H})$ and $\dim \Omega^{N}(\mathcal{H})=1$. Suppose in addition that $\mathcal{H}$ has an Haar state $h:\mathcal{H}\to\IC$, i.e. a unital functional, which is
invariant, i.e. $(\id\otimes h)\Delta x=(h\otimes \id)\Delta x=h(x)1$ for any $x\in \mathcal{H}$, and positive, i.e. $h(x^{*}x)\geq0$ for all $x\in \mathcal{H}$. An Haar state so defined is unique and automatically faithful: $h(x^{*}x)=0$ implies $x=0$.
Upon fixing an inner product on a left invariant basis of forms, the state $h$ is then used to endow
the whole exterior algebra with a left and a right inner product, when requiring left or right invariance,
\begin{align}
&\hs{x\,\omega}{x^{\prime}\,\omega^{\prime}}^{L}:=h(x^{*}x^{\prime})\hs{\omega}{\omega^{\prime}} ,
\nonumber \\
&\hs{\omega\,x}{\omega^{\prime}\,x^{\prime}}^{R}:=h(x^*x^{\prime})\hs{\omega}{\omega^{\prime}}
\label{inpo}
\end{align}
for any $x,x^{\prime}\in \mathcal{H}$ and $\omega,\omega^{\prime}$ in $\Omega_{inv}(\mathcal{H})$.
The spaces $\Omega^{k}(\mathcal{H})$ are taken to be pairwise orthogonal (this is stated by saying that
the inner product is graded).
The differential calculus is said to be non-degenerate if, whenever $\eta\in \Omega^{k}(\mathcal{H})$ and $\eta^{\prime}\wedge\eta=0$ for any $\eta^{\prime}\in \Omega^{N-k}(\mathcal{H})$, then necessarily $\eta=0$.
Choose in $\Omega^{N}(\mathcal{H})$ a left invariant hermitian basis element $\mu=\mu^{*}$, referred to as the volume form of the calculus. For the sake of the present paper, we assume that the differential calculus has a volume form such that $\mu\,x=x\,\mu$ for any $x\in\,\mathcal{H}$ (this condition is satisfied by the $4D_{+}$ bicovariant calculus on $\SU$ that we are considering). Then one defines an `integral'
\begin{align*}
&\int _{\mu} \, : \, \Omega(\mathcal{H})\to\IC, \qquad\qquad
\int _{\mu} x\, \mu =h(x) , \qquad \mathrm{for} \quad x\in \mathcal{H} ,
\end{align*}
and $\int_{\mu} \eta=0$ for any $k$-form
$\eta$ with $k<N$. For a non-degenerate calculus the functional $\int_{\mu}$ is left-faithful if $\eta \in \Omega^{k}(\mathcal{H})$ is such that $\int_{\mu}\eta^{\prime}\wedge\eta=0$ for all $\eta^{\prime}\in \Omega^{N-k}(\mathcal{H})$, then $\eta=0$.
The central result is \cite{kmt}:
\begin{prop}
\label{Lge}
Consider a left covariant, non-degenerate differential calculus on a $*$-Hopf algebra, whose corresponding exterior algebra is
such that $\dim\, \Omega^{N-k}(\mathcal{H})=\dim\, \Omega^{k}(\mathcal{H})$ and $\dim\, \Omega^{N}(\mathcal{H})=1$, with a left-invariant volume form $\mu=\mu^*$ satisfying $x\,\mu=\mu\,x$ for any $x\in\,\mathcal{H}$. If $\Omega(\mathcal{H})$ is endowed with inner products and integrals as before, there exists a unique left $\mathcal{H}$-linear bijective operator
$L:\Omega^{k}(\mathcal{H})\to\Omega^{N-k}(\mathcal{H})$ for $k=0,\ldots,N$ (resp. a unique right $\mathcal{H}$-linear bijective operator $R$) such that
\begin{equation}\label{Lop}
\int_{\mu}\eta^{*}\wedge L(\eta^{\prime})=\hs{\eta}{\eta^{\prime}}^L, \qquad\qquad
\int_{\mu}\eta^{*}\wedge R(\eta^{\prime})=\hs{\eta}{\eta^{\prime}}^R
\end{equation}
for any $\eta,\eta^{\prime}\in \Omega^{k}(\mathcal{H})$.
\end{prop}
We mention that there is no $R$ operator in \cite{kmt}. It is just to prove its right $\mathcal{H}$-linearity that one needs the condition $x\mu=\mu x$ for the volume form $\mu$ with $x\in\,\mathcal{H}$.
We are now ready to make contact with the previous \S\ref{sub:H-1}. The $4D_{+}$ differential calculus on
$\SU$ is easily seen to be non degenerate. On the other hand, the Haar state functional $h$ is given by (cf. \cite{KS97}):
\begin{equation}
h(1)=1;\qquad h((cc^{*})^{k})=(\sum_{j=0}^{k}\,q^{2j})^{-1}=\frac{1}{1+q^{2}+\ldots+q^{2k}},
\label{Has}
\end{equation}
with $k\in \IN$, all other generators mapping to zero.
Now, use the sesquilinear map \eqref{p4q} for an inner product
$\hs{\omega}{\omega^{\prime}}:=\{\omega,\omega^{\prime}\}$ on generators of $\Omega_{inv}(\SU)$ and extend it to a left invariant and a right invariant ones to the whole of $\Omega_{inv}(\SU)$ as in \eqref{inpo} using the state $h$. The uniqueness of the operators $L$ and $R$ from Proposition~\ref{Lge} then implies that the extended left and right inner products are related to the left and right Hodge operators \eqref{Hlr} by
\begin{equation}
\label{Holr}
\int_{\mu}\eta^{*}\wedge (\star^{L}\,\eta^{\prime})=\hs{\eta}{\eta^{\prime}}^L, \qquad\qquad
\int_{\mu}\eta^{*}\wedge (\star^{R}\,\eta^{\prime})=\hs{\eta}{\eta^{\prime}}^R
\end{equation}
for any $\eta,\eta^{\prime}\in \Omega^{k}(\mathcal{H})$.
\section{Hodge operators on $\Omega(S^{2}_{q})$} \label{s:hl3}
From the previous section,
the procedure to introduce Hodge operators on the quantum sphere appears outlined. Inner products on $\Omega(\SU)$ naturally induce inner products on $\Omega(S^{2}_{q})$, and we shall explore the use of relations like the \eqref{Holr} above to define a class of Hodge operators.
The exterior algebra $\Omega(S^{2}_{q})$ over the quantum sphere $S^{2}_{q}$ is described in \S\ref{se:cals2}. In particular, we recall its description in terms of the $\Asq$-bimodules $\mathcal{L}_{n}$ given in \eqref{libu}:
\begin{align}
&\Omega^{0}(S^{2}_{q}) \simeq \Asq \simeq \mathcal{L}_{0} \, \nonumber \\
&\Omega^{1}(S^{2}_{q}) \simeq \mathcal{L}_{-2}\,\omega_{-}\oplus\mathcal{L}_{2}\,\omega_{+}\oplus\mathcal{L}_{0}\,\omega_{0}\,\simeq \omega_{-}\,\mathcal{L}_{-2}\oplus\omega_{+}\,\mathcal{L}_{2}\oplus \omega_{0}\,\mathcal{L}_{0} \nonumber \\
&\Omega^{2}(S^{2}_{q}) \simeq \mathcal{L}_{-2}\,(\omega_{-}\wedge\omega_{0})\oplus\mathcal{L}_{0}\,(\omega_{-}\wedge\omega_{+})\oplus\mathcal{L}_{2}\,(\omega_{0}\wedge\omega_{+})\nonumber \\
&\qquad\qquad\qquad\qquad \simeq
(\omega_{-}\wedge\omega_{0})\,\mathcal{L}_{-2}\oplus(\omega_{-}\wedge\omega_{+})\,\mathcal{L}_{0}\oplus(\omega_{0}\wedge\omega_{+})\,\mathcal{L}_{2} \, \nonumber \\
&\Omega^{3}(S^{2}_{q}) \simeq \mathcal{L}_{0}\,\omega_{-}\wedge\omega_{+}\wedge\omega_{0} \label{isoc2} \,\simeq\,\omega_{-}\wedge\omega_{+}\wedge\omega_{0}\,\mathcal{L}_{0} \, .
\end{align}
In the rest of this section, to be consistent with the notation introduced in \S\ref{se:cals2}, we shall consider elements $\phi,\psi\in\,\mathcal{L}_{-2}$, elements $\phi^{\prime},\psi^{\prime}\in\,\mathcal{L}_{2}$ and elements $\phi^{\prime\prime},\psi^{\prime\prime}\in\,\mathcal{L}_{0}$.
\begin{lemm}
The above left covariant 3D calculus on $S^{2}_{q}$ is non-degenerate.
\label{nddc}
\begin{proof}
Given $\theta\in \Omega^{k}(S^{2}_{q})$ the condition of non degeneracy, namely $\theta^{\prime}\wedge\theta=0$ for any $\theta^{\prime}\in \Omega^{3-k}(S^{2}_{q})$ only if $\theta=0$, is trivially satisfied for $k=0,3$.
From \eqref{isoc2} take the 1-form
$\theta=\phi\,\omega_{-}$ and a 2-form $\theta^{\prime}=\psi\,\omega_{-}\wedge\omega_{0}+\psi^{\prime}\omega_{+}\wedge\omega_{0}+\psi^{\prime\prime}\omega_{-}\wedge\omega_{+}$. Using the commutation properties \eqref{bis2} between 1-forms and elements in $\ASU$, one has $\theta^{\prime}\wedge\theta=\{\psi^{\prime}(K^{2}\lt\phi)-\psi^{\prime\prime}(q^{-\frac{1}{2}} KE\lt\phi)\}\omega_{-}\wedge\omega_{+}\wedge\omega_{0}$, so that the equation $\theta^{\prime}\wedge\theta=0$ for any $\theta^{\prime}\in \Omega^{2}(S^{2}_{q})$ is equivalent to the condition $\{\psi^{\prime}(K^{2}\lt\phi)-\psi^{\prime\prime}(q^{-\frac{1}{2}} KE\lt\phi)\}=0$ for any $\theta^{\prime}\in \Omega^{2}(S^{2}_{q})$; taking $\theta^{\prime}=\psi^{\prime}\omega_{+}\wedge\omega_{0}$, one shows that this condition is satisfied only if $\phi=0$. A similar conclusion is reached with a 1-form $\theta=\phi^{\prime}\omega_{+}$, and with a 1-form $\theta=\phi^{\prime\prime}\omega_{0}$.
Consider then a 2-form $\theta=\phi\,\omega_{-}\wedge\omega_{0}$, and a 1-form $\theta^{\prime}=\psi\omega_{-}+\psi^{\prime}\omega_{+}+\psi^{\prime\prime}\omega_{0}$. Their product is $\theta^{\prime}\wedge\theta=(\psi^{\prime}\phi)\omega_{+}\wedge\omega_{-}\wedge\omega_{0}$, so that the condition $\theta^{\prime}\wedge\theta=0$, for all $\theta^{\prime}\in \Omega^{1}(S^{2}_{q})$ is equivalent to the condition $\psi^{\prime}\phi=0$ for any $\psi^{\prime}$; this condition is obviously satisfied only by $\phi=0$. It is clear that a similar analysis can be performed for any 2-form $\theta\in \Omega^{2}(S^{2}_{q})$.
\end{proof}
\end{lemm}
The Haar state $h$ of $\ASU$ given in \eqref{Has} yields a faithful and invariant state when restricted to
$\Asq$. As a volume form we take $\check{\mu}=\check{m}\,\omega_{-}\wedge\omega_{+}\wedge\omega_{0}=\check{\mu}^*$ with $\check{m}\in \IR$. It commutes with every algebra element, $f\,\check{\mu}=\check{\mu}\,f$ for $f\in\,\Asq$, so the integral on the exterior algebra $\Omega(S^{2}_{q})$ can be defined by
\begin{align}
&\int_{\check{\mu}}\theta=0,\qquad &\mathrm{on}\quad \theta\in \Omega^{k}(S^{2}_{q}),\,\mathrm{for}\,k=0,1,2 \, ,
\nonumber \\
&\int_{\check{\mu}}f\,\check{\mu}=h(f) ,\qquad &\mathrm{on}\quad f\,\check{\mu} \in \Omega^{3}(S^{2}_{q}) \, .
\label{ints2}
\end{align}
\begin{lemm}
\label{lef}
The integral $\int_{\check{\mu}}:\Omega(S^{2}_{q})\to\IC$ defined by \eqref{ints2} is left-faithful.
\begin{proof}
The proof of the left-faithfulness of the integral can be easily established from a direct analysis, using the faithfulness of the Haar state $h$. \end{proof}
\end{lemm}
The restriction to $\Omega(S^{2}_{q})$ of the left and right $\ASU$-linear graded inner products on $\Omega(\SU)$ in \eqref{Holr} gives left and right
$\Asq$-linear
graded inner products on $\Omega(S^{2}_{q})$:
\begin{equation}
\hs{\theta}{\theta^{\prime}}_{S^{2}_{q}}^{L}:=
\hs{\theta}{\theta^{\prime}}^{L}; \qquad\qquad\qquad
\hs{\theta}{\theta^{\prime}}_{S^{2}_{q}}^{R}:=
\hs{\theta}{\theta^{\prime}}^{R}
\label{prosca}
\end{equation}
with $\theta,\theta^{\prime}\in\,\Omega(S^{2}_{q})$. The analogue result to relation \eqref{Holr} is given in the following
\begin{prop}
\label{LH}
On the exterior algebra on the sphere $S^{2}_{q}$ endowed with the above graded left (resp. right) inner product, there exists a unique invertible left
$\Asq$-linear
Hodge operator $\check{L}:\Omega^{k}(S^{2}_{q})\to\Omega^{3-k}(S^{2}_{q})$, (resp. a unique invertible right $\Asq$-linear Hodge operator $\check{R}$) for $k=0,1,2,3$, satisfying
\begin{equation}\label{Les}
\int_{\check{\mu}}\theta^{*}\wedge \check{L}(\theta^{\prime})=\hs{\theta}{\theta^{\prime}}^{L}_{S^{2}_{q}} \, , \qquad\qquad\qquad
\int_{\check{\mu}}\theta^{*}\wedge \check{R}(\theta^{\prime})=\hs{\theta}{\theta^{\prime}}^{R}_{S^{2}_{q}}
\end{equation}
for any $\theta,\theta^{\prime}\in \Omega^{k}(S^{2}_{q})$.
They can be written in terms of the sesquilinear map \eqref{p4q} as:
\begin{equation} \label{cLo}
\begin{array}{lcl}
\check{L}(1)=\check{\mu} \, , &
\quad\quad & \check{L}(\check{\mu})= \{\check{\mu},\check{\mu}\} \\
\check{L}(\phi\,\omega_{-})=\check{m}\alpha\,\phi\,\omega_{-}\wedge\omega_{0} \, , &
\quad \quad & \check{L}(\phi\,\omega_{-}\wedge\omega_{0})=\check{m}\,\{\omega_-\wedge\omega_{0},\omega_-\wedge\omega_0\}\phi\,\omega_{-} \, ,\\
\check{L}(\phi^{\prime}\omega_{+})=\check{m}\,q^2\alpha\,\phi^{\prime}\omega_{0}\wedge\omega_{+} \, , &
\quad\quad & \check{L}(\phi^{\prime}\omega_{0}\wedge\omega_{+})=\check{m}\,\{\omega_{+}\wedge\omega_{0},\omega_{+}\wedge\omega_{0}\}\,\phi^{\prime}\omega_{+} \, , \\
\check{L}(\omega_{0})=-\check{m}\nu\,\omega_{-}\wedge\omega_{+} \, , &
\quad\quad& \check{L}(\omega_{-}\wedge\omega_{+})=-\check{m}\,\{\omega_{-}\wedge\omega_{+},\omega_{-}\wedge\omega_{+}\}\,\omega_{0} \,
\end{array}
\end{equation}
and
\begin{equation}
\begin{array}{lcl}
\check{R}(1)=\check{\mu} \, , & \quad\quad & \check{R}(\check{\mu})= \{\check{\mu},\check{\mu}\} \\
\check{R}(\omega_{-}\,\phi)=\check{m}q^2\alpha\,\omega_{-}\wedge\omega_{0}\,\phi \, , & \quad\quad
& \check{R}(\omega_{-}\wedge\omega_{0}\,\phi)=\check{m}\,q^2\{\omega_-\wedge\omega_{0},\omega_-\wedge\omega_0\}\,\omega_{-} \,\phi ,\\
\check{R}(\omega_{+}\,\phi^{\prime})=\check{m}\,\alpha\,\omega_{0}\wedge\omega_{+} \,\phi^{\prime} , & \quad\quad & \check{R}(\omega_{0}\wedge\omega_{+}\,\phi^{\prime})=\check{m}\,q^{-2}\{\omega_{+}\wedge\omega_{0},\omega_{+}\wedge\omega_{0}\}\,\omega_{+} \,\phi^{\prime} , \\
\check{R}(\omega_{0})=-\check{m}\nu\,\omega_{-}\wedge\omega_{+} \, , & \quad\quad& \check{R}(\omega_{-}\wedge\omega_{+})=-\check{m}\,\{\omega_{-}\wedge\omega_{+},\omega_{-}\wedge\omega_{+}\}\,\omega_{0} \, .
\end{array}
\label{cRo}
\end{equation}
\begin{proof}
For the rather technical proof we refer to \cite{ale09}, where the same strategy has been adopted for the analysis of an Hodge operator on a two dimensional exterior algebra on $S^{2}_{q}$. Here we only observe that the uniqueness follows from the result in Lemma~\ref{lef}. Given two operators $\check{L},\check{L}^{\prime}:\Omega^{k}(S^{2}_{q})\to\Omega^{3-k}(S^{2}_{q})$ satisfying \eqref{Les} (or equivalently $\check{R},\check{R}^{\prime}$), their difference must satisfy the relation $\int_{\check{\mu}}\theta^{\prime*}\wedge (\check{L}(\theta)-\check{L}^{\prime}(\theta))=0$ for any $\theta,\theta^{\prime}\in \Omega^{k}(S^{2}_{q})$. The left-faithfulness of the integral allows one then eventually to get $\check{L}(\theta)=\check{L}^{\prime}(\theta)$.
\end{proof}
\end{prop}
From \eqref{p13} and \eqref{p15-} it is $\check{\mu}=\check{m}\,\chi_{z}$, so we define
\begin{align}
\det\check{\Gamma}:=\frac{\Gamma(\chi_z,\chi_z)}{2(1+q^2+q^{-2})} ,
\qquad sgn(\check{\Gamma}):=\frac{\det(\check{\Gamma})}{|\det\check{\Gamma}|}
\label{pxiv22}
\end{align}
and set
$$
\check{m}^2\det\check{\Gamma}:=sgn(\check{\Gamma})
$$ as a definition for the scale factor $\check{m}\in\,\IR$. Clearly this choice gives $\check{L}^2(1)=\check{R}^2(1)=sgn(\check{\Gamma})$.
Analogously to what happened for $\SU$ before, the sign in \eqref{pxiv22} turns out to be positive for the class of contractions we are considering, i.e. $sgn(\check{\Gamma})=1$.
We conclude by noticing that the Hodge operators \eqref{cLo} and \eqref{cRo}
are diagonal, but still there is no choice for the parameters \eqref{piv22} and \eqref{pv22} of a real and hermitian contraction map such that a relation like \eqref{p8} is satisfied.
\section{Laplacian operators}
\label{se:L}
Given the Hodge operators constructed in the previous sections, the corresponding Laplacian operators on the quantum group $\SU$,
\begin{align*}
&\Box^L_{\SU}:\ASU\to\ASU , \qquad\qquad \Box^{L}_{\SU}(x) := - \star^L{\rm d}\star^L{\rm d} x , \\
&\Box^R_{\SU}:\ASU\to\ASU , \qquad\qquad \Box^{R}_{\SU}(x) := - \star^R{\rm d}\star^R{\rm d} x
\end{align*}
can be readily written in terms of the basic derivations \eqref{Lq} and \eqref{Rder} for the first order differential calculus as
\begin{align}
\label{elp4}
\Box^L_{\SU} x =
\left\{\alpha \left( L_{+}L_{-}+q^2 L_{-}L_{+}\right) + \nu\,L_{0}L_{0}+\gamma\,L_{z}L_{z}+2\epsilon L_{0}L_{z}\right\}\lt x,
\end{align}
and
\begin{align}
\Box^R_{\SU} x =
\left\{\alpha \left( q^2 R_{+}R_{-}+R_{-}R_{+} \right) + \nu\,R_{0}R_{0}+\gamma\,R_{z}R_{z}+2\epsilon R_{0}R_{z}\right\}\lt x,
\label{erp4}
\end{align}
with parameters given in \eqref{piv22}.
From the decomposition \eqref{dcmp} and the action \eqref{rellb} it is immediate to see that such Laplacians restrict to operators $:\mathcal{L}_{n}\to\mathcal{L}_{n}$. In order to diagonalise them, we recall the decomposition \eqref{decoln}.
The action of each term of the Laplacians on the basis elements $\{\phi_{n,J,l} \}$ in \eqref{bsphi} can be explicitly computed by \eqref{lact}, giving:
\begin{align}
&L_{-}L_{+}\lt\,\phi_{n,J,l}=q^{-1-n} \, [J-{\tfrac{1}{2}} n][J+1- {\tfrac{1}{2}} n ]\,\phi_{n,J,l} \, , \nonumber \\
&L_{+}L_{-}\lt\,\phi_{n,J,l}=q^{1-n} \, [J+{\tfrac{1}{2}} n][J+1-{\tfrac{1}{2}} n]\,\phi_{n,J,l} \, , \nonumber \\
&L_{z}\lt\,\phi_{n,J,l}=-q^{-\frac{1}{2} n } \, [{\tfrac{1}{2}} n]\,\phi_{n,J,l} \, , \nonumber \\
&L_{0}\lt\,\phi_{n,J,l}=([J+{\tfrac{1}{2}}]^{2}-[{\tfrac{1}{2}}]^2)\,\phi_{n,J,l}=[J][J+1]\,\phi_{n,J,l} \, .
\label{eopep}
\end{align}
Here for the labels one has $n\in\IN$ with $J=\tfrac{|n|}{2}+\IZ$ and $l=0,\ldots,2J$.
The Laplacians on the quantum sphere are, with $f\in\,\Asq$:
\begin{align}
\Box_{S^{2}_{q}}^L f:=-\check{L}{\rm d}\check{L}{\rm d} f=
\left\{\alpha\,L_{+}L_{-}+q^2\alpha\,L_{-}L_{+}+\nu\,L_{0}L_{0}\right\}\lt f,
\label{sfl}
\end{align}
and
\begin{align}
\Box_{S^{2}_{q}}^R f:=-\check{R}{\rm d}\check{R}{\rm d} f=
\left\{q^2 \alpha\,R_{+}R_{-}+\alpha\,R_{-}R_{+}+\nu\,R_{0}R_{0}\right\}\lt f.
\label{sfr}
\end{align}
They both are the restriction to $S^{2}_{q}$ of the Laplacian on $\SU$, the left and right one respectively.
Their actions can be written in terms of the action of the Casimir element $C_{q}$ of $\su$, immediately giving their spectra. They coincide on $S^{2}_{q}$:
\begin{align*}
\Box_{S^{2}_{q}}^{L,R} &= 2q\alpha(C_{q}+\tfrac{1}{4}-[\tfrac{1}{2}]^2)
+\nu(C_{q}+\tfrac{1}{4}-[\tfrac{1}{2}]^2)^2 , \nonumber \\
&= 2q\alpha \,L_{0} +\nu \, L^2_{0} \qquad \mathrm{on} \quad \Asq .
\end{align*}
Using \eqref{eopep}, spectra are readily found:
\begin{equation}
\Box_{S^{2}_{q}}^{L,R} (\phi_{0,J,l}) = \left( 2q\alpha[J][J+1]+\nu[J]^2[J+1]^2 \right) \phi_{0,J,l} ,
\label{specL}
\end{equation}
with $J\in\,\IN, l=0,\ldots,2J$.
We end this section by comparing these spectra to the spectrum of $D^2$, the square of the Dirac operator on $S^{2}_{q}$ studied in \cite{gs10}. Some straightforward computation leads to:
\begin{align}\label{cosp}
\text{spec}( \,\Box_{S^{2}_{q}}^{L,R})=\text{spec}(D^2-[\tfrac{1}{2}]^2)\quad\quad\Leftrightarrow\quad\quad2q\alpha=1, \quad \nu=q^{-2}(q-q^{-1})^4 .
\end{align}
\section{A digression: connections on the Hopf fibration over $S^{2}_{q}$ } \label{s:ccd}
A monopole connection for the quantum fibration $\Asq\hookrightarrow\ASU$ on the standard Podle\'s sphere -- with a left-covariant 3d calculus on $\SU$ and the (corresponding restriction to a) 2d left-covariant calculus on $S^{2}_{q}$ -- was explicitly described in \cite{bm93}.
A slightly different, but to large extent equivalent \cite{durcomm} formulation of this and of a fibration constructed on the same topological data $\Asq\hookrightarrow\ASU$, but with $\SU$ equipped with a bicovariant 4D calculus inducing on
$S^{2}_{q}$ a left-covariant 3d calculus, are presented in \cite{durII}.
The general problem of finding the conditions between the differential calculi on a base space algebra and on a `structure' group, in a way giving a principal bundle structure with compatible calculi and a consistent definition of connections on it has been deeply studied \cite{bm98,ps97,dur98,haj97}. The slightly different perspective of this digression is to follow the path reviewed in appendix~\S\ref{ap:qpb}, namely to recall from \cite{gs10} the formulation of a Hopf bundle on the standard Podle\'s sphere starting from the 4D bicovariant calculus \`a la Woronowicz on the total space $\SU$, in order to fully describe the set of its connections. The first step in this analysis consists in describing how the differential calculus on $\SU$ naturally induces a 1 dimensional bicovariant calculus on the structure group $\U(1)$, and in which sense these two calculi are compatible.
\subsection{A 1D bicovariant calculus on $\U(1)$}
\label{se:cu1}
The Hopf projection \eqref{qprp} allows one to define an ideal $\mathcal{Q}_{\U(1)}\subset\ker\varepsilon_{\U(1)}$ as the projection
$\mathcal{Q}_{\U(1)}=\pi(\mathcal{Q}_{\SU})$. Then $\mathcal{Q}_{\U(1)}$ is generated by the three elements
\begin{align*}
\xi_{1}&=(z^{2}-1)+q^{2}(z^{-2}-1), \\
\xi_{2}&=(q^{2}z+z^{-1}-(q^{3}+q^{-1}))(q^{2}z+z^{-1}-(1+q^{2})), \\
\xi_{3}&=(q^{2}z+z^{-1}-(q^{-1}+q^{3}))(z^{-1}-z),
\end{align*}
and, since $\mathrm{Ad}(\mathcal{Q}_{\U(1)})\subset\mathcal{Q}_{\U(1)}\otimes\mathcal{A}(\U(1))$, it corresponds to a bicovariant differential calculus on $\U(1)$. The identity
$$
-q(1+q^{4})^{-1}(1+q^{2}+q^{3}+q^{5})^{-1}\{(q^{6}-1)\xi_{3}+(1+q^{4})\xi_{2}-q^{2}(1+q^{2})\xi_{1}\}=(z-1)+q(z^{-1}-1)
$$
shows that $\xi=(z-1)+q(z^{-1}-1)$ is in $\mathcal{Q}_{\U(1)}$. By induction one also sees that
\begin{align}
&j>0:\qquad z^{j}(z-1)=\xi(\sum_{n=0}^{j-1}\,q^{n}z^{j-n})+q^{j}(z-1), \nonumber \\
&j<0:\qquad z^{-\abs{j}}(z-1)=-\xi(\sum_{n=1}^{\abs{j}-1}\,q^{-n}z^{n-\abs{j}})+q^{-\abs{j}}(z-1).
\label{idzj}
\end{align}
From these relations it is immediate to prove (as in \cite{gs10}) that there is a complex vector space isomorphism
$\ker\varepsilon_{\U(1)}/\mathcal{Q}_{\U(1)} \simeq\IC$. The differential calculus induced by $\mathcal{Q}_{\U(1)}$ is 1-dimensional, and the projection $\pi_{\mathcal{Q}_{\U(1)}}:\ker\varepsilon_{\U(1)}\to\ker\varepsilon_{\U(1)}/\mathcal{Q}_{\U(1)}$ can be written as
\begin{equation}
\label{lam4}
\pi_{\mathcal{Q}_{\U(1)}}:\quad z^{j}(z-1)\to q^{j}[z-1],
\end{equation}
on the vector space basis $\varphi(j)=z^{j}(z-1)$ in $\ker\varepsilon_{\U(1)}$, with notation $[z-1]\,\in\,\ker\varepsilon_{\U(1)}/\mathcal{Q}_{\U(1)}$. The projection \eqref{lam4} will be used later on to define connection 1-forms on the fibration.
As a basis element for the quantum tangent space $\mathcal{X}_{\mathcal{Q}_{\U(1)}}$ we take
\begin{equation}
X=L_z=\frac{K^{-2}-1}{q-q^{-1}}.
\label{X1}
\end{equation}
The $*$-Hopf algebras $\mathcal{A}(\U(1))$ and $\cu(1)\simeq\{K,K^{-1}\}$ are dually paired via the pairing, induced by the one in \eqref{ndp} between $\ASU$ and $\su$, with
\begin{equation}
\hs{K^{\pm1}}{z}=q^{\mp\frac{1}{2}},\qquad\hs{K^{\pm1}}{z^{-1}}=q^{\pm\frac{1}{2}},
\label{bp1}
\end{equation}
on the generators. Thus, the exterior derivative ${\rm d}:\mathcal{A}(\U(1))\to\Omega^{1}(\U(1))$ can be written,
for any $u\in\,\mathcal{A}(\U(1))$,
as $ {\rm d} u=(X\lt u)\ \theta $ on the left invariant basis 1-form $\theta\sim[z-1]$. On the generators of the coordinate algebra one has
\begin{equation}
{\rm d} z=\frac{q-1}{q-q^{-1}}\,z \ \theta,
\qquad {\rm d} z^{-1}=\frac{q^{-1}-1}{q-q^{-1}}\,z^{-1} \ \theta,
\label{dz1}
\end{equation}
so to have $\theta=(q-1)(q-q^{-1})^{-1}z^{-1}{\rm d} z$. From the coproduct $\Delta X=1\otimes X+X\otimes K^{-2}$ the
$\mathcal{A}(\U(1))$-bimodule structure in $\Omega^{1}(\U(1))$ is
$$
\theta \ z^{\pm}=q^{\pm}z^{\pm}\ \theta.
$$
\subsection{Connections on the principal bundle}\label{conHf}
The compatibility -- as described in appendix~\S\ref{ap:qpb} and expressed by the exactness of the sequence \eqref{des} -- of the differential calculus $\U(1)$ presented above with the 4D differential calculus on $\SU$ presented in \S\ref{se:4dc}, has been proved in \cite{gs10}. As a consequence, collecting the various terms, the data
$$
\left(\ASU, \Asq, \mathcal{A}(\U(1)); \mathcal{N}_{\SU}=r^{-1}(\SU\otimes\mathcal{Q}_{\SU}), \mathcal{Q}_{\U(1)}\right)
$$
is a quantum principal bundle with the described calculi.
In order to obtain connections on this bundle, that is maps \eqref{si} splitting the sequence \eqref{des}, we need to compute the action of the map
$$\sim_{\mathcal{N}_{\SU}}:\Omega^{1}(\SU)\to\ASU\otimes(\ker\varepsilon_{\U(1)}/\mathcal{Q}_{\U(1)})$$ defined via the diagram \eqref{qdia}.
Since it is left $\ASU$-linear, we take as representative universal 1-forms corresponding to the left invariant 1-forms \eqref{om4} in $\Omega^1(\SU)$:
\begin{align*}
&\pi^{-1}_{\mathcal{N}_{\SU}}(\omega_{+})=(a\delta c-qc\delta a) \\
&\pi^{-1}_{\mathcal{N}_{\SU}}(\omega_{-})= (c^* \delta a^*- qa^*\delta c^*) \\
&\pi^{-1}_{\mathcal{N}_{\SU}}(\omega_{0})=\{a^{*}\delta a+c^{*}\delta c+ q(a\delta a^{*}+q^{2}c\delta c^{*})\}/ (q+1)\lambda_{1} \\%\in[\pi_{\mathcal{Q}_{\SU}}]^{-1}(\omega_{0}),
&\pi^{-1}_{\mathcal{N}_{\SU}}(\omega_{z})=a^{*}\delta a+c^*\delta c - (a\delta a^*+q^2 c\delta c^*) .
\end{align*}
On them the action of the canonical map \eqref{chimap} is found to be:
\begin{align*}
&\chi(a\delta c-qc\delta a)=(ac-qca)\otimes(z-1)=0 \\
&\chi(c^* \delta a^*- qa^*\delta c^*) = (c^*a^* - qa^*c^*)\otimes(z^{*}-1)=0 \\
&\chi\left((1+q)^{-1}\lambda_{1}^{-1}\{a^{*}\delta a+c^{*}\delta c+ q(a\delta a^{*}+q^{2}c\delta c^{*})\}\right)=1\otimes\{(z-1)+q(z^{-1}-1)\}=1\otimes\xi \\
&\chi(a^{*}\delta a+c^*\delta c - (a\delta a^*+q^2 c\delta c^*))=1\otimes(z-z^{-1})
\end{align*}
with $\xi\in\mathcal{Q}_{\SU}$ introduced in \S\ref{se:cu1}. From the isomorphism \eqref{lam4} one finally has:
\begin{align}
&\sim_{\mathcal{N}_{\SU}}(\omega_{\pm})= \sim_{\mathcal{N}_{\SU}}(\omega_{0})=0 \nonumber \\
&\sim_{\mathcal{N}_{\SU}}(\omega_{z})=1\otimes(1+q^{-1})[z-1].
\label{omzv}
\end{align}
{}From these one recovers $\Omega^{1}_{\mathrm{hor}}(\SU)=\ker\sim_{\mathcal{N}_{\SU}}$ with, using \eqref{bis2},
\begin{equation}
\label{le:hf}
\ker\sim_{\mathcal{N}_{\SU}}\simeq\ASU\{\omega_{\pm},\omega_{0}\}\simeq\{\omega_{\pm},\omega_{0}\}\ASU.
\end{equation}
\begin{rema} \label{horv}
\textup{
From \eqref{omzv}, for the generator $X=L_z$ in \eqref{X1} one gets that
$$
\widetilde{X}(\omega_z) = \langle{X},{\sim_{\mathcal{N}_{\SU}}(\omega_{z})}\rangle
=1 ,
$$
which identifies $L_{z}\in\,\mathcal{X}_{\mathcal{Q}}$ as a vertical vector for the fibration. In turn it is used to extend the notion of horizontality to higher order forms in $\Omega(\SU)$. One defines \cite{KS97} a contraction operator $i_{L_{z}}:\Omega^{k}(\SU)\to\Omega^{k-1}(\SU)$, giving $i_{L_{z}}(\omega_{\pm})=i_{L_{z}}(\omega_{0})=0$, and $i_{L_{z}}(\omega_{z})=1$ on 1-forms, so that $\ker i_{L_{z}}\simeq\Omega^{1}_{\mathrm{hor}}(\SU)$. Then one defines
\begin{equation}
\Omega^{k}_{\mathrm{hor}}(\SU):=\left.\ker\right|_{\Omega^{k}(\SU)}i_{L_{z}},
\label{khor}
\end{equation}
that is the kernel of the contraction map when restricted to the bimodule of $k$-forms.
}
\end{rema}
Given the explicit expression \eqref{omzv} for the canonical map compatible with the differential calculi we are using, and the $\mathcal{A}(\U(1))$-coaction
\begin{equation}
\delta_{R}^{(1)}\omega_{z}=\omega_{z}\otimes 1, \qquad
\delta_{R}^{(1)}\omega_{0}=\omega_{0}\otimes 1, \qquad
\delta_{R}^{(1)}\omega_{\pm}=\omega_{\pm}\otimes z^{\pm2},
\label{rifo}
\end{equation}
using the vector space basis $\varphi(j)$ in $\ker\varepsilon_{\U(1)}$ of \S\ref{se:cu1},
a connection \eqref{si} is given by
\begin{equation}
\tilde{\sigma}(\phi\otimes[\varphi(j)])=q^{-2j}(1+q^{-1})^{-1}\phi(\omega_{z}+\mathrm{a})
\label{si3}
\end{equation}
for any $\phi\in\,\ASU$ and any element $\mathrm{a}\in\,\Omega^1(S^{2}_{q})$. On vertical forms, the projection $\Pi$ associated to this connection turns out to be
\begin{align}
&\Pi(\omega_{\pm})=0=\Pi(\omega_{0}),\nonumber \\
&\Pi(\omega_{z})=\tilde{\sigma}(\sim_{\mathcal{N}_{\SU}}(\omega_{z}))=\tilde{\sigma}(1\otimes[\varphi(0)])=\omega_{z}+\mathrm{a},
\label{Pi3}
\end{align}
while the corresponding connection 1-form $\omega:\mathcal{A}(\U(1))\to\Omega^{1}(\SU)$ is given by
\begin{equation}
\omega(z^{n})=\tilde{\sigma}(1\otimes[z^{n}-1])
=q^{n/2}[\tfrac{n}{2}] (\omega_{z}+\mathrm{a}).
\label{ome3}
\end{equation}
Connections corresponding to $\mathrm{a}=s\omega_{0}$ with $s\in\,\IR$ were already considered in \cite{durII}.
The vertical projector \eqref{Pi3} allows one to define a covariant derivative
$$
\mathfrak{D}:\ASU\to\Omega^{1}_{\mathrm{hor}}(\SU),
$$
given (as usual) as the horizontal projection of the exterior derivative:
\begin{equation}
\label{mfD}
\mathfrak{D}\phi:=(1-\Pi){\rm d}\phi.
\end{equation}
Covariance here clearly refers to the right coaction of the structure group $\U(1)$ of the bundle, since it is that $\delta_{R}\phi=\phi\otimes z^{-n} \,\Leftrightarrow\,\delta_{R}^{(1)}(\mathfrak{D}\phi)=(\mathfrak{D}\phi)\otimes z^{-n}$. From \eqref{ome} the action of this operator can be written as
\begin{equation}
\label{mfDo}
\mathfrak{D}\phi={\rm d}\phi-\phi\wedge\omega(z^{-n})
\end{equation}
for any $\phi\in\,\mathcal{L}_{n}$. From the bimodule structure \eqref{bis2} it is easy to check that all the above connections are \emph{strong} connections in the sense of \cite{hajsc}.
\medskip
The analysis in this section allows us to prove the results in Proposition~\ref{lea}. The exterior algebra $\Omega(S^{2}_{q})$ is defined to be the set of horizontal and $\U(1)$-coinvariant elements in $\Omega(\SU)$, with respect to the extension $\delta_{R}^{(k)}$ (introduced in App.\ref{ap:qpb}) of the canonical coaction \eqref{cancoa} to higher order forms in $\Omega(\SU)$. It is then easy to check, from \eqref{khor} and \eqref{rifo}, that the isomorphisms given in expressions \eqref{ntd} for $\Omega(S^{2}_{q})$ do hold.
\section{Gauged Laplacians on line bundles} \label{GLoLB}
Each $\Asq$-bimodule $\mathcal{L}_{n}$ defined in \eqref{libu} is a bimodule of co-equivariant elements in $\ASU$ for the right $\U(1)$-coaction \eqref{cancoa}, and as such can be thought of as a module of `sections of a line bundle' over the quantum sphere $S^{2}_{q}$. Without requiring any compatibility with additional structures, any $\mathcal{L}_{n}$ can be realized both as a projective right or left $\Asq$-module (of rank 1 and winding number $-n$). One of such structures is that of a connection
on the quantum principal bundle $\Asq\hookrightarrow\ASU$. By transporting the covariant derivative \eqref{mfD} on the principal bundle to a derivative on sections forces to break the symmetry between the left or the right $\Asq$-module realization of $\mathcal{L}_{n}$.
With the choice in
\S\ref{s:qsb} for the principal bundle, we need an isomorphism $\mathcal{L}_{n}\simeq\mathcal{F}_n$ with $\mathcal{F}_{n}$ a projective left $\Asq$-module \cite{HM98}.
This isomorphism is constructed in terms of a projection operator $\mathfrak{p}^{\left(n\right)}$.
Given this identification, in \S\ref{se:lbss} we shall describe the complete equivalence between covariant derivatives on $\mathcal{F}_{n}$ (associated to the $3{\rm d}$ left covariant differential calculus over
$S^{2}_{q}$) and connections (as described in \S\ref{s:ccd}) on the principal bundle $\Asq\hookrightarrow\ASU$, corresponding to compatible $4D_{+}$ bicovariant calculus over $\SU$ and $3{\rm d}$ left covariant calculus over $S^{2}_{q}$. We shall then move to a family of gauged Laplacian operators on $\mathcal{F}_{n}$, obtained by coupling the Laplacian operator over the quantum sphere with a set of suitable gauge potentials. We
finally show that among them there is one whose action extends to $\mathcal{L}_n$ the action of the Laplacian \eqref{sfr} on $\mathcal{L}_0\simeqS^{2}_{q}$. As we noticed in \S\ref{se:L}, the action of the (right) Laplacian \eqref{sfr} on $S^{2}_{q}$ is given by the restriction of the action \eqref{erp4} of the (right) Laplacian $\Box^R_{\SU}$. Here we obtain that the action of such gauged Laplacian can be written in terms of the ungauged (right) Laplacian on $\SU$, in parallel to what happens on a classical principal bundle (see e.g. \cite[Prop.~5.6]{bgv}) and on the Hopf fibration of the sphere $S^{2}_{q}$ with calculi coming from the left covariant one on $\SU$ as shown in \cite{lareza,ale09}.
\subsection{Line bundles as projective left $\Asq$-modules}\label{se:lbss}
Every (equivariant, the only ones we use in this paper) finitely generated projective (left or right) $\Asq$-module is a direct sum of $\mathcal{L}_{n}$'s (cf. \cite{SWPod}). As said, these are line bundles of degree $-n$ on the quantum sphere and to describe them all one needs is a collection of idempotents $\mathfrak{p}^{\left(n\right)}$, which we are going to introduce.
With $n\in\,\IZ$, we consider the projective left
$\Asq$-module $\mathcal{F}_{n}=(\Asq)^{\mid n\mid+1}\mathfrak{p}^{\left(n\right)}$, with projections \cite{BM98,HM98} (cf. also \cite{lareza})
\begin{equation}
\mathfrak{p}^{\left(n\right)}=\ket{\Psi^{\left(n\right)}}\bra{\Psi^{\left(n\right)}}
\label{dP},
\end{equation}
written in terms of elements
$\ket{\Psi^{\left(n\right)}}\in \ASU^{\mid n\mid+1}$ and their duals $\bra{\Psi^{\left(n\right)}}$ as follows. One has:
\begin{align}
&n\leq0:\qquad\ket{\Psi^{\left(n\right)}}_{\mu}=\sqrt{\alpha_{n,\mu}} ~ c^{\mid n\mid-\mu}a^{\mu}\,\in \mathcal{L}_{n}, \nonumber \\
&\mathrm{where}\qquad\alpha_{n,\mid n\mid}=1;\qquad\alpha_{n,\mu}=\prod\nolimits_{j=0}^{\mid n\mid-\mu-1}\left(\frac{1-q^{2\left(\mid n\mid-j\right)}}
{1-q^{2\left(j+1\right)}}\right), \quad \mu = 0, \ldots, \mid n\mid-1
\label{ketpi}
\end{align}
\begin{align}
&n\geq 0:\qquad\ket{\Psi^{\left(n\right)}}_{\mu}=\sqrt{\beta_{n,\mu}} ~ c^{* \mu}a^{* n-\mu}\,\in \mathcal{L}_{n}, \nonumber\\
&\mathrm{where} \qquad \beta_{n,0}=1;\qquad
\beta_{n,\mu}=q^{2\mu}\prod\nolimits_{j=0}^{\mu-1}\left(\frac{1-q^{-2\left(n-j\right)}}{1-q^{-2\left(j+1\right)}}\right),
\quad \mu = 1, \ldots, n
\label{ketni} .
\end{align}
The coefficients are chosen so that $\hs{\Psi^{\left(n\right)}}{\Psi^{\left(n\right)}}=1$, as a consequence $(\mathfrak{p}^{\left(n\right)})^{2}=\mathfrak{p}^{\left(n\right)}$. Also by construction it holds that $(\mathfrak{p}^{\left(n\right)})^{\dagger}=\mathfrak{p}^{\left(n\right)}$.
The isomorphism $\mathcal{L}_n \simeq \mathcal{F}_{n}=(\Asq)^{\mid n\mid+1}\mathfrak{p}^{\left(n\right)}$ is realized as follows:
\begin{equation}
\mathcal{L}_{n} ~\xrightarrow{~\simeq~}~ \mathcal{F}_{n}, \quad \phi \mapsto \bra{\sigma_{\phi}} = \phi \bra{\Psi^{(n)} },
\label{iso1n}
\end{equation}
with inverse
$$
\mathcal{F}_{n} ~\xrightarrow{~\simeq~}~ \mathcal{L}_{n}, \quad \bra{\sigma_{\phi}} \mapsto \phi =\hs{\sigma_{\phi}}{ \Psi^{(n)} }.
$$
Given the exterior algebra $(\Omega(S^{2}_{q}),{\rm d})$ on the quantum sphere we are considering,
a covariant derivative on the left $\Asq$-modules $\mathcal{F}_n$ is a $\IC$-linear map
\begin{equation}
\nabla:\Omega^k(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}\,\to\,\Omega^{k+1}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}
\label{cdev}
\end{equation}
that satisfies the \emph{left} Leibniz rule
$$
\nabla(\xi\wedge\bra{\sigma})=({\rm d}\xi)\wedge\bra{\sigma}+(-1)^m \xi\wedge \nabla\bra{\sigma}
$$
for any $\xi\,\in\,\Omega^{m}(S^{2}_{q})$ and $\bra{\sigma}\,\in\,\Omega^{k}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}$.
The curvature associated to a covariant derivative is
$\nabla^{2}:\mathcal{F}_{n}\to\Omega^{2}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}$, that is $\nabla^{2}(\xi\,\bra{\sigma})=\xi\,\nabla^2(\bra{\sigma})=\xi\,F_{\nabla}(\bra{\sigma})$ with the last equality defining the curvature 2-form $F_{\nabla}\in\,\textup{Hom}_{\Asq}(\mathcal{F}_{n},\Omega^{2}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n})$.
Any covariant derivative -- an element in $C(\mathcal{F}_{n})$ -- and its curvature
can be written as
\begin{align}
\nabla\bra{\sigma}&=({\rm d}\bra{\sigma})\mathfrak{p}^{\left(n\right)}\,+(-1)^k\,\bra{\sigma}\mathrm{A^{(n)}}, \nonumber
\\
\nabla^{2}\bra{\sigma}&=\bra{\sigma}\{-{\rm d}\mathfrak{p}^{\left(n\right)}\wedge{\rm d}\mathfrak{p}^{\left(n\right)}\,+\,{\rm d}\mathrm{A^{(n)}}\,-\,\mathrm{A^{(n)}}\wedge\mathrm{A^{(n)}}\}\mathfrak{p}^{\left(n\right)} .
\label{cocug}
\end{align}
with $\bra{\sigma}\in\,\Omega^{k}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}$. The negative signs in the second expression above come from the left Leibniz rule, since form valued sections are elements of projective left $\Asq$-modules.
For the `gauge potential' $\mathrm{A^{(n)}}$ one has
\begin{equation}
\mathrm{A^{(n)}}=\mathfrak{p}^{\left(n\right)} \mathrm{A^{(n)}}=\mathrm{A^{(n)}}\mathfrak{p}^{\left(n\right)}=\ket{\Psi^{\left(n\right)}}\mathrm{a}^{(n)}\bra{\Psi^{\left(n\right)}}\,\in\,\mathrm{Hom}_{\Asq}(\mathcal{F}_n, \Omega^{1}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}),
\label{apA}
\end{equation}
with $\mathrm{a}^{(n)}\in\,\Omega^{1}(S^{2}_{q})$. The monopole (Grassmann) connection corresponds to $\mathrm{a}^{(n)}=0$.
In analogy with the identification \eqref{iso1n}, the covariant derivative $\nabla$ naturally induces an operator
$D:\mathcal{L}_{n}\to\mathcal{L}_{n}\otimes_{\Asq}\Omega^{1}(S^{2}_{q})$ that can be written as
\begin{equation}
D\phi:=(\nabla\bra{\sigma_\phi})\ket{\Psi^{\left(n\right)}}={\rm d}\phi-\phi\,\{\hs{\Psi^{\left(n\right)}}{{\rm d}\Psi^{\left(n\right)}}-\mathrm{a}^{(n)}\}.
\label{domeg}
\end{equation}
We refer to the 1-form
\begin{equation}
\label{coAp}
\Omega^{1}(\SU)\,\ni\,\varpi^{(n)}=\left(\hs{\Psi^{\left(n\right)}}{{\rm d}\Psi^{\left(n\right)}}-\mathrm{a}^{(n)}\right)
\end{equation}
as the connection 1-form of the gauge potential. It allows to express the curvature as
\begin{equation}
F_{\nabla}=-\ket{\Psi^{\left(n\right)}}\left( {\rm d}\varpi^{(n)}+\varpi^{(n)}\wedge\varpi^{(n)} \right)\bra{\Psi^{\left(n\right)}}
\label{Fp}
\end{equation}
where $({\rm d}\varpi^{(n)}+\varpi^{(n)}\wedge\varpi^{(n)})\in\,\Omega^{2}(S^{2}_{q})$.
The covariant derivatives defined above on the left modules $\mathcal{F}_n$ fit in the general theory of connections on the quantum Hopf bundle as described in the \S\ref{conHf}: any covariant vertical projector, as in \eqref{Pi3}, induces a gauge potential $\mathrm{A^{(n)}}$ as in \eqref{apA}. The notion \eqref{coAp} of connection 1-form of a given gauge potential in $C(\mathcal{F}_{n})$ matches the notion \eqref{ome3} of connection 1-form $\omega:\mathcal{A}(\U(1))\to\Omega^{1}(\SU)$ on the Hopf bundle. From the $\Asq$-bimodule isomorphism $\oplus_{n\in\,\IZ}\mathcal{L}_{n}\otimes_{\Asq}\Omega^{1}(S^{2}_{q})\,\simeq\,\Omega^{1}_{\mathrm{hor}}(\SU)$ (see Remark~\ref{horv}), this matching amounts to equate the actions of the covariant derivative operators \eqref{domeg} and \eqref{mfD},
\begin{equation}
\forall\,\phi\in\,\mathcal{L}_{n}: \quad D\phi=\mathfrak{D}\phi\qquad\Leftrightarrow\qquad\varpi^{(n)}=\omega(z^{-n}).
\label{omom}
\end{equation}
From formula \eqref{ome3}, this correspondence can be written as
\begin{equation}
\mathrm{a}^{(n)}=\lambda_{n}\omega_{0}-\xi_{-n}\mathrm{a},
\label{figi}
\end{equation}
where the coefficients refer to the eigenvalue equations:
\begin{align}
&L_{z}\lt\ket{\Psi^{\left(n\right)}}:=\xi_{n}\ket{\Psi^{\left(n\right)}} \qquad\Rightarrow\qquad\xi_{n}=-q^{-\tfrac{n}{2}}\,[\frac{n}{2}]\nonumber \\
&L_{0}\lt\ket{\Psi^{\left(n\right)}}:=\lambda_{n}\ket{\Psi^{\left(n\right)}}\qquad\Rightarrow\qquad\lambda_{n}=[\frac{\mid n\mid}{2}] \label{lamni}
[\frac{\mid n\mid}{2}+1].
\end{align}
Finally, the equivalence \eqref{omom} allows one to introduce a covariant derivative
$$ D:\Omega^{k}_{\mathrm{hor}}(\SU)\to\Omega^{k+1}_{\mathrm{hor}}(\SU),
$$ thus extending to horizontal forms on the total space of the quantum Hopf bundle the covariant derivative operator on $\ASU$ as given in \eqref{mfD}. This follows the formulation described in \cite{hajsc}, since any connection on the principal bundle is strong. Upon defining
$$
\mathcal{L}_{n}^{(k)}:=\{\phi\in\,\Omega^{k}_{\mathrm{hor}}(\SU)\,:\,\delta_{R}^{(k)}\phi=\phi\otimes z^{-n}\},
$$
where $\delta_{R}^{(k)}$ is the natural right $\U(1)$-coaction on $\Omega^{k}(\SU)$, one obtains:
\begin{equation}
D\phi={\rm d}\phi-(-1)^{k}\phi\wedge\omega(z^{-n}).
\label{esD}
\end{equation}
A further extension to the whole exterior algebra $\Omega(\SU)$ is proposed in \cite{durII}: a generalisation of the analysis in \cite[\S9]{ale09} shows how this extension is far from being unique.
We restrict our analysis again to covariant derivatives $\nabla_{s}\ket{\sigma}$ in \eqref{cocug}
whose gauge potential and corresponding connection 1-form
are of the form:
\begin{align}
\mathrm{A}_{s}^{(n)}=s\ket{\Psi^{\left(n\right)}}\omega_{0}\bra{\Psi^{\left(n\right)}}, \qquad\qquad \varpi_{s}^{(n)}=\xi_{n}\omega_{z}\,+\,(\lambda_{n}-s)\omega_{0} ,
\label{osn}
\end{align}
for $s\in\IR$ and coefficients as in \eqref{lamni}, since they reduce in the classical limit to the monopole connection on line bundles associated to the classical Hopf bundle $\pi:S^3\to S^{2}$.
Relations \eqref{2fw} and \eqref{d2f} allow to compute the curvature 2-form \eqref{Fp} as
\begin{align*}
&{\rm d}\varpi_{s}^{(n)}=\left( (q+q^{-1})\xi_{n}-(s-\lambda_{n})(q-q^{-1}) \right) \omega_+\wedge\omega_- \nonumber \\
&\varpi_{s}^{(n)}\wedge\varpi_{s}^{(n)}=(q-q^{-1})\xi_{n} \left( (q+q^{-1})\xi_{n}+(q-q^{-1})(s-\lambda_{n}) \right)
\omega_{+}\wedge\omega_{-}.
\end{align*}
\subsection{Gauged Laplacians}
In order to introduce an Hodge operator
\begin{equation}
\star^{\mathcal{R}}:\Omega^{k}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n}\to\Omega^{3-k}(S^{2}_{q})\otimes_{\Asq}\mathcal{F}_{n},
\label{scR}
\end{equation}
we use the right $\Asq$-linear Hodge operator \eqref{cRo} on $\Omega(S^{2}_{q})$:
\begin{equation}
\star^{\mathcal{R}}(\xi\,\bra{\sigma}):=(\check{R}\xi)\bra{\sigma}
\label{scRe}
\end{equation}
so that a gauged Laplacian operator is defined as
$$
\Box^{\mathcal{R}}_{\nabla}:\mathcal{F}_{n}\to\mathcal{F}_{n},\qquad\qquad\qquad\Box^{\mathcal{R}}_{\nabla}\bra{\sigma}:=-\star^{\mathcal{R}}\nabla(\star^{\mathcal{R}}\nabla\bra{\sigma}) .
$$
Equivalently we have an operator on $\mathcal{L}_{n}\simeq\mathcal{F}_{n}$ via the left $\Asq$-modules isomorphism \eqref{iso1n}. With $\phi=\hs{\sigma}{\Psi^{\left(n\right)}}$, it holds that
\begin{equation}
\Box^{\mathcal{R}}_{\nabla}:\mathcal{L}_{n}\to\mathcal{L}_{n}, \qquad\qquad\qquad \Box^{\mathcal{R}}_{\nabla}\phi=(\Box^{\mathcal{R}}_{\nabla}\bra{\sigma})\ket{\Psi^{\left(n\right)}}.
\label{glp}
\end{equation}
With the family of connections \eqref{osn} and using the identities
\begin{align}
&\left(R_{\pm}\lt\,\bra{\sigma}\right)\ket{\Psi^{\left(n\right)}}=q^{-n}\,R_{\pm}\lt \phi ,\nonumber \\
&\left(R_{0}\lt\,\bra{\sigma}\right)\ket{\Psi^{\left(n\right)}}=q^{-n}\left(R_{0}-[\frac{\mid n\mid}{2}][1-\frac{\mid n\mid}{2}]\right)\lt\phi
\label{idL}
\end{align}
one readily computes:
\begin{equation}
\label{rfB}
\Box_{\nabla_{s}}^{\mathcal{R}}\phi=
q^{-2n}\left\{ \alpha\, ( q^2 R_{+}R_{-}+ R_{-}R_{+} ) +\nu\,(R_{0}+sq^{-n}-[\frac{\mid n\mid}{2}][1-\frac{\mid n\mid}{2}])^2\right\}\lt \phi.
\end{equation}
Finally, fixing the parameter to be
\begin{equation}
\label{sCr}
s(n)= q^{n} [\frac{\mid n\mid}{2}][1-\frac{\mid n\mid}{2}],
\end{equation}
the action of the gauged Laplacians extends, apart from a multiplicative factor depending on the label $n$, to elements in the line bundles $\mathcal{L}_{n}$ the action of the Laplacian operator \eqref{sfr} on the quantum sphere, that is,
\begin{equation}
q^{2n}\left(\Box^{\mathcal{R}}_{\nabla_{s}}\phi\right)
=\left\{ \alpha ( q^2 R_{+}R_{-}\,+\,R_{-}R_{+} )\,+\,\nu R^2_{0} \right\}\lt\phi.
\label{finr}
\end{equation}
From \eqref{Rder}, the above action can be written on $\phi\in\,\mathcal{L}_{n}$ as the left action \eqref{d4} of a polynomial in $\su$. We get
\begin{align}\label{zed}
\Box^{\mathcal{R}}_{\nabla_{s}} &= (- 2q \alpha\, R_0 K^{2}+\nu\,R_{0}^2) K^{-4} - q\alpha
\frac{(q+q^{-1})(K-K^{-1})^2}{(q-q^{-1})^2}\,K^{-2} \\
&=2q \alpha\, (C_{q} - [\tfrac{1}{2}]^{2}+\tfrac{1}{4} ) K^{-2}+\nu\,(C_{q} - [\tfrac{1}{2}]^{2}+\tfrac{1}{4} )^2) K^{-4} - q\alpha
\frac{(q+q^{-1})(K-K^{-1})^2}{(q-q^{-1})^2}\,K^{-2} \nonumber ,
\end{align}
having used \eqref{casbis}. This relation is the counterpart of what happens on a classical principal bundle (see e.g. \cite[Prop.~5.6]{bgv}) and on the Hopf fibration of the sphere $S^{2}_{q}$ with calculi coming from the left covariant one on $\SU$ as shown in \cite{lareza,ale09}.
|
{
"timestamp": "2011-07-12T02:01:00",
"yymm": "1009",
"arxiv_id": "1009.3738",
"language": "en",
"url": "https://arxiv.org/abs/1009.3738"
}
|
\section*{Acknowledgments}
ST is supported by the JSPS under Contract No. 20-10616.
|
{
"timestamp": "2010-09-21T02:01:01",
"yymm": "1009",
"arxiv_id": "1009.3568",
"language": "en",
"url": "https://arxiv.org/abs/1009.3568"
}
|
\section{Introduction}
PKS~1934$-$638 is the archetype of a class of radio galaxies known as GHz-Peaked Spectrum (GPS) radio galaxies. The properties of this class are well described by \citet{odea98}.
PKS~1934$-$638 was first noted in terms of its strongly peaked radio spectrum at 1.4 GHz by \citet{bol63} and \citet{kel66}, indicating the likely existence of compact structure in the radio source \citep{sli63}. Subsequent interferometric observations confirmed this \citep{gub71} and high angular resolution observations of PKS~1934$-$638 were made periodically over the next 20 years \citep{pre89,tzi5ghz,tzi02,kin94}, at a range of frequencies. Most recently, \citet{oja04} presented dual-epoch 8.4 GHz VLBI observations of PKS~1934$-$638 as part of a combined analysis of $\sim$30 years of observations.
\citet{oja04} estimated an expansion rate of 23$\pm10\mu$as/yr between the two compact components of radio emission that dominate the structure of PKS~1934$-$638, at a redshift of $z=0.183$ \citep{pen78} giving an apparent separation speed of $\sim 0.2\pm0.1c$, an order of magnitude higher than what is generally assumed for the hot spot advance speed of Cygnus A \citep{rea96}. Throughout the history of VLBI observations of PKS~1934$-$638 it has been clear that the structure and flux density of the object evolve slowly, if at all. This stability is consistent with the general properties of GPS radio galaxies \citep{odea98}. The flux density stability of PKS~1934$-$638, in particular, has led it to be used as the primary flux density calibrator for the Australia Telescope Compact Array at centimetre wavelengths.
Because of the proposed physical nature of GPS sources, as the young progenitors of FR-I and FR-II radio galaxies \citep{odea98}, studies of the evolution of GPS sources are very important, in order to estimate their ages via measurement of expansion of the radio structure \citep{pol03}. Long time series observations are required because of the slow rate of evolution in GPS radio galaxies and further, optical depth effects may complicate the interpretation of multi-epoch observations made at different frequencies. Mechanisms such as free-free absorption and synchrotron self-absorption have been proposed as providing significant optical depth in GPS sources at radio wavelengths and these different mechanisms are potentially difficult to disentangle \citep{tin03}. GPS radio spectra peak at GHz frequencies and the optical depths change rapidly over the frequency range 1 - 10 GHz. One must therefore be wary when interpreting multi-epoch data obtained at different frequencies, as is the case for the historical VLBI data for PKS~1934$-$638. Frequency dependent structure has been clearly seen in another GPS radio galaxy, CTD 93 \citep{sha99, nag06}.
\citet{oja04}, in their analysis of historical PKS~1934$-$638 VLBI data, raise the possibility that frequency dependent structure effects complicate their analysis. In this paper we further examine the possibility of frequency dependent structure in PKS~1934$-$638 and find evidence that the apparent expansion of the source noted by \citet{oja04} can be explained as a consequence of these frequency dependent effects. In section 2 we describe new VLBI observations at 1.4 GHz that use the Australian Long Baseline Array (LBA) augmented by two new radio telescopes. In section 3 we discuss our new results within the context of the historical results (both those presented by Ojha et al. and other data we have extracted from the literature), in particular the possibility of frequency dependent structure in PKS~1934$-$638 and the impact on historical measurements of the expansion rate of PKS~1934$-$638.
\section{New VLBI observations and results}
VLBI observations of PKS~1934$-$638 were undertaken on 2010 April 29, using the array of radio telescopes listed in Table 1 and shown in Figure 1. Two of these telescopes were used for the first time for scientific observations, the first ASKAP (Australian SKA Pathfinder) antenna in Western Australia \citep{joh07}, and Warkworth, a new facility of the Auckland University of Technology in New Zealand \citep{gul09}. The addition of ASKAP and Warkworth greatly increase the angular resolution (factor of $\sim$4) and $(u,v)$ coverage for observations of radio sources by the Australian Long Baseline Array (LBA). This is particularly useful at relatively low frequencies, where long baselines are required to obtain high angular resolution.
\begin{deluxetable}{lllcr}
\tablecaption{List of radio telescopes used in the VLBI observations. ASKAP $=$ Australian Square Kilometre Array Pathfinder; ATCA $=$ Australia Telescope Compact Array. Longitudes are listed as east of Greenwich. SEFD $=$ System Equivalent Flux Density.}
\tablewidth{0pt}
\startdata
Telescope&Long. (deg)&Lat. (deg)&Diameter (m)&SEFD (Jy) \\ \hline
ASKAP& 116.64 &$-$26.69& 12& 6000 \\
Hobart& 147.44 &$-$42.80& 26& 600 \\
Parkes& 148.26 &$-$33.00& 64& 25 \\
Mopra& 149.07 &$-$31.30& 22& 350 \\
ATCA& 149.57 &$-$30.31& 5 $\times$ 22& 70 \\
Warkworth& 174.66 &$-$36.43& 12& 8000 \\ \hline
\enddata
\end{deluxetable}
\begin{figure}
\epsscale{0.5}
\plotone{f1.eps}
\vspace{2cm}
\caption{Schematic geographical distribution of antennas used in the VLBI observations of PKS~1934$-$638}
\end{figure}
The VLBI observations were made by recording right and left circularly polarised signals at each antenna in the frequency range 1368 $-$ 1432 MHz (a 64 MHz bandwidth). Data at the ATCA, Parkes, Mopra and Hobart antennas were recorded with the standard LBA systems \citep{phi09}. The observation occurred over a period of six hours.
The data from the ASKAP and Warkworth antennas were obtained from a custom-made single-pixel feed and receiver system that delivered the band limited signals (downconverted to the frequency range 256 $-$ 320 MHz) at both polarisations to a custom-made digital system. The digital system utilises a commodity Signatec sampler/digitiser PCI-based card (Signatec PX14400) to sample the band limited analog signals at the Nyquist-Shannon rate \citep{sha49}
and digitise the signals with 14-bit precision. These 14-bit samples are then recorded to a 32 TB RAID disk. The Signatec card and RAID are hosted in a server class PC. After recording, the 14-bit samples are converted to 2-bit precision, to decrease data storage and transport requirements, and to be more compatible with the 2-bit samples recorded with the standard LBA system. A full description of the digital system will appear in \citet{sta10}. A hydrogen maser clock was available at Warkworth and a rubidium clock was used at ASKAP, to drive frequency synthesisers for the local oscillator and the digital system sample clock. Global Positioning System receivers at Warkworth and ASKAP were used to determine ($x,y,z$) positions for the antennas and generate 1PPS signals for the digital system.
Following the observations, data from most telescopes were transferred via fast network connections to Curtin University of Technology in Western Australia to be correlated using the DiFX software correlator \citep{del07}. The data from the ASKAP telescope were transported to Perth by car, as fast network connections are still under construction. The data were correlated using an integration period of 2 seconds and 32 frequency channels per 64 MHz band. All Stokes parameters were correlated. PKS~1934$-$638 was detected with high signal to noise on all baselines in the array. The resultant $(u,v)$ coverage for the observation is shown in Figure 2. All baselines above $6M\lambda$ are due to the new antennas at ASKAP and Warkworth, increasing the maximum baseline to ~$24M\lambda$ and thus the array resolution by a factor of ~4.
\begin{figure}
\epsscale{0.5}
\plotone{f2.eps}
\caption{$(u,v)$ coverage of the VLBI observation of PKS~1934$-$638. All baselines larger than $6M\lambda$ are contributed by the new antennas at ASKAP and Warkworth.}
\end{figure}
The correlated data were imported into AIPS for initial processing using standard techniques such as fringe-fitting. The data were also amplitude calibrated using system temperature and gain values. For the new antennas at ASKAP and Warkworth these calibration parameters were measured at the telescopes by performing on-off pointings toward strong radio sources of known strength. Once fringe-fitted and amplitude calibrated, the data were exported to DIFMAP \citep{she94} for excision of bad data, imaging and model-fitting.
Figure 3 shows the best representation of the Stokes I data in the image plane, obtained by model-fitting the visibility data, using the model specified in Table 2, determined using the modelfit task in DIFMAP. A uniform weighting of the $(u,v)$ data was used in the transformation to the image plane. Phase self-calibration was used in early iterations of model-fitting, with amplitude self-calibration used in the final iterations of model-fitting. Amplitude self-calibration led to changes in individual antenna gains of the order 15\%, relative to the {\it a priori} amplitude calibration.
\begin{figure}
\plotone{f3.eps}
\caption{Stokes I VLBI image of PKS~1934$-$638 at 1.4 GHz. Contour levels are $-$4 ,4, 8, 16, 32 and 64\% of the peak surface brightness of 2.4 Jy/beam. The restoring beam is 10.8 $\times$ 5.4 mas at a position angle of 45.7${^\circ}$. Note, the phase centre of the image has been shifted relative to the positions of the components indicated in Table 2 by (-20 mas, 0 mas), for convenience of display.}
\end{figure}
\begin{deluxetable}{crrccr}
\tablecaption{Model for the structure of PKS~1934$-$638 at 1.4 GHz. $S =$ flux density of model component; $R =$ distance of model component from phase centre of image; $\theta =$ position angle of model component centroid from the phase centre of image; $a =$ major axis length of model component; $r =$ minor axis to major axis ratio; $\phi =$ position angle of model component major axis}
\tablewidth{0pt}
\startdata
$S$ (Jy) &$R$ (mas) &$\theta$ (deg) &$a$ (mas) &$r$ &$\phi$ (deg) \\ \hline
5.6 & 0.0 & 0.0 & 8.7 & 0.9 & 6.4 \\
3.7 & 41.0 & $-$90.7 & 8.2 & 0.0 & $-$62.5 \\ \hline
\enddata
\end{deluxetable}
The structure seen in Figure 3 agrees very well with all previous published images of PKS~1934$-$638 in this angular resolution regime with synthesised beams of $\approx$ 5-10 mas \citep{tzi89,kin94,tzi5ghz,tzi02,oja04,vsop}. The two components have been interpreted as the terminal hot spots from oppositely directed jets emerging from a black hole accretion disk system, a smaller and younger version of an FR-I or FR-II type radio galaxy, into which GPS radio galaxies are postulated to evolve. This is the general interpretation of these double component structures, which are typical of GPS radio galaxies \citep{odea98}.
We derived errors on the main parameter of interest in this analysis, the separation between the two components in the image, in order to compare our results to historical results. We did this by keeping one model component fixed at the phase centre of the image and forcing variation of the position of the second component, then assessing the degradation of the fit of data to model as a function of this variation. This technique for estimating errors has been used extensively previously \citep{tin98}. In the case of PKS~1934$-$648, which is an almost equal strength double at this angular resolution, the visibility amplitudes and phases vary strongly, allowing this technique to tightly constrain the error bars on the separation, especially using the long and sensitive baselines from the east coast Australian antennas to ASKAP and Warkworth. We estimated that the separation of the two components at 1.4 GHz is 41.0 mas $\pm$ 0.6 mas (3$\sigma$).
\section{Discussion}
To place our new measurement of the separation between the two compact components in PKS~1934$-$638 within the context of the historical measurements, we reproduce in Table 3 the data compiled by \citet{oja04}, with the addition of our new datum at 1.4 GHz. Also listed in Table 3 are the results of other historical observations that were not included in the analysis of \citet{oja04}, namely a 5 GHz observation from the Southern Hemisphere VLBI Experiment (a precursor of the Australian LBA) \citep{tzi5ghz} and a 1.6 GHz observation from the VLBI Space Observatory Programme \citep{vsop}. In addition, the parameter measurements for the images of 1998 and 1991 \citep{kin94, tzi5ghz, tzi02} have been repeated using the original published images, and new estimates of separation and position angle are reported, together with conservative error estimates. Table 3 now includes all historical VLBI observations of PKS~1934$-$638, with all parameters reported in a consistent way.
\begin{deluxetable}{cllcc}
\tablewidth{0pt}
\tablecaption{Measured component separations in PKS~1934$-$638. References are: 1=\citet{gub71}; 2=\citet{tzi89}; 3=\citet{kin94}; 4=\citet{tzi02}; 5=\citet{tzi5ghz}; 6=\citet{vsop}; 7=\citet{oja04}; 8=This work.}
\startdata
Epoch &Separation& Position Angle & Freq (GHz) & references \\ \hline
1970.8 &41.9$\pm$0.2 &$-$90$\pm$0.1 & 2.3 & 1 \\
1982.3 &42.0$\pm$0.2 &$-$90.5$\pm$0.1 & 2.3 & 2 \\
1988.9 &41.4$\pm$0.5 &$-$92$\pm$0.5 & 2.3 & 3,4,5 \\
1991.9 &42.2$\pm$0.5 &$-$92$\pm$0.5 & 8.4 & 3,4,5 \\
1992.9 &41.9$\pm$0.5 &$-$92$\pm$0.5 &4.85 & 5 \\
1999.3 &41.4$\pm$0.5 &$-$92$\pm$0.5 &1.65 & 6 \\
2002.5 &42.7$\pm$0.4 &$-$92$\pm$0.5 &8.4& 7 \\
2002.9 &42.6$\pm$0.3 &$-$92$\pm$0.4 &8.4& 7 \\
2010.3 &41.0$\pm$0.6 &$-$91$\pm$1.0 & 1.4& 8 \\ \hline
\enddata
\end{deluxetable}
We reproduce Figure 2 of \citet{oja04} below in Figure 4, including the new datum and the additional historical data. Also plotted are lines showing the expansion rate of 23$\pm10\mu$as/yr between the two compact components of radio emission in PKS~1934$-$638, as estimated in \citet{oja04}.
As can be seen from Table 3 and Figure 4, the new measurement of separation at 1.4 GHz lies significantly below the extrapolation of the previously measured values of separation at 2.3 and 8.4 GHz. The same can be said of the 1.6 GHz VSOP datum. However, if the separations are plotted as a function of observing frequency, as shown in Figure 5, a consistent trend of separation with frequency is revealed, with larger separations measured at higher frequencies and smaller separations measured at lower frequencies.
\begin{figure}
\plotone{f4.eps}
\caption{Separation between the two components in PKS~1934$-$638, as a function of time. The solid lines show the predictions of the expansion rate estimated in \citet{oja04}.}
\end{figure}
\begin{figure}
\epsscale{1}
\plotone{f5.eps}
\caption{Separation between the two components in PKS~1934$-$638, as a function of observing frequency. The results of the model described in the text is plotted for a viewing angle $\theta=0$ (solid line), to explain the variation in separation as a function of frequency. }
\end{figure}
We suggest that this is a signature of frequency dependent structure in PKS~1934$-$638 and is caused by optical depth effects. To investigate the origin of the apparent increase of lobe separation with frequency, we explored a two-dimensional synchrotron source model which takes into account opacity effects associated with synchrotron self-absorption. We modelled the electron density profiles of the two lobes in terms of a ``cut-off'' gaussian profile as a means of representing the discontinuity associated with the shock front at the jet working surface and the corresponding backflow from the shock region. In the plane of the jet axis the lobe edges are located at $\pm x_0$, measured relative to the core at $x=0$, so that the electron density profiles of the two lobes are:
\begin{equation}
\rho(x,z) = \left\{ \begin{array}{ll} \rho_0 \exp \left[ - \frac{(|x| -
x_0)^2 + z^2}{r_0^2} \right], & |x| < x_0, \\
0, & \hbox{otherwise}, \\
\end{array} \right. \label{rhoProfile} \\
\end{equation}
A constant magnetic field was assumed, and a spectral index of $-3.7$ for the electron energy distribution was chosen to match that implied by the $\nu^{-1.35}$ spectrum of the optically thin regime at frequencies above $2\,$GHz. The spectral index was assumed constant across the extent of the source; the model does not take into account any spectral steepening due to synchrotron cooling in the regions removed from the shock front. Correction for an arbitrary viewing angle, $\theta$, was made by effecting a rotation of the co-ordinate axes $(x',z') \rightarrow (x \cos \theta + z \sin \theta, -x \sin \theta + z \cos \theta)$, and integrating along the $z'$ axis the equations of radiative transfer \citep{saz}:
\begin{eqnarray}
{d\over d z'}\pmatrix{I\cr Q \cr
U}= \pmatrix{\alpha_I\cr \alpha_Q\cr 0}
+\pmatrix{-\mu_I&-\mu_Q&0\cr -\mu_Q&-\mu_I&-\rho_V\cr
0&\rho_V&-\mu_I}\pmatrix{I\cr Q\cr U}. \label{IQUmatrix}
\end{eqnarray}
Here, the emission and absorption coefficients are $\alpha_{I,Q}$ and $\mu_{I,Q}$ respectively \citep{mel91}. We neglect the contribution of Faraday rotation associated with any ambient thermal plasma and set $\rho_V=0$.
The lobe separation was computed by calculating the separation between the intensity-weighted means of the emission around each peak in the emission profile. The value of $x_0$ was fixed by requiring that, for a given viewing angle, the radiative transfer code reproduced the observed separation between the lobes at high frequency, where the lobe separation asymptotes to the true lobe separation.
This simple model explains the data to the satisfaction of the error bars.We find that models within $\sim 20^\circ$ of the plane of the sky account well for the observed frequency dependence of the lobe separation. Thus, we can demonstrate that our suggestion of frequency-dependent structure has a reasonable basis in physics. A corollary of this suggestion is that the expansion rate calculated by \citet{oja04} should be viewed with considerable caution, until more VLBI measurements can be made at a single frequency, to minimise frequency dependent structure effects. We note that the sequence of 2.3 GHz observations listed in Table 3, and shown in Figure 4, by themselves do not constitute evidence for a significant expansion rate. Likewise, the sequence of 8.4 GHz observations, taken by themselves, do not show evidence for significant expansion. Given the evidence for frequency dependent source structure, we consider the evidence for source expansion to be weak. A further consequence of the frequency dependent source structure is that estimates of the radio source age derived from the apparent expansion must also be considered with caution.
\section{Conclusions}
We have obtained new high angular resolution observations of the archetype GPS radio galaxy, PKS~1934$-$638, by using the Australian Long Baseline Array enhanced with two new radio telescopes, the first antenna of the Australian SKA Pathfinder in Western Australia and a new 12 m antenna at Warkworth, New Zealand. The addition of these two new antennas have greatly improved the angular resolution obtainable at the low frequency end of the the LBA's operating range, 1.4 GHz. These new capabilities have been used to examine the frequency dependent structure of PKS~1934$-$638 on the parsec scale, within the context of historical VLBI observations of the source at various frequencies.
We show that frequency dependent structure effects are important in PKS~1934$-$638 and present a simple two-dimensional synchrotron source model in which opacity effects due to synchrotron self-absorption are taken into account. We find that previous estimates of the expansion rate of the radio source must be viewed with caution, and therefore that calculations of the source age and limits of the onset of jet production must also be considered with care.
\acknowledgments
The International Centre for Radio Astronomy Research is a Joint Venture between Curtin University of Technology and The University of Western Australia, funded by the State Government of Western Australia and the Joint Venture Partners. The Australian Long Baseline Array is part of the Australia Telescope which is funded by the Commonwealth of Australia for operation as a National Facility managed by CSIRO. Mr Bruce Stansby is supported by a PhD scholarship provided by Curtin University of Technology. Prof. Steven Tingay is a Western Australian Premier's Fellow. The authors acknowledge the role of the Australia New Zealand SKA Coordination Committee in helping to foster and coordinate radio astronomy developments between Australia and New Zealand. This scientific work uses data obtained from the Murchison Radio-astronomy Observatory. We acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site.
|
{
"timestamp": "2010-09-23T02:01:53",
"yymm": "1009",
"arxiv_id": "1009.4339",
"language": "en",
"url": "https://arxiv.org/abs/1009.4339"
}
|
\section{Introduction}
One of the main achievements of quantum field theory in curved spacetime is the verification of the equilibrium thermodynamic description of black hole mechanics \cite{wald:1994}. In using this formalism \cite{hawking:1974,hawking:1975}, Hawking was able to provide a physical interpretation of the black hole temperature through the discovery of particle pair production at the horizon, while also solidifying the connection between black hole entropy and horizon surface area predicted a few years earlier \cite{bekenstein:1972}. Subsequently, there has been a large body of work devoted to understanding this thermodynamic description of black holes and its deeper implications \cite{jacobson:1995, bousso:2002,susskind:2006}. Yet at the same time, the \textit{non-equilibrium} thermodynamic properties of black holes, namely the steady-state flow, or transport, of energy and entropy via Hawking radiation, has received markedly less attention \cite{saida:2006,saida:2007}.
When viewed by an observer at spatial infinity, the metric of a non-rotating, uncharged black hole is given by the (1+3)-dimensional Schwarzschild metric. Therefore, in the thermodynamic description of black holes, it is natural to assume that the emission of Hawking radiation corresponds to that of a three-dimensional (3D) thermal body obeying the Stefan-Boltzmann law. However, recently there has been an increasing body of evidence suggesting that black hole emission is instead a 1D radiative process. One indicator is the well-known near-horizon approximation under which the Schwarzschild metric of a black hole can be reduced to a (1+1)-dimensional Rindler space possessing infinite-dimensional conformal symmetry \cite{fabbri:2005}. The ability to calculate the stress-energy tensor using conformal symmetry is the basis for standard derivations of the Hawking flux \cite{birrell:1982,brout:1995}. More recently, it has been suggested that this conformal symmetry is responsible for the Hawking effect \cite{agullo:2010}, as it has been shown that this symmetry alone is sufficient to determine both the Hawking thermal spectrum \cite{jacobson:1993,iso:2007} and radiation flux \cite{robinson:2005}; the Hawking radiation is an inherently (1+1)-dimensional process. This near-horizon conformal symmetry also reproduces the Bekenstein-Hawking form of the black hole entropy \cite{carlip:2002}, thus connecting to the other familiar dimensional reduction in black hole physics, namely the holographic principle \cite{bousso:2002}.
The first to focus on the entropic and information implications of a 1D evaporation process was Bekenstein \cite{bekenstein:2001}, who proposed that the entropy flow rate from a black hole is of the same form as that of a 1D quantum channel \cite{pendry:1983}, thus constraining the information flow from a black hole. This same 1D channel description applies in the context of laboratory analogues of Hawking radiation \cite{schutzhold:2005,philbin:2008,nation:2009}, and it was noted that the power output from the analogue Hawking process coincides with the optimal energy current through a single quantum channel \cite{nation:2009}. The concept of a 1D quantum channel was first considered by Landauer and others \cite{imry:1999,imry:2008} in the modeling of electrical transport in mesoscopic circuits. The Landauer approach expresses the conductance of a 1D system $G_{c}$ in terms of its scattering properties \cite{imry:2008} via the relation
\begin{equation}\label{eq:landauer}
G_{c}=\frac{I}{\mu_{1}-\mu_{2}}=\frac{e^{2}}{\pi\hbar}T,
\end{equation}
where $I$ is the current through the 1D channel, $\mu_{1}$ and $\mu_{2}$ are the chemical potentials of the channel reservoirs, and $T$ is the transmission coefficient. For perfect transmission, $T=1$, the channel conductance is given by $e^2/(\pi\hbar)$, a value that is independent of the microscopic, material nature of the channel, due to the mutual cancellation of the group velocity and density of states factors entering the current formula in 1D. This Landauer formalism was subsequently extended to describe multiple channels \cite{buttiker:1985,sivan:1986}, as well as thermal transport \cite{sivan:1986,rego:1998,blencowe:1999,schwab:2000,meshke:2006}, where the currents are generated by temperature differences rather than by chemical potential differences. Quantum mechanics places upper limits on the energy and entropy currents in 1D channels. These upper limits are attained in the absence of backscattering for bosonic channels \cite{pendry:1983,blencowe:2000}, and are again independent of the material properties of the channel. Furthermore, for thermal transport, these upper limits can be independent of whether the particles are bosons or fermions, and thus are termed ``universal" \cite{blencowe:2000,blencowe:2004}.
Motivated by these connections, in this paper we argue that a non-equilibrium Landauer-transport model can be applied to black hole entropy flow and energy production rates, describing the Hawking effect in terms of currents flowing in 1D quantum channels connecting thermal reservoirs at each end. We thus relate the emission of Hawking radiation of astrophysical black holes to 1D thermal transport in mesoscopic devices; systems that differ by orders of magnitude in energy. In particular, we emphasize the conditions under which the 1D currents are independent of particle statistics. In contrast to the emitted power, the black hole entropy current cannot be obtained directly from the stress-energy tensor, and is rarely touched on in the literature without a priori assuming the validity of the 3D Stefan-Boltzmann law \cite{saida:2006,saida:2007,zurek:1982}. Therefore, a theory that is capable of providing \textit{both} the black hole energy and entropy currents is required for the correct description of black hole evaporation \cite{thooft:1993}.
Assuming the validity of 1D Landauer transport theory enables the description of certain non-equilibrium, steady state emission processes for black holes, without necessarily requiring knowledge of their microscopic physics. In essence, the Landauer approach allows us to extend the methodology of applying thermodynamic principles to black holes \cite{bekenstein:1974}. Moreover, the Landauer model gives a physical insight into the transport of energy and entropy from a black hole that is lacking in existing field-theoretic derivations.
This paper is organized as follows. In Sec.~\ref{sec:near} we review the well-known near-horizon approximation and the resulting conformal symmetry that leads to the standard derivation of the stress-energy tensor and the energy flow rate for Hawking radiation. Next, in Sec.~\ref{sec:1d} we introduce the Landauer transport description for 1D quantum channels, and highlight the statistics-independent properties of the energy and entropy transport in these channels. Section~\ref{sec:app} establishes the Landauer transport model to the emission of Hawking radiation, for both bosonic and fermionic particles, to a black hole in vacuum. Charged and rotating black holes are also addressed. As an application of the 1D Landauer approach, In Sec.~\ref{sec:entropy} we obtain the net entropy production of a black hole and compare with the standard 3D calculation given in Ref.~\cite{zurek:1982}. The special case of a black hole near thermal equilibrium with its environment is also highlighted. Finally, Sec.~\ref{sec:conclusion} ends with a brief discussion of the results.
\section{Near-horizon conformal symmetry and the Hawking flux}\label{sec:near}
For an observer near the horizon of a spherically symmetric Schwarzschild black hole of mass $M$, the original 4D metric $(G=c=1)$,
\begin{equation}\label{eq:schwar}
ds^{2}=-\left(1-\frac{2M}{r}\right)dt^{2}+\frac{dr^{2}}{\left(1-\frac{2M}{r}\right)}+r^{2}d\Omega^{2},
\end{equation}
can be reduced to that of a (1+1)-dimensional spacetime through the coordinate transformation $r=2M+x^{2}/8M$, where near $x=0$, $1-2M/r\approx x^{2}/16M^{2}$. Near the horizon, excitations and dimensional quantities transverse to the $t$-$x$ plane are redshifted and can be ignored \cite{carlip:2007} (i.e. effective potentials for partial wave modes vanish exponentially fast at the horizon \cite{robinson:2005}). Thus, the near-horizon form of the metric is given by \cite{fabbri:2005}
\begin{equation}\label{eq:rindler}
ds^{2}=-\left(\kappa x\right)^{2}dt^{2}+dx^{2},
\end{equation}
where $\kappa=1/4M$ is the surface gravity and the $t$-$x$ portion of the metric defines the flat (1+1)-dimensional Rindler spacetime. Equation~(\ref{eq:rindler}) can be brought into conformal form by defining the coordinate $x=\kappa^{-1}\exp\left(\kappa \xi\right)$ and forming null coordinates, $u=t-\xi$ and $v=t+\xi$, under which the metric takes the form
\begin{equation}\label{eq:conformal}
ds^{2}=-C(u,v)du\,dv=-e^{\kappa\left(v-u\right)}du\,dv,
\end{equation}
where $C(u,v)$ is the conformal factor. Here we ignore the effects of the radial potential as it is blue-shifted away by the conformal symmetry \cite{agullo:2010}. The regularized expectation values for the stress-energy tensor can be immediately evaluated from the conformal structure of Eq.~(\ref{eq:conformal}): \cite{padmanabhan:2005}
\begin{equation}\label{eq:conformal-expect}
\left<T^{2\mathrm{D}}_{ii}\right>=-\frac{1}{12\pi}C^{1/2}\partial_{i}^{2}C^{-1/2},
\end{equation}
for $i=u,v$. For a Schwarzschild black hole, the expectation value with respect to the Unruh vacuum at the horizon, for a single photon polarization, is given as \cite{brout:1995}
\begin{equation}\label{eq:influx}
\left.\left<T^{2\mathrm{D}}_{vv}\right>_{U}\right|_{r=2M}=-\frac{1}{12\pi}\left(\frac{1}{64M^{2}}\right)=-\frac{\pi}{12\hbar}T_{\mathrm{H}}^{2},
\end{equation}
where $T_{\mathrm{H}}=\kappa/2\pi$. This represents the influx of negative energy across the horizon, responsible for the evaporation of the black hole, corresponding to the outgoing Hawking flux, as may be checked using the conformal factor for the t-r sector of the Schwarzschild metric, $C(r)=\left(1-M/r\right)$, and Eq.~(\ref{eq:conformal-expect})
\begin{equation}\label{eq:outflux}
\left<T^{2\mathrm{D}}_{uu}\right>_{U}=\frac{\pi}{12}T_{\mathrm{H}}^{2}\left[1-\frac{2M}{r}\right]^{2}\left[1+\frac{4M}{r}+\frac{12M^{2}}{r^{2}}\right] .
\end{equation}
The power emitted through Hawking radiation as seen by an inertial observer at $r=\infty$ is obtained from Eq.~(\ref{eq:outflux}) as
\begin{equation}\label{eq:hflux}
\left<T^{2\mathrm{D}}_{uu}\right>_{U}=\frac{\pi k_{\mathrm{B}}^{2}}{12\hbar}T_{\mathrm{H}}^{2},
\end{equation}
where, reintroducing dimensional constants for later convenience, we have $T_{\mathrm{H}}=\hbar c^{3}/8\pi k_{\mathrm{B}}GM$. With $\sim 98\%$ of photons, and likewise $\sim 96\%$ of neutrinos, emitted in the radial direction (s-wave) \cite{page:1976}, Eq.~(\ref{eq:hflux}) is approximately valid in the full 4D spacetime as well, where the stress-energy tensor in the $r$-$t$ plane is given as \cite{brout:1995}
\begin{equation}
\left<T^{4\mathrm{D}}_{\mu\nu}\right>=\frac{1}{4\pi r^{2}}\left<T^{2\mathrm{D}}_{\mu\nu}\right>.
\end{equation}
The net flux across a spherical surface of radius $r$ is then given by $4\pi r^{2}\left<T^{4\mathrm{D}}_{\mu\nu}\right>$, which results in a net flux that is again expressed though Eq.~(\ref{eq:hflux}) \cite{padmanabhan:2005}.
\section{One-dimensional quantum channels}\label{sec:1d}
As a model for a single 1D quantum channel, we will consider two thermal reservoirs characterized by the temperatures $T_{\mathrm{L}}$ and $T_{\mathrm{R}}$ and with chemical potentials $\mu_{\mathrm{R}}$ and $\mu_{\mathrm{L}}$, respectively. The reservoirs are coupled adiabatically through an effectively 1D connection supporting the bidirectional propagation of particles. The subscripts L and R denote the left and right thermal reservoirs respectively. Here we will assume $T_{\mathrm{L}}>T_{\mathrm{R}}$ and that the transport through the 1D-connection is ballistic.
Although our focus is on fundamental fields/particles, for complete generality we will assume interpolating fractional statistics where the distribution function is \cite{wu:1994}
\begin{equation}
f_{g}\left(E\right)=\left[w\left(\frac{E-\mu}{k_{\mathrm{B}}T}\right)+g\right]^{-1},
\end{equation}
where $w(x)^{g}\left[1+w(x)\right]^{1-g}=\exp(x)$ with $x\equiv\left(E-\mu\right)/k_{\rm B}T$. Here, $g=0$ and $g=1$ describe bosons and fermions respectively. The individual single-channel energy and entropy currents flowing from the left (L) and right (R) reservoirs may be written as \cite{blencowe:2000,blencowe:2004}
\begin{equation}\label{eq:edot}
\dot{E}_{\mathrm{L(R)}}=\frac{\left(k_{\mathrm{B}}T_{\mathrm{L(R)}}\right)^{2}}{2\pi\hbar}\int_{x_{\mathrm{L(R)}}^{0}}^{\infty}\!\!\!\!\!\!\!\!dx\left(x+\frac{\mu_{\mathrm{L(R)}}}{k_{\mathrm{B}}T_{\mathrm{L(R)}}}\right)f_{g}(x)
\end{equation}
and
\begin{eqnarray}\label{eq:sdot}
\dot{S}_{\mathrm{L(R)}}=&&-\frac{k_{\mathrm{B}}^{2}T_{\mathrm{L(R)}}}{2\pi\hbar}\int_{x_{\mathrm{L(R)}}^{0}}^{\infty}\!\!\!\!\!\!\!\!dx\left\{f_{g}\ln f_{g}+\left(1-gf_{g}\right)\ln(1-gf_{g})\right. \cr
&&\left.-\left[1+(1-g)f_{g}\right]\ln\left[1+(1-g)f_{g}\right]\right\},
\end{eqnarray}
where $x_{\mathrm{L(R)}}^{0}=-\mu_{\mathrm{L(R)}}/k_{\mathrm{B}}T_{\mathrm{L(R)}}$. We define the zero of energy with respect to the longitudinal component of the kinetic energy. For the case of bosons with $\mu_{\mathrm{L}}=\mu_{\mathrm{R}}=0$ (e.g. photons), the net power and entropy flow through the quantum channel, $\dot{E}^{\leftrightarrow}_{1D}=\dot{E}_{\mathrm{L}}-\dot{E}_{\mathrm{R}}$ and $\dot{S}^{\leftrightarrow}_{1D}=\dot{S}_{\mathrm{L}}-\dot{S}_{\mathrm{R}}$ respectively, become
\begin{equation}\label{eq:power}
\dot{E}^{\leftrightarrow}_{\mathrm{1D}}=\frac{\pi k_{\mathrm{B}}^{2}}{12\hbar}\left(T_{\mathrm{L}}^{2}-T_{\mathrm{R}}^{2}\right)
\end{equation}
and
\begin{equation}\label{eq:Sflow}
\dot{S}^{\leftrightarrow}_{\mathrm{1D}}=\frac{\pi k_{\mathrm{B}}^{2}}{6\hbar}\left(T_{\mathrm{L}}-T_{\mathrm{R}}\right).
\end{equation}
The emitted power Eq.~(\ref{eq:power}) holds for all bosonic quantum channels since the group velocity and density of states mutually cancel in 1D.
The unidirectional power
\begin{equation}\label{eq:unip}
\dot{E}^{\rightarrow}_{1\mathrm{D}}=\frac{\pi k_{\mathrm{B}}^{2}T^{2}_{\mathrm{L}}}{12\hbar}
\end{equation}
and the entropy current
\begin{equation}\label{eq:uniS}
\dot{S}^{\rightarrow}_{\mathrm{1D}}=\frac{\pi k_{\mathrm{B}}^{2}}{6\hbar}T_{\mathrm{L}}
\end{equation}
are the maximum possible rates for single-channel bosonic flow.
The unidirectional entropy current (\ref{eq:uniS}) is in fact the maximum possible rate for single-channel fermionic flow as well, i.e., it is independent of the particle statistics \cite{anghel:2002,blencowe:2004}. To see this, we make a change of integration variables in Eq.~(\ref{eq:sdot}), $x=\left(E-\mu\right)/k_{\mathrm{B}}T\rightarrow w$, upon which the entropy current can be simplified to \cite{blencowe:2004}
\begin{equation}\label{eq:sw}
\dot{S}_{\mathrm{L}}=\frac{k_{\mathrm{B}}^{2}T_{\mathrm{L}}}{2\pi\hbar}\int_{w_g\left(\frac{-\mu_{\mathrm{L}}}{k_{\mathrm{B}}T_{\mathrm{L}}}\right)}^{\infty}dw\left[\frac{\ln\!\left(1+w\right)}{w}-\frac{\ln\! w}{1+w}\right].
\end{equation}
We can see that the statistics of the particles shows up only in the lower integration bound of Eq.~(\ref{eq:sw}). The maximum current (\ref{eq:uniS}) is obtained in the degenerate limit where the statistics-dependence vanishes, since $-\mu_{\mathrm{L}}/k_{\mathrm{B}}T_{\mathrm{L}}\rightarrow 0^{+}$, $w_{g=0}(0)=0$ for bosons, and $-\mu_{\mathrm{L}}/k_{\mathrm{B}}T_{\mathrm{L}}\rightarrow -\infty$, $w_{g=1}(-\infty)=0$ for fermions. However, this same statistics independence in the degenerate limit does not hold for the unidirectional power Eq.~(\ref{eq:edot}). If one instead considers bidirectional current flow for fermions with $\mu_{\mathrm{R}}=\mu_{\mathrm{L}}$ and $T_{\mathrm{R}}=0$, then in the degenerate limit one recovers the same maximum rate (\ref{eq:unip}) as for bosons \cite{blencowe:2000}. If the maximum energy and entropy current expressions, Eqs.~(\ref{eq:unip}) and (\ref{eq:uniS}) respectively, are combined by eliminating $T_{\mathrm{L}}$, then one obtains equality for the bound
\begin{equation}\label{eq:bound}
\left(\dot{S}^{\rightarrow}_{\mathrm{1D}}\right)^{2}\le\frac{\pi k_{\mathrm{B}}^{2}}{3\hbar}\dot{E}^{\rightarrow}_{\mathrm{1D}},
\end{equation}
which holds for 1D quantum channels with arbitrary reservoir temperatures, chemical potentials, and particle statistics \cite{pendry:1983,blencowe:2000}. We note in passing that this bound is similar in form to the conjectured Bekenstein holographic bound \cite{bekenstein:1981}.
\section{Hawking radiation from a black hole in vacuum}\label{sec:app}
The Landauer description of Hawking radiation is not limited to 1D, but also applies equally well to the 3D black hole spacetime viewed by an observer at infinity. There, the entropy and energy flow rates can be characterized by a large ensemble of quantum channels, each labeled by a transverse spatial (i.e. angular momentum) quantum number, with interactions between channels described via a scattering matrix \cite{buttiker:1985}. Therefore, scattering due to the potential barrier away from the horizon can be accounted for in the Landauer description through its known multichannel generalization with the inclusion of intra and inter-channel scattering (see, e.g., Ref.~\cite{sivan:1986}). Although this seems to suggest that Hawking radiation flows through a vast number of quantum channels, the near horizon region, where Hawking radiation is emitted and absorbed, is not 3D but rather given by the Rindler metric, Eq.~(\ref{eq:rindler}). With only a single spatial dimension remaining, the (1+1)-dimensional conformal symmetry of the metric near the horizon allows for a single 1D-quantum channel description of the Hawking process (see Fig.~\ref{fig:fig1}), where the remaining quantum channel corresponds to the lowest possible angular momentum mode. Comparing Eq.~(\ref{eq:power}) with Eq.~(\ref{eq:hflux}), we can see immediately that the Landauer 1D channel formula for the zero chemical potential, bosonic power flow coincides with the Hawking radiation flux where $T_{\mathrm{L}}=T_{\mathrm{H}}$ and $T_{\mathrm{R}}=T_{\mathrm{E}}=0$, with $T_{\mathrm{E}}$ defined to be the temperature of the thermal environment surrounding the black hole. The mutual cancellation of the group velocity and density of states factors in the 1D Landauer formula should make Eq.~(\ref{eq:unip}) valid not just in flat but in arbitrary curved spacetimes \cite{bekenstein:2001-2}, although the conformal symmetry of the near-horizon region suggests that the production of Hawking radiation is itself essentially a flat-space process.
\begin{figure}[t]
\begin{center}
\includegraphics[width=5.5in]{figure1}
\caption{(Color online) (a) In the equilibrium thermodynamic description of a Schwarzschild black hole, both the entropy $S$ and temperature $T_{\mathrm{H}}$ of the black hole are given by the properties of the two-dimensional horizon surface, a section of which is highlighted, being proportional to the surface area $A$ and surface gravity $\kappa$ respectively. (b) Near the horizon surface, the conformal symmetry results in an effectively (1+1)-dimensional spacetime, allowing for a 1D Landauer description. Here, the power and entropy flow is through the 1D channel formed by the radial Schwarzschild coordinate $r$. For a black hole in a thermal environment with temperature $T_{\mathrm{E}}>0$, the channel supports the bi-directional propagation of energy and entropy to and from the black hole. The net energy $\dot{E}^{\leftrightarrow}_{1\mathrm{D}}$ and entropy $\dot{S}^{\leftrightarrow}_{1\mathrm{D}}$ flow, Eqs.~(\ref{eq:power}) and (\ref{eq:Sflow}) respectively, is away from the black hole when $T_{\mathrm{H}}>T_{\mathrm{E}}$.}
\label{fig:fig1}
\end{center}
\end{figure}
Although we have appealed to conformal symmetry, these 1D emission properties of Hawking radiation are evident in the full 3D spacetime as well. Following the original argument of Bekenstein \cite{bekenstein:2001} we note that the flat spacetime entropy emission rate for a blackbody in D-dimensions scales with the output power as
\begin{equation}\label{eq:D}
\dot{S}_{\mathrm{D}}\propto \left(\dot{E}_{\mathrm{D}}\right)^{\mathrm{D}/(\mathrm{D}+1)}.
\end{equation}
As a result, if a black hole were to radiate as a 3D object, one should expect the emitted entropy to scale as the $3/4$ power of the energy flow rate. However, substitution of the Hawking temperature $T_{\mathrm{H}}$ into the Stefan-Boltzmann law, and making use of the black hole surface area $A=16\pi(GM)^{2}/c^{4}$, one finds that the emitted entropy
\begin{equation}\label{eq:bhineq}
\dot{S}^{2}=\frac{1}{90}\frac{\pi k_{\rm B}^{2}}{3\hbar}\dot{E}
\end{equation}
goes as the $1/2$ power of the energy rate, just as one would expect for a 1D emitter. In fact, Eq.~(\ref{eq:bhineq}) is identical to Eq.~(\ref{eq:bound}) up to a numerical factor arising from the assumption of a 3D, as opposed to 1D, emitter. This result is attributable to the inverse dependence of the Hawking temperature on the black hole mass $M$, a property of black holes not shared by other blackbody emitters. Therefore, the thermodynamic properties of a black hole correspond to that of a 1D blackbody emitter, as one might suspect given the ability to derive both the Hawking temperature and black hole entropy from the 1+1-dimensional conformal symmetry in the near horizon region.
In what follows, we will assume the validity of Eqs.~(\ref{eq:edot}) and (\ref{eq:sdot}) [equivalently Eq.~(\ref{eq:sw})] in describing the net energy and entropy outflow rates, respectively, for particles radiating from a black hole into the vacuum (i.e., $T_{\mathrm{E}}=0)$. With the goal of introducing the Landauer description of Hawking radiation in the near horizon region, we will ignore scattering due to the radial potential barrier. However, the full Landauer approach, relating transport to scattering processes \cite{imry:1999,imry:2008}, can incorporate inter-channel scattering due to particle interactions and back scatter from the radial potential barrier not considered here.
The electrochemical potential of the black hole reservoir is $\mu_{\mathrm{L}}=\mu_{\mathrm{BH}}=q\Phi$, where $q$ is the electric charge of the field under consideration and $\Phi$ is the electrostatic potential corresponding to the charge of the black hole \cite{fabbri:2005}. For a Schwarzschild black hole with $\Phi=0$, and hence $\mu_{\mathrm{BH}}=0$, bosons such as photons and gravitons have maximum rates given by Eq.~(\ref{eq:unip}) and (\ref{eq:uniS}) with $T_{\mathrm{L}}=T_{\mathrm{H}}$. For fermions such as neutrinos and electrons (i.e. leptons), setting $\mu_{\mathrm{BH}}=0$ gives a lower integration limit of $w_{g=1}(0)=1$ in Eq.~(\ref{eq:sw}), resulting in entropy and energy rates that are reduced by a factor of $1/2$ from the maximum values (\ref{eq:uniS}) and (\ref{eq:unip}). This result for the energy rate was established in earlier calculations for massless fermions \cite{davies:1977}, and shows up in the relative values of the conformal and gravitational anomalies \cite{robinson:2005}. However, as explicitly pointed out in Ref.~\cite{davies:1977}, the physical reason behind this result could not be established. In contrast, the Landauer model presented here shows that these reduced fermionic currents are a direct consequence of the vanishing chemical potential of a Schwarzschild black hole and the 1D nature of the emission process. Subsequently, it was pointed out \cite{davies:1978} that in a (1+1)-dimensional curved spacetime, the fermionic field describing a massless particle plus its antiparticle is equivalent to a single massless bosonic field. From the Landauer viewpoint, the combined fermionic particle/antiparticle single channel currents can therefore be thought of as a single effective bosonic channel that satisfies the maximum rates, Eqs.~(\ref{eq:unip}) and (\ref{eq:uniS}), when $\mu_{\mathrm{BH}}=0$. Although leptons are massive particles, the conformal symmetry removes the length scale set by the particle mass \cite{agullo:2010}; the particles are effectively massless. In the case of ballistic transport, multiple channels can be treated independently. Thus, the net Schwarzschild black hole energy and entropy outflow rates are bounded by $N\left(T_{\mathrm{H}}\right)$ times the single channel rates given by Eq.~(\ref{eq:unip}) and Eq.~(\ref{eq:uniS}), respectively; a Schwarzschild black hole in vacuum radiates energy and entropy at the maximum rate allowed by quantum mechanics in 1D, saturating the bound in Eq.~(\ref{eq:bound}). Here, $N\left(T_{\mathrm{H}}\right)$ is the total number of effective bosonic channels spontaneously produced by a black hole at temperature $T_{\mathrm{H}}$; a quantity limited by the number of particle species emitted and their corresponding number of polarizations. The temperature dependence of the effective channel number arises due to the requirement that $k_{\mathrm{B}}T_{\mathrm{H}}\gtrsim 2mc^{2}$ for pair production of particles with mass $m$.
For a black hole with nonzero electrochemical potential, charged particle/antiparticle rates differ so as to cause the black hole net charge to decrease over time. The maximum entropy rate for a single charged fermionic channel coincides with the maximum rate for a single bosonic channel as shown above, giving Eq.~(\ref{eq:uniS}). The extent to which these maximum rates can be achieved depends on how close to degenerate is the thermal Hawking reservoir of the black hole for charged particles. A special case is provided by extremal charged black holes \cite{fabbri:2005} satisfying $Q^{2}/M^{2}\approx1$, where $Q$ is the non-dimensional black hole charge. In this limit, $T_{\mathrm{H}}\rightarrow 0$ giving $-\mu_{\mathrm{BH}}/k_{\mathrm{B}}T_{\mathrm{H}}\rightarrow -\infty$, the degenerate limit for fermions. Charged fermions then satisfy Eq.~(\ref{eq:uniS}). It may be possible to reach the degenerate limit for other choices of black hole parameters. Similar reasoning applies to a black hole with angular momentum where, although spherical symmetry is broken, the emission of Hawking radiation is still governed by (1+1)-dimensional conformal symmetry \cite{agullo:2010}. Here, the $U(1)$ gauge symmetry corresponding to the angular isometry in the (1+1)-dimensional theory may be written as a chemical potential in the same manner as that of a charged black hole \cite{iso:2006,iso:2006b}. Therefore, the Landauer model presented here is quite general, being valid for black holes both with or without charge and angular momentum. Finally, we point out that the cancellation of the density of states and group velocity in 1D quantum channels suggests that Eq.~(\ref{eq:unip}) should also be valid for black holes in other spacetimes, such as BTZ black holes \cite{banados:1992} in anti de-Sitter space, where conformal methods may still be applied \cite{carlip:2005}.
\section{Net entropy production in (1+1)-dimensions}\label{sec:entropy}
Originally considered by Zurek \cite{zurek:1982}, the rate of net entropy production by a Schwarzschild black hole due to the emission of Hawking radiation into a thermal environment, neglecting backscattering due to the radial potential barrier \cite{page:1983}, is given by
\begin{equation}\label{eq:R}
R=\frac{dS}{dS_{\mathrm{BH}}}=T_{\mathrm{H}}\frac{dS}{dE}=T_{\mathrm{H}}\frac{\dot{S}}{\dot{E}}
\end{equation}
where we have used the first law of thermodynamics $dE_{\mathrm{BH}}=T_{\mathrm{H}}dS_{\mathrm{BH}}$ and assumed energy conservation, $dE=dE_{\mathrm{BH}}$. For a 3D black hole radiating into a thermal environment with temperature $T_{\mathrm{E}}$, the power and entropy currents are
\begin{eqnarray}
\dot{E}^{\leftrightarrow}_{\mathrm{3D}}&\sim& a\left(T_{\mathrm{H}}^{4}-T_{\mathrm{E}}^{4}\right)\\
\dot{S}^{\leftrightarrow}_{\mathrm{3D}}&\sim& \frac{4a}{3}\left(T_{\mathrm{H}}^{3}-T_{\mathrm{E}}^{3}\right),
\end{eqnarray}
where $a$ is a constant. Upon substitution into Eq.~(\ref{eq:R}), this yields the 3D black hole entropy production ratio
\begin{equation}\label{eq:3d}
R_{\mathrm{3D}}=\frac{4}{3}\frac{1-\left(T_{\mathrm{E}}/T_{\mathrm{H}}\right)^{3}}{1-\left(T_{\mathrm{E}}/T_{\mathrm{H}}\right)^{4}},
\end{equation}
which gives $R_{\mathrm{3D}}=4/3$ for a black hole in vacuum: $T_{\mathrm{E}}/T_{\mathrm{H}}\rightarrow 0$ \cite{zurek:1982}.
However, as we have shown above, the emission properties of a black hole are better characterized as a 1D Landauer process. Therefore, we compare the entropy produced via our 1D model to the standard calculation presented by Zurek. Our focus in this paper is on the near-horizon region, and hence we do not include scattering. The conformal symmetry in this region removes any inherent length scales, allowing the scattering barrier to be blue shifted away \cite{agullo:2010}. Moreover, it is important to note that the entropy current Eq.~(\ref{eq:sw}), like the 1D energy flow Eq.~(\ref{eq:unip}), should hold for a black hole in any spacetime where conformal symmetry may be invoked, even though the corresponding scattering properties may be markedly different. Likewise, the Landauer approach is valid for analogue black hole models \cite{schutzhold:2005,philbin:2008,nation:2009}, as well as a moving mirror in (1+1)-dimensional spacetime \cite{fulling:1976} that can reproduce the emission properties of Hawking radiation from a Schwarzschild black hole in vacuum, even though no scattering barrier is present. The effects of any scattering potential can be incorporated into a multichannel Landauer model \cite{buttiker:1985,sivan:1986}, and for the current case of a Schwarzschild black hole, will be presented elsewhere. Since scattering can serve to only increase the net entropy produced from a Schwarzschild black hole \cite{page:1983}, the entropy production rates considered in this section may be viewed as lower bounds. However, we note that for 1D transport, scattering will reduce the individual unidirectional energy $\dot{E}_{\mathrm{L(R)}}$ and entropy currents $\dot{S}_{\mathrm{L(R)}}$, Eq.~(\ref{eq:edot}) and Eq.~(\ref{eq:sdot}) respectively, to values below the ballistic bosonic channel limits, Eqs.~(\ref{eq:unip}) and (\ref{eq:uniS}).
For comparison, in our Landauer model we set $\mu_{\mathrm{E}}=\mu_{\mathrm{BH}}=0$, and the net energy and entropy currents are given by Eqs.~(\ref{eq:power}) and (\ref{eq:Sflow}) respectively. The factors of 1/2 in the fermion rates will drop out when evaluating the ratio Eq.~(\ref{eq:R}). The 1D entropy production ratio is then
\begin{equation}\label{eq:1d}
R_{\mathrm{1D}}=2 \frac{1-\left(T_{\mathrm{E}}/T_{\mathrm{H}}\right)^{\phantom{1}}}{1-\left(T_{\mathrm{E}}/T_{\mathrm{H}}\right)^{2}},
\end{equation}
which yields a larger value of $R_{1D}=2$ when radiating into vacuum; the net entropy production by Hawking radiation into vacuum is 50\% larger than that of a corresponding 3D thermal body at the Hawking temperature. Again, this is due to the 1D properties of the near-horizon region, and the emitted radiation, for which Eq.~(\ref{eq:3d}) is no longer valid. The difference between the 3D and 1D entropy rates, Eqs.~(\ref{eq:3d}) and (\ref{eq:1d}) respectively, for various ratios of $T_{\mathrm{E}}/T_{\mathrm{H}}$ is presented in Fig.~\ref{fig:fig2}.
\begin{figure}[htbp]
\begin{center}
\includegraphics[width=3.0in]{figure2}
\caption{(Color online) Entropy production ratio for a black hole characterized as 1D quantum channel $R_{\mathrm{1D}}$ (dashed-blue) compared to the standard 3D answer $R_{\mathrm{3D}}$ (red). Both results agree near thermal equilibrium $T_{\mathrm{H}}\approx T_{\mathrm{E}}$. }
\label{fig:fig2}
\end{center}
\end{figure}
In the case where $T_{\mathrm{H}}\approx T_{\mathrm{E}}$, both Eqs.~(\ref{eq:3d}) and (\ref{eq:1d}) give approximately $R\approx 1+\delta/T_{\mathrm{H}}$ to first order in $\delta=\left(T_{\mathrm{H}}-T_{\mathrm{E}}\right)/2$. As to be expected, in equilibrium ($\delta=0$), there is no net entropy production ($R=1$).
Near thermal equilibrium we can make use of linear response for small temperature differences, $\left(T_{\mathrm{H}}-T_{\mathrm{E}}\right)\ll\bar{T}$ where $\bar{T}=\left(T_{\mathrm{H}}+T_{\mathrm{E}}\right)/2$, to relate the energy and entropy flows by $\dot{S}_{1\mathrm{D}}=\dot{E}_{1\mathrm{D}}/\bar{T}$. In this regime the unidirectional entropy rate Eq.~(\ref{eq:uniS}) allows us to recover the quantum of thermal conductance for a single effective bosonic channel \cite{blencowe:2004}:
\begin{equation}\label{eq:conductance}
G_{Q}=\frac{\dot{E}_{1\mathrm{D}}}{T_{\mathrm{H}}-T_{\mathrm{E}}}=\frac{\left(\dot{S}_{\mathrm{H}}-\dot{S}_{\mathrm{E}}\right)\bar{T}}{T_{\mathrm{H}}-T_{\mathrm{E}}}=\frac{\pi k_{\mathrm{B}}^{2}}{6\hbar}\bar{T},
\end{equation}
that, like Landauer's original expression Eq.~(\ref{eq:landauer}), relates conductance to transmission via only fundamental constants. From the statistics independence of Eq.~(\ref{eq:uniS}), it follows that Eq.~(\ref{eq:conductance}) provides a general upper bound on the thermal conductance of a black hole that is independent of the particle statistics, as discussed in \cite{krive:1999,rego:1999}.
\section{Conclusion}\label{sec:conclusion}
Using the conformal symmetry in the near-horizon region of a black hole, we have presented a 1D Landauer transport model for the non-equilibrium transport of both energy and entropy flow from a black hole, valid for particles with arbitrary statistics, and which clarifies the independence of the underlying microscopic physics. Although our focus is on the near horizon region, the 1D nature of the emission properties are evident in the full (1+3)-dimensional spacetime seen by an observer at infinity, and may be derived from the inverse relationship between black hole mass and Hawking temperature. For a Schwarzschild black hole in vacuum, conformal symmetry results in a Hawking radiation energy flux that is identical to the power flowing in a single 1D quantum channel connected to a thermal bath with the Hawking temperature at one end and zero temperature at the other. Including multiple particle-species and polarizations, a Schwarzschild black hole in vacuum radiates power and entropy at the optimal rate, as a collection of effective bosonic channels. This is a direct result of the statistics independence of unidirectional energy and entropy flow in 1D highlighted by the Landauer formalism, and has not been discussed previously. Furthermore, we have shown that the reduced emission rates for fermions from a Schwarzschild black hole are due to the absence of a black hole chemical potential, giving a physical interpretation that is lacking in previous derivations. Moreover, in contrast to field-theory derivations using the stress-energy tensor, our Landauer model directly yields the entropy current from a black hole without assuming the validity of the 3D Stefan-Boltzmann law. Both the charge and angular momentum of a black hole may be represented as an effective black hole chemical potential, and can be fully incorporated into the Landauer description presented here. The unidirectional entropy current leads to a statistics independent heat flow near thermal equilibrium characterized by the quantum of thermal conductance. Again, this property of black hole transport has not been addressed earlier. In addition, the energy and entropy currents in 1D give a Hawking radiation entropy production ratio that is twice the corresponding value lost by the black hole when radiating into vacuum: a 50\% higher value when compared to the currently accepted 3D blackbody rate. These results are a direct consequence of the reduced dimensionality in the near-horizon region and its conformal symmetry. Given the intimate connection between entropy and information, the present findings, in particular Eq.~(\ref{eq:bound}), place strict limits on the rate of information transfer into and out of a black hole \cite{lloyd:2004}, and therefore will play a role in addressing the information loss problem in black hole evaporation \cite{hawking:1976,bekenstein:2004}. However, we note that the reliance on conformal symmetry means that the Landauer model, in its present form, is incapable of describing non-thermal Hawking spectrum transport properties required for unitary black hole evaporation \cite{page:1993,nation:2010}.
\ack
PDN thanks S. Carlip and H. Kang for helpful discussions. PDN was supported by JSPS Postdoctoral Fellowship P11202. MPB was partially supported by the National Science Foundation (NSF) under grant No. DMR-0804477. FN acknowledges partial support from DARPA, AFOSR, Laboratory of Physical Science, National Security Agency, Army Research Office, NSF grant No. 0726909, JSPS-RFBR contract No. 09-02-92114, Grant-in-Aid for Scientific Research (S), MEXT Kakenhi on Quantum Cybernetics, and JSPS through the FIRST program.
\bibliographystyle{unsrt}
|
{
"timestamp": "2012-03-14T01:01:22",
"yymm": "1009",
"arxiv_id": "1009.3974",
"language": "en",
"url": "https://arxiv.org/abs/1009.3974"
}
|
\section{Introduction}
Let $S^d$ be the unit sphere in $\R^{{d+1}}$ with the Lebesgue
measure $\mu_d$ normalized by $\mu_d(S^d)=1$.
A set of points $x_1,\ldots,x_N\in S^d$ is called a {\it spherical
$t$-design} if
$$ \int_{S^d}P(x)\,d\mu_d(x)=\frac 1N\sum_{i=1}^N P(x_i) $$ for all
algebraic polynomials in $d+1$ variables, of total degree at most
$t$. The concept of a spherical design was introduced by Delsarte,
Goethals, and Seidel~\cite{DGS}. For each $t, d\in {\mathbb N}$
denote by $N(d,t)$ the minimal number of points in a spherical
$t$-design in $S^d$. The following lower bound
\begin{equation}
\label{hh} N(d,t)\ge \begin{cases}\displaystyle {{d+k}\choose{d}}+{{d+k-1}\choose{d}}&\text{if $
t=2k$,}\\&\\ \displaystyle 2\,{{d+k}\choose{d}}& \text{if $ t=2k+1$,}\end{cases}
\end{equation}
is proved in \cite{DGS}.
Spherical $t$-designs attaining this bound are called tight. The
vertices of a regular $t+1$-gon form a tight spherical $t$-design in
the circle, so $N(1,t)=t+1$. Exactly eight tight spherical designs
are known for $d\ge 2$ and $t\ge 4$. All such configurations of
points are highly symmetrical, and optimal from many different
points of view (see Cohn, Kumar~\cite{CK} and Conway,
Sloane~\cite{CS}). Unfortunately, tight designs rarely exist. In
particular, Bannai and Damerell~\cite{Bannai1, Bannai2} have shown
that tight spherical designs with $d\ge 2$ and $t\ge 4$ may exist
only for $t=4$, $5$, $7$ or $11$. Moreover, the only tight
$11$-design is formed by minimal vectors of the Leech lattice in
dimension $24$. The bound~\eqref{hh} has been improved by Delsarte's
linear programming method for most pairs $(d,\,t)$; see~\cite{Yu}.
On the other hand, Seymour and Zaslavsky~\cite{SZ} have proved that
spherical $t$-designs exist for all $d$, $t\in {\mathbb N}$.
However, this proof is nonconstructive and gives no idea of how big
$N(d,t)$ is. So, a natural question is to ask how $N(d,t)$ differs
from the tight bound~\eqref{hh}. Generally, to find the exact value
of $N(d,t)$ even for small $d$ and $t$ is a surprisingly hard
problem. For example, everybody believes that 24 minimal vectors of
the $D_4$ root lattice form a $5$-design with minimal number of
points in $S^3$, although it is only proved that $22\le N(3,5)\le
24$; see~\cite{BDN}. Further, Cohn, Conway, Elkies, and
Kumar~\cite{CCEK} conjectured that every spherical $5$-design
consisting of $24$ points in $S^3$ is in a certain $3$-parametric
family. Recently, Musin~\cite{M} has solved a long standing problem
related to this conjecture. Namely, he proved that the kissing
number in dimension $4$ is $24$.
In this paper we focus on asymptotic upper bounds on $N(d,t)$ for
fixed $d\ge 2$ and $t\to\infty$. Let us give a brief history of this
question. First, Wagner~\cite{Wag} and Bajnok~\cite{B} proved that
$N(d,t)\le C_dt^{Cd^4}$ and $N(d,t)\le C_dt^{Cd^3}$, respectively.
Then, Korevaar and Meyers \cite{KM} have improved these inequalities
by showing that $N(d,t)\le C_dt^{(d^2+d)/2}$. They have also
conjectured that
$$
N(d,t)\le C_dt^d.
$$
Note that \eqref{hh} implies $N(d,t)\ge c_dt^d$. Here and in what
follows we denote by $C_d$ and $c_d$ sufficiently large and
sufficiently small positive constants depending only on $d$,
respectively.
The conjecture of Korevaar and Meyers attracted the interest of many
mathematicians. For instance, Kuijlaars and Saff~\cite{SK}
emphasized the importance of this conjecture for $d=2$, and revealed
its relation to minimal energy problems. Mhaskar, Narcowich, and
Ward~\cite{MNW} have constructed positive quadrature formulas in
$S^d$ with $C_dt^d$ points having {\em almost} equal weights. Very
recently, Chen, Frommer, Lang, Sloan, and Womersley~\cite{CFL,CW}
gave a computer-assisted proof that spherical $t$-designs with
$(t+1)^2$ points exist in $S^2$ for $t\le 100$.
For $d=2$, there is an even stronger conjecture by Hardin and
Sloane~\cite{HS} saying that $N(2,t)\le\frac12t^2+o(t^2)$ as
$t\to\infty$. Numerical evidence supporting the conjecture was also
given.
In~\cite{BV1}, we have suggested a nonconstructive approach for
obtaining asymptotic bounds for $N(d,t)$ based on the application of
the Brouwer fixed point theorem. This led to the following result:
\\{\it For each $N\ge C_dt^\frac{2d(d+1)}{d+2}$ there
exists a spherical $t$-design in $S^d$ consisting of $N$ points.}\\
Instead of the Brouwer fixed point theorem we use in this paper the
following result from the Brouwer degree theory~\cite[Th. 1.2.6, Th. 1.2.9]{OCC}.\\
{\sc Theorem A. }{\it Let $f: \R^n\to \R^n$ be a continuous mapping
and $\Omega$ an open bounded subset, with boundary $\partial\Omega$,
such that $0\in\Omega\subset \R^n$. If $(x, f(x))> 0$ for all
$x\in\partial \Omega$, then there exists $x\in \Omega$ satisfying $f(x)=0$.}\\
We employ this theorem to prove the conjecture of Korevaar and
Meyers.
\begin{theorem}\label{main}
For each $N\ge C_dt^d$ there exists a spherical $t$-design in $S^d$
consisting of $N$ points.
\end{theorem}
Note that Theorem 1 is slightly stronger than the original
conjecture because it guarantees the existence of spherical
$t$-designs for {\it each} $N$ greater than $C_dt^d$.
This paper is organized as follows. In Section 2 we explain the main
idea of the proof. Then in Section 3 we present some auxiliary
results. Finally, we prove Theorem 1 in Section 4.
\section{Preliminaries and the main idea}
Let $\mathcal{P}_t$ be the Hilbert space of polynomials $P$ on
$S^{d}$ of degree at most $t$ such that $$
\int_{S^d}P(x)d\mu_d(x)=0,
$$
equipped with the usual inner product
$$
(P,Q)=\int_{S^d}P(x)Q(x)d\mu_d(x).
$$
By the Riesz representation theorem, for each point $x\in S^d$ there
exists a unique polynomial $G_x\in \p_t$ such that
$$(G_x,Q)=Q(x) \;\;\mbox{for all}\;\;Q\in\p_t.$$
Then a set of points $x_1,\ldots,x_N\in S^d$ forms a spherical
$t$-design if and only if
\begin{equation}
\label{sph} G_{x_1}+\cdots+G_{x_N}=0.
\end{equation}
For a differentiable function $f: \R^{d+1}\to \R$ denote by
$$ \frac{\partial f}{\partial x}(x_0):=\left(\frac{\partial
f}{\partial \xi_1}(x_0),\ldots, \frac{\partial f}{\partial
\xi_{d+1}}(x_0)\right) $$ the gradient of $f$ at the point $x_0\in
\R^{d+1}$.
For a polynomial $Q\in\p_t$ we define the spherical gradient as
follows:
\begin{equation}
\label{grad} \nabla Q(x):=\frac{\partial}{\partial x}Q\left(\frac
{x}{|x|}\right),
\end{equation}
where $|\cdot|$ is the Euclidean norm in $\R^{d+1}$.
We apply Theorem A to the open subset $\Omega$ of a vector
space~$\p_t$,
\begin{equation}\label{omega}\Omega:=\left\{P\in\mathcal{P}_t\left|\,\int_{S^d}|\nabla
P(x)|d\mu_d(x)<1\right.\right\}.
\end{equation}
Now we observe that the existence of a continuous mapping $F:
\mathcal{P}_t\to (S^d)^N$, such that for all $P\in\partial\Omega$
\begin{equation}\label{positive}\sum_{i=1}^N
P(x_i(P))>0,\;\mathrm{where}\;F(P)=(x_1(P),...,x_N(P)),\end{equation}
readily implies the existence of a spherical
$t$-design in $S^d$ consisting of $N$ points. Consider a mapping
$L:(S^d)^N\to \p_t$ defined by
$$(x_1,\ldots,x_N)\ \,{\mathop{{\longrightarrow}}\limits^{L}}\ \,
G_{x_1}+\cdots+G_{x_N},$$ and the following composition mapping
$f=L\circ F: \mathcal{P}_t\to\mathcal{P}_t$. Clearly
$$
(P,f(P))=\sum_{i=1}^N P(x_i(P))
$$
for each $P\in\mathcal{P}_t$. Thus, applying Theorem A to the
mapping $f$, the vector space $ \mathcal{P}_t$, and the subset
$\Omega$ defined by \eqref{omega}, we obtain that $f(Q)=0$ for some
$Q\in\mathcal{P}_t$. Hence, by~\eqref{sph}, the components of
$F(Q)=(x_1(Q),...,x_N(Q))$
form a spherical $t$-design in $S^d$ consisting of $N$
points.
The most naive approach to construct such $F$ is to start with a
certain well-distributed collection of points $x_i$
($i=1,\ldots,N$), put $F(0):=(x_1,\ldots,x_N)$, and then move each
point along the spherical gradient vector field of $P$. Note that
this is the most greedy way to increase each $P(x_i(P))$ and make
$\sum_{i=1}^N P(x_i(P))$ positive for each $P\in\partial\Omega$.
Following this approach we will give an explicit construction of $F$
in Section 4, which will immediately imply the proof of Theorem 1.
\section{Auxiliary results}
To construct the corresponding mapping $F$ for each $N\ge C_dt^d$ we
extensively use the following notion of an area-regular partition.
Let $\mathcal{R}=\left\{R_1,\ldots, R_N\right\}$ be a finite
collection of closed sets $R_i\subset S^d$ such that $\cup_{i=1}^N
R_i=S^d$ and $\mu_d(R_i\cap R_j)=0$ for all $1\le i<j\le N$. The
partition $\mathcal{R}$ is called area-regular if $\mu_d(R_i)=1/N$,
$i=1,\ldots,N$. The partition norm for $\mathcal{R}$ is defined by
$$ \|\mathcal{R}\|:=\max_{R\in\mathcal{R}}\mathrm{diam}\, R,$$
where $\mathrm{diam}\,R$ stands for the maximum geodesic distance
between two points in $R$. We need the following fact on
area-regular partitions (see Bourgain,
Lindenstrauss~\cite{BL} and Kuijlaars, Saff~\cite{SK2}):\\
{\sc Theorem B. }{\it For each $N\in\mathbb{N}$ there exists an
area-regular partition $\mathcal{R}=\left\{R_1,\ldots, R_N\right\}$
with $\|\mathcal{R}\|\leq B_dN^{-1/d}$ for some constant $B_d$ large
enough.}
We will also use the following spherical Marcinkiewicz--Zygmund type
inequality: \\
{\sc Theorem C. }{\it There exists a constant $r_d$ such that for
each area-regular partition $\mathcal{R}=\{R_1,\ldots,R_N\}$ with
$\|\mathcal{R}\|<\frac{r_d}m$, each collection of points $x_i\in
R_i$ ($i=1,\ldots,N$), and each algebraic polynomial $P$ of total
degree $m$, the inequality
\begin{equation}
\label{Mhaskar}
\frac12\int_{S^d}|P(x)|d\mu_d(x)\le\frac1N\sum_{i=1}^N|P(x_i)|\le\frac32\int_{S^d}|P(x)|d\mu_d(x)
\end{equation}
holds.}\\
Theorem C follows naturally from the proof of Theorem 3.1
in~\cite{MNW}.
\begin{corollary} For each area-regular partition $\mathcal{R}=\{R_1,\ldots,R_N\}$
with $\|\mathcal{R}\|<\frac{r_d}{m+1}$, each collection of points
$x_i\in R_i$ ($i=1,\ldots,N$), and each algebraic polynomial $P$ of
total degree $m$,
\begin{equation}
\label{Mhaskar-mod} \frac{1}{3\sqrt{d}}\int_{S^d}|\nabla
P(x)|d\mu_d(x)\le\frac1N\sum_{i=1}^N|\nabla P(x_i)|\le
3\sqrt{d}\int_{S^d}|\nabla P(x)|d\mu_d(x).
\end{equation}
\end{corollary}
\begin{proof}
Since $|\nabla P|=\sqrt{P_1^2+\ldots+P_{d+1}^2}$ in $S^d$, where
$P_j$ are polynomials of total degree $m+1$, Corollary 1 is an
immediate consequence of~\eqref{Mhaskar} applied to $P_j$,
$j=1,\ldots,d+1$.
\end{proof}
\section{Proof of Theorem 1}
In this section we construct the map $F$ introduced in Section 2 and
thereby finish the proof of Theorem 1.
For $d, t\in{\mathbb N}$, take $C_d>(54 d B_d/r_d)^d$, where $B_d$
is as in Theorem B and $r_d$ is as in Theorem C, and fix $N\ge
C_dt^d$. Now we are in a position to give an exact construction of
the mapping $F: \mathcal{P}_{t}\to(S^d)^N$ which satisfies condition
\eqref{positive}. Take an area-regular partition
$\mathcal{R}=\left\{R_1,\ldots, R_N\right\}$ with
\begin{equation}
\label{eee}
\|\mathcal{R}\|\le B_d N^{-1/d}<\frac{r_d}{54dt}
\end{equation}
as provided by Theorem B, and choose an arbitrary $x_i\in R_i$ for
each $i=1,\ldots, N$. Put $\epsilon=\frac{1}{6\sqrt d}$ and consider
the function
$$ h_\epsilon(u):=\begin{cases} u & \text{ if $u>\epsilon$},\\ \epsilon& \mbox{otherwise.}\end{cases} $$
Take a mapping $U:\p_t\times S^d\to{\mathbb R}^{d+1}$ such that
$$
U(P, y)=\frac{\nabla P(y)}{h_\epsilon (|\nabla P(y)|)}.
$$
For each $i=1,\ldots,N$ let $y_i:\p_t\times[0,\infty)\to S^d$ be the
map satisfying the differential equation
\begin{equation}
\label{diffur} \frac{d }{ds}y_i(P,s)=U(P,y_i(P,s))
\end{equation}
with the initial condition
$$
y_i(P,0)=x_i,
$$
for each $P\in\p_t$. Note that each mapping $y_i$ has its values in
$S^d$ by definition of spherical gradient~\eqref{grad}. Since the
mapping $U(P,y)$ is Lipschitz continuous in both $P$ and $y$, each
$y_i$ is well defined and continuous in both $P$ and $s$, where the
metric on $\p_t$ is given by the inner product.
Finally put
\begin{equation}
\label{map} F(P)=(x_1(P),\ldots,x_N(P)):=\big(y_1(P,
\frac{r_d}{3t}),\ldots,y_N(P,\frac{r_d}{3t})\big ).
\end{equation}By definition the mapping $F$ is continuous on $\p_t$.
So, as explained in Section~2, to finish the proof of Theorem 1 it
suffices to prove
\begin{lemma}
Let $F: \mathcal{P}_{t}\to(S^d)^N$ be the mapping defined
by~\eqref{map}. Then for each $P\in\partial\Omega$,
$$
\frac1N\sum_{i=1}^N P(x_i(P))>0,
$$
where $\Omega$ is given by~\eqref{omega}.
\end{lemma}
\begin{proof}Fix $P\in\partial\Omega$. For the sake of
simplicity we write $y_i(s)$ in place of $y_i(P,s)$. By the
Newton-Leibniz formula we have
$$
\frac1N\sum_{i=1}^N P(x_i(P))=\frac
1N\sum_{i=1}^NP(y_i(r_d/3t))
$$
\begin{equation}
\label{ee}=\frac 1N\sum_{i=1}^NP(x_i) +\int_0^{r_d/3t} \frac{d}{d s}\left[\frac 1N\sum_{i=1}^N
P(y_i(s))\right]ds.
\end{equation}
Now to prove Lemma 1, we first estimate the value
$$
\left|\frac 1N\sum_{i=1}^NP(x_i)\right|
$$
from above, and then estimate the value
$$
\frac{d}{d s}\left[\frac 1N\sum_{i=1}^N P(y_i(s))\right]
$$
from below, for each $s\in [0, r_d/3t]$. We have
$$
\left|\frac
1N\sum_{i=1}^NP(x_i)\right|=\left|\sum_{i=1}^N\int_{R_i}P(x_i)-P(x)\,d\mu_d(x)\right|\le
\sum_{i=1}^N\int_{R_i}|P(x_i)-P(x)|d\mu_d(x)
$$
$$
\le\frac {\|\mathcal{R}\|}{N}\sum_{i=1}^N\max_{z\in
S^d:\,\mathrm{dist}(z,x_i)\le\|\mathcal{R}\|}|\nabla P(z)|
$$
where $\mathrm{dist}(z,x_i)$ denotes the geodesic distance between
$z$ and $x_i$. Hence, for $z_i\in S^d$ such that
$\mathrm{dist}(z_i,x_i)\le\|\mathcal{R}\|$ and
$$|\nabla P(z_i)|=\max_{z\in
S^d:\,\mathrm{dist}(z,x_i)\le\|\mathcal{R}\|}|\nabla P(z)|,$$ we
obtain
$$
\left|\frac 1N\sum_{i=1}^NP(x_i)\right|\le\frac{\|\mathcal{R}\|}{N}
\sum_{i=1}^N|\nabla P(z_i)|.
$$
Consider another area-regular partition
$\mathcal{R'}=\left\{R'_1,\ldots, R'_N\right\}$ defined by
$R'_i=R_i\cup\{z_i\}$. Clearly $\|\mathcal{R'}\|\le
2\|\mathcal{R}\|$ and so, by~\eqref{eee}, we get
$\|\mathcal{R'}\|<r_d/(27\, d\,t)$. Applying
inequality~\eqref{Mhaskar-mod} to the partition $\mathcal{R'}$ and
the collection of points $z_i$ we obtain that
\begin{equation}
\label{e1} \left|\frac 1N\sum_{i=1}^NP(x_i)\right|\le
3\sqrt{d}\,\|\mathcal{R}\|\,\int_{S^d}|\nabla P(x)|d\mu_d(x)<\frac{
r_d}{18\,\sqrt{d} \, t}
\end{equation}
for any $P\in\partial\Omega$. On the other hand, the differential
equation \eqref{diffur} implies \begin{align} \frac{d}{d
s}\left[\frac 1N\sum_{i=1}^N P(y_i(s))\right]=
& \frac 1N\sum_{i=1}^N\frac{|\nabla P(y_i(s))|^2}{h_\epsilon (|\nabla P(y_i(s))|)
} \notag \\ \geq & \frac 1N\sum_{i:\,|\nabla P(y_i(s))|\geq \epsilon} |\nabla P(y_i(s))|\notag \\
\label{epsilon}\geq & \frac 1N\sum_{i=1}^N |\nabla
P(y_i(s))|-\epsilon.
\end{align}
Since $$\left|\frac{\nabla P(y)}{h_\epsilon (|\nabla
P(y)|)}\right|\le1$$ for each $y\in S^d$, it follows again from
\eqref{diffur} that $\left|\frac{d y_i(s)}{d s}\right|\le1$. Hence
we arrive at
$$
\mathrm{dist}(x_i,y_i(s))\leq s.
$$
Now for each $s\in [0, r_d/3t]$ consider the area-regular partition
$\mathcal{R''}=\left\{R''_1,\ldots, R''_N\right\}$ given by
$R''_i=R_i\cup\{y_i(s)\}$. By~\eqref{eee} we have
$$
\|\mathcal{R''}\|<\frac{r_d}{54dt}+\frac{r_d}{3t};
$$
so we can apply~\eqref{Mhaskar-mod} to the partition $\mathcal{R''}$
and the collection of points $y_i(s)$. This and
inequality~\eqref{epsilon} yield
$$
\frac{d}{d s}\left[\frac 1N\sum_{i=1}^N P(y_i(s))\right]\ge \frac
1N\sum_{i=1}^N |\nabla P(y_i(s))|-\frac 1{6\sqrt{d}}
$$
\begin{equation}
\label{ee2} \ge\frac 1{3\sqrt{d}}\int_{S^d}|\nabla
P(x)|d\mu_d(x)-\frac 1{6\sqrt{d}}=\frac 1{6\sqrt{d}},
\end{equation}
for each $P\in\partial\Omega$ and $s\in [0,r_d/3t]$. Finally,
equation~\eqref{ee} and inequalities \eqref{e1} and \eqref{ee2}
imply
\begin{equation}
\label{cond3} \frac1N\sum_{i=1}^N P(x_i(P))> \frac
1{6\sqrt{d}}\,\frac{r_d}{3t}-\frac{ r_d}{18\,\sqrt{d} \, t}=0.
\end{equation}
Lemma 1 is proved.
\end{proof}
|
{
"timestamp": "2011-03-08T02:04:26",
"yymm": "1009",
"arxiv_id": "1009.4407",
"language": "en",
"url": "https://arxiv.org/abs/1009.4407"
}
|
\section{Introduction}
Recently, there is a convincing evidence that neutrinos have a tiny non-zero mass. The evidence of neutrino mass is based on the experimental facts that both solar and atmospheric neutrinos undergo oscillations.\cite{Fukuda98}-\cite{Ahmad} A global analysis of neutrino oscillations data gives the best fit value to solar neutrino mass-squared differences,\cite{Gonzales-Garcia04}
\begin{equation}
\Delta m_{21}^{2}=(8.2_{-0.3}^{+0.3})\times 10^{-5}~{\rm eV^2}~
\label{11}
\end{equation}
with
\begin{equation}
\tan^{2}\theta_{21}=0.39_{-0.04}^{+0.05},
\label{21}
\end{equation}
and for the atmospheric neutrino mass-squared differences
\begin{equation}
\Delta m_{32}^{2}=(2.2_{-0.4}^{+0.6})\times 10^{-3}~{\rm eV^2}~
\label{22}
\end{equation}
with
\begin{equation}
\tan^{2}\theta_{32}=1.0_{-0.26}^{+0.35},
\end{equation}
where $\Delta m_{ij}^2=m_{i}^2-m_{j}^2~ (i,j=1,2,3)$ with $m_{i}$ is the neutrino mass in eigenstates basis $\nu_{i}~(i=1,2,3)$, and $\theta_{ij}$ is the mixing angle between $\nu_{i}$ and $\nu_{j}$. The mass eigenstates basis are related to the weak (flavor) eigenstates basis $(\nu_{e},\nu_{\mu},\nu_{\tau})$ as follows,
\begin{equation}
\bordermatrix{& \cr
&\nu_{e}\cr
&\nu_{\mu}\cr
&\nu_{\tau}\cr}=V\bordermatrix{& \cr
&\nu_{1}\cr
&\nu_{2}\cr
&\nu_{3}\cr}
\label{5}
\end{equation}
where $V$ is the mixing matrix.
It is also known that neutrino masses are very small compared to its corresponding charged lepton masses and mixing does exist in neutrino sector. Charged lepton mass has a normal hierarchy, but neutrino mass can have either a normal or an inverted hierarchy. Thus, neutrinos have some different properties from charged leptons. From the theoretical side, it has been a guiding principle that the presence of hierarchies or of tiny quantities imply a certain protection symmetry in underlying physics. The candidates of such symmetry in neutrino physics may include $U(1)_{L'}$ based on the conservation of $L_{e}-L_{\mu}-L_{\tau}=L'$ and a $\mu-\tau$ symmetry based on the invariance of flavor neutrino mass term underlying the interchange of $\nu_{\mu}$ and $\nu_{\tau}$.
To accommodate a tiny non-zero neutrino mass that can produce the mass-squared differences and the neutrino mixing, several models for the neutrino mass matrices together with the responsible mechanisms for generating it patterns have been proposed by many authors. One of the interesting mechanism which can generate a small neutrino mass is the seesaw mechanism, in which the right-handed neutrino $\nu_{R}$ has a large Majorana mass $M_{N}$ and the left-handed neutrino $\nu_{L}$ obtain a mass through leakage of the order of $~(m/M_{N})$ with $m$ is the Dirac mass.\cite{Fukugita03}
According to seesaw mechanism,\cite{Gell-Mann79} the neutrino mass matrix $M_{\nu}$ is given by,
\begin{eqnarray}
M_{\nu}\approx -M_{D}M_{N}^{-1}M_{D}^T
\label{Mnu}
\end{eqnarray}
where $M_{D}$ and $M_{N}$ are the Dirac and Majorana mass matrices respectively. The mass matrix model of a massive Majorana neutrino $M_{N}$ which is constrained by the solar and atmospheric neutrinos deficit and incorporating the seesaw mechanism and Peccei-Quinn symmetry have been reported by Fukuyama and Nishiura.\cite{Fukuyama97} Neutrino mass matrix patterns together with its underlying symmetry become an interesting research topic during the last few years. In related to the seesaw mechanism, Ma\cite{Ma05} pointed out that it is more sense to consider the structure of $M_{N}$ for its imprint on $M_{\nu}$.
In order to consider the structure of the $M_{N}$ for its imprint on $M_{\nu}$, in this paper we construct the neutrino mass matrices $M_{\nu}$ arise from a seesaw mechanism with both heavy Majorana and Dirac neutrino mass matrix are invariant under a cyclic permutation. As we have already knew that neutrino mass matrix which is invariant under a cyclic permutation gives $m_{1}=m_{3}$ and then it fails to predict mass-squared difference $\Delta m_{31}^{2}\neq 0$. The charged-lepton mass matrix which is invariant under a cyclic permutation have been analyzed by Koide\cite{Koide05} that also suggested to break the invariant under a cyclic permutation if we want to obtain the charged-lepton mass spectrum compatible with the empirical fact.
To overcome the weakness of cyclic permutation on predicting mass-squared differences, in this paper we introduce a perturbation into neutrino mass matrix with the assumption that the perturbed cyclic permutation mass matrix has the same trace with the unperturbed neutrino mass matrix. This paper is organized as follows: In Section 2, we construct the heavy Majorana and Dirac neutrino mass matrices which are invariant under a cyclic permutation. The resulted neutrino mass matrices to be used for obtaining the neutrino mass matrix $M_{\nu}$ in the scheme of seesaw mechanism. In Section 3, we use a seesaw mechanism for obtaining neutrino mass matrix and evaluate its phenomenological consequences. Finally, the Section 4 is devoted to a conclusion.
\section{Neutrino Mass Matrix with Invariant under Cyclic Permutation}
As we have already stated above, the aim of this paper is to study the phenomenological consequences of the perturbed cyclic permutation neutrino mass matrices arise from a seesaw mechanism with both heavy Majorana and Dirac neutrino mass matrices are invariant under a cyclic permutation. The seesaw mechanism to be considered in this paper is the type-I seesaw. In order to realize the goals of this section, first we write down the the possible patterns for heavy Majorana neutrino mass matrices $M_{N}$ is invariant under a cyclic permutation. Second, we write down the possible patterns for Dirac neutrino mass matrices by taking into account the same constraints that we have imposed on heavy Majorana neutrino mass matrix.
We consider the Majorana neutrino mass matrix $M_{N}$ in Eq.~(\ref{Mnu}) to be symmetric in form and that matrix is given by
\begin{equation}
M_{N}=\bordermatrix{& & &\cr
&A &B &C\cr
&B &D &E\cr
&C &E &F\cr}.
\label{MN}
\end{equation}
In order to obtain the $M_{N}$ that invariant under a cyclic permutation among neutrino fields: $\nu_{1}\rightarrow \nu_{2}\rightarrow \nu_{3}\rightarrow \nu_{1}$, first we define
\begin{equation}
\nu'_{i}=U_{ij}\nu_{j},
\label{T}
\end{equation}
where $U_{ij}$ are the entries of the cyclic permutation matrix $U$. From Eq.~(\ref{T}), one can see that the $M_{N}$ matrix satisfy
\begin{equation}
M'_{N}=U^{T}M_{N}U.
\label{M}
\end{equation}
If the $M_{N}$ matrix is invariant under a cyclic permutation, then the pattern of the $M'_{N}$ is the same with the $M_{N}$ pattern.
By imposing the requirement that the form of the $M_{N}$ matrix in Eq. (\ref{MN}) must be invariant under a cyclic permutation together with the requirement that the $M_{N}$ is non-singular matrix such that the $M_{N}$ has a $M_{N}^{-1}$, then we have three possible patterns for heavy Majorana neutrino mass matrix $M_{N}$ as follow
\begin{eqnarray}
M_{N}=\bordermatrix{& & &\cr
&A &B &B\cr
&B &A &B\cr
&B &B &A\cr},
\label{MN11}\\
M_{N}=\bordermatrix{& & &\cr
&A &0 &0\cr
&0 &A &0\cr
&0 &0 &A\cr},
\label{MN12}\\
M_{N}=\bordermatrix{& & &\cr
&0 &B &B\cr
&B &0 &B\cr
&B &B &0\cr}.
\label{MN13}
\end{eqnarray}
From Eqs.~(\ref{MN11}),~(\ref{MN12}), and~(\ref{MN13}), one can see that the patterns of neutrino mass matrices in Eqs.~(\ref{MN12}) and (\ref{MN13}) are special cases of the neutrino mass matrix pattern in Eq.~(\ref{MN11}). Thus, the neutrino mass matrix given by Eq.~(\ref{MN11}) is the most general pattern, and we will consider it as a good candidate for neutrino mass matrix $M_{N}$.
To obtain the neutrino mass matrices $M_{\nu}$ arise from a seesaw mechanism (using Eq.~(\ref{Mnu})), we should know the patterns of the Dirac neutrino mass matrices $M_{D}$. Because the heavy neutrino fields are part of the Dirac mass term, according to Eqs.~(\ref{T}) and (\ref{M}), the possible patterns for Dirac neutrino mass matrices are given by
\begin{eqnarray}
M_{D}=\bordermatrix{& & &\cr
&a &a &a\cr
&a &a &a\cr
&a &a &a\cr},
\label{MD0}\\
M_{D}=\bordermatrix{& & &\cr
&a &b &b\cr
&b &a &b\cr
&b &b &a\cr},
\label{MD1}\\
M_{D}=\bordermatrix{& & &\cr
&a &0 &0\cr
&0 &a &0\cr
&0 &0 &a\cr},
\label{MD2}\\
M_{D}=\bordermatrix{& & &\cr
&0 &b &b\cr
&b &0 &b\cr
&b &b &0\cr}.
\label{MD3}
\end{eqnarray}
It is apparent from Eqs.~(\ref{MD0})-(\ref{MD3}) that the pattern of neutrino mass matrix $M_{D}$ in Eq. (\ref{MD1}) is the most general pattern. Thus, we will consider neutrino mass matrix $M_{D}$ in Eq.~(\ref{MD1}) as a good candidate for Dirac neutrino matrix.
\section{Neutrino Mass matrix via a Seesaw Mechanism}
Using the seesaw mechanism in Eq.~(\ref{Mnu}), the heavy Majorana neutrino mass matrix in Eq.~(\ref{MN11}), and the Dirac neutrino mass matrix in Eq.~(\ref{MD1}), we then obtain a neutrino mass matrix with pattern,
\begin{eqnarray}
M_{\nu}=\bordermatrix{& & &\cr
&P &Q &Q\cr
&Q &P &Q\cr
&Q &Q &P\cr}.
\label{MD01}
\end{eqnarray}
The eigenvalues of the neutrino mass matrix in Eq.~(\ref{MD01}) are given by
\begin{eqnarray}
\lambda_{1}=\lambda_{2}=P-Q,\ \lambda_{3}=P+2Q.
\label{as}
\end{eqnarray}
It is easy to see that one of the eigenvectors of the $M_{\nu}$ is $(1,1,1)^{T}$ and this eigenvector corresponds to eigenvalue $\lambda_{3}$. Thus, the eigenvalue $\lambda_{3}$ should be identified as neutrino mass $m_{2}$, meanwhile $\lambda_{1}$ and $\lambda_{2}$ correspond to neutrino masses $m_{1}$ and $m_{2}$. Finally, the neutrino mass matrix in Eq.~(\ref{MD01}) gives neutrino masses,
\begin{equation}
m_{1}=m_{3}=P-Q,\ m_{2}=P+2Q.
\label{as1}
\end{equation}
From Eq.~(\ref{as1}), we have $\Delta m_{21}^{2}= \left|\Delta m_{32}^{2}\right|=6PQ+3Q^{2}$ which is contrary to the experimental fact. Thus, the neutrino mass matrix in Eq.~(\ref{MD01}) could not reproduce the mass-squared difference $\Delta m_{21}^{2}<< \Delta m_{32}^{2}$. It is also apparent that the resulted neutrino mass matrix in this scenario gives $m_{1}+m_{2}+m_{3}=3P$ is which is equal to $Tr(M_{\nu})$.
Even though neutrino mass matrix that invariant under a cyclic permutation, as one can see in Eq.~(\ref{MD01}), could not predict correctly the experimental data, we can still use it as a neutrino mass matrix. In order to obtain neutrino mass matrix that can give correct predictions on mass-squared differences and mixing parameters, we modify the neutrino mass matrix $M_{\nu}$ in Eq.~(\ref{MD01}) by introducing one parameter $\delta$ to perturb the diagonal elements of $M_{\nu}$ such that the perturbed mass matrix satisfies the requirement $Tr(M_{\nu})=3P$. In this scenario, we then can put the neutrino mass matrix $M_{\nu}$ in form,
\begin{equation}
M_{\nu}=\bordermatrix{& & &\cr
&P+2\delta &Q &Q\cr
&Q &P-\delta &Q\cr
&Q &Q &P-\delta\cr}.
\label{MD011}
\end{equation}
The eigenvalues of the neutrino mass matrix in Eq.~(\ref{MD011}) read
\begin{eqnarray}
\beta_{1,2}=P+\frac{Q}{2}+\frac{\delta}{2}\mp\frac{\sqrt{9\delta^{2}-6Q\delta+9Q^{2}}}{2},\\
\beta_{3}=P-Q-\delta.
\label{ec}
\end{eqnarray}
If the neutrino mass matrices $M_{\nu}$ in Eq.~(\ref{MD011}) is diagonalized by mixing matrix $V$ in Eq.~(\ref{5}) with $V$ given by\cite{Ma05}
\begin{equation}
V=\bordermatrix{& & &\cr
&\cos\theta &-\sin\theta &0\cr
&\sin\theta/\sqrt{2} &\cos\theta/\sqrt{2} &-1/\sqrt{2}\cr
&\sin\theta/\sqrt{2} &\cos\theta/\sqrt{2} &1/\sqrt{2}\cr},
\label{511}
\end{equation}
then we obtain,
\begin{equation}
\tan^{2}(2\theta)=\frac{8Q^{2}}{(Q-3\delta)^{2}},
\label{Teta}
\end{equation}
and neutrino masses as follow,
\begin{eqnarray}
m_{1}=P+\frac{Q}{2}+\frac{\delta}{2}-\frac{\sqrt{9\delta^{2}-6Q\delta+9Q^{2}}}{2},
\label{m1}\\
m_{2}=P+\frac{Q}{2}+\frac{\delta}{2}+\frac{\sqrt{9\delta^{2}-6Q\delta+9Q^{2}}}{2},
\label{m2}\\
m_{3}=P-Q-\delta.
\label{m3}
\end{eqnarray}
One can see that the obtained neutrino masses in this scenario is an inverted hierarchy with masses: $\left|m_{3}\right|<\left|m_{1}\right|<\left|m_{2}\right|$.
If $\theta$ is the $\theta_{21}$ in Eq.~(\ref{21}), then from Eq.~(\ref{Teta}) we have $\delta=-0.1271Q$. If we insert this $\delta$ value into Eqs.~(\ref{m1})-(\ref{m3}), then we have the neutrino masses as follow
\begin{eqnarray}
m_{1}=P-0.1374Q,
\label{m11}\\
m_{2}=P+2.0103Q,
\label{m22}\\
m_{3}=P-0.8729Q.
\label{m33}
\end{eqnarray}
The plot of $m_{1}$, $m_{2}$, and $m_{3}$ as function of parameters $P$ and $Q$ are shown in Fig.1.
\begin{figure}[h]
\centering
\includegraphics[width=0.3\textwidth]{graph1.eps}
\includegraphics[width=0.3\textwidth]{graph2.eps}
\includegraphics[width=0.3\textwidth]{graph3.eps}
\caption{Neutrino masses $m_{1}$, $m_{2}$, and $m_{3}$ as function of parameters $P$ and $Q$.}
\label{figure1}
\end{figure}
From Fig.~1 we can see that the neutrino mass can have normal, degenerate, or inverted hierarchy which it depends on the sign and values of of parameters $P$ and $Q$. For example, if we put the values of $P=-0.8$ and $Q=1$, then we have neutrino masses in normal hierarchy: $\left|m_{1}\right|<\left|m_{2}\right|<\left|m_{3}\right|$. The degenerate hierarchy: $\left|m_{1}\right|\approx\left|m_{2}\right|\approx\left|m_{3}\right|$ can be obtained if we put the value of parameter $Q\approx 0$, and the inverted hierarchy: $\left|m_{3}\right|<\left|m_{1}\right|<\left|m_{2}\right|$ is produced for $P>0.8729Q$ and $Q>0$.
If we use the advantages of the experimental data of neutrino oscillation in Eqs.~(\ref{11}) and (\ref{22}), from Eqs.~(\ref{m1}),~(\ref{m2}), and~(\ref{m3}), then we obtain the neutrino masses,
\begin{equation}
\left|m_{1}\right|=0.101023 ~{\rm eV}~, \; \left|m_{2}\right|=0.101428 ~{\rm eV}~,\;\left|m_{3}\right|=0.084413 ~{\rm eV},
\label{NM}
\end{equation}
for $Q=-0.06432$ eV and $P=0.02827$ eV. One can see that the value of $\delta=0.008176$ eV is smaller than the values of $P$ and $Q$. Inserting the obtained values of $P, Q$, and $\delta$ into Eq.~(\ref{MD011}), we finally have neutrino mass matrix in eV unit as follow
\begin{equation}
M_{\nu}=\bordermatrix{& & &\cr
&0.04463 &-0.06432 &-0.06432\cr
&-0.06432 &0.02010 &-0.06432\cr
&-0.06432 &-0.06432 &0.02010\cr}.
\end{equation}
\section{Conclusion}
Neutrino mass matrix $M_{\nu}$ arise from a seesaw mechanism, with both heavy Majorana and Dirac neutrino mass matrices are invariant under a cyclic permutation, can not be used to explain the present experimental data of neutrino oscillation. By introducing one parameter $\delta$ to perturb the diagonal elements of $M_{\nu}$ with the assumption that the value of the trace of $M_{\nu}$ remain constant, we then have a neutrino mass matrix that can be used to explain mass-squared differences. In this scenario, the possible hierarchy of neutrino mass that can be used to explain mass-squared differences is normal or inverted hierarchy. By using the mass-squared differences and mixing parameters which obtained from neutrino oscillation experiments, we then have neutrino masses in inverted hierarchy with masses: $\left|m_{1}\right|=0.101023$ eV, $\left|m_{2}\right|=0.101428$ eV, and $\left|m_{3}\right|=0.084413$ eV.
\section*{Acknowledgments}
Author would like to thank reviewers for their useful comments and suggestions.
|
{
"timestamp": "2011-05-11T02:01:15",
"yymm": "1009",
"arxiv_id": "1009.4080",
"language": "en",
"url": "https://arxiv.org/abs/1009.4080"
}
|
\section{Introduction and statement of the main result}
\subsection{The Boolean model}
Let $d\ge 2$.
Let $\mu$ be a finite measure on $]0,+\infty[$.
We assume that the mass of $\mu$ is positive.
Let $\xi$ be a Poisson point process on $\mathbb{R}^d \times ]0,+\infty[$ whose intensity is the product of the Lebesgue measure on $\mathbb{R}^d$ by $\mu$.
With $\xi$ we associate a random set $\Sigma(\mu)$ defined as follows:
$$
\Sigma(\mu)=\bigcup_{(c,r) \in \xi} B(c,r)
$$
where $B(c,r)$ is the open Euclidean ball of radius $r$ centered at $c$.
The random set $\Sigma(\mu)$ is the Boolean model with parameter $\mu$.
When shall sometimes write $\Sigma$ to simplify the notations.
The following description may be more intuitive.
Let $\chi$ denote the projection of $\xi$ on $\mathbb{R}^d$.
With probability one this projection is one-to-one.
We can therefore write:
$$
\xi=\{(c,r(c)), c \in \chi\}.
$$
Write $\mu=m\nu$ where $\nu$ is a probability measure.
Then, $\chi$ is a Poisson point process on $\mathbb{R}^d$ with density $m$.
Moreover, given $\chi$, the sequence $(r(c))_{c\in\chi}$ is a sequence of independent random variable with common distribution $\nu$.
We shall not use this point of view.
\subsection{Percolation in the Boolean model}
\label{s:boolean}
Let $C$ denote the connected component of $\Sigma$ that contains the origin.
We say that $\Sigma$ percolates if $C$ is unbounded with positive probability.
We refer to the book by Meester and Roy \cite{Meester-Roy-livre} for background on continuum percolation.
Set:
$$
\lambda_c(\mu) = \inf \{\lambda>0 : \Sigma(\lambda\mu) \hbox{ percolates}\}.
$$
One easily check that $\lambda_c(\mu)$ is finite as soon as $\mu$ has a positive mass.
In \cite{G-perco-boolean-model} we proved that $\lambda_c(\mu)$ is positive if and only if:
$$
\int r^d\mu(dr) < \infty.
$$
The only if part had been proved earlier by Hall \cite{Hall-continuum-percolation}.
For all $A,B \subset \mathbb{R}^d$, we write $\perco{A}{B}{\Sigma}$ if there exists a path in $\Sigma$ from $A$ to $B$.
We denote by $S(c,r)$ the Euclidean sphere or radius $r$ centered at $c$ :
$$
S(c,r)=\{x \in \mathbb{R}^d : \|x-c\|_2=r\}.
$$
We write $S(r)$ when $c=0$.
The critical parameter $\lambda_c(\mu)$ can also be defined as follows:
$$
\lambda_c(\mu)
= \sup \left\{ \lambda>0 : P\left(\perco{\{0\}}{S(r)}{\Sigma(\lambda\mu)}\right) \to 0 \hbox{ as } r \to \infty\right\}, \\
$$
We shall need two other critical parameters:
\begin{eqnarray*}
\widehat{\lambda_c}(\mu)
& = & \sup \left\{ \lambda>0 : P\left(\perco{S(r/2)}{S(r)}{\Sigma(\lambda\mu)}\right) \to 0 \hbox{ as } r \to \infty\right\}, \\
\widetilde{\lambda_c}(\mu)
& = & \sup \left\{ \lambda>0 : r^dP\left(\perco{\{0\}}{S(r)}{\Sigma(\lambda\mu)}\right) \to 0 \hbox{ as } r \to \infty\right\}.
\end{eqnarray*}
We have (see Lemma \ref{l:lambdacinegalite}) :
\begin{equation}\label{e:lambdacinegalite}
\widetilde{\lambda_c}(\mu) \le \widehat{\lambda_c}(\mu) \le \lambda_c(\mu).
\end{equation}
When the support of $\mu$ is bounded,
$$
P\left(\perco{\{0\}}{S(r)}{\Sigma(\lambda\mu)}\right)
$$
decays exponentially fast to $0$ as soon as $\lambda<\lambda_c(\mu)$
(see for example \cite{Meester-Roy-livre}, Section 12.10 in \cite{Grimmett-percolation} in the case of constant radii
or the papers \cite{Meester-Roy-Sarkar}, \cite{Menshikov-Sidorenko-coincidence}, \cite{Zuev-I} and \cite{Zuev-II}).
Therefore:
\begin{equation}\label{e:lambdacborne}
\widetilde{\lambda_c}(\mu)=\widehat{\lambda_c}(\mu)=\lambda_c(\mu) \hbox{ as soon as the support of }\mu\hbox{ is bounded}.
\end{equation}
\prg{Remarks}
\begin{itemize}
\item
The treshold parameter $\widehat{\lambda_c}(\mu)$ is positive if and only if $\int r^d \mu(dr)$ is finite
(i.e., if and only if $\lambda_c(\mu)$ is positive). See Lemma \ref{l:hat}.
\item Using ideas of \cite{G-perco-boolean-model},
we can check that $\widetilde{\lambda_c}(\mu)$ is positive if and only if
$$
x^d \int_x^\infty r^d \mu(dr) \to 0 \hbox{ as } x\to\infty.
$$
If we only use results stated in \cite{G-perco-boolean-model}, we can easily get the following weaker statements.
Let $D(\lambda\mu)$ denote the Euclidean diameter of the connected component of $\Sigma(\lambda\mu)$ that contains the origin.
Note that $\widetilde{\lambda_c}(\mu)$ is positive if and only if there exists $\lambda$ such that:
\begin{equation}\label{e:decay}
r^dP(D(\lambda\mu) \ge r) \to 0, \hbox{ as } r\to\infty.
\end{equation}
If $E(D(\lambda\mu)^d)$ is finite then \eqref{e:decay} holds.
If \eqref{e:decay} holds then $E(D(\lambda\mu)^{d-\epsilon})$ is finite for any small enough $\epsilon>0$.
By Theorem 2.2 of \cite{G-perco-boolean-model} we thus get the following implications:
$$
\int_0^{+\infty} r^{2d} \mu(dr)<\infty \; \hbox{ implies } \; \widetilde{\lambda_c}(\mu)>0
\; \hbox{ implies }\; \forall\epsilon>0: \int_0^{+\infty} r^{2d-\epsilon} \mu(dr)<\infty.
$$
\end{itemize}
\subsection{A multiscale Boolean model}
Let $\rho>1$ be a scale factor.
Let ${(\Sigma_n)}_{n\ge 0}$ be a sequence of independent copies of $\Sigma(\mu)$.
In this paper, we are interested in percolation properties of the following multiscale Boolean model:
\begin{equation}\label{e:definition-multi}
\Sigma^{\rho}(\mu)=\bigcup_{n \ge 0} \rho^{-n} \Sigma_n.
\end{equation}
We shall sometimes write $\Sigma^{\rho}$ to simplify the notations.
As before,
we say that $\Sigma^{\rho}$ percolates if the connected component of $\Sigma^{\rho}$ that contains the origin is unbounded with positive probability.
This model seems to have been first introduced as a model of failure in geophysical medias in the $80'$.
We refer to the paper by Molchanov, Pisarenko and Reznikova \cite{Molchanov-al-failure} for an account of those studies.
For more recent results we refer to
\cite{Broman-Camia-self-similar}, \cite{Meester-Roy-livre}, \cite{Meester-Roy-Sarkar}, \cite{Menshikov-al-multi},
\cite{Menshikov-al-multi-unbounded} and \cite{Popov-V-sticks}.
This model is related to a discrete model introduced by Mandelbrot \cite{Mandelbrot-fractale-perco}.
We refer to the survey by L. Chayes \cite{Chayes-aspects-fractale-perco}
and, for more recent results, to \cite{Broman-Camia-mandelbrot}, \cite{O-fractale-perco} and \cite{White}.
In \cite{Menshikov-al-multi}, Menshikov, Popov and Vachkovskaia considered the case where the radii of the unscaled process $\Sigma_0$ equal $1$.
They proved the following result.
\begin{theorem}[\cite{Menshikov-al-multi}] \label{th:MPV1}
If $\lambda<\lambda_c(\delta_1)$ then, for all large enough $\rho$, $\Sigma^{\rho}(\lambda\mu)$ does not percolate.
\end{theorem}
In \cite{Menshikov-al-multi-unbounded} the same authors considered the case where the radii are random and can be unbounded.
They considered the following sub-autosimilarity assumption on the measure $\mu$:
\begin{equation}\label{e:MPV2}
\lim_{a\to\infty} \sup_{r\ge 1/2} \frac{a^d\mu([ar,+\infty[)}{\mu([r,+\infty[)}=0
\end{equation}
with the convention $0/0=0$.
They proved the following result.
\begin{theorem}[\cite{Menshikov-al-multi-unbounded}] \label{th:MPV2}
Assume that the measure $\mu$ satisfies \eqref{e:MPV2}.
Assume that $\widetilde{\lambda_c}(\mu)$ is positive.
If $\lambda < \widetilde{\lambda_c}(\mu)$ then, for all large enough $\rho$, $\Sigma^{\rho}(\lambda\mu)$ does not percolate.
\end{theorem}
Note that \eqref{e:MPV2} is fulfilled for any measure with bounded support.
Because of \eqref{e:lambdacborne}, Theorem \ref{th:MPV2} is then a generalization of Theorem \ref{th:MPV1}.
In \cite{G-perco-generale} we proved the following related result in which $\rho$ is fixed.
\begin{theorem}[\cite{G-perco-generale}] \label{th:Gmulti}
Let $\rho>1$.
There exists $\lambda>0$ such that $\Sigma^{\rho}(\lambda\mu)$ does not percolate if and only if:
\begin{equation}\label{e:th}
\int_{[1,+\infty[} \beta^d \ln(\beta) \mu(d\beta)<\infty.
\end{equation}
\end{theorem}
The main result of this paper is the first item of the following theorem.
The second item is easy and already contained in Theorem \ref{th:Gmulti}.
Recall that, by Lemma \ref{l:hat}, $\widehat{\lambda_c}(\mu)$ is positive as soon as $\int r^d\mu(dr)$ is finite and therefore as soon as
\eqref{e:th} holds.
\begin{theorem} \label{th:0}
$ $
\begin{enumerate}
\item Assume \eqref{e:th}.
Then, for all $\lambda<\widehat{\lambda}_c(\mu)$, there exists $\rho(\lambda)>1$ such that, for all $\rho\ge\rho(\lambda)$:
\begin{equation}\label{e:pitilde0}
P\left(\perco{S(r/2)}{S(r)}{\Sigma^{\rho}(\lambda\mu)}\right) \to 0 \hbox{ as } r \to \infty
\end{equation}
and therefore $\Sigma^{\rho}(\lambda\mu)$ does not percolate.
\item
Assume that \eqref{e:th} does not hold.
Then, for all $\lambda>0$ and for all $\rho>1$, $\Sigma^{\rho}(\lambda\mu)$ percolates.
\end{enumerate}
\end{theorem}
The proof is given in Section \ref{s:preuve-th:0}.
The ideas of its proof and the ideas of the proofs of Theorems \ref{th:MPV1} and \ref{th:MPV2} are given in Subsection \ref{s:idees}.
The first item of Theorem \ref{th:0} is a generalization of Theorem \ref{th:MPV2} and thus of Theorem \ref{th:MPV1}.
Indeed, by \eqref{e:lambdacinegalite}, one has $\lambda<\widehat{\lambda_c}$ as soon as $\lambda<\widetilde{\lambda_c}$.
Moreover, by the second item of Theorem \ref{th:0}, \eqref{e:th} has to be a consequence of the assumptions of Theorem \ref{th:MPV2}.
For example, one can check that \eqref{e:th} is a consequence of \eqref{e:MPV2}
\footnote{From \eqref{e:MPV2} one gets the existence of $a>1$ such that, for all $r \ge a$, one has $\mu([r,+\infty[)\le 2^{-1}a^{-d}\mu([r/a,+\infty[)$.
By induction and standard computations this yields, for all $r \ge a$, $\mu([r,+\infty[) \le Ar^{-\ln(2)/\ln(a)-d}$.
Therefore, for a small enough $\eta>0$, one has $\int r^{d+\eta} \mu(dr)<\infty$.}.
Alternatively, one can check that \eqref{e:th} is a consequence of $\widetilde{\lambda_c}(\mu)>0$ (see the remarks at the end of Section \ref{s:boolean}).
Let us denote by $\lambda_c(m^{\rho}_{\infty})$ and $\widehat{\lambda}_c(m^{\rho}_{\infty})$ the $\lambda_c$ and $\widehat{\lambda}_c$
critical tresholds for the multiscale model with scale parameter $\rho$.
Theorems \ref{th:Gmulti} and \ref{th:0} yield the following result:
\begin{enumerate}
\item If \eqref{e:th} holds then $\lambda_c(m^{\rho}_{\infty})>0$ (and actually the proof of Theorem \ref{th:Gmulti} yields
$\widehat{\lambda}_c(m^\rho_{\infty})>0$) for all $\rho>1$ and
$\widehat{\lambda}_c(m^{\rho}_{\infty}) \to \widehat{\lambda}_c(\mu) > 0$ as $\rho \to \infty$.
\item Otherwise, $\widehat{\lambda}_c(m^{\rho}_{\infty})=\lambda_c(m^{\rho}_{\infty}) = 0$ for all $\rho>1$.
\end{enumerate}
Let us denote by $D^{\rho}(\lambda\mu)$ the diameter of the connected component of $\Sigma^{\rho}(\lambda\mu)$ that contains the origin.
The following result is an easy consequence of Theorem \ref{th:0} above and Theorems 2.9 and 1.2 in \cite{G-perco-generale}.
\begin{theorem} \label{th:1}
Let $s>0$, $\lambda>0$ and $\rho>1$.
\begin{enumerate}
\item If $\int_{[1,+\infty[} \beta^{d+s} \mu(d\beta)<\infty$ and \eqref{e:pitilde0} holds, then $E\big((D^{\rho}(\lambda\mu))^s\big)<\infty$.
\item If $\int_{[1,+\infty[} \beta^{d+s} \mu(d\beta)=\infty$ then $E\big((D^{\rho}(\lambda\mu))^s\big)=\infty$.
\end{enumerate}
\end{theorem}
The proof is given is Section \ref{s:preuve-th:1}.
\subsection{Superposition of Boolean models with different laws}
Using the same arguments as in the proof of Theorem \ref{th:0}, we could prove similar results for infinite superpositions
$$
\bigcup_{n \ge 0} \rho^{-n} \Sigma_n
$$
where the Boolean models $\Sigma_n$ are independent but not identically distributed.
We will not give such a result here.
However, we wish to give a weaker result for the superposition of two independent Boolean models at different scales.
As we consider only two scales the proof is easier than the proof of Theorem \ref{th:0}.
The proof uses Lemmas \ref{l:eta}, \ref{l:dichotomie} and \ref{l:carre} and is given in Section \ref{s:preuve-p}.
The result gives some insight on the critical treshold in the case of balls of random radii.
This result, in the case where the supports of $\nu_1$ and $\nu_2$ are bounded, is already implicit in \cite{Meester-Roy-Sarkar}
in their proof of non universality of critical covered volume (see \eqref{e:phic} below).
See also \cite{Molchanov-al-failure}.
\begin{prop} \label{p}
Let $\nu_1$ and $\nu_2$ be two finite measures on $]0,+\infty[$.
We assume that the masses of $\nu_1$ and $\nu_2$ are positive.
Let $0<\alpha<1$.
Then, for all $\rho>1$,
$$
\widehat{\lambda}_c(\alpha \nu_1 + (1-\alpha) H_{\rho} \nu_2)
\le
\min\big( \widehat{\lambda}_c(\alpha \nu_1),\widehat{\lambda}_c((1-\alpha) H_{\rho} \nu_2)\big)
=
\min\left(\frac{\widehat{\lambda}_c(\nu_1)}{\alpha}, \frac{\widehat{\lambda}_c(\nu_2)}{1-\alpha}\right).
$$
Moreover,
$$
\widehat{\lambda}_c(\alpha \nu_1 + (1-\alpha) H_{\rho} \nu_2)
\to
\min\left(\frac{\widehat{\lambda}_c(\nu_1)}{\alpha}, \frac{\widehat{\lambda}_c(\nu_2)}{1-\alpha}\right)
\hbox{ as } \rho\to\infty.
$$
The above convergence is uniform in $\alpha$.
\end{prop}
We now make some remarks about this result and about some related numerical results.
For a finite measure $\mu$ on $]0,+\infty[$, we denote by $\phi_c(\mu)$ the critical covered volume:
\begin{equation}\label{e:phic}
\phi_c(\mu) = P\big( 0 \in \Sigma(\lambda_c(\mu)\mu)\big) = 1-\exp\left(-\lambda_c(\mu)\int v_dr^d\mu(dr)\right)
\end{equation}
where $v_d$ is the volume of the unit Euclidean ball in $\mathbb{R}^d$.
This is the mean volume occupied by the critical Boolean model and this is scale invariant.
Let us assume that $\nu_1=\nu_2=\delta_1$.
By \eqref{e:lambdacborne}, by Proposition \ref{p} and with the above notation we have:
\begin{equation}\label{e:fatigue}
\phi_c(\alpha \delta_1 + (1-\alpha) H_{\rho} \delta_1)
\to
1-\exp\left(-v_d \lambda_c(\delta_1)\min\left(\frac{1}{\alpha}, \frac{1}{1-\alpha}\right)\right).
\end{equation}
There are several numerical studies of the above critical covered volume when $d=2$ and $d=3$.
To the best of our knowledge, the most acccurate values when $d=2$ are given in \cite{QZ-PRE-2007}.
Let us assume henceforth that $d=2$.
In \cite{QZ-PRE-2007}, the authors give:
\begin{equation}\label{e:approx}
\phi_c(\delta_1)=1-\exp(-v_2 \lambda_c(\delta_1)) \approx 0.6763475(6).
\end{equation}
In Figure \ref{f} we reproduce the graph of critical covered volume $\phi(\alpha,\rho)$ as a function of $\alpha$ when $\rho=2$, $\rho=5$ and $\rho=10$
(see \cite{QZ-PRE-2007} for more results).
We also represent the graph of the right-hand side of \eqref{e:fatigue}, that we denote by $\phi(\alpha,\infty)$, as a function of $\alpha$.
We use \eqref{e:approx} to get an approximate value of $v_2\lambda_c(\delta_1)$.
\begin{figure}[h!]
\centering
\includegraphics[width=0.95\textwidth]{multiscale-figure.eps}
\caption{Critical covered volume as a function of $\alpha$ for different values of $\rho$.
From bottom to top: $\rho=2, \rho=5, \rho=10$ and the limit as $\rho \to \infty$.}
\label{f}
\end{figure}
\paragraph{Remarks}
\begin{itemize}
\item
When $\rho \to \infty$,
the critical covered volume $\phi(\cdot,\rho)$ converges to $\phi(\cdot,\infty)$ which is symmetric: $\phi(\alpha,\infty)=\phi(1-\alpha,\infty)$.
When $\rho$ is finite, the critical covered volume may also look symmetric but Quintanilla and Ziff showed,
based on their numerical simulations and statistical analysis, that this was not the case.
\item
When $\rho$ is finite, the critical covered volume looks concave as a function of $\alpha$.
However $\phi(\cdot,\infty)$ is not concave as soon as $\phi_c(\delta_1)<1-\exp(-2)$.
Based on \eqref{e:approx}, $\phi(\cdot,\infty)$ is therefore not concave.
As a consequence, at least for large enough $\rho$, $\phi(\cdot,\rho)$ is not concave.
\item
The numerical results suggests that the minimum of the critical covered fraction is reached when all the disks have the same radius.
(Equivalently, for all $\rho$ and all $\alpha$, $\phi(\alpha,\rho) \ge \phi(0,\rho)=\phi(1,\rho)=\phi_c(\delta_1)$.)
However there is neither a proof nor a disproof of such a result.
\item
The numerical results also suggest some monotonicity in $\rho$.
This has not been proven nor disproven.
\end{itemize}
\section{Proof of Theorem \ref{th:0}}
\label{s:preuve-th:0}
\subsection{Some notations}
In the whole of Section \ref{s:preuve-th:0}, we make the following assumptions:
\begin{itemize}
\item $\mu$ satisfies \eqref{e:th}.
\item $1<\widehat{\lambda_c}(\mu)$.
\end{itemize}
For all $\eta>0$,
we denote by $T_{\eta}\mu$ the measure defined by $T_{\eta}\mu(A)=\mu(A-\eta)$.
In other words, we can built $\Sigma(T_{\eta}\mu)$ from $\Sigma(\mu)$ by adding $\eta$ to each radius.
For all $\rho>1$,
we denote by $H^{\rho}\mu$ the measure defined by $H^{\rho}\mu(A)=\rho^d\mu(\rho A)$.
With this definition, $\rho^{-1}\Sigma(\mu)$ is a Boolean model driven by the measure $H^{\rho}\mu$.
For all $n \ge 0$, we let:
$$
m_n^{\rho} = \sum_{k=0}^n H^{\rho^k}\mu.
$$
With this definition and the notations of \eqref{e:definition-multi},
$$
\bigcup_{k=0}^n \rho^{-k} \Sigma_k
$$
is a Boolean model driven by $m_n^{\rho}$.
We also let:
$$
m_{\infty}^{\rho} = \sum_{k \ge 0} H^{\rho^k}\mu.
$$
So, $\Sigma^{\rho}(\mu)$ is a Boolean model driven by the locally finite measure $m_{\infty}^{\rho}$.
Let $p(a,\mu)$ denote the probability of existence of a path from $S(a/2)$ to $S(a)$ in $\Sigma(\mu)$:
$$
p(a,\mu) = P(\perco{S(a/2)}{S(a)}{\Sigma(\mu)}).
$$
We aim at proving that, for large enough $\rho$, $p(a,m_{\infty}^{\rho})\to 0$ as $a$ tends to infinity and $\Sigma^{\rho}(\mu)$ does not percolate.
The first item of Theorem \ref{th:0} follows by applying this result to the measure $\lambda\mu$.
Recall that the second item of Theorem \ref{th:0} is contained in Theorem \ref{th:Gmulti}.
\subsection{Ideas}
\label{s:idees}
In this subsection we first sketch the proof of the existence of $\rho$ and $a$ such that $p(a,m_{\infty}^{\rho})$ is small.
This gives the main ingredients of the proof of the first item of Theorem \ref{th:0}.
A full proof is given in Subsection \ref{ss:proof}.
We then give the ideas of the proof of Theorems \ref{th:MPV1} and \ref{th:MPV2} by Menshikov, Popov and Vachkovskaia.
Their basic strategy is similar but the implementation of the proofs are different.
\subsubsection*{Sketch of the proof of the first item of Theorem \ref{th:0}}
Consider a small $\epsilon_1>0$.
Fix a small $\eta>0$ and a large $a$ such that (see Lemma \ref{l:eta}):
\begin{equation}\label{e:maj0}
p(a,T_{\eta}\mu) \le \epsilon_1/2.
\end{equation}
For all $n \ge 1$, write:
$$
m_n^{\rho}=H^{\rho}m_{n-1}^{\rho}+\mu.
$$
If the event $\{\perco{S(a/2)}{S(a)}{\Sigma(m_n^{\rho})}\}$ occurs,
then either the event $\{\perco{S(a/2)}{S(a)}{T_{\eta}\mu}\}$ occurs (with a natural coupling between the Boolean models)
either in $\Sigma(H^{\rho}m_{n-1}^{\rho}) \cap B(a)$ one can find a component of diameter at least $\eta$.
We use this observation through its following crude consequence (see Lemma \ref{l:dichotomie}):
$$
p(a,m_n^{\rho}) \le p(a,T_{\eta}\mu) + Ca^d\eta^{-d}p(\eta/2,H^{\rho}m_{n-1}^{\rho}).
$$
By scaling and by \eqref{e:maj0}, this yields:
\begin{equation} \label{e:majn}
p(a,m_n^{\rho}) \le \epsilon_1/2 + Ca^d\eta^{-d}p(\rho\eta/2,m_{n-1}^{\rho}).
\end{equation}
But for any $\epsilon_2$, any small enough $\epsilon_1$ and any large enough $a$ we can find $\tau$ such that
(see Lemmas \ref{l:carre} and \ref{l:poitiers}):
\begin{equation}\label{e:turbo}
p(\tau a, m_{n-1}^{\rho}) \le \epsilon_2 \hbox{ as soon as } p(a, m_{n-1}^{\rho}) \le \epsilon_1.
\end{equation}
An important fact is that $\tau$ does not depends on $n$ nor on $\rho$,
provided $\rho \ge \rho_0$ where $\rho_0$ is an arbitrary constant strictly larger than $1$.
Here we use assumption \eqref{e:th} to bound error terms due to the existence of large balls.
We choose $\epsilon_2$ such that:
$$
Ca^d\eta^{-d}\epsilon_2 = \epsilon_1/2.
$$
We set $\rho=2\tau a/\eta$.
Then, \eqref{e:majn} and \eqref{e:turbo} can be rewritten as follows:
\begin{eqnarray}
p(a,m_n^{\rho}) & \le & \epsilon_1/2 + Ca^d\eta^{-d}p(\tau a,m_{n-1}^{\rho}) \label{e:majn2} \\
Ca^d\eta^{-d}p(\tau a,m_{n-1}^{\rho}) & \le & \epsilon_1/2 \hbox{ as soon as } p(a, m_{n-1}^{\rho}) \le \epsilon_1 \label{e:turbo2}.
\end{eqnarray}
As moreover \eqref{e:maj0} implies $p(a,m_0^{\rho}) \le \epsilon_1$ we get, by induction and then sending $n$ to infinity
(see Lemma \ref{l:compacite}):
$$
p(a,m_{\infty}^{\rho}) \le \epsilon_1.
$$
The convergence of $p(a,m_{\infty}^{\rho})$ to $0$ is then extracted from the above result for a small enough $\epsilon_2$ and from
arguments behind \eqref{e:turbo} applied to $m_{\infty}^{\rho}$ and other $\epsilon$.
\subsubsection*{Sketch of the proofs of Theorems \ref{th:MPV1} and \ref{th:MPV2} by Menshikov, Popov and Vachkovskaia}
\label{s:MPV}
Let us quickly describe the ideas of the proofs of Menshikov, Popov and Vachkovskaia.
Those ideas are used in their papers \cite{Menshikov-al-multi} and \cite{Menshikov-al-multi-unbounded} through a discretization of space ;
we describe them in a slightly more geometric way.
For simplicity we only consider two scales: $\rho^{-1}\Sigma_1$ and $\Sigma_0$.
For simplicity, we also assume that the radius is one in the unscaled model ($\mu=\lambda\delta_1$).
We assume that the scale factor $\rho$ is large enough.
Assume that $C$ is a connected component of $\rho^{-1}\Sigma_1 \cup \Sigma_0$ whose diameter is a least $\alpha$
(it can be much larger) for a small enough constant $\alpha>0$.
Then, $C$ is included in the union of the following kind of sets:
\begin{enumerate}
\item connected components of $\rho^{-1}\Sigma_1$ whose diameter is at least $\alpha$ ;
\item balls of $\Sigma_0$ enlarged by $\alpha$ (same centers but the radii are $1+\alpha$ instead of $1$).
\end{enumerate}
Then, they show that the union of all those sets is stochastically dominated by a Boolean model similar to $\Sigma_0$
but with radii enlarged by a factor $\alpha$ and with density of centers $1+\alpha'$ times the corresponding density for $\Sigma_0$
for a suitable $\alpha'>0$.
This part uses $\lambda<\widehat{\lambda_c}$.
In some sense,
one can therefore control percolation in the union of two models by percolation in one model.
Iterating the argument with some care in the constants $\alpha$ and $\alpha'$,
one sees that -- for large enough $\rho$ -- one can control percolation in the multiscale model by percolation in a subcritical model.
This yields the result.
\subsection{Proof of Theorem \ref{th:0}}
\label{ss:proof}
As $1<\widehat{\lambda_c}(\mu)$, we know that $p(a,\mu)$ tends to $0$ as $a$ tends to infinity.
We need the following slightly stronger consequence.
\begin{lemma} \label{l:eta} There exists $\eta>0$ such that $p(a,T_{\eta}\mu)$ tends to $0$.
\end{lemma}
\prg{Proof} Let $\epsilon>0$ and $x>0$.
We have:
\begin{eqnarray}
H^{1+\epsilon}T_{\epsilon^2}\mu([x,+\infty[)
& = & (1+\epsilon)^d T_{\epsilon^2}\mu([x(1+\epsilon),+\infty[) \nonumber \\
& = & (1+\epsilon)^d \mu([x(1+\epsilon)-\epsilon^2,+\infty[) \nonumber \\
& \le & \kappa(\epsilon) (1+\epsilon)^d \mu([x,+\infty[) \label{e:domination}
\end{eqnarray}
where
$$
\kappa(\epsilon)=\frac{\mu(]0,+\infty[)}{\mu([\epsilon,+\infty[)}.
$$
The inequality is proven as follows.
If $x \ge \epsilon$, then $[x(1+\epsilon)-\epsilon^2,+\infty[ \subset [x,+\infty[$ and the result follows from $\kappa(\epsilon) \ge 1$.
If, on the contrary, $x<\epsilon$,
then the left hand side is bounded above by $(1+\epsilon)^d \mu(]0,+\infty[)$ which is itself bounded above
by the right hand side.
Note that $\kappa(\epsilon)(1+\epsilon)^d$ tends to $1$ as $\epsilon$ tends to $0$.
Let us say that a measure $\nu$ is subcritical if $\widehat{\lambda_c}(\nu)>1$.
As $\mu$ is subcritical, we get that
$\kappa(\epsilon)(1+\epsilon)^d \mu$ is subcritical for small enough $\epsilon$.
We fix such an $\epsilon$.
By \eqref{e:domination} we can couple a Boolean model driven by $H^{1+\epsilon}T_{\epsilon^2}\mu$
and a Boolean model driven by $\kappa(\epsilon)(1+\epsilon)^d \mu$ in such a way that the first one is contained in the second one.
Therefore the first one is subcritical.
By scaling, a Boolean model driven by $T_{\epsilon^2}\mu$ is then subcritical.
We take $\eta=\epsilon^2$. \hspace{\stretch{1}}{$\square$}
\begin{lemma} \label{l:dichotomie} Let $\nu_1$ and $\nu_2$ be two finite measures on $]0,+\infty[$.
One has, for all $\eta>0$ and $a \ge 4\eta$:
$$
p(a,\nu_1+\nu_2) \le p(a,T_{\eta}\nu_1)+C_1 a^d\eta^{-d}p(\eta/2,\nu_2)
$$
where $C_1=C_1(d)>0$ depends only on the dimension $d$.
\end{lemma}
\prg{Proof}
Let $(x_i)_{i\in I}$ be a family of points such that :
\begin{itemize}
\item The balls $B(x_i,\eta/4)$, $i\le I$, cover $B(a)$.
\item There are at most $C_1 a^d\eta^{-d}$ points in the family where $C_1=C_1(d)$ depends only on the dimension $d$.
\end{itemize}
We couple the different Boolean model as follows.
Let $\Sigma(\nu_1)$ be a Boolean model driven by $\nu_1$.
Let $\Sigma(\nu_2)$ be a Boolean model driven by $\nu_2$.
Assume that $\Sigma(\nu_1)$ and $\Sigma(\nu_2)$ are independent.
Then $\Sigma(\nu_1) \cup \Sigma(\nu_2)$ is a Boolean model driven by $\nu_1+\nu_2$.
We set $\Sigma(\nu_1 + \nu_2)=\Sigma(\nu_1) \cup \Sigma(\nu_2)$.
We also consider $\Sigma(T_{\eta}\nu_1)$, the Boolean model obtained by adding $\eta$ to the radius of each ball of $\Sigma(\nu_1)$.
Thus $\Sigma(T_{\eta}\nu_1)$ is driven by $T_{\eta}\nu_1$.
Let us prove the following property:
\begin{equation} \label{e:key-eta}
\{\perco{S(a/2)}{S(a)}{\Sigma(\nu_1+\nu_2)}\}
\subset \{\perco{S(a/2)}{S(a)}{\Sigma(T_{\eta}\nu_1)}\} \cup \bigcup_{i\in I} \{\perco{S(x_i,\eta/4)}{S(x_i,\eta/2)}{\Sigma(\nu_2)}\}.
\end{equation}
Assume that $\Sigma(\nu_1+\nu_2)=\Sigma(\nu_1) \cup \Sigma(\nu_2)$ connects $S(a/2)$ with $S(a)$.
Recall $a \ge 4\eta$.
If the diameter of all connected components of $\Sigma(\nu_2) \cap B(a)$ are less or equal to $\eta$,
then $\Sigma(T_{\eta}\nu_1)$ connects $S(a/2)$ with $S(a)$.
Otherwise, let $C$ be a connected component of $\Sigma(\nu_2) \cap B(a)$ with diameter at least $\eta$.
Let $x,y$ be two points of $C$ such that $\|x-y\|>\eta$.
The point $x$ belongs to a ball $B(x_i,\eta/4)$.
As $y$ does not belong to $B(x_i,\eta/2)$,
the component $C$ connects $S(x_i,\eta/4)$ to $S(x_i,\eta/2)$.
Therefore, $\Sigma(\nu_2)$ connects $S(x_i,\eta/4)$ to $S(x_i,\eta/2)$.
We have proven \eqref{e:key-eta}.
The lemma follows. \hspace{\stretch{1}}{$\square$}
\medskip
The following lemma is essentially the first item of Proposition 3.1 in \cite{G-perco-boolean-model}.
For the sake of completeness we nevertheless provide a proof.
\begin{lemma} \label{l:carre}
Let $\nu$ be a finite measure on $]0,+\infty[$.
There exists a constant $C_2=C_2(d)>0$ such that, for all $a>0$:
$$
p(10a,\nu) \le C_2 p(a, \nu)^2+ C_2 \int_{[a,+\infty[} r^d \nu(dr).
$$
\end{lemma}
\prg{Proof}
Let $K$ be a finite subset of $S(5)$ such that $K+B(1/2)$ covers $S(5)$.
Let $L$ be a finite subset of $S(10)$ such that $L+B(1/2)$ covers $S(10)$.
Let $A$ be the following event: there exists a random ball $B(c,r)$ of $\Sigma(\nu)$ such that $r\ge a$ and $B(c,r)\cap B(10a)$ is non empty.
We have:
$$
\{\perco{S(5a)}{S(10a)}{\Sigma(\nu)}\} \setminus A \subset \{\percorayon{S(5a)}{S(10a)}{\Sigma(\nu)}{a}\}
$$
where, in the last event, we ask for a path using only balls of $\Sigma(\nu)$ of radius at most $a$.
Let us prove the following:
\begin{multline}\label{e:key-carre}
\{\perco{S(5a)}{S(10a)}{\Sigma(\nu)}\} \setminus A \\
\subset
\bigcup_{k \in K, l \in L}
\{\percorayon{S(ak,a/2)}{S(ak,a)}{\Sigma(\nu)}{a}\} \cap \{\percorayon{S(al,a/2)}{S(al,a)}{\Sigma(\nu)}{a}\}.
\end{multline}
Assume that the event on the left hand side occurs.
Then, by the previous remark, there exists a path from a point $x\in S(5a)$ to a point $y\in S(10a)$
that is contained in balls of $\Sigma(\nu)$ of radius at most $a$.
As $Ka+B(a/2)$ covers $S(5a)$, there exists $k\in K$ such that $x$ belongs to $B(ka,a/2)$.
Using the previous path, one gets that the event
$$
\{\percorayon{S(ak,a/2)}{S(ak,a)}{\Sigma(\nu)}{a}\}
$$
occurs.
By a similar arguments involving $y$ we get \eqref{e:key-carre}.
Observe that, for all $k \in K$ and $l\in L$, the events
$$
\{\percorayon{S(ak,a/2)}{S(ak,a)}{\Sigma(\nu)}{a}\} \hbox{ and }
\{\percorayon{S(al,a/2)}{S(al,a)}{\Sigma(\nu)}{a}\}
$$
are independent.
Indeed, the first one depends only on balls with centers in $B(ak,2a)$, the second one depends only on balls with centers in $B(al,2a)$, and
$\|ak-al\| \ge 5a$.
Using this independence, stationarity and \eqref{e:key-carre}, we then get:
$$
P(\{\perco{S(5a)}{S(10a)}{\Sigma(\nu)}\}) \le CP(\percorayon{S(a/2)}{S(a)}{\Sigma(\nu)}{a}\})^2+P(A)
$$
where $C$ is the product of the cardinality of $K$ by the cardinality of $L$.
The probability $P(A)$ is bounded above by standard computations. \hspace{\stretch{1}}{$\square$}
\medskip
From the previous lemma, we deduce the following result.
\begin{lemma} \label{l:poitiers}
Let $\epsilon>0$.
There exists $C_3=C_3(d)>0$, $a_0=a_0(d,\mu)$ and $k_0=k_0(d,\mu,\epsilon)$ such that, for all $N$, all $\rho \ge 2$ and all $a \ge a_0$:
if $p(a,m^{\rho}_N) \le C_3$ then for all $k \ge k_0$, $p(a10^k,m^{\rho}_N) \le \epsilon$.
\end{lemma}
\prg{Proof}
For all $\rho \ge 2$ and all $a \ge 1$ we have:
\begin{eqnarray*}
\int_{[a,+\infty[} r^d m^{\rho}_{\infty}(dr)
& = & \sum_{k\ge 0} \rho^{kd} \int_{]0,+\infty[} 1_{[a,+\infty[}(r \rho^{-k}) (r \rho^{-k})^d \mu(dr) \\
& = & \int_{]0,+\infty[} \sum_{k\ge 0} 1_{[a,+\infty[}(r \rho^{-k}) r^d \mu(dr) \\
& = & \int_{[a,+\infty[} \big(\left\lfloor \ln(r/a)\ln(\rho)^{-1} \right\rfloor +1\big) r^d \mu(dr) \\
& \le & \int_{[a,+\infty[} (\ln(r)\ln(2)^{-1}+1) r^d \mu(dr).
\end{eqnarray*}
Let $C_2$ be the constant given by Lemma \ref{l:carre}.
By \eqref{e:th} we can chose $a_0=a_0(d,\mu) \ge 1$ such that
\begin{equation}\label{e:defa0}
C_2^2 \int_{[a_0,+\infty[} (\ln(r)\ln(2)^{-1}+1) r^d \mu(dr) \le \frac{1}{4}.
\end{equation}
Let $C_3=(2C_2)^{-1}$.
Let $N$, $\rho$ and $a$ be as in the statement of the lemma.
From Lemma \ref{l:carre} we get:
\begin{eqnarray}
C_2 p(10a,m^{\rho}_N)
& \le & (C_2 p(a,m^{\rho}_N))^2+C_2^2 \int_{[a,+\infty[} r^d m^{\rho}_N(dr) \\
& \le & (C_2 p(a,m^{\rho}_N))^2+C_2^2 \int_{[a,+\infty[} (\ln(r)\ln(2)^{-1}+1) r^d \mu(dr) \label{e:angouleme}
\end{eqnarray}
Let $(u_k)$ be a sequence defined by $u_0=1/2$ and, for all $k\ge 0$:
\begin{equation}\label{e:defu}
u_{k+1} = u_k^2+C_2^2 \int_{[a_0 10^k,+\infty[} (\ln(r)\ln(2)^{-1}+1) r^d \mu(dr).
\end{equation}
Note that the sequence $(u_k)$ only depends on $d$ and $\mu$.
Assume that $p(a,m^{\rho}_N)\le C_3$.
We then have $C_2p(a,m^{\rho}_N) \le u_0$.
Using $a\ge a_0$ and \eqref{e:angouleme}, we then get $C_2p(a10^k,m^{\rho}_N) \le u_k$ for all $k$.
Therefore, it sufficies to show that the sequence $(u_k)$ tends to $0$.
Using \eqref{e:defu}, \eqref{e:defa0} and $u_0=1/2$ we get $0 \le u_k \le 1/2$ for all $k$.
Therefore, $0 \le \limsup u_k \le 1/2$.
By \eqref{e:defu} and by the convergence of the integrale we also get $\limsup u_k \le (\limsup u_k)^2$.
As a consequence, $\limsup u_k=0$ and the lemma is proven. \hspace{\stretch{1}}{$\square$}
\begin{lemma} \label{l:compacite} For all $a>0$ and $\rho>1$ the following convergence holds:
$$
p(a,m^{\rho}_{\infty})=\lim_{N\to\infty} p(a,m^{\rho}_N) .
$$
\end{lemma}
\prg{Proof}
The sequence of events
$$
A_N=\{\perco{S(a/2)}{S(a)}{\Sigma(m^{\rho}_N)}\}
$$
is increasing (we use the natural coupling between our Boolean models).
Therefore, it suffices to show that the union of the previous events is
$$
A=\{\perco{S(a/2)}{S(a)}{\Sigma(m^{\rho}_{\infty})}\}.
$$
If $A$ occurs, then there is is path from $S(a/2)$ to $S(a)$ that is contained in $\Sigma(m^{\rho}_{\infty})$.
By a compactness argument, this path is included in a finite union of ball of $\Sigma(m^{\rho}_{\infty})$.
Therefore, there exists $N$ such that the path is included in $\Sigma(m^{\rho}_N)$ and $A_N$ occurs.
This proves $A \subset \cup A_N$.
The other inclusion is straightforward. \hspace{\stretch{1}}{$\square$}
\proofof{the second item of Theorem \ref{th:0}}
By Lemma \ref{l:eta}, we can fix $\eta_1>0$ such that $p(a,T_{10 \eta_1}\mu)$ tends to $0$ as $a$ tends to $\infty$.
Let $C_1$ be given by Lemma \ref{l:dichotomie}.
Let $a_0$ and $C_3$ be as given by Lemma \ref{l:poitiers}.
Fix $a_1 \ge \max(40\eta_1,a_0,1)$ such that $p(a,T_{10\eta_1}\mu) \le C_3/2$ for all $a \ge a_1$.
Let $k_0$ be given by Lemma \ref{l:poitiers} with the choice:
$$
\epsilon=C_1^{-1}(10a_1)^{-d}\eta_1^dC_3/2.
$$
Therefore, for all $\rho \ge 2$, all $N$, all $a\in [a_1, 10a_1]$ and all $\eta\in [\eta_1, 10\eta_1]$:
$$
C_1 a^d \eta^{-d} p(a 10^k,m_N^{\rho}) \le \frac{C_3}{2} \hbox{ for all } k \ge k_0 \hbox{ as soon as } p(a,m_N^{\rho}) \le C_3.
$$
Fix $k \ge k_0$, $a \in [a_1, 10a_1]$ and $\eta \in [\eta_1, 10\eta_1]$.
Set:
$$
\rho=2a 10^k \eta^{-1}.
$$
Note $\rho \ge 8 \ge 2$ as $a \ge a_1 \ge 40\eta_1 \ge 4\eta$.
By Lemma \ref{l:dichotomie} we have, for all $N$:
\begin{eqnarray*}
p(a,m_{N+1}^{\rho})
& \le & p(a,T_{\eta}\mu) + C_1 a^d \eta^{-d} p(\eta/2, m^{\rho}_{N+1}-\mu) \\
& = & p(a,T_{\eta}\mu) + C_1 a^d \eta^{-d} p(\eta/2, H^{\rho} m_N^{\rho}).
\end{eqnarray*}
By definition of $a_1$, by $a_1 \le a$, by $\eta \le 10\eta_1$, by scaling and by definition of $\rho$ we get, for all $N$:
\begin{eqnarray*}
p(a,m_{N+1}^{\rho})
& \le & \frac{C_3}{2}+C_1 a^d \eta^{-d} p(\rho\eta/2, m_N^{\rho}) \\
& = & \frac{C_3}{2}+C_1 a^d \eta^{-d} p(a 10^k, m_N^{\rho}).
\end{eqnarray*}
Combining this inequality with the property defining $k_0$, we get that
$p(a,m_N^{\rho}) \le C_3$ implies $p(a,m_{N+1}^{\rho}) \le C_3$.
As $p(a,m_0^{\rho}) = p(a,\mu) \le p(a,T_{10\eta_1}\mu) \le C_3/2$ we get $p(a,m_N^{\rho}) \le C_3$ for all integer $N$.
Let $\epsilon>0$.
Using again Lemma \ref{l:poitiers} we get the existence of an integer $k'_0$ such that
$p(a 10^{k'}, m_N^{\rho}) \le \epsilon$ for all $k' \ge k'_0$ as soon as $p(a,m_N^{\rho}) \le C_3$.
But we have proven the latter property.
Therefore $p(a 10^{k'}, m_N^{\rho}) \le \epsilon$ for all $N$ and all $k' \ge k'_0$.
By Lemma \ref{l:compacite}, we get $p(a 10^{k'}, m_{\infty}^{\rho}) \le \epsilon$ for all $k' \ge k'_0$.
Using the freedom on the choice of $k \ge k_0$ and $\eta \in [\eta_1, 10\eta_1]$, we get that the previous result holds for all
$\rho \ge 2a 10^{k_0-1} \eta_1^{-1}$ and then for all $\rho \ge 2a_1 10^{k_0} \eta_1^{-1}$.
Moreover, using the freedom on the choice of $a \in [a_1, 10a_1]$ and $k' \ge k'_0$, we get:
$$
p(r,m_{\infty}^{\rho}) \le \epsilon \hbox{ for all } r \ge a_1 10^{k'_0} \hbox{ and all } \rho\ge 2a_1 10^{k_0} \eta_1^{-1}.
$$
Therefore, $p(r,m_{\infty}^{\rho})$ tends to $0$ as $r$ tends to infinity.
As a consequence, $\Sigma^{\rho}(\mu)$ does not percolate for any $\rho \ge 2a_1 10^{k_0} \eta_1^{-1}$.
\hspace{\stretch{1}}{$\square$}
\section{Proof of Theorem \ref{th:1}}
\label{s:preuve-th:1}
\begin{lemma} \label{l:s}
Let $s>0$ and $\rho>1$.
The following assumptions are equivalent:
\begin{enumerate}
\item $\int_{]0,+\infty[} r^{d+s} \mu(dr) < \infty$.
\item $\int_{[1,+\infty[} r^{d+s} m_{\infty}^{\rho}(dr) < \infty$.
\end{enumerate}
\end{lemma}
\prg{Proof}
We have:
\begin{eqnarray*}
\int_{[1,+\infty[} r^{d+s} m^{\rho}_{\infty}(dr)
& = & \sum_{k\ge 0} \rho^{kd} \int_{]0,+\infty[} 1_{[1,+\infty[}(r \rho^{-k}) (r \rho^{-k})^{d+s} \mu(dr) \\
& = & \int_{[1,+\infty[} \sum_{k\ge 0} 1_{[1,+\infty[}(r \rho^{-k}) \rho^{-ks} r^{d+s} \mu(dr).
\end{eqnarray*}
Therefore:
$$
\int_{[1,+\infty[} r^{d+s} \mu(dr) \le \int_{[1,+\infty[} r^{d+s} m^{\rho}_{\infty}(dr) \le \frac{1}{1-\rho^{-s}} \int_{[1,+\infty[} r^{d+s} \mu(dr).
$$
This yields the result.
\hspace{\stretch{1}}{$\square$}
\proofof{the first item of Theorem \ref{th:1}}
By the discussion at the beginning of Section 1.5 in \cite{G-perco-generale},
$\Sigma^{\rho}(\lambda\mu)$ is driven by a a Poisson point process whose intensity is the product
of the Lebesgue measure by the locally finite measure $\lambda m_{\infty}^{\rho}$.
Let us check the three items of Theorem 2.9 in \cite{G-perco-generale} with $\rho=10$
($\rho$ is not use in the same way in \cite{G-perco-generale}).
We refer to Section 2.1 of \cite{G-perco-generale} for definitions.
\begin{enumerate}
\item
The first item is fulfilled thanks to \eqref{e:pitilde0}
\item
For all $\beta>0$ and all $x \in \mathbb{R}^d$,
the event $G(x,0,\beta)$ only depends on balls $B(c,r) \in \Sigma^{\rho}(\lambda\mu)$ such that $c$ belongs to $B(x,3\beta)$.
By the independance property of Poisson point processes, we then get that $G(0,0,\beta)$ and $G(x,0,\beta)$ are independent whenever $\|x\| \ge 10\beta$.
Therefore $I(10,0,\beta)=0$ and the second item of Theorem 2.9 is fulfilled.
\item
The third item (note that $\mu$ in \cite{G-perco-generale} is $m_{\infty}^{\rho}$ in this paper) is fulfilled thanks to Lemma \ref{l:s}
\end{enumerate}
Theorem 2.9 in \cite{G-perco-generale} yields the result. \hspace{\stretch{1}}{$\square$}
\proofof{the second item of Theorem \ref{th:1}}
If $\int r^d\mu(dr)$ is infinite then, $\Sigma(\lambda\mu)$ percolates for all $\lambda>0$ (see the dicussion of Section \ref{s:boolean}).
Therefore $\Sigma^{\rho}(\lambda\mu)$ percolates for all $\rho>1$ and $\lambda>0$.
Therefore $D^{\rho}(\lambda\mu)=\infty$ with positive probability for all $\rho>1$ and $\lambda>0$.
Now, assume that $\int r^d \mu(dr)$ is finite.
Then, by the discussion at the beginning of Section 1.5 in \cite{G-perco-generale},
$\Sigma^{\rho}(\lambda\mu)$ is driven by a a Poisson point process whose intensity is the product
of the Lebesgue measure by the locally finite measure $\lambda m_{\infty}^{\rho}$.
We can therefore apply Theorem 1.2 in \cite{G-perco-generale}.
By Lemma \ref{l:s}, assumption (A3) of Theorem 1.2 in \cite{G-perco-generale} is not fulfilled
(note that $\mu$ in \cite{G-perco-generale} is $m_{\infty}^{\rho}$ in this paper).
Theorem 1.2 in \cite{G-perco-generale} then yields the result. \hspace{\stretch{1}}{$\square$}
\section{Proof of Proposition \ref{p}}
\label{s:preuve-p}
We first need a lemma, which is a consequence of Lemmas \ref{l:dichotomie} and \ref{l:carre}.
\begin{lemma} \label{l:resume}
Let $\nu_1$ and $\nu_2$ be two finite measures on $]0,+\infty[$.
Let $\eta>0$ and $a_0 \ge 4\eta$.
Let $\rho>1$.
There exists $C_4=C_4(d)>0$ such that $\widehat{\lambda}_c(\nu_1+H_{\rho}(\nu_2)) \ge 1$ as soon as the following conditions hold:
\begin{enumerate}
\item $p(a,T_{\eta}\nu_1) \le C_4$ for all $a \in [a_0, 10a_0]$.
\item $a_0^d \eta^{-d} p(\rho\eta/2, \nu_2) \le C_4$.
\item $\int_{[a_0,+\infty[} r^d \nu_1(dr) \le C_4$ and $\int_{[a_0,+\infty[} r^d \nu_2(dr) \le C_4$.
\end{enumerate}
\end{lemma}
\prg{Proof} Let $C_4=C_4(d)>0$ be such that $C_4C_2(1+10^dC_1) \le 1/2$ and $2C_2^2C_4 \le 1/4$,
where $C_1$ appears in Lemma \ref{l:dichotomie} and $C_2$ appears in Lemma \ref{l:carre}.
Set $\nu=\nu_1+H_{\rho}(\nu_2)$.
For all $a \in [a_0,10a_0]$ we have,
by Lemma \ref{l:dichotomie} applied to $\nu_1$ and $H_{\rho}(\nu_2)$, by scaling and by the assumptions of the lemma:
\begin{eqnarray}
p(a,\nu)
& \le & p(a,T_{\eta}\nu_1) + C_1 a^d\eta^{-d}p(\eta/2,H_{\rho}\nu_2) \nonumber \\
& = & p(a,T_{\eta}\nu_1) + C_1 a^d\eta^{-d}p(\rho\eta/2,\nu_2) \nonumber \\
& \le & C_4(1+10^dC_1) \nonumber \\
& \le & 1/(2C_2). \label{e:fevrier0}
\end{eqnarray}
But for all $a \ge a_0$ we have, by Lemma \ref{l:carre} and by the assumptions of the lemma:
\begin{eqnarray}
C_2p(10a,\nu)
& \le & (C_2p(a,\nu))^2 + C_2^2\int_{[a,+\infty[}r^d \nu(dr) \nonumber \\
& = & (C_2p(a,\nu))^2 + C_2^2\int_{[a,+\infty[}r^d \nu_1(dr) + C_2^2\int_{[a\rho,+\infty[}r^d \nu_2(dr) \label{e:fevrier1} \\
& \le & (C_2p(a,\nu))^2 + 2C_2^2C_4 \nonumber \\
& \le & (C_2p(a,\nu))^2 + 1/4.\label{e:fevrier2}
\end{eqnarray}
By \eqref{e:fevrier0} and \eqref{e:fevrier2} we get $C_2p(a,\nu) \le 1/2$ for all $a \ge a_0$ and therefore $0 \le \limsup C_2p(a,\nu) \le 1/2$.
By \eqref{e:fevrier1} and the third assumption of the lemma, we get $\limsup C_2p(a,\nu) \le (\limsup C_2p(a,\nu))^2$.
Therefore, we must have $\limsup C_2p(a,\nu)=0$
and the lemma is proven. \hspace{\stretch{1}}{$\square$}
\proofof{Proposition \ref{p}}
The inequality is straightforward.
To prove the inequality, we note that, by scaling, $\widehat{\lambda}_c(H_{\rho}\nu_2)=\widehat{\lambda}_c(\nu_2)$.
Let us prove the convergence.
We can assume $\widehat{\lambda}_c(\nu_1)>0$ and $\widehat{\lambda}_c(\nu_2)>0$,
otherwise the convergence is obvious.
Therefore, by Lemma \ref{l:hat}, the integrals $\int r^d \nu_1(dr)$ and $\int r^d \nu_2(dr)$ are finite.
Let $C_4$ be the constant given by Lemma \ref{l:resume}.
Let $0<\epsilon<1$.
Note:
$$
\widehat{\lambda_c}\big((1-\epsilon)\widehat{\lambda}_c(\nu_1)\nu_1\big) = (1-\epsilon)^{-1}\widehat{\lambda}_c(\nu_1)^{-1}\widehat{\lambda}_c(\nu_1)>1,
$$
Therefore, by Lemma \ref{l:eta} (in which \eqref{e:th} is not used), we can fix $\eta>0$ such that
$$
p(a,T_{\eta}(1-\epsilon)\widehat{\lambda}_c(\nu_1)\nu_1) \to 0.
$$
We can then fix $a_0 \ge 4\eta$ such that:
\begin{equation}\label{e:annexe1}
p(a,T_{\eta}(1-\epsilon)\widehat{\lambda}_c(\nu_1)\nu_1) \le C_4 \hbox{ for all } a \ge a_0
\end{equation}
and such that
\begin{equation}\label{e:annexe3.1}
\int_{[a_0,+\infty[} r^d \widehat{\lambda}_c(\nu_1)\nu_1(dr) \le C_4
\end{equation}
and
\begin{equation}\label{e:annexe3.2}
\int_{[a_0,+\infty[} r^d \widehat{\lambda}_c(\nu_2)\nu_2(dr) \le C_4.
\end{equation}
Now we fix $\rho_0>1$ such that :
\begin{equation}\label{e:annexe2}
a_0^d\eta^{-d} p(\rho\eta/2, (1-\epsilon)\widehat{\lambda}_c(\nu_2)\nu_2) \le C_4 \hbox{ for all } \rho\ge \rho_0.
\end{equation}
Now, let $0<\alpha<1$ and let
$$
\lambda = \min\left(\frac{\widehat{\lambda}_c(\nu_1)(1-\epsilon)}{\alpha}, \frac{\widehat{\lambda}_c(\nu_2)(1-\epsilon)}{1-\alpha}\right).
$$
By \eqref{e:annexe1}, \eqref{e:annexe2}, \eqref{e:annexe3.1} and \eqref{e:annexe3.2}
we get that Assumptions 1
, 2 and 3 of Lemma \ref{l:resume} are fulfilled
for the measures $\alpha\lambda\nu_1$ and $(1-\alpha) \lambda \nu_2$ and for $\rho \ge \rho_0$.
Therefore, we get
$$
\widehat{\lambda}_c(\alpha \lambda \nu_1 + (1-\alpha) \lambda H_{\rho} \nu_2) \ge 1
$$
and thus:
$$
\widehat{\lambda}_c(\alpha \nu_1 + (1-\alpha) H_{\rho} \nu_2) \ge \lambda = (1-\epsilon)
\min\left(\frac{\widehat{\lambda}_c(\nu_1)}{\alpha}, \frac{\widehat{\lambda}_c(\nu_2)}{1-\alpha}\right).
$$
Therefore, as soon as $\rho \ge \rho_0$, we have:
\begin{eqnarray*}
0
& \le &
\min\left(\frac{\widehat{\lambda}_c(\nu_1)}{\alpha}, \frac{\widehat{\lambda}_c(\nu_2)}{1-\alpha}\right)
- \widehat{\lambda}_c(\alpha \nu_1 + (1-\alpha) H_{\rho} \nu_2) \\
& \le &
\epsilon \min\left(\frac{\widehat{\lambda}_c(\nu_1)}{\alpha}, \frac{\widehat{\lambda}_c(\nu_2)}{1-\alpha}\right) \\
& \le &
\epsilon \max(2\widehat{\lambda}_c(\nu_1),2\widehat{\lambda}_c(\nu_2)).
\end{eqnarray*}
This yields the proposition. \hspace{\stretch{1}}{$\square$}
|
{
"timestamp": "2011-03-14T01:01:14",
"yymm": "1009",
"arxiv_id": "1009.3719",
"language": "en",
"url": "https://arxiv.org/abs/1009.3719"
}
|
\section{Introduction}
The past decades has been tremendous advances in cosmology.
The discovery of dark energy (Perlmutter et al \cite{ref1}$-$\cite{ref3}; Riess et al \cite{ref4}$-$\cite{ref5}) has crushed widely-held expectations
that some unknown mechanism might set the cosmological constant to zero.
At the same time, substantial theoretical progress in string theory has brought
forth a diverse new generation of cosmological models, some of which are subject to
direct observational tests. One key advance in the emergence of methods of moduli stabilization.
Compactification of string theory from the total dimension D down to four dimensions introduces many
gravitationally-coupled scalar fields moduli from the point of view of the four dimensional theory.
Recently we have studied inhomogeneous string cosmological model
formed by geometric string and use this model as a source of gravitational field \cite{ref6,ref7}. We had two main reason
to study the above mentioned model. First, as a test of consistency, for some particular field theories based on string
models and second we point out the Universe can be represented by a collection of extended galaxies. It is generally
assumed that after the big bang, the Universe may have undergone a series of phase transitions as its temperature
cooled below some critical
temperature as predicted by grand unified theories \cite{ref8}$-$\cite{ref12}. At the very early stage of evolution
of universe, it is believed that during the phase transition, the
symmetry of Universe was broken spontaneously. That could have given rise to topologically-stable defects such as
domain walls, strings and monopoles \cite{ref12}. Among all the three cosmological structures, only cosmic strings have excited
the most interesting consequence \cite{ref13}, because it gives rise the density perturbations which leads
to the formation of galaxies. The cosmic string can be closed (like loops) and open (like a hair) which move through
time and trace out a tube or a sheet, according to whether it is closed or open. These cosmic strings have stress
energy and couple to the gravitational field. Therefore, it is interesting to study the gravitational effect
which arises from strings by using Einstein's equations. \\
The general treatment of strings was initiated by Letelier \cite{ref14,ref15} and
Stachel \cite{ref16}. Letelier \cite{ref14} obtained the general solution of Einstein's
field equations for a cloud of strings with spherical, plane and a particular
case of cylindrical symmetry. Letelier \cite{ref15} also obtained massive string
cosmological models in Bianchi type-I and Kantowski-Sachs space-times. Benerjee
et al \cite{ref17} have investigated an axially symmetric Bianchi type I string dust
cosmological model in presence and absence of magnetic field using a supplementary
condition $\alpha = a \beta$ between metric potential where $\alpha = \alpha(t)$ and
$\beta = \beta(t)$ and $a$ is constant. Exact solutions of string cosmology for
Bianchi type-II, $-VI_{0}$, -VIII and -IX space-times have been studied by Krori et al
\cite{ref18} and Wang \cite{ref19}. Wang \cite{ref20}$-$\cite{ref23} has investigated bulk viscous string
cosmological models in different space-times. Bali and Anjali \cite{ref24}, Yadav \cite{ref25},
Pradhan et al \cite{ref26,ref27} and
Yadav et al \cite{ref28} have studied string cosmological models in different physical contexts.
The string cosmological models with a magnetic field are discussed by
Chakraborty \cite{ref29}, Tikekar and Patel \cite{ref30,ref31}, Patel and Maharaj \cite{ref32}. Singh and Singh \cite{ref33} investigated string
cosmological models with magnetic field in the context of space-time with $G_{3}$
symmetry. Singh \cite{ref34,ref35} has studied string cosmology with electromagnetic fields in
Bianchi type-II, -VIII and -IX space-times. Lidsey, Wands and Copeland \cite{ref36} have
reviewed aspects of super string cosmology with the emphasis on the cosmological
implications of duality symmetries in the theory. Recently Saha and Visinescu \cite{ref37} and Saha et al \cite{ref38}
have investigated Bianchi I string cosmological model in presence of magnetic flux. They have found that the present
of cosmic string does not allow the anisotropic Universe to evolve into an isotropic one \cite{ref37}.\\
In this paper we have studied locally rotationally symmetric (LRS) Bianchi type I string cosmological model
with time varying deceleration parameter (DP). The paper has following structure. In section 2, the metric and field equations are described.
In section 3, we introduce a few plausible solutions consistent with observations. At the end we shall summarize the
findings.\\
\section{The Metric and Field Equations}
We consider the LRS Bianchi type I metric of the form
\begin{equation}
\label{eq1}
ds^2 = -dt^2 + A^2dx^2 + B^2 \left(dy^2 + dz^2\right)
\end{equation}
where, A and B are functions of t only. This ensures that the model is spatially homogeneous.\\
The energy-momentum tensor $T^{i}_{j}$ for a cloud
of massive strings and perfect fluid distribution is taken as
\begin{equation}
\label{eq2} T^{i}_{j} = (\rho + p)v^{i}v_{j} + p g^{i}_{j} -\lambda
x^{i}x_{j},
\end{equation}
where $p$ is the isotropic pressure; $\rho$ is the proper energy density for a cloud strings with particles
attached to them; $\lambda$ is the string tension density; $v^{i}=(0,0,0,1)$ is the four-velocity of the
particles, and $x^{i}$ is a unit space-like vector representing the direction of string. The vectors $v^{i}$
and $x^{i}$ satisfy the conditions
\begin{equation}
\label{eq3} v_{i}v^{i}=-x_{i}x^{i}=-1,\;\; v^{i}x_{i}=0.
\end{equation}
Choosing $x^{i}$ parallel to $\partial/\partial x$, we have
\begin{equation}
\label{eq4} x^{i} = (A^{-1},0,0,0).
\end{equation}
If the particle density of the configuration is denoted by
$\rho_{p}$, then
\begin{equation}
\label{eq5} \rho = \rho_{p}+\lambda.
\end{equation}
The Einstein's field equations (in gravitational units $c = 1$, $8\pi G = 1$) read as
\begin{equation}
\label{eq6}
R_{j}^i - \frac{1}{2}g_{j}^{i}R = -T_{j}^i
\end{equation}
The Einstein's field equations (\ref{eq6}) for the line-element (\ref{eq1})
lead to the following system of equations
\begin{equation}
\label{eq7}
2\frac{B_{44}}{B} + \frac{B_{4}^2}{B^2} = -p +\lambda
\end{equation}
\begin{equation}
\label{eq8}
\frac{A_{44}}{A} + \frac{B_{44}}{B} + \frac{A_{4}B_{4}}{AB} =-p
\end{equation}
\begin{equation}
\label{eq9}
\frac{B_{4}^2}{B^2} + 2\frac{A_{4}B_{4}}{AB}
= \rho
\end{equation}
Here, and in what follows, sub in-dices 4 in $A$, $B$ and elsewhere indicates
differentiation with respect to $t$. The energy conservation
equation $\;T^{ij}_{\;\;\;;j}=0$, leads to the following expression:
\begin{equation}
\label{eq10} \rho_{4} + (\rho + p)\left(\frac{A_{4}}{A} +
2\frac{B_{4}}{B}\right) -
\lambda\frac{A_{4}}{A} = 0\;,
\end{equation}
which is consequence of the field equations (\ref{eq7})-(\ref{eq9}).\\
The average scale factor (R) of LRS Bianchi type I model is defined as
\begin{equation}
\label{eq11}
R=(AB^{2})^{\frac{1}{3}}
\end{equation}
The spatial volume (V) is given by
\begin{equation}
\label{eq12}
V = R^{3} = AB^{2}
\end{equation}
We define the mean Hubble parameter (H) for LRS Bianchi I space-time as
\begin{equation}
\label{eq13}
H=\frac{R_{4}}{R}=\frac{1}{3}\left(\frac{A_{4}}{A}+2\frac{B_{4}}{B}\right)
\end{equation}
The expansion scalar ($\theta$), shear scalar ($\sigma$) and mean anisotropy parameter ($A_{m}$) are defined as
\begin{equation}
\label{eq14}
\theta =3H = \frac{A_{4}}{A}+2\frac{B_{4}}{B}\;,
\end{equation}
\begin{equation}
\label{eq15}
\sigma^{2}=\frac{1}{2}\left(\sum_{i=1}^{3} H_{i}^{2}-\frac{1}{3}\theta^{2}\right)\;,
\end{equation}
\begin{equation}
\label{eq16}
A_{m} = \frac{1}{3}\sum_{i=1}^{3}\left(\frac{H_{i}-H}{H}\right)^{2}
\end{equation}
\section{Solutions of the Field Equations}
Observations of type Ia supernovae \cite{ref5} allow to probe the expansion history
of Universe. In literature it is common to use a constant deceleration parameter, as it duly
gives a power law for metric function or corresponding quantity. But at present the expansion of Universe
is accelerating and decelerating in the past. Also the transition redshift from decelerating phase to accelerating
phase is about $0.5$. Now for the Universe which was decelerating in the past and accelerating
at present time, the DP must show signature flipping \cite{ref39}. So, in general, DP is not constant but variable.
Following, Virey et al
\cite{ref40}, we consider the DP to be variable i. e.
\begin{equation}
\label{eq17}
q=-\frac{RR_{44}}{R_{4}^{2}} = b\;(variable)
\end{equation}
where R is average scale factor. In this paper, we show, how the variable DP models with metric (\ref{eq1})
behave in presence of string fluid as a source of matter. \\
Pradhan et al \cite{ref6} and recently Yadav and Yadav \cite{ref41},
have obtained cosmological models with proportionality relation between
shear scalar ($\sigma$) and expansion scalar ($\theta$). This condition
leads to the following relation between the metric potentials:
\begin{equation}
\label{eq18} A = B^{n},
\end{equation}
where $n$ is a positive constant.\\
From equation (\ref{eq17}), we obtain
\begin{equation}
\label{eq19}
\frac{R_{44}}{R_{4}}+b\frac{R_{4}^{2}}{R^{2}} = 0
\end{equation}
In order to solve equation (\ref{eq19}), we have to assume $\;b = b\;(R)$. It is important to note here that one can assume
$b = b\;(t) = b\;(R(t))$, as R is also a time dependent function. But this is possible only when one avoid singularity like big bang
or big rip because both $t$ and $R$ are increasing function.\\
Thus the general solution of equation (\ref{eq19}) with assumption $\;b = b\;(R)$, is given by
\begin{equation}
\label{eq20}
\int{e^{\int\frac{b}{R}dR}} = t+m
\end{equation}
where m is the constant of integration.\\
One can not solve eq. (\ref{eq20}) in general as $b$ is variable. So, in order to solve the problem completely,
we have to choose $\int\frac{b}{R}\;dR$ in such a manner that eq. (\ref{eq20}) be integrable with out any loss of generality,
we consider
\begin{equation}
\label{eq21}
\int\frac{b}{R}dR = \emph{ln}\;L(R)
\end{equation}
Which does not effect the nature of generality of the solution.\\
Hence, from equations (\ref{eq20}) and (\ref{eq21}), we obtain
\begin{equation}
\label{eq22}
\int{L(R)dR}=t+m
\end{equation}
Of course, the choice of $L(R)$, in eq. (\ref{eq22}), is quite arbitrary but, since we are looking for a physically
viable models of Universe consistent with observations. We consider the following cases
\subsection{Solution in the polynomial form}
Let us consider $L(R) = \frac{1}{2k_{1}\sqrt{R+k_{2}}}\;$, where $k_{1}$ and $k_{2}$ are constants.\\
In this case, on integrating, eq.(\ref{eq22}) gives the exact solution
\begin{equation}
\label{eq23}
R=\alpha_{1}T^{2} + \alpha_{2}T + \alpha_{3}
\end{equation}
where\\
$T=t+m$, $\alpha_{1} = k_{1}^{2}$, $\alpha_{2} = 2c_{1}k_{1}^{2}$, $\alpha_{3} = c_{1}^{2}k_{1}^{2}-k_{2}$\\
Here, $c_{1}$ is constant of integration.
Solving equations (\ref{eq11}), (\ref{eq18}) and (\ref{eq23}), we obtain the metric function as
\begin{equation}
\label{eq24}
A=\left(\alpha_{1}T^{2} + \alpha_{2}T + \alpha_{3}\right)^{\frac{3n}{n+2}}\;,
\end{equation}
\begin{equation}
\label{eq25}
B=\left(\alpha_{1}T^{2} + \alpha_{2}T + \alpha_{3}\right)^{\frac{3}{n+2}}\;,
\end{equation}
Hence the metric (\ref{eq1}) is reduced to
\begin{equation}
\label{eq26}
ds^2 = -dT^2 + \left(\alpha_{1}T^{2} + \alpha_{2}T + \alpha_{3}\right)^{\frac{6m}{m+2}}dx^2 +
\left(\alpha_{1}T^{2} + \alpha_{2}T + \alpha_{3}\right)^{\frac{6}{m+2}} \left(dy^2 + dz^2\right)
\end{equation}
The expressions for the isotropic pressure ($p$), the proper energy
density ($\rho$), the string tension ($\lambda$) and the particle
density ($\rho_{p}$) for the model (\ref{eq26}) are obtained as
\begin{equation}
\label{eq27}
p=\frac{3(3n-2-5n^{2})(2\alpha_{1}T+\alpha_{2})^{2}}{(n+2)^{2}(\alpha_{1}T^{2} + \alpha_{2}T + \alpha_{3})^{2}}-
\frac{6\alpha_{1}(n+1)}{(n+2)(\alpha_{1}T^{2} + \alpha_{2}T + \alpha_{3})}\;,
\end{equation}
\begin{equation}
\label{eq28}
\rho = \frac{9(2n+1)(2\alpha_{1}T+\alpha_{2})^{2}}{(n+2)^{2}(\alpha_{1}T^{2} + \alpha_{2}T + \alpha_{3})^{2}}\;,
\end{equation}
\begin{equation}
\label{eq29}
\lambda = \frac{3(n+4-5n^{2})(2\alpha_{1}T+\alpha_{2})^{2}}{(n+2)^{2}(\alpha_{1}T^{2} + \alpha_{2}T + \alpha_{3})^{2}} -
\frac{6\alpha_{1}(n-1)}{(n+2)(\alpha_{1}T^{2} + \alpha_{2}T + \alpha_{3})}\;,
\end{equation}
\begin{equation}
\label{eq30}
\rho_{p} = \frac{3(5n^{2}+5n-1)(2\alpha_{1}T+\alpha_{2})^{2}}{(n+2)^{2}(\alpha_{1}T^{2} + \alpha_{2}T + \alpha_{3})^{2}} -
\frac{6\alpha_{1}(1-n)}{(n+2)(\alpha_{1}T^{2} + \alpha_{2}T + \alpha_{3})}\;,
\end{equation}
The energy conservation equation (\ref{eq10}) is satisfied identically by the above solutions, as expected.\\
We observe that all the parameters diverge at $T=\frac{-\alpha_{2}\pm\sqrt{\alpha_{2}^{2}-4\alpha_{1}\alpha_{3}}}{2\alpha_{1}}$.
Therefore, the model has singularity
at $T=\frac{-\alpha_{2}\pm\sqrt{\alpha_{2}^{2}-4\alpha_{1}\alpha_{3}}}{2\alpha_{1}}$, which can be shifted to $T=0$ by
choosing $\alpha_{2} = \alpha_{3} = 0$. This singularity is of Point Type
as all the scale factors vanish at $T=\frac{-\alpha_{2}\pm\sqrt{\alpha_{2}^{2}-4\alpha_{1}\alpha_{3}}}{2\alpha_{1}}$.
The parameters $p$, $\rho$, $\rho_{p}$ and $\lambda$ start off
with extremely large values. In particular, the large values of $\rho_{p}$ and $\lambda$ in the beginning suggest
that strings dominate the early Universe. For sufficiently large time, $\rho_{p}$ and $\lambda$ become negligible.
Therefore, the strings disappear from Universe for large time that is why, the strings are not observable in the present
Universe.
From equation (\ref{eq28}), it is observed that the proper energy density $\rho$ is
decreasing function of time. $\textbf{Fig. 1}$ depicts the variation of rest energy density versus time.
The proper energy density $\rho$ and particle energy density $\rho_{p}$ have been graphed in $\textbf{Fig. 2}$.
it is evident that $\rho_{p} > \rho$, i. e. the particle energy density remains larger than the proper energy density
during the cosmic expansion, as expected.\\
From equation (\ref{eq29}) and (\ref{eq30}), we have $ \frac{\rho_{p}}{\arrowvert\lambda\arrowvert} > 1 $ i. e. particle
energy density $(\rho_{p})$ remains larger than the string tension density $(\lambda)$ during the cosmic expansion, especially
in early Universe. This behaviour of $\rho_{p}$ and $\arrowvert\lambda\arrowvert$ is clearly shown in $\textbf{Fig. 3}$.
According to the ref.
(Letelier \cite{ref15}; Krori et al \cite{ref18}),
when $ \frac{\rho_{p}}{\arrowvert\lambda\arrowvert} > 1 $, in the process of evolution,
the Universe is dominated by massive strings, and when $ \frac{\rho_{p}}{\arrowvert\lambda\arrowvert} < 1 $, the
universe is dominated by strings. From $\textbf{Fig. 3}$, we see that $ \frac{\rho_{p}}{\arrowvert\lambda\arrowvert} > 1 $.
Thus in derived model, the early Universe is
dominated by massive string. According to Ref. \cite{ref42}, since there is no direct evidence of strings in the present day Universe, we are in
general, interested in constructing models of Universe that evolves purely from the era dominated by either geometric
strings or massive strings and end up in the particle dominated era with or without remnants of strings. Therefore,
the above model describes the evolution of the Universe consistent with the present-day observations.\\
\begin{figure}
\begin{center}
\includegraphics[width=4.0in]{anil26RF2.eps}
\caption{The plot of proper energy density $(\rho)$ vs. time (T).}
\label{fg:anil26RF2.eps}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=4.0in]{anil26RF3.eps}
\caption{Proper energy density $(\rho)$ and particle energy density $(\rho_{p})$ vs. time (T).}
\label{fg:anil26RF3.eps}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=4.0in]{anil26RF4.eps}
\caption{String tension density $(\lambda)$, particle energy density $(\rho_{p})$ and $ \frac{\rho_{p}}{\arrowvert\lambda\arrowvert}$
vs. time (T).}
\label{fg:anil26RF4.eps}
\end{center}
\end{figure}
The rate of expansion in the direction of $x$, $y$ and $z$ are given by
\begin{equation}
\label{eq31}
H_{x} = \frac{A_{4}}{A} = \frac{3n(2\alpha_{1}T+\alpha_{2})}{(n+2)(\alpha_{1}T^{2}+\alpha_{2}+\alpha_{3})}
\end{equation}
\begin{equation}
\label{eq32}
H_{y} = H_{z} = \frac{3(2\alpha_{1}T+\alpha_{2})}{(n+2)(\alpha_{1}T^{2}+\alpha_{2}+\alpha_{3})}
\end{equation}
The mean Hubble's parameter ($H$), expansion scalar $(\theta)$ and shear scalar $(\sigma^{2})$ of model (\ref{eq26})
are given by
\begin{equation}
\label{eq33}
H = \frac{(2\alpha_{1}T+\alpha_{2})}{(\alpha_{1}T^{2}+\alpha_{2}+\alpha_{3})}
\end{equation}
\begin{equation}
\label{eq34}
\theta = \frac{3(2\alpha_{1}T+\alpha_{2})}{(\alpha_{1}T^{2}+\alpha_{2}+\alpha_{3})}
\end{equation}
\begin{equation}
\label{eq35}
\sigma^{2} = \frac{3(n-1)^{2}(2\alpha_{1}T+\alpha_{2})^{2}}{(n+2)^{2}(\alpha_{1}T^{2}+\alpha_{2}+\alpha_{3})^{2}}
\end{equation}
The spatial volume $(V)$, mean anisotropy parameter $(A_{m})$ and DP $(q)$ are found to be
\begin{equation}
\label{eq36}
V = (\alpha_{1}T^{2}+\alpha_{2}+\alpha_{3})^{3}
\end{equation}
\begin{equation}
\label{eq37}
A_{m} = \frac{2(n-1)^{2}}{(n+2)^{2}}
\end{equation}
\begin{equation}
\label{38}
q = -\frac{2\alpha_{1}(\alpha_{1}T^{2}+\alpha_{2}T+\alpha_{3})}{(2\alpha_{2}T+\alpha_{2})^{2}}
\end{equation}
The variation of DP versus time has been graphed in $\textbf{Fig. 4}$. It is observed that DP evolves with in the
range predicted by present-day observations. From eq. (\ref{eq36}), it can be seen that the spatial volume is zero
at $T=\frac{-\alpha_{2}\pm\sqrt{\alpha_{2}^{2}-4\alpha_{1}\alpha_{3}}}{2\alpha_{1}}$, and it increases with the cosmic time.
The parameter $H_{x}$, $H_{y}$, $H_{z}$, $H$, $\theta$ and $\sigma^{2}$ diverge at initial singularity. These parameters
decrease with evolution of Universe and finally drops to zero at late time. For $n=1$, the mean anisotropy parameter
vanishes and the directional scale factors vary as\\
$$A = B = \left(\alpha_{1}T^{2} + \alpha_{2}T + \alpha_{3}\right)$$
Therefore, isotropy is achieved in the derived model for $n = 1$. For this particular values of $n$, we see that $A(T) = B(T) = R(T) $,
therefore, metric (\ref{eq1}) reduces to flat FRW space-time. Thus, the derived model acquires flatness for $n = 1$. But in the
same spirit, the string tension density $(\lambda)$ vanishes for $n=1$. Hence we can not choose $n=1$ in presence of string fluid
as a source of matter in LRS Bianchi I space-time.
\begin{figure}
\begin{center}
\includegraphics[width=4.0in]{anil26RF1.eps}
\caption{The plot of DP (q) vs. time (t).}
\label{fg:anil26RF1.eps}
\end{center}
\end{figure}
\subsection{Solution in sine hyperbolic form}
We consider, $L(R) = \frac{1}{k_{3}\sqrt{1+R^{2}}}\;$, where $k_{3}$ is an arbitrary constant.\\
In this case, on integrating, eq.(\ref{eq22}) gives the exact solution
\begin{equation}
\label{eq39}
R=sinh\;(k_{3}T)
\end{equation}
where $T=t+m$, and the constant of integration has been omitted by assuming that $R=0$ at $T=0$.\\
Solving equations (\ref{eq11}), (\ref{eq18}) and (\ref{eq39}), we obtain the metric function as
\begin{equation}
\label{eq40}
A=sinh^{\frac{3n}{n+2}}\;(k_{3}T)
\end{equation}
\begin{equation}
\label{41}
B=sinh^{\frac{3}{n+2}}\;(k_{3}T)
\end{equation}
Hence the metric (\ref{eq1}) is reduced to
\begin{equation}
\label{eq42}
ds^2 = -dT^2 + sinh^{\frac{6n}{n+2}}(\;k_{3}T)dx^2 + sinh^{\frac{6}{n+2}}(\;k_{3}T) \left(dy^2 + dz^2\right)
\end{equation}
The expressions for the isotropic pressure ($p$), the proper energy
density ($\rho$), the string tension ($\lambda$) and the particle
density ($\rho_{p}$) for the model (\ref{eq42}) are obtained as
\begin{equation}
\label{eq43}
p=\frac{3(n+1)k_{3}^{2}}{n+2}cosech^{2}(\;k_{3}T)-\frac{9(n^{2}+n+1)k_{3}^{2}}{(n+2)^{2}}coth^{2}(\;k_{3}T)\;,
\end{equation}
\begin{equation}
\label{eq44}
\rho=\frac{9(2n+1)k_{3}^{2}}{(n+2)^{2}}coth^{2}(\;k_{3}T)\;,
\end{equation}
\begin{equation}
\label{eq45}
\lambda = \frac{3(n-1)k_{3}^{2}}{n+2}cosech^{2}(\;k_{3}T)-\frac{9(n^{2}+n-2)k_{3}^{2}}{(n+2)^{2}}coth^{2}(\;k_{3}T)\;,
\end{equation}
\begin{equation}
\label{eq46}
\rho_{p}=\frac{9(n^{2}+3n-1)k_{3}^{2}}{(n+2)^{2}}coth^{2}\;(k_{3}T)-\frac{3(n-1)k_{3}^{2}}{n+2}cosech^{2}(\;k_{3}T)\;,
\end{equation}
The energy conservation equation (\ref{eq10}) is satisfied identically by the above solutions, as expected.\\
We observe that all the parameters diverge at $T=0$.
Therefore, the model has big bang singularity
at $T=0$. This singularity is of Point Type
as all the scale factors vanish at $T=0$.
The parameters $p$, $\rho$, $\rho_{p}$ and $\lambda$ start off
with extremely large values.
\begin{figure}
\begin{center}
\includegraphics[width=4.0in]{anil26RF6.eps}
\caption{Proper energy density $(\rho)$, string tension density $(\lambda)$, particle energy density $(\rho_{p})$
and $ \frac{\rho_{p}}{\arrowvert\lambda\arrowvert}$ vs. time (T).}
\label{fg:anil26RF6.eps}
\end{center}
\end{figure}
The proper energy density $\rho$, string tension density $\lambda$ and particle energy density $\rho_{p}$ have
been graphed versus time $T$ in $\textbf{Fig. 5}$. It is evident that string tension density becomes negligible
for sufficient large time. Therefore, the strings disappear from Universe at late time that is why, the strings are not
observable in present universe. From $\textbf{Fig. 5}$, we see that $ \frac{\rho_{p}}{\arrowvert\lambda\arrowvert} > 1 $.
Therefore, the early Universe was
dominated by massive string.\\
\begin{figure}
\begin{center}
\includegraphics[width=4.0in]{anil26RF5.eps}
\caption{The plot of DP (q) vs. time (T).}
\label{fg:anil26RF5.eps}
\end{center}
\end{figure}
The rate of expansion in the direction of $x$, $y$ and $z$ are given by
\begin{equation}
\label{eq47}
H_{x} = \frac{3nk_{3}}{n+2}coth\;(k_{3}T)\;,
\end{equation}
\begin{equation}
\label{eq48}
H_{y}=H_{z}= \frac{3k_{3}}{n+2}coth\;(k_{3}T)\;,
\end{equation}
The mean Hubble's parameter ($H$), expansion scalar $(\theta)$ and shear scalar $(\sigma^{2})$ of model (\ref{eq42})
are given by
\begin{equation}
\label{eq49}
H=k_{3}coth\;(k_{3}T)\;,
\end{equation}
\begin{equation}
\label{eq50}
\theta=3k_{3}coth\;(k_{3}T)\;,
\end{equation}
\begin{equation}
\label{eq51}
\sigma^{2}=\frac{3(n-1)^{2}k_{3}^{2}}{(n+2)^{2}}coth^{2}\;(k_{3}T)
\end{equation}
The spatial volume $(V)$, mean anisotropy parameter $(A_{m})$ and DP $(q)$ are found to be
\begin{equation}
\label{eq52}
V=sinh^{3}\;(k_{3}T)\;,
\end{equation}
\begin{equation}
\label{eq53}
A_{m}=\frac{2(n-1)^{2}}{(n+2)^{2}}\;,
\end{equation}
\begin{equation}
\label{eq54}
q=-tanh^{2}\;(k_{3}T)
\end{equation}
The variation of DP versus time has been graphed in $\textbf{Fig. 6}$. It is observed that DP evolves with in the range
predicted by SN Ia \cite{ref1}$-$\cite{ref5} and CMBR \cite{ref43} observations. We observe that at $T=0$, the spatial
volume vanishes and it increases with cosmic time. For $n=1$, the mean anisotropy parameter vanishes and the directional
scale factors vary as
$$A(T) = B(T) = sinh\;(k_{3}T)$$
Therefore, $n=1$, turns out to condition of isotropy but in the same spirit, the string tension density $(\lambda)$ vanishes.
Therefore, the presence of cosmic string does not allow to choose $n = 1$.
\section{Concluding Remarks}
In this paper, we have studied LRS Bianchi type I string cosmological models in general relativity.
The Einstein's field equations have been solved exactly with suitable physical assumptions and the solutions satisfy
the energy conservation equation identically. Therefore, exact and physically viable LRS Bianchi I string cosmological models
have been obtained. The derived models have singular origin i. e. the Universe starts expanding with a big bang singularity.\\
The main features of the models are as follows:\\
\begin{itemize}
\item The models are based on exact solution of Einstein's field equations for LRS Bianchi I space-time
in presence of string fluid as a source of matter.\\
\item It has been found that massive strings dominate the early Universe, which is eventually disappear from
the Universe for sufficiently large time. This is in agreement with current astronomical observations.\\
\item In the derived models, $n=1$, turns out to be a condition of isotropy and flatness of Universe.
It is important to mention here that for $n = 1$, the string tension density $(\lambda)$ vanishes in both cases.
So, we conclude that presence of cosmic string does not allow to choose $n = 1$. Thus, the
anisotropic LRS Bianchi I Universe may not evolve into isotropic one in presence of cosmic string.
The same is predicted by Saha and Visinescu \cite{ref37}
with different approach in Bianchi I space-time. \\
\item The DP $(q)$ is evolving with negative value and the existing range of $q$ is in nice agreement with SN Ia
data and CMBR observations. Thus the derived models are realistic.
\end{itemize}
\section*{Acknowledgements}
Author is thankful to The Institute of Mathematical Science (IMSc), Chennai, India
for providing facility and support where part of this work was carried out. Also Author thanks to B. Saha for helpful
discussions.
|
{
"timestamp": "2011-08-24T02:03:42",
"yymm": "1009",
"arxiv_id": "1009.3867",
"language": "en",
"url": "https://arxiv.org/abs/1009.3867"
}
|
\section{Introduction}
HR~7355 (HD~182180) is a bright B2Vn helium-strong star originally classified as a Be star due to H$\alpha$ emission
present in its spectrum (\cite[Abt \& Cardona 1984]{Abt_Cardona1984}). Previous studies of this star show a $v \sin i \sim$ 300 km s$^{-1}$
(\cite[Abt et al. 2002]{Abt_etal2002}) with a P$_{\rm{rot}} \sim$ 0.52 d (\cite[Koen \& Eyer 2002]{Koen_Eyer2002}),
as well as variation in helium, H$\alpha$, and brightness, suggesting the presence of a magnetosphere (\cite[Rivinius et al. 2008]{Rivinius_etal2008}).
HR~7355 is the most rapidly rotating helium-strong star, rotating near its critical velocity, providing an excellent testbed for magnetospheres under the effects of rapid rotation.
\vspace*{-0.5 cm}
\section{Method}
Least-Squares Deconvolution (LSD) describes the stellar spectrum as the
convolution of a mean Stokes I or V profile, representative of the average
shape of the line profile, and a line mask, describing the position, strength
and magnetic sensitivity of all lines in the spectrum. From the LSD mean Stokes I and V profiles, we calculate the
longitudinal magnetic field, B$_{\ell}$:
\begin{equation}
B_{\ell} = -2.14 \times 10^{11} \frac{ \int v V(v) dv}{\lambda g c \int [1-I(v)]dv}
\end{equation}
(\cite[Wade et al. 2000]{Wade_etal2000}), where $\lambda$ is the average wavelength and g is the average Land\'e factor in the mask. I$_{c}$ is
the continuum value of the intensity profile. The integral is evaluated over the full
velocity range of the mean profile.
\vspace*{-0.25 cm}
\section{Results}
We detect a strong magnetic field on HR~7355, the most rapidly rotating helium-strong star discovered thus far. A simultaneous independent confirmation of the field detection has been obtained with FORS at the VLT by Rivinius et al. 2010. The longitudinal magnetic field varies sinusoidally with the rotation period, with extrema -2 to 2.5 kG.
Assuming a dipole magnetic field, the polar value of the magnetic field is $\sim$ 13-17 kG. The photometric (brightness) light curve constructed from HIPPARCOS archival data and new CTIO measurements shows two minima separated by 0.5 in rotational phase and occurring 0.25 cycles before/after the magnetic extrema. Using the Scargle periodogram, eclipse-like photometric variations give a highly precise P$_{rot}$ = 0.5214404(6) days. We confirm spectral variability of helium and metal lines, as well as variability of H$\alpha$ emission. H$\alpha$ emission indicates circumstellar material extending out to 5 R$_{\star}$ from the star, rotating rigidly with the stellar surface.
We conclude that HR 7355 is a magnetic oblique rotator with a magnetosphere, mirroring the
physical picture for $\sigma$ Ori E (\cite[Townsend et al. 2005]{Townsedn_etal2008}).
\begin{figure}[ht]
\begin{center}
\includegraphics[width=3.4in]{s2-16_oksala_fig1.eps}
\caption{ Top: Longitudinal magnetic field measurements for HR 7355 and the best-fit first order
sine curve. \cite[Oksala et al. (2010)]{Oksala_etal2010} (asterisks) and \cite[Rivinius et al. (2010)]{Rivinius_etal2010} (diamonds) with 1 $\sigma$ error bars.
Bottom: The V-band photometric light curve for HR 7355 including both HIPPARCOS photometry (asterisks) and new CTIO data (plus signs).}
\label{fig1}
\end{center}
\end{figure}
\vspace*{-0.5 cm}
|
{
"timestamp": "2010-09-22T02:02:02",
"yymm": "1009",
"arxiv_id": "1009.4083",
"language": "en",
"url": "https://arxiv.org/abs/1009.4083"
}
|
\section{Introduction}
The physical motivation
behind studying Hermitian random matrix ensembles is as a
model of the Hamiltonian for complex systems, where the eigenvalues of the
random Hermitian matrix represent the energy levels of a system without
time-reversal invariance \cite{Wigner:1951}.
Let ${\bf A}$ be a fixed Hermitian matrix. We consider the set of all
$n\times n$ Hermitian matrices ${\bf M}$ endowed with the probability measure
\begin{equation}
\label{prob-measure}
\mu_n(d{\bf M}) = \frac{1}{Z_n}e^{-n\text{Tr}(V({\bf M})-{\bf A}{\bf M})}d{\bf M}; \quad Z_n:=\int e^{-n\text{Tr}(V({\bf M})-{\bf A}{\bf M})}d{\bf M},
\end{equation}
where $d{\bf M}$ is the entry-wise Lebesgue measure
and the integration is over all Hermitian matrices.
When ${\bf A}={\boldsymbol 0}$ (no external source) and
$V({\bf M})={\bf M}^2/2$, \eqref{prob-measure}
describes the Gaussian Unitary Ensemble, or GUE. For ${\bf A}\neq {\bf 0}$
and $V({\bf M})={\bf M}^2/2$, this measure arises in the study of
Hamiltonians that can be written as the sum of a random matrix and a
deterministic source matrix \cite{Brezin:1996}.
We are specifically interested in small-rank sources of the form
\begin{equation}
\label{A}
{\bf A} = \text{diag}(\underbrace{a,\dots,a}_r,\underbrace{0,\dots,0}_{n-r})
\end{equation}
assuming that either $r=\mathcal O(n^\gamma)$, $0<\gamma<1$ or $r$ is finite
(in which case we define $\gamma:=0$). The ratio of $r$ to $n$, which is
asymptotically small, will be denoted as
\begin{equation}
\kappa:=\frac{r}{n}.
\end{equation}
P\'ech\'e \cite{Peche:2006} studied the limiting distribution of the largest
eigenvalue in the Gaussian case ($V({\bf M})={\bf M}^2/2$) under these
assumptions and found three distinct behaviors. In
the supercritical case, $r$ eigenvalues are expected to exit the
bulk and are found to distribute as the eigenvalues of an
$r\times r$ GUE matrix. For the subcritical case, the
largest eigenvalue is expected to lie at the right band endpoint and
behave as the largest eigenvalue of an $n\times n$ GUE matrix.
In the critical case, when the outliers
lie at the band endpoint, the distribution for the largest eigenvalue
is an extension of the standard GUE Tracy-Widom function
\cite{Tracy:1994} which arises when $r=0$ (see also \cite{Adler:2009a, Baik:2005,Baik:2006}).
One of the primary goals of random matrix theory is to determine universality
classes of matrix ensembles, that is, find different probability measures
on the space of matrices for which the spectral properties are the same in
the large-$n$ limit. With this goal in mind we consider more general
functions $V({\bf M})$, with our specific assumptions listed in Section
\ref{assumptions}. Basically, we assume $V({\bf M})$ is a generic
single-gap analytic potential with sufficient growth at infinity. We
show that P\'ech\'e's results \cite{Peche:2006} hold for these general potentials in the
supercritical and subcritical cases. The universality of the critical case
will be considered elsewhere \cite{Bertola:2010}.
\subsection{Definition of the supercritical, subcritical, and critical regimes}
Let $g(z)$ be the $g$-function associated with the orthogonal
polynomials with potential $V(z)$ (see, for instance, \cite{Deift:1998-book}
or \cite{Deift:1999a}).
It may be written as
\begin{equation}
\label{g-form}
g(z):=\int_{\mathbb{R}}\log(z-s)\rho_\text{min}(s)ds,
\end{equation}
where $\rho_\text{min}$ is the unique probability measure minimizing the
functional
\begin{equation}
\mathcal{F}[\rho]:=\int_\mathbb{R}V(s)\rho(s)ds-\int_\mathbb{R}\int_\mathbb{R}\rho(s)\rho(s')\log|s-s'|dsds'.
\end{equation}
We will assume this equilibrium measure $\rho_\text{min}$ is supported on
a single band $[\alpha,\beta]$ (see Assumption \ref{assumptionAV} (v) in Section
\ref{assumptions}). Define
\begin{eqnarray}
\label{P1-nolog}
P_1(z) & := & -V(z)+2g(z)+l_1, \\
\label{P2-nolog}
P_2(z) & := & -V(z)+az+g(z)+l_2, \\
\label{P3-nolog}
P_3(z) & := & -P_1(z) + P_2(z) = az-g(z)-l_1+l_2.
\end{eqnarray}
Here $g(z)$ and $l_1$ are uniquely determined by the conditions
that $P_1(z)_\pm$ is purely imaginary on
{the support of the equilibrium measure}
and has negative real part on its complement in $\mathbb{R}$ (here the subscripts $_\pm$ denote the boundary values from above/below the real axis). How
$l_2$ is chosen will be described at the end of this section.
It is also known that $\Re g(z)$ is a {\bf continuous} function on $\mathbb{R}$ and {\bf harmonic} on the complement of the support of $\rho_{\text{min}}$ (up to a sign it is also known as the {\em logarithmic potential} in potential theory).
\begin{defn}
\label{acrit}
Define $a_c$ to be the (unique) value of $a$ so that $P_2'(\beta)=0$.
\end{defn}
The uniqueness is promptly seen because $P_2'(\beta) = -V'(\beta) + a +g'(\beta)$;
in fact the effective potential $P_1$ is known \cite{Deift:1998a} to satisfy
\begin{equation}
P_1'(z)= \mathcal O{(z-\beta)^\frac 12}
\end{equation}
and in particular $P_1'(\beta)=0$ and hence $P_2'(\beta) = a-g'(\beta) = a-\frac 12 V'(\beta)$. Thus the critical value of $a$ is given by
\begin{equation}
a_c = g'(\beta) = \frac{1}{2} V'(\beta)\,.\label{110}
\end{equation}
We first have
\begin{lemma}
\label{acpositive}
The critical value $a_c = g'(\beta)$ is positive. Moreover $g'(\alpha)<0$.
\end{lemma}
\begin{proof}
From \eqref{g-form} we see that $g'(z) = \int_{\alpha}^\beta \frac 1{z-s} \rho_{\text{min}}(s){\rm d} s$ is {\em positive} for $z>\beta$. It is also known that the density $\rho_{\text{min}}$ vanishes like a square root at the endpoints $\alpha, \beta$ and hence the integral representation of $g'(\beta)$ is convergent and immediately shows it to be positive.
Similarly $g'(\alpha)$ is {\em negative}. Note that this proof does not require the support to consists of a single band as long as we understand $\beta = \sup {\rm supp}\, \rho_{\text{min}}$ and $\alpha = \inf {\rm supp} \,\rho_{\text{min}}$.
\end{proof}
Lemma \ref{acpositive} implies that there is no loss of generality in studying only the case $a>0$ since there is always a positive critical $a_c$ (and a negative one); the negative case ($a<0$) is equivalent to the positive case by replacing $a\mapsto -a$, $V(z)\mapsto V(-z)$.
\begin{lemma}\label{lemmaP3}
The critical point structure of $\Re\, P_3(z)$ is:
\begin{itemize}
\item For $a > a_c$, $\Re\, P_3(z)$ is strictly increasing on $\mathbb{R}\setminus[\alpha,\beta]$;
\item For $a=a_c$, $\Re\, P_3(z)$ is strictly increasing on $\mathbb{R}\setminus[\alpha,\beta]$ and $\Re\,P_3'(\beta)=0$;
\item For $0<a < a_c$, $\Re\, P_3(z)$ has unique local minimum on $\mathbb{R}\setminus(\alpha,\beta)$. This minimum occurs at a point $b^\star\in (\beta,\infty)$.
\end{itemize}
\end{lemma}
\begin{proof}
From the representation \eqref{g-form} of $g$ one sees immediately that
\begin{equation}
g''(z) =- \int_\mathbb{R} \frac 1{(z-s)^2} \rho_{\text{min}}(s) {\rm d} s\label{11}
\end{equation}
which shows clearly that for $z\in \mathbb{R} \setminus {\rm supp\,} \rho_{\text{min}}$ the real part of $g$ is concave downward. Thus $\Re g'(z)$ is decreasing in $\mathbb{R} \setminus {\rm supp\,} \rho_{\text{min}}$; moreover, from
\begin{equation}
g'(z) = \int_\mathbb{R} \frac 1{(z-s)} \rho_{\text{min}}(s) {\rm d} s \label{12}
\end{equation}
we see that $\Re g'$ is negative for $z<\inf {\rm supp\,} \rho_{\text{min}} = \alpha$ and positive for $z>\sup {\rm supp\,} \rho_{\text{min}}=\beta$.
From the definition we see that $P_3'(z) = a - g'(z)$ and hence we infer:
\begin{itemize}
\item
For $a > a_c=g'(\beta)>0$, $ {\Re}\,(P_3'(z))=a-g'(z)>g'(\beta)-g'(z)$ is positive on $[\beta, \infty)$ therefore $\Re\,P_3$ is strictly increasing. On the other hand $a- g'(z)$ is clearly positive on $(-\infty, \alpha]$ because $a>0$ and $- g'>0$ from \eqref{12};
\item For $a=a_c$, $ {\Re}\,P_3'(\beta) = 0$ and $ {\Re}\, P_3'(z)$ is a monotonically increasing positive function on $(\beta, \infty)$, and a monotonically increasing positive function on $(-\infty, \alpha]$. Therefore there is a single critical point of $ {\Re}\, P_3(z)$ at $z=\beta$. As $ {\Re}\, P_3''(z) = - {\Re}\, g''(z) > 0$ this must be a minimum;
\item For $0<a < a_c$, $ {\Re}\, P_3'(\beta) < 0$ and $P_3'(z)\to a > 0$ for $z \to \infty$; moreover $ {\Re}\, P_3'(z)$ is a monotonically increasing function on $[\beta , \infty)$, and a (monotonically increasing) positive function on $(-\infty, \alpha]$.
Since $P_3'(\beta)<0$ there must be a unique point $b^\star>\beta$ where $P_3'(b^\star)=0$. As $ {\Re}\, P_3''(z) = - {\Re}\, g''(z) > 0 $ this must be the local minimum (or, equivalently, the global minimum on $(\beta,\infty)$).
\end{itemize}
\end{proof}
We can now define four regimes: supercritical, subcritical, critical, and
jumping outliers. We define the subcritical and critical regimes first.
\begin{defn}\label{subcritical}
The matrix model specified by \eqref{prob-measure} is in the {\bf subcritical regime} if $a < a_c$ and
$P_2(x)<P_3(b^\star)$ for all $x\geq b^\star$.
\end{defn}
\begin{defn}\label{critical}
The matrix model specified by \eqref{prob-measure} is in the {\bf critical regime} if $a=a_c$ and $P_2(x)<P_2(\beta)$ for all $x>\beta$.
\end{defn}
Now the supercritical regime can be efficiently defined as the remaining cases, except the small---codimension one---cases that we distinguish as the ``jumping outlier regime."
\begin{defn}
\label{supercritical}
The model is in the {\bf supercritical regime} if $P_2$ has a unique point of global maximum on $\{x > \max\{\beta, b^\star\}\}$ at a point $x=a^\star\in \mathbb{R}$ and any of the three conditions below is satisfied:
\begin{itemize}
\item $a>a_c$.
\item $a=a_c$ and $P_2(\beta)<P_2(x)$ for some $x>\beta$.
\item $0<a< a_c$ and $P_3(b^{\star})<P_2(x)$ for some $x>b^{\star}$.
\end{itemize}
Note that $a^\star$ is always greater than $\beta$ and $b^\star$.
If the global maximum of $P_2$ on $(\max\{\beta,b^\star\}, \infty)$ is attained at several distinct points then we will say that we are in the {\bf jumping outlier regime} that also includes the following remaining case.
\begin{itemize}
\item $0<a<a_c$ and $P_2(x)=P_3(b^\star)$ for some $x> b^\star$. (The case $x=b^\star$ cannot occur for {\bf regular} $V$.)
\end{itemize}
\end{defn}
In the present paper we consider the supercritical and subcritical regimes. The critical
and jumping outlier regimes will be considered elsewhere
\cite{Bertola:2010}.
The definition of the supercritical regime is complicated and the reader may wonder whether the above definitions ever hold in actual examples.
It is however not difficult to engineer a situation where they do occur, explained in the following example
\begin{example} [Second and third bullet in Definition \ref{supercritical}]
Consider a potential $V$ such that a new spectral band (i.e. interval of support of $\rho_{\text{min}}$) is about to emerge. Then $P_1$ has a local maximum $-E$ at $x_0>\beta$ outside of the main band which is slightly negative but small in absolute value. It is simple to arrange examples where $E$ is arbitrarily small.
Since $a=a_c$ we have $\Re\, P_3'(\beta)=0$ (see \eqref{110}) and since $\Re\,P_3$ is concave upwards, it must be increasing for $x>\beta$. On the other hand $P_2 = P_1+P_3$ and thus
\begin{equation}
\Re\left(P_2(x_0)- P_2(\beta)\right) = -E + \Re(P_3(x_0)-P_3(\beta)).
\end{equation}
Since $E$ can be chosen arbitrarily small, and since $\Re(P_3(x_0)-P_3(\beta))>0$, we see that necessarily we can have the situation described in the second bullet.
By a continuity argument on $a<a_c$, this also provides an example for the third bullet since $P_3(b^\star)<P_3(\beta)\approx P_2(\beta)< P_2(x_0)$.
\end{example}
Though we do not consider in this paper, one can also create an example in the jumping outlier regime.
We show in Proposition \ref{suffcond} that if $V$ is convex, then we are either in the super or subcritical depending on $a>a_c$ or $0\leq a <a_c$, respectively; in particular, in this case, the situation described in the second and third bullet points of Definition \ref{supercritical} cannot occur.
\begin{prop}
\label{suffcond}
Suppose that $V(z)$ satisfies Assumptions \ref{assumptionAV} and in addition it is convex ($V''>0$). Then
\begin{enumerate}
\item[{\bf (i)}] for $a>a_c$ the model is supercritical and the maximum of $P_2$ at $a^\star$ is nondegenerate;
\item[{\bf (ii)}] for $0<a<a_c$ the model is subcritical;
\item[{\bf (iii)}] there is no jumping outlier regime.
\end{enumerate}
\end{prop}
\begin{proof}
It is known that the convexity of $V$ is a sufficient condition for the support of the equilibrium measure to be a single band $[\alpha, \beta]$.
Moreover from Lemma \ref{acpositive} we see $V'(\beta)=2g'(\beta)>0>V'(\alpha)=2g'(\alpha)$.
Note that $P_j$ are all real-valued in $[\beta,\infty)$ (the cut of the logarithm runs in $(-\infty, \beta]$).\\
{\bf (i)} We then observe that
\begin{equation}
P_2'' = -V'' + g''<0
\end{equation}
since both $-V$ (by assumption) and $g$ (by \eqref{11}) are concave downward. $P_2$ may have at most a single global (nondegenerate) maximum in $[\beta, \infty)$ because $P_2'(\beta) = a - \frac 1 2 V'(\beta) = a-a_c>0$. This also proves ({\bf iii}). \\
{\bf (ii)} If $0<a<a_c$, $P_2$ strictly decreases on $[\beta,\infty)$ because $P_2'(\beta) = a - \frac 1 2 V'(\beta) = a-a_c<0$.;
Also we have $P_2(b^\star)=P_1(b^\star)+P_3(b^\star)<P_3(b^\star)$. Therefore $P_2(x)<P_3(b^\star)$ for all $x\geq b^\star$ as in Definition \ref{subcritical}.
\end{proof}
It should be noted here that, contrary to the work done for $V = z^2/2$, the position of $a$ relative to $a_c$ is not sufficient (for general $V$) to define the critical and subcritical regimes. If, however, $V$ is convex (for example an even monomial with positive coefficient) then by Proposition \ref{suffcond} the position of $a$ relative to $a_c$ determines the supercritical/subcritical regime as in \cite{Bleher:2004b, Aptekarev:2005, Bleher:2007, Adler:2009a, Peche:2006}.
The secondary conditions in Definition \ref{supercritical} are dealing with whether the Lagrange multiplier $\ell_2$ in the effective potentials $P_j$ can be chosen such that the off diagonal entries of the jump matrices for the deformed Riemann-Hilbert problems that we will construct in Sections 2, 3, and 4, decay to zero at an exponential rate.
The problem of finding necessary and sufficient conditions on $V(x)$ and $a$ for the matrix model to be in the supercritical/subcritical regime is quite difficult, as much as it is difficult to find necessary and sufficient conditions for $V(x)$ to be a single-band potential.
{We now specify the constant $l_2$:
\begin{defn}The constant $l_2$ in Definition \ref{P2-nolog} will be chosen as follows:
\label{defell2}
\begin{itemize}
\item In the supercritical case, the constant $l_2$ is chosen so that
the unique global maximum of $P_2(z)$ on $(\beta,\infty)$ is zero (i.e. $\Re P_2(a^\star)=0$).
\item In the subcritical case, the constant
\begin{equation}
l_3:=-l_1+l_2
\end{equation}
is chosen so that $P_3(b^{\star})=0$.
\end{itemize}
\end{defn}}
\subsection{The kernel and its connection to multiple orthogonal polynomials}
Let $\rho_m(\lambda_1,\dots,\lambda_m)$ be the probability density that
the $n\times n$ matrix ${\bf M}$ chosen using \eqref{prob-measure} has
eigenvalues
$\{\lambda_1,\dots,\lambda_m\}$ (here $m\leq n$). Then, the $m$-point correlation function is
$R_m(\lambda_1,\dots,\lambda_m):=\frac{n!}{(n-m)!}\rho_m(\lambda_1,\dots,\lambda_m)$. Br\'ezin and Hikami
\cite{Brezin:1996, Brezin:1997a, Brezin:1997b, Brezin:1998} showed that,
in the Gaussian case, the $m$-point correlation functions can
all be expressed in terms of a single kernel $K(x,y)$:
\begin{equation}
R_m(\lambda_1,\dots,\lambda_m) = \det(K(\lambda_i,\lambda_j))_{i,j=1,\dots,n}.
\end{equation}
Zinn-Justin \cite{Zinn-Justin:1997,Zinn-Justin:1998} extended this result to
the case of more general $V({\bf M})$. Bleher and Kuijlaars \cite{Bleher:2004a}
rewrote the kernel in terms of {\it multiple orthogonal polynomials}, a
significant result because it allows one to analyze the asymptotic behavior of
these polynomials via the associated Riemann-Hilbert problem.
This approach was followed by Aptekarev, Bleher, and
Kuijlaars \cite{Bleher:2004b, Aptekarev:2005, Bleher:2007} in the Gaussian
case when the matrix ${\bf A}$ has two eigenvalues $\pm a$, each of
multiplicity
$n/2$. When $a$ is sufficiently large the eigenvalues of ${\bf M}$
accumulate on two disjoint intervals (the supercritical case). As $a$
decreases the two bands collide (the critical case). Below this
critical value of $a$, the eigenvalues accumulate on a single interval
(the subcritical case). Related behavior also appears in the
theory of nonintersecting one-dimensional Brownian motions; see, for
instance, Adler, Orantin, and van Moerbeke \cite{Adler:2009b} for the
critical case.
In general, the existence and number of bands on which eigenvalues accumulate
for large-rank sources for general $V({\bf M})$ is a complicated problem.
For more on this question
see McLaughlin \cite{McLaughlin:2007} in which the quartic case
$V({\bf M}) = {\bf M}^4/4$ is worked out. Bleher, Delvaux, and Kuijlaars
\cite{Bleher:2010} have studied the external source problem with
two eigenvalues of equal multiplicity and where $V({\bf M})$
is a sum of even-degree monomials with positive coefficients.
The external source with a finite number of different eigenvalues with various multiplicity
for supercritical case has been considered in \cite{Delvaux:2010}.
The starting point of our analysis is the Riemann-Hilbert problem
associated to the multiple
orthogonal polynomials. Suppose ${\bf Y}(z)$ is a $3\times 3$
matrix-valued function of the complex variable $z$ satisfying
\begin{equation}
\label{rhp}
\begin{cases}
{\bf Y}(z) \text{ is analytic for } z\notin\mathbb{R}, \\
{\bf Y}_+(x) = {\bf Y}_-(x)\begin{pmatrix} 1 & e^{-nV(x)} & e^{-n(V(x)-ax)} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \text{ for } x\in\mathbb{R}, \\
{\bf Y}(z) = \left({\bf I}+O\left(\frac{1}{z}\right)\right)\begin{pmatrix} z^n & 0 & 0 \\ 0 & z^{-(n-r)} & 0 \\ 0 & 0 & z^{-r} \end{pmatrix} \text{ as } z\to\infty.
\end{cases}
\end{equation}
Here ${\bf Y}_\pm(x) := \lim_{\varepsilon\to 0}{\bf Y}(x\pm i\varepsilon)$ denote the
non-tangential limits of ${\bf Y}(z)$ as $z$ approaches the real axis from the
upper and lower half-planes. Whenever posing a Riemann-Hilbert problem
we assume (unless otherwise stated) that the solution has continuous boundary
values along the jump contour when approached from either side.
Under our assumption (iv) in Section \ref{assumptions}, the unique
solution ${\bf Y}(z)$ can be written
explicitly in terms of multiple orthogonal polynomials of the second
kind (see \cite{Bleher:2004b}, Section 2).
In the case of two distinct
eigenvalues $a$ and $0$,
the kernel $K_n(x,y)$ may be written in terms of the function ${\bf Y}(z)$ as
\begin{equation}
\label{mop-kernel}
K_n(x,y) = \frac{e^{-\frac{1}{2}n(V(x)+V(y))}}{2\pi i(x-y)} \begin{pmatrix} 0 & 1 & e^{nay} \end{pmatrix} {\bf Y}(y)^{-1}{\bf Y}(x)\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}.
\end{equation}
In the technical analysis of this Riemann-Hilbert problem we use and improve
certain
ideas introduced by Bertola and Lee \cite{Bertola:2009a} to study the
first finitely many eigenvalues in the birth of a new spectral band for
the random Hermitian matrix model without source.
We note here that Baik \cite{Baik:2008} has recently expressed the
kernel $K_n(x,y)$ in terms of the standard (not multiple) orthogonal
polynomials.
This offers an alternative method for approaching the
problem we consider here.
Based on this approach, Baik and Dong \cite{private} have obtained the universality result similar to ours, for the case of finite $r$ but possibly for non-degenerate eigenvalues, i.e. ${\bf A}=\text{diag}(a_1,a_2,\cdots,a_r,0,\cdots,0)$.
Since the rank of the matrices involved in
this alternative formulation grow with $r$, analyzing the Riemann-Hilbert
problem \eqref{rhp} seems more feasible if $r$ is allowed to grow sublinearly
with $n$.
\subsection{Assumptions on \texorpdfstring{${\bf A}$ and ${\bf V(z)}$}{AV} and results}
\label{assumptions}
First we gather the assumptions we will make in the rest of the paper.
\begin{assumption}
\label{assumptionAV}
We make the following requirements
\begin{itemize}
\item[(i)] $a>0$.
\item[(ii)] ${\bf A}$ is a \emph{small-rank} external source of the form
\eqref{A} with either $r$ a fixed positive integer or $r=\mathcal O(n^\gamma)$, with $0\leq \gamma<1$. When $r$
is fixed we say $\gamma=0$.
\item[(iii)] $V(z)$ is real-analytic.
\item[(iv)] $\displaystyle\lim_{|z|\to\infty}\frac{V(z)}{\log(1+z^2)} = \infty \quad \text{and} \quad \lim_{|z|\to\infty}\frac{V(z)-az}{\log(1+z^2)} = \infty.$
\item[(v)] $V(z)$ is a {\em single-band} potential (for example it can be convex).
\item[(vi)] The density of the equilibrium measure of $V(z)$ has square root decay at its
two endpoints (i.e. it is {\em regular} in the sense of \cite{Deift:1999a}).
\item[(vii)] For the supercritical regime, $P_2(z)$ behaves quadratically
near $a^{\star}$. Specifically,
\begin{equation}
\label{P2-near-a*}
-V'(z) + a + g'(z) = -c(z-a^{\star}) + O( (z-a^{\star})^2) \text{ as } z\to a^{\star}
\end{equation}
for some constant $c>0$.
\end{itemize}
\end{assumption}
Assumption (i) is for convenience, as the case when $a<0$ is equivalent by
sending $a\to-a$ and $V(z)\to V(-z)$. Regarding assumption (ii),
in the general case when ${\bf A}$ has $m>2$ distinct eigenvalues the
kernel can be written in terms of multiple orthogonal polynomials associated
to an $(m+1)\times (m+1)$ Riemann-Hilbert problem, which is beyond the scope of
this paper. Assumption (iii) allows us
to use the nonlinear steepest-descent method for Riemann-Hilbert problems,
while (iv) guarantees the existence of the multiple orthogonal polynomials
needed to ensure the Riemann-Hilbert problem has a solution.
Assumption
(v) avoids the necessity of using Riemann-theta functions for the solution
of the outer model Riemann-Hilbert problem. We expect similar results to
hold generically in the multi-band case.
Both (vi) and (vii) are
genericity assumptions. Assumption (vi) allows us to use Airy parametrices near the
band endpoints. Assumption (vii) produces Hermite (or Gaussian) behavior
of the outlying zeros.
Our results are computations of the large $n$ behavior of the kernel function
\eqref{mop-kernel}. We explicitly compute the kernel in a neighborhood of $a^{\star}$ for the supercritical regime and in a neighborhood of $b^{\star}$ for the subcritical regime. In the remaining portions of the complex plane, our result is that the kernel function converges to the kernel for the classical orthogonal polynomial problem with respect to $V(x)$. In particular, our results include that, away from the $a^{\star}$ and $b^\star$, the standard universality classes apply (i.e. a sine kernel in the bulk of the spectrum, and Airy kernels at the edges).
\begin{shaded}
\begin{thm}\label{theorem-super-kernel}
Suppose $V(z)$ and $a$ satisfy assumptions (i)--(vii) and definition
\ref{supercritical} of the supercritical regime.
Let $\zeta_x$ and $\zeta_y$ be the local coordinates
corresponding to $x$ and $y$ near $a^{\star}$ as defined in
\eqref{super-zeta}.
Uniformly for $\zeta_x, \zeta_y$ in compact sets we have the following asymptotics for $r= C n^\gamma$ for some $C>0$, $0\leq\gamma<1$.
\begin{equation}
\label{kernel-thm-form}
K_n(x(\zeta_x),y(\zeta_y)) = e^{-\frac{n}{2}P_3(x)+\frac{n}{2}P_3(y)}\frac{\sqrt{f''(0)}}{k_{r-1}^{(r)}}\kappa^{-1/2} \left(K_r^\text{GUE}(\zeta_x,\zeta_y)+\mathcal{O}(n^{-(1-\gamma)/2}) \right),
\end{equation}
where $P_3(x)$ is given by \eqref{P3-nolog}, $f(z;\kappa)$ is defined in \eqref{super-def-of-f}, $k_{r-1}^{(r)}$ is
defined by \eqref{def-of-kr}, $\kappa=r/n$, and
\begin{equation}
K_r^\text{GUE}(\zeta_x,\zeta_y):=\frac{H_r^{(r)}(\zeta_x-\zeta_0)H_{r-1}^{(r)}(\zeta_y-\zeta_0)-H_{r-1}^{(r)}(\zeta_x-\zeta_0)H_r^{(r)}(\zeta_y-\zeta_0)}{\zeta_x-\zeta_y}e^{-\frac{r}{4}(\zeta_x-\zeta_0)^2-\frac{r}{4}(\zeta_y-\zeta_0)^2}
\end{equation}
is the kernel for $r$ eigenvalues of the Gaussian Unitary Ensemble of scale
$r$ centered at $\zeta_0$, which is defined by the change of variables
\eqref{super-P2-zeta}. Here $H_i^{(r)}(\zeta)$ are the rescaled monic
Hermite polynomials satisfying the orthogonality condition \eqref{def-of-kr}.
\end{thm}
\end{shaded}
The presence of $
\exp(-n P_3(x) / 2 )
$ in (\ref{kernel-thm-form})
does not affect spectral properties of the kernel (because it amounts to a conjugation of the kernel by a diagonal operator) and therefore the implication is that asymptotically the eigenvalues near $a^{\star}$ are equivalent to those of a scaled $r\times r$ GUE problem; if $V(x)$ is a quadratic potential this agrees with the results of \cite{Peche:2006}.
\begin{remark}
If the critical point of $P_2$ at $a^\star$ is more degenerate, $P_2(z) = \mathcal O((z-a^\star)^{2k})$, then one may follow similar steps as in \cite{Bertola:2009a} and \cite{Bertola:2009} and conclude that the relevant statistics of the outliers are determined by the kernel of a unitary ensemble with potential given by a polynomial of degree $2k$ instead of a Gaussian, namely, obtained from a deformation of the Freud orthogonal polynomials.
\end{remark}
Theorem \ref{theorem-super-kernel} shows that, as expected, the equilibrium
measure has no mass near $a^{\star}$. We have
the mean density of states
\begin{equation}
\rho_n(x(\zeta_x) ) = \lim_{\zeta_y \to \zeta_x} \frac{1}{n} K_n( x(\zeta_x), y(\zeta_y) ) = \frac{\sqrt{f''(0)}}{k_{r-1}^{(r)}} \kappa^{1/2} \left( \rho_r^{(r)}( \zeta_x ) + \mathcal{O}(n^{-(1-\gamma)/2} r^{-1} ) \right)
\end{equation}
where $\rho_r^{(r)}(\zeta)$ is the mean density of eigenvalues for the $r \times r$ GUE ensemble. If $r$ is fixed, this quantity is $\mathcal{O}(n^{-1/2})$ for large $n$, and if $r= n^\gamma$, using that $\rho_r^{(r)}(\zeta_x) \to \frac{1}{2\pi} \sqrt{ 4 - \zeta_x^2} $ as $r \to \infty$, we find
\begin{equation}
\lim_{n\to \infty} \rho_n(x(\zeta_x)) = \frac{\sqrt{f''(0)}}{2 \pi k_{r-1}^{(r)}} \kappa^{1/2} \sqrt{ 4 - \zeta_x^2} \,.
\end{equation}
In either case, our conclusion is that for large $n$ the mean density of eigenvalues is asymptotically small (of order $\kappa^{1/2}$) in the neighborhood of $a^{\star}$ chosen in the theorem. See Chapters 5 and 6 of \cite{Deift:1998-book}, Theorem 1.1 of \cite{Bleher:2004b}, and Theorem 8.1 of \cite{McLaughlin:2007} for similar results regarding the derivation of the asymptotic mean distribution of eigenvalues from the kernel.
\begin{shaded}
\begin{thm}
\label{theorem-sub-kernel}
Suppose the pair $(V(z),a)$ satisfies Definition \ref{subcritical} of
the subcritical regime.
There exists a closed disk of fixed radius centered at $b^{\star}$ such that,
for $x$ and
$y$ in this disk, for large $n$, and for $r=C n^\gamma$ for some $C>0$ and $0\leq \gamma<1$, there is a $c>0$ such that
\begin{equation}
K_n(x,y) = \mathcal{O}(n^{-(1-\gamma)/2} e^{-cn}).
\end{equation}
\end{thm}
\end{shaded}
\bigskip
\noindent
{\bf Acknowledgments.} The authors would like to thank Jinho Baik,
Ken McLaughlin, Sandrine P\'ech\'e and Dong Wang for several illuminating discussions. We thank Baik and Wang for sharing their unpublished results.
M. Bertola
was supported by NSERC. R. Buckingham was
supported by the Taft Research Foundation. V. Pierce was supported by
NSF grant DMS-0806219.
\section{The supercritical regime}
\subsection{Modified equilibrium problem}
\label{supercrit}
In this section we will use a positive integer $K$; the general statements are valid for any $K$ but we will choose (for future use)
\begin{equation}
\label{super-k-def}
K\geq \max \left\{\frac{3\gamma-1}{1-\gamma},0\right\}
\end{equation}
where, we recall, $\gamma$ is the exponent of growth of $r=Cn^\gamma$ for some $C$ and $0\leq \gamma<1$.
We will need to build a {\em perturbation} of the equilibrium problem that leads to the definition of $g(z)$; we will denote by $\mathfrak g$ the resulting $g$--function of this perturbation scheme.
The construction, rather involved, will be broken down in steps.
The unperturbed equilibrium measure is supported on the single interval $[\alpha, \beta]$ (by assumption) with external field $V(z)$. Recall that $a^\star$ is lying {\em outside} of $[\alpha,\beta]$ and we fix a compact interval $J$ containing $[\alpha,\beta]$ in its interior and $a^\star\not \in J$.
\begin{prop}
\label{propdeform}
For any $K\in \mathbb N$ there is a neighborhood of the origin in $(\kappa,\vec \delta) \in \mathbb{C}^{1+K}$ such that the equilibrium measure $\widetilde \sigma(x){\rm d} x$ of total mass $1-\kappa$ for the external field
\begin{equation}
\widetilde V(z):=V(z) - \delta V(z), \ \ \ \delta V(z):= \kappa \ln (z-a^\star ) + \kappa \sum_{j=1}^K \frac {\delta_j}{2(z-a^\star)^j}
\label{deformV}
\end{equation}
is supported on a single interval $[\alpha(\kappa,\vec \delta), \beta (\kappa,\vec \delta)]$ still contained in the interior of $J$: the endpoints $\alpha(\kappa,\vec \delta), \beta (\kappa,\vec \delta)$ are analytic functions of the specified variables.
Furthermore the (normalized) $g$--function of this problem
\begin{equation}
\mathfrak g(z):= \int \ln(z-w) \frac{\widetilde \sigma(w)}{1-\kappa}{\rm d} w
\end{equation}
converges uniformly over closed subsets not containing $[\alpha,\beta]$ to the unperturbed $g$--function.
\end{prop}
\begin{remark}
In this proposition we treat the deformation parameters $\kappa, \vec \delta$ as independent from each other; later on, in Proposition \ref{propdeltas}, they will be uniquely determined in terms of the sole parameter $\kappa$.
\end{remark}
\begin{proof}
It is well known (see, for example, \cite{Deift:1998a}) that
\begin{itemize}
\item $\mathfrak{g}(z)$ is analytic for $z\notin (-\infty, \widetilde \beta] $, where $\widetilde \beta = \sup\ \text{supp}(\widetilde \sigma)$;
\item $\mathfrak{g}(z)$ has continuous boundary values and satisfies
\begin{equation}
\label{super-g-rhp2}
\begin{split}
& \mathfrak{g}_+(z) - \mathfrak{g}_-(z) = 2 \pi i, \; z \in (-\infty, \widetilde \alpha), \\
& (1-\kappa) \left( \mathfrak{g}_+(z; \kappa) + \mathfrak{g}_-(z; \kappa) \right) = V(z) - \kappa \log(z - a^{\star}) -\sum_{j=1}^{2k}\frac{\delta_j}{2(z-a^{\star})^j} - \ell_1, \; z \in (\widetilde \alpha,\widetilde \beta)
\end{split}
\end{equation}
where $\widetilde \alpha = \inf\ \text{supp}(\widetilde \sigma)$
with the real axis oriented left to right;
\item $\mathfrak{g}(z; \kappa) = \log(z) + \mathcal{O}\left(\displaystyle\frac{1}{z}\right) $ as $z \to \infty$.
\end{itemize}
Vice versa, the $g$--function may be characterized by the scalar Riemann--Hilbert problem (\ref{super-g-rhp2}) with the additional requirement that $\Im \mathfrak g_+$ is a nondecreasing function.
To show the analytic dependence of $\mathfrak g$ on $\kappa,\vec \delta$ we proceed as follows:
define the function
\begin{equation}
\label{super-R-def}
R(z):=((z-\widetilde \alpha)(z-\widetilde \beta))^{1/2},
\end{equation}
where the principal branch of the square root is chosen so
$R(z;\kappa)=z+\mathcal{O}(1)$ as $z\to\infty$.
Taking the derivative of \eqref{super-g-rhp2} with respect to $z$ and using the
Plemelj formula gives
\begin{equation}
\label{super-gprime}
{\mathfrak g}'(z) = \frac{R(z)}{2\pi i(1-\kappa)}\int_{\widetilde \alpha}^{\widetilde \beta} \frac{V'(s) - \kappa/(s-a^{\star}) + \sum_{j=1}^{2k}j\delta_j/(2(s-a^{\star})^{j+1}) }{(s-z)R_+(s)} ds
\end{equation}
where $R_+(s)$ refers to the limit in $s$ from the upper half-plane.
The large-$z$ expansion of (\ref{super-gprime}) along with the condition
\begin{equation}
\mathfrak{g}'(z; \kappa) = \frac{1}{z} + \mathcal{O}\left( \frac{1}{z^2}\right)
\end{equation}
gives two conditions on $\alpha(\kappa), \beta (\kappa)$:
\begin{equation} \label{super-Jacobian1}
\int_{\widetilde \alpha}^{\widetilde \beta} \frac{V'(s) - \kappa/(s-a^{\star}) + \sum_{j=1}^{2k}j\delta_j/(2(s-a^{\star})^{j+1}) }{ R_+(s)} ds = 0
\end{equation}
and
\begin{equation} \label{super-Jacobian2}
\frac{1}{2\pi i} \int_{\widetilde \alpha}^{\widetilde \beta} \frac{V'(s) - \kappa/(s-a^{\star}) + \sum_{j=1}^{2k}j\delta_j/(2(s-a^{\star})^{j+1}) }{ R_+(s) } s \,ds = 1-\kappa \,.
\end{equation}
These two equations uniquely determine $\widetilde \alpha, \widetilde \beta$ as analytic functions of the parameters by the implicit function theorem.
The inequality $\Im \mathfrak g_+'>0$ remains valid, using a continuity argument, for suitably small values of $\kappa, \vec \delta$ because it is valid (with the strict inequality) for the unperturbed $g$--function (by our initial assumption).
Therefore the expression (\ref{super-gprime}) yields a {\em bona fide} $g$--function for the modified external field $\widetilde V$ in a neighborhood of $(\kappa, \vec \delta ) = (0,\vec 0)$.
The expression for $\mathfrak g$ may be obtained by integration; specifically
\begin{equation}
\mathfrak g(z) = \int_{\widetilde\alpha}^z \mathfrak g'(s){\rm d} s - \ell_1
\end{equation}
and $\ell_1$ is also determined by the requirement that $\mathfrak g(z) = \ln (z) + \mathcal O(z^{-1})$ (without the constant term). Explicitly
\begin{equation}
\ell_1 = \lim_{z\to\infty} \left(\int_{\widetilde\alpha}^z \mathfrak g'(s){\rm d} s - \ln z\right)
\end{equation}
which expression shows that $\ell_1$ is also analytic in the parameters, given the already proven analyticity of $\mathfrak g'$.
The statement about the convergence follows easily by noticing that $\mathfrak g'$ converges to $g'(z)$ as desired (note that they both have behavior $1/z + \mathcal O(z^{-2})$). This is best seen from the integral representations and the already proven analytic dependence on the deformation parameters.
\end{proof}
In parallel with the definition of the functions $P_j$ we shall define
\begin{shaded}
\begin{eqnarray}
\label{super-mathcalP1}
{\mathcal P}_1(z;\kappa) & := & -\widetilde V(z) + 2(1-\kappa){\mathfrak g}(z;\kappa) + \ell_1, \\
\label{super-mathcalP2}
{\mathcal P}_2(z;\kappa) & := & -\widetilde V(z) + az +\delta V + (1-\kappa){\mathfrak g}(z;\kappa) +
{l_2}, \\
\label{super-mathcalP3}
{\mathcal P}_3(z;\kappa) & := & -{\mathcal P}_1(z;\kappa) + {\mathcal P}_2(z;\kappa)\\
\widetilde V(z)& : =&V(z) - \delta V(z), \ \ \ \delta V(z):= \kappa \ln (z-a^\star ) + \kappa \sum_{j=1}^K \frac {\delta_j}{2(z-a^\star)^j}
\end{eqnarray}
\end{shaded}
\noindent where $l_2$ as in Definition \ref{defell2} and is independent of the deformation.
Using deformation and continuity arguments we have that (for a sufficiently small value of the deformation parameters $\kappa, \vec \delta$) the real part of $\mathcal P_2$ has a global maximum in a neighborhood of $a^\star$. The main tool in the analysis of the supercritical case when $r$ grows shall be the next theorem.
\begin{thm}
\label{rhoprop}
There exists a conformal change of coordinate $\rho = \rho(z;\kappa,\vec \delta)$ { fixing $z=a^\star$} (i.e. $\rho(a^\star;\kappa,\vec \delta) \equiv 0$) that depends analytically on the parameters $\kappa, \vec \delta$ such that
\begin{equation}
\mathcal P_2 (z):= -V(z) + a z + (1-\kappa)\mathfrak g + l_2 +2\kappa\ln(z-a^\star) + \sum_{j=1}^K \frac {\delta_j}{(z-a^\star)^j}
\end{equation}
can be written as
\begin{equation}
\mathcal P_2(z;\kappa,\vec \delta) = -\frac 1 2 (\rho- {\mathfrak a} )^2 + 2\kappa \ln \rho+\mathfrak b + \sum_{j=1}^{K} \frac
{ \gamma_j}{\rho^j}\label{mainid}
\end{equation}
where the parameters ${\mathfrak a} = {\mathfrak a} (\kappa,\vec \delta)$, $\mathfrak b = \mathfrak b (\kappa,\vec\delta)$ and $\vec \gamma = \vec \gamma(\kappa;\vec \delta)$ are analytic functions of the indicated parameters.
Furthermore the Jacobian
\begin{equation}
\frac {\partial \vec \gamma}{\partial \vec \delta}
\end{equation}
is nonsingular in a neighborhood of the origin (i.e. for $\kappa$ and $\vec\delta$ sufficiently small).
\end{thm}
\begin{proof}
To simplify the notation we set $a^\star =0$ (up to a translation this entails no loss of generality).
We can write $\mathcal P_2$ as
\begin{equation}
\mathcal P_2 = - f(z;\kappa,\delta) +2\kappa\ln(z) + \sum_{j=1}^K \frac {\delta_j}{z^j} \,.
\end{equation}
By the definition of $a^\star$ (which is now translated to $0$), the function $f(z;\kappa,\vec \delta)$ has the property that
\begin{equation}
f(z; 0, \vec 0) = \frac C2 z^2(1 + \mathcal O(z))\ , C>0
\end{equation}
and hence
\begin{equation}
f(z; \kappa, \vec \delta) = \frac C2 z^2(1 + \mathcal O(z)) + \mathcal O(\kappa,\vec \delta).
\end{equation}
Let us fix any (smooth) curve in the parameter space $\kappa(t), \vec \delta(t)$ and denote by a $\partial $ its tangent vector; we then must show the identity (\ref{mainid}) for $t$ near 0. We suppress the notation of the dependence on $\kappa, \vec \delta$ for brevity in what follows, with the understanding that $f(z), {\mathfrak a} , \gamma_j, \mathfrak b $ all depend on them.
In this part we only sketch the main idea, leaving a full proof for Appendix \ref{confmap-appendix}.
Let $\mathbb D(r)$ be the open disk of radius $r>0$ and let $\Omega_{1}$ be the Banach {\bf manifold} of {\bf univalent, analytic functions} $\rho:\mathbb D(r)\to \mathbb{C}$ which fix the origin $\rho(0)=0$; this is a closed Banach submanifold of all univalent analytic functions because the evaluation map is continuous.
Define now
\begin{equation}
\mathcal M:= \Omega_1 \times \mathbb{C}^{K+1} = \left \{{\bf p} = (\rho,{\mathfrak a} , \mathfrak b ,\vec \gamma),\ \ \rho\in \Omega_1, \ \ {\mathfrak a} ,\mathfrak b \in \mathbb{C} \right \}
\end{equation}
which is naturally also an {\em infinite dimensional} Banach manifold.
We are going to show that the ordinary differential equation in $\partial$ that derives from (\ref{mainid}) is integrable on $\mathcal M$; taking the implicit differentiation of (\ref{mainid}) we obtain
\begin{eqnarray}\nonumber
&
-\partial f(z) +2\partial \kappa\ln(z) + \sum_{j=1}^K \frac { \partial \delta_j }{z^j} = \left({\mathfrak a} - \rho + 2 \frac \kappa \rho - \sum_{j=1}^{K} \frac {j \delta_j}{\rho^{j+1}} \right) \partial \rho
+\partial \mathfrak b
- \partial {\mathfrak a} (\rho-{\mathfrak a} ) + 2\partial \kappa \ln \rho + \sum_{j=1}^K \frac { \partial \gamma_j }{\rho^j}
\\\nonumber
&\displaystyle\Longrightarrow\quad
\partial \rho =
\frac{
-\partial f(z) +2\partial \kappa\ln\left( \frac {z}{\rho}\right) + \sum_{j=1}^K \frac {\partial \delta _j}{z^j} +\partial \mathfrak b + \partial {\mathfrak a} (\rho-{\mathfrak a} )
- \sum_{j=1}^K \frac { \partial \gamma_j }{\rho^j}}{
{\mathfrak a} - \rho + \frac \kappa \rho - \sum_{j=1}^{K} \frac {j \gamma_j}{\rho^{j+1}}
}\\
&\displaystyle \Longrightarrow\quad
\partial \rho =
\rho\frac{
- \rho^{K} \partial f(z) +2\rho^{K} \ln\left( \frac {z}{\rho}\right)\partial \kappa + \sum_{j=1}^K \frac {\rho^{K} \partial \delta_j}{z^j} +\rho^{K} (\rho-{\mathfrak a} ) \partial {\mathfrak a}
- \sum_{j=1}^K \partial \gamma_j \rho^{K-j} } {
{\mathfrak a} \rho^{K+1} - \rho^{K+2} + 2 \kappa \rho^K - \sum_{j=1}^{K} j \gamma_j\rho^{K-j}
}\label{49}.
\end{eqnarray}
Formula (\ref{49}) should be regarded as defining a vector field on $\mathcal M$, and this flow together with $\rho(z;0,\vec 0) = \sqrt{2f(z;0,\vec 0)}$ gives an initial value problem. To see this we have to remember that the tangent space to $\mathcal M$ consists of
\begin{equation}
T\mathcal M:= \Omega_0\times \mathbb{C}^{K+2}
\end{equation}
where $\Omega_0$ stands for the Banach vector space of bounded analytic functions on $\mathbb D(r)$ (without the requirement of being univalent) mapping $0$ to $0$.
The denominator vanishes generically at $K+2$ values $\rho_j$; since $\partial \rho$ must be an analytic function, the numerator must vanish at the same points and this yields a linear system for the $K+2$ values $\partial \mathfrak b , \partial {\mathfrak a} , \partial \gamma_1,\dots, \partial \gamma_K$. To see how this works in more detail, let $\rho_j$ be the roots of the denominator in (\ref{49})
\begin{equation}
- \rho^{K+2} + {\mathfrak a} \rho^{K+1} + 2 \kappa \rho^K - \sum_{j=1}^{K} j \gamma_j\rho^{K-j} =- \prod_{j=1}^{K+2} (\rho-\rho_j)
\label{52}\ .
\end{equation}
For $\kappa, {\mathfrak a} , \vec \gamma$ sufficiently small all the roots $\rho_j$ belong to the disk $\mathbb D(r)$ where $\rho(z)$ is univalent and therefore there are corresponding values $z_1,\dots, z_{K+2}$.
The linear system that determines $\partial{\mathfrak a} , \partial \vec \gamma$ is then
\begin{equation}
\rho_{\ell}^{K}\left( - \partial f(z_\ell) +2\ln\left( \frac {z_\ell }{\rho_\ell }\right)\partial \kappa\right) + \sum_{j=1}^K \frac {\rho_\ell^{K}}{z_\ell ^j} {\partial \delta_j}+\rho_\ell ^{K}(\rho_\ell -{\mathfrak a} ) \partial {\mathfrak a} +\rho_{\ell}^{K} \partial \mathfrak b +
\sum_{j=1}^K{\rho_\ell^{K-j}}{ \partial \gamma_j }=0\ ,\ \ \ell=1,\dots, K + 2 .\label{linsys}
\end{equation}
What we want to see is that this system determines $\partial {\mathfrak a}, \partial {\mathfrak b},\partial \vec \gamma$ as {\em analytic} functions of $\kappa, {\mathfrak a} , \vec \gamma$; to see this we observe that the coefficient matrix of the linear system (\ref {linsys}) is
\begin{eqnarray}
\begin{bmatrix}
\rho_1^{K}(\rho_1-{\mathfrak a} ) & \rho_1^K & \rho_1^{K-1} & \dots & 1\\
\rho_2^{K}(\rho_2-{\mathfrak a} ) & \rho_2^K & \rho_2^{K-1} & \dots & 1\\
\vdots&&&\\
\rho_{K+2}^{K}(\rho_{K+2}-{\mathfrak a} ) & \rho_{K+2}^K & \rho_{K+2}^{K-1} & \dots & 1
\end{bmatrix}\begin{bmatrix}
\partial {\mathfrak a} \\
\partial \mathfrak b \\
\partial \gamma_K\\
\vdots \\
\partial \gamma_1
\end{bmatrix} = - \begin{bmatrix}
H(z_1)\\
H(z_2)\\
\vdots\\
H(z_{K+2})
\end{bmatrix}\ ,\\
H(z_{\ell}):= \rho_{\ell}^{K}\left( 2\ln\left( \frac {z_\ell }{\rho_\ell }\right)\partial \kappa-\partial f(z_\ell) \right) + \sum_{j=1}^K \frac {\rho_\ell^{K}}{z_\ell ^j} {\partial \delta_j}\,.
\end{eqnarray}
Solving this system by Cramer's rule yields $\partial{\mathfrak a} ,\partial \vec \gamma$ as {\bf symmetric} functions of the roots $\rho_\ell$; moreover it is seen that the determinant of the linear part is simply the Vandermonde determinant $\Delta(\vec \rho):= \prod_{j<\ell\leq K+2} (\rho_j-\rho_\ell)$ and since the determinants in the numerators of Cramer's formula will also vanish whenever two roots coincide, it follows that the ratio is actually {\em analytic} on the diagonals $\rho_\ell = \rho_k, \ \ell\neq k$.
We have thus proved that $\partial {\mathfrak a} , \partial \vec \gamma, \partial \mathfrak b $ are analytic symmetric functions of the $\rho_\ell$'s; it is well known that the ring of analytic symmetric functions is generated by the elementary symmetric polynomials in the roots, namely, the coefficients of the polynomial (\ref{52}). This means that $\partial {\mathfrak a} , \partial \vec \gamma, \partial \mathfrak b $ are actually expressible in terms of analytic expressions of ${\mathfrak a} , \kappa, \vec \gamma$.
In order to complete the proof we should check that the vector field determined by (\ref{49}) is {\bf Lipshitz} with respect to the Banach norm of $T\mathcal M$; the check is rather straightforward but lengthy and a detailed analysis is deferred to App. \ref{confmap-appendix} in the simplified case $K=0$. After this, the existence and uniqueness of the integrated flow follows from standard theorems in Banach spaces.
\paragraph {\bf Jacobian at the origin.} To compute the Jacobian at the origin $(\kappa, \vec \delta)=(0,\vec 0)$ we have to set
\begin{equation}
\rho = \sqrt {2f(z; 0,\vec 0)}\ ,\ \ {\mathfrak a} =\kappa = \delta_j = \gamma_j =\mathfrak b =0\ .
\end{equation}
Taking now $\partial_\ell $ to mean $\partial_{\delta_\ell}$ we find the equations
\begin{eqnarray}
\partial_\ell \rho = \frac {-\partial_\ell f + \frac 1 {z^\ell } + \partial_\ell \mathfrak b + \rho\partial_\ell {\mathfrak a} - \sum_{j=1}^K \frac {\partial_\ell \gamma_{j }}{\rho^j}}{-\rho}\ .
\label{50}
\end{eqnarray}
Since we want $\rho(0;\kappa,\vec \delta)\equiv 0$ we must impose that $\partial_\ell \rho$ in (\ref{50}) vanishes at least of order $z$ at $z=0$; this yields a linear system for the coefficients $\partial_\ell \mathfrak b ,\partial_\ell {\mathfrak a}, \partial_\ell \gamma_{j}$ and in particular
\begin{eqnarray}
\left\{\begin{array}{ccl}
\partial_\ell \gamma_{j} (0,\vec 0) &=& 0\ , j> \ell\\
\partial_j \gamma_{j} (0,\vec 0) &=& 1 \\
\partial_\ell \gamma_{j}(0,\vec 0) &=& \star\ , j<\ell.
\end{array}\right.\qquad
\frac {\partial \vec \gamma}{\partial \vec \delta} = \begin{bmatrix}
1& \star& \star& \dots\\
&1& \star & \dots\\
&&\ddots&\\
&&&1
\end{bmatrix}
\end{eqnarray}
with the $\star$ denoting some expression which is not relevant to us; the above Jacobian is then triangular with $1$ on the diagonal, and hence it is invertible in a neighborhood of $\kappa=0,\ \vec \delta =\vec 0$.
\end{proof}
\subsection{Determination of the \texorpdfstring{$\delta_j$'s}{deltaj}}
We now introduce the rescaled variable ($\kappa>0$)
\begin{equation}
\zeta = \frac \rho {\sqrt\kappa} \ ,\ \ \ \zeta_0:= \frac {{\mathfrak a} }{\sqrt\kappa}\,.
\end{equation}
Let
\begin{eqnarray}
\label{super-gH-def}
g_H(\zeta):= -\frac \zeta 4 \sqrt{\zeta^2 -4} + \ln (\zeta + \sqrt{\zeta^2-4})+ \frac {\zeta^2}2 + \frac {\ell_H} 2\\
\label{super-ellH}
\ell_H:=-1-2\log 2
\end{eqnarray}
be the $g$--function for the Gaussian Unitary Ensemble. It admits an asymptotic expansion of the form
\begin{equation}
g_H(\zeta):=\ln \zeta + \sum_{\ell=1}^\infty \frac {c^{(0)}_\ell}{\zeta^{2\ell}}\ .
\end{equation}
We define the constants $c_j^{(H)}$ by the requirement
\begin{equation}
\label{super-cH-def}
g_H(\zeta-\zeta_0) - \ln \zeta + \sum_{j}^{K} \frac {c_j^{(H)}}{\zeta^j} = \mathcal O(\zeta^{-K-1})\ , \ \ \zeta\to\infty.
\end{equation}
It is easily verified that the $c_j^{(H)}$ are polynomials in $\zeta_0$.
\begin{prop}
\label{propdeltas}
The parameters $\vec \delta$ are uniquely determined as Puiseux series of $\sqrt{\kappa}$ by the requirement
\begin{eqnarray}
\mathcal P_2(z) = -\frac \kappa 2 (\zeta-\zeta_0)^2 + 2 \kappa \ln (\sqrt{\kappa} \zeta) + \kappa \sum_{j=1}^K \frac {\kappa^{-\frac j 2} \gamma_j}{\zeta^j} + \mathfrak b = -\frac \kappa 2 (\zeta-\zeta_0)^2 + 2 \kappa \ln (\sqrt{\kappa} \zeta) +\mathfrak b +2 \kappa \sum_{j=1}^K \frac {c^{(H)}_j}{\zeta^j}
\end{eqnarray}
Moreover we have
\begin{equation}
\zeta_0 = \mathcal O(\sqrt{\kappa}), \ \ \mathfrak b = \mathcal O(\kappa)\ ,\ \ \ \vec \delta = \mathcal O(\kappa^\frac 32).
\end{equation}
\end{prop}
\begin{proof}
Recall that $ \zeta_0$ depends on both $\kappa, \vec \delta$ analytically; although it is possible to give more detailed information about this dependence, it will not be necessary to the end of establishing the present proposition.
We need to solve the nonlinear system
\begin{eqnarray}
\kappa^{-\frac j2 }\gamma_j(\kappa,\vec \delta) = 2\kappa c_j^{(H)}( \zeta_0) \ ,\ \ \ j=1,\dots, K
\end{eqnarray}
for the unknowns $\vec \delta$. The local solvability of the system around $\vec \delta =\vec 0$ is guaranteed if we can guarantee the nonsingularity of the Jacobian matrix. But this system can be rewritten
\begin{equation}
\vec \gamma - 2\kappa^D \vec c^{(H)} =0\ ,\ \ D:= {\rm diag}\left(\frac 32 , 2,\frac 5 2, \dots \frac {K+2}2\right)\label{64}.
\end{equation}
Since $\vec c^{(H)}$ is analytic in $\vec \delta$ one promptly sees that the Jacobian is
\begin{equation}
J:= \frac {\partial \vec \gamma}{\partial \vec \delta} -2 \kappa^D \frac{\partial \vec c^{(H)}}{\partial \vec \delta}
\end{equation}
and hence $\det J = 1 + \mathcal O(\kappa^{\frac 32})$. This guarantees that there is a polydisk $|\kappa|<C_1,\ \|\vec \delta\|<C_2$ (for suitable constants) where the system admits a solution in Puiseux series (i.e. analytic in $\sqrt\kappa$). It is also clear from \eqref{64} that $\vec \delta = \mathcal O(\kappa ^\frac 3 2)$. Thus, $\zeta_0(\kappa, \delta(\kappa))$ is still of order $\mathcal O(\sqrt\kappa)$ and $\mathfrak b(\kappa, \delta(\kappa))$ is still $\mathcal O(\kappa)$, since all these depend analytically on $\vec \delta$, which is not of higher order than $\kappa$.
\end{proof}
For future reference and definiteness we collect the result of the above discussion in the theorem below
\begin{shaded}
\begin{thm}
\label{confchange}
There exists a conformal change of coordinate $\zeta(z;\kappa)$ of the form
\begin{equation}
\zeta(z;\kappa)= \frac {\rho(z;\kappa)}{\sqrt{\kappa}} = \frac 1{\sqrt{\kappa}} C (z-a^\star) (1+\mathcal O(z-a^\star))\ , C>0
\label{super-zeta}
\end{equation}
and a choice of $\vec \delta = \vec \delta(\kappa)$ for the deformed potential (\ref{deformV}) in Puiseux series of $\sqrt{\kappa}$
such that
\begin{equation}
\mathcal P_2(z; \kappa, \vec \delta(\kappa)) = - \frac \kappa 2 (\zeta - \zeta_0)^2 + 2 \kappa \ln (\sqrt \kappa \zeta) +\mathfrak b + 2 \kappa \sum_{j=1}^K \frac {c^{(H)}}{\zeta^j}\ .
\label{mainidzeta}
\end{equation}
The functions $\zeta_0(\kappa), \beta(\kappa), \vec \delta(\kappa)$ admit a Puiseux expansion and are of orders
\begin{equation}
\zeta_0 = \mathcal O(\sqrt\kappa)\ ,\ \ \beta = \mathcal O(\kappa)\ ,\ \ \vec \delta = \mathcal O(\kappa).
\end{equation}
The expressions $c^{(H)}_j$ are polynomials of degree $j$ in $\zeta_0$ determined by the formula \eqref{super-cH-def}.
\end{thm}
\end{shaded}
\subsection{Steepest descent analysis (supercritical case)}
We make the following change of variables from ${\bf Y}(z)$ to
${\bf W}(z)$:
\begin{eqnarray}
\label{super-Y-to-W}
{\bf W}(z;\kappa)&\hspace{-40pt}:=\left(\begin{array}{ccc}e^{\frac{n}{2}\ell_1} & 0 & 0 \\0 & e^{-\frac{n}{2}\ell_1} & 0 \\0 & 0 & e^{\frac{n}{2}
(\ell_1-2l_2+
2\eta)
}\end{array}\right)
{\bf Y}(z)\left(\begin{array}{ccc}e^{-\frac{n}{2}V} & 0 & 0 \\0 & e^{\frac{n}{2}V} & 0 \\0 & 0 & e^{\frac{n}{2}(V-2a z)}\end{array}\right)\times
\\ &\times
\left.\begin{cases}
{\bf I}, &z\in \Omega_1\cup\Omega_6
\\ \left(\begin{array}{ccc}1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{array}\right) , &z\in \Omega_2 \cup \Omega_5
\\ \left(\begin{array}{ccc}1 & 0 & 0 \\ -1 & 1 & -1 \\0 & 0 & 1\end{array}\right) , &z\in \Omega_3
\\ \left(\begin{array}{ccc}1 & 0 & 0 \\ 1 & 1 & -1 \\0 & 0 & 1\end{array}\right) , &z\in \Omega_4
\end{cases}\right\}
\left(\begin{array}{ccc}
e^{-\frac{n}{2}{\mathcal P}_1} & 0 & 0 \\
0 & e^{\frac{n}{2}{\mathcal P}_1 } & 0 \\
0 & 0 & e^{\frac{n}{2}(2\mathcal P_2 - \mathcal P_1 -
2\eta ) }
\end{array}\right) e ^{-\frac n2 \delta V(z)} \\
&
\delta V(z):= \kappa \ln (z-a^\star ) + \kappa \sum_{j=1}^K \frac {\delta_j}{2(z-a^\star)^j}
\\
&\eta:= \kappa \ln \kappa + \mathfrak b + \kappa \ell_H\ ,\ \ \ (\ell_H:= -1-2\ln2).
\label{defeta}
\end{eqnarray}
The constant (in $z$) $\eta$ is chosen to carefully balance other constants later on in the study of the local parametrix; we recall that $\mathfrak b = \mathcal O(\kappa)$ has appeared in Theorem \ref{confchange}, which is also the source of the $\kappa \ln \kappa $ term. The constant $\ell_H$ was introduced in \eqref{super-ellH} and it is the Robin's constant for the equilibrium problem associated to the quadratic potential.
See Figure \ref{fig_biglensSuper} for a visual on the different regions $\Omega_j$'s. The exact choice of the outer lenses
is given below in the proof of Lemma \ref{P-nolog-lemma}(d).
The inner lenses are chosen in the standard way for the $2\times 2$
Riemann-Hilbert problem for (non-multiple) orthogonal polynomials.
The new matrix ${\bf W}(z)$ satisfies a new Riemann Hilbert Problem which can be directly evinced from the one for ${\bf Y}$ and is of the form ${\bf W}_+(z)={\bf W}_-(z){\bf V^{(W)}}(z)$ with jumps
\begin{equation} \label{super-Wjumps}
{\bf V^{(W)}}(z) = \left\{\begin{array}{ll}
\begin{pmatrix} 1 & e^ {n{\mathcal P}_1(z)} & e^{n({\mathcal P}_2(z)
-\eta)
} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, & z\in \partial\Omega_1\cap\partial\Omega_6,
\\\begin{pmatrix} 1&0&0\\0&1&-e^{n({\mathcal P}_3(z)-\eta)}\\0&0&1\end{pmatrix}, & z\in (\partial\Omega_1\cap\partial\Omega_2) \cup (\partial\Omega_5\cap\partial\Omega_6),
\\\begin{pmatrix} 1 & 0 & 0 \\ e^{-n{\mathcal P}_1(z)} & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, & z\in (\partial\Omega_2\cap\partial\Omega_3)\cup(\partial\Omega_4\cap\partial\Omega_5),
\\\begin{pmatrix} 0&1&0\\-1&0&0\\0&0&1\end{pmatrix}, & z\in[\alpha,\beta],
\\\begin{pmatrix} 1& e^{n{\mathcal P}_1(z)} &0\\0&1&0\\0&0&1\end{pmatrix}, & z\in\partial\Omega_2\cap\partial\Omega_5,
\end{array}\right.
\end{equation}
and the asymptotic conditions
\begin{eqnarray}
\label{super-W-infinity}
&& {\bf W}(z)={\bf I}+{\cal O}\left(\displaystyle\frac{1}{z}\right), \quad z\to\infty\\
\label{super-Wzero}
&& \begin{split}
{\bf W}(z) = \text{(analytic)}\begin{pmatrix}
{ {\rm e}^{- n \delta V(z)}}
& 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 &
{ {\rm e}^{n \delta V(z)}}
\end{pmatrix}& \\
\text{as }z\to a^{\star}.&
\end{split}
\end{eqnarray}
The orientation of the contours is given in Figure \ref{fig_biglensSuper}.
Here we have used the factorization
\begin{equation}
\begin{pmatrix} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 1 \end{pmatrix}.
\end{equation}
\begin{figure}
\setlength{\unitlength}{2.7pt}
\begin{center}
\begin{picture}(100,50)(-50,-25)
\put(-10,0){\circle*{1}}
\put(-11.5,2){$\alpha$}
\put(10,0){\circle*{1}}
\put(9.5,2){$\beta$}
\put(28,0){\circle*{1}}
\put(27,2){$a^{\star}$}
\put(-50,0){\line(1,0){90}}
\put(-42,14){$\Omega_1$}
\put(-22,8.5){$\Omega_2$}
\put(-1.5,3){$\Omega_3$}
\put(-1.5,-4.5){$\Omega_4$}
\put(-22,-9.5){$\Omega_5$}
\put(-42,-15){$\Omega_6$}
\qbezier(-10,0)(0,16),(10,0)
\qbezier(-10,0)(0,-16),(10,0)
\qbezier(15,0)(15,25)(-10,25)
\qbezier(15,0)(15,-25)(-10,-25)
\qbezier(-35,0)(-35,25)(-10,25)
\qbezier(-35,0)(-35,-25)(-10,-25)
\thicklines
\put(-42,0){\vector(1,0){1}}
\put(-22,0){\vector(1,0){1}}
\put(0,0){\vector(1,0){1}}
\put(22,0){\vector(1,0){1}}
\put(35,0){\vector(1,0){1}}
\put(-10,-25){\vector(1,0){1}}
\put(-10,25){\vector(-1,0){1}}
\put(0,8){\vector(1,0){1}}
\put(0,-8){\vector(1,0){1}}
\end{picture}
\end{center}
\caption{The oriented jump contour $\Gamma$ for {\bf W}(z) and the regions $\Omega_i$ in the supercritical case.\label{fig_biglensSuper}}
\end{figure}
\subsubsection{The outer parametrix}
We will show below in Section \ref{super-error} that the jump matrices for ${\bf W}(z)$ decay uniformly to constant jump matrices
as $n\to\infty$ outside of small fixed neighborhoods of $\alpha$, $\beta$,
and $a^{\star}$. These limiting jump matrices are the identity except on the
band $[\alpha,\beta]$. We therefore define the outer parametrix
${\bf \Psi}(z)$ to be the solution to the following Riemann-Hilbert problem:
\begin{equation} \label{Vpsi}
{\bf \Psi}_+(z) = {\bf \Psi}_-(z)\begin{pmatrix} 0 & 1 & 0 \\ -1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix} \text{ for } z\in(\alpha,\beta)\ , \qquad {\bf \Psi}(z)={\bf I}+{\cal O}\left(\frac{1}{z}\right).
\end{equation}
It is well known that the solution to this Riemann-Hilbert problem is
\begin{equation}
{\bf \Psi}(z) = {\bf U}^{-1}\begin{pmatrix} \displaystyle \left(\frac{z-\beta}{z-\alpha}\right)^{-1/4} & 0 & 0 \\ 0 & \displaystyle \left(\frac{z-\beta}{z-\alpha}\right)^{1/4} & 0 \\ 0 & 0 & 1 \end{pmatrix} {\bf U}, \quad {\bf U}:=\begin{pmatrix} \frac{1}{2} & \frac{i}{2} & 0 \\ -\frac{1}{2} & \frac{i}{2} & 0 \\ 0 & 0 & 1 \end{pmatrix},
\end{equation}
where
\begin{equation}
\lim_{z\to\infty}\left(\frac{z-\beta}{z-\alpha}\right)^{1/4} = 1
\end{equation}
and this function is cut along $[\alpha,\beta]$.
\subsubsection{The local parametrix near \texorpdfstring{$\boldsymbol{a^{\star}}$}{astar} }
Special attention is needed near the point $z=a^{\star}$ as the jump
matrices do not decay
uniformly to the identity near this point.
According to the definition of $a^\star$ as the point of maximum for $P_2$ and given that $\mathcal P_2$ is a deformation of $P_2$ we have
\begin{equation}
\label{super-def-of-f}
{\mathcal P}_2(z;\kappa) = - \overbrace{ \frac{C}{2}(z-a^{\star})^2(1+ \mathcal O(z-a^\star)) +\mathcal O(\delta) }^{:=f(z;\kappa)} + 2\kappa\log(z-a^{\star}) + \sum_{j=1}^{2k}\frac{\delta_j(\kappa)}{(z-a^{\star})^j}
\end{equation}
where $C>0$ and the deformation $\mathcal O(\delta)$ is some analytic function of $z$ of the indicated order in $\kappa$.
Let $\mathbb{D}_{a^{\star}}$
be a fixed-size
circular disk centered at $a^{\star}$ chosen small enough so that
\begin{equation}
\label{condition-on-Da*}
\left|\Re\left[
\frac {f(z;\kappa)}2\right]\right|<|\Re\, P_1(z)|
\end{equation}
(recall that $\Re P_1>0$ outside of the support of the equilibrium measure)
inside the disk, and such that the disk does not intersect the outer lenses.
Orient $\partial\mathbb{D}_{a^{\star}}$ clockwise.
We now apply Theorem \ref{confchange}:
let the $k$ constants $c_j^{(H)}$, $j=1,\dots,k$ be specified by
\eqref{super-cH-def}:
in the local scaling coordinate $\zeta$ we have
\begin{equation}
\label{super-P2-zeta}
n\mathcal P_2 = -\frac r 2 (\zeta- \zeta_0)^2 + 2r \ln \zeta + 2 r \sum_{j=1}^K \frac {c^{(H)}} {\zeta^j} + r \ln \kappa + n \mathfrak b.
\end{equation}
where the scaling coordinate $\zeta$ has the following behavior on the boundary of the disk $\mathbb D_{a^\star}$
\begin{equation}
\label{super-zeta-order}
\zeta = \mathcal{O}(n^{(1-\gamma)/2}) \quad \text{when } z\in\partial\mathbb{D}_{a^{\star}}.
\end{equation}
We also recall (Theorem \ref{confchange}) that $\mathfrak b =\mathcal O(\kappa)$ and hence $n\mathfrak b = \mathcal O(r)$.
This suggests the following definition for the model Riemann Hilbert Problem of the local parametrix.
\begin{defn} The local parametrix within the disk $\mathbb D_{a^\star}$ shall be the unique solution ${\bf R}(z)$ to the following model Riemann-Hilbert:
\begin{eqnarray}
\label{R-RHP}
\begin{cases}
& {\bf R}_+(\zeta) = {\bf R}_-(\zeta)\begin{pmatrix} 1 & 0 & \zeta^{2r}\exp\left(-\frac{r}{2}(\zeta-\zeta_0)^2
+
r\ell_H+2r\sum_{j=1}^K\frac{c_j^{(H)}}{\zeta^{j}}\right) \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}
\\ & \phantom{{\bf R}_+(\zeta)} = {\bf R}_-(\zeta)\begin{pmatrix} 1 & 0 & e^{n(\mathcal P_2 - \eta)} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \quad \zeta\in\mathbb{R},
\\ &{\bf R}(\zeta)={\bf I}+{\cal O}\left(\displaystyle\frac{1}{\zeta}\right) \text{ as } \zeta\to\infty,
\\&{\bf R}(\zeta)=(\mbox{analytic})\begin{pmatrix} \zeta^{-r}\exp\left(-r\sum_{j=1}^K \frac{c_j^{(H)}}{\zeta^{j }}\right) & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & \zeta^r\exp\left(r\sum_{j=1}^K \frac{c_j^{(H)}}{\zeta^{j }}\right) \end{pmatrix} \text{ as } \zeta\to 0.
\end{cases}\\
\eta := \kappa\ln \kappa + \mathfrak b - \kappa\ell_H
\end{eqnarray}
\end{defn}
The behavior at $\zeta=0$ ($z=a^\star$) is dictated by \eqref{super-Wzero}.
We point out that the problem is essentially $2\times 2$; moreover it will be shown below that it is a slight modification of the Fokas-Its-Kitaev Riemann-Hilbert
problem for Hermite orthogonal polynomials (see \cite{Fokas:1992} and
\cite{Deift:1999b}, Section 3).
\begin{prop}
\label{propHparametrix}
Let the rescaled Hermite polynomials
$H^{(r)}_i(\zeta)$ be the family of monic polynomials satisfying the
orthogonality condition
\begin{equation}
\label{def-of-kr}
\int_{-\infty}^\infty H^{(r)}_i(\zeta)H^{(r)}_j(\zeta) e^{-\frac r2 \zeta^2}d\zeta=r^{j-\frac 1 2} j! \sqrt{2\pi}\delta_{ij} =k^{(r)}_j\delta_{ij},
\end{equation}
where the $k^{(r)}_i$ are normalization constants.
Then the solution to \eqref{R-RHP} is
\begin{equation}
\label{super-R}
{\bf R}(\zeta) = \exp\left(-\frac{r}{2}
\ell_H
{\bf\Lambda}_{13}\right) \,{\bf H}_{13}(\zeta) \,\zeta^{-r{\bf\Lambda}_{13}}\,\exp\left(\left(\frac{r}{2}
\ell_H
- r\sum_{j=1}^K \frac{c_j^{(H)}}{\zeta^{j}}\right){\bf\Lambda}_{13}\right),
\end{equation}
where
\begin{equation}
\label{super-H}
{\bf H}_{13}(\zeta):=\begin{pmatrix} H^{(r)}_r(\zeta-\zeta_0) & 0 &\displaystyle \frac{1}{2\pi i}\int_{-\infty}^\infty \frac{H^{(r)}_r(s-\zeta_0) e^{-\frac{r}{2}s^2}}{s-\zeta}ds \\ 0 & 1& 0 \\ \displaystyle\frac{2\pi i}{-k^{(r)}_{r-1}}H^{(r)}_{r-1}(\zeta-\zeta_0)& 0 & \displaystyle \frac{-1}{k^{(r)}_{r-1}}\int_{-\infty}^\infty \frac{H^{(r)}_{r-1}(s-\zeta_0) e^{-\frac{r}{2}s^2}}{s-\zeta}ds \end{pmatrix}
\end{equation}
and
\begin{equation}
{\bf\Lambda}_{13}:=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & -1 \end{pmatrix}.
\end{equation}
\end{prop}
The proof is a direct manipulation and it is left to the reader.\\
Now the well known asymptotics of Hermite polynomials can be written as a {\bf joint} asymptotic expansion for $\zeta $ {\bf and} $r$ large as follows
\begin{eqnarray}
{\bf H}_{13}(\zeta)=e^{\frac{r}{2}\ell_H {\bf\Lambda}_{13}} \left({\bf I}+{\cal O}\left(\frac{1}{r(|\zeta|+1)}\right)\right)
\mathbf U^{-1}
\left(\begin{array}{ccc}
\sqrt[4]{\frac {\zeta-\zeta_0 -2} {\zeta-\zeta_0 +2}} &0&0 \\0&1&0\\
0&0&\sqrt[4]{\frac {\zeta-\zeta_0 +2} {\zeta-\zeta_0 -2}} \end{array}\right)
\mathbf U
e^{r g_H(\zeta-\zeta_0){\bf\Lambda}_{13}-\frac{r}{2}\ell_H{\bf\Lambda}_{13}}\ ,
\label{Hasympt}
\\
\mathbf U:= \frac 1 {\sqrt{2}} \begin{bmatrix}
1 & 0 & i\\
0 & 1 & 0\\
-1 & 0 & i
\end{bmatrix}
\end{eqnarray}
and $g_H(\zeta)$ is given by \eqref{super-gH-def}.
For large $\zeta$, we have the expansion of $g_H$ as in \eqref{super-cH-def}.
The error term of ${\cal O}(1/r(|\zeta|+1))$ in \eqref{Hasympt}
is from the standard Airy parametrix used in the Riemann-Hilbert problem
for Hermite orthogonal polynomials at
$\zeta=\pm 2$. For more details on this calculation see equation (7.72) in
\cite{Deift:1998-book} or equation (4.16) and Appendix B in \cite{Deift:1999b},
noting that the variable $\zeta$ is rescaled by a constant factor.
Then for large $\zeta$ (such as on $\partial\mathbb{D}_{a^{\star}}$) we can estimate
\begin{equation}
\begin{split}
\label{super-R-error}
{\bf R}(\zeta) & = \left({\bf I}+{\cal O}\left(\frac{1}{r\zeta}\right)\right) \left({\bf I}+\mathcal{O}\left(\frac{1}{\zeta}\right) \right) \exp\left[r \left( g_H(\zeta-\zeta_0) -\log\zeta - \sum_{j=1}^K \frac{c_j^{(H)}}{\zeta^{j}} \right) {\bf\Lambda}_{13} \right]\\
& = {\bf I} + \mathcal{O}\left(\frac{1}{\zeta}\right) + \mathcal{O}\left(\frac{r}{\zeta^{K+1}}\right).
\end{split}
\end{equation}
\subsection{Error analysis in the supercritical case}
\label{super-error}
Let $\mathbb{D}_\alpha$ and $\mathbb{D}_\beta$ be
small, fixed, closed disks centered at $\alpha$ and $\beta$
that are bounded away from the outer lenses.
Orient the boundaries $\partial\mathbb{D}_\alpha$ and
$\partial\mathbb{D}_{\beta}$
clockwise. For $z\in\mathbb{D}_\alpha$,
let ${\bf P}_{\mbox{Ai}}^{(\alpha)}(z)$ be the {\it Airy parametrix}
satisfying
\begin{itemize}
\item ${\bf P}_{\mbox{Ai}}^{(\alpha)}(z)$ has the same jumps as
${\bf W}(z)$ for $z\in\mathbb{D}_\alpha$,
\item $ \displaystyle {\bf P}_{\mbox{Ai}}^{(\alpha)}(z){\bf \Psi}(z)^{-1} = {\bf I} + \mathcal{O}\left(\frac{1}{n}\right)$ for $z\in\mathbb{D}_\alpha$.
\end{itemize}
The construction of the Airy parametrix is standard, involving
Airy functions and a local change of variables. See \cite{Bleher:2004b}
Section 7, for example, for an Airy parametrix for a $3\times3$ Riemann-Hilbert
problem. The Airy parametrix ${\bf P}_{\mbox{Ai}}^{(\beta)}(z)$ is
defined analogously for $z\in\mathbb{D}_\beta$.
We now define the global parametrix ${\bf \Psi}^\infty(z)$ by
\begin{equation}
{\bf \Psi}^\infty(z):=
\begin{cases}
{\bf \Psi}(z), & z\notin \mathbb{D}_\alpha \cup \mathbb{D}_\beta \cup \mathbb{D}_{a^{\star}}, \\
{\bf \Psi}(z) {\bf R}(\zeta(z)), & z\in\mathbb{D}_{a^{\star}}, \\
{\bf P}_{\mbox{Ai}}^{(\alpha)}(z), & z\in\mathbb{D}_\alpha \\
{\bf P}_{\mbox{Ai}}^{(\beta)}(z), & z\in\mathbb{D}_\beta.
\end{cases}
\end{equation}
The \emph{error matrix} ${\bf E}(z)$ is given by
\begin{equation}
\label{super-error-matrix}
{\bf E}(z) := {\bf W}(z){\bf \Psi}^\infty(z)^{-1}.
\end{equation}
Let $\Gamma$ denote the contours given by the boundaries of the regions $\Omega_j$ in Figure \ref{fig_biglensSuper}.
The error matrix satisfies a Riemann-Hilbert problem with the following
jumps:
$\bullet$ For $z$ outside the disks $\mathbb{D}_\alpha$, $\mathbb{D}_\beta$,
and $\mathbb{D}_{a^{\star}}$, and excluding the band $[\alpha,\beta]$:
\begin{equation}
\label{super-VE-outside}
{\bf V^{(E)}}(z) =
{\bf \Psi}(z) {\bf V^{(W)}}(z) {\bf \Psi}(z)^{-1}, \quad z\in \Gamma\cap\left( \mathbb{D}_{\alpha} \cup \mathbb{D}_{\beta} \cup \mathbb{D}_{a^{\star}} \right)^c\cap[\alpha,\beta]^c,
\end{equation}
where ${\bf V^{(W)}}(z)$ is given by (\ref{super-Wjumps}).
$\bullet$ For $z$ on the boundaries of the disks $\partial\mathbb{D}_\alpha$,
$\partial\mathbb{D}_\beta$, and $\partial\mathbb{D}_{a^{\star}}$:
\begin{equation}
\label{super-VE-boundaries}
{\bf V^{(E)}}(z) =
\begin{cases}
{\bf \Psi}(z){\bf R}(\zeta) {\bf \Psi}(z)^{-1}, & z\in\partial\mathbb{D}_{a^{\star}}, \\
{\bf P}_{\mbox{Ai}}^{(\alpha)}(z) {\bf \Psi}(z)^{-1}, & z\in\partial\mathbb{D}_\alpha, \\
{\bf P}_{\mbox{Ai}}^{(\beta)}(z) {\bf \Psi}(z)^{-1}, & z\in\partial\mathbb{D}_\beta.
\end{cases}
\end{equation}
$\bullet$ For $z$ inside the disk $\mathbb{D}_{a^{\star}}$:
\begin{equation}
\label{super-VE-a*}
{\bf V^{(E)}}(z) = {\bf \Psi}(z) {\bf R}_-(\zeta)
\begin{pmatrix}
1 & e^{n\mathcal{P}_1} & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{pmatrix} {\bf R}_-(\zeta)^{-1} {\bf \Psi}(z)^{-1},
\;\; z \in \Gamma\cap\mathbb{D}_{a^{\star}}.
\end{equation}
$\bullet$ Furthermore, ${\bf V^{(E)}}(z)={\bf I}$ on the contours
$$[\alpha,\beta]\cap(\mathbb{D}_\alpha\cup\mathbb{D}_\beta)^c, \quad \Gamma\cap\mathbb{D}_\alpha, \quad \text{and} \quad \Gamma\cap\mathbb{D}_\beta.$$
The jump contours $\Gamma^{\bf(E)}$ for ${\bf E}(z)$ are shown in Figure
\ref{super-error-fig}.
\begin{figure}
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\begin{center}
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\put(-10,0){\circle{6}}
\put(-12.2,-1){$\mathbb{D}_\alpha$}
\put(10,0){\circle{6}}
\put(7.9,-1){$\mathbb{D}_\beta$}
\put(28,0){\circle{6}}
\put(20,-6){$\mathbb{D}_{a^{\star}}$}
\put(23,-4){\line(2,1){5}}
\put(-50,0){\line(1,0){37}}
\put(13,0){\line(1,0){30}}
\qbezier(-9,3)(0,13),(9,3)
\qbezier(-9,-3)(0,-13),(9,-3)
\qbezier(15,0)(15,25)(-10,25)
\qbezier(15,0)(15,-25)(-10,-25)
\qbezier(-35,0)(-35,25)(-10,25)
\qbezier(-35,0)(-35,-25)(-10,-25)
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\put(-22,0){\vector(1,0){1}}
\put(20,0){\vector(1,0){1}}
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\put(0,8){\vector(1,0){1}}
\put(0,-8){\vector(1,0){1}}
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\put(10.2,2.8){\vector(1,0){1}}
\put(28,2.8){\vector(1,0){1}}
\end{picture}
\end{center}
\caption{The jump contours $\Gamma^{\bf(E)}$ for the Riemann-Hilbert problem for ${\bf E}(z)$ in the supercritical case.\label{super-error-fig}}
\end{figure}
We now show that all of the jump matrices in
\eqref{super-VE-outside}-\eqref{super-VE-a*}
are uniformly close to the identity as $n\to\infty$.
For the error bounds it will be convenient to split $\Gamma^{\bf(E)}$
into a compact component $\Gamma_C^{\bf(E)}$ and a
noncompact component $\Gamma_N^{\bf(E)}$:
\begin{equation}
\begin{split}
\Gamma_C^{\bf(E)}:=&\partial\mathbb{D}_\alpha\cup\partial\mathbb{D}_\beta\cup\partial\mathbb{D}_{a^{\star}}\cup(\Gamma\cap\mathbb{D}_{a^{\star}}),\\
\Gamma_N^{\bf(E)}:=&\Gamma^{\bf(E)}\backslash\Gamma_C^{\bf(E)}.
\end{split}
\end{equation}
We now gather the results we will need on the functions $P_1(z)$, $P_2(z)$,
and $P_3(z)$ defined by \eqref{P1-nolog}--\eqref{P3-nolog}.
\begin{lemma}
\label{P-nolog-lemma}
In the supercritical regime, the inner and outer lenses can be chosen so that:
\begin{itemize}
\item[(a)] On the inner lenses outside of the disks around $\alpha$ and
$\beta$: The real part of $P_1(z)$ is positive and bounded away from zero
for
$z\in[(\partial\Omega_2\cap\partial\Omega_3)\cup(\partial\Omega_4\cap\partial\Omega_5)]\cap(\mathbb{D}_\alpha\cup\mathbb{D}_\beta)^c$.
\item[(b)] On the real axis outside of $[\alpha,\beta]$ and the disks around
$\alpha$ and $\beta$: The real part of $P_1(z)$ is negative and bounded away
from zero for
$z\in[(\partial\Omega_1\cap\partial\Omega_6)\cup(\partial\Omega_2\cap\partial\Omega_5)]\cap(\mathbb{D}_\alpha\cup\mathbb{D}_\beta)^c$.
\item[(c)] On the real axis outside of the band $[\alpha,\beta]$ and a fixed
distance away from $\alpha$, $\beta$, and $a^{\star}$: The real part of $P_2(z)$
is negative and bounded away from zero
for $z\in[(\partial\Omega_2\cap\partial\Omega_5)\cap(\mathbb{D}_\alpha\cup\mathbb{D}_\beta\cup\mathbb{D}_{a^{\star}})^c]\cup(\partial\Omega_1\cap\partial\Omega_6)$.
\item[(d)] On the outer lenses: For $\kappa$ sufficiently small, the real
part of $P_3(z)$ is negative and bounded away from zero for
$z\in(\partial\Omega_1\cap\partial\Omega_2)\cup(\partial\Omega_5\cap\partial\Omega_6).$
\end{itemize}
\end{lemma}
\begin{proof}
Statements (a) and (b) follow from the analysis of the Riemann-Hilbert
problem for the standard (not multiple) orthogonal polynomials
(see, for instance, \cite{Deift:1998-book}). Statement (c) follows from the
definitions of the supercritical region and $a^{\star}$.
For (d), we begin
by choosing the outer lenses used to define ${\bf W}(z)$. Fix
$\kappa=0$. Note that
\begin{equation}
P_3(\beta) = -P_1(\beta)+P_2(\beta) = P_2(\beta) < P_2(a^{\star}) = 0.
\end{equation}
The second equality uses the fact that $ {\Re}\, P_1$ is zero on
$[\alpha,\beta ]$ and $P_1$ is real for $x>\beta$;
the inequality follows since $a^\star$ is the location of the global maximum
of $P_2(z)$; and the final equality is true by the choice of the Lagrange
multiplier $l_2$. Thus, there is a fixed radius neighborhood of $\beta$
in which ${\rm Re}P_3(z)<0$ for real $z$. We choose the outer lenses to
be a circle centered below $\alpha$ whose right-most endpoint passes through
the real axis at some point on $(\beta,a^\star)$. We choose the circle big enough such that $\Re P_2$ is negative on the real axis to the left of the circle. This is always possible due to Assumption \ref{assumptionAV} (iv).
We now show that the outer lenses are descent lines of
${\rm Re}\,P_3(z)$. Clearly the real part of $az$ decreases as we move to
the left along the lenses. Note that
${\rm Re}\,g(z) = \int_\alpha^\beta\log|z-s|\rho_{\text{min}}(s)ds$ where
$\rho_{\text{min}}(s)$ is the associated equilibrium measure. Now
for any $s\in(\alpha,\beta)$, $\log|z-s|$ is increasing as $z$ moves to the
left along the lens (one can see this clearly by drawing a circle that is
centered at $s$ and is tangent to the outer lenses at the right-most point).
Since $\rho_{\text{min}}(s)$ is positive for
$s\in(\alpha,\beta)$, ${\rm Re}\,g(z)$ increases as $z$ moves to the left
along the lenses. This shows (d) for $\kappa=0$. Since
$\widetilde \alpha(\kappa)=\alpha+\mathcal{O}(\kappa)$ and
$\widetilde \beta=\beta+\mathcal{O}(\kappa)$, (d) also holds for
$\kappa$ sufficiently small.
\end{proof}
We now present the results we will need for the modified functions
$\mathcal{P}_1(z;\kappa)$, $\mathcal{P}_2(z;\kappa)$, and
$\mathcal{P}_3(z;\kappa)$ defined by
\eqref{super-mathcalP1}--\eqref{super-mathcalP3}.
\begin{lemma}
\label{super-Pmathcal-lemma}
For $\kappa$ sufficiently small (refer to Figure \ref{fig_biglensSuper} and Figure \ref{super-error-fig}):
\begin{itemize}
\item[(a)] On the inner lenses outside of
the disks around $\alpha$ and $\beta$: The real part of
${\mathcal P}_1(z;\kappa)$
is positive and bounded away from zero for
$z\in[(\partial\Omega_2\cap\partial\Omega_3)\cup(\partial\Omega_4\cap\partial\Omega_5)]\cap(\mathbb{D}_\alpha\cup\mathbb{D}_\beta)^c$.
\item[(b)] On the real axis outside of $[\alpha,\beta]$ and a fixed distance
away from $\alpha$, $\beta$, and $a^{\star}$: The real part of
${\mathcal P}_1(z;\kappa)$ is negative and bounded away from zero for
$z\in[(\partial\Omega_1\cap\partial\Omega_6)\cup(\partial\Omega_2\cap\partial\Omega_5)]\cap(\mathbb{D}_\alpha\cup\mathbb{D}_\beta\cup\mathbb{D}_{a^{\star}})^c$.
\item[(c)] On the real axis outside of the outer
lenses and a fixed distance away from $a^{\star}$: The real part of
$\mathcal{P}_2(z;\kappa)$ is negative and bounded
away from zero for
$z\in(\partial\Omega_1\cap\partial\Omega_6)\cap(\mathbb{D}_{a^{\star}})^c$.
\item[(d)] On the outer lenses: The real part of
$\mathcal{P}_3(z;\kappa)$ is negative and bounded away from zero for
$z\in(\partial\Omega_1\cap\partial\Omega_2)\cup(\partial\Omega_5\cap\partial\Omega_6).$
\item[(e)] For real $z$ inside $\mathbb{D}_{a^{\star}}$: Let $g_H(\zeta)$ be
defined by \eqref{super-gH-def}. Then the real part of
$$\mathcal{P}_1(z;\kappa) +\kappa g_H(\zeta) - \kappa\log\zeta - \kappa\sum_{j=1}^K \frac{c_j^{(H)}}{\zeta^{j}}$$
is negative and bounded away from zero for
$z\in\Gamma\cap\mathbb{D}_{a^{\star}}$.
\end{itemize}
\end{lemma}
\begin{proof}
Parts (a) through (d) follow from Lemma \ref{P-nolog-lemma}(a)--(d)
together
with the convergence $\mathfrak g \to g$ guaranteed in Proposition \ref{propdeform},
the boundedness
of $\log(z-a^{\star})$ and $(z-a^{\star})^{-j}$, $j=1,\dots,K $ outside of
$\mathbb{D}_{a^{\star}}$,
and $\delta_j(0)=0$.
For part (e), first note that comparing the two expressions
\eqref{super-mathcalP2} and \eqref{super-P2-zeta} for
$\mathcal{P}_2(z;\kappa)$ gives
\begin{equation}
\label{rearranging-P2}
\kappa\log(z-a^{\star}) - \kappa\log\zeta + \sum_{j=1}^{K }\frac{\delta_j}{2(z-a^{\star})^j} - \kappa \sum_{j=1}^{K } \frac{c_j^{(H)}}{\zeta^{j}} =
\frac 1 2 f(z;\kappa) -\frac 12 \mathfrak b
-\kappa\ln \sqrt\kappa
-\frac{\kappa}{4}(\zeta-\zeta_0)^2 .
\end{equation}
By the choice of $\mathbb{D}_{a^{\star}}$ (see \eqref{condition-on-Da*}), for $\kappa$
sufficiently small we have
\begin{equation}
\left|\Re\left[
\frac 1 2 f(z;\kappa) - \frac {\mathfrak b}2
- \kappa\ln \sqrt\kappa
-\frac{\kappa}{4}(\zeta-\zeta_0)^2\right]\right| < |\Re \, P_1(z)| \text{ for } z\in\mathbb{D}_{a^{\star}}.
\end{equation}
Write
\begin{equation}
\begin{split}
& \hspace{-.5in} \mathcal{P}_1(z;\kappa) +\kappa g_H(\zeta) - \kappa\log\zeta - \kappa\sum_{j=1}^k\frac{c_j^{(H)}}{\zeta^{2j}} \\
& = \underbrace{-V + 2(1-\kappa)\mathfrak{g} + \ell_1 + \kappa g_H}_{=P_1 + \mathcal{O}(\kappa)} + \underbrace{\kappa\log(z-a^{\star}) - \kappa\log\zeta + \sum_{j=1}^{
K
}\frac{\delta_j}{2(z-a^{\star})^j} - \kappa \sum_{j=1}^{
K
}\frac{c_j^{(H)}}{\zeta^{j
}}}_{\text{Has real part bounded above by }|\Re\, P_1|}.
\end{split}
\end{equation}
For $\kappa$ small, by
the convergence $\mathfrak g\to g$ in Proposition \ref{propdeform},
the first group of terms on the
left-hand side,
$-V + 2(1-\kappa)\mathfrak{g} + \ell_1 + \kappa g_H$, is within $\mathcal{O}(\kappa)$
of $P_1$, which has strictly negative real part for $z\in\mathbb{D}_{a^{\star}}$. Since
the real part of the second group of terms is strictly less than the magnitude of
the real part of $P_1$, part (e) follows.
\end{proof}
We are now in a position to bound the jumps ${\bf V^{(E)}}(z)$ of the
error problem.
\begin{lemma}
\label{super-VE-lemma}
In the supercritical regime, for large $n$,
\begin{itemize}
\item[(a)] Outside the disks $\mathbb{D}_\alpha$, $\mathbb{D}_\beta$, and
$\mathbb{D}_{a^{\star}}$: There is a constant $c>0$ such that
$${\bf V^{(E)}}(z;\kappa) = {\bf I} + \mathcal{O}(e^{-cn}), \quad z\in\Gamma_N^{\bf(E)}.$$
\item[(b)] On the boundary of $\mathbb{D}_{a^{\star}}$:
$$
{\bf V^{(E)}}(z;\kappa) = {\bf I} + \mathcal{O}\left(\displaystyle n^{-(1-\gamma)/2}\right) + \mathcal{O}\left(\displaystyle n^{
\gamma - \frac {1-\gamma}2 (K+1)} \right), \quad z\in\partial\mathbb{D}_{a^{\star}}.
$$
\item[(c)] On the boundaries of $\mathbb{D}_\alpha$ and $\mathbb{D}_\beta$:
$${\bf V^{(E)}}(z;\kappa) = {\bf I} + \mathcal{O}\left(n^{-1}\right), \quad z\in\partial\mathbb{D}_\alpha\cup\partial\mathbb{D}_\beta.$$
\item[(d)] Inside $\mathbb{D}_{a^{\star}}$:
$${\bf V^{(E)}}(z) = {\bf I} + \mathcal{O}(e^{-cn}), \quad z\in\Gamma\cap\mathbb{D}_{a^{\star}}.$$
\end{itemize}
\end{lemma}
\begin{proof}
Part (a) follows from \eqref{super-VE-outside}, Lemma
\ref{super-Pmathcal-lemma}(a)--(d), and the boundedness of ${\bf \Psi}(z)$.
Part (b) follows from \eqref{super-zeta-order}, \eqref{super-R-error}, and the
boundedness of ${\bf \Psi}(z)$.
Part (c) comes from the construction of the parametrices
${\bf P}_\text{\bf Ai}^{(\alpha)}(z)$ and
${\bf P}_\text{\bf Ai}^{(\beta)}(z)$ (see, for instance, \cite{Deift:1999b}).
For part (d) we consider the jumps
\eqref{super-VE-a*} inside the disk $\mathbb{D}_{a^{\star}}$.
Looking at the formula
\eqref{super-R} for ${\bf R}(\zeta)$, it appears there may be a problem at
$\zeta=0$. However, note that
\begin{equation}
\label{VE-bound-Gamma-a*}
\begin{split}
{\bf V^{(E)}}(z;\kappa) = & \Psi(z) e^{-\frac{r}{2}\ell_H{\bf\Lambda}_{13}} {\bf H}_{13-}(\zeta) e^{-r(g_H(\zeta)-\frac{\ell_H}{2}){\bf\Lambda}_{13}} \begin{pmatrix} 1 & (*)_{12} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \times \\
& \times e^{r(g_H(\zeta)-\frac{\ell_H}{2}){\bf\Lambda}_{13}}{\bf H}_{13-}(\zeta)^{-1}e^{\frac{r}{2}\ell_H{\bf\Lambda}_{13}}\Psi(z)^{-1}, \quad z\in\Gamma\cap\mathbb{D}_{a^{\star}},
\end{split}
\end{equation}
wherein
\begin{equation}
(*)_{12} = \exp\left(n\mathcal{P}_1(z;\kappa)+rg_H(\zeta)-r\log\zeta - r\sum_{j=1}^{K}
\frac{c_j^{(H)}}{\zeta^{j}}\right).
\end{equation}
Now \eqref{VE-bound-Gamma-a*} together with Lemma
\ref{super-Pmathcal-lemma}(e) and the boundedness of ${\bf \Psi}(z)$ inside
$\mathbb{D}_{a^{\star}}$ establishes (d).
\end{proof}
We can now show that the error matrix ${\bf E}(z)$ is uniformly close
to the identity.
\begin{lemma}
\label{super-E}
In the supercritical regime, for $n$ large,
$${\bf E}(z) = {\bf I}+\mathcal{O}\left(n^{-(1-\gamma)/2}\right)$$
uniformly in $z$.
\end{lemma}
\begin{proof}
From Lemma \ref{super-VE-lemma}(b)--(d),
\begin{equation}
\label{super-VE-compacta}
{\bf V^{(E)}}(z) = {\bf I} + \mathcal{O}\left(n^{-(1-\gamma)/2}\right) + \mathcal{O}\left(n^{
\gamma -\frac {1-\gamma}2 (K+1)
}\right), \quad z\in\Gamma_C^{\bf(E)}.
\end{equation}
The first error term always dominates or matches the second term if the
nonnegative integer $K$ is chosen so \eqref{super-k-def} is satisfied.
Then, for $n$ sufficiently large there
exists a constant $c$ such that
\begin{equation}
||{\bf V^{(E)}}-{\bf I}||_{L^2\left(\Gamma_C^{\bf(E)}\right)} + ||{\bf V^{(E)}}-{\bf I}||_{L^\infty\left(\Gamma_C^{\bf(E)}\right)} \leq c n^{-(1-\gamma)/2}.
\end{equation}
Also, from Lemma \ref{super-VE-lemma}(a), for $n$ sufficiently large there is
a constant $c$ such that
\begin{equation}
||{\bf V^{(E)}}-{\bf I}||_{L^2\left(\Gamma_N^{\bf(E)}\right)} + ||{\bf V^{(E)}}-{\bf I}||_{L^\infty\left(\Gamma_N^{\bf(E)}\right)} \leq c e^{-cn},
\end{equation}
The result follows by a standard technique that consists of writing the
solution to the Riemann-Hilbert
problem in terms of a Neumann series involving ${\bf V^{(E)}}-{\bf I}$
(see, for instance, \cite{Deift:1999b} Section 7.2 or
\cite{Ercolani:2003} Section 3.5).
\end{proof}
\subsection{The supercritical kernel and proof of Theorem \ref{theorem-super-kernel}}
\label{super-kernel-proof}
\begin{proof}[Proof of Theorem \ref{theorem-super-kernel}]
Recall that the kernel is defined by (\ref{mop-kernel}):
\begin{equation}
\label{super-Kn}
K_n(x, y) = \frac{e^{-\frac{1}{2} n ( V(x) + V(y) )}}{2\pi i(x-y)} \left( \left[ {\bf Y}(y)^{-1} {\bf Y}(x) \right]_{21} + e^{n a y} \left[ {\bf Y}(y)^{-1}{\bf Y}(x)\right]_{31} \right).
\end{equation}
We consider local coordinates $\zeta_x$ and $\zeta_y$ in $\mathbb{D}_{a^{\star}}$.
While the function ${\bf Y}(z)$ has a
jump in this region, the first column of ${\bf Y}(z)$ does not (see the
Riemann-Hilbert problem \eqref{rhp}). Therefore we can pick $x$ and $y$ to
be in a convenient region. We choose $x$ and $y$ to be in $\Omega_1$ as
defined in Figure \ref{fig_biglensSuper}.
From the transformation \eqref{super-Y-to-W}, we see that
\begin{eqnarray}
\label{super-Yinv-Y21}
&& \begin{split}
& \left[{\bf Y}(y)^{-1}{\bf Y}(x)\right]_{21} = \left[{\bf W}(y)^{-1}{\bf W}(x)\right]_{21} \exp\left(n\left((1-\kappa){\mathfrak g}(y)+(1-\kappa){\mathfrak g}(x)+
{\delta V(x)}
+\ell_1\right)\right),
\end{split}
\\
\label{super-Yinv-Y31}
&& \begin{split}
\left[{\bf Y}(y)^{-1}{\bf Y}(x)\right]_{31} = \left[{\bf W}(y)^{-1}{\bf W}(x)\right]_{31} \exp\left(n\left((1-\kappa){\mathfrak g}(x)+
{\delta V(x)} +
{\delta V(y)} +
l_2 - \eta
\right)\right)
\end{split}
\end{eqnarray}
for $x$ and $y$ in $\Omega_1$. We have
\begin{equation}
\label{super-W-Psiinf}
{\bf W}(z) = {({\bf 1} +\mathcal{O}(n^{-(1-\gamma)/2}) }{\bf \Psi}^\infty(z) =
{({\bf 1} +\mathcal{O}(n^{-(1-\gamma)/2}) }
{\bf \Psi}(z){\bf R}(\zeta(z))
\end{equation}
for $z\in\mathbb{D}_{a^{\star}}$. From \eqref{super-zeta}, we have
\begin{equation}
\label{Psiinv-Psi}
{\bf \Psi}(y)^{-1} {\bf \Psi}(x) = {\bf I} + \mathcal{O}\left((\zeta_x - \zeta_y)\kappa^{1/2}\right).
\end{equation}
We define the functions
$\mathcal{Q}_i(z;\kappa)$ to be:
\begin{eqnarray}
{\mathcal Q}_1(z;\kappa) & := & -V(z) + 2(1-\kappa){\mathfrak g}(z;\kappa) + \ell_1(\kappa), \\
{\mathcal Q}_2(z;\kappa) & := & -V(z) + az + (1-\kappa){\mathfrak g}(z;\kappa) +
l_2, \\
{\mathcal Q}_3(z;\kappa) & := & az - (1-\kappa){\mathfrak g}(z;\kappa) -\ell_1(\kappa) +
{l_2}.
\end{eqnarray}
Now combining \eqref{super-Yinv-Y21},
\eqref{super-W-Psiinf}, \eqref{Psiinv-Psi}, \eqref{super-R},
$\det{\bf R}(\zeta)=1$, and
noting the $\mathcal{O}(\kappa)$ error terms from \eqref{Psiinv-Psi}
are subsumed by the $\mathcal{O}(n^{-(1-\gamma)/2})$ error terms from
\eqref{super-W-Psiinf} gives
\begin{equation}
\label{super-YinvY21}
e^{-\frac{n}{2}(V(x)+V(y))}\left[{\bf Y}(y)^{-1}{\bf Y}(x)\right]_{21} = \left(\mathcal{O}\left((\zeta_x-\zeta_y)\kappa^{1/2}\right)\cdot H_r^{(r)}(\zeta_x-\zeta_0) + \mathcal{O}(n^{-(1-\gamma)/2}) \right) e^{n(*)},
\end{equation}
where
\begin{equation}
\begin{split}
(*) = & \frac{1}{2}\mathcal{Q}_1(x) + \frac{1}{2}\mathcal{Q}_1(y) +
{\delta V(x)}
-\kappa\log\zeta_x - \kappa\sum_{j=1}^{K} \frac{c_j^{(H)}}{\zeta_x^j} \\
= & \frac{1}{2}P_1(x) + \frac{1}{2}P_1(y) + \mathcal{O}(\kappa\ln \kappa).
\end{split}
\end{equation}
The last equality is shown by noticing that rearranging the terms in \eqref{mainidzeta} we have
\begin{equation}
{\delta V(x)}
-\kappa\log\zeta_x - \kappa\sum_{j=1}^{K} \frac{c_j^{(H)}}{\zeta_x^j} =
\frac 1 2\left(V(x) - a x - (1-\kappa)\mathfrak g(x) - \ell_2 - \frac \kappa 2(\zeta-\zeta_0)^2 + \kappa \ln \kappa + \mathfrak b \right)
\label{125}
\end{equation}
Since the real part of $P_1(z)$ is negative for $z$ near $a^{\star}$ (Lemma
\ref{P-nolog-lemma}(b)), for $\kappa$ sufficiently
small the real part of the exponent in \eqref{super-YinvY21} is negative.
Define
\begin{equation}
F_r^\text{GUE}(\zeta_x,\zeta_y):=\frac{2\pi i}{k_{r-1}^{(r)}}\left(H_r^{(r)}(\zeta_x-\zeta_0)H_{r-1}^{(r)}(\zeta_y-\zeta_0)-H_{r-1}^{(r)}(\zeta_x-\zeta_0)H_r^{(r)}(\zeta_y-\zeta_0)\right).
\end{equation}
From \eqref{super-Yinv-Y31},
\begin{equation}
e^{-\frac{n}{2}(V(x)+V(y))+nay}\left[{\bf Y}(y)^{-1}{\bf Y}(x)\right]_{31} = \left(F_r^\text{GUE}(\zeta_x,\zeta_y)+\mathcal{O}(n^{-(1-\gamma)/2}) \right) e^{n(**)},
\end{equation}
where from \eqref{super-Yinv-Y31}
\begin{eqnarray}
\begin{split}
(**) &= -\frac {V(x)}2 - \frac {V(y)}2 + ay + \delta V(y) + \delta V(x)
- \kappa \ln \zeta_x - \kappa \sum_{j=1}^K \frac {c_j^{(H)}}{\zeta_x^j}
- \kappa \ln \zeta_y - \kappa \sum_{j=1}^K \frac {c_j^{(H)}}{\zeta_y^j} +
\\ &\phantom{=} + (1-\kappa) \mathfrak g(x) + \kappa \ell_H\,.
\end{split}
\end{eqnarray}
Using now \eqref{125} (for both $x$ and $y$) and rearranging the terms we find
\begin{eqnarray}
(**) =&& \frac 1 2 \left[ ay - (1-\kappa) \mathfrak g(y) - \frac \kappa 2 (\zeta_y-\zeta_0)^2 - l_2 + \kappa \ln \kappa + \mathfrak b \right]
+ \nonumber\\
&&+
\frac 1 2 \left[ (1-\kappa) \mathfrak g(x) -ax - \frac \kappa 2 (\zeta_x-\zeta_0)^2 - l_2 + \kappa \ln \kappa + \mathfrak b \right]
+ \kappa \ell_H - \eta \\
=&& -\frac{\kappa}{4}(\zeta_x-\zeta_0)^2-\frac{\kappa}{4}(\zeta_y-\zeta_0)^2-\frac{1}{2}\mathcal{Q}_3(x)+\frac{1}{2}\mathcal{Q}_3(y).
\end{eqnarray}
where we have used the definition of $\eta := \kappa \ln \kappa + \mathfrak b + \kappa \ell_H$ \eqref{defeta}.
From this and the
fact that \eqref{super-YinvY21} is exponentially decaying in $n$ shows
\begin{equation}
\label{super-kernel-formula}
K_n(x,y) = \frac{e^{-\frac{n}{2}\mathcal{Q}_3(x)+\frac{n}{2}\mathcal{Q}_3(y)}}{2\pi i (x-y)}\left(F_r^\text{GUE}(\zeta_x,\zeta_y)+\mathcal{O}(n^{-(1-\gamma)/2})\right)e^{-\frac{r}{4}(\zeta_x-\zeta_0)^2-\frac{r}{4}(\zeta_y-\zeta_0)^2}.
\end{equation}
One final application of \eqref{super-zeta} (to switch $x-y$ to
$\zeta_x-\zeta_y$) and
Proposition \ref{propdeform} (to show convergence of $\mathcal{Q}_3$ to $P_3$) gives
\eqref{kernel-thm-form}.
\end{proof}
\section{The subcritical regime}
We now take $V(x)$ and $a$ so Definition
\ref{subcritical} of the subcritical regime is satisfied; that is $a<a_c$ and the function $\Re\,P_2$ has no global maximum on $\mathbb R \setminus [\alpha,\beta]$. In this case $\Re\, P_3 $ has a (unique) global minimum at $z=b^\star$ (with value zero, as per our choice of $l_2$ in Definition \ref{defell2}).
We will show that almost surely there are no outliers.
We will freely reuse the same notation from Section \ref{supercrit}
for new objects which played a similar role in the analysis of the
supercritical regime. To begin, fix $\gamma\in[0,1)$ and again set $K$
to be the smallest nonnegative integer satisfying
\begin{equation}
\label{sub-k-def}
K\geq \max \left\{\frac{3\gamma-1}{1-\gamma},0\right\}
\end{equation}
\subsection{Modified equilibrium problem (subcritical case)}
The procedure here parallels closely the one followed in the supercritical case, and hence we will only state the results since their proof does not differ significantly from the other case.
Let $J$ be a closed subset not containing the point $b^\star$ and containing $[\alpha,\beta]$ in its interior; we recall that $b^\star(a)>\beta$ for $0<a<a_c$.
\begin{prop}
\label{propdeform-sub}
For any $K\in \mathbb N$ there is a neighborhood of the origin in $(\kappa,\vec \delta) \in \mathbb{C}^{1+K}$ such that the equilibrium measure $\widetilde \sigma(x){\rm d} x$ of {\bf unit} total mass for the external field
\begin{equation}
\widetilde V(z):=V(z) + \delta V(z), \ \ \ \delta V(z):= \kappa \ln (z-b^\star ) + \kappa \sum_{j=1}^K \frac {\delta_j}{2(z-b^\star)^j}
\label{deformV-sub}
\end{equation}
is supported on a single interval $[\alpha(\kappa,\vec \delta), \beta (\kappa,\vec \delta)]$ still contained in the interior of $J$: the endpoints $\alpha(\kappa,\vec \delta), \beta (\kappa,\vec \delta)$ are analytic functions of the specified variables.
Furthermore the $g$--function of this problem
\begin{equation}
\mathfrak g(z):= \int \ln(z-w) \widetilde \sigma(w) {\rm d} w
\end{equation}
converges uniformly over closed subsets not containing $[\alpha,\beta]$ to the unperturbed $g$--function.
\end{prop}
The proof is identical to that of Proposition \ref{propdeform}: the only difference is that now the modified equilibrium measure is of unit total mass, rather than of mass $1-\kappa$.
We next re-define the three functions $\mathcal P_j$'s; the definition is subtly different from the previous \eqref{super-mathcalP1}, \eqref{super-mathcalP2},\eqref{super-mathcalP3} and hence there is a possibility of confusion for the reader. The advantage is that we will be able to recycle many of the previous computations.
\begin{shaded}
\begin{eqnarray}
\label{sub-mathcalP1}
{\mathcal P}_1(z) & := & -\widetilde V(z) + 2{\mathfrak g}(z) + \ell_1, \\
\label{sub-mathcalP2}
{\mathcal P}_2(z) & := & {\mathcal P}_1(z) + {\mathcal P}_3(z) = -V(z)+\delta V(z) + a z + \mathfrak g(z) +l_2 \\
\label{sub-mathcalP3}
{\mathcal P}_3(z) & := & az - {\mathfrak g}(z) + 2\delta V(z) + l_2 - \ell_1 = \mathcal P_2(z)- \mathcal P_1(z)\\
\widetilde V(z)&:=& V(z) + \delta V(z)\ ,\qquad
\delta V(z) := \kappa\log(z-b^{\star}) + \sum_{j=1}^{2k}\frac{\delta_j}{2(z-b^{\star})^j} + \ell_1
\end{eqnarray}
\end{shaded}
It is {\bf important} to point out the change of sign in the definition of $\widetilde V$, relative to the supercritical case. We also remind that $\ell_1, \mathfrak g$ are analytic functions of $\kappa, \vec \delta$, while $l_2$ is the constant mandated in Definition \ref{defell2}.
\begin{thm}
\label{rhoprop-sub}
There exists a conformal change of coordinate $\rho = \rho(z;\kappa,\vec \delta)$ {\em fixing $z=b^\star$} ($\rho(b^\star;\kappa,\vec \delta) \equiv 0$) that depends analytically on the parameters $\kappa, \vec \delta$ such that
\begin{equation}
\mathcal P_3 (z):= az - \mathfrak g(z;\kappa,\vec \delta) +l_2-\ell_1 +2\kappa\ln(z-b^\star) + \sum_{j=1}^K \frac {\delta_j}{(z-b^\star)^j}
\label{38}
\end{equation}
can be written as
\begin{equation}
\mathcal P_3(z;\kappa,\vec \delta) = \frac 1 2 (\rho- {\mathfrak a} )^2 + 2\kappa \ln \rho+\mathfrak b + \sum_{j=2}^{K} \frac
{ \gamma_j}{\rho^j}\label{mainid-sub}
\end{equation}
where the parameters ${\mathfrak a} = {\mathfrak a} (\kappa,\vec \delta)$, $\mathfrak b = \mathfrak b (\kappa,\vec\delta)$ and $\vec \gamma = \vec \gamma(\kappa;\vec \delta)$ are analytic functions of the indicated parameters.
Furthermore the Jacobian
\begin{equation}
\frac {\partial \vec \gamma}{\partial \vec \delta}
\end{equation}
is nonsingular in a neighborhood of the origin (for $\kappa$ sufficiently small).
\end{thm}
\begin{thm}
\label{confchange-sub}
There exists a conformal change of coordinate $\zeta(z;\kappa)$ of the form
\begin{equation}
\zeta(z;\kappa)= \frac {\rho(z;\kappa)}{i \sqrt{\kappa}} = \frac 1{i \sqrt{\kappa}} C (z-b^\star) (1+\mathcal O(z-b^\star))\ , C>0
\label{sub-zeta}
\end{equation}
and a choice of $\vec \delta = \vec \delta(\kappa)$ for the deformed potential (\ref{deformV}) in Puiseux series of $\sqrt{\kappa}$
such that
\begin{equation}
\mathcal P_3(z; \kappa, \vec \delta(\kappa)) = - \frac \kappa 2 (\zeta - \zeta_0)^2 + 2 \kappa \ln (\sqrt \kappa \zeta) +\mathfrak b + 2 \kappa \sum_{j=1}^K \frac {c^{(H)}_j}{\zeta^j}\label{312}\,.
\end{equation}
The functions $\zeta_0(\kappa), \beta(\kappa), \vec \delta(\kappa)$ admit a Puiseux expansion and are of orders
\begin{equation}
\zeta_0 = \mathcal O(\sqrt\kappa)\ ,\ \ \beta = \mathcal O(\kappa)\ ,\ \ \vec \delta = \mathcal O(\kappa).
\end{equation}
The expressions $c^{(H)}_j$ are polynomials of degree $j$ in $\zeta_0$ determined by the formula \eqref{super-cH-def}.
\end{thm}
\begin{figure}
\setlength{\unitlength}{2.7pt}
\begin{center}
\begin{picture}(100,50)(-60,-25)
\put(-10,0){\circle*{1}}
\put(-11.5,2){$\alpha$}
\put(10,0){\circle*{1}}
\put(9.5,2){$\beta$}
\put(15,0){\circle*{1}}
\put(16,1){$b^{\star}$}
\put(-50,0){\line(1,0){80}}
\put(-42,14){$\Omega_1$}
\put(-22,8.5){$\Omega_2$}
\put(-1.5,3){$\Omega_3$}
\put(-1.5,-4.5){$\Omega_4$}
\put(-22,-9.5){$\Omega_5$}
\put(-42,-15){$\Omega_6$}
\qbezier(-10,0)(0,16),(10,0)
\qbezier(-10,0)(0,-16),(10,0)
\qbezier(15,0)(15,25)(-10,25)
\qbezier(15,0)(15,-25)(-10,-25)
\qbezier(-35,0)(-35,25)(-10,25)
\qbezier(-35,0)(-35,-25)(-10,-25)
\thicklines
\put(-42,0){\vector(1,0){1}}
\put(-22,0){\vector(1,0){1}}
\put(0,0){\vector(1,0){1}}
\put(22,0){\vector(1,0){1}}
\put(-10,-25){\vector(1,0){1}}
\put(-10,25){\vector(-1,0){1}}
\put(0,8){\vector(1,0){1}}
\put(0,-8){\vector(1,0){1}}
\end{picture}
\end{center}
\caption{The regions $\Omega_i$ and the oriented contour $\Gamma$ for the subcritical case.\label{sublens}}
\end{figure}
\subsection{Steepest descent analysis (subcritical case)}
\label{sub-steepest}
The regions $\Omega_i$, $i=1,...6$ are defined in Figure \ref{sublens}.
Following the example of the supercritical case, we introduce the {\bf same} new matrix $\mathbf W$ as in \eqref{super-Y-to-W}. Note, however that the definition of the regions $\Omega_j$'s now follows Figure \ref{sublens}, the constant $l_2$ follows Definition \ref{defell2} in the subcritical case and $\delta V$ is given in \eqref{deformV-sub} instead.
The new matrix ${\bf W}(z;\kappa)$ satisfies the same jump conditions \eqref{super-Wjumps}
as in the supercritical case (with, of course, the new definitions of
the $\mathcal{P}_j(z;\kappa)$'s and $\Omega_j$'s) as well as the same
asymptotic condition \eqref{super-W-infinity}.
Due to the new definitions of $\mathcal P_j$'s \eqref{sub-mathcalP1}, \eqref{sub-mathcalP2}, \eqref{sub-mathcalP3} , the behavior near $z=b^{\star}$ is now different:
\begin{equation}
\label{newWasympt}
\begin{split}
{\bf W} = \text{(analytic)}\begin{pmatrix} 1 & 0 & 0 \\ 0 &
{ {\rm e}^{-n \delta V(z)}}
& 0 \\ 0 & 0 &
{ {\rm e}^{n \delta V(z)}}
\end{pmatrix} \\
\text{as } z\to b^{\star}.
\end{split}
\end{equation}
The outer parametrix problem is the same as in the supercritical case, so we
again define the outer parametrix solution ${\bf \Psi}(z;\kappa)$ as in
\eqref{Vpsi}.
\subsubsection{The local parametrix near \texorpdfstring{$\boldsymbol{b^{\star}}$}{bstar}}
Define $\mathbb{D}_{b^{\star}}$ to be
a fixed-size circular disk centered at $b^{\star}$ which is small enough
so that it does not intersect the
inner lenses, and
\begin{equation}
\label{condition-on-Db*}
\Re(P_2)<0 \quad \text{for} \quad z\in\mathbb{D}_{b^{\star}}.
\end{equation}
This last condition is possible for $\kappa$ sufficiently small because --due to the definition of $l_2$ in Definition \ref{defell2} for the subcritical case--
$P_2(b^{\star})<0$.
We now use Theorem \ref{confchange-sub}; in a local coordinate centered at $b^{\star}$, the analytic part of
$\mathcal{P}_3(z;\kappa)$ behaves the same way (quadratically with a
maximum at the origin) along the imaginary axis
as $\mathcal{P}_2(z;\kappa)$ did along the real axis in the supercritical
regime.
The scaling of $\zeta$ is analogous to \eqref{super-zeta-order};
\begin{equation}
\label{sub-zeta-order}
\zeta=\mathcal{O}\left(n^{(1-\gamma)/2}\right) \quad \text{when } z\in\partial\mathbb{D}_{b^{\star}}.
\end{equation}
\begin{equation}
{\mathcal P}_3 = f(z; \kappa) + 2 \delta V(z) +l_2-\ell_1= -\frac{\kappa}{2}(\zeta-\zeta_0)^2 + 2\kappa\log\zeta + 2\kappa\sum_{j=1}^k\frac{c_j^{(H)}}{\zeta^{2j}} + \kappa \ln \kappa + \mathfrak b.
\end{equation}
\begin{defn}
\label{localparam-sub}
The local parametrix within the disk $\mathbb D_{b^\star}$ shall be the unique solution ${\bf R}(z)$ to the following model Riemann--Hilbert problem:
\begin{equation}
\label{sub-R-RHP}
\begin{cases}
& {\bf R}_+(\zeta) = {\bf R}_-(\zeta)\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & -\zeta^{2r}\exp\left(-\frac{r}{2}(\zeta-\zeta_0)^2+r\ell_H +2r\sum_{j=1}^k\frac{c_j^{(H)}}{\zeta^{2j}}\right) \\ 0 & 0 & 1 \end{pmatrix}
\\ & \phantom{{\bf R}_+(\zeta)} = {\bf R}_-(\zeta)\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & -e^{n\mathcal{P}_3} \\ 0 & 0 & 1 \end{pmatrix}, \quad \zeta\in\mathbb{R},
\\ &{\bf R}(\zeta)={\bf I}+{\cal O}\left(\displaystyle\frac{1}{\zeta}\right) \text{ as } \zeta\to\infty,
\\&{\bf R}(\zeta)=(\mbox{analytic})\begin{pmatrix} 1 & 0 & 0 \\ 0 & \zeta^{-r}\exp\left(-r\sum_{j=1}^k\frac{c_j^{(H)}}{\zeta^{2j}}\right) & 0 \\ 0 & 0 & \zeta^r\exp\left(r\sum_{j=1}^k\frac{c_j^{(H)}}{\zeta^{2j}}\right) \end{pmatrix} \text{ as } \zeta\to 0.
\end{cases}
\end{equation}
\end{defn}
This problem is almost the same as \eqref{R-RHP}. Analogously to
\eqref{super-R}, the solution is ($\ell_H = -1 -2\ln 2$ as in \eqref{super-ellH})
\begin{equation}
\label{sub-R}
{\bf R}(\zeta) = \exp\left(-\frac{r}{2}\ell_H{ \bf\Lambda}_{23}\right) \,{\bf H}_{23}(\zeta) \,\zeta^{-r{\bf\Lambda}_{23}}\,\exp\left(\left(\frac{r}{2}
\ell_H - r\sum_{j=1}^k\frac{c_j^{(H)}}{\zeta^{2j}}\right){\bf\Lambda}_{23}\right),
\end{equation}
where
\begin{equation}
\label{sub-H23}
{\bf H}_{23}(\zeta):=\begin{pmatrix} 1 & 0 & 0 \\ 0 & H^{(r)}_r(\zeta-\zeta_0) & \displaystyle \frac{-1}{2\pi i}\int_{-\infty}^\infty \frac{H^{(r)}_r(s-\zeta_0) e^{-\frac{r}{2}s^2}}{s-\zeta}ds \\ \\ 0 & \displaystyle\frac{2\pi i}{-k^{(r)}_{r-1}}H^{(r)}_{r-1}(\zeta-\zeta_0)& \displaystyle \frac{1}{k^{(r)}_{r-1}}\int_{-\infty}^\infty \frac{H^{(r)}_{r-1}(s-\zeta_0) e^{-\frac{r}{2}s^2}}{s-\zeta}ds \end{pmatrix} \quad \text{and} \quad
{\bf\Lambda}_{23}:=\begin{pmatrix} 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & -1 \end{pmatrix}.
\end{equation}
Again the polynomials $H_m^{(r)}(\zeta)$ and the normalization constants
$k_m^{(r)}$ are defined by \eqref{def-of-kr}. The analysis in the
supercritical regime leading to \eqref{super-R-error}
applies here as well, leading to
\begin{equation}
\label{sub-R-error}
{\bf R}(\zeta) = {\bf I} + \mathcal{O}\left(\frac{1}{\zeta}\right) + \mathcal{O}\left(\frac{r}{\zeta^{2k+2}}\right).
\end{equation}
\subsection{The subcritical error analysis}
Let $\mathbb{D}_\alpha$ and $\mathbb{D}_\beta$ be small, closed disks of
fixed radii centered at $\alpha$ and $\beta$ that are
bounded
away from the outer lenses and $\mathbb{D}_{b^{\star}}$. Orient the boundaries
$\partial\mathbb{D}_\alpha$ and $\partial\mathbb{D}_\beta$ clockwise.
Let ${\bf P}_{\mbox{Ai}}^{(\alpha)}$ and ${\bf P}_{\mbox{Ai}}^{(\beta)}$
be the Airy parametrices constructed in $\mathbb{D}_\alpha$
and $\mathbb{D}_\beta$, respectively (see Section \ref{super-error}).
Define the global parametrix ${\bf \Psi}^\infty(z)$ by
\begin{equation}
{\bf \Psi}^\infty(z):=
\begin{cases}
{\bf \Psi}(z), & z\notin \mathbb{D}_\alpha \cup \mathbb{D}_\beta \cup \mathbb{D}_{b^{\star}}, \\
{\bf \Psi}(z) {\bf R}(\zeta(z)), & z\in\mathbb{D}_{b^{\star}}, \\
{\bf P}_{\mbox{Ai}}^{(\alpha)}(z), & z\in\mathbb{D}_\alpha, \\
{\bf P}_{\mbox{Ai}}^{(\beta)}(z), & z\in\mathbb{D}_\beta.
\end{cases}
\end{equation}
The error matrix ${\bf E}(z)$ is given by
\begin{equation}
{\bf E}(z) := {\bf W}(z){\bf \Psi}^\infty(z)^{-1}.
\end{equation}
Let $\Gamma$ denote the contours given by the boundaries of the regions
$\Omega_j$ in Figure \ref{sublens}.
The error matrix satisfies a Riemann-Hilbert problem with jump matrix
${\bf V^{(E)}}(z)$ on the contours $\Gamma^{\bf(E)}$ shown in
Figure \ref{sub-error-fig}. The form of the jump matrix is as follows: \\
\indent
$\bullet$ For $z$ outside the disks $\mathbb{D}_\alpha$, $\mathbb{D}_\beta$,
and $\mathbb{D}_{b^{\star}}$, and excluding the band $[\alpha,\beta]$:
\begin{equation}
\label{sub-VE-outside}
{\bf V^{(E)}}(z) =
{\bf \Psi}(z) {\bf V^{(W)}}(z) {\bf \Psi}(z)^{-1}, \quad z\in \Gamma\cap\left( \mathbb{D}_{\alpha} \cup \mathbb{D}_{\beta} \cup \mathbb{D}_{b^{\star}} \right)^c\cap[\alpha,\beta]^c,
\end{equation}
where ${\bf V^{(W)}}(z)$ is given by the formulas in (\ref{super-Wjumps}).\\
\indent
$\bullet$ For $z$ on the boundaries of the disks
$\partial\mathbb{D}_\alpha$,
$\partial\mathbb{D}_\beta$, and $\partial\mathbb{D}_{b^{\star}}$:
\begin{equation}
\label{sub-VE-boundaries}
{\bf V^{(E)}}(z) =
\begin{cases}
{\bf \Psi}(z){\bf R}(\zeta) {\bf \Psi}(z)^{-1}, & z\in\partial\mathbb{D}_{b^{\star}}, \\
{\bf P}_{\mbox{Ai}}^{(\alpha)}(z) {\bf \Psi}(z)^{-1}, & z\in\partial\mathbb{D}_\alpha, \\
{\bf P}_{\mbox{Ai}}^{(\beta)}(z) {\bf \Psi}(z)^{-1}, & z\in\partial\mathbb{D}_\beta.
\end{cases}
\end{equation}
\indent
$\bullet$ For $z$ inside the disk $\mathbb{D}_{b^{\star}}$:
\begin{equation}
\label{sub-VE-b*1}
{\bf V^{(E)}}(z) =
\begin{cases}
{\bf \Psi}(z) {\bf R}(\zeta)
\begin{pmatrix}
1 & e^{n\mathcal{P}_1(z)} & e^{n \mathcal{P}_2(z)} \\
0 & 1 & 0 \\
0 & 0 & 1 \end{pmatrix} {\bf R}(\zeta)^{-1} {\bf \Psi}(z)^{-1},
& z \in (\partial\Omega_1\cap\partial\Omega_6)\cap\mathbb{D}_{b^{\star}}, \\
{\bf \Psi}(z) {\bf R}(\zeta)
\begin{pmatrix}
1 & e^{n\mathcal{P}_1(z)} & 0 \\
0 & 1 & 0 \\
0 & 0 & 1 \end{pmatrix} {\bf R}(\zeta)^{-1} {\bf \Psi}(z)^{-1},
& z \in (\partial\Omega_2\cap\partial\Omega_5)\cap\mathbb{D}_{b^{\star}}.
\end{cases}
\end{equation}
\indent
$\bullet$ Furthermore, ${\bf V^{(E)}}(z)={\bf I}$
on the contours
$$[\alpha,\beta]\cap(\mathbb{D}_\alpha\cup\mathbb{D}_\beta)^c, \quad \Gamma\cap\mathbb{D}_\alpha, \quad \Gamma\cap\mathbb{D}_\beta, \quad (\partial\Omega_1\cap\partial\Omega_2)\cap\mathbb{D}_{b^{\star}}, \quad \text{and} \quad (\partial\Omega_5\cap\partial\Omega_6)\cap\mathbb{D}_{b^{\star}}.$$
\begin{figure}
\setlength{\unitlength}{2.7pt}
\begin{center}
\begin{picture}(100,50)(-60,-25)
\put(-9.5,0){\circle{4}}
\put(-16,3){$\mathbb{D}_\alpha$}
\put(-14,2.25){\line(2,-1){4.5}}
\put(9.5,0){\circle{4}}
\put(9,5){$\mathbb{D}_\beta$}
\put(11.5,4){\line(-1,-2){2}}
\put(15,0){\circle{4}}
\put(18,3){$\mathbb{D}_{b^{\star}}$}
\put(20,2.25){\line(-3,-1){4.5}}
\put(-50,0){\line(1,0){38.5}}
\put(11.5,0){\line(1,0){16}}
\qbezier(-9,2)(0,14),(9,2)
\qbezier(-9,-2)(0,-14),(9,-2)
\qbezier(15,0)(15,25)(-10,25)
\qbezier(15,0)(15,-25)(-10,-25)
\qbezier(-35,0)(-35,25)(-10,25)
\qbezier(-35,0)(-35,-25)(-10,-25)
\thicklines
\put(-42,0){\vector(1,0){1}}
\put(-22,0){\vector(1,0){1}}
\put(22,0){\vector(1,0){1}}
\put(-10,-25){\vector(1,0){1}}
\put(-10,25){\vector(-1,0){1}}
\put(0,8){\vector(1,0){1}}
\put(0,-8){\vector(1,0){1}}
\end{picture}
\end{center}
\caption{The jump contours $\Gamma^{\bf(E)}$ for the Riemann-Hilbert problem for ${\bf E}(z)$ in the subcritical case.\label{sub-error-fig}}
\end{figure}
We now show that all of the jump matrices in
\eqref{sub-VE-outside}--\eqref{sub-VE-b*1} are
uniformly close to the identity as $n\to\infty$.
Here are the results we will need for $P_1$, $P_2$, and $P_3$, analogous
to Lemma \ref{P-nolog-lemma}:
\begin{lemma}
\label{sub-nolog-lemma}
In the subcritical regime, the inner and outer lenses can be chosen such
that
\begin{itemize}
\item[(a)] On the inner lenses outside of the disks around $\alpha$ and
$\beta$: The real part of $P_1(z)$ is positive and bounded away from zero
for
$z\in[(\partial\Omega_2\cap\partial\Omega_3)\cup(\partial\Omega_4\cap\partial\Omega_5)]\cap(\mathbb{D}_\alpha\cup\mathbb{D}_\beta)^c$.
\item[(b)] On the real axis outside of $[\alpha,\beta]$ and the disks
around
$\alpha$ and $\beta$: The real part of $P_1(z)$ is negative and bounded away from zero for $z\in[(\partial\Omega_1\cap\partial\Omega_6)\cup(\partial\Omega_2\cap\partial\Omega_5)]\cap(\mathbb{D}_\alpha\cup\mathbb{D}_\beta)^c$.
\item[(c)] On the outer lenses outside of the disk around $b^{\star}$: For
$\kappa$ sufficiently small, the real part of $P_3(z)$ is negative and
bounded away from zero for
$z\in[(\partial\Omega_1\cap\partial\Omega_2)\cup(\partial\Omega_5\cap\partial\Omega_6)]\cap\mathbb{D}_{b^{\star}}^c$.
\item[(d)] On the real axis outside of the outer lenses or on the
real axis inside
$\mathbb{D}_{b^{\star}}$: For $\kappa$
sufficiently small, the real part of $P_2(z)$ is negative and bounded away
from zero for $z\in(\partial\Omega_1\cap\partial\Omega_6) \cup [(\partial\Omega_2\cap\partial\Omega_5)\cap\mathbb{D}_{b^{\star}}]$.
\end{itemize}
\end{lemma}
\begin{proof}
Parts (a) and (b) follow from the analysis of the Riemann-Hilbert problem
for the standard (non-multiple) orthogonal polynomials
(for example, \cite{Deift:1998-book}).
\\
\indent
For (c), first note $P_3(b^{\star})=0$. Following the proof of Lemma
\ref{P-nolog-lemma}(d),
the outer lenses (defined in this regime to be a
circle centered below $\alpha$ and passing through $b^{\star}$, that is big enough such that $\Re P_2$ is negative on the real axis to the left of the circle) are descent lines of
${\rm Re}\,P_3(z)$ for $\kappa$ sufficiently small. The result follows.
\\
\indent
For (d), start with $\kappa=0$. Consider real $z$ to the left of the outer
lenses. From Lemma \ref{sub-nolog-lemma}(c), ${\rm Re}\,P_3<0$
at the left-most point of the outer lenses. Thus ${\rm Re}\,P_3(z)<0$
for such $z$
since ${\rm Re}\,P_3$ is a strictly increasing function for
$z\in(-\infty,\alpha)$. Since ${\rm Re}\,P_1(z)$ is also negative here
by construction, this means
${\rm Re}\,P_2(z)={\rm Re}(P_1(z)+P_3(z))$ is also
negative. This is also true for real $z$ inside $\mathbb{D}_{b^{\star}}$ by
\eqref{condition-on-Db*}.
Next, consider real $z$ to the right of the outer lenses.
By Definition \ref{subcritical}
we have
${\rm Re}\,P_2(z)<{\rm Re}\,P_3(b^{\star})=0$.
Along with the fact that ${\rm Re}\,P_1(z)<0$ here shows the desired
result.
\end{proof}
Next come the necessary results for $\mathcal{P}_1(z;\kappa)$,
$\mathcal{P}_2(z;\kappa)$, and $\mathcal{P}_3(z;\kappa)$ defined by
\eqref{sub-mathcalP1}--\eqref{sub-mathcalP3}. This lemma is analogous
to Lemma \ref{super-Pmathcal-lemma} for the supercritical regime.
\begin{lemma}
\label{sub-Pmathcal-lemma}
For $\kappa$ sufficiently small:
\begin{itemize}
\item[(a)] On the two inner lenses outside of
the disks around $\alpha$ and $\beta$: The real part of
${\mathcal P}_1(z;\kappa)$ is positive and bounded away from zero for
$z\in[(\partial\Omega_2\cap\partial\Omega_3)\cup(\partial\Omega_4\cap\partial\Omega_5)]\cap(\mathbb{D}_\alpha\cup\mathbb{D}_\beta)^c$.
\item[(b)] On the real axis outside of
$[\alpha,\beta]$ and the disks around
$\alpha$, $\beta$, and $b^{\star}$: The real part of
${\mathcal P}_1(z;\kappa)$ is negative and bounded away from zero for
$z\in[(\partial\Omega_1\cap\partial\Omega_6)\cup(\partial\Omega_2\cap\partial\Omega_5)]\cap(\mathbb{D}_\alpha\cup\mathbb{D}_\beta\cup\mathbb{D}_{b^{\star}})^c$.
\item[(c)] On the outer lenses outside of $\mathbb{D}_{b^{\star}}$: The real
part of $\mathcal{P}_3(z;\kappa)$ is negative and bounded away from zero
for
$z\in[(\partial\Omega_1\cap\partial\Omega_2)\cup(\partial\Omega_5\cap\partial\Omega_6)]\cap\mathbb{D}_{b^{\star}}^c.$
\item[(d)] On the real axis outside of the outer lenses and
$\mathbb{D}_{b^{\star}}$: The real part of
$\mathcal{P}_2(z;\kappa)$ is negative and bounded away from zero for
$z\in(\partial\Omega_1\cap\partial\Omega_6)\cap\mathbb{D}_{b^{\star}}^c$.
\item[(e)] On the real axis inside $\mathbb{D}_{b^{\star}}$: The real parts of
$$\mathcal{P}_1(z;\kappa)-\kappa g_H(\zeta)+\kappa\log\zeta + \kappa\sum_{j=1}^k\frac{c_j^{(H)}}{\zeta^{2j}} \quad \text{and} \quad \mathcal{P}_2(z;\kappa) + \kappa g_H(\zeta) - \kappa\log\zeta - \kappa\sum_{j=1}^k\frac{c_j^{(H)}}{\zeta^{2j}}$$
are negative and bounded away from zero for
$z\in[(\partial\Omega_1\cap\partial\Omega_6)\cup(\partial\Omega_2\cap\partial\Omega_5)]\cap\mathbb{D}_{b^{\star}}$.
\end{itemize}
\end{lemma}
\begin{proof}
Parts (a) and (b) come from Proposition \ref{propdeform-sub} along with the fact that
$\log(z-b^{\star})$ and $(z-b^{\star})^{-j}$, $j=1,\dots,2k$,
is bounded outside of $\mathbb{D}_{b^{\star}}$. Part (c) comes from
combining Lemma \ref{sub-nolog-lemma}(c) with Proposition \ref{propdeform-sub} and the
boundedness of $\log(z-b^{\star})$ and $(z-b^{\star})^{-j}$,
$j=1,\dots,2k$.
Part (d) follows from Lemma \ref{sub-nolog-lemma}(d),
Proposition \ref{propdeform-sub} , and the boundedness of
$\log(z-b^{\star})$ and $(z-b^{\star})^{-j}$, $j=1,\dots 2k$.
Finally, part (e) comes from (b) and (d) of Lemma
\ref{sub-nolog-lemma} along with \eqref{sub-zeta} and Proposition \ref{propdeform-sub}.
\end{proof}
These results allow us to now bound the jumps ${\bf V^{(E)}}$ of the
error problem. Divide $\Gamma^{\bf(E)}$ into a compact component $\Gamma_C^{\bf(E)}$ and a
noncompact component $\Gamma_N^{\bf(E)}$:
\begin{equation}
\begin{split}
\Gamma_C^{\bf(E)}:=&\partial\mathbb{D}_\alpha\cup\partial\mathbb{D}_\beta\cup\partial\mathbb{D}_{b^{\star}}\cup(\Gamma\cap\mathbb{D}_{b^{\star}}),\\
\Gamma_N^{\bf(E)}:=&\Gamma^{\bf(E)}\backslash\Gamma_C^{\bf(E)}.
\end{split}
\end{equation}
\begin{lemma}
\label{sub-VE-lemma}
In the subcritical regime, for large $n$:
\begin{itemize}
\item[(a)] Outside the disks $\mathbb{D}_\alpha$, $\mathbb{D}_\beta$,
and $\mathbb{D}_{b^{\star}}$: There is a constant $c>0$ such that
$${\bf V^{(E)}}(z;\kappa) = {\bf I} + \mathcal{O}(e^{-cn}), \quad z\in\Gamma_N^{\bf(E)}.$$
\item[(b)] On the boundary of $\mathbb{D}_{b^{\star}}$:
$${\bf V^{(E)}}(z;\kappa) = {\bf I} + \mathcal{O}\left(\displaystyle n^{-(1-\gamma)/2}\right) + \mathcal{O}\left(\displaystyle n^{-k-1+(k+2)\gamma} \right), \quad z\in\partial\mathbb{D}_{b^{\star}}.$$
\item[(c)] On the boundaries of $\mathbb{D}_\alpha$ and $\mathbb{D}_\beta$:
$${\bf V^{(E)}}(z;\kappa) = {\bf I} + \mathcal{O}\left(\frac{1}{n}\right), \quad z\in\partial\mathbb{D}_\alpha\cup\partial\mathbb{D}_\beta.$$
\item[(d)] Inside $\mathbb{D}_{b^{\star}}$: There is a constant $c>0$ such that
$${\bf V^{(E)}}(z;\kappa) = {\bf I} + \mathcal{O}\left(e^{-cn}\right), \quad z\in\Gamma\cap\mathbb{D}_{b^{\star}}.$$
\end{itemize}
\end{lemma}
\begin{proof}
Part (a) is the result of Lemma \ref{sub-Pmathcal-lemma}(a)--(d) and the
boundedness of ${\bf \Psi}(z)$. Part (b) is from
\eqref{sub-zeta-order}, \eqref{sub-R-error}, and the
boundedness of ${\bf \Psi}(z)$. Part (c) is from the construction
of the parametrices ${\bf P}_\text{\bf Ai}^{(\alpha)}(z)$ and
${\bf P}_\text{\bf Ai}^{(\beta)}(z)$ (see, for instance,
\cite{Deift:1999b}).
\\
\indent
For part (d), consider the jumps \eqref{sub-VE-b*1}.
By \eqref{sub-R} for ${\bf R}(\zeta)$,
\begin{equation}
\label{VE-bound-Gamma-b*1}
\begin{split}
{\bf V^{(E)}}(z;\kappa) = & {\bf \Psi}(z) e^{-\frac{r}{2}\ell_H{\bf\Lambda}_{23}} {\bf H}_{23-}(\zeta) e^{-r(g_H(\zeta)-\frac{\ell_H}{2}){\bf\Lambda}_{23}} \begin{pmatrix} 1 & (*)_{12} & (*)_{13} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \times \\
& \times e^{r(g_H(\zeta)-\frac{\ell_H}{2}){\bf\Lambda}_{23}}{\bf H}_{23-}(\zeta)^{-1}e^{\frac{r}{2}\ell_H{\bf\Lambda}_{23}}{\bf \Psi}(z)^{-1}, \quad z\in(\partial\Omega_1\cap\partial\Omega_6)\cap\mathbb{D}_{b^{\star}},
\end{split}
\end{equation}
and
\begin{equation}
\label{VE-bound-Gamma-b*2}
\begin{split}
{\bf V^{(E)}}(z;\kappa) = & {\bf \Psi}(z) e^{-\frac{r}{2}\ell_H{\bf\Lambda}_{23}} {\bf H}_{23-}(\zeta) e^{-r(g_H(\zeta)-\frac{\ell_H}{2}){\bf\Lambda}_{23}} \begin{pmatrix} 1 & (*)_{12} & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \times \\
& \times e^{r(g_H(\zeta)-\frac{\ell_H}{2}){\bf\Lambda}_{23}}{\bf H}_{23-}(\zeta)^{-1}e^{\frac{r}{2}\ell_H{\bf\Lambda}_{23}}{\bf \Psi}(z)^{-1}, \quad z\in(\partial\Omega_2\cap\partial\Omega_5)\cap\mathbb{D}_{b^{\star}},
\end{split}
\end{equation}
wherein
\begin{equation}
\label{sub-VE-entries}
\begin{split}
(*)_{12} = & \exp\left(n\mathcal{P}_1(z;\kappa) - rg_H(\zeta) + r\log\zeta + r\sum_{j=1}^k\frac{c_j^{(H)}}{\zeta^{2j}}\right), \\
(*)_{13} = & \exp\left(n\mathcal{P}_2(z;\kappa)+rg_H(\zeta)-r\log\zeta - r\sum_{j=1}^k\frac{c_j^{(H)}}{\zeta^{2j}}\right).
\end{split}
\end{equation}
This along with Lemma \ref{sub-Pmathcal-lemma}(e) and the boundedness of
${\bf \Psi}(z)$ in $\mathbb{D}_{b^{\star}}$ gives the result (d).
\end{proof}
We can now show that ${\bf E}(z)$ is asymptotically close to the identity.
The proof of the following lemma follows that of Lemma \ref{super-E}:
\begin{lemma}
\label{sub-E}
In the subcritical regime, for $n$ large,
$${\bf E}(z) = {\bf I} + \mathcal{O}\left(n^{-(1-\gamma)/2}\right)$$
uniformly in $z$.
\end{lemma}
\subsection{The subcritical kernel and proof of Theorem \ref{theorem-sub-kernel}}
\begin{proof}[Proof of Theorem \ref{theorem-sub-kernel}]
Once again, recall the kernel (\ref{mop-kernel}):
\begin{equation}
K_n(x, y) = \frac{e^{-\frac{1}{2} n ( V(x) + V(y) )}}{2\pi i(x-y)} \left( \left[ {\bf Y}(y)^{-1} {\bf Y}(x) \right]_{21} + e^{n a y} \left[ {\bf Y}(y)^{-1}{\bf Y}(x)\right]_{31} \right).
\end{equation}
While the function ${\bf Y}(z)$ has a
jump for $z\in\mathbb{D}_{b^{\star}}$, the first column of ${\bf Y}(z)$ does not
(see the
Riemann-Hilbert problem \eqref{rhp}); observing the Riemann Hilbert Problem for ${\bf Y}^{-1}$ we also note that the second and third {\em rows} of ${\bf Y}^{-1}$ are entire functions. Therefore we can pick $x$ and $y$ to
be in a convenient region. We choose $x$ and $y$ to be in $\Omega_1$ as
defined in Figure \ref{sublens}.
From the transformation \eqref{super-Y-to-W} (which is the same in the subcritical case as noted at the beginning of Section \ref{sub-steepest}) and using the new definitions \eqref{sub-mathcalP1}, \eqref{sub-mathcalP2}, \eqref{sub-mathcalP3}, we see that
\begin{equation}
\label{Yinv-Y21}
\left[{\bf Y}(y)^{-1}{\bf Y}(x)\right]_{21} = \left[{\bf W}(y)^{-1}{\bf W}(x)\right]_{21}\exp\left(n\left({\mathfrak g}(y)+{\mathfrak g}(x)
{-\delta V(y)}
+\ell_1\right)\right),
\end{equation}
\begin{equation}
\label{Yinv-Y31}
\left[{\bf Y}(y)^{-1}{\bf Y}(x)\right]_{31} = \left[{\bf W}(y)^{-1}{\bf W}(x)\right]_{31}\exp\left(n\left({\mathfrak g}(x)+
{ \delta V(y)}
+
{l_2-\eta}\right)\right)
\end{equation}
for $x$ and $y$ in $\Omega_1$. As in the supercritical case, we have
\begin{equation}
{\bf W}(z) ={\left ({\bf I} + \mathcal{O}\left(n^{-(1-\gamma)/2}\right)\right)} {\bf \Psi}(z){\bf R}(\zeta(z))
\end{equation}
and
\begin{equation}
\label{sub-PsiPsiinf}
{\bf \Psi}(y)^{-1}{\bf \Psi}(x) = {\bf I} + \mathcal{O}\left((\zeta_x-\zeta_y)\kappa^{1/2}\right).
\end{equation}
Define $\mathcal{Q}_i(z;\kappa)$ to be
$\mathcal{P}_i(z;\kappa)$ without the logarithm or pole terms:
\begin{eqnarray}
{\mathcal Q}_1(z;\kappa) & := & -V(z) + 2{\mathfrak g}(z;\kappa) + \ell_1, \\
{\mathcal Q}_2(z;\kappa) & := & -V(z) + az + {\mathfrak g}(z;\kappa) +
{l_2}, \\
{\mathcal Q}_3(z;\kappa) & := & az - {\mathfrak g}(z;\kappa) +
{l_2-\ell_1}.
\end{eqnarray}
Recall \eqref{sub-H23} that
\begin{equation}
\label{342}
{\bf R}(\zeta) = \mathcal O(1) {\rm e}^{-n \left( \kappa \sum_{j=1}^K \frac {c_j^{(H)}}{\zeta^{j}} + \kappa \ln \zeta - \frac \kappa 2 \ell_H\right)\mathbf \Lambda_{23}} \ \text{as} \ \zeta\to 0.
\end{equation}
Now combining \eqref{Yinv-Y21}, \eqref{sub-PsiPsiinf}, \eqref{342}
gives
\begin{equation}
e^{-\frac{n}{2}(V(x)+V(y))}\left[{\bf Y}(y)^{-1}{\bf Y}(x)\right]_{21} = \mathcal{O}((x-y)n^{-(1-\gamma)/2})e^{n(*)}
\end{equation}
and
\begin{equation}
e^{-\frac{n}{2}(V(x)+V(y))+nay}\left[{\bf Y}(y)^{-1}{\bf Y}(x)\right]_{31} = \mathcal{O}((x-y)n^{-(1-\gamma)/2}) e^{n(**)}.
\end{equation}
Rearranging the terms from \eqref{38} and \eqref{312} we find
\begin{equation}
\delta V(z) - \kappa \ln \zeta - \kappa\sum_{j=1}^K \frac {c^{(H)}_j} {\zeta^j} = \frac 1 2 \left(
\mathfrak g(z) - a z - \frac \kappa 2 (\zeta - \zeta_0)^2 + \kappa \ln \kappa + \mathfrak b -l_2 + \ell_1
\right)\label{346}.
\end{equation}
Using \eqref{346} we can rewrite
\begin{eqnarray}
(\star) =
\frac 1 2 \mathcal Q_1(x) + \frac 1 2 \mathcal Q_2(y) + \frac \kappa 4 (\zeta_y-\zeta_0)^2 - \kappa \ln \sqrt \kappa -\frac {\mathfrak b }{2} =
\frac 1 2 P_1(x) + \frac 1 2 P_2(y) + \mathcal O(\kappa \ln \kappa)
\end{eqnarray}
and
\begin{eqnarray}
(\star \star) = \frac 1 2 \mathcal Q_1(x) + \frac 12 \mathcal Q_2(y) - \frac \kappa 4 (\zeta_y - \zeta_0)^2 - \frac \eta 2 + \kappa \ell_H= \frac 12 P_1(x) + \frac 1 2 P_2(y) + \mathcal O(\kappa \ln \kappa).
\end{eqnarray}
Here we have used Proposition \ref{propdeform-sub} to convert $\mathfrak{g}$ to $g$.
Since $\Re P_1 (b^\star)< 0$ and also $\Re P_2(b^\star)<0$, the theorem follows.
\end{proof}
|
{
"timestamp": "2010-09-21T02:04:03",
"yymm": "1009",
"arxiv_id": "1009.3894",
"language": "en",
"url": "https://arxiv.org/abs/1009.3894"
}
|
\section*{Introduction}
The {\it principle} that the Hilbert scheme of
lines contained in a (smooth) projective
variety $X\subset{\mathbb P}^N$ and passing through a (general) point can inherit intrinsic and extrinsic geometrical properties
of the variety, has emerged recently. This principle allowed
to attack some problems in a {\it unified way}, provided
non trivial connections between different theories and put some basic questions in a new light. A typical example is
the Hartshorne Conjecture on complete intersections, see \cite{HC, DD} and also \cite{QEL1, QEL2}. The technique
of studying, or even reconstructing, $X$ from the {\it variety of minimal rational tangents} introduced
in the work of Hwang, Mok and others (a generalization of the Hilbert scheme of lines
passing through a point) was applied to the theory of Fano manifolds (see e.g. \cite{HM, HM2, HM3, Hwang, HK, FHw}). On the other hand, Landsberg and others investigated some possible characterizations of special homogeneous
manifolds via the projective second fundamental form (see e.g. \cite{Lan3, Lan4, HY, LR}).
The Hilbert schemes of lines through a general point
of many homogeneous varieties with notable geometrical properties are also somehow {\it nested}, see Tables \eqref{hermitian}
and \eqref{contact}, or {\it part of a matrioska}. For this class of varieties, or more generally for classes
where the principle holds,
one
starts an induction process which sometimes stops after only a few steps, see e.g.
\cite[Theorem 2.8, Corollary 3.1 and 3.2]{QEL1} and also \cite{FHw}.
An example of this kind is the following: if $X\subset{\mathbb P}^N$ is a $LQEL$-manifold of type $\delta\geq 3$, then the Hilbert scheme of lines $\mathcal L_{x,X}\subset{\mathbb P}^{n-1}$, $n=\dim(X)$, passing through a general point $x\in X$
is a $QEL$-manifold of type $\delta-2$, \cite[Theorem 2.3]{QEL1}. Then starting the induction with $X\subset{\mathbb P}^N$ a $LQEL$-manifold of type $\frac{n}{2}$, one deduces
immediately $n=2,4,8$ or 16, yielding as a consequence a quick proof that Severi varieties appear only in these dimensions (see \cite[Corollary 3.2]{QEL1}, also for the definitions of $(L)QEL$-variety and of Severi variety, introduced by Zak, see e.g. \cite{Zak}).
The Hilbert scheme of lines through a point is closely related to the base locus of the (projective) second fundamental form, a classical tool used in projective differential geometry and reconsidered
in modern algebraic geometry by Griffiths and Harris (see \cite{GH} and also \cite{IL}). In this theory one tries to reconstruct a (homogeneous) variety
from its second fundamental form (see e.g \cite{Lan3, Lan4, HY, LR}) by integrating local differential equations and obtaining global results. We note that the base locus of the second fundamental form at a general point of a smooth variety
is typically not smooth, while this property is preserved by the Hilbert scheme of lines, see Proposition \ref{Yx}.
An important class where the two previous objects coincide is that of {\it quadratic varieties},
that is varieties $X\subset{\mathbb P}^N$ scheme theoretically defined by quadratic equations.
All known {\it prime Fano manifolds of high index},
other than complete intersections (for example many homogeneous manifolds), are quadratic; moreover, they are embedded with {\it small codimension}.
For quadratic varieties the Hilbert scheme of lines through a smooth point is also quadratic,
see Proposition \ref{quadraticLx}. Moreover, since it coincides with the base locus scheme of the second fundamental form, it may be scheme theoretically defined by at most $c=\operatorname{codim}(X)$
(quadratic) equations, see Corollary \ref{quadraticformLx}.
If $X\subset{\mathbb P}^N$ is smooth and $x\in X$ is general,
then $\mathcal L_{x,X}\subset{\mathbb P}^{n-1}$ is also smooth, see Proposition \ref{Yx}. Thus for quadratic manifolds, if $\mathcal L_{x,X}$ is also irreducible, a beautiful {\it matrioska} naturally appears.
From this point of view, a quadratic manifold
$X\subset{\mathbb P}^N$ with $3n>2N$ is a complete intersection
{\it because} $\mathcal L_{x,X}\subset{\mathbb P}^{n-1}$ is a smooth irreducible non-degenerate
complete intersection, defined exactly by $c$ quadratic equations, so that it has
the {\it right dimension}, \cite[Theorems 4.8 and 2.4]{DD} and Remark \ref{ci}.
The aim of this note is twofold:
In Section~ \ref{Lx} we study in detail the intrinsic and extrinsic properties of the Hilbert
scheme of lines passing through a smooth point of an equidimensional connected variety
$X\subset{\mathbb P}^N$, providing an almost self contained treatment. In Section~ 2 we illustrate another incarnation of the principle presented above by studying the problem of extending smooth varieties uniruled by lines as hyperplane sections of irreducible varieties.
First we describe the possible singularities of $\mathcal L_{x,X}$,
proving that a singular point of the Hilbert scheme of lines
passing through a general point $x$ of an irreducible variety produces a line joining $x$ to a singular point of $X$, a stronger condition
than the mere existence of a singular point on $X$, see Proposition \ref{Yx}. Then
we relate the equations defining $X\subset{\mathbb P}^N$ with those of
$\mathcal L_{x,X}\subset{\mathbb P}((t_{x,X})^*)$, see \eqref{eqLxE}. This is applied to quadratic varieties showing that the Hilbert scheme of lines passing through a smooth point is
a quadratic scheme, which coincides with the projectivized tangent cone at $x$ to the scheme $T_xX\cap X$,
see Proposition \ref{quadraticLx}. After introducing the base locus of the second fundamental
form of $X$ at $x$, $B_{x,X}\subset {\mathbb P}((t_{x,X})^*)$, we show that in general
$\mathcal L_{x,X}\subseteq B_{x,X}$ as schemes with equality holding, as schemes, if $X\subset{\mathbb P}^N$ is quadratic, see Corollary
\ref{quadraticformLx},
\cite[Theorem 2.4 and Section~ 4]
{DD} and also Proposition \ref{settheoretically} here. Then we recall some results about lines on prime Fano manifolds to illustrate further
how geometric properties of $X\subset{\mathbb P}^N$ are transferred to $\mathcal L_{x,X}\subset{\mathbb P}((t_xX)^*)$, see Proposition \ref{Fano}
and Example \ref{excomp}.
In Section~ \ref{ext} we consider the classical problem of
the existence of projective extensions $X\subset{\mathbb P}^{N+1}$
of a subvariety $Y\subset {\mathbb P}^N\subset{\mathbb P}^{N+1}$.
It is well known that some special manifolds cannot be hyperplane sections of smooth varieties
and that in some cases only the trivial extensions exist. These are given by cones over
$Y$ with vertex a point $p\in{\mathbb P}^{N+1}\setminus{\mathbb P}^N$ (see e.g. \cite{CSegre}, \cite{Scorza1}, \cite{Scorza2},
\cite{Terracini} and also Section~ \ref{ext} for precise definitions).
Recently the interest in the above problem
(and further generalizations of it) was renewed.
Complete references, many results and
a lot of interesting connections with other
areas, such as deformation theory of isolated singularities, can be found in the monograph \cite{Badescu}, especially relevant for this
problem being Chapters 1 and 5. One could also look at the survey \cite{Zakdual}.
Many sufficient conditions for the non-existence of non-trivial extensions of smooth varieties are known.
These conditions are usually expressed, in the more general setting of extensions as ample divisors, by the vanishing of (infinitely many) cohomology groups
of the twisted tangent bundle of $Y$ (or of its normal bundle in ${\mathbb P}^N$). These results are general and concern a lot of applications, see {\it loc. cit.}, but even in the simplest cases the computation of these
cohomology groups can be quite complicated. In any case their geometrical meaning is not so obvious to the non-expert in the field.
Here we prove a simple geometrical sufficient condition for non-extendability, Theorem \ref{criterion}, for smooth projective
complex varieties uniruled by lines. The simplest version
states that $Y\subset{\mathbb P}^N$
admits only trivial extensions $X\subset{\mathbb P}^{N+1}$ as soon as $\mathcal L_{y,X}\subset {\mathbb P}((t_yX)^*)$ admits no smooth extension (a weaker condition than the thesis!). Indeed, one easily shows in Proposition \ref{extlines}, via the results of Section~ 1, that also
$\mathcal L_{y, X}\subset{\mathbb P}((t_yX)^*)$ is a projective extension of
$\mathcal L_{y,Y}\subset{\mathbb P}((t_yY)^*)$ for $y\in Y$ general. Then under the hypothesis of Theorem \ref{criterion} one deduces the existence of a line through $y$ and a singular point $p_y\in X$. Then $p_y=p$ does not vary with $y\in Y$ general since $X$ has at most a finite number of singular points so that $X\subset{\mathbb P}^{N+1}$ is a cone of vertex $p$.
The range of applications of Theorem \ref{criterion}
is quite wide, see Corollary \ref{Segre}, \ref{exthermitian}, \ref{extcontact}, allowing us to recover
some results previously obtained differently, see \cite{Badescu} and Remark \ref{dualhom}.
We were led to the analysis of the problem of extending smooth varieties by the desire of understanding geometrically why in some well-known examples the geometry of $Y\subset{\mathbb P}^N$ forces that every extension is trivial and
by the curiosity of explicitly constructing the cones extending $Y$.
Moreover, this approach reveals that Scorza's result about the non-extendability of ${\mathbb P}^a\times{\mathbb P}^b\subset {\mathbb P}^{ab+a+b}$ for $a+b\geq 3$, originally proved in \cite{Scorza2} and recovered later by many authors (see e.g. \cite{Badescu} and Corollary \ref{Segre} here), implies the non-extendability of a lot of homogeneous varieties via the
description of their Hilbert scheme of lines.
From this perspective the Pl\" ucker embedding of $Y=\mathbb G(r,m)$, with $1\leq r<m-1$ and for $r=1$
with $m\geq 4$, admits only trivial extensions because $\mathcal L_{y,Y}={\mathbb P}^r\times{\mathbb P}^{m-r-1}$ admits only trivial extensions (see \cite{Fiore} for an ad-hoc proof following Scorza's approach).
Besides the applications contained in Corollary \ref{Segre} and \ref{exthermitian}, we
also show that our analysis can be used to provide a direct proof that
$\nu_2({\mathbb P}^n)\subset{\mathbb P}^{\frac{n(n+3)}{2}}$ admits only trivial extensions, see Proposition \ref{Veronese}, a well-known classical fact originally proved by Scorza in \cite{Scorza1} and later obtained differently by many authors.
{\bf Acknowledgements}.
I am indebted to Paltin Ionescu for his useful remarks and comments leading to an improvement
of the exposition and especially for various discussions on these subjects. Giovanni Staglian\` o read carefully the text
and made useful comments on a preliminary version.
A special thank to Prof. Markus Brodmann for
his invitation to give a talk at the Oberseminar at Z\" urich University in May 2010, for his kind hospitality and for his interest in my work. On that occasion I began to organize the material contained in Section~ 1.
\section{Geometry of (the Hilbert scheme of) lines contained in a variety and passing through a (general) point}\label{Lx}
\subsection{Notation, definitions and preliminary results}\label{prel}
Let $X\subset{\mathbb P}^N$ be a (non-degenerate) connected equidimensional projective variety of dimension $n\geq 1$,
defined over a fixed algebraically closed field of characteristic zero, which
from now on will be simply called a {\it projective variety}. If $X$ is smooth and irreducible, we shall call $X$ a {\it manifold}.
Let $X_{\operatorname{reg}}=X\setminus\operatorname{Sing}(X)$ be the smooth locus of $X$. Let $t_xX$ denote {\it the affine tangent space to $X$ at $x$}, let $T_xX\subset{\mathbb P}^N$ denote
{\it the projective tangent space to $X$ at $x$ of $X\subset{\mathbb P}^N$} and for an arbitrary scheme
$Z$ and for a closed point $z\in Z$ let $C_zZ$ denote {\it the affine tangent cone to $Z$ at $z$}. Let $\mathcal L_{x,X}$
denote the Hilbert scheme of lines contained in $X$ and passing through the point $x\in X$. For a line $L\subset X$
passing through $x$, we let $[L]\in \mathcal L_{x,X}$ be the corresponding point.
Let $\pi_x: \mathcal{H}_x \to \mathcal{L}_{x,X}$ denote the universal family
and let $\phi_x: \mathcal{H}_x \to X$ be the tautological morphism.
From now on we shall always suppose that $x\in X_{\operatorname{reg}}$.
Note that $\pi_x$ admits a section $s_x:\mathcal L_{x,X}\to {\mathcal E}_x\subset\mathcal H_x$, which is contracted by $\phi_x$ to the point $x$. Consider the blowing-up $\sigma_x: \operatorname{Bl}_xX\to X$ of $X$ at $x$. For every $[L]\in\mathcal L_{x,X}$ the line $L=\phi_x(\pi_x^{-1}([L]))$ is smooth at $x$ so that \cite[Lemma 4.3]{IN} and the universal property of the blowing-up ensure the existence of a morphism $\psi_x : {\mathcal H}_x \to \operatorname{Bl}_xX$ such that $\sigma_x \circ \psi_x = \phi_x$. So we have the following diagram
\begin{equation}\label{joindiagram1}\raisebox{.7cm}{\xymatrix{
&\mathcal{H}_x \ar[d]_{\pi_x} \ar[dr]^{\phi_x}\ar[r]^{\psi_x}&\operatorname{Bl}_xX\ar[d]^{\sigma_x}\\
&\mathcal{L}_{x,X}&X.
}}
\end{equation}
In particular, $\psi_x$ maps the section ${\mathcal E}_x$
to $E_x$, the exceptional divisor of $\sigma_x$.
Let $\widetilde \psi _x : {\mathcal E}_x \to E_x$ be the restriction of $\psi_x$
to $\mathcal E_x$.
We can define the morphism
\begin{equation}\label{taux}
\tau_x=\tau_{x,X}=\widetilde \psi_x\circ s_x:\mathcal L_{x,X}\to {\mathbb P}((t_xX)^*)=E_x={\mathbb P}^{n-1},
\end{equation}
which associates to each line $[L]\in \mathcal L_{x,X}$ the corresponding tangent direction through
$x$, i.e. $\tau_x([L])={\mathbb P}((t_xL)^*)$. The morphism $\tau_x$ is clearly injective and we claim that $\tau_x$ is a closed immersion.
Indeed, by taking in the previous construction $X={\mathbb P}^N$ the corresponding morphism $\tau_{x,{\mathbb P}^N}:\mathcal L_{x,{\mathbb P}^N}\to {\mathbb P}((t_x{\mathbb P}^N)^*)={\mathbb P}^{N-1}$ is an isomorphism between $\mathcal L_{x,{\mathbb P}^N}$ and the exceptional divisor of $\operatorname{Bl}_x{\mathbb P}^N$. By definition
the inclusion $X\subset{\mathbb P}^N$ induces a closed embedding $i_x:\mathcal L_{x,X}\to\mathcal L_{x,{\mathbb P}^N}$. If $j_x:{\mathbb P}((t_xX)^*)\to
{\mathbb P}((t_x{\mathbb P}^N)^*)$ is the natural closed embedding, then we have the following commutative diagram
\begin{equation}\label{diagram2}{\xymatrix{
&\mathcal{L}_{x,X} \ar[d]_{i_x} \ar[r]^{\tau_{x,X}}&{\mathbb P}((t_xX)^*)\ar[d]^{j_x}\\
&\mathcal{L}_{x,{\mathbb P}^N}\ar[r]^{\tau_{x,{\mathbb P}^N}}&{\mathbb P}((t_x{\mathbb P}^N)^*),
}}
\end{equation}
proving the claim.
For $x\in X_{\operatorname{reg}}$ such that $\mathcal L_{x,X}\neq\emptyset$, we shall always identify $\mathcal L_{x,X}$ with $\tau_x(\mathcal L_{x,X})$ and we shall naturally consider $\mathcal L_{x,X}$ as a subscheme of ${\mathbb P}^{n-1}={\mathbb P}((t_xX)^*)$.
We denote by $\mathcal C_x$ the scheme theoretic image of $\mathcal H_x$, that is $\phi_x(\mathcal H_x)=\mathcal C_x\subset X$. Via \eqref{joindiagram1}
we deduce the following relation:
\begin{equation}\label{tangentconeCx}
{\mathbb P}(C_x(\mathcal C_x))=\mathcal L_{x,X},
\end{equation}
as subschemes of ${\mathbb P}((t_xX)^*)$, where ${\mathbb P}(C_x(\mathcal C_x))$
is the {\it projectivized tangent cone to $\mathcal C_x$ at $x$}, see \cite[II,Section~ 3]{Mumford}.
\subsection{Singularities of $\mathcal L_{x,X}$}\label{sing}
We begin by studying the intrinsic geometry of $\mathcal L_{x,X}\subset{\mathbb P}^{n-1}$. When it is clear from the context which variety $X\subset{\mathbb P}^N$ we are considering we shall write $\mathcal L_x$ instead of $\mathcal L_{x,X}$.
The normal bundle $N_{L/X}$ is locally free being a subsheaf of the locally free sheaf $N_{L/{\mathbb P}^N}\simeq\O_{{\mathbb P}^1}(1)^{N-1}$.
If $L\cap X_{\operatorname{reg}}\neq\emptyset$, then $N_{L/X}$ is locally free of rank $n-1$ and more precisely
\begin{equation}\label{ai1}
N_{L/X}\simeq\bigoplus_{i=1}^{n-1} \O_{{\mathbb P}^1}(a_i),
\end{equation}
with $a_i\leq 1$ since $N_{L/X}\subset N_{L/{\mathbb P}^N}$.
If $N_{L/X}$ is
also generated by global sections, then
\begin{equation}\label{split}
N_{L/X}\simeq \O_{{\mathbb P}^1}(1)^{s(L,X)}\oplus\O_{{\mathbb P}^1}^{n-1-s(L,X)}.
\end{equation}
Therefore if $N_{L/X}$ is generated by global sections, then $\mathcal L_x$ is unobstructed at $[L]$, that is $h^1(N_{L/X}(-1))=0$,
$\mathcal L_x$ is smooth at $[L]$ and $\dim_{[L]}(\mathcal L_x)=h^0(N_{L/X}(-1))=s(L,X)$,
where $s(L,X)\geq 0$ is the integer defined in \eqref{split}.
For $x\in X_{\operatorname{reg}}$, let
$$S_x=S_{x,X}=\{[L]\in \mathcal L_x\text{ such that } L\cap \operatorname{Sing}(X)\neq \emptyset\;\}\subseteq \mathcal L_x.$$
Then $S_{x,X}$ has a natural scheme structure and the previous inclusion holds at the scheme theoretic level.
If $X$ is smooth, then $S_{x,X}=\emptyset$. Moreover, if $L\subset X$ is a line passing through $x\in X_{\operatorname{reg}}$, clearly
$[L]\not\in S_{x,X}$ if and only if $L\subset X_{\operatorname{reg}}$.
\medskip
We now prove that a singular point of $\mathcal L_x$
produces a line passing through $x$ and through a singular point of $X$, a stronger condition than
the mere existence of a singular point on $X$. These results are well known to experts, at least for manifolds, see \cite[Proposition 1.5]{Hwang} and also
\cite[Proposition 2.2]{QEL1}. In \cite{DG}, the singularities of the Hilbert scheme of lines contained in a projective variety are related to some geometrical
properties of the variety.
\medskip
\begin{Proposition}\label{Yx}
Let notation be as above and let $X\subset{\mathbb P}^N$ be an irreducible projective variety of dimension $n\geq 2$. Then for $x\in X_{\operatorname{reg}}$ general:
\begin{enumerate}
\item $\mathcal L_x\subset{\mathbb P}^{n-1}$ is smooth outside $S_{x,X}$, that is $\operatorname{Sing}(\mathcal L_x)\subseteq S_x.$
In particular if $X\subset{\mathbb P}^N$ is smooth and if $x\in X$ is general, then $\mathcal L_x\subset{\mathbb P}^{n-1}$
is a smooth variety.
\medskip
\item If $\mathcal L_x^j$, $j=1,\ldots,m$, are the irreducible components
of $\mathcal L_x$ and if
\[\dim(\mathcal L_x^l)+\dim(\mathcal L_x^p)\geq n-1 \quad \mbox{for some } l\neq
p,\]
then $\mathcal L_x$ is singular, $X$ is singular and there exists a line $[L]\in \mathcal L_x$ such
that $L\cap \operatorname{Sing}(X)\neq\emptyset$.
\end{enumerate}
\end{Proposition}
\begin{proof}
There exists an open dense subset $U\subseteq X_{\operatorname{reg}}$ such that for
every line $L\subset X_{\operatorname{reg}}$ such that $L\cap U\neq \emptyset$
the normal bundle $N_{L/X}$ is generated by global sections,
see for example \cite[Proposition 4.14]{Debarre}. Combining this result
with the above discussion, we deduce that for every $x\in U$ the variety
$\mathcal L_x\subset{\mathbb P}^{n-1}$ is smooth outside $S_x$, proving the first assertion.
The condition on
the dimensions of two irreducible components of $\mathcal L_x$ in (2) ensures that these
components have to intersect in ${\mathbb P}^{n-1}$. A point of intersection is a singular
point of $\mathcal L_x\subset{\mathbb P}^{n-1}$. This forces $X$ to be singular by the first part and
also the existence of
a line $[L]\in S_x$, which by definition cuts $\operatorname{Sing}(X)$.
\end{proof}
\subsection{Equations for $\mathcal L_{x,X}\subset{\mathbb P}((t_xX)^*)$}\label{equations}
\medskip
We now follow and expand the treatment outlined in \cite[Theorem 2.4]{DD} by looking at the equations defining $\mathcal L_x\subset{\mathbb P}^{n-1}$ for $x\in X_{\operatorname{reg}}$.
Let $$ X=V(f_1,\ldots, f_m)\subset{\mathbb P}^{N}\hskip 3cm (\ast),$$
be a projective equidimensional connected variety, not necessarily irreducible, let $x\in X_{\operatorname{reg}}$, let $n=\dim(X)$ and let $c=\operatorname{codim}(X)=N-n$. Thus we are assuming that
$X\subset{\mathbb P}^N$ is scheme theoretically the intersection of $m\geq 1$
hypersurfaces of degrees $d_1\geq d_2\geq\ldots\geq d_m\geq 2.$ Moreover it is implicitly assumed that $m$ is minimal, i.e. none of the hypersurfaces contains the intersection of the others.
Define, following \cite{DD}, the integer
$$d:=\min\{\sum_{i=1}^{c}(d_i-1)\text{ for expressions $(\ast)$ as above}\}\geq c.$$
With these definitions $X\subset{\mathbb P}^N$ (or more generally a scheme $Z\subset{\mathbb P}^N$) is called {\it quadratic} if it is scheme theoretically an intersection of quadrics, which means that we can assume $d_1=2$. In particular $X\subset{\mathbb P}^N$ is quadratic if and only if $d=c$.
\medskip
We can choose homogeneous coordinates $(x_0:\ldots:x_N)$ on ${\mathbb P}^N$ such that $x=(1:0:\ldots:0),$ $T_xX=V(x_{n+1},\ldots, x_N).$ Let $\mathbb A^N={\mathbb P}^N\setminus V(x_0)$ with affine coordinates $(y_1,\ldots, y_N)$, that is $y_l=\frac{x_l}{x_0}$ for every $l=1,\ldots, N$. Let $\widetilde {\mathbb P}^N=\operatorname{Bl}_x{\mathbb P}^N$ with exceptional divisor $E'\simeq{\mathbb P}((t_x{\mathbb P}^N)^*)={\mathbb P}^{N-1}$ and let $\widetilde X=\operatorname{Bl}_xX$ with exceptional
divisor $E={\mathbb P}((t_xX)^*)={\mathbb P}^{n-1}$. Looking at the graph of the projection from $x$ onto $V(x_0)$ we can naturally
identify the projectivization of $\mathbb A^N\setminus \mathbf 0=\mathbb A^N\setminus x$ with $E'$ and with the projective hyperplane $V(x_0)={\mathbb P}^{N-1}$.
Let $f_i=f_i^1+f_i^2+\cdots+f_i^{d_i}$, with $f_i^j$ homogeneous of degree $j$ in the variables $(y_1,\ldots, y_N)$. So
$f_1^1=\ldots=f_m^1=0$ are the equations of $t_xX=T_xX\cap\mathbb A^N\subset\mathbb A^N$,
which reduce to $y_{n+1}=\ldots=y_N=0$ by the previous choice of coordinates, yielding
$$V(f_1^1,\cdots, f_m^1)={\mathbb P}((t_xX)^*)\subset {\mathbb P}(( t_x{\mathbb P}^N)^*)={\mathbb P}^{N-1}.$$
With the previous identifications $\mathcal L_{x,{\mathbb P}^N}=E'={\mathbb P}^{N-1}={\mathbb P}((t_x{\mathbb P}^N)^*)$. We now write a set of equations defining $\mathcal L_x\subset E\subset E'$ as a subscheme of $E'$ and of $E$. By definition $\mathbf y=(y_1:\ldots:y_n)$ are homogeneous coordinates on
$E\subset E'$. For every $j=2,\ldots, m$ and for every $i=1,\ldots, m$, let
$$\widetilde f_i^j(\mathbf y)=f_i^j(y_1,\ldots,y_n,0,0,\ldots,0,0).$$
Then we have that $\mathcal L_x\subset E'$ is the scheme $$
V(f_1^1,f_1^2,\cdots,f_1^{d_1}, \cdots, f_m^1,f_m^2,\cdots,f_m^{d_m})\subset E',$$ while $\mathcal L_x\subset E$ is the scheme
\begin{equation}\label{eqLxE}
V(\widetilde f_1^2,\cdots,\widetilde f_1^{d_1}, \cdots, \widetilde f_m^2,\cdots,\widetilde f_m^{d_m}),
\end{equation}
so that it is scheme theoretically defined by at most $\sum_{i=1}^m(d_i-1)$ equations.
The equations of $T_xX\cap X\cap \mathbb A^N=t_xX\cap X\cap \mathbb A^N$, as a subscheme of $\mathbb A^N$, are
$$V(f_1^1, \ldots, f_m^1, f_1^1+f_1^2+\cdots+f_1^{d_1},\ldots, f_m^1+f_m^2+\cdots+f_m^{d_m})=$$
\begin{equation}\label{eqTxAN}
V(f_1^1, \ldots, f_m^1, f_1^2+\cdots+f_1^{d_1},\ldots, f_m^2+\cdots+f_m^{d_m})\subset\mathbb A^N.
\end{equation}
Thus the equations of $T_xX\cap X\cap\mathbb A^N=t_xX\cap X\cap\mathbb A^N$ as a subscheme of $t_x(X\cap \mathbb A^N)=t_xX$ are
\begin{equation}\label{eqTxtx}
V(\widetilde f_1^2+\cdots+\widetilde f_1^{d_1},\ldots, \widetilde f_m^2+\cdots+\widetilde f_m^{d_m})\subset t_xX=\mathbb A^n.
\end{equation}
Let $I=\langle \widetilde f_1^2+\cdots+\widetilde f_1^{d_1},\ldots, \widetilde f_m^2+\cdots+\widetilde f_m^{d_m}\rangle\subset\mathbb C[y_1,\ldots,y_n]=S$ and let $I^*$ be the ideal generated by the {\it initial forms} of elements of $I$. Remark that if $I$ is homogeneous and generated by forms of the same degree, then clearly $I=I^*$. Then the affine tangent cone to $T_xX\cap X$ at $x$ is $C_x(T_xX\cap X)={\rm Spec}(\frac{S}{I^*})$ so that
\begin{equation}\label{exp1}
{\mathbb P}(C_x(T_xX\cap X))={\rm Proj} (\frac{S}{I^*}),
\end{equation}
see \cite[III, Section~\,3]{Mumford}.
Let $J\subset S$ be the homogeneous ideal generated by the polynomials in
\eqref{eqLxE} defining $\mathcal L_{x,X}$ scheme theoretically, that is
$\mathcal L_{x,X}={\rm Proj}(\frac{S}{J})\subset{\mathbb P}((t_xX)^*)$. Clearly $I^*\subseteq J$, yielding the closed embedding of schemes
\begin{equation}\label{inclLxBx}
\mathcal L_{x,X}\subseteq {\mathbb P}(C_x(T_xX\cap X)).
\end{equation}
If $X\subset{\mathbb P}^N$ is quadratic, then $I=I^*=J$. In conclusion we have proved the following results.
\medskip
\begin{Proposition}\label{quadraticLx} Let $X\subset{\mathbb P}^N$ be a (non-degenerate) projective variety, let $x\in X_{\operatorname{reg}}$ be a point and let notation be as above.
If $X\subset{\mathbb P}^N$ is quadratic, then
\begin{equation}\label{fund0}
T_xX\cap X\cap\mathbb A^N=C_x(T_xX\cap X)\subset t_xX
\end{equation}
and
\begin{equation}\label{fund}
\mathcal L_{x,X}={\mathbb P}(C_x(T_xX\cap X))\subset{\mathbb P}((t_xX)^*).
\end{equation}
In particular if $X\subset{\mathbb P}^N$ is quadratic, then the scheme $\mathcal L_{x,X}\subset{\mathbb P}((t_xX)^*)$
is quadratic.
\end{Proposition}
\subsection{$\mathcal C_x$ versus $T_xX\cap X$ for a quadratic variety}\label{conesex}
The closed embedding \eqref{inclLxBx} holds at the scheme theoretic level.
If $\mathcal L_{x,X}$ were reduced, or better smooth, it would be enough to prove that there exists an inclusion as sets.
Since $x\in X_{\operatorname{reg}}$ was arbitrary we cannot control
a priori the structure of $\mathcal L_{x,X}$ even if $X\subset{\mathbb P}^N$ is a manifold.
Recall that by Proposition \ref{Yx} $\mathcal L_{x,X}$ is smooth as soon as $X$ is a manifold and
$x\in X$ is a general point.
{\it From now on we shall suppose $X\subset{\mathbb P}^N$ quadratic}. Then
\begin{enumerate}
\item $(\mathcal C_x)_{\operatorname{red}}=(T_xX\cap X)_{\operatorname{red}}$;
\item if $X\subset{\mathbb P}^N$ is a manifold and if
$x\in X$ is a general point, then $\mathcal C_x=(T_xX\cap X)_{\operatorname{red}}$;
\item the strict transforms of $\mathcal C_x$ and of $T_xX\cap X$ on $\operatorname{Bl}_xX$ cut the exceptional
divisor $E={\mathbb P}((t_xX)^*)$ of $\operatorname{Bl}_xX$ in the same scheme $\mathcal L_{x,X}$ (see \eqref{tangentconeCx} and \eqref{fund});
\item if $x\in X$ is a general point on
a quadratic manifold $X\subset{\mathbb P}^N$ and if $I^*$ is saturated, then $T_xX\cap X$ is reduced in a neighborhood of $x$ so
that it coincides with $\mathcal C_x$ in a neighborhood of $x$. Indeed since $T_xX\cap X\cap \mathbb A^n={\rm Spec}(\frac{S}{I})$,
with $I=I^*=J$ homogeneous and saturated, it follows that $T_xX\cap X$ is reduced at $x$; therefore it is reduced also in a neighborhood of $x$.
\end{enumerate}
Already for quadratic manifolds there exist many important differences between ${\mathbb P}(C_x(T_xX\cap X))\subset{\mathbb P}((t_xX)^*)$
and $C_x(T_xX\cap X)=T_xX\cap X\cap \mathbb A^N\subset t_xX$ and also between $T_xX\cap X$ and the cone $\mathcal C_x\subseteq T_xX\cap X$. We shall discuss some examples in order to analyze closer these important schemes containing a lot of
geometrical information.
\begin{Example} ({\it $T_xX\cap X$ non-reduced only at $x$})\label{exscrolls}
Remark that $t_x(T_xX\cap X)=t_xX$ so that
$\langle C_x(T_xX\cap X)\rangle=t_xX$, while in some cases ${\mathbb P}(C_x(T_xX\cap X))$
is degenerate in ${\mathbb P}((t_xX)^*)$. Consider a rational normal scroll $X\subset{\mathbb P}^N$, different from the Segre varieties ${\mathbb P}^1\times{\mathbb P}^{n-1}$, $n\geq 2$, and a general point $x\in X$. It is well known that $X\subset{\mathbb P}^N$ is quadratic so that $\mathcal L_{x,X}={\mathbb P}(C_x(T_xX\cap X))\subset{\mathbb P}((t_xX)^*)$ by \eqref{inclLxBx}. On the other hand, if ${\mathbb P}^{n-1}_x$ is the unique ${\mathbb P}^{n-1}$ of the ruling passing through $x\in X$, it is easy to see, letting notation as above, that in this case
$$\mathcal L_{x,X}={\mathbb P}({\mathbb P}^{n-1}_x\cap\mathbb A^n)={\mathbb P}^{n-2}\subset{\mathbb P}((t_xX)^*)={\mathbb P}^{n-1}.$$
This is possible because in this example $T_xX\cap X$ and $C_x(T_xX\cap X)$ are not reduced at $x$. Indeed, the point $x\in C_x(T_xX\cap X)$ corresponds to the irrelevant ideal of $S$. $I^*$ is not saturated, because the equation defining the hyperplane $\mathcal L_{x,X}$ belongs to the saturation of $I^*$, but is not in $I^*$ ($I^*$ is generated
by quadratic polynomials!).
\end{Example}
\medskip
In the case of rational normal scrolls discussed in Example \ref{exscrolls} we saw that $T_xX\cap X\setminus x=\mathcal C_x\setminus x$ as schemes, the affine tangent cones are different affine schemes, but the projectivized tangent cones coincide.
By choosing suitable quadrics $Q_1,\ldots, Q_c$
we shall see in subsection \ref{implicit} that the complete intersection $Y=Q_1\cap\ldots\cap Q_c$ coincides locally with $X$ around $x$. Thus $T_xY\cap Y$ and $T_xX\cap X$ coincide locally around $x$.
In particular the intersection of their strict transform on $\operatorname{Bl}_xX$ with the exceptional
divisor is the same, so that $\mathcal L_{x,X}=\mathcal L_{x,Y}$ and the last scheme
can be defined scheme theoretically by $r\leq c$ linearly independent quadrics by \eqref{eqLxE}.
In any case the double nature of $T_xX\cap X$ as a subscheme of $T_xX$ and $X$
plays a central role for its infinitesimal properties at $x$, measured exactly by
${\mathbb P}(C_x(T_xX\cap X))\subset{\mathbb P}((t_xX)^*)$.
It is useful to think of ${\mathbb P}(C_x(T_xX\cap X))\subset{\mathbb P}((t_xX)^*)$ as being the base locus scheme of the restriction to the exceptional divisor over $x$ of the projection of $X$ from $T_xX$, as we shall
do in the next section. We shall provide in this way another reason why $\mathcal L_{x,X}$
can be defined scheme theoretically by at most $c$ quadratic equations for an arbitrary point
$x\in X_{\operatorname{reg}}$.
\subsection{Tangential projection and second fundamental form}\label{second}
There are several possible equivalent
definitions of the projective second fundamental form
$|II_{x,X}|\subseteq{\mathbb P}(S^2(t_xX))$ of a connected equidimensional
projective variety $X\subset{\mathbb P}^N$ at $x\in X_{\operatorname{reg}}$, see
for example \cite[3.2 and end of Section~3.5]{IL}. We use the
one related to tangential projections, as in \cite[Remark
3.2.11]{IL}.
Suppose $X\subset{\mathbb P}^N$ is non-degenerate, as always,
let $x\in X_{\operatorname{reg}}$ and
consider the projection from $T_xX$ onto a disjoint ${\mathbb P}^{c-1}$
\begin{equation}\label{tangentdef}
\pi_x:X\dasharrow W_x\subseteq{\mathbb P}^{c-1}.
\end{equation}
The map $\pi_x$ is not defined along the scheme $T_xX\cap X$, which contains $x$,
and it is associated to the linear system of hyperplane
sections cut out by hyperplanes containing $T_xX$, or equivalently
by the hyperplane sections singular~at~$x$.
Let $\phi:\operatorname{Bl}_xX\to X$ be the blow-up of $X$ at $x$, let
\[E={\mathbb P}((t_xX)^*)={\mathbb P}^{n-1}\subset\operatorname{Bl}_xX\] be the exceptional
divisor and let $H$ be a hyperplane section of $X\subset{\mathbb P}^N$. The
induced rational map $\widetilde{\pi}_x:\operatorname{Bl}_xX\dasharrow{\mathbb P}^{c-1}$ is
defined as a rational map along $E$ since $X\subset{\mathbb P}^N$ is not a linear space,
see also the discussion below.
The restriction of
$\widetilde{\pi}_x$ to $E$ is given by a linear system in
$|\phi^*(H)-2E|_{|E}\subseteq|-2E_{|E}|=|\O_{{\mathbb P}((t_xX)^*)}(2)|={\mathbb P}(S^2(t_xX))$, whose base locus
scheme will be denoted by $B_{x,X}$.
Consider the strict transform scheme of $T_xX\cap X$ on $\operatorname{Bl}_xX$, denoted from now on by
$\widetilde T=\operatorname{Bl}_x(T_xX\cap X)$. Then $\widetilde T$ is the base locus scheme of $\widetilde{\pi}_x$ and the restriction of
$\widetilde{\pi}_x$ to $E$ has base locus scheme equal to
\begin{equation}\label{Base}
\widetilde T\cap E={\mathbb P}(C_x(T_xX\cap X))=B_{x,X}\subset {\mathbb P}((t_xX)^*).
\end{equation}
\begin{Definition}
The {\it second fundamental form}
$|II_{x,X}|\subseteq{\mathbb P}(S^2(t_xX))$ of a connected equidimensional
non-degenerate projective variety $X\subset{\mathbb P}^N$ of dimension $n\geq 2$ at a
point $x\in X_{\operatorname{reg}}$ is the non-empty linear system of quadric
hypersurfaces in ${\mathbb P}((t_xX)^*)$ defining the restriction of
$\widetilde{\pi}_x$ to $E$ and $B_{x,X}\subset{\mathbb P}((t_xX)^*)$ is the so called {\it base locus scheme
of the second fundamental form of $X$ at $x$}.
\end{Definition}
Clearly $\dim(|II_{x,X}|)\leq c-1$ and
$\widetilde{\pi}_x(E)\subseteq W_x\subseteq{\mathbb P}^{c-1}$.
Let $\widetilde I\subset S$ be the homogeneous ideal generated by the $r\leq c$ linearly independent quadratic forms in the second fundamental form of $X$ at $x$. Then via \eqref{Base} we obtain
\begin{equation}\label{projBx}
{\rm Proj (\frac{S}{\widetilde I})}=B_{x,X}={\mathbb P}(C_x(T_xX\cap X))={\rm Proj (\frac{S}{I^*})}\subset{\mathbb P}((t_xX)^*).
\end{equation}
In conclusion we have proved the following results by combining \eqref{Base} with \eqref{fund} and \eqref{projBx}.
\medskip
\begin{Corollary}\label{quadraticformLx} Let $X\subset{\mathbb P}^N$ be a non-degenerate projective variety, let $x\in X_{\operatorname{reg}}$ be a point and let notation be as above. Then:
\begin{enumerate}
\item $\mathcal L_{x,X}\subseteq B_{x,X}$;
\item if $X\subset{\mathbb P}^N$ is quadratic, then equality holds and
$\mathcal L_{x,X}\subset{\mathbb P}((t_xX)^*)$ can be defined scheme theoretically by the $r\leq c$ quadratic equations defining the second fundamental form of $X$ at $x$.
\end{enumerate}
\end{Corollary}
\medskip
\begin{Remark}\label{ci} The previous result has many important applications. We recall that, as proved in \cite{DD}, if $X\subset{\mathbb P}^N$ is
a quadratic manifold and if $c\leq\frac{n-1}{2}$, then, for $x\in X$ general, $\mathcal L_{x,X}\subset{\mathbb P}((t_xX)^*)$ is the complete intersection
of the $c$ linearly independent quadratic polynomials defining $|II_{x,X}|$. Then $\mathcal L_{x,X}$ has dimension $n-1-c$
from which it follows that $X\subset{\mathbb P}^N$ is a complete intersection. This proves the Hartshorne Conjecture
on complete intersections in the quadratic case and also leads to the classification of quadratic Hartshorne manifolds, see \cite[Theorem 2.4 and Section 4]{DD} for details.
The paper \cite{PR} considers also irreducible projective varieties $X\subset{\mathbb P}^{2n+1}$ which are
3--covered by twisted cubics, i.e. such that through three general points of $X\subset{\mathbb P}^{2n+1}$ there passes a twisted cubic contained in $X$. A key remark for the classification of these varieties is \cite[Theorem 5.2]{PR}, which among other things shows that for such an $X$ the equality $\mathcal L_{x,X}=B_{x,X}$ holds for $x\in X$ general.
A posteriori all the known examples of varieties 3--covered by twisted cubics are projectively equivalent to the so called {\it twisted cubics over Jordan algebras}, which are quadratic, see {\it loc. cit} for definitions and details.
This fact has also many important consequences for the theory of Jordan algebras and for the classification of {\it quadro-quadric}
Cremona transformations, as shown in the forthcoming paper \cite{PR2}.
\end{Remark}
\subsection{Approach to $B_{x,X}=\mathcal L_{x,X}$ via \cite{BEL}}\label{implicit}
For manifolds $X\subset{\mathbb P}^N$ there is another approach
based on a construction of \cite{BEL} elaborating and generalizing an idea due to Severi, see {\it loc. cit.}
It can be used to give a proof of a weaker form of Corollary \ref{quadraticformLx}
(in the sense that we shall prove it only for $x\in X$ general); this approach illustrates the local nature of the second fundamental form.
Let us remark that the treatment in the general setting developed in the previous sections
is unavoidable because the point $x\in X$ is not necessarily general on the complete intersection $Y\supseteq X$ we now construct.
It was proved in \cite{BEL} that given a manifold $X=V(f_1,\ldots, f_m)\subset{\mathbb P}^N$ as above, we can choose $g_i\in H^0({\mathcal I}_X(d_i))$, $i=1,\ldots, c$ such that
\begin{equation}\label{YX}
Y=V(g_1,\ldots, g_c)=X\cup X',
\end{equation}
where $X'$ (if nonempty) meets $X$ in a divisor $D$. Moreover from \eqref{YX} it follows
\begin{equation}\label{chernD}
\O_X(D)\simeq\det(\frac{{\mathcal I}_X}{{\mathcal I}_X^2})\otimes\O_X(\sum_{i=1}^{c}d_i)\simeq
\O_X(d-n-1)\otimes\omega_X^*,
\end{equation}
see also \cite[pg. 597]{BEL}. We now illustrate the usefulness of this construction
by proving some facts and results contained
in \cite[Theorem 2.4]{DD}.
\medskip
Suppose that $X\subset {\mathbb P}^N$ is a quadratic manifold and consider a point $x\in U=X\setminus\operatorname{Supp}(D)$. By definition $Y\setminus\operatorname{Supp} (D)=U\amalg V$, where $V=X'\setminus\operatorname{Supp}(D)$.
Consider the two schemes $T_xX\cap X\cap U$ and $T_xY\cap Y\cap U$. Since $t_xX=t_xY$ and since $Y\cap U=X\cap U$
by the above construction, we obtain the equality as schemes
$$C_x(T_xX\cap X)=C_x(T_xX\cap X\cap U)=C_x(T_xY\cap Y\cap U)=C_x(T_xY\cap Y).$$
Via \eqref{fund} we deduce the following equality as subschemes of ${\mathbb P}((t_xX)^*)$:
\begin{equation}\label{asinDD}
\mathcal L_{x,Y}={\mathbb P}(C_x(T_xY\cap Y))={\mathbb P}(C_x(T_xX\cap X))=\mathcal L_{x,X}.
\end{equation}
Since $\mathcal L_{x,Y}$ can be scheme-theoretically defined by $r\leq c$ linearly independent quadratic equations, the same is true for $\mathcal L_{x,X}$.
Now, without assuming anymore that $X$ is quadratic, since $x\in X$ is general, $\mathcal L_{x,X}$ is smooth and hence reduced. Clearly a line $L$ passing through $x$ is contained in $X$
if and only if it is contained in $Y$, yielding $\mathcal L_{x,X}=(\mathcal L_{x,Y})_{\operatorname{red}}$, see \cite[Theorem 2.4]{DD}. We proved:
\medskip
\begin{Proposition}\label{settheoretically} Let $X\subset{\mathbb P}^N$ be a manifold, let notation be as above and let $x\in U$ be a general point. Then:
\begin{enumerate}
\item $\mathcal L_{x,X}=(\mathcal L_{x,Y})_{\operatorname{red}}$ so that $\mathcal L_{x,X}$ can be defined
set theoretically by the $r\leq d$ equations defining $\mathcal L_{x,Y}$ scheme theoretically.
In particular, if $d\leq n-1$, then $\mathcal L_{x,X}\neq \emptyset$.
\item If $X\subset{\mathbb P}^N$ is quadratic, then
$\mathcal L_{x,X}=\mathcal L_{x,Y}$ so that $\mathcal L_{x,X}\subset{\mathbb P}((t_xX)^*)$ is a quadratic manifold defined
scheme theoretically by at most $c$ quadratic equations.
\end{enumerate}
\end{Proposition}
\subsection{Lines on prime Fano manifolds}\label{Fano}
Let $X\subset{\mathbb P}^N$ be a (non-degenerate) manifold of dimension $n\geq 2$.
For a general point $x\in X$ we know that $\mathcal L_x\subset{\mathbb P}^{n-1}$ is smooth, Proposition \ref{Yx}.
There are well-known examples when $\mathcal L_x\subset{\mathbb P}^{n-1}$ is not irreducible, such as $X={\mathbb P}^a\times{\mathbb P}^b\subset{\mathbb P}^{ab+a+b}$ Segre embedded, and also examples where $\mathcal L_x\subset{\mathbb P}^{n-1}$ is degenerate,
see Example \ref{exscrolls} and also table \eqref{contact} below. A relevant class of manifolds where the properties of smoothness, irreducibility
and non-degeneracy of $X\subset{\mathbb P}^N$ are transfered to $\mathcal L_x\subset{\mathbb P}^{n-1}$ consists of prime Fano manifolds of high index, which we now define.
A manifold $X\subset{\mathbb P}^N$ is called a {\it prime Fano manifold} if $-K_X$ is ample and if
$\operatorname{Pic}(X)\simeq\mathbb Z\langle\O(1)\rangle$. The {\it index of $X$} is the positive integer
defined by $-K_X=i(X)H$, with $H$ a hyperplane section of $X\subset{\mathbb P}^N$.
Let us recall some fundamental facts. Part (1) below is well known and follows from the previous discussion except for a fundamental
Theorem of Mori which implies that for prime Fano manifolds of index greater than $\frac{n+1}{2}$, necessarily $\mathcal L_x\neq\emptyset$, see \cite{Mori} and \cite[Theorem V.1.6]{Kollar}.
\medskip
\begin{Proposition} Let $X\subset{\mathbb P}^N$ be a projective manifold and let $x\in X$ be a general
point. Then
\begin{enumerate}
\item If $\mathcal L_x\neq\emptyset$, then for every $[L]\in\mathcal L_x$
we have $\dim_{[L]}(\mathcal L_x)=-K_X\cdot L-2.$
In particular for prime Fano manifolds of index $i(X)\geq \frac{n+3}{2}$
the variety $\mathcal L_x\subset{\mathbb P}^{n-1}$ is irreducible (and in particular non-empty!).
\item {\rm (\cite{Hwang})} If $X\subset{\mathbb P}^N$ is a prime Fano manifold
of index $i(X)\geq \frac{n+3}{2}$, then $\mathcal L_x\subset{\mathbb P}^{n-1}$ is a
non-degenerate manifold of dimension $i(X)-2$.
\end{enumerate}
\end{Proposition}
\medskip
Let us finish this section by looking at another significant example in which meaningful geometrical properties of $X\subset{\mathbb P}^N$ are reflected in
similar properties of $\mathcal L_x\subset{\mathbb P}^{n-1}$, when this is non-empty.
\medskip
\begin{Example}\label{excomp}
Let $X\subset{\mathbb P}^N$ be a smooth complete intersection of type $(d_1,d_2,\ldots,d_c)$ with $d_c\geq 2$. Then:
\begin{itemize}
\item if $n+1-d>0,$ then $X$ is a Fano manifold and $i(X)=n+1-d$;
\item if $n\geq 3$, then $\operatorname{Pic}(X)\simeq\mathbb Z\langle\O(1)\rangle$;
\item if $i(X)\geq 2$, then $\mathcal L_x\neq\emptyset$ and for every $[L]\in\mathcal L_x$ we have $$\dim_{[L]}(\mathcal L_x)=(-K_X\cdot L)-2=i(X)-2=n-1-d\geq 0 ,$$ so that $\mathcal L_x\subset{\mathbb P}^{n-1}$ is a smooth complete intersection of type $$(2,\ldots,d_1;
2,\ldots, d_2;\ldots;2,\ldots d_{c-1}; 2,\ldots,d_c)$$ since it is scheme theoretically defined by the $d$ equations in \eqref{eqLxE}.
\end{itemize}
\end{Example}
\section{A condition for non-extendability}\label{ext}
\begin{Definition} Let us consider $H={\mathbb P}^N$ as a hyperplane in ${\mathbb P}^{N+1}$. Let $Y\subset{\mathbb P}^N=H$
be a smooth (non-degenerate) irreducible variety of dimension $n\geq 1$. An irreducible variety
$X\subset{\mathbb P}^{N+1}$ will be called {\it an extension of $Y$} if
\medskip
\begin{enumerate}
\item $\dim(X)=\dim(Y)+1$;
\medskip
\item $Y=X\cap H$ as a scheme.
\end{enumerate}
\end{Definition}
\medskip
For every $p\in {\mathbb P}^{N+1}\setminus H$, the irreducible cone
$$X=S(p,Y)=\bigcup_{y\in Y}<p,y>\subset{\mathbb P}^{N+1}$$
is an extension of $Y\subset{\mathbb P}^N=H$, which will be called {\it trivial}. Let us observe that for any extension $X\subset{\mathbb P}^{N+1}$ of $Y\subset{\mathbb P}^N$ we necessarily have $\#(\operatorname{Sing}(X))<\infty$ since $X$ is smooth along the very ample divisor $Y=X\cap H$.
We also remark that in our definition $Y$ is a fixed hyperplane section. In the classical approach
usually it was required that $H$ was a general hyperplane section of $X$, see for example \cite{Scorza1}.
Under these more restrictive hypotheses one can always suppose that a general point on $Y$ is also a general point on $X$.
\medskip
\subsection{Extensions of $\mathcal L_{x,Y}\subset{\mathbb P}^{n-1}$ via $\mathcal L_{x,X}\subset{\mathbb P}^n$}
Let $y\in Y$ be a general point and let us consider an extension $X\subset{\mathbb P}^{N+1}$ of $Y$ and an irreducible component $\mathcal L_{y,Y}^j$ of $\mathcal L_{y,Y}\subset{\mathbb P}^{n-1}$,
which is a smooth irreducible variety by Proposition \ref{Yx}. The results of Section~ 1 yield that this property is immediately translated in terms of Hilbert schemes of lines. Indeed we deduce the following result, where part (4) requires an ad hoc proof
since in our hypotheses the point $y\in Y$ is general on $Y$, but not necessarily on $X$, so that we cannot apply
Proposition \ref{Yx}.
\medskip
\begin{Proposition}\label{extlines} Let $X\subset{\mathbb P}^{N+1}$ be an irreducible projective variety which is an extension of
the non-degenerate manifold $Y\subset{\mathbb P}^N$. Let $n=\dim(Y)\geq 1$ and let $y\in Y$ be an arbitrary point such that
$\mathcal L_{y,Y}\neq\emptyset$. Then:
\begin{enumerate}
\item $\mathcal L_{y,X}\cap {\mathbb P}((t_yY)^*)=\mathcal L_{y,Y}$ as schemes.
\item if $y\in Y$ is general, then $\dim_{[L]}(\mathcal L_{y,X})=\dim_{[L]}(\mathcal L_{y,Y})+1$ and $[L]$ is a smooth
point of $\mathcal L_{y,X}$ for every $[L]\in\mathcal L_{y,Y}.$
\item if $y\in Y$ is general and if $\mathcal L_{y,Y}^j$ is an irreducible component of positive dimension,
then there exists an irreducible component $\mathcal L_{y,X}^j$ such that $\mathcal L_{y,Y}^j=\mathcal L_{y,X}^j\cap{\mathbb P}((t_yY)^*)$
as schemes.
\item If $y\in Y$ is general, then $\operatorname{Sing}(\mathcal L_{y,X})\subseteq S_{y,X}$.
\end{enumerate}
\end{Proposition}
\begin{proof} Let $Y=X\cap H$, with $H={\mathbb P}^N\subset{\mathbb P}^{N+1}$ a hyperplane and let notation be as in subsection \ref{equations}.
The conclusion in (1) immediately follows from \eqref{eqLxE}.
Let us pass to (2) and consider an arbitrary line $[L]\in \mathcal L_{y,Y}^j$, an irreducible component of the smooth not necessarily irreducible variety $\mathcal L_{y,Y}$.
We have an exact sequence
of normal bundles
\begin{equation}\label{tangentspace}
0\to N_{L/Y}\to N_{L/X}\to N_{Y/X|L}\simeq\O_{{\mathbb P}^1}(1)\to 0.
\end{equation}
Since $y\in Y$ is general, $N_{L/Y}$ is generated by global sections, see the proof of Proposition \ref{Yx},
so that \eqref{split} yields
\begin{equation}\label{normale}
N_{L/X}\simeq N_{L/Y}\oplus\O_{{\mathbb P}^1}(1)\simeq \O_{{\mathbb P}^1}(1)^{s(L,Y)+1}\oplus\O_{{\mathbb P}^1}^{n-s(L,Y)-1}.
\end{equation}
Thus also $N_{L/X}$ is generated by global sections, $\mathcal L_{y,X}$ is smooth at $[L]$ and
$\dim_{[L]}(\mathcal L_{y,X})=\dim_{[L]}(\mathcal L_{y,Y})+1$, proving (2).
Therefore if $y\in Y$ is general,
there exists a unique irreducible component of $\mathcal L_{y,X}\subset{\mathbb P}((t_yX)^*)$, let us say $\mathcal L_{y,X}^j$, containing $[L]$ and by the previous calculation
$\dim(\mathcal L^j_{y,X})=s(L,Y)+1=\dim(\mathcal L_{y,Y}^j)+1$. Recall that by part (1) we have $t_{[L]}\mathcal L_{y,Y}=t_{[L]}\mathcal L_{y,X}\cap {\mathbb P}((t_yY)^*)$ so that
\begin{equation}\label{extYx}
\mathcal L_{y,Y}^j\subseteq \mathcal L_{y,X}^j\cap {\mathbb P}((t_yY)^*)\subseteq \mathcal L_{y,Y}\subset {\mathbb P}^{n-1}={\mathbb P}((t_yY)^*),
\end{equation}
yielding that $\mathcal L_{y,Y}^j$ is an irreducible component of $\mathcal L_{y,X}^j\cap {\mathbb P}((t_yY)^*)$ as well as an irreducible component of the smooth variety $\mathcal L_{y,Y}$.
Hence, if $\dim(\mathcal L_{y,Y}^j)\geq 1$, we have the equality $\mathcal L^j_{y,Y}= \mathcal L_{y,X}^j\cap {\mathbb P}((t_yY)^*)$ as schemes,
i.e. under this hypothesis $\mathcal L_{y,X}^j\subset{\mathbb P}((t_yX)^*)$ (or better $(\mathcal L_{y,X}^j)_{\operatorname{red}}$) is a projective extension of the smooth positive dimensional irreducible variety $\mathcal L_{y,Y}^j\subset{\mathbb P}((t_xY)^*)$.
Indeed, $\dim(\mathcal L^j_{y,Y})\geq 1$ forces $\dim(\mathcal L_{y,X}^j)\geq 2$ so that it is sufficient to recall that $\mathcal L_{y,X}$ is smooth along $\mathcal L_{y,Y}$ by the previous discussion and also that an
arbitrary hyperplane section of the irreducible variety $(\mathcal L_{y,X}^j)_{\operatorname{red}}$ is connected by the Fulton-Hansen Theorem, \cite{FH}. More precisely, if $\dim(\mathcal L_{y,Y}^j)\geq 1$, then equality as schemes holds in \eqref{extYx}, proving part (3).
By \cite[Proposition 4.9]{Debarre} there exists a non-empty open subset $U\subseteq X$ such that
$N_{\widetilde L/X}$ is generated by global sections for every line $\widetilde L\subset X_{\operatorname{reg}}$
intersecting $U$. If $U\cap Y\neq \emptyset$, then (4) clearly holds. Suppose $Y\cap U=\emptyset$.
Let $[\widetilde L]\in \mathcal L_{y,X}\setminus S_{y,X}$. If $\widetilde L\cap U\neq \emptyset$, then $[\widetilde L]$ is
a smooth point of $\mathcal L_{y,X}$ by the previous analysis. If $\widetilde L\cap U=\emptyset$, then $\widetilde L\subset Y$
by the generality of $y\in Y$ and $N_{\widetilde L/X}$ is generated by global sections by \eqref{normale},
concluding the proof of (4).
\end{proof}
\medskip
Now we are in position to prove the main result of this section and to deduce some applications.
\medskip
\begin{Theorem}\label{criterion}
Let notation be as above and let $y\in Y$ be a general point. Then:
\begin{enumerate}
\item Suppose there exist two distinct irreducible components $\mathcal L_{y,X}^1$ and $\mathcal L_{y,X}^2$ of $\mathcal L_{y,X}\subset{\mathbb P}((t_yX)^*)$, extending two irreducible components $\mathcal L_{y,Y}^1$, respectively $\mathcal L_{y,Y}^2$, of $\mathcal L_{y,Y}$ in the sense specified above. If $\mathcal L_{y,X}^1\cap \mathcal L_{y,X}^2\neq\emptyset$,
then $X\subset{\mathbb P}^{N+1}$ is a cone over $Y\subset{\mathbb P}^N$ of vertex a point $p\in {\mathbb P}^{N+1}\setminus {\mathbb P}^N$.
\item If $\mathcal L_{y,Y}\subset{\mathbb P}((t_yY)^*)$ is a manifold whose extensions are singular, then
every extension of $Y\subset{\mathbb P}^N$ is trivial.
\end{enumerate}
\end{Theorem}
\begin{proof} By the above discussion, we get that in both cases, for $y\in Y$ general, the variety $S_{y,X}\subseteq \mathcal L_{y,X}$ is not empty so that for $y\in Y$ general there exists a line $L_y\subseteq X$ passing through $y$ and through a singular point $p_y\in L_y\cap \operatorname{Sing}(X)$. Since $Y$ is irreducible and since $\operatorname{Sing}(X)$ consists of a finite
number of points, there exists $p\in \operatorname{Sing}(X)$ such that $p\in L_y$ for $y\in Y$ general. This implies
that $X=S(p,Y)$ is a cone over $Y$ with vertex $p$.
\end{proof}
\medskip
The first easy consequence is a result due to Scorza (see \cite{Scorza2} and also \cite{Zakdual}, \cite{Badescu}), proved by him under the stronger assumption that $Y=X\cap H$ is a general hyperplane section of $X$.
Under these more restrictive hypotheses, the analysis before the proof of Theorem \ref{criterion} could be simplified via Proposition \ref{Yx}, since we may assume that the general point
$y\in Y$ is also general on $X$.
\medskip
\begin{Corollary}\label{Segre} Let $1\leq a\leq b$ be integers, let $n=a+b\geq 3$ and let
$Y\subset{\mathbb P}^{ab+a+b}$ be a smooth irreducible variety projectively equivalent to the
Segre embedding ${\mathbb P}^a\times{\mathbb P}^b\subset{\mathbb P}^{ab+a+b}$. Then every extension
of $Y$ in ${\mathbb P}^{ab+a+b+1}$ is trivial.
\end{Corollary}
\begin{proof} For $y\in Y$ general, it is well known that $\mathcal L_{y,Y}=\mathcal L_{y,Y}^1\amalg \mathcal L_{y,Y}^2\subset{\mathbb P}^{a+b-1}={\mathbb P}^{n-1}$ with $\mathcal L_{y,Y}^1={\mathbb P}^{a-1}$ and $\mathcal L_{y,Y}^2={\mathbb P}^{b-1}$, both linearly embedded. Observe that
$b-1\geq 1$. By \eqref{extYx} and the discussion following it,
there exist two irreducible components $\mathcal L_{y,X}^j$, $j=1,2$, of $\mathcal L_{y,X}\subset{\mathbb P}^{n}={\mathbb P}^{a+b}$ with $\dim(\mathcal L_{y,X}^1)=a$ and $\dim(\mathcal L_{y,X}^2)=b$. If $a\neq b$ then clearly $\mathcal L_{y,X}^1\neq \mathcal L_{y,X}^2$. If $a=b\geq 2$, then $\mathcal L_{y,X}^1\neq \mathcal L_{y,X}^2$ because an arbitrary hyperplane section of a variety of dimension at least
2 is connected, see \cite{FH}.
Since $a+b=n$, $\mathcal L_{y,X}^1\cap \mathcal L_{y,X}^2\neq\emptyset$ and the conclusion follows from the first part of Theorem \ref{criterion}.
\end{proof}
\medskip
The previous result
has some interesting consequences via iterated
applications of the second part of Theorem \ref{criterion}. Indeed, let us consider the following homogeneous varieties
(also known as
irreducible hermitian symmetric spaces), in their homogeneous embedding, and the description of the Hilbert scheme
of lines passing through a general point, see \cite[Section~ 1.4.5]{Hwang} and also \cite{Strick}.
\medskip
\begin{equation}\label{hermitian}
\begin{tabular}{|c|c|c|c|}
\hline &$Y$ & $\mathcal L_{y,Y}$ & $\tau_y:\mathcal L_{y,Y}\to {\mathbb P}((t_yY)^*)$\\
\hline 1 &$\mathbb G(r,m)$ & ${\mathbb P}^r\times{\mathbb P}^{m-r-1}$ & \text{Segre embedding}\\
\hline 2 &$SO(2r)/U(r)$& $\mathbb G(1,r-1)$ & \text{Pl\" ucker embedding}\\
\hline 3 &$E_6$ & $SO(10)/U(5)$& \text{miminal embedding}\\
\hline 4 &$E_7/E_6\times U(1)$ & $E_6$ & \text{Severi embedding}\\
\hline 5 &$Sp(r)/U(r)$ & ${\mathbb P}^{r-1}$ & \text{quadratic Veronese embedding}\\
\hline
\end{tabular}
\end{equation}
\medskip
There are also the following homogeneous contact manifolds with Picard number one associated
to a complex simple Lie algebra $\mathbf g$, whose Hilbert scheme of lines passing through
a general point is known. Let us observe that in these examples the variety
$\mathcal L_{y,Y}\subset{\mathbb P}^{n-1}={\mathbb P}((t_yY)^*)$ is degenerate and its linear span is exactly ${\mathbb P}((D_y)^*)={\mathbb P}^{n-2}$, there $D_y$ is the tangent
space at $y$ of the distribution associated to the contact structure on $Y$, i.e. there is the following factorization $\tau_y:\mathcal L_{y,Y}\to {\mathbb P}((D_y)^*)\subset{\mathbb P}((t_yY)^*)$. For more details one can consult \cite[Section~ 1.4.6]{Hwang}.\medskip
\begin{equation}\label{contact}
\begin{tabular}{|c|c|c|c|}
\hline &$\mathbf g$ & $ \mathcal L_{y,Y}$ & $\tau_x:\mathcal L_{x,Y}\to {\mathbb P}((D_y)^*$\\
\hline 6 &$F_4$ & $Sp(3)/U(3)$ & \text{Segre embedding}\\
\hline 7 &$E_6$ & $\mathbb G(2,5)$& \text{Pl\" ucker embedding}\\
\hline 8 & $E_7$ & $SO(12)/U(6)$ & \text{minimal embedding}\\
\hline 9 &$E_8$ & $E_7/E_6\times U(1)$& \text{minimal embedding}\\
\hline 10&${\mathbf so}_{m+4}$& ${\mathbb P}^1\times Q^{m-2}$ & \text{Segre embedding}\\
\hline
\end{tabular}
\end{equation}
\medskip
By case 1')
we shall denote a variety as in 1) of \eqref{hermitian} satisfying the following numerical conditions:
$r<m-1$; if $r=1$, then $m\geq 4$. By 2') we shall denote a variety as in 2) with $r\geq 5$.
\medskip
\begin{Corollary}\label{exthermitian} Let $Y\subset{\mathbb P}^N$ be a manifold as in Examples 1'), 2'), 3), 4), 7),
8), 9) above. Then every extension of $Y$ is trivial.
\end{Corollary}
\begin{proof} In cases 2'), 3), 4) and 9) in the statement the variety $\mathcal L_{y,Y}\subset{\mathbb P}^{n-1}$ of one example is the variety $Y\subset{\mathbb P}^N$ occurring in the next one. Thus for these cases, by the second part of Theorem \ref{criterion}, it is sufficient to prove the result for case 1'). For this variety the conclusion follows from Corollary \ref{Segre}. For the remaining cases, the variety $\mathcal L_{y,Y}\subset{\mathbb P}^{n-1}$ is either
as in case 1') with $(r,m)=(2,5)$ or as in case 2) with $r=6$ and the conclusion follows once
again by the second part of Theorem \ref{criterion}.
\end{proof}
\medskip
The next result is also classical and well-known but we provide a direct geometric proof. Under the assumption that the hyperplane section $H\cap X=Y$ is general, it was proved by C. Segre for $n=2$ in \cite{CSegre} and by Scorza in \cite{Scorza1}, see also \cite{Terracini},
for arbitrary $n\geq 2$ (and also for arbitrary Veronese embeddings $\nu_d({\mathbb P}^n)\subset{\mathbb P}^{N(d)}$, with $n\geq 2$ and $d\geq 2$; modern proofs of this general case are contained in \cite{Badescu} and in \cite{Zakdual}).
\medskip
\begin{Proposition}\label{Veronese} Let $n\geq 2$ and let $Y\subset{\mathbb P}^{\frac{n(n+3)}{2}}$ be
a manifold projectively equivalent to the quadratic Veronese embedding
$\nu_2({\mathbb P}^n)\subset{\mathbb P}^{\frac{n(n+3)}{2}}$. Then every extension of $Y$ is trivial.
\end{Proposition}
\begin{proof} Let $y\in Y$ be a general point and let $N=\frac{n(n+3)}{2}$. Since $\mathcal L_{y,Y}=\emptyset$, then $\mathcal L_{y,X}\subset{\mathbb P}^n$, if
not empty, consists of at most a finite number of points and through $y\in X$ there passes at most a finite number of
lines contained in $X$. Consider a conic $C\subset Y$ passing through $y$. Then $N_{C/Y}\simeq\O_{{\mathbb P}^1}(1)^{n-1}$.
The exact sequence
of normal bundles
$$0\to N_{C/Y}\to N_{C/X}\to N_{Y/X|C}\simeq\O_{{\mathbb P}^1}(2)\to 0,$$
yields
$$N_{C/X}\simeq N_{C/Y}\oplus\O_{{\mathbb P}^1}(2)\simeq \O_{{\mathbb P}^1}(1)^{n-1}\oplus\O_{{\mathbb P}^1}(2).$$
Thus there exists a unique irreducible component $\mathcal C_{y,X}$ of the Hilbert scheme of conics contained in $X\subset{\mathbb P}^{N+1}$ passing through $y\in X$ to which $[C]$ belongs. Moreover $\dim(\mathcal C_{y,X})=n+1$ and the conics
parametrized by $\mathcal C_{y,X}$ cover $X$. Hence there exists a one dimensional family of conics
through $y$ and a general point $x\in X$. By Bend and Break, see for example \cite[Proposition 3.2]{Debarre}, there is at least a singular conic through $y$ and $x$. Since $X\subset{\mathbb P}^{N+1}$ is not a linear space, there exists no line joining $y$ and a general $x$, i. e. the singular conics through $x$ and $y$ are reduced. Thus given a general point $x$ in $X$, there
exists a line $L_x\subset X$ through $x$, not passing through $y$, and a line $L_y\subset X$ through $y$ such that
$L_y\cap L_x\neq \emptyset$. Since there are a finite number of lines contained in $X$ and passing through $y$, we can conclude that given a general point $x\in X$, there exists a fixed line passing through $y$, $\widetilde{L}_y$,
and a line $L_x$ through $x$ such that $L_x\cap \widetilde{L}_y\neq \emptyset.$
Moreover, a general conic $[C_{x,y}]\in\mathcal C_{y,X}$ and passing through a general point $x$ is irreducible, does not pass through the finite set $\operatorname{Sing}(X)$ and has ample normal bundle
verifying $h^0(N_{C_{x,y}/X}(-1))=h^0(N_{C/X}(-1))=n+1$. This means that the deformations of $C_{x,y}$ keeping $x$ fixed cover an open subset of $X$ and also that through general
points $x_1,x_2\in X$ there passes a one dimensional family of irreducible conics. The plane spanned by one of these conics contains $x_1$ and $x_2$ so that it has to vary
with the conic. Otherwise the fixed plane would be contained in $X$ and $X\subset{\mathbb P}^{N+1}$ would be a linearly embedded ${\mathbb P}^{N+1}$, which is contrary to our assumptions. In conclusion
through a general point $z\in <x_1,x_2>$ there passes at least a one dimensional family of secant lines to $X$ so that
\begin{equation}\label{dimSX}
\dim(SX)\leq 2(n+1)-1=2n+1<N+1=\frac{n(n+3)}{2}+1,
\end{equation}
yielding $SX\subsetneq{\mathbb P}^{N+1}$.
Suppose the point $p_x=\widetilde{L}_y\cap L_x$, for $y\in Y$ general, varies on $\widetilde{L}_y$. Then the linear span of two general tangent spaces $T_{x_1}X$ and $T_{x_2}X$ would
contain the line $\widetilde{L}_y$. Since $T_zSX=<T_{x_1}X,T_{x_2}X>$ by the Terracini Lemma, we deduce that a general tangent space to $SX$ contains
$\widetilde{L}_y$ and a fortiori $y$. Since $SX\subsetneq{\mathbb P}^{N+1}$, the variety $SX\subset{\mathbb P}^{N+1}$ would be a cone whose vertex, which is a linear space,
contains $\widetilde{L}_y$ and a fortiori $y\in Y$. By the generality of $y\in Y$ we would deduce that $Y\subset{\mathbb P}^N$ is degenerate.
Thus $p_x=\widetilde{L}_y\cap L_x$ does not vary with $x\in X$ general. Let us denote this point by
$p$. Then clearly $X\subset{\mathbb P}^{N+1}$ is a cone with vertex $p$ over $Y$.
\end{proof}
\medskip
\begin{Corollary}\label{extcontact} Let $Y\subset{\mathbb P}^N$ be a manifold either as in 5) above with $r\geq 3$ or as in 6)
above. Then every extension of $Y$ is trivial.
\end{Corollary}
\begin{proof} By \eqref{hermitian} we know that in case 5) with $r\geq 3$ we have $n-1=\frac{(r-1)(r+2)}{2}$ and the variety $\mathcal L_{y,Y}\subset{\mathbb P}^{n-1}$
is projectively equivalent to $\nu_2({\mathbb P}^{r-1})\subset{\mathbb P}^{\frac{(r-1)(r+2)}{2}}$. To conclude we apply Proposition \ref{Veronese} and the second part of Theorem \ref{criterion}. Case 6) follows
from case 5) with $r=3$ by the second part of Theorem \ref{criterion}.
\end{proof}
\medskip
\begin{Remark}\label{dualhom} There is a different and interesting approach to Corollary \ref{exthermitian} and Corollary \ref{extcontact}
based on the theory of dual varieties and proposed by Zak in \cite{Zakdual}, which also avoids direct computations
of vanishing of cohomology groups in each case. This approach is less direct and less elementary
than ours and it is based on the following facts. By a result of Kempf the dual variety of any homogeneous variety is normal, see e.g. \cite[Theorem III.1.2]{Zak}. Then in \cite[Corollary 1]{Zakdual} it is stated that a smooth variety $X\subset{\mathbb P}^N$ whose dual variety is normal admits only trivial
extensions. As far as we know, to establish this result one first shows that the normality of $X^*$ implies its linear normality, which seems to follow from some well known but not trivial results. Finally one applies \cite[Theorem ]{Zakdual}, which is a general criterion for admitting only trivial extension. For us Theorem \ref{criterion} is simply another incarnation of the Principle described in the Introduction while Corollary \ref{exthermitian} and Corollary \ref{extcontact}, surely well known to everybody, were included only to show that they are an immediate consequences of Scorza's result in \cite{Scorza2}, a fact which seems to have been overlooked till now.
\end{Remark}
\def\bibaut#1{{\sc #1}}
|
{
"timestamp": "2011-05-06T02:02:41",
"yymm": "1009",
"arxiv_id": "1009.3637",
"language": "en",
"url": "https://arxiv.org/abs/1009.3637"
}
|
\section{Introduction}
\subsection*{Anders\'en-Lempert theory and density property}
\hspace{1cm}
Let $X$ be a complex manifold. We say that a holomorphic vector field $\mu$ on $X$ is complete if the solution of the associated first order ODE exists
for all complex time, for any choice of initial point on $X$.
In other words, $\mu$ induces a holomorphic action of the additive group $\ensuremath{\mathbb{C}}_+$ on $X$; conversely, given an holomorphic $\ensuremath{\mathbb{C}}_+$-action $\psi_t$ on $X$
there is a unique complete holomorphic vector field $\mu$ such that its flow coincides with $\psi_t$.
For $n\geq 2$, the abundance (in the sense of the next definition)
of complete vector fields on $\ensuremath{\mathbb{C}}^n$ was a crucial observation
in the work of Anders\' en and Lempert on the group $ \operatorname{{\rm Aut}}_{hol}(\ensuremath{\mathbb{C}}^n)$ of holomorphic automorphisms of $\ensuremath{\mathbb{C}}^n$ \cite{AL}, further developed by Forstneric and Rosay \cite{FR}.
\begin{definition}\label{DP}
A complex manifold $X$ is said to have the density property if
the Lie algebra $ \operatorname{{\rm Lie}_{hol}} (X)$ generated by
complete holomorphic vector fields on $X$ is dense, with respect to the compact-open topology, in
the Lie algebra $ \operatorname{{\rm VF}_{hol}} (X)$ of all holomorphic vector fields on
$X$.
\end{definition}
The density property for $\ensuremath{\mathbb{C}}^n$ has many implications.
An example is the existence of non-equivalent holomorphic embeddings of $\ensuremath{\mathbb{C}}^k$ into $\ensuremath{\mathbb{C}}^n$ \cite{BK}, a result that was used by Kutzschebauch and Derksen to construct non-linearizable holomorphic actions of reductive Lie groups on affine spaces \cite{DK1, DK2}.
The density property was introduced by Varolin \cite{V1},
who was the first to extend the results of Anders\'en and Lempert
to Stein manifolds different from $\ensuremath{\mathbb{C}}^n$. An interesting application
from the prospective of the present paper concerns $k$-homogeneity.
\begin{definition}\label{tr}
Let $X$ be a complex manifold, $k$ a positive integer.
We say that a group $G$ of holomorphic automorphisms of $X$
acts $k$-transitively on $X$ if given two collections $(x_1,...,x_k)$, $(y_1,...,y_k)$ of pairwise distinct points, there exists an automorphism $f\in G$ such that $f(x_i)=y_i$.
\end{definition}
\begin{proposition}\label{dptr}
If a Stein manifold $X$ has the density property, then the group
$ \operatorname{{\rm Aut}}_{hol}X$ of holomorphic automorphisms of $X$ is acts $k$-transitively for all $k$.
\end{proposition}
Varolin found various examples of Stein manifolds with the density property,
such as some homogeneous spaces of complex semisimple Lie groups
(see the paper with Toth, \cite{V2}).
If we are interested in affine algebraic varieties, we can refine the definition as follows.
\begin{definition}\label{ADP}
A smooth affine algebraic variety $X$ is said to have the algebraic density property if
the Lie algebra $ \operatorname{{\rm Lie}_{alg}} (X)$ generated by
complete algebraic vector fields on $X$ coincides with
the Lie algebra $ \operatorname{{\rm VF}_{alg}} (X)$ of all algebraic vector fields on
$X$.
\end{definition}
Since the ring of regular functions of an affine variety is dense in the ring
of the holomorphic functions, the algebraic density property implies the density property.
Kaliman and Kutzschebauch proved that all complex linear algebraic groups with the exception of the tori $(\ensuremath{\mathbb{C}}^*)^n$ and $\ensuremath{\mathbb{C}}_+$ have the algebraic density property \cite{KK1}. The tori do not have the algebraic density property since Anders\'en \cite{A}, by using some results from Nevanlinna theory, proves that all complete polynomial vector fields on $(\ensuremath{\mathbb{C}}^*)^n$ must have divergence zero with respect to the canonical volume form, but we do not know if the tori have the (not necessarily algebraic) density property.
Dvorsky, Kaliman and the author \cite{DDK} prove the algebraic density property for homogeneous spaces not isomorphic to $(\ensuremath{\mathbb{C}}^*)^n$, of dimension at least three that are quotient of linear algebraic groups by a reductive subgroup.
The proof relies on the methods developed in \cite{KK1}, but it also requires some non trivial facts from Lie theory, and an application of the Luna's slice theorem.
Finally, we mention the affine surface $X$ in $\ensuremath{\mathbb{C}}^3$ given by the equation $x+y+xyz=1$. By using results of Brunella on holomorphic foliations on rational surfaces \cite{Br1, Br2}, Kaliman and Kutzschebauch \cite{KK3} prove that a complete vector field on $X$ must have divergence zero with respect to the volume form $\frac{dx}{x}\wedge\frac{dy}{y}$,
and hence that $X$ does not have the density property.
In the same preprint, the authors show that $ \operatorname{{\rm Aut}}_{hol} (X)$
acts $k$-transitively for all $k$. Since we do not know if $X$ has the density property, $X$ could be counterexample to the converse of Proposition \ref{dptr}
(together with the tori).
For a good survey of Anders\'en-Lempert Theory and it applications, we refer the reader to \cite{KK3}.
\subsection*{Danilov-Gizatullin surfaces}
\hspace{1cm}
A Danilov-Gizatullin surface is an affine surface which is the complement of an ample section $S$ in a Hirzebruch surface $\Sigma_d$.
The ampleness of $S$ implies that $n>d$ and $n\geq 2$.
We have the following remarkable result of Danilov and Gizatullin \cite{DG}
(see also \cite{FKZ} for a short proof ):
\begin{theorem}\label{DGT}
Let $S$ ($S'$) be an ample section of $\Sigma_d$ ($\Sigma_{d'}$).
Then
$\Sigma_d-S$ is isomorphic to $\Sigma_{d'}-S'$ if and only if
$S^2=S'^2$.
\end{theorem}
Consequently, for all $n\geq 2$
we will denote by $V_n$ the surface complementary to a section $S$ with $S^2=n$.
The Danilov-Gizatullin surfaces belong to the class of the affine rational
surfaces with trivial Makar-Limanov invariant
(the suburing of regular functions that are invariant with respect to all
$\ensuremath{\mathbb{C}}_+$-actions consists of constants only) \cite{GMMR}.
Moreover, $V_n$ is a flexible affine variety, that is for any point $x\in V_n$
there are pairs of $\ensuremath{\mathbb{C}}_+$-actions, such that the corresponding complete vector
fields span the tangent space at $x$.
On a forthcoming paper of Arzhantsev, Flenner, Kaliman, Kutzschebauch and Zaidenberg \cite{AFKKZ}, it is proven that the flexibility implies that the subgroup $G$ of $ \operatorname{{\rm Aut}} (V_n)$ generated by $\ensuremath{\mathbb{C}}_+$-actions acts $k$-transitively on $V_n$, for all $k$ (see Definition \ref{tr} ).
Moreover, the set of complete vector fields whose flow is an algebraic $\ensuremath{\mathbb{C}}_+$-action generates an infinite-dimensional Lie algebra.
In this paper (Theorem \ref{main}) we prove that for all $n\geq 2$, $V_n$ has the algebraic density property.
\subsection*{Sketch of the proof of Theorem \ref{main}}
\hspace{1cm}
By presenting the Hirzebruch
surface $\Sigma_d$ as a quotient of an open toric variety in $\ensuremath{\mathbb{C}}^4$ by a two dimensional torus $T^2$, we can describe any section $S$ of the ruling
as the zero locus of a $T^2$-invariant polynomial in $\ensuremath{\mathbb{C}}^4$.
The Danilov-Gizatullin theorem now plays a crucial role:
by choosing to embed $V_n$ in $\Sigma_{n-2}$, and by making a specific choice for a section $S$ with $S^2=n$, we prove that $V_n$ is isomorphic to the algebraic quotient by a one-dimensional torus $T$ of a smooth affine threefold $F_n$
(Theorem \ref{quotient}), and we find a set of generators for
$\ensuremath{\mathbb{C}} [F_n]^T$, the ring of $T$-invariant regular functions on $F_n$ (Proposition \ref{ring}), which is isomorphic to $\ensuremath{\mathbb{C}} [V_n]$. Next, we construct some complete $T$-invariant vector fields on $F_n$, that descends to complete vector fieds on $V_n$ (Proposition \ref{fields}). We then perform some computations, involving those fields and the generators of $\ensuremath{\mathbb{C}} [F_n]^T\cong\ensuremath{\mathbb{C}} [V_n]$, until we construct a non-zero $\ensuremath{\mathbb{C}} [V_n]$-submodule $\mathfrak{N}$ of all algebraic vector fields that is contained in the Lie algebra generated by the complete ones (Theorem \ref{M}). We then
extend $\mathfrak{N}$ to a $\ensuremath{\mathbb{C}} [V_n]$-submodule $\mathfrak{M}$ such that the fiber of $\mathfrak{M}$ at a point
$x$ of $V_n$ generates the tangent space at $x$.
The transitivity of the action of the group
${\rm Aut}_{alg}{V_n}$ of algebraic automorpshisms of $V_n$ allows then to apply to $\mathfrak{M}$ a technical principle (Theorem \ref{principle}) to prove the algebraic density property
of $V_n$ (Theorem \ref{main}).
\section{Construction of $V_n$ as a quotient by a torus action}
A Hirzebruch surface can be realized as a quotient
of a toric variety by a two-dimensional torus (\cite{K}, Section 1):
consider the action of $T^2\cong\ensuremath{\mathbb{C}}_{t_1}^*\times \ensuremath{\mathbb{C}}_{t_2}^*$ on $\ensuremath{\mathbb{C}}^4_{(a_1,a_2,a_3,a_4)}$ given by
\begin{align}
\nonumber (t_1,t_2).(a_1,a_2,a_3,a_4)=(t_1t_2^da_1,t_2a_2,t_1a_3,t_2a_4)
\end{align}
and let
\begin{align}
\nonumber Z=\{a_1=a_3=0\}\cup \{a_2=a_4=0\}.
\end{align}
Then we have that
\begin{align}
\nonumber \Sigma_d\cong (\ensuremath{\mathbb{C}}^4-Z)/T^2.
\end{align}
The isomorphism above can be understood by defining coordinates
\begin{align}
&\nonumber t_0=\frac{a_1}{a_2^da_3}\quad \text{for}\quad a_2\neq 0 ,\\
&\nonumber t_\infty=\frac{a_1}{a_4^da_3}\quad\text{for}\quad a_4\neq 0 ,\\
&\nonumber v=\frac{a_2}{a_4}\in\mathbb{P}^1
\end{align}
(since they are $T^2$-invariant, they are well defined).
The quotient space then is isomorphic to the Hirzerbruch surface $\Sigma_d$,
with ruling
\begin{align}
&\nonumber \pi : \Sigma_d\rightarrow \mathbb{P}^1\\
&\nonumber \pi (a_1,a_2,a_3,a_4)=(a_2,a_4)
\end{align}
and transition function
\begin{align}
\nonumber t_\infty=v^dt_0.
\end{align}
Recall now (\cite{B}, chapter IV)
that the Picard group of $\Sigma_d$ is generated by the linear equivalence class $F$ of the fiber of $\pi$, and by the class of the unique irreducible curve $C$ with self-intersection $C^2=-d$.
It follows that $S$ is linear equivalent to $C+bF$, and that $n=2b-d$.
Theorem \ref{DGT} implies that $\Sigma_d-S\cong V_n$ for any section $S$ with $S^2=n$, and we claim that
\begin{align}
\nonumber t_\infty=v^b
\end{align}
is a local equation (for $v\neq \infty$) of a section
$S$ of the ruling $\pi$ with self intersection $n=2b-d$. In fact, the base locus of the
family of section $t_\infty=cv^b$ consists of two multiple points, namely
the point $A(v=0,t_\infty=0)$ with multiplicity $b$, and $B(v=\infty, t_0=0)$ with multiplicity $b-d$: if we sum the intersection multiplicities at $A$ and
$B$, we obtain that $S^2=2b-d=n$.
In the coordinates $(a_1,a_2,a_3,a_4)$ the defining equation of $S$ takes the
form
\begin{align}\label{a1}
a_1a_4^{b-d}-a_2^ba_3=0
\end{align}
(the ampleness of $S$ implies that $b>d$).
Let $S'$ be the hypersurface in $\ensuremath{\mathbb{C}}^4$ given by equation (\ref{a1}).
Then the inverse image $\rho^{-1}(S)=S'-Z$ under the quotient map is the closed
$T$-invariant subset of $\ensuremath{\mathbb{C}}^4$ given by the equation (\ref{a1}), and
\begin{align}\label{a2}
\nonumber V_n\cong\rho (\ensuremath{\mathbb{C}}^4-(S'\cup Z))=(\ensuremath{\mathbb{C}}^4-(S'\cup Z))/T^2.
\end{align}
Since $Z\subset S'$, the variety $\ensuremath{\mathbb{C}}^4-(S'\cup Z)$ is isomorphic to the affine manifold
$V'=\ensuremath{\mathbb{C}}-S'$.
\begin{lemma}\label{gq=aq}
The geometric quotient $V'/T^2$ is isomorphic to the algebraic quotient
$V'//T^2= \operatorname{{\rm Spec}} \ensuremath{\mathbb{C}} [V']^{T^2}$, where $\ensuremath{\mathbb{C}} [V']^{T^2}$ denotes the subring of the $T^2$-invariant regular functions of $V'$.
\begin{proof}
The orbits of the $T^2$-action restricted to $V'$ are all closed.
Therefore the lemma follows from the Luna's slice theorem (\cite{Dr}, Theorem 5.4).
\end{proof}
\end{lemma}
We can invoke Theorem \ref{DGT} again and fix
from now on that $V_n$ is embedded in $\Sigma_{n-2}$, that is $n=d+2$, and $b=d+1$. Under this assumption, the equation of $S'$ becomes
\begin{align}
a_1a_4-a_2^ba_3=0.
\end{align}
\begin{proposition}
Consider the smooth affine threefold
\begin{align}\label{equation}
F_n=\{ a_1a_4-a_2^ba_3=1 \}\subset \ensuremath{\mathbb{C}}^4.
\end{align}
Then $V'$ is equivarianlty isomorphic to
$\ensuremath{\mathbb{C}}^*\times F_n$, where the action of $T^2$ on $\ensuremath{\mathbb{C}}^*\times F_n$ is defined by:
\begin{align}\label{action}
(t_1,t_2).(w,a_1,a_2,a_3,a_4)=
(t_1^{-1}t_2^{-b}w,t_2^{-1}a_1,t_2a_2,t_2^{-b}a_3,t_2a_4).
\end{align}
\begin{proof}
The isomorphism (as affine varieties)
is given by
\begin{align}
(a_1,a_2,a_3,a_4)\mapsto \left( a_1a_4-a_3^ba_2,\frac{a_1}{a_1a_4-a_3^ba_2}, a_2,\frac{a_3}{a_1a_4-a_3^ba_2},a_4 \right)
\end{align}
Then it is easy to check that the induced $T^2$ action on $\ensuremath{\mathbb{C}}^*\times F_n$ by the isomorphism is given by (\ref{action}).
\end{proof}
\end{proposition}
When we pass to the quotient, we get rid of the factor $\ensuremath{\mathbb{C}}^*$, as follows.
\begin{theorem}\label{quotient}
For any $n\geq 2$, the Danilov-Gizatullin surface $V_n$ is isomorphic to the algebraic quotient $F_n//T$, where $F_n$ is the affine threefold given
by the equation
\begin{align}
a_1a_4-a_2^ba_3=1
\end{align}
and the torus $T\cong \ensuremath{\mathbb{C}}^*$ acts on $F_n$ via
\begin{align}
t.(a_1, a_2,a_3,a_4)= (t^{-1}a_1, ta_2, t^{-b}a_3,ta_4)
\end{align}
\begin{proof}
Lemma \ref{gq=aq} implies that $V'/T^2\cong V'//T^2$.
Observe that the $T^2$-action of $V'$ (equation \ref{action}) is such that
the $t_1$-variable acts non-trivially only on the $w$-coordinate.
Therefore there are no non-constant $T^2$-invariant polynomials depending
on $w$, and the result follows.
\end{proof}
\end{theorem}
Perhaps, the above theorem has been already estabilished, but we do not know a reference for it.
\begin{remark}
Theorem \ref{quotient} presents $V_n$ as a quotient of a hypersurface in
$\ensuremath{\mathbb{C}}^4$ given by an equation of the type $uv=p(y,z)$, for $p$ a polynomial
with smooth zero locus. The affine manifolds of this type have
the algebraic density property, as shown by Kaliman and Kutzschebauch \cite{KK2}. Furthermore, it is clear from its defining equation that
$F$ is a ramified $b$-sheeteed covering of $SL_2$, another manifold with the algebraic density property \cite{V2}.
The behavior of the density property with respect to
quotients or coverings is not known in general.
\end{remark}
\begin{proposition}\label{ring}
$\ensuremath{\mathbb{C}} [V_n]\cong \ensuremath{\mathbb{C}} [F_n]^T$ is generated by the monomials
\begin{align}
y= a_1a_2,\quad z=a_1a_4,\quad \quad
x_k=a_2^{b-k}a_3a_4^k \quad (0\leq k\leq b).
\end{align}
\begin{proof}
Since the action is monomial, it is sufficient to look
for monomial generators. The action of $t$ on $a_1^Xa_2^Ya_3^Za_4^W$ is
\begin{align}
\nonumber t.(a_1^Xa_2^Ya_3^Za_4^W)=t^{-X+Y-bZ+W}a_1^Xa_2^Ya_3^Za_4^W,
\end{align}
therefore a monomial is invariant if and only if
\begin{align}\label{lattice}
-X+Y-bZ+W=0.
\end{align}
We show that every solution (with non-negative integers) of equation (\ref{lattice})
is a linear combination with non-negative integer coefficients of the vectors
\begin{align}
&\nonumber (1,1,0,0)\quad (1,0,0,1),\\
&\nonumber (0,b-k,1,k)\quad (\text{for}\quad 0\leq k\leq b).
\end{align}
Write $W=NbZ+r$ and $Y=MbZ+r'$, for $0\leq r,r'\leq bZ$.
Suppose that $W<bZ$: choose $Z$ integers $k_i$, with $0\leq k_i\leq b$,
such that $\sum_i^Zk_i=W$. Then
\begin{align}
\nonumber (X,Y,Z,W)=X(1,1,0,0)+ \sum_i^Z(0,b-k_i,1,k_i)
\end{align}
Suppose instead that $W\geq bZ$. Then $X-MbZ=W-bZ+r'\geq 0$; choose $Z$ integers $k_i$, with $0\leq k_i\leq b$, such that $\sum _i^Z(b-k_i)=r'$. Then we can write
\begin{align}
\nonumber (X,Y,Z,W)=(X-MbZ)(1,0,0,1)+\sum_i^Z(0,b-k_i,1,k_i)+MbZ(1,1,0,0).
\end{align}
\end{proof}
\end{proposition}
\section{Algebraic density property of $V_n$}
In this section we prove the main result of the paper.
\begin{theorem}\label{main}
The Danilov-Gizatullin surfaces $V_n$ have the algebraic density property.
\end{theorem}
We make use of the following principle \cite{KK1}.
\begin{theorem}\label{principle}
Let $X$ be a smooth affine algebraic variety, such that the group ${\rm Aut}_{alg} (X)$
of algebraic automorphisms acts transitively on it. Let $\mathfrak{M}$ be a submodule
of the $\ensuremath{\mathbb{C}} [X]$-module of all algebraic vector fields such that $\mathfrak{M}\subset \operatorname{{\rm Lie}_{alg}} (X)$. Suppose that the fiber of $\mathfrak{M}$
at a point $x_0\in X$ generates the tangent space $T_{x_0}M$.
Then $X$ has the algebraic density property.
\end{theorem}
In our case the transitivity of the action of ${\rm Aut}_{alg} (V_n)$ follows from the following result of Gizatullin \cite{G}.
\begin{theorem}
Let $X$ be an irreducible smooth affine variety over an algebraically closed field of characteristic zero. If $X$ can be completed by a smooth rational curve, then ${\rm Aut}_{alg} (X)$ acts transitively on $X$.
\end{theorem}
We start by listing the complete vector fields that will be used to construct the module $\mathfrak{M}$ of the theorem above.
\begin{proposition}\label{fields}
The following vector fields on $F_n$ are complete and $T$-invariant, and they descend to non-identically zero complete vector fields on $V_n$.
\begin{align}
&\delta =ba_2^{b-1}a_3\frac{\partial}{\partial a_1}+a_4\frac{\partial}{\partial a_2}\\
&\delta'=a_1^{b-1}a_2^b\frac{\partial}{\partial a_4}+a_1^b\frac{\partial}{\partial a_3}\\
&\epsilon=a_1\frac{\partial}{\partial a_1}-a_4\frac{\partial}{\partial a_4}
\end{align}
Moreover, $\delta$ and $\delta'$ are locally nilpotent derivations
(their flow is an algebraic action of $\ensuremath{\mathbb{C}}_+$).
\begin{proof}
The vector fields are tangent to $F_n$, since they annihilate the defining equation of the threefold. The invariance and their completeness is a straightforward check left to the reader.
Finally, the vector fields descends non-trivially on $V_n$ because they are generically transversal to the vector field $
-a_1\frac{\partial}{\partial a_1}+a_2\frac{\partial}{\partial a_2}
-ba_3\frac{\partial}{\partial a_3}+a_4\frac{\partial}{\partial a_4}
$ that defines the $T$-action on $F_n$.
\end{proof}
\end{proposition}
We collect in the next two lemmas the formulas upon which the
proofs of lemmas 4.5-4.9 are based.
\begin{lemma}\label{on functions}
For the surface $V_n$:
\begin{align}
&\epsilon (x_k)=-kx_k\\
&\epsilon (y)=y\\
&\delta (y)=1+nx_0\\
&\delta (x_k)=(b-k)x_{k+1}\\
&\delta' (x_0)=y^b\\
&\delta'(y)=0
\end{align}
\end{lemma}
\begin{lemma}\label{commutation}
For the surface $V_n$:
\begin{align}
&[\epsilon ,\delta ] = -\delta\\
&[\epsilon , \delta '] = b\delta' \quad (n=b+1)
\end{align}
\end{lemma}
We will also need a well-known fact (\cite{V4}, Proposition 3.1) about differential equations that will be used implicitly in some of the remaining statements.
\begin{proposition}
Let $\mu$ be a complete vector field on a complex manifold $X$, and $f$ be a holomorphic function.
Then the vector field $f\mu$ is complete if and only if $\mu^2(f)=0$.
\end{proposition}
\begin{lemma}\label{epsilon}
The following algebraic vector fields belong to $ \operatorname{{\rm Lie}_{alg}} (V_n)$:
(1) $x_0\epsilon$, (2) $x_b\epsilon$, (3) $x_0x_b\epsilon$.
\begin{proof}
Statement (1) follows from $\epsilon (x_0)=0$ (in particular, $x_0\epsilon$ is complete).
As for (2), let $X_1= [\delta , x_0\epsilon]$, $X_s= [\delta , X_{s-1}]$.
By definition $X_s\in \operatorname{{\rm Lie}_{alg}} (V_n)$, for all $s$.
Then by induction on $s\geq 2$ it follows that
\begin{align}
\nonumber X_s=sb(b-1)...(b-s+2)x_{s-1}\delta +b(b-1)...(b-s+1)x_s\epsilon.
\end{align}
In particular, for $s=b$, one obtains
$X_b=b!x_b\epsilon + b^2(b-1)...(2)x_{b-1}\delta$.
Since $\delta^2(x_{b-1})=0$, $x_{b-1}\delta$ is complete, and the statement (2)
is proven.
From the first two facts, one has that (3) $[x_0\epsilon, x_b\epsilon ]=-bx_0x_b\epsilon\in \operatorname{{\rm Lie}_{alg}} (V_n)$.
\end{proof}
\end{lemma}
\begin{lemma}\label{delta}
For $0\leq k \leq b$, for all $N>0$, $x_kx_b^N\delta\in \operatorname{{\rm Lie}_{alg}} (V_n)$.
\begin{proof}
Now we calculate that
\begin{align}
\label{s1}
[x_0\epsilon , x_b\delta ]&=-(1+b)x_0x_b\delta-bx_1x_b\epsilon\\
\label{s2}[\delta , x_0x_b\epsilon ]&=x_0x_b\delta+bx_1x_b\epsilon
\end{align}
The sum of (\ref{s1}) and (\ref{s2}) shows that
$x_0x_b\delta\in \operatorname{{\rm Lie}_{alg}} (V_n)$ (use Lemma \ref{epsilon}).
Then, for all $M\geq 0,$ we have $[x_b^M\delta , x_0x_b\delta ]=bx_1x_b^{M+1}\delta\in \operatorname{{\rm Lie}_{alg}} (V_n)$.
By induction on $k$ we then prove the statement for all positive $k$.
As for $k=0$, observe that since $yx_b=(1+x_0)x_{b-1}$,
we obtain $yx_b\delta =x_0x_{b-1}\delta + x_{b-1}\delta$.
This equation shows that $yx_b\delta\in \operatorname{{\rm Lie}_{alg}} (V_n)$: in fact, $x_{b-1}\delta$
is complete, and $x_0x_{b-1}\delta\in \operatorname{{\rm Lie}_{alg}} (V_n)$ as follows from Lemma \ref{epsilon} and Lemma \ref{delta} applied to
\begin{align}
\nonumber [x_0\epsilon , x_{b-1}\delta ]=-bx_0x_{b-1}\delta-bx_1x_{b-1}\epsilon = -bx_0x_{b-1}\delta-bx_0x_b\epsilon,
\end{align}
the second equality being true since $x_1x_{b-1}=x_0x_b$.
Hence, for $M\geq 0$ we obtain
\begin{align}
\nonumber [x_b^M\delta , yx_b\delta ]=x_b^{M+1}(1+nx_0)\delta =x_b^{M+1}\delta+nx_0x_b^{M+1}\delta
\end{align}
which shows the case $k=0$.
\end{proof}
\end{lemma}
\begin{lemma}\label{xe}
For $1\leq k \leq b$, $N>0$, $M\geq 0$, $x_0^Mx_kx_b^N\epsilon\in \operatorname{{\rm Lie}_{alg}} (V_n)$.
\begin{proof}
We know that $x_0x_b\epsilon\in \operatorname{{\rm Lie}_{alg}} (V_n)$ (Lemma \ref{epsilon}).
To show that $x_kx_b\epsilon\in \operatorname{{\rm Lie}_{alg}} (V_n)$ for all $k$, apply induction on $k$
and Lemma \ref{delta} to the equation
\begin{align}
\nonumber [\delta ,x_{k-1}x_b\epsilon ]=x_{k-1}x_b\delta + (b-k+1)x_kx_b\epsilon.
\end{align}
Next, the result for $N'\geq 0$ $M=0$ follows from the equation
\begin{align}
\nonumber [x_b^{N'}\delta , x_kx_b\epsilon ]= (1+bN')x_kx_b^{N'+1}\delta+x_b^{N'+1}(b-k)x_{k+1}\epsilon.
\end{align}
Finally, the case $M\geq 0$, $N>0$ follows from
\begin{align}
\nonumber [x_0^M\epsilon , x_kx_b^N\epsilon ]=(-Nb-k)x_0^Mx_kx_b^N\epsilon.
\end{align}
\end{proof}
\end{lemma}
The next computations deal with the $y$-variable.
\begin{lemma}\label{ey}
For all $R\geq 0$,
$y^{b+R}\epsilon\in \operatorname{{\rm Lie}_{alg}} (V_n)$.
\begin{proof}
Use the formula
\begin{align}
\nonumber [x_0\epsilon , y^R\delta' ]= (b+j)x_0y^R\delta'-y^{b+R}\epsilon
\end{align}
and notice that $\delta'^{2}(y^Rx_0)=0$.
\end{proof}
\end{lemma}
\begin{proposition}\label{d}
For $N>0$, $M>0$, $0<k<b$, $R\geq 0$,
$y^{b+R}x_0^Mx_kx_b^N\epsilon\in \operatorname{{\rm Lie}_{alg}} (V_n)$
\begin{proof}
Apply Lemma \ref{xe} and Lemma \ref{ey} to the equation
\begin{align}
&\nonumber [y^{b+R}\epsilon ,x_0^Mx_kx_b^N\epsilon ]=\\
&\nonumber (1-k-bN-b-R) y^{b+R}x_0^Mx_kx_b^N\epsilon.
\end{align}
\end{proof}
\end{proposition}
Next, we make use of some relations between the invariant monomials.
\begin{proposition}\label{monomial}
For any collection $k_1,...,k_r$, there are integers $N>0$ and $M>0$,
$0\leq h\leq b$, such that
\begin{align}
x_0x_{k_1}x_{k_2}...x_{k_r}x_b=x_0^Mx_hx_b^N,
\end{align}
\begin{proof}
We observe that:
\begin{align}
&if \quad h+k\leq b\quad x_kx_h=x_{h+k}x_0\label{red1}\\
&if \quad h+k\geq b\quad x_kx_h=x_bx_{h+k-b}\label{red2}
\end{align}
Write the monomial in the form
\begin{align}
\nonumber x_0^{N'}x_{k_1}...x_{k_s}x_b^{M'}
\end{align}
for $x_{k_1},...,x_{k_{s}}\neq x_0,x_b$.
Then, by applying (\ref{red1}) and (\ref{red2})
we can eliminate the factors $x_k$ not equal to $x_0$ and $x_b$
untill there will be left at most one.
\end{proof}
\end{proposition}
\begin{theorem}\label{M}
Let $\mathfrak{N}$ be the
$\ensuremath{\mathbb{C}} [V_n]$-submodule of $ \operatorname{{\rm VF}_{alg}} (V_n)$ generated by the vector field
$x_0x_1...x_by^b\epsilon$. Then $\mathfrak{N}\subset \operatorname{{\rm Lie}_{alg}} (V_n)$.
\begin{proof}
We need to show that, for all monomials $z^Ty^Rx_{k_1}...x_{k_r}$,
$z^Ty^Rx_{k_1}...x_{k_r}x_0x_1...x_by^b\epsilon\in \operatorname{{\rm Lie}_{alg}} (V_n)$.
Observe first that we can assume $T=0$, since we can eliminate $z$ via the relation $z=1+x_0$.
Then, according to Proposition \ref{monomial}, we can write
\begin{align}
\nonumber y^Rx_{k_1}...x_{k_r}x_0x_1...x_by^b=y^{b+R}x_0x_{h_1}...x_{h_{r'}}x_b=
\nonumber x_0^Nx_hx_b^M,
\end{align}
for some $N,M>0$, $0\leq h\leq b$
The theorem follows from Proposition \ref{d}.
\end{proof}
\end{theorem}
\subsection*{Conclusion of the proof of Theorem \ref{main}}
We follow the idea of \cite{KK1} to produce a module $\mathfrak{M}$
satisfying the hypothesis of Theorem \ref{principle}.
Consider the regular function $f=x_b$.
The flow of the vector field $f\delta$ is an algebraic action of $\ensuremath{\mathbb{C}}_+$:
this can be checked directly by solving the associated ODE, or
by using the fact that $\delta$ is a locally nilpotent derivation and
$\delta (x_b)=0$ (\cite {Fre}, Principle 7 page 24).
Let $p\in V_n$ be a point with $f(p)=0$, $\delta_p\neq 0$,
$\epsilon_p\neq 0$, $\epsilon_p (f)\neq 0$:
then (\cite{KK1}, Claim page 4) the flow $\phi$ of $f\delta$ at time one
induces an isomorphism $\phi_*$
on the tangent space at $p$,
that map $\epsilon_p$ to $\epsilon_p+\epsilon_p(f)\delta_p$.
Let $\mathfrak{M}=\mathfrak{N}+\phi_*\mathfrak{N}$: then $\mathfrak{M}\subset \operatorname{{\rm Lie}_{alg}} (V_n)$, and since $\epsilon_p$ and $\delta_p$ are linearly independent vectors,
its fiber at $p$ spans the tangent space at $p$.
Since $V_n$ is homogeneous with respect to ${\rm Aut}_{alg} (V_n)$,
we can then apply Theorem \ref{principle} to conclude the proof of Theorem \ref{main}.
\subsection*{Acknowledgments}
I would like to thank Shulim Kaliman and Dror Varolin
for reading the paper and giving a lot of useful comments;
Patrick Clarke and Andrew Young for useful conversations.
\vfuzz=2pt
\providecommand{\bysame}{\leavevmode\hboxto3em{\hrulefill}\thinspace}
|
{
"timestamp": "2010-09-23T02:00:23",
"yymm": "1009",
"arxiv_id": "1009.4209",
"language": "en",
"url": "https://arxiv.org/abs/1009.4209"
}
|
\section{Introduction}
Supersymmetric solutions in supergravity
have played an important r\^ole in the development of string theory
and the anti-de Sitter(AdS)/conformal field theory
(CFT)-correspondence. A pioneer work in this direction was the
great success of microscopic deviation of black hole entropy from
the viewpoint of intersecting D-branes.
By virtue of the saturation of Bogomol'nyi-Prasad-Sommerfield (BPS) bound,
the supersymmetric solutions can provide arena for exploring
the non-perturbative limits of string theory.
The BPS equality constraints the supersymmetry variation spinor
to satisfy the 1st-order differential equation.
Such a covariantly constant spinor is called a Killing spinor, which ensures that the
energy is positively bounded by central charges, guaranteeing the
stability of the theory. The relationship between vacuum stability and
BPS states was suggested by Witten's positive energy theorem~\cite{Witten:1981mf}, and
later validated firmly by~\cite{GH1982,PET}.
From a standpoint of pure gravitating object, black hole solutions
admitting a Killing spinor are sharply distinguished from non-BPS black
hole solutions. These BPS configurations are dynamically very simple.
First of all, BPS black hole solutions necessarily have zero Hawking
temperature (the converse is not true), implying that
the horizon is degenerate. Accordingly they are free
from thermal excitation. Such a non-bifurcating horizon
universally admits a throat infinity and
enhanced isometries of ${\rm SO}(2, 1)$~\cite{Kunduri:2007vf}.
Secondly, most BPS solutions satisfy ``no-force'' condition. For example, we
are able to superpose the extreme Reissner-Nordstr\"om solutions at our
disposal due to the delicate compensation between the gravitational attractive
force and the electromagnetic repulsive force. The resulting multicenter metric,
originally found by Majumdar and Papapetrou, maintains static
equilibrium and describes collection of charged black holes~\cite{Hartle:1972ya}.
This property can be ascribed to the complete linearization of field
equations. Besides these, all the BPS black holes are known to be
strictly stationary, viz, the ergoregion does not exist even if the
black hole has nonvanishing angular momentum. Dynamically evolving
states are not compatible with supersymmetry.
Then, to what extent these known intuitive properties continue to hold?
Motivated by this inquiry
it is important to explore general properties and classify BPS solutions.
A first progress was made by
Tod, who catalogued all the BPS solutions admitting nontrivial
Killing spinors of 4-dimensional $\ma N=2$ supergravity~\cite{Tod}, inspired by the
early study of Gibbons and Hull~\cite{GH1982}.
Recently, Gauntlett {\it et al.}~\cite{GGHPR}
were able to obtain general supersymmetric solutions in 5-dimensional
minimal supergravity exploiting bilinears constructed from a Killing
spinor. Since their technique has no restriction upon the spacetime
dimensionality, reference~\cite{GGHPR} has sparked a considerable development in the
classifications of supersymmetric solutions in various
supergravities~\cite{Gauntlett:2003fk,Gutowski:2003rg,Caldarelli:2003pb,GR}.
This formalism is useful for finding supersymmetric black holes~\cite{Gutowski:2004ez,Kunduri:2006ek}
and black rings~\cite{Elvang:2004rt,Elvang:2004ds,Gauntlett:2004qy}, and
for proving uniqueness theorem of certain black holes~\cite{Reall2002,Gutowski:2004bj}.
It turns out that all the BPS black holes fulfill above mentioned
properties except for the equilibrium condition which is valid
only in the ungauged case.
On the other hand, non-BPS black hole solutions--especially the
time-dependent black hole solutions--have been much less understood.
In this paper we address some properties of cosmological black-hole
solutions which have an interpretation as arising from the gauged
supergravity with non-compact R-symmetry gauged. The most simplest
theory is the 4-dimensional minimal de Sitter supergravity consisting of
the graviton, the Maxwell fields and a {\it positive} cosmological constant~\cite{Pilch:1984aw}.
A time-dependent solution in this theory was found by Kastor and Traschen~\cite{KT,London},
which is the generalization of Majumdar-Papapetrou solution in the
de Sitter background. The Kastor-Traschen solution describes
coalescing black holes in the contracting de Sitter universe (or
splitting white holes in the expanding de Sitter universe) and
inherits some salient characteristics from the Majumdar-Papapetrou solution.
The reason why multicenter metric is in mechanical equilibrium
irrespective of the time-dependence is attributed to the
first-order ``BPS equation'' that extremizes the action,
allowing the complete linearization of field equations. Since
these ``BPS'' states are not truly supersymmetric in the usual sense,
they are referred to as pseudo-supersymmetric and the corresponding
theory is called a ``fake'' supergravity.
Recently, all pseudo-supersymmetric solutions in 4- and
5-dimensional fake de Sitter fake supergravity were classified using the
spinorial geometry method~\cite{Gutowski:2009vb,Grover:2008jr}
(see~\cite{Meessen:2009ma} for a non-Abelian generalization).
In this paper, we discuss properties of pseudo-supersymmetric solutions
of 5-dimensional fake supergravity with arbitrary number of
${\rm U}(1)$ gauge fields and scalar fields. Some time-dependent black
hole solutions in this theory have been available so
far~\cite{Behrndt:2003cx,Klemm:2000gh}, but their properties and
causal structures are yet to be explored. Even for the simplest case in
which the harmonic function is sourced by a single point mass,
the spacetime is highly dynamical except in the de Sitter supergravity.
In the present case the background spacetime is the
Friedmann-Lema\^itre-Robertson-Walker (FLRW) cosmology.
(In the context of fake supergravity,
it is argued that the FLRW cosmologies are duals of supersymmetric
domain walls. See~\cite{DW_FRW} for details.)
A series of recent papers of present authors~\cite{MN,MNII} revealed
that the solution of a single point source
found in~\cite{MOU,GMII} actually describes a charged black hole in the
FLRW cosmology. Though the metric in~\cite{MOU,GMII} were shown
to be the exact solutions of Einstein-Maxwell-dilaton system,
we show in this paper that the 5-dimensional solutions
of~\cite{MOU,MNII} in fact satisfy the 1st-order BPS equation in fake
supergravity. The pseudo-supersymmetry is indeed consistent with an
expanding universe.
This work will establish new insights for black holes in
time-dependent and non-supersymmetric backgrounds.
The main concern in this paper is to see the effects of black-hole rotation
in 5-dimensions by restricting to the single point mass case.
As it turns out, rotation makes the properties of spacetime much richer.
Our work is organized as follows. In the next section we describe
a fake supergravity model and derive (in a gauge different
from~\cite{Klemm:2000vn}) a rotating, time-dependent solution
preserving the pseudo-supersymmetry. Section~\ref{sec:bhs} is devoted
to explore physical and geometrical properties of the spacetime.
We establish that the black hole horizon is generated by a rotating
Killing horizon, in sharp contrast with the supersymmetric black-hole
horizon which admits a non-rotating degenerate Killing horizon without
an ergoregion.
It is also demonstrated that the solution generally admits closed timelike
curves in the vicinity of timelike singularities (with trivial
fundamental group).
Combining the analysis of the near-horizon geometries, we
shall elucidate the causal structures by illustrating Carter-Penrose diagrams.
In section~\ref{sec:liftup} the liftup and reduction scheme of the
5-dimensional solution is accounted for. It is shown that the
5-dimensional solutions derived in \cite{MOU,MNII} and in section~\ref{sec:bhs}
are elevated to describe the non-BPS dynamically
intersecting M2/M2/M2-branes in 11-dimensional supergravity.
Upon dimensional reduction the 4-dimensional black hole~\cite{MOU,GM} is
obtainable. We shall also present some Kaluza-Klein black holes in the FLRW
universe. Section~\ref{conclusion} gives final remarks.
We will work in mostly plus metric signature and the standard
curvature conventions
$2\nabla_{[\rho }\nabla_{\sigma] }V^\mu={R^\mu}_{\nu\rho\sigma}V^\nu$.
Gamma matrix conventions are such that
$\gamma_{\mu\nu\rho\sigma\tau }=i\epsilon_{\mu\nu\rho\sigma\tau }$
with $\epsilon_{01234}=1$ and $\bar \psi:=i\psi^\dagger \gamma^0$.
\section{Five dimensional solutions in minimal supergravity}
\label{5D_sol_SG}
The metrics obtained in~\cite{MOU,GMII,MN,MNII} are the exact solutions
of Einstein's equations sourced by two-${\rm U}(1)$ fields and a scalar
field coupled to the gauge fields. Since the solution involves two kinds of harmonic
functions, it manifests mechanical equilibrium regardless of
time-evolving spacetime. When each harmonic has a point source at the center,
the solution in~\cite{MOU,GMII,MN,MNII} describes a spherically
symmetric black hole embedded in the FLRW cosmology.
In this section, we consider
a five-dimensional supergravity-type Lagrangian and present more
general (pseudo) BPS solutions, which encompass the
5-dimensional solution in~\cite{MOU,MNII} as special limiting cases.
Let us start from the
minimal 5-dimensional gauged supergravity coupled to
$N$ abelian vector multiplets. The bosonic
action involves graviton, ${\rm U}(1)$ gauge fields $A^{(I)}$
($I=1,..., N$)
with real scalars $\phi^A$ ($A=1, ..., N-1$)~\cite{Gunaydin:1984ak},
\begin{align}
S
=&\frac{1}{2\kappa_5^2 }\int \left[\left({}^5 R
+2 \mathfrak g^2 V \right)\star _51
-\ma G_{AB} {\rm d} \phi^A \wedge \star_5 {\rm d} \phi^B
\right. \nonumber \\ & \left.
- G_{IJ} F^{(I)}\wedge \star_5F^{(J)}-\frac 16 C_{IJK}
A^{(I)}\wedge F^{(J)}\wedge F^{(K)}
\right]\,,
\label{5Daction}
\end{align}
where $F^{(I)}={\rm d} A^{(I)}$ are the field strengths of gauge fields and
$\mathfrak g$ is the coupling constants corresponding to the
reciprocal of the AdS curvature radius.
$C_{IJK}$ are
constants symmetric in $(IJK)$ and obey the ``adjoint identity''
\begin{align}
C_{IJK}C_{J'(LM}C_{PQ)K'}\delta ^{JJ'}\delta ^{KK'}
=\frac 43 \delta _{I(L}C_{MPQ)}\,,
\end{align}
where the round brackets denote symmetrization of the suffixes.
The potential $V$ can be expressed in terms of a superpotential $W$ as
\begin{align}
V=6W^2-\frac 92 \ma G^{AB} (\partial_A W)( \partial_B W)\,,
\end{align}
where $\partial_AX^I:={\rm d} X^I(\phi)/{\rm d} \phi^A$.
The superpotential takes the form,
\begin{align}
W=V_IX^I\,,
\end{align}
where $V_I$ are constants arising from an Abelian gaugings of the
${\rm SU}(2)$-R symmetry with the gauge field $A=V_IA^I$~\cite{Gunaydin:1984ak}.
The $N$-scalars $X^I$ are constrained by
\begin{align}
\ma V:= \frac 16C_{IJK}X^IX^JX^K =1\,.
\label{constraint_XI}
\end{align}
It is convenient to define
\begin{align}
X_I= \frac 16 C_{IJK}X^JX^K\,,
\end{align}
in terms of which Eq.~(\ref{constraint_XI})
is simply $X_IX^I=1$.
The coupling matrix
$G_{IJ}$ is the metric of the
``very special geometry''~\cite{deWit:1992cr} defined by
\begin{align}
G_{IJ}:=-\frac 12\left.\frac{\partial^2 }{\partial X^I\partial X^J}\ln \ma V
\right|_{\mathcal V=1}=
\frac 92 X_IX_J-\frac 12 C_{IJK}X^K\,,
\label{G_IJ}
\end{align}
with its inverse
\begin{align}
G^{IJ}=2X^IX^J-6 C^{IJK}X_K\,,
\end{align}
where
$C^{IJK}=\delta^{IL}\delta ^{JP}\delta^{KQ}C_{LPQ}$.
The other coupling matrix $\ma G_{AB}$ is given by
\begin{align}
\ma G_{AB}=G_{IJ} \partial _AX^I\partial_B X^J\,,
\end{align}
It follows that
\begin{align}
X^I= \frac 92C^{IJK}X_JX_K\,,
\end{align}
and
\begin{align}
X_I=\frac 23 G_{IJ}X^J\,, \qquad
X^I=\frac 32 G^{IJ}X_J\,.
\end{align}
From these relations, we obtain useful expressions
\begin{align}
&{\rm d} X_I=-\frac 23G_{IJ}{\rm d} X^J\,,\qquad
{\rm d} X^I=-\frac 32 G^{IJ}{\rm d} X_J\,, \nonumber \\
&
X^I{\rm d} X_I=X_I{\rm d} X^I=0\,, \nonumber \\
&\ma G^{AB}\partial_A X^I\partial_BX^J=G^{IJ}-\frac 23X^IX^J.
\label{useful_relation}
\end{align}
Using these formulae, the potential reads
\begin{align}
V=27C^{IJK}V_IV_JX_K\,.
\end{align}
If this theory is derived via gauging the supergravity derived from
the Calabi-Yau compactification of M-Theory, $\ma V$ is
the intersection form, $X^I$ and $X_I$ correspond respectively to
the size of the two- and four-cycles. The constants $C_{IJK}$ are the
intersection numbers of the Calabi-Yau threefold and $N$ denotes the
Hodge number $h_{1,1}$~\cite{Papadopoulos:1995da}.
\begin{widetext}
The governing equations are the Einstein equations (varying $g^{\mu\nu }$),
\begin{align}
{}^5R_{\mu \nu } -\frac 12 \left({}^5 R +2\mathfrak g^2 V\right)
g_{\mu \nu } =G_{IJ}
\left[(\nabla_\mu X^I)(\nabla_\nu X^J)-\frac 12
(\nabla^\rho X^I)(\nabla_\rho X^J)g_{\mu\nu }
+F^{(I)\rho }_\mu F^{(J)}_{\nu\rho }-\frac 14
F^{(I)}_{\rho\sigma }F^{(J)\rho\sigma }g_{\mu \nu }
\right]\,,
\end{align}
the electromagnetic field equations (varying $A^{(I)}$),
\begin{align}
\nabla_\nu \left(
G_{IJ}F^{(J)\mu \nu }
\right)-\frac{1}{16}C_{IJK}\epsilon^{\mu\nu\rho\sigma\tau }
F^{(J)}_{\nu\rho }F^{(K)}_{\sigma\tau }=0\,,
\label{Maxwell_eq}
\end{align}
where $\epsilon_{\mu\nu\rho\sigma\tau }$
is the metric-compatible volume element,
and the scalar field equations (varying $\phi^A$),
\begin{align}
&\biggl[
\nabla^\mu \nabla_\mu X_I+
6\mathfrak g^2 V_LV_MC_{IJK}C^{KLM}X^J\nonumber \\
&~~~
+\left(C_{IJL}X_KX^L-\frac 16C_{IJK}\right)
\left((\nabla_\mu X^J)(\nabla^\mu X^K)
+\frac 12 F^{(J)}_{\mu\nu }F^{(K)\mu\nu }\right)
\biggr]
\partial_A X^I=0\,.
\label{scalar_eq}
\end{align}
From the condition $X_I{\rm d} X^I=0$, the terms in square-bracket in the
above equation must be proportional to $X_I$. Denoting it by $LX_I$,
one obtains the expression of $L$ using the relation~$X_IX^I=1$.
Then the scalar equations are rewritten as
\begin{align}
\nabla^\mu \nabla_\mu X_I+
\left(\frac 12C_{JKL}X_IX^L-\frac 16C_{IJK}\right)
(\nabla^\mu X^J) (\nabla_\mu X^K)
+6\mathfrak g^2C^{JLM}V_LV_M
\left(6X_IX_J-C_{IJK}X^K\right)&\nonumber \\
+\frac 12
\left(C_{IJL}X_KX^L-\frac 16 C_{IJK}-6X_IX_JX_K+\frac
16C_{JKL}X_IX^L\right)F^{(J)}_{\mu\nu }
F^{(K)\mu\nu }
&=0\,.
\end{align}
The supersymmetric transformations for the gravitino $\psi_\mu $ and
gauginos $\lambda_A $ are given by
\begin{align}
\delta \psi_\mu
&=\left[\ma D_\mu -\frac {3i}2 \mathfrak g V_IA^{(I)}_\mu
+\frac{i}{8}X_I\left({\gamma _\mu }^{\nu\rho }
-4{\delta_\mu }^\nu \gamma^\rho \right)F_{\nu\rho }^{(I)}
+\frac 12 \mathfrak g \gamma_\mu X^IV_I\right]\epsilon \,,
\label{KS_STU1}\\
\delta \lambda _A &= \left[\frac{3}{8}\gamma^{\mu\nu }F_{\mu\nu }^{(I)}
\partial_A X_I
-\frac i{2}\ma G_{AB}\gamma ^\mu \partial_\mu \phi^B
+\frac{3i}2 \mathfrak g V_I\partial_A X^I
\right]\epsilon \,,
\label{KS_STU2}
\end{align}
\end{widetext}
where $\epsilon $ is a spinor generating an infinitesimal supersymmetry
transformation.
Here and throughout the paper,
$\ma D_\mu $ will be used for a gravitationally-covariant derivative
defined by
\begin{align}
\ma D_\mu \epsilon =\left(\partial_\mu +\frac 14{\omega _\mu
}^{ab}\gamma_{ab} \right)\epsilon \,,
\end{align}
where ${\omega_\mu }^{ab}$ is a spin-connection without torsion.
We have used the Dirac spinor
instead of the symplectic-Majorana spinor.
The supersymmetric solutions in this theory have been
analyzed~\cite{GR}. One recovers ungauged supergravity
by $\mathfrak g\to 0$.
\subsection{Pseudo-supersymmetric solutions in fake supergravity}
If we consider a non-compact gauging of R-symmetry,
an imaginary coupling arises, $\mathfrak g \to i k ~(k\in \mathbb R)$.
Since only the R-symmetry is gauged, the imaginary coupling reflects the
non-compactness of R-symmetry. The Lagrangian~(\ref{5Daction}) is neutral
under the R-symmetry, so that the theory is free from
the ghost-like contribution.
This theory is called a fake supergravity.
The fake ``Killing spinor'' equations reduce to
\begin{widetext}
\begin{align}
\left[\ma D_\mu+\frac {3k}2 V_IA^{(I)}_\mu +\frac{i}{8}X_I\left({\gamma _\mu }^{\nu\rho }
-4{\delta_\mu }^\nu \gamma^\rho \right)F_{\nu\rho }^{(I)}
+\frac i2 k \gamma_\mu X^IV_I\right]\epsilon =0
\label{fakeKS1}\,, \\
\left[\frac{3}{8}\gamma^{\mu\nu }F_{\mu\nu }^{(I)}
\partial_A X_I
-\frac i{2}\ma G_{AB}\gamma ^\mu \partial_\mu \phi^B
-\frac{3}2 k V_I\partial_A X^I
\right]\epsilon =0 \,.\label{fakeKS2}
\end{align}
\end{widetext}
Here, the supercovariant derivative operator is no longer
hermitian for $k\in \mathbb R$. This implies that we are unable to use
$\epsilon $ to prove the positive energy theorem in the usual manner. Still, we presume
that the above equations~(\ref{fakeKS1}) and (\ref{fakeKS2}) continue to
be valid for $k\in \mathbb R$.
Inferring from the supersymmetric solutions in~\cite{GR},
we assume the standard metric ansatz,
\begin{align}
{\rm d} s_5^2 =-f^2 ({\rm d} t+\omega )^2 +f^{-1}h_{mn}{\rm d} x^m {\rm d} x^n\,,
\label{SUSYmetric}
\end{align}
where the 4-metric $h_{mn}$ is orthogonal to
$V^\mu =(\partial/\partial t)^\mu $ ($i_Vh_{\mu\nu }=0$)
and supposed to be independent of $t$ ($\mas L_V h_{\mu\nu}=0$).
The one-form $\omega $ corresponds to the ${\rm U}(1)$
fibration of the transverse base space $(\ma B, h_{mn})$.
In what follows, indices $m, n,...$ are raised and lowered by $h_{mn}$ and its inverse
$h^{mn}$. The connection $\omega $ is orthogonal to the timelike vector
field $V^\mu $ and assumed to be independent of $t$
($\mas L_{V}\omega =0$).
We further suppose that the lapse function is given by
\begin{align}
f^{-3}=\frac 16C^{IJK}H_IH_JH_K\,,
\label{f_cube}
\end{align}
where $H_I$'s are some functions.
We also assume the profiles of the
electromagnetic and the scalar fields as
\begin{align}
A^{(I)} =fX^I ({\rm d} t+\omega )\,, \qquad
X_I =\frac 13 f H_I\,.
\label{SUSYgauge}
\end{align}
In the ungauged supersymmetric case (when $\mathfrak g=0$),
the condition~(\ref{f_cube}) is obtained as a special case of
the general supersymmetric
solutions, as referred to hereinafter in section~\ref{sec:liftup}.
In this section, we just assume (\ref{f_cube}).
Taking the orthonormal frame
\begin{align}
e^0=f({\rm d} t+\omega ) \,, \qquad e^i =f^{-1/2} \hat e^i\,,
\end{align}
where $\hat e^i$ is the orthonormal frame for $h_{mn}$, one can calculate
the time and spatial components of ``Killing spinor'' equation (\ref{fakeKS1}), which are
given by
\begin{widetext}
\begin{align}
& \biggl[\partial_t
+kfV_IX^I+\left\{\frac{1}{2}fkV_IX^I+\frac{1}{4}f^3\partial_{[m}\omega_{n]}
\hat \gamma^{mn}+\frac{i}{2}f^{1/2}(\partial _mf-\omega_m
\partial_t f)\hat \gamma^{m}\right\}(1-i\gamma^0)\biggr]\epsilon =0 \,, \\
&\biggl[{}^h\ma D_m-\omega_m\partial_t -\frac{1}{2f}(\partial_mf-\omega_m
\partial_t f )i\gamma^0
+\frac{f^{3/2}}{2}\left(\frac{1}{2}{{}^h\epsilon_{mn}}^{pq}\partial_{[p}\omega_{q]} +\partial_{[m}
\omega_{n]}\right)\hat \gamma^{n}\gamma^0
\nonumber \\
&~
+\frac{i}{2f^{1/2}}\hat \gamma_m\left(kV_IX^I+\frac{i\gamma^0 \partial_t f}{2f^2}\right)
+\left\{
-\frac{1}{4f}(\partial_nf-\omega_n \partial_t f){\hat{\gamma}_m}^{~n}-
i f^{3/2}\partial_{[m}\omega_{n]}
\right\}(1-i\gamma^0 )\biggr]\epsilon =0\,,
\end{align}
\end{widetext}
where $\hat \gamma ^m={\hat{e}}^{~m}_i \gamma^i $.
${}^h \ma D$ and ${}^h\epsilon$ are, respectively, the Lorentz-covariant derivative and the
volume-element with respect to $h_{mn}$. From Eqs.~(\ref{f_cube}) and
(\ref{SUSYgauge}), we have a useful relation
\begin{align}
kV_IX^I+ \frac 12f^{-2}{\partial_t f}=\frac{1}{2}f^2C^{IJK}H_IH_J
\left(kV_I-\frac{1}{6}\partial_t H_I\right)\,.
\end{align}
Thus, if ${\rm d} \omega $ satisfies the anti-self duality
condition,
\begin{align}
{\rm d} \omega +\star_h {\rm d} \omega =0\,,
\label{domega}
\end{align}
where
$\star_h$ denotes the Hodge dual operator with respect
to the base space metric $h_{mn}$
and if $H_I$'s satisfy the differential equations
$\partial _t H_I=6kV_I$,
the Killing spinor equations are solved by
\begin{align}
i\gamma ^0 \epsilon &=\epsilon \label{sol_KS1} \,, \\
\epsilon &=f^{1/2}\zeta \,.
\label{sol_KS2}
\end{align}
Here $\zeta $ is a covariantly constant Killing spinor
with respect to the 4-dimensional metric $h_{mn}$,
\begin{align}
{}^h\ma D_m \zeta =0\,,
\label{Killingsp_4}
\end{align}
satisfying
\begin{align}
\hat \gamma ^{ 1 2 3 4}\zeta =\zeta \,.
\label{chiral}
\end{align}
It follows that $H_I$'s take the form,
\begin{align}
H_I(t, x^m)=6kV_I t +\bar H_I(x^m)\,,
\label{tI}
\end{align}
where $H_I$'s are functions on the base space.
The integrability condition of Eq.~(\ref{Killingsp_4}) is
${}^hR _{mnpq}\hat\gamma^{pq}\epsilon =0$.
From the chirality condition~(\ref{chiral}), one can find that
$\hat\gamma_{mn}\epsilon $ is anti-self dual on the base space.
This implies that the Riemann tensor of $h_{mn}$ is self-dual
$\star_h (^hR_{mnpq})={}^hR_{mnpq}$.
Hence, the base space $(\ma B, h_{mn})$ turns out to be
the hyper-K\"ahler manifold whose complex structures $\mathfrak J^{(i)}$ are
anti-self-dual $\star_h\mathfrak J^{(i)}=-\mathfrak J^{(i)}$.
The chirality condition~(\ref{chiral})
is a direct consequence of $i\gamma ^0 \epsilon =\epsilon $, which is
the only projection imposed on the Killing spinor. It follows that
the solution preserves at least half of pseudo-supersymmetries.
If Eqs.~(\ref{tI}) and (\ref{sol_KS1}) are satisfied, one verifies
that the dilatino equation~(\ref{fakeKS2}) is satisfied automatically.
Let us next turn to the Maxwell equations (\ref{Maxwell_eq}). Only the
0th component is nontrivial, giving
\begin{align}
{}^h \Delta \bar H_I=0\,,
\end{align}
where ${}^h \Delta $ is the Laplacian operator with respect to
$h_{mn}$. This equation manifests the complete linearization.
All the metric components are obtained by use of Killing spinor and
Maxwell equations under our ansatz.
We have nowhere solved the scalar and Einstein's equations so far.
Nevertheless, these equations are automatically satisfied if the Bianchi
identities ${\rm d} F^{(I)}=0$ and Maxwell equations~(\ref{Maxwell_eq}) are
satisfied, on account of the integrability conditions for the pseudo-Killing
spinor equations.
The procedure for generating time-dependent backgrounds presented here
was previously given in~\cite{Behrndt:2003cx}.
It is however observed that the above metric-form is not fully general.
According to the analysis for the de Sitter supergravity~\cite{Grover:2008jr}, the base space is
allowed to have a torsion. We expect that the general classification in this
theory is also possible following the same fashion as~\cite{Grover:2008jr}.
\subsection{Rotating black hole in STU theory}
To be concrete, let us consider the ``STU-theory,''
which is defined by
the conditions such that
$C_{123}=C_{(123)}=1$ and the other $C_{IJK}$'s vanish. In this theory,
one has three Abelian gauge fields and two unconstrained scalars.
For simplicity, let us choose the flat space as a base space
$(\ma B, h_{mn})$,
\begin{align}
{\rm d} s_{\mathcal B}^2 ={\rm d} r^2+r^2 \left({\rm d} \vartheta^2 +\sin^2\vartheta
{\rm d} \phi_1^2+\cos ^2\vartheta {\rm d} \phi_2^2 \right)\,.
\end{align}
Then, the equation for $\omega $~(\ref{domega}) is easily solved to give
\begin{align}
\omega =\frac{J}{r^2 }\left(\sin^2\vartheta {\rm d} \phi_1 +\cos^2\vartheta
{\rm d}\phi_2 \right)\,,
\label{omegasol}
\end{align}
where the volume form of $(\ma B, h_{mn})$ is taken as
${\rm d} r\wedge(r{\rm d}\vartheta)\wedge(r\sin\vartheta {\rm d}\phi_1)\wedge(r\cos\vartheta {\rm d}\phi_2)$
and $J$ is a constant representing the rotation of the spacetime.
In what follows we shall specialize to the case where
each harmonic function has a point source at the origin $\propto
Q_I/r^2$.
Denoting
\begin{align}
t_I=(6kV_I)^{-1}\,,
\end{align}
we classify the solutions into the following four cases
depending on how many $V_I$'s vanish~\footnote{
We shall not examine the cases in which some charges vanish $Q_I=0$
since these cases will provide nakedly singular spacetimes without
regular horizons.
}.
\bigskip\noindent (i)
$V_1=V_2=V_3=0$ for which
\begin{align}
H_1=1+\frac{Q_1}{r^2} \,,\qquad
H_2=1+\frac{Q_2}{r^2} \,, \qquad
H_3=1+\frac{Q_3}{r^2} \,.
\label{sol_i}
\end{align}
This is nothing but the solution in
the ungauged true supergravity in which
the scalar field potential vanishes. The
supersymmetric solutions have been completely
classified in~\cite{GR,Elvang:2004ds}.
This theory can be uplifted to 11-dimensional supergravity as described
later. The 11-dimensional solution describes the rotating
M2/M2/M2-branes preserving 1/8-supersymmetry.
In the following, we do not elaborate this case unless otherwise stated
since its physical properties have been widely discussed
in the existing literature~\cite{BMPV,BLMPSV,GMT}.
\bigskip\noindent (ii)
$V_1 \ne 0 $, $V_2=V_3=0$ for which
\begin{align}
H_1=\frac{t}{t_1}+\frac{Q_1}{r^2} \,,\qquad
H_2=1+\frac{Q_2}{r^2} \,, \qquad
H_3=1+\frac{Q_3}{r^2} \,.
\label{sol_ii}
\end{align}
This case corresponds also to the zero potential
$V=27C^{IJK}V_IV_JX_K =0$ due to $C_{11K}=0$.
It is notable that the potential height $V_1$ makes a contribution
to the pseudo-Killing spinor equations~(\ref{fakeKS1}) and (\ref{fakeKS2}).
This pseudo-supersymmetric solution can be oxidized
to 11-dimensions, but
the resultant spacetime is not pseudo-supersymmetric since
11-dimensional supergravity has no potential term.
The oxidized solution is interpreted as the intersecting M2/M2/M2-branes
in the background rotating Kasner universe.
The detail is described in section~\ref{M2M2M2}.
\bigskip\noindent (iii) $V_1, V_2 \ne 0 $, $V_3=0$ for which
\begin{align}
H_1=\frac{t}{t_1}+\frac{Q_1}{r^2} \,,\qquad
H_2=\frac{t}{t_2}+\frac{Q_2}{r^2} \,, \qquad
H_3=1+\frac{Q_3}{r^2} \,.
\label{sol_iii}
\end{align}
These two cases (ii) and (iii) have not been discussed in~\cite{Behrndt:2003cx} although
the authors arrived at the same equation as~(\ref{tI}).
\bigskip\noindent (iv) $V_1, V_2, V_3 \ne 0 $ for which
\begin{align}
H_1=\frac{t}{t_1}+\frac{Q_1}{r^2} \,,\qquad
H_2=\frac{t}{t_2}+\frac{Q_2}{r^2} \,, \qquad
H_3=\frac{t}{t_3}+\frac{Q_3}{r^2} \,.
\label{sol_iv}
\end{align}
When $t_1=t_2=t_3$ and $Q_1=Q_2=Q_3$, all scalar fields are trivial.
This case corresponds to the fake de Sitter supergravity for which the potential is constant
$\mathfrak g^2 V=-3/(2t_1^2)$. The complete classification of
timelike class for the de Sitter supergravity was done
in~\cite{Gutowski:2009vb,Grover:2008jr}.
Even if $t_I$'s and $Q_I$'s are not all identical, this solution inherits many
properties of that in de Sitter supergravity, irrespective of nontrivial
scalar fields $X^I$. In fact, by a coordinate transformation
\begin{align}
&
r' = r \left(\frac{t}{t_0}\right)^{1/2} \,, \qquad
\ln \left(\frac{t}{t_0}\right)=\frac{t'}{t_0}+\int ^{r'}
\frac{h_2(r')}{h_1(r')}{\rm d} r' \,,
\nonumber \\
&
\phi_{1,2}= \phi_{1,2}'+\int ^{r'} h_2(r'){\rm d} r'\,,
\end{align}
where $t_0\equiv (t_1t_2t_3)^{1/3} $ and
\begin{align}
&
h_1( r'):=\frac{J r'^2 t_0}{H^3 r'^6-J^2} \,, \qquad
h_2( r'):=\frac{2Jr' t_0}{J^2-H^3r'^6+4r'^4 t_0^2 } \,,
\nonumber \\
&
H^3:=\left(\frac{t_0}{t_1}+\frac{Q_1}{r'^2}\right)
\left(\frac{t_0}{t_2}+\frac{Q_2}{
r'^2}\right)\left(\frac{t_0}{t_3}+\frac{Q_3}{r'^2}\right)\,,
\end{align}
the metric~(\ref{sol_iv}) can be brought to the stationary form,
\begin{widetext}
\begin{align}
{\rm d} s^2 =&\frac{{r'}^2 H}{4 t_0^2}{\rm d} {t'}^2-H^{-2}
\left[{\rm d} t'+\frac{J}{{r'}^2} \left(\sin^2\vartheta{\rm d} \phi_1'
+\cos^2\vartheta{\rm d} \phi_2' \right)\right]^2 \nonumber \\
&+H \left[
\frac{{\rm d} {r'}^2}{1-H^3 {r'}^2/(4t_0^2)+J^2/(4t_0^2{r'}^4)}+{r'}^2 \left({\rm d}
\vartheta^2 +\sin^2\vartheta {\rm d} {\phi_1'}^2 +\cos^2\vartheta {\rm d}
{\phi_2'}^2\right)\right]\,.
\label{KSsol}
\end{align}
\end{widetext}
This is asymptotically de Sitter with curvature radius
$\ell =2t_0$~\cite{Klemm:2000vn}.
When the rotation vanishes ($\omega=0$),
these solutions reduce to the ones considered in our previous
papers~\cite{MN,MNII}, describing a spherically symmetric black hole
in 5-dimensional FLRW universe.
It is then expected that the present solution
describes a rotating black hole in the expanding universe.
To see this more concretely,
let us consider the asymptotic limit $r\to \infty $ of the solutions.
Let $n$ denote the number of time-dependent harmonics, i.e.,
$n=1, 2$ and 3 are the cases (ii), (iii) and (iv), respectively.
Changing to the new time slice
\begin{align}
\frac{\bar t}{\bar t_0}=\left(\frac{t}{t_0}\right)^{1-n/3}\,, \qquad
\bar t_0=\frac{3t_0}{3-n}\,,
\end{align}
for $n=1,2$ and
$\bar t=t_0 \ln (t/t_0)$ for $n=3$, one easily finds that
each solution (\ref{sol_i})--(\ref{sol_iv})
approaches to the 5-dimensional flat FLRW universe,
\begin{align}
{\rm d} s_5^2=-{\rm d} \bar t^2+a^2\delta_{mn}{\rm d} x^m {\rm d} x^n \,.
\end{align}
Here, $\bar t$ measures the cosmic time at infinity and the scale factor
obeys
\begin{align}
a = (\bar t/\bar t_0) ^{n/[2(3-n)]}\,,
\label{scale_factor1}
\end{align}
for $n=1,2$ and
\begin{align}
a= e^{\bar t/2 t_0} \,,
\label{scale_factor2}
\end{align}
for $n=3$,
which are respectively the same expansion law
as the stiff-matter dominant universe ($n=1$), the universe filled by
fluid with equation of state $P=-\rho/2$
($n=2$), and the de Sitter universe with curvature radius $2t_0$ ($n=3$).
In either case, the solution tends to be spatially
homogeneous and isotropic in the asymptotic region
$r\to \infty $.
On the other hand, when one takes the limit in
which $r$ goes to zero {\it with $t$ kept finite}, the
solution~(\ref{sol}) approaches to a deformed AdS$_2 \times$ $S^3$:
\begin{align}
{\rm d} s^2_{r\to 0}=&- \left(\frac{r^2}{\bar Q}\right)^2 \left[{\rm d}
t+\frac{j}{r^2}(\sin^2\vartheta {\rm d} \phi_1+\cos^2\vartheta {\rm d} \phi_2)
\right]^2
\nonumber \\
&
+\left(\frac{\bar Q}{r^2}\right)^2 {\rm d} r^2+\bar Q {\rm d} \Omega_3^2\,,
\label{throat}
\end{align}
where $\bar Q\equiv (Q_1Q_2Q_3)^{1/3}$ and ${\rm d} \Omega_3^2$
denotes the unit line-element of $S^3$.
This is the same as the near-horizon geometry of a BMPV black
hole~\cite{Reall2002,GMT},
implying that $r=0$ is a point at the tip of an infinite throat.
Note that when all harmonics are
time-independent, the solution reduces to the
BMPV black hole with a degenerate horizon at $r=0$.
It is noteworthy, however, that this metric~(\ref{throat}) does {\it not} describe
the geometry of a neighborhood of ``would-be horizon'' since we
have fixed the time-coordinate when taking the $r\to 0$ limit.
As pointed out in~\cite{MN,MNII}
the null surfaces piercing the throat correspond to the infinite
redshift ($t\to +\infty $) and blueshift ($t\to -\infty $) surfaces.
The structures of these null surfaces can be analyzed by taking
appropriate ``near-horizon'' limit, as we will discuss later.
As it turns out, the horizon, if it exists, is not extremal in general,
contrary to the na\"ive expectations from (\ref{throat}).
The reason why we consider rotating black holes in 5-dimensions
is that rotation is compatible with
supersymmetry in 5-dimensions. In $D$-dimensions, the gravitational
attractive force and centrifugal force behave respectively
as $-M/r^{D-3}$ and $J^2/M^2r^2$, so that
the balance is maintained only in
$D=5$.
The spinning cosmological solution in the Einstein-Maxwell-axion gravity
is obtained via dimensional reduction of a chiral null
model in 5-dimensions~\cite{Shiromizu:1999xj}.
Incidentally, let us mention the issue of the fact that
the action involved several gauge fields.
This is a necessary price in order to obtain the finite sized horizon area.
Just with a single gauge field, the
spacetime becomes nakedly singular unless the scalar field potential is
a pure cosmological constant. A specific example is given in Appendix~\ref{appA}
within the framework of the Einstein-Maxwell-dilaton gravity.
\section{Physical properties of 5-dimensional rotating black holes}
\label{sec:bhs}
Let us explore the
physical properties of the solutions~(\ref{sol_i})--(\ref{sol_iv}).
For further simplicity of our argument, we shall confine ourselves to
the case in which all charges are identical $(Q_1=Q_2=Q_3\equiv Q>0)$
and all the potential height are the same ($t_1=t_2=t_3\equiv t_0>0$). Then,
the metric~(\ref{SUSYmetric}) is described in a unified way as
\begin{align}
f=
H_T^{-n/3} H_S^{-1+n/3}
\,,
\label{sol}
\end{align}
with
\begin{align}
H_T:=\frac{t}{t_0}+\frac{Q}{r^2}\,,
\qquad
H_S:= 1+\frac{Q}{r^2}\,,
\label{sol_H}
\end{align}
where $ n\,(=0, 1, 2, {\rm or}~3)$ counts the number of time-dependent
harmonics. This section is devoted to explore physical properties of
the solution (\ref{sol}) with (\ref{sol_H}).
Here and hereafter, the subscript ``$T$'' and ``$S$'' will be used
consistently for the
time-dependent and time-independent quantities.
The time-dependent and static scalar fields $X_I$ are given by
\begin{align}
X_T=\frac 13\left(\frac{H_T}{H_S}\right)^{1-n/3}\,, \qquad
X_S=\frac 13\left(\frac{H_T}{H_S}\right)^{-n/3}\,.
\label{XTXS}
\end{align}
Similarly, the gauge fields $A^{(I)}$ are
\begin{align}
&
A^{(T)}=H_T^{-1}\left({\rm d} t+\frac{J}{2r^2}\sigma_3^R\right)\,,
\nonumber \\
&
A^{(S)}=H_S^{-1}\left({\rm d} t+\frac{J}{2r^2}\sigma_3^R\right)\,.
\end{align}
The solution reduces to the BMPV solution describing an asymptotically flat
rotating black hole for $n=0$~\cite{BMPV,BLMPSV},
the Klemm-Sabra solution describing a rotating black hole in the de Sitter
universe for $n=3$~\cite{Klemm:2000vn}.
Our previous solution describing a spherically symmetric
black hole in the FLRW universe is recovered when the rotation
vanishes $\omega =0$~\cite{MNII}. To make contact with the notation of
the reference~\cite{MNII}, let us define a canonical scalar field
\begin{align}
\Phi = \sqrt{\frac{n (3-n)}{6}} \ln
\left(\frac{H_T}{H_S}\right)\,,
\label{Phi}
\end{align}
and make the replacements the electromagnetic fields as
\begin{align}
(A^{(T)}, A^{(S)}) \to \frac{1}{\sqrt{2\pi}}(A^{(T)},
A^{(S)})\,.
\label{em_MNII}
\end{align}
Then the solution (\ref{sol}) with (\ref{sol_H}), (\ref{Phi}) and
(\ref{em_MNII}) solves the field equations derived from the action,
\begin{align}
S_5 =&\frac{1}{2\kappa_5^2 }
\int {\rm d} ^5 x\sqrt{-g_5}\left[{}^5 R - (\nabla \Phi )^2
-\frac{n(n-1)}{2t_0^2}e^{-\lambda_T\Phi}
\right.\nonumber \\& \left.
-
\sum_{A=T,S}n_A e^{\lambda_A \Phi }
F^{(A)}_{\mu\nu }F^{(A)\mu \nu }
+2\epsilon^{\mu\nu\rho\sigma\tau }
A_\mu^{(T)}F_{\nu\rho }^{(S)}F_{\sigma\tau}^{(S)}
\right]\,,
\label{5Deffaction}
\end{align}
where $n_T=3-n_S=n$ and
\begin{align}
\lambda_T =2\sqrt{\frac{2n_S}{3n_T}}\,,\qquad
\lambda_S =-2\sqrt{\frac{2n_T}{3n_S}} \,,
\end{align}
which is the $D=5$ action considered
in~\cite{MNII} when the Chern-Simons term does not contribute, i.e.,
there is no rotation.
When the theory is motivated by supergravity,
the parameter $n$ takes an integer value.
We should stress that even if $n$ is not an integer,
the aforementioned metric~(\ref{SUSYmetric})
with (\ref{sol}) and (\ref{sol_H}) is still an exact solution of
the Einstein-Maxwell-scalar system, in which we have two ${\rm U}(1)$
fields coupled to the scalar field with an Liouville-type exponential
potential~(\ref{5Deffaction}).
The solution with non-integral values of $0<n<2$ is qualitatively similar to
the one with $n=1$. (The case $2<n<3$ has no representative
in this paper.)
The geometrical properties with $n=1$ discussed in what follows
are also applied to the solution with $0<n<2$.
\subsection{Symmetries}
At first sight, one might expect that the metric admits
${\rm U}(1)\times {\rm U}(1)$ spatial symmetries generated by
$\partial/\partial \phi_1$ and $\partial/\partial \phi_2$.
In order to see that the solution indeed admits much larger symmetry,
let us introduce the Euler angles
$(\theta , \phi, \psi)$ by
\begin{align}
\theta =2\vartheta \,,\qquad
\phi=\phi_2-\phi_1\,,\qquad
\psi=\phi_2+\phi_1\,,
\end{align}
which take ranges in
$0\le \theta\le \pi$,
$0\le \phi\le 2\pi$ and
$0\le \psi\le 4\pi$.
In terms of above coordinates,
the left-invariant one-forms $\sigma^R_i$
($i=1,2,3$) on ${\rm SU}(2)\simeq S^3$
are given by
\begin{align}
\sigma_1^R &=-\sin\psi {\rm d}\theta+\cos \psi \sin\theta {\rm d} \phi \,,\\
\sigma_2^R &=\cos\psi {\rm d} \theta +\sin\psi \sin \theta {\rm d} \phi \,,\\
\sigma_3^R &={\rm d} \psi+\cos \theta {\rm d} \phi \,.
\end{align}
These one-forms satisfy
\begin{align}
{\rm d} \Omega_3^2=\frac 14\sum_i(\sigma_i^R)^2\,,
\qquad
{\rm d}\sigma_i^R=\frac 12\sum_k\epsilon_{ijk}\sigma_j^R\wedge\sigma_k^R\,.
\end{align}
The right-invariant vector fields $\xi ^L_i$
are the spacetime Killing fields. They are given by
\begin{align}
\xi^L_1&=-\frac{\cos\phi }{\sin \theta }\partial_\psi+\sin\phi
\partial_\theta +\cot \theta \cos\phi \partial_\phi \,,\\
\xi^L_2&=\frac{\sin\phi }{\sin\theta }\partial_\psi+\cos\phi
\partial_\theta -\cot\theta \sin \phi \partial_\phi \,,\\
\xi^L_3&=\partial_\phi \,,
\end{align}
which are the generators of the left transformations of ${\rm SU}(2)$.
These Killing vectors satisfy
\begin{align}
&
\mas L_{\xi_i^L} \sigma_j^R=0 \,, \qquad
[\xi_i^L, \xi_j^L]=\sum_k\epsilon_{ijk}\xi_k^L\,,
\nonumber \\
&
\left(\frac{\partial }{\partial \Omega_3}\right)^2 =
4 \sum_i\xi_i^L \xi_i^L\,.
\end{align}
Besides these,
there exists an additional U$(1)$-Killing field
\begin{align}
\xi_3^R&=\partial_\psi \,.
\end{align}
The orbits of $\xi_3^R$ are the fibres of Hopf fibration of $S^3$.
It follows that the metric is invariant under the
action of ${\rm U}(2)\simeq {\rm SU}(2)\times {\rm U}(1)$,
acting on the 3-dimensional orbits which are spacelike at infinity.
Thus, the metric is expressed as
\begin{align}
{\rm d} s^2=-f ^2 \left({\rm d} t+\frac{J}{2r^2}\sigma_3^R\right)^2
+f^{-1}
\left(
{\rm d} r^2+r^2{\rm d} \Omega_3^2 \right)\,.
\label{metirc_Euler}
\end{align}
As discussed in~\cite{GibbonsBMPV},
the metric with ${\rm U}(2)$-symmetry admits a {\it reducible} Killing tensor
\begin{align}
\nabla _{(\mu }K_{\nu \rho )}=0\,, \qquad
K^{\mu \nu }=\sum_i (\xi_i^L)^\mu (\xi^L_i)^\nu \,,
\label{Killing_tensor}
\end{align}
which enables us to separate angular variables for the
geodesic motion and scalar field equation.
It should be remarked, however, that the solution does not admit a
timelike Killing field, so that the geodesic motion is not
immediately solved.
\subsection{Singularities}
One can immediately find that the scalar fields $X_I$~(\ref{XTXS}) blow up at
\begin{align}
t=t_s(r):=-\frac{t_0Q}{r^2} \quad {\rm and } \quad
r^2=-Q\,.
\label{sing}
\end{align}
Straightforward calculations reveal that all the curvature invariants are divergent
at these spacetime points, i.e., they are
spacetime curvature singularities. For example, the Ricci scalar
curvature is given by
\begin{widetext}
\begin{align}
{}^5 R= \frac{f^4 }{6r^8 H_T^2}\Biggl[&\frac{2n(3n-4)r^8}{t_0^2 f^6}
+J^2 \left(24 H_T^2+\frac{n(2-n)r^2}{t_0^2f^3}\right)
\nonumber \\ &
-4 Q^2r^2
H_T^nH_S^{1-n}\left\{2(nH_S^2+(3-n)H_T^2)-(nH_S+(3-n)H_T)^2\right\}
\Biggr]\,,
\end{align}
\end{widetext}
which diverges at the above spacetime points, as expected.
Note that the $t=0$ surface and the surface $r=0$ with
$t$ kept finite are not the curvature singularities, where the curvature
invariants are bounded. Hence,
the big-bang singularity at $t=0$ is completely smoothed out due to
electromagnetic charges. As in the case (i), the surface
$r=0$ is a plausible candidate of event horizon.
\subsection{Closed timelike curves}
Since the vector field $\xi_3^R=\partial_\psi$
generates closed orbits of the period
$4\pi$, there appear closed timelike curves if an orbit of $\xi_3^R$
becomes timelike.
Rewrite the metric~(\ref{metirc_Euler}) as
\begin{align}
{\rm d} s^2 =&-\frac{f^2}{\Delta_L}{\rm d} t^2 +\frac{{\rm d}
r^2}{f}
+\frac{r^2}{4f}
\biggl[(\sigma_1^R)^2+(\sigma_2^R)^2
\nonumber \\ & +\Delta_L \left(\sigma_3^R-
\frac{2 J f^3 }{r^4\Delta_L}{\rm d} t\right)^2\biggl]\,,
\label{metric_CP}
\end{align}
where
\begin{align}
\Delta_L:=1-\frac{J^2f^3}{r^6}\,.
\label{Delta_L}
\end{align}
Inspecting
\begin{align}
g (\xi_3^R, \xi_3^R)=\frac{f^2}{4}\left(
H_T^nH_S^{3-n}-\frac{J^2}{r^6}
\right)\,,
\end{align}
we can see that the
first term on the right hand side vanishes
at the singularities. It follows that
the Hopf fibres become timelike, i.e.,
closed timelike curves inevitably emerge in the vicinity of
singularities for all values of $J (\ne 0)$.
$\Delta_L=0$ defines the velocity of light surface
(VLS), where closed causal curves appear for $\Delta _L<0$.
For $n=0$ (the BMPV spacetime without time-dependence),
the VLS is located at
$r^2 =J^{2/3}-Q$ which is inside the horizon for the small rotation
$J^{2/3}<Q$, otherwise it is outside the horizon.
For $n\ne 0$, the VLS has the time-dependent profile
\begin{align}
t_{\rm VLS}(r):= \frac{t_0}{r^2}\left[\left\{\frac{J^2}
{(r^2+Q)^{3-n}}\right\}^{1/n}-Q\right]\,.
\label{VLS}
\end{align}
Since $S^3$ is ${\rm U}(1)$-fibration over $S^2$,
one can introduce the radius of $S^2$ by
\begin{align}
R= |r |f^{-1/2}\,.
\end{align}
In terms of $R$, the VLS is positioned at the constant radius,
\begin{align}
R_{\rm L}=J^{1/3}\,.
\end{align}
We shall say the region $R<R_L$ ($R>R_L$) inside (outside) the VLS.
Inside the VLS,
$\xi_3^R$ is pointing into the future direction
for $J>0$ and into the past one for $J<0$.
It is obvious that the singularity $t_s(r)$ exists for $r>0$.
As we will see in the next subsection,
a horizon is positioned at $r=0$ (with
$t=\infty$), so
that these closed timelike curves yield naked time machines--the
causally anomalous region that is not hidden behind the
event horizon--for every
choice of parameters.
Since the spacetime is simply connected, these causal pathologies cannot
be circumvented by extending to a universal covering space.
Hence the fake supersymmetry fails to get rid of causal pathologies,
as occurred for the present BPS rotating solutions.
Figure~\ref{fig:VLS} plots the typical behaviors of VLS.
When the angular momentum $J$ is smaller than the critical
value $Q^{2/3}$,
$t_{\rm VLS}(r)<0$ is satisfied and
$t_{\rm VLS}(r)\to -\infty $
as $r\to 0$. On the other hand,
when the angular momentum is larger than $Q^{3/2}$,
$t_{\rm VLS}(r) \to +\infty $ as $r\to 0$ and for $n=3$
$t_{\rm VLS}(r) $ is always positive,
whereas it is negative at large value of $r$ for $n=1,2$.
Using the radius $R$ one finds that the singularities~(\ref{sing}) are
both central $R=0$. So the VLS completely encloses the spacetime
singularities.
In the neighborhood of the VLS,
$(t-t_{\rm VLS}(r))/t_0\ll 1$, one finds
$f^{-3}\simeq J^2/r^6$. Hence the neighborhood of the VLS
in the present spacetime may be approximated by that
in the near-horizon geometry of the
BMPV black hole~(\ref{throat})
with $J^2=Q^3$.
In this case, $\xi_3^R =\partial/\partial \psi $
becomes a hypersurface-orthogonal null Killing vector. Moreover $\psi $
corresponds to the affine parameter of the null geodesics
$(\xi_3^R)^\nu \nabla_\nu (\xi_3^R)^\mu =0$, so that the spacetime
describes the
plane-fronted wave (not the plane-fronted wave with parallel rays
(pp-wave) since $\xi_3^R $ is not covariantly
constant $\nabla_\mu (\xi_3^R)^\nu \ne 0$)~\cite{Stephani:2003tm}.
We can expect that properties of the VLS in the present spacetime
is captured by that in the near-horizon geometry of the BMPV black hole
with $J=Q^{3/2}$.
\begin{widetext}
\begin{center}
\begin{figure}[h]
\includegraphics[width=14cm]{VLS.eps}
\caption{Plots of velocity of light surface $t_{\rm VLS}$
against $r$ for
$n=1, 2$ with $J^{2/3}>Q$ (left), for
$n=3$ with $J^{2/3}>Q$ (middle) and for $J^{2/3}<Q$. }
\label{fig:VLS}
\end{figure}
\end{center}
\end{widetext}
\subsection{Scaling limit}
Since the event horizon of a black hole is a global concept,
it is a difficult task to identify its locus especially in a
time-dependent spacetime. Following the previous papers we
shall argue the ``near-horizon geometry'' of the present metric and
demonstrate that the null surface of the event-horizon candidate
is described by a Killing horizon. By solving null geodesics
numerically, we can verify that when $J<Q^{2/3}$ these Killing horizons are indeed event horizons
in the original spacetimes. (The reason of the restriction $J<Q^{2/3}$
will be discussed later.)
For convenience, we define dimensionless parameters
\begin{align}
\tau:=\frac{t_0}{Q^{1/2}}\,,\qquad
j :=\frac{J}{Q^{3/2}}\,,
\end{align}
and denote dimensionless variables (normalized by $Q$) with tilde,
e.g., $\tilde x^\mu :=Q^{-1/2} x^\mu $. Then we can
work with the dimensionless metric,
\begin{align}
{\rm d} \tilde s^2 =&- f^2
\left(\tau {\rm d} \tilde t+\frac{j}{2\tilde r^2 }\sigma_3^R \right)^2
+f^{-1}\left({\rm d} \tilde r^2+\tilde r^2 {\rm d} \Omega_3^2\right)
\,, \nonumber \\
f=& \tilde r^2 \left(\tilde t\tilde r^2+1
\right)^{-n/3}
\left(\tilde r^2+1\right)^{(n-3)/3}\,.
\end{align}
The parameter $j$ is the reduced angular momentum and $\tau $
denotes the ratio of energy densities of the scalar fields and the Maxwell
fields evaluated on the horizon, respectively~\cite{MN,MNII}.
To simplify the notation, we shall omit tilde in the following.
We have seen in Eq.~(\ref{throat}) that the surface $r\to 0$
with $t$ being finite
corresponds to the throat infinity. Hence the null surfaces
``intersecting'' at the throat should be a candidate of
future and past horizons. These surfaces are described by the
infinite redshift and blueshift surfaces, respectively.
We shall focus on the geometry of the very neighborhood
of these horizon candidates.
The only well-defined ``near-horizon'' limit is given by
\begin{align}
t~\to~\frac{t}{\epsilon ^2}\,, \qquad r~\to ~\epsilon r \,,
\qquad \epsilon~\to ~ 0\,,
\label{NHlimit}
\end{align}
under which the metric is free from the scaling parameter $\epsilon $.
The above scaling limit gives rise to the near-horizon geometry,
if a horizon exists, the metric of which is given by
\begin{align}
{\rm d} s^2_{\rm NH} =&- r^4\left( t r^2+1\right)^{-2n/3}
\left(\tau {\rm d} t+\frac{j}{2 r^2}\sigma_3^R\right)^2 \nonumber \\
&+ r^{-2}
\left( t r^2+1\right)^{n/3}\left({\rm d} r^2+ r^2{\rm d} \Omega_3^2
\right)\,.
\label{NHmetric}
\end{align}
The scalar and gauge fields are also well-defined and given by
\begin{align}
X_T=\frac 13\left(tr^2+1\right)^{1-n/3}\,,
\qquad
X_S=\frac 13\left(tr^2+1\right)^{-n/3}\,,
\label{NHscalar}
\end{align}
and
\begin{align}
&
A^{(T)}=r^2 \left(tr^2+1\right)^{-1}\left(\tau {\rm d}
t+\frac{j}{2r^2}\sigma_3^R\right)\,,
\nonumber \\
&
A^{(S)}=r^2 \left(\tau {\rm d}
t+\frac{j}{2r^2}\sigma_3^R\right)\,.
\end{align}
This spacetime is pseudo-supersymmetric in its own right since
it admits a nonvanishing
Killing spinor of the form (\ref{sol_KS1}) and (\ref{sol_KS2})
with $f=r^2(tr^2+1)^{-n/3}$.
The above near-horizon metric~(\ref{NHmetric}) is still time-evolving and
spatially inhomogeneous. Nevertheless,
as a consequence of the scaling limit~(\ref{NHlimit}),
the near-horizon metric~(\ref{NHmetric}) admits a
Killing vector
\begin{align}
\xi^\mu = t\left(\frac{\partial }{\partial t}\right)^\mu
-\frac{ r}{2}\left(\frac{\partial }{\partial r}\right)^\mu \,.
\label{xi}
\end{align}
It is then convenient to take $\xi^\mu $ to be a coordinate vector so that the
metric is independent of that coordinate.
A possible coordinate choice ($T, R, \psi'$) is given by
\begin{align}
T =&\ln | t|
+\int ^R \frac{6 R^{6/n-1}(R^6-j^2){\rm d} R}{n (R^{6/n}-1)\Delta }\,,
\nonumber \\
R=&( t r^2+1)^{n/6}\,,
\nonumber \\
\psi' =&\psi+\int^R \frac{12 j\tau R^{n/6-1}}{n \Delta }{\rm d} R\,,
\label{coord_change}
\end{align}
where
\begin{align}
\Delta :=&4 R^4 F( R)+j^2 \,, \nonumber \\
F( R):= &\tau^2 R^{-4} ( R^{6/n}-1)^2 -\frac 14 R^2 \,,
\end{align}
In this new coordinate system, the Killing field is simply given by
$\xi^\mu =(\partial/\partial T)^\mu $, as we desired.
After some algebra the near-horizon metric~(\ref{NHmetric}) is cast
into an apparently stationary form,
\begin{align}
{\rm d} s^2_{\rm NH}=&
-F( R)\left[{\rm d} T+\frac{ j\tau ( R^{6/n}-1)}{2 R^4 F(R)}
{\sigma _3'}^R\right]^2 +\frac{j^2 R^2({\sigma_3'}^R)^2}{16F( R)}
\nonumber \\ &
+\frac{36\tau^2 R^{12/n}{\rm d} R^2 }{n^2 \Delta }
+\frac{ R^2}4 \left[(\sigma_1^R)^2+(\sigma_2^R)^2+({\sigma_3'}^R)^2\right]\,.
\label{NHmetric2}
\end{align}
Here, ${\sigma_3'}^R={\rm d} \psi '+\cos\theta {\rm d} \phi$.
Although its asymptotic structure is highly nontrivial, it is easy to recognize
that this spacetime has Killing horizons (if any) at $\Delta =0$.
The Killing horizon is generated by a linear combination of stationary and
angular Killing vectors,
\begin{align}
\zeta =\frac{\partial }{\partial T}+2\Omega_h \frac{\partial
}{\partial \psi'}\,,
\label{generator}
\end{align}
where
\begin{align}
\Omega_h=\left.\frac{j}{2 \sqrt{R ^6-j^2 }}\right|_{\rm horizon}\,.
\end{align}
Here, $\Omega _h$ is the angular velocity of the horizon
(associated with
$2\partial/\partial \psi'=\partial/\partial\phi_2+\partial/\partial{\phi_1}$).
The horizon angular velocity $\Omega_h$ is
constant anywhere on the horizon, which is a generic feature of
a Killing horizon~\cite{Carter}.
Contrary to (truly) supersymmetric black holes, the angular velocity
of the horizon is nonvanishing, i.e., the horizon is {\it rotating}.
In other words, the generator of the event horizon of a supersymmetric
black hole is tangent to the stationary Killing field at infinity.
Equation~(\ref{generator}) shows that $\partial/\partial t $ is not
the generator of the event horizon. This is a distinguished property
not shared by the BPS black holes.
Since $\Delta $ fails to have a double root in general, it follows that
the horizon is not extremal unless parameters $(\tau, j)$ are
fine-tuned. The reason of the appearance of the ``throat'' geometry at $r\to 0$ lies
in the fact that $(t, r)$-coordinates cover the ``white-hole region''
as well as the outside region of a black hole (see Figure~5 in~\cite{MNII}).
Equations~(\ref{NHscalar}) and~(\ref{coord_change}) imply
that the values of scalar fields $X_I$ on the horizon
are determined by the horizon radius,
which is expressed in terms of the charge $Q$, (inverse of) potential
height $t_0$ and the angular momentum $j$. This situation is closely
analogous to the attractor mechanism~\cite{attractor}, according to
which the values of scalar fields on the horizon are expressed by
charges and independent of the asymptotic values of the scalar
fields at infinity. But as it stands
it appears hard to say whether such a mechanism always works
in the time-dependent case.
In the following subsections we shall clarify various physical
features of the near-horizon metric~(\ref{NHmetric2}).
\subsubsection{Horizons}
The loci of Killing horizons $\Delta=0$ can be classified according to the
values of $\tau $ and $j^2$. We shall say ``under-rotating'' when the
spacetime~(\ref{NHmetric}) admits horizons. Otherwise it is said to be
``over-rotating.'' The quantity $R_-$ will be consistently used
when $\Delta >0$ for $R<R_-$.
\bigskip\noindent
(i) $n=1$.
When the angular momentum parameter $|j|$ is less
than the critical value $j_{(1)}$, i.e.,
\begin{align}
j^2< j_{(1)}^2:=\frac{1+16\tau^2 }{16\tau^2 }\,,
\end{align}
the near-horizon spacetime admits two horizons,
\begin{align}
R_\pm^6 =\frac{1+8\tau^2 \pm \sqrt{1+16\tau ^2 (1-j^2)}}{8\tau^2 }\,.
\label{Rpm_1}
\end{align}
For the over-rotating case $j^2>j_{(1)}^2$,
there exist no horizons.
We find the similar results for the case of non-integer values of $n<2$.
\bigskip\noindent
(ii) $n=2$. This case is further categorized into the following three cases.
\begin{enumerate}
\item $0<\tau<1/2$. For any values of $j$, a single horizon occurs at
\begin{align}
R_-^3 =\frac{\sqrt{4\tau^2 +(1-4\tau^2 )j^2}-4\tau^2 }{1-4\tau^2 }\,.
\label{Rm2}
\end{align}
\item $\tau =1/2$. For any values of $j$, a single horizon occurs at
\begin{align}
R_-^3 =\frac{1+j^2}{2}\,.
\label{Rm3}
\end{align}
\item $\tau >1/2$. When the angular momentum parameter $|j|$ is less
than the critical value $j_{(2)}$, i.e.,
\begin{align}
j^2< j_{(2)}^2:=\frac{4\tau^2 }{4\tau^2-1 }\,,
\end{align}
two horizons exist at
\begin{align}
R_\pm^3 =\frac{4\tau^2 \pm \sqrt{4\tau^2
-(4\tau^2-1)j^2} }{4\tau^2-1 }\,.
\label{Rm4}
\end{align}
For the over-rotating case $j^2>j^2_{(2)}$,
no horizons develop.
\end{enumerate}
\bigskip\noindent
(iii) $n=3$.
In this case,
the metric~(\ref{NHmetric2}) is not the ``near-horizon''
geometry, but is the original metric itself written in the stationary
coordinates. This metric describes a charged rotating black hole in de Sitter
space derived by Klemm and Sabra~\cite{Klemm:2000vn}.
Let us discuss its horizon structure in detail.
There exists at least one horizon corresponding to the cosmological
horizon. For $\tau \le \sqrt{3/2}$ there appears only a cosmological horizon
$R_c$. For $\tau>\sqrt{3/2}$, the number of horizons depend on the value of $j^2$.
Three distinct horizons ($R_-<R_+<R_c$) exist for
$j^2_{(3)-}<j^2<j^2_{(3)+}$, where
\begin{align}
j^2_{(3)\pm}&:=\frac{4\tau^2 }{27}\left[\pm
8\sqrt 2\tau (2\tau^2-3)^{3/2}
\nonumber \right.
\\
&
\left.~~~~~~~~~~
-32\tau^4
+9(8\tau^2-3)\right]\,.
\end{align}
For $j^2 =j^2_{(3)+}$, inner and outer black-hole horizons are
degenerate. While,
for $j^2 =j^2_{(3)-}$ the outer black-hole horizon and the
cosmological horizon are degenerate.
$j^2_{(3)-}$ takes real positive values for $\sqrt{3/2}<\tau <3\sqrt{3}/4$,
otherwise the inner horizon does not exist.
\bigskip
A simple calculation reveals that the spacetime~(\ref{NHmetric}) is
regular on and outside the Killing horizon (if any). Only the existing
curvature singularity is at $R=0$. It is almost clear to construct the
local coordinate systems that pass through the Killing horizon
$\Delta =0$.
In hindsight,
we can understand why the horizon in the present spacetime
is not extremal as follows.
In the case of the time-independent (truly) BPS solutions such as a BMPV
black hole, the Killing horizon lies at $f=0$
since $V^\mu =(\partial/\partial t)^\mu $ is an everywhere causal
Killing field constructed
by a Killing spinor $\epsilon $ as
$V^\mu =i \bar \epsilon \gamma^\mu\epsilon$ (see~\cite{GGHPR}).
For the present time-dependent pseudo-supersymmetric black hole,
on the other hand,
the vector field $V^\mu =(\partial/\partial t)^\mu$ is not the
Killing-horizon generator: the horizon is generated by
$
\xi ^\mu =t(\partial/\partial t)^\mu -(r/2)(\partial/\partial r)^\mu
+\Omega_h(\partial/\partial \psi)^\mu
$
given in Eq.~(\ref{generator}).
The vector field $V^\mu$ does not give rise to any (asymptotic)
symmetry.
Physically speaking, the degeneracy of the horizon is broken by
introducing of the time-dependent scalar fields (which do not contribute
to the total mass when the spacetime is stationary) or the positive
cosmological constant. These ingredients destroy the fine balance
between the mass energy and the charges.
When the rotation is also added the centrifugal force
gives a negative contribution to the mass energy $M\to M-J^2$--which
takes place only in $D=5$ as discussed before--thus it exceeds the
extremal threshold value if the rotation becomes too large.
\bigskip
\subsubsection{Ergoregion}
An obvious major difference from our previous non-rotating
solutions~\cite{MN,MNII} is that
the near-horizon metric possesses the ergosurface at $F(R)=0$.
Since $\Delta >4R^4 F(R)$, the ergosurface
lies strictly outside the horizon, contrary to the 4-dimensional
Kerr black hole for which the ergosurface touches the
horizon at the rotation axis.
When the rotating vanishes ($j=0$),
the roots of $F=0$ correspond to the loci of horizons~\cite{MN,MNII}.
Since $\Delta =0$ reduces to $F(R)=0$ when $j=0$,
the explicit expression of the ergosphere is given by setting $j=0$ of the
horizon radius. For $n=1,2$ they are given by Eqs.~(\ref{Rpm_1}),
(\ref{Rm2}), (\ref{Rm3}) and (\ref{Rm4}) with $j=0$.
Note, however, that since the asymptotic structures are quite
peculiar when $n=1,2$, there may arise an ambiguity concerning the definition
of the energy~\footnote{For example, the AdS spacetime has various
Killing fields which are timelike outside the AdS degenerate horizon.
The definition of energy in asymptotically AdS spacetime
is different depending on which Killing field is chosen.
}.
It may therefore equivocal whether $R_{\rm erg}$ has a definitive meaning
in the $n=1,2$ cases.
When $n=3$ the asymptotic region is described by de Sitter space,
so that we can use the standard time translation with respect to the
observer at the cosmological horizon to define the energy.
Hence the notion of ergoregion is meaningful in this sense.
There exist three distinct roots,
$ R_{\rm erg, -}< R_{\rm erg, +}< R_{\rm erg, c}$, for
$\tau >\tau _{\rm cr}:=3\sqrt 3/4$, two roots
for $\tau=\tau_{\rm cr}$ and a single root $ R_{\rm erg, -}$
for $\tau <\tau _{\rm cr}$.
The ergoregion does not arise for the
supersymmetric black hole, which inevitably forbids the ergoregion
inside which the stationary Killing field becomes spacelike.
The ergoregion is intrinsic to a rotating black hole and allows
particles to have a negative energy. This means that the rotation energy
of a black hole can be subtracted via the Penrose process and the
superradiant scattering process. We shall demonstrate in
Appendix~\ref{sec:superradiance} that this is
indeed the case for the $n=3$ Klemm-Sabra solution.
\subsubsection{Closed timelike curves}
Write the near-horizon metric~(\ref{NHmetric2}) as
\begin{widetext}
\begin{align}
{\rm d} s_{\rm NH}^2 =- \frac{\Delta }{4 R^4\varDelta_L }{\rm d} T^2
+\frac{36\tau^2 R^{12/n}{\rm d} R^2}{n^2\Delta }+\frac{R^2}{4}\left[
(\sigma_1^R)^2 +(\sigma_2^R)^2 +\varDelta_L\left({\sigma_3'}^R-
\frac{2j\tau (R^{6/n}-1)}{R^6\varDelta_L}{\rm d} T\right)^2 \right]\,,
\end{align}
\end{widetext}
where we have also used $\varDelta_L$ as the near-horizon
limit of (\ref{Delta_L}):
\begin{align}
\varDelta_L= 1-\frac{j^2}{ R^6}\,.
\label{Delta_LNH}
\end{align}
Consequently
the Hopf fibres become timelike inside the VLS ($\varDelta_L<0$), viz,
the near-horizon metric~(\ref{NHmetric2}) is also causally unsound.
In terms of $\varDelta_L$, $\Delta $ is
\begin{align}
\Delta =4\tau^2 (R^{6/n}-1)^2 -R^6\varDelta_L \,.
\end{align}
It follows that the event horizon ($\Delta =0$) is outside the
VLS~\cite{Matsuno:2007ts}.
Hence {\it the causality violating region
is always hidden behind the horizon} ($ R_L< R_+$) in the near-horizon
geometry~(\ref{NHmetric2})~\footnote{
Since $R_-^6-R_L^6=4\tau ^2(R_-^2-1)^2>0$, $R_L<R_-$ also holds.}.
On the other hand, it is naked in the
over-rotating case where the horizon does not exist.
This should be contrasted with the BMPV or asymptotically AdS
($\mathfrak g\in \mathbb R$)
Klemm-Sabra black hole. In the former case
the VLS is outside the event horizon if the angular momentum is
large $J>Q^{2/3}$.
In the latter case a naked time machine
inevitably appears outside the event horizon.
However, it allows no geodesics
to penetrate, so that the horizon exterior is geodesically complete.
In the present case the area of the horizon is given by
\begin{align}
{\rm Area} &=2\pi ^2 R^3\left.\sqrt{\varDelta_L}\right|_{\rm horizon}
=4\pi^2\tau (R_+^{6/n}-1)
\,.
\end{align}
which always makes sense contrary to the BMPV or the asymptotically AdS
Klemm-Sabra black hole: the latter two spacetimes have
an ``imaginary horizon area'' in the
over-rotating case. These formal horizons in the over-rotating case
are ``repulsons'' into which no freely falling orbits
penetrate~(see e.g.,~\cite{GibbonsBMPV,Caldarelli:2001iq,Herdeiro:2000ap,Dyson:2006ia}).
\subsubsection{Geodesic motions}
It is illustrative to consider geodesics in the near-horizon
metric~(\ref{NHmetric2}). For $n=3$, the following analysis
yields the geodesic motion in the exact Klemm-Sabra geometry, not
restricted in the neighborhood of its horizons.
The particle motion in asymptotically AdS Klemm-Sabra solution
($\mathfrak g \in \mathbb R $) was
previously examined in~\cite{Caldarelli:2001iq}.
Although the
behavior of the particle motion in asymptotically
de Sitter case is of course considerably different from that case,
the technical method is similar.
The analysis in this subsection unveils that the horizon can be reached
within a finite affine time from outside.
The Hamilton-Jacobi equation in the near-horizon
geometry~(\ref{NHmetric2}) reads
\begin{align}
-
\frac{\partial S}{\partial \lambda }
=
\frac 12g^{\mu\nu}_{\rm NH}\left(\frac{\partial S}{\partial x^\mu}\right)
\left(\frac{\partial S}{\partial x^\nu}\right)\,,
\end{align}
where the right-hand side of this equation defines a geodesic
Hamiltonian and $\lambda $ is an affine parameter.
Assume the separable form of Hamilton's principal
function,
\begin{align}
S=& \frac 12m^2\lambda -E T +L_L\phi+L_R \psi'
+S_R ( R)+S_\theta (\theta)\,,
\label{pri_fun}
\end{align}
where $E, L_R, L_L$ and $m$ are constants of motion corresponding to
energy, right-rotation, left-rotation and rest mass of a
particle.
Since the near-horizon metric keeps the ${\rm U}(2)$-symmetry,
there exists a reducible Killing tensor of the
form~(\ref{Killing_tensor}), which reads
in the coordinates (\ref{NHmetric2}) as
\begin{align}
K_{\mu\nu }{\rm d} x^\mu {\rm d} x^\nu =&\left[
\frac{j\tau }{2R^4}(R^{6/n}-1){\rm d} T-\frac{R^2\varDelta _L}{4}
({\sigma'_3}^R)\right]^2 \nonumber \\
&
+\frac{R^4}{16}\left[(\sigma_1^R)^2+(\sigma_2^R)^2 \right]\,.
\\
\nonumber
\end{align}
Accordingly, besides obvious constants of motion ($E, L_R,L_L, m$)
generated by Killing vectors,
we have an additional integration constant $L^2$ with dimensions of
angular momentum squared such that
\begin{align}
L^2:= \sum _i(\xi^R_i S)^2 \,.
\label{Lsq}
\end{align}
This constant of motion enables us to separate the variables as
\begin{align}
\left(\frac{{\rm d} }{{\rm d} \theta }S_\theta
\right)^2+\frac{1}{\sin ^2 \theta }
\left(L_L^2+L_R^2-2\cos\theta L_RL_L\right)=L^2\,.
\label{Lsq2}
\end{align}
The constant $L^2$ represents the left and right Casimir invariant
of the ${\rm SU}(2)$ subgroup of ${\rm SO}(4)$ rotation group.
These two Casimirs turn out to be the same for the scalar representation.
It follows that the particle motion and the scalar-field equation are
Liouville-integrable.
The governing equations are obtainable by differentiating the
principal function~(\ref{pri_fun}) by corresponding constants of
motion. Using the relation for angular variable~(\ref{Lsq2}),
we obtain a set of useful 1st-order equations
\begin{widetext}
\begin{align}
\frac{{\rm d} R}{{\rm d} \lambda }&=\pm
\frac{n\sqrt {\Delta }}{6\tau R^{6/n}}
\left[\frac{E}{F}-m^2-\frac{4 (L^2-L_R^2)}{ R^2}
-\frac{16 R^2 F}
{\Delta }\left(L_R+\frac{j\tau ( R^{6/n}-1)E}{2 R^4
F}\right)^2\right]^{1/2}\,,\label{NH_radeq}\\
\frac{{\rm d} \theta }{{\rm d} \lambda } &= \pm \frac{4}{R^2}\left[
L^2-\frac{1}{\sin ^2 \theta }
\left(L_L^2+L_R^2-2\cos\theta L_RL_L\right)
\right]^{1/2} \,,\\
\frac{{\rm d} T}{{\rm d} \lambda }&=\frac{4 R^4 \varDelta_L}{\Delta
}E -\frac{8j\tau (
R^{6/n}-1)L_R}{\Delta R^2}\,,\label{NH_timeeq}\\
\frac{{\rm d} \phi}{{\rm d} \lambda }&=\frac{4}{ R^2 \sin^2\theta }
(L_L-L_R\cos \theta )\,,\label{NH_phieq}\\
\frac{{\rm d} \psi' }{{\rm d} \lambda }&=\frac{4(L_R-L_L\cos\theta )}{
R^2\sin^2\theta }
-\frac{4}{R^2\Delta }\left[j^2 L_R-2j\tau (R^{6/n}-1)E\right]\,.
\label{NH_psieq}
\end{align}
\end{widetext}
If there are no angular momenta of a particle ($L=L_R=L_L=0$),
one sees that there is no motion in directions
$\theta $ and $\phi$, but there is a nonvanishing motion along $\psi'$,
encoding the frame dragging due to the black-hole rotation.
By virtue of high degree of symmetries, the problem reduces to the one
dimensional radial equation~(\ref{NH_radeq}), which is arranged to give
\begin{align}
\left(\frac{{\rm d} R}{{\rm d} \lambda }\right)^2=&
\frac{n^2}{9 \tau^2 R^{4(3/n-1)}}\biggl[
\varDelta_L (E-2\Omega L_R)^2
\nonumber \\ &
-\frac{j^2L_R^2\Delta }{R^{12}\varDelta_L}-\Delta
\left(\frac{L^2}{R^6}+\frac{m^2}{4R^4}\right)\biggl]\,,\label{NH_radeq2}\\
=&
\frac{n^2\varDelta _L}{9 \tau^2
R^{4(3/n-1)}}(E-V^+)(E-V^-)\,,\label{NH_radeq3}
\end{align}
where $\Omega $ and $V^\pm$ are the angular velocity of a locally
nonrotating observer and the effective potentials,
which are defined by
\begin{align}
\Omega :=&\frac{j\tau (R^{6/n}-1)}{R^6\varDelta_L}\,,\\
V^\pm :=& 2\Omega L_R\pm \frac{\sqrt{
\Delta \left[j^2L_R^2+\varDelta _LR^6\left(L^2+{m^2R^2}/4\right)\right]
}}{\varDelta_LR^6} \,.\label{V_pm}
\end{align}
The allowed region is $E>V^+$ or $E<V^-$ for $\varDelta_L>0$, whereas it is
${\rm min}[V^\pm]<E<{\rm max}[V^\pm]$ for $\varDelta_L<0$.
Equation~(\ref{NH_timeeq}) becomes
\begin{align}
\frac{{\rm d} T}{{\rm d} \lambda }=\frac{4R^4\varDelta_L}{\Delta
}\left(E-2\Omega L_R\right)\,.
\end{align}
When $\Delta>0$ and $\varDelta_L>0$,
$E>V^0:=2\Omega L_R$ follows. Thus,
$E$ must be positive for a particle with $\Omega L_R>0$
moving forwards with respect to the time coordinate $T$.
Inside the VLS ($\varDelta_L<0$) where $\Delta>0$,
a particle with $E>V^0$
moves backwards with respect to the coordinate $T$.
One also verifies that the horizon $\Delta=0$ is an infinite redshift
surface for the time coordinate $T$, which is of course a coordinate artifact.
From~(\ref{Lsq}) one finds $L^2\ge L_R^2$. When the equality holds,
$L_L=0$ is satisfied. Thus Eq.~(\ref{Lsq2}) implies that the
particle motion is confined on the equatorial plane $\theta =\pi/2$ and
Eq.~(\ref{NH_phieq}) implies $\phi={\rm constant}$.
The same remark applies to the original metric~(\ref{sol}) since
this assertion only comes from the ${\rm U}(2)$-symmetries of the
solution.
It is clear from Eq.~(\ref{NH_radeq2}) that massless particles with $L=L_R=0$
cannot cross the VLS.
In the over-rotating case, the geodesics with $L_R=0$ cannot cross the VLS
either, since the right-hand side of (\ref{NH_radeq2})
becomes negative before the VLS is reached.
In the case of $L_R \ne 0$, it is dependent on the parameters whether
the geodesic particle moving forwards can cross the VLS or not.
When $j<1 $,
$\Omega L_R$ diverges positively (negatively) as $R\to R_L+0$ for the
particle having the opposite (same) spin as the black hole. Hence the
particle with opposite angular momentum ($jL_R<0$) cannot penetrate the VLS
for $j<1$.
Similarly, when $j>1$
$\Omega L_R$ diverges positively (negatively) as $R\to R_L+0$ for the
particle having the same (opposite) spin as the black hole. Thus the
particle with $j>1$ never penetrate the VLS when it has the same spin as
the hole $jL_R>0$.
Though causal geodesics may cross the VLS, it is shown that
they never encounter the singularity at $R=0$ at least for $n=2, 3$.
For $L>|L_R|$, the function inside the square-root of $V^\pm$~(\ref{V_pm})
becomes negative around $R=0$, so that $V^\pm$ does not exist around
$R=0$ and has a confluent point inside the VLS, which
prohibits geodesics to enter inside.
For $L=|L_R|$, it can be easily shown that
$V^+<V^0<V^-$ holds around $R=0$ and they take value $2\tau L_R/j$ at
$R=0$. It follows that geodesics with $E= 2\tau L_R/j$ may reach $R=0$.
For $n=1$ this is indeed the case. By contrast, for $n=2, 3$
${\rm d} V^0/{\rm d} R <(>)0$ holds around $R=0$ for $jL_R>(<)0$,
which forbids the geodesics to hit the singularity since $E<V^0$ and
$V^+<E<V^-$ are the allowed region for the future-pointing particles.
Accordingly, the singularity $R=0$ has a repulsive nature.
We can expect that geodesics rarely reach the singularity
also in the dynamical settings.
\subsection{Global structure}
We are now ready to discuss the global structures of the
time-dependent and rotating spacetime~(\ref{sol}).
The most useful visualization of the causal structure of a spacetime is
the conformal diagram. To this end it is necessary to find a
two-dimensional (totally geodesic) integrable submanifold.
Now the spacetime is regarded as an $\mathbb R^2$ bundle over $S^3$.
Unfortunately, the distribution spanned by
$\partial/\partial t$ and $\partial/\partial r$ is not integrable,
forbidding us to have a foliation by a two-dimensional conformal diagram.
The frame-dragging effect inevitably drives the $\psi$-motion.
Nevertheless, the two-dimensional metric
\begin{align}
{\rm d} s_2 ^2 =-\frac{\tau^2 f^2}{\Delta _L}{\rm d} t^2 +\frac{{\rm d} r^2 }{f} \,,
\label{2dmetric}
\end{align}
still contains some information about the spacetime structure and
gives us useful visualization~\footnote{For $n=3$, the 2-dimensional
metric~(\ref{2dmetric}) is locally isometric to the static metric
${\rm d} s_2^2=-(\Delta/\Delta_L){\rm d} T'^2 +\Delta ^{-1}{\rm d} R^2$,
where $R=(tr^2+1)^{1/2}$ and ${\rm d} T' ={\rm d} t/t+R^3\Delta_L{\rm d} R/[2\Delta (R^2-1)]$.
}.
The above metric~(\ref{2dmetric}) is associated with the null geodesics
with $\theta =\pi/2$ and $\phi={\rm constant}$ corresponding to
$L=L_R=L_L=0$: hence they cannot penetrate the VLS, which is found to be
a timelike or null surface.
As in the BMPV case, the 2-dimensional metric~(\ref{2dmetric}) is not
Lorentzian inside the VLS.
From the analysis of the previous subsection,
we found that the original time-independent metric~(\ref{NHmetric2}) admits
Killing horizons at $\Delta=0$. In the non-rotating case ($j=0$),
the null surfaces $r=0$ with $t=\pm \infty $ are
Killing horizons also for the original spacetime~\cite{MNII},
since the Killing vector is parallel to the generators of horizons.
When a rotation is present we must be careful.
Now there exists a VLS~(\ref{VLS}), which is bounded below
when $j>1$ (see left and middle plots in figure~\ref{fig:VLS}), so that
the past horizon $t\to -\infty $ may not exist (since we are focusing on
the 2-dimensional metric~(\ref{2dmetric}), no causal geodesics penetrate
the VLS: inside the VLS is not the physical region of spacetime).
Even if the near-horizon metric~\cite{MNII} admits some Killing
horizons, we cannot immediately conclude that they are also Killing
horizons in the original metric.
The analysis of singularities, asymptotic infinity,
behaviors of VLS (figure~\ref{fig:VLS}) and the near-horizon geometries
have provided us sufficient information to deduce Carter-Penrose diagrams.
As a striking confirmation we have solved the geodesic equations numerically and obtained the
conformal diagrams displayed in figure~\ref{fig:PD}, which may be
summarized as follows (we have excluded the special case of the
degenerate horizons).
\noindent
(i) $n=1$.
The asymptotic region is approximated by an FLRW universe obeying a
decelerating expansion
$a =(\bar t/\bar t_0)^{1/4}$ caused by a massless scalar field.
Then the null infinity $\mas I^-$
possesses an ingoing null structure. When $j<j_{(1)}$,
two Killing horizons $R_\pm$ arise~(\ref{Rpm_1}). Since
the VLS diverges negatively as $r\to 0$ when $j<1$ (right plots in
figure~\ref{fig:VLS}), the conformal diagram is (I).
Even if two horizons exist in the near-horizon geometry for $1<j<j_{(1)}$, the VLS
conceals the past horizon $R_-$ (corresponding to $t\to-\infty $)
since the VLS diverges positively as $r\to $ (left plots in
figure~\ref{fig:PD}).
Then diagram (II) is obtained.
Note that the $R_-={\rm constant}$ surface
asymptotically approaches null as $t\to\infty $, and $R_{L}$ is
timelike almost everywhere (it happens to be null precisely at one point).
For the over-rotation $j>j_{(1)}$, no Killing horizons arise.
Hence the conformal diagram is (V).
\bigskip
\noindent
(ii) $n=2$.
The spacetime approaches to the marginally accelerating universe,
expanding linearly with cosmic time
$a =\bar t/\bar t_0$. This is caused by the fluid with equations of
state $P=-\rho /2$. For $\tau >1/2$ there exists two Killing
horizons~(\ref{Rpm_1}), so that conformal diagram is the same as case
(i); it is (I) for $0<j<1$, (II) for $1<j<j_{(2)}$ and (V) for $j>j_{(2)}$.
An essential difference from the $n=1$ case arises when $\tau \le 1/2$,
in which case there exists an internal null infinity $\mas I^+_{\rm in}$ where
$R\to \infty $ with $r\to 0$ and
$t\to \infty $. Only ingoing null particles can get to
$\mas I^+_{\rm in}$.
The existence of internal null infinity can be shown by solving the
geodesics asymptotically as in 4-dimensions~\cite{MNII}.
It follows that conformal diagrams for $\tau \le 1/2$ are
(III) when $j<1$ and (IV) when $j>1$.
\bigskip\noindent
(iii) $n=3$.
The conformal diagrams are similar to the Kerr-de Sitter spacetime.
Infinity $\mas I^+$ consists of a spacelike slice due to the
acceleration of the universe. First, consider the case in which
the near-horizon metric~(\ref{NHmetric2}) admits three distinct horizons
$R_\pm$ and $R_c$. This occurs when $\sqrt{3/2}<\tau <3\sqrt 3/4$ with
$j_{(3)-}<j<j_{(3)+}$ and $\tau>3\sqrt{3}/4$ with $(0\le)j<j_{(3)+}$.
Taking into account the fact that
for $j<1$ the VLS $t_{\rm VLS}(r)$ diverges negatively as $r\to 0$, which
removes past horizons ($t\to-\infty $ and $r\to 0$ with $t r^2$ finite)
in the near-horizon geometry~(\ref{NHmetric}).
Therefore when $\tau>3\sqrt{3}/4$ with $(0\le)j<1$
the conformal diagram is (VI), whereas it is
(VI') when $1<j<j_{(3)+}$ with $\tau>3\sqrt 3/4$, or
$(1<)j_{(3)-}<j<j_{(3)+}$ with $\sqrt{3/2}<\tau <3\sqrt 3/4$.
These two are essentially the same: they constitute the different
coordinate patches depending on the value of $j$.
In (V') the slice $t=0$ and $r\to \infty $ with $tr^2 $
finite comprises a null boundary.
When there appears only a cosmological horizon $R_c$ (i.e.,
$\tau <\sqrt{3/2}$, $\sqrt{3/2}<\tau <3\sqrt 3/4$ with
$j<j_{(3)-}$ or $j>j_{(3)+}$ and $\tau>3\sqrt{3}/4$ with $j>j_{(3)+}$),
the spacetime diagram is (VII) for $j<1$ and (VII') otherwise.
Again, (VII) and (VII') are essentially identical.
In (VII') the slice $t=0$ and $r\to \infty $ with $tr^2$
finite is also a null surface.
To summarize, the cases (I), (II), (VI) and (VI') correspond to the
rotating black-hole geometry.
\begin{widetext}
\begin{center}
\begin{figure}[h]
\includegraphics[width=15cm]{PD_5dim.eps}
\caption{Conformal diagrams of the 2-dimensional spacetime
(\ref{2dmetric}), by which null geodesics with zero angular momentum
is described.
$R_+$, $R_-$ and $R_c$ are all Killing horizons corresponding
respectively to the black-hole event horizon, the white-hole horizon and
the cosmological horizon. The thick dotted curves represent the VLS.
Thin black and gray dotted curves are $t={\rm constant}$ and
$r={\rm constant}$ surfaces, respectively.
White and black circle are infinities (including throat) and bifurcation
surfaces.
Since the 2-dimensional metric~(\ref{2dmetric}) becomes Riemannian inside
the VLS,
the diagrams come to an end at $R_L$. Remark that we are formally
writing the 2-dimensional figures, there still remains the angular
motion because of the frame dragging: these figures do not display all
the causal information.
Though these diagrams are restricted to the $r^2>0$ region,
the spacetime can be extended across the null surfaces $R_\pm$
and $R_c$, which are nothing but the ordinary chart boundaries.
The conformal diagrams are
(I) for $n=1$ with $j<1$, and for $n=2$ with $j<1$ and $\tau >1/2$,
(II) for $n=1$ with $1<j<j_{(1)}$, and for $n=2$ with $1<j<j_{(2)}$ and
$\tau >1/2$,
(III) for $n=2$ with $\tau \le 1/2$ and $j<1$,
(IV) for $n=2$ with $\tau \le 1/2$ and $j>1$,
(V) for $n=1$ with $j>j_{(1)}$ and for $n=2$ with
$\tau >1/2$ and $j>j_{(2)}$, whereas diagrams
(VI)--(VII') correspond to $n=3$:
(VI) for $\tau >3\sqrt 3/4$ with $j<1$,
(VI')
for $\tau >3\sqrt 3/4$ with $1<j<j_{(3)+}$, and
for $\sqrt{3/2}<\tau<3\sqrt 3/4$ with $j_{(3)-}<j<j_{(3)+}$,
(VII) for $\tau <\sqrt{3/2}$ with $j<1$ and
for $\sqrt{3/2}<\tau<3\sqrt 3/4$ with $j<1$,
and
(VII') for $\tau <\sqrt{3/2}$ with $j>1$,
for $\sqrt{3/2}<\tau<3\sqrt 3/4$ with $1<j<j_{(3)-}$ or
$j>j_{(3)+}$ and
for $\tau>3\sqrt 3/4$ with $j>j_{(3)+}$.
}
\label{fig:PD}
\end{figure}
\end{center}
\end{widetext}
\section{Dimensional oxidization and reduction}
\label{sec:liftup}
In the previous sections, some black hole solutions in the STU theory
have been elaborated in the framework of the 5-dimensional theory.
We shall discuss in this section the liftup and compactification
procedure to other number of dimensions.
\subsection{Lift up to M-theory}
The time-evolving and spatially-inhomogeneous solutions in 4-
and 5- dimensions were originally derived from the dimensional reduction
of intersecting M-branes in 11-dimensional supergravity. Now we argue
the solutions of case (ii)--where
two of $V_I$'s vanish--can be embedded in 11-dimensional supergravity.
The 11-dimensional supergravity action is given by
\begin{align}
S_{11}=\frac{1}{2\kappa_{11}^2} \int\left(
{}^{11}R\star_{11} 1 -\frac 12 {\ma F}\wedge
\star_{11}{\ma F}-\frac 16 \ma A \wedge \ma F\wedge \ma F \right)\,,
\label{11Daction}
\end{align}
where $\ma F={\rm d} \ma A$ is the 4-form field strength.
The equations of motion are Einstein's equations,
\begin{align}
{}^{11}R_{AB }-\frac 12 {}^{11}R g_{AB}=&\frac{1}{2\cdot 3!}\left(
\ma F_{ACDE}{\ma F_B }^{CDE}
\right. \nonumber \\& \left.
-\frac 18
g_{AB }\ma F_{CDEF}\ma F^{CDEF
}\right)\,,
\label{11DEOM}
\end{align}
and the gauge-field equations
\begin{align}
{\rm d} \star_{11}\ma F +\frac 12 \ma F\wedge \ma F&=0\,.
\label{11DMaxwell}
\end{align}
In this section $A,B,...$ denote the 11-dimensional indices.
Let us consider the ``intersecting M2/M2/M2 metric'' of the following
form~\cite{Elvang:2004ds},
\begin{align}
{\rm d} s_{11}^2= & {\rm d} s_5^2+X^1 \left({\rm d} y_1^2+{\rm d} y_2^2 \right)
+X^2 \left({\rm d} y_3^2+{\rm d} y_4^2 \right)
\nonumber \\
& +X^3 \left({\rm d} y_5^2+{\rm d} y_6^2
\right)\,,\label{11Dmetric}\\
\ma A=& A^{(1)} \wedge {\rm d} y_1 \wedge {\rm d} y_2 +
A^{(2)}\wedge {\rm d} y_3\wedge {\rm d} y_4 \nonumber \\
&+A^{(3)}\wedge {\rm d} y_5\wedge {\rm d} y_6\,,
\end{align}
where the metric is independent of the brane coordinates
$y_1, ...,y_6$.
This solution is specified by 5-dimensional metric,
\begin{align}
{\rm d} s_5^2 = &-\left(H_1H_2H_3\right)^{-2/3} \left({\rm d} t+\omega \right)^2
\nonumber \\
& +\left(H_1H_2H_3\right)^{1/3}h_{mn }{\rm d} x^m{\rm d} x^n \,,
\label{11Dmetric2}
\end{align}
as well as
three scalars $X^I~(I=1,2,3)$ and three one-forms $A^{(I)}$ which are
given by
\begin{align}
A^{(I)} =& H_I^{-1} \left({\rm d} t+\omega \right)\,,\qquad
X^I =H_I^{-1}\left(H_1H_2H_3\right)^{1/3}\,.
\end{align}
Here, $h_{mn}$ is the metric on the 4-dimensional base space.
$\omega=\omega_m {\rm d} x^m$ is viewed as a one-form on the base space,
i.e., $\omega_\mu V^\mu =0$ where $V^\mu =(\partial /\partial t)^\mu $.
Since the metric ansatz~(\ref{11Dmetric2}) is independent of the
coordinates $y_1, ..., y_6$, the solution can be dimensionally reduced
to 5-dimensions.
Noting that the six-torus ${T}^6$ has a constant volume
$X^1X^2X^3=1$, it turns out that the 5-dimensional metric ${\rm d} s_5^2 $
is the 5-dimensional Einstein-frame metric.
Thus, the metric ansatz~(\ref{11Dmetric}) gives
the 5-dimensional action of gravity sector as
\begin{align}
S_g &=\frac{1}{2\kappa_5^2 }\int {\rm d} ^{5}x\sqrt{-g_5}
\left[{}^5 R -\frac 12 \sum_I(\nabla^\mu \ln X^I)
(\nabla_\mu \ln X^I)\right]\,,
\label{5Deffection}
\end{align}
where we have used $X^1X^2X^3=1$.
We can proceed the form field sector analogously. Letting
$F^{(I)}:={\rm d} A^{(I)}$ denote the two-form field strengths, we find
\begin{align}
&\ma F_{ABCD }\ma F^{ABCD}
= 12\sum_I (X^I)^{-2 } F^{(I)}_{\mu \nu }F^{(I)\mu \nu }\,,\\
&\ma A\wedge \ma F \wedge \ma F =2\left(A^{(1)}\wedge F^{(2)}\wedge F^{(3)}
+A^{(2)}\wedge F^{(3)} \wedge F^{(1)}
\right.\nonumber \\ &\qquad \qquad \qquad \left.
+A^{(3)}\wedge F^{(1)} \wedge F^{(3)} \right)\wedge {\rm Vol}(T^6)\,,
\end{align}
then the Lagrangian for the gauge fields reads
\begin{widetext}
\begin{align}
S_F =\frac{1}{2\kappa_5^2 } \int {\rm d} ^5 x \sqrt{-g_5}
\left[
-\frac{1}{4} \sum_I (X^I)^{-2 } F^{(I)}_{\mu \nu }F^{(I)\mu \nu }
+\frac{1}{12}\epsilon^{\mu\nu\rho\sigma\tau}
\left(A^{(1)}_{\mu }F^{(2)}_{\nu\rho }F^{(3)}_{\sigma\tau}+
A^{(2)}_{\mu }F^{(3)}_{\nu\rho }F^{(1)}_{\sigma\tau }+
A^{(3)}_{\mu }F^{(1)}_{\nu\rho }F^{(2)}_{\sigma\tau }\right)
\right]\,,
\end{align}
\end{widetext}
where
$\epsilon_{\mu\nu\rho\sigma\tau}$
is the volume-element
compatible with the 5-dimensional metric ${\rm d} s^2_5$ and
$\kappa_5^2:=\kappa_{11}^2/{\rm Vol}(T^6)$.
It follows that the reduced action $S_5=S_g+S_F$
exactly coincides with that of the STU-theory; the
5-dimensional minimal ungauged ($\mathfrak g=0$) ${\rm U}(1)^3$-supergravity~(\ref{5Daction})
with the metric of the potential space given by
\begin{align}
G_{IJ}=\frac 12 {\rm diag}\left[(X^1)^{-2}, (X^2)^{-2}, (X^3) ^{-2} \right]\,,
\end{align}
and the constants $C_{IJK}$ are totally-symmetric in ($IJK$)
with $C_{123}=1$ and $0$ otherwise.
If we consider three equal harmonics $H_1=H_2=H_3:=H$
(i.e., $X^I=1$, $A^1=A^2=A^3=:(2/\sqrt 3)A$ and $F={\rm d} A$),
all scalar fields are trivial. Then
the action $S_5=S_g+S_F$ reduces to
that of the minimal supergravity in 5-dimensions~\cite{GGHPR},
the action of which is given by
\begin{align}
S_5=\frac{1}{2\kappa_5^2}
&\int {\rm d} ^5x \sqrt{-g_5}\left({}^5 R -F_{\mu \nu }F^{\mu \nu }
\right. \nonumber\\&\left.
+\frac{2}{3\sqrt 3}\epsilon^{\mu\nu\rho\sigma\tau }A_\mu F_{\nu \rho }
F_{\sigma\tau }\right)\,.
\label{5DminimalSUGRA}
\end{align}
\subsubsection{Supersymmetric solution in ungauged theory}
Let us first consider the case where the 5-dimensional spacetime is supersymmetric,
i.e., there exists a nontrivial Killing spinor satisfying
(\ref{KS_STU1}) and (\ref{KS_STU2}) with $\mathfrak g=0$~\cite{GR,Elvang:2004ds}.
For the timelike family of solutions for which $V=\partial/\partial t$
is a timelike Killing vector,
the supersymmetry requires that the base space is hyper-K\"ahler and
the Maxwell fields are expressed as
\begin{align}
F^{(I)}={\rm d} [fX^I({\rm d} t+\omega )]+\Theta ^{I}\,,
\end{align}
where $\Theta^{I}$ are self-dual 2-forms on the base space satisfying
$X_I\Theta ^{I}=-f({\rm d} \omega+\star_h{\rm d} \omega )/3$. The Bianchi
identity for $F^{(I)}$ requires ${\rm d} \Theta^{I}=0$, and the Maxwell
equation leads to
\begin{align}
{}^h\Delta (f^{-1}X_I) =\frac 1{12} C_{IJK}\Theta ^{(J)mn}\Theta^{(K)}_{mn}\,.
\end{align}
For $\Theta^I=0$, the solution reduces precisely to the one
assumed for the pseudo-supersymmetric solutions (\ref{f_cube}).
If we set $h_{mn}=\delta _{mn}$, $\omega =0$ and $H_I=1+Q_I/r^2$,
the metric describes the standard static intersecting M2/M2/M2-branes
with corresponding charges $Q_I$.
In this case
the 11-dimensional solution admits a Killing spinor
$\varepsilon =(H_1H_2H_3)^{-1/6}\varepsilon_\infty $ with
\begin{align}
i\Gamma ^{0\hat y_1\hat y_2}\varepsilon_\infty =\varepsilon_\infty \,,~
i\Gamma ^{0\hat y_3\hat y_4}\varepsilon_\infty =\varepsilon_\infty \,,~
i\Gamma ^{0\hat y_5\hat y_6}\varepsilon_\infty =\varepsilon_\infty \,.
\end{align}
satisfying
\begin{align}
\left[
{}^{11} \ma D_A +\frac{i}{288}\left({\Gamma_A }^{BCDE}-8
{\delta_A }^B \Gamma^{CDE }\right) \ma F_{BCDE
}\right]\varepsilon =0\,,
\end{align}
where $\Gamma $ is the 11-dimensional gamma matrices.
It deserves to mention that
the fact that ${\rm d} s_5^2$ in~(\ref{11Dmetric}) is the 5-dimensional
Einstein-frame metric means that the causal pathologies are not cured
by lifting up to M-theory.
\subsubsection{Dynamically intersecting M2/M2/M2-branes}
\label{M2M2M2}
Let us next consider the non-supersymmetric case
where the metric is time-dependent.
The importance of dynamically intersecting branes in supergravity theory
lies in their applications to cosmology and dynamical black holes.
The dynamically intersecting branes without rotation are analyzed in
detail in~\cite{MOU}. We are going to discuss its rotating version.
The potential
$V=27C^{IJK}V_IV_JX_K$ vanishes identically for the STU theory with
the case (ii) $V_1\ne 0, V_2=V_3=0$.
This lies at the heart of why the pseudo-supersymmetric solution
of the case (ii) derived in section~\ref{5D_sol_SG} [see Eq. (\ref{sol_ii})]
can be embedded in to 11-dimensional supergravity.
Note, however, that the 11-dimensional
configuration is no-longer (true nor fake) supersymmetric.
Nevertheless, the 5-dimensional pseudo-supersymmetry
justifies the mechanical equilibrium of dynamically intersecting branes.
If $\bar H_1, H_2$ and $H_3$ represent harmonics with a single point
source on the Euclid 4-space,
the solution describes the dynamically intersecting rotating M2/M2/M2
branes obeying the harmonic superposition rule.
For the vanishing charges $\bar H_1=0$ and $H_2=H_3=1$,
the background metric is obtained, which is
the 11-dimensional ``rotating'' Kasner universe,
\begin{widetext}
\begin{align}
{\rm d} s_{11}^2=&-\left[{\rm d} \bar t+\frac{J}{2r^2(\bar t/\bar
t_0)^{1/2}}(\sin^2 \vartheta {\rm d} \phi_1+\cos^2\vartheta {\rm d} \phi_2)\right]^2+
(\bar t/\bar t_0)^{1/2}\left[{\rm d} r^2+r^2({\rm d} \vartheta^2+\sin^2\vartheta {\rm d} \phi_1^2+
\cos^2\vartheta {\rm d} \phi_2^2)\right] \nonumber \\
&+
(\bar t/\bar t_0)^{-1}\left({\rm d} y_1^2+{\rm d} y_2^2\right)+(\bar t/\bar t_0)^{1/2}
\left({\rm d} y_3^2+\cdots +{\rm d} y_6^2\right)\,.
\end{align}
\end{widetext}
Here $\bar t\propto t^{2/3}$ measures the cosmic time. The 11-dimensional universe
collapses into the $y_1$-$y_2$ directions and expand other
directions~\cite{Gibbons:2005rt}.
It follows that the three
kinds of branes are intersecting in the background of Kasner universe.
The case of $J=0$ recovers the conventional vacuum Kasner solution.
\subsubsection{The cases {\rm (iii)} and {\rm (iv)}}
For the cases (iii) and (iv), there exists a non-zero potential
in the fake supergravity theory.
It might be reasonable to expect that the FLRW universe may be realized
from the viewpoint of intersecting branes, which are the fundamental constituents of
supergravity.
Assuming the brane intersection rule~\cite{MOU} and
making the nonzero vacuum expectation values of the 4-form $\ma F$,
we have tried to uplift the
solutions~(\ref{sol}) with (\ref{sol_H}) into 11-dimensions but failed.
Whether the present solutions are obtainable from the brane
picture is an outstanding issue at present.
We leave this possibility to the future work.
\subsection{Compactification to 4-dimensions}
When discussing the FLRW spacetime, it is much more reasonable to
argue within the 4-dimensional effective theory.
In this section we shall show how to achieve this.
\subsubsection{Dimensional reduction via Gibbons-Hawking space
and Kaluza-Klein black hole}
One can obtain the 4-dimensional solutions in~\cite{GMII,MNII} via
dimensional reduction of 5-dimensional solutions~(\ref{SUSYmetric}) as follows.
We employ the Gibbons-Hawking
space~\cite{Gibbons:1979zt} as a 4-dimensional base space,
\begin{align}
{\rm d} s^2_{\mathcal B} = {h}^{-1}\left({\rm d} x^5 +\chi_i{\rm d} x^i \right)^2 +h
\delta_{ij }{\rm d} x^i {\rm d} x^j\,,
\label{GHmetric}
\end{align}
where $i,j,...$ denote 3-dimensional indices (hence no distinction is
made for upper and lower indices) and
\begin{align}
\vec \nabla \times \vec \chi =\vec \nabla h\,.\label{chi}
\end{align}
$\vec \nabla$ is the derivative operator on the flat
Euclid 3-space and usual vector convention will be
used for the quantities on the Euclid space henceforth. The integrability condition of (\ref{chi})
implies that $h$ is a harmonic function on the Euclid space
$\vec \nabla^2 h=0$.
In the Gibbons-Hawking base space,
$\partial/\partial x^5$ is a Killing vector preserving the
three complex structures, which are given by~\cite{Gibbons:1987sp}
\begin{align}
\mathfrak J^{(i)} =({\rm d} x^5+\chi )\wedge {\rm d} x^i-\frac 12 h\epsilon_{ijk}{\rm d}
x^j\wedge {\rm d} x^k \,.
\end{align}
The orientation is chosen in such a way that the complex structures are
anti-self-dual, viz, the volume form is given by
$h{\rm d} x^5 \wedge {\rm d} x^1 \wedge {\rm d} x^2\wedge {\rm d} x^3$.
Under the change of Killing coordinates $x^5 \to x^5 +g(x^i)$ where $g$
is an arbitrary function of $x^i$, $\chi_i$ transforms as
$\chi_i \to \chi_i-\partial_i g$ and $h$ is unchanged
in order to preserve the metric form.
Prime examples of Gibbons-Hawking space are the flat space
($h=1$ or $M/|{\vec x}|$), the Taub-NUT space ($h=1+M/|{\vec x}|$)
and the Eguchi-Hanson space ($h=M/|{\vec x-\vec x}_1|+M/|{\vec x-\vec x}_2|$).
Assuming that the vector field $\partial/\partial x^5$ is also a
Killing vector for the whole 5-dimensional spacetime,
it turns out that functions $H_I$ are also harmonics on the Euclid
3-space $\vec \nabla^2 H_I=0$ (and the linear time-dependence remains
intact). Let $\omega $ decompose as
\begin{align}
\omega =\omega_5({\rm d} x^5+\chi_i{\rm d} x^i)+\omega_i{\rm d} x^i \,,
\end{align}
and let us write the metric as
\begin{align}
{\rm d} s_5^2= &\Lambda \left[{\rm d} x^5+\chi_i {\rm d} x^i-f^2
\omega_5\Lambda^{-1} ({\rm d} t+\omega_i{\rm d} x^i )\right]^2
\nonumber \\ &
-
fh^{-1}\Lambda ^{-1}({\rm d} t+\omega_i {\rm d} x^i)^2 +f^{-1}h\delta_{ij}{\rm d} x^i{\rm d} x^j\,,\\
=&:e^{-4\sigma /\sqrt 3}({\rm d} x^5+B_\alpha {\rm d} x^\alpha )^2 +e^{2\sigma/\sqrt
3}g_{\alpha\beta }{\rm d} x^\alpha {\rm d} x^\beta \,,
\label{DD}
\end{align}
where $\Lambda=f^{-1}h^{-1}-f^2\omega_5^2 $,
$g_{\alpha\beta }$ is the 4-dimensional Einstein frame metric,
$
B_\alpha {\rm d} x^\alpha {\rm d} x^\alpha
=\chi_i {\rm d} x^i-f^2\omega_5\Lambda^{-1}({\rm d} t+\omega_i{\rm d} x^i )
$
is the Kaluza-Klein gauge field and
$\sigma=-{\sqrt{3}\over 4} \ln \Lambda$ is a dilaton field.
The anti-self duality of Sagnac curvature
${\rm d} \omega+\star_h{\rm d} \omega=0 $~\cite{GibbonsBMPV} reduces to
\begin{align}
\vec \nabla \times \vec \omega =h^2 \vec \nabla (h^{-1}\omega_5 )\,.
\label{3D_omega}
\end{align}
The integrability condition of this equation is
$\vec\nabla^2 \omega_5=0$, i.e., $\omega_5 $ is another harmonic
function.
The Einstein frame metric $g_{\alpha \beta }$ is given by
\begin{align}
{\rm d} s_4^2&
=-\Xi ({\rm d} t+\omega_i {\rm d} x^i)^2+ \Xi^{-1} \delta_{ij}{\rm d} x^i {\rm d} x^j\,,
\end{align}
with
\begin{align}
\Xi :=fh^{-1}\Lambda ^{-1/2}\,,
\end{align}
where $\vec\omega $ is determined by~(\ref{3D_omega}) up to a gradient.
When $\omega_5$ is proportional to $h$, Eq.~(\ref{3D_omega}) implies
that $\vec \omega $ is written as a gradient of some scalar function,
which can be made to vanish by redefinition of $t$ and harmonic
functions if we work in a ``Coulomb gauge''
$\vec \nabla \cdot \vec \omega =0$. Thus
the 4-dimensional rotation vanishes ($\vec \omega=0$) in this case. If
two harmonics are equal ($H_2=H_3$) in the STU-theory and $\omega_5=0$,
the 4-dimensional solutions given in~\cite{MN,MNII} except the
$n_T=4$ case are recovered.
Since the dimensional reduction does not spoil the fraction of
supersymmetries, it turns out that the 4-dimensional
solutions in~\cite{MN,MNII} are also pseudo-supersymmetric
in the context of fake supergravity.
The resulting 4-dimensional theory involves many scalar and vector
multiplets. To see this we consider the general Kaluza-Klein ansatz~(\ref{DD}).
Defining
\begin{align}
&H_{\alpha\beta }=2 \partial_{[\alpha }B_{\beta ]}\,,
\qquad A^{(I)}=A_\alpha ^{'(I)}{\rm d} x^\alpha +\theta ^{(I)}{\rm d} x^5\,,
\nonumber \\
&F^{'(I)}_{\alpha\beta }=2\partial_{[\alpha }A_{\beta ]}^{'(I)}\,,
\quad
{}^4F_{\alpha\beta }^{(I)}=F^{'(I)}_{\alpha\beta }-2\partial_{[\alpha
}\theta^{(I)}B_{\beta ]}\,,
\end{align}
one finds that the 5-dimensional theory~(\ref{5Daction}) leads to
the following 4-dimensional effective Lagrangian,
\begin{widetext}
\begin{align}
L_4 =& {}^4R-2k^2 Ve^{2\sigma/\sqrt 3}
-2g^{\alpha \beta }\partial _\alpha \sigma\partial_\beta \sigma-\frac{1}{4}e^{-2\sqrt
3\sigma }H_{\alpha\beta }H^{\alpha\beta }
-\ma G_{AB}g^{\alpha\beta }\partial _\alpha \phi^A\partial_\beta \phi^B
\nonumber \\
&
-\frac 12 e^{-2\sigma /\sqrt 3}
G_{IJ}{}^4 F^{(I)}_{\alpha\beta }{}^4 F^{(J)\alpha\beta }
-e^{4\sigma/\sqrt 3}G_{IJ}g^{\alpha\beta }\partial _\alpha \theta ^{(I)}
\partial_\beta \theta^{(J)}
\nonumber \\
&
-\frac 18{\epsilon^{\alpha\beta\gamma\delta }}C_{IJK}\theta ^{(I)}\left(
{}^4 F^{(J)}_{\alpha\beta }{}^4F^{(K)}_{\gamma\delta }-\theta^{(J)}\cdot
{}^4F_{\alpha\beta }^{(K)}H_{\gamma\delta }
+\frac 13 \theta^{(J)}\theta^{(K)}H_{\alpha\beta }H_{\gamma\delta}
\right)\,.
\end{align}
\end{widetext}
Thus the 4-dimensional effective theory derived from the
Lagrangian~(\ref{5Daction}) comprises $2N$ scalars
$(\sigma, \phi^A, \theta^{(I)})$ and
$N+1$ gauge fields $(A'^{(I)}_\mu , B_\mu )$ in general.
While, its supersymmetric solution is specified by $N+2$ harmonics
($H_I, h, \omega_5$).
As an obvious application let us consider the case where the 4-dimensional base space
($\ma B, h_{mn}$) is the Taub-NUT space. The Taub-NUT metric can be written as a
Gibbons-Hawking form~(\ref{GHmetric}) as,
\begin{align}
{\rm d} s^2_{\rm TN}=&\left(\varepsilon +\frac{M}{\rho }\right)^{-1} M^2 (\sigma_R^3)^2
\nonumber \\ &+
\left(\varepsilon +\frac{M}{\rho }\right) \left[{\rm d} \rho ^2 +
\rho ^2
\left\{(\sigma_R^1)^2 +(\sigma_R^2)^2 \right\}
\right]\,.
\label{TaubNUT}
\end{align}
where $\rho:=|\vec x|$ and $M (>0) $ corresponds to the NUT parameter.
For later convenience, we have introduced a parameter $\varepsilon$,
which is unity for the Taub-NUT space.
A natural 5-dimensional background ($|\vec x| \to \infty $) in this case
is
\begin{align}
{\rm d} s_{\rm GPS}^2 = -{\rm d} \bar t^2 +a(\bar t) ^2 {\rm d} s_{\rm TN}^2 \,.
\label{GPS}
\end{align}
where the scale factor $a(\bar t)$ is given by (\ref{scale_factor1}) and
(\ref{scale_factor2}).
This is the Gross-Perry-Sorkin type monopole~\cite{Gross:1983hb} immersed in the
FLRW universe. At large distance $|\vec x|\to \infty $
it may be rewritten as a U(1)-fibration over the FLRW
universe $M_5 \simeq M_4\times S^1$,
\begin{align}
{\rm d} s_{\rm GPS}^2 ={\rm d} s^2 _{\rm FLRW} +M a(\bar t)^2 \rho (\sigma
_R^3)^2 \,.
\end{align}
Thus the spacetime is effectively 4-dimensional at infinity.
Since the metric~(\ref{GPS}) admits a homothetic Killing field,
one can analyze its causal structures analytically. The conformal
diagrams are the same as the 5-dimensional FLRW universe.
Reminding the fact that the flat Euclid space is
recovered when $\varepsilon =0$ in the metric~(\ref{TaubNUT}) (note that
in this case $M$ is not the NUT charge),
the spacetime structure as $\rho \to 0$
(with or without $t\to \pm \infty $)
is identical to that for the solution~(\ref{sol}).
Then the vicinity of horizons is indeed 5-dimensional.
Therefore this geometry describes a Kaluza-Klein type black
hole~\cite{Ida:2007vi}.
\subsubsection{A caged black hole}
As discussed in~\cite{Myers:1986rx,Maeda:2006hd} for the supersymmetric case,
a caged black-hole geometry is obtained by
superimposing an infinite number of black holes
aligned in one direction
with an equal separation.
Since the present time-dependent solution found in
section \ref{5D_sol_SG} is linearized in space, we can
construct similar configurations easily.
Decomposing the Euclid 4-space coordinates as $x^m=(x,y,z,w)$ with the
orientation ${\rm d} x\wedge {\rm d} y \wedge {\rm d} z\wedge {\rm d} w$ and
putting the same point sources along $w$-axis with an equal-spacing of
$2\pi R_5$, we obtain
\begin{widetext}
\begin{eqnarray}
H_S&=&1
+Q_{S} \sum_{k=-\infty}^{\infty} \frac{1}{\rho^2+(w+2\pi k R_5)^2}
=1+{Q_S\over 2 R_5^2}
{\sinh \bar \rho \over \bar \rho\left(
\cosh \bar \rho-\cos \bar{w} \right)}
\label{ha1}\,,
\\
H_T&=&{t\over t_0}
+Q_{T} \sum_{k=-\infty}^{\infty} \frac{1}{\rho^2+(w+2\pi k R_5)^2}
={t\over t_0}+{Q_T\over 2 R_5^2}
{\sinh \bar \rho\over \bar \rho\left(
\cosh \bar\rho-\cos \bar{w} \right)}
\label{ha2}\,,
\\
\omega _{\phi_1}&=& {J}
\sum_{k=-\infty}^{\infty} \frac{x^2+y^2}{[\rho^2+(w+ 2\pi k R_5)^2]^2}
={J\over 4R_5^2}
{\left(\bar{x}^2+\bar{y}^2\right)\over \bar\rho^2}
\left[
{\left(\cosh \bar\rho\cos \bar{w}-1
\right)\over \left(\cosh \bar \rho-\cos \bar{w}\right)^2}
+{\sinh \bar \rho\over \bar{\rho }
\left(\cosh \bar\rho-\cos \bar{w}\right)}
\right]\,, \\
\omega _{\phi_2}&=& {J}
\sum_{k=-\infty}^{\infty}
\frac{z^2+(w+ 2\pi k R_5)^2}{[\rho^2+(w+ 2\pi k R_5)^2]^2}
\nonumber \\
&=&
{J\over 4R_5^2}
\left[
-{(\bar{x}^2+\bar{y}^2)\over \bar{\rho }^2}
{\left(\cosh \bar\rho\cos \bar w-1
\right)\over \left(\cosh \bar \rho-\cos \bar{w}\right)^2}
+{\left(\bar \rho^2+\bar z^2 \right)\over \bar{\rho }^2}
{\sinh \bar \rho
\over \bar\rho
\left(\cosh \bar\rho-\cos \bar{w}\right)}
\right]
\,,
\end{eqnarray}
\end{widetext}
where $\rho^2\equiv x^2+y^2+z^2$, and we have introduced
dimension free coordinates $\bar x^m=x^m/R_5$ and
$\bar \rho =\rho/R_5$.
To derive these expressions we have used a series expansion
\begin{align}
\sum_{k=-\infty }^\infty \frac{1}{\xi^2+(\eta +2\pi k)^2}=\frac{\sinh \xi
}{2\xi (\cosh \xi-\cos \eta )}\,.
\end{align}
Since this solution is periodic in the $w$-direction
by identifying $w=0$ and $2\pi R_5$,
it can be regarded as a deformed BMPV ``black hole''
in a compactified five-dimensional spacetime ($0 \leq w \leq 2 \pi R_5$)
with pseudo-supersymmetry.
Introducing the 3-dimensional spherical coordinates
($\rho,\Theta,\Phi$),
which are defined by
\begin{align}
&&
x=\rho \sin \Theta \cos \Phi, \quad
y=\rho \sin \Theta \sin \Phi , \quad
z=\rho \cos \Theta
\,,~~~~~~~~
\end{align}
the 4-dimensional Einstein frame metric
in the asymptotic region ($\rho\gg \pi R_5$)
reads
\begin{align}
{\rm d} \bar{s}_4^2=&-f^{3/2}\left(
{\rm d} \bar{t}+\bar \omega_\Phi {\rm d} \Phi
\right)^2
\nonumber \\
& +f^{-3/2}\left[{\rm d} \bar \rho^2+\bar\rho^2\left({\rm d} \Theta^2+
\sin^2\Theta {\rm d} \Phi^2\right)
\right]\,,
\label{CBH_4D_metric}
\end{align}
where
\begin{eqnarray}
&&
f=\left(1+{1\over 2R_5^2}{Q_S\over \bar\rho}\right)^{-n/3}
\left({t\over t_0}+{1\over 2R_5^2}{Q_T\over \bar\rho}\right)^{-1+n/3}\,,
\nonumber \\
&&
\bar \omega_\Phi= { 1 \over 4 R_5^3} {J\over \bar\rho}
\sin^2\Theta
\,.
\end{eqnarray}
In the asymptotic limit $\rho\rightarrow \infty$,
the metric~(\ref{CBH_4D_metric}) describes an FLRW universe with
the power exponent of the scale factor being $p=1/(4-n)$.
One might therefore expect that this solution describes a caged black hole
in the effective 4-dimensional FLRW universe.
However, we have to be careful to judge whether it is a black hole
or not.
A two-black hole system in the Kastor-Traschen spacetime
[the case (iv) without rotation] will collide and merge to form a single
black hole in the contracting universe ($t_0<0$).
In the expanding universe, the solution describes the time reversal one.
Namely it corresponds to the two-white hole system,
since one object disrupt into two objects, which is
possible for a white hole but not for a black hole.
In the present case we have infinite numbers of point sources
before identification, so that we can expect a similar result.
It therefore appears that the object in the expanding universe corresponds to
a splitting ``white string'' into an array of white holes. In order to clarify this rigorously,
we have to analyze (numerically) the horizons of
a multi-object system in the expanding universe.
Especially one important question to be answered
is whether black holes will collide
in a contracting universe for any value of $n$.
\section{Concluding remarks}
\label{conclusion}
We have presented pseudo-supersymmetric solutions to 5-dimensional
``fake'' supergravity coupled to arbitrary ${\rm U}(1)$
gauge fields and scalar fields. The non-compact gaugings of
R-symmetry correspond to the Wick-rotation of gauge coupling constant
($\mathfrak g \to ik$).
Since the bosonic action is not charged with respect to R-symmetry, no
ghosts appear in this sector, i.e., all kinetic terms possess the
correct sign. The net effect of imaginary coupling
produces a positive potential for the scalar fields. Hence the background spacetime is
generally dynamical, contrary to the supersymmetric case.
The metric solves 1st-order Killing spinor equation,
which automatically guarantees that the Einstein equations and the
scalar field equations are satisfied
if the Maxwell equations are solved.
The solution is specified by time-dependent and time-independent
harmonics $H_I$ on a hyper-K\"ahler base space.
This encodes the balances of forces of the solution: the gravitational
attraction is adjusted to cancel the electromagnetic repulsive force
(the scalar fields can contribute both sides depending on the potential).
We specialized to the case in which a single point source on the Euclid
4-space and explored its physical properties.
The solutions we found are the rotating generalizations of
our previous solutions~\cite{GMII,MNII} describing a black hole in the FLRW universe.
The present metric has four parameters: the Maxwell charge $Q$, the
angular momentum $J$, the number of time-dependent harmonics $n$
and the ratio of energy densities of the Maxwell field
and the scalar field at the horizon $\tau $.
The spacetime approaches to the rotating ${\rm AdS}_2\times S^3$ for small radii,
while it asymptotes to the FLRW cosmology for large radii.
So, the solution is a BMPV black hole immersed in the time-dependent background
cosmology. Except the asymptotic de Sitter case, one cannot introduce a
stationary coordinate patch even in the single centered case.
Though we have made some simplification, it turns out that
the solution enjoys much richer physical properties than stationary ones.
The analysis of near-horizon geometry uncovers that the horizon is
described by a Killing horizon. Hence the ambient materials fail to
accrete onto the black hole irrespective of the dynamical background.
This property may be attributed to the pseudo-supersymmetry.
The ``BPS'' solution maintains equilibrium, forbidding the horizon to grow.
An important issue to be noted is that the event horizon is not
extremal in general. This is due to the fact that the
event horizon is not generated by the coordinate vector field
in the metric~(\ref{SUSYmetric}).
Furthermore the event horizon is rotating, i.e.,
the event horizon is generated by a linear combination of
time and angular Killing vectors~(\ref{generator}).
This is in sharp contrast to the supersymmetric BMPV black hole
with vanishing angular velocity.
The nonvanishing angular velocity of the horizon indicates that there
exists an ergoregion lying strictly outside the horizon.
The presence of an ergoregion implies the possibility of rotating energy
removal process via the Penrose process and the superradiant
scattering~\cite{Nozawa:2005eu}.
We can find that this is indeed the case for $n=3$ as shown
in Appendix~\ref{sec:superradiance}.
For other values of $n$, the energy of a particle and a wave is not
conserved, so it is not a straightforward issue to conclude whether such an energy
extraction process is actually realizable under a dynamical setting.
This is an interesting future
work to be argued.
We have also revealed that rotating solutions generically suffer from
causal violation in the neighborhood of singularities. The pseudo-supersymmetry
cannot elude naked time machines. The reason is obvious: the (pseudo-)supersymmetry
variations~(\ref{fakeKS1}) and (\ref{fakeKS2}) are local, so that
they make no direct mention of global structure of spacetime such as
closed timelike curves. In particular, the timelike singularity
$t=t_s(r)$ in the $r^2>0$ domain is repulsive.
The original time-dependent equilibrium solution was derived via
compactification of M2/M2/M5/M5-branes in 11-dimensional
supergravity~\cite{MOU}. We discussed in section~\ref{M2M2M2} that
the present metric with a single time-dependent harmonic function
can be embedded into 11-dimensions, describing a dynamically intersecting
M2/M2/M2-branes in a rotating Kasner universe.
It is shown that the 4-dimensional solution~\cite{MOU} was also derived
from compactification of 5-dimensional solution on the Gibbons-Hawking
space. Unfortunately,
such a liftup procedure fails to act as a chronology protector.
It is of particular interest to see whether it oxidizes to a causally well-behaved
solution in 10-dimensional supergravity, as in~\cite{Herdeiro:2000ap}.
It appears appealing to examine if the occurrence of closed timelike
curves corresponds to the loss of unitarity in the
context of de Sitter/CFT correspondence.
{\it Note added.}
During the completion of this work, we noticed the work
of~\cite{Gutowski:2010sx}, which classifies all the
pseudo-supersymmetric solution of the theory~(\ref{5Daction}).
It is intriguing to examine if more general classes of solutions admit
black hole horizons in the expanding universe.
\acknowledgments{
This work was partially supported
by the Grant-in-Aid for Scientific Research
Fund of the JSPS (No.22540291) and
and by the Waseda University Grants for Special Research Projects.
}
|
{
"timestamp": "2010-12-30T02:02:29",
"yymm": "1009",
"arxiv_id": "1009.3688",
"language": "en",
"url": "https://arxiv.org/abs/1009.3688"
}
|
\section{Introduction}
In the framework of integrable systems, the study of open spin chains
with integrable boundaries have been developed a long time ago
\cite{cherednik,sklyanin}. There, it has been shown that the model is
integrable provided the two matrices characterizing the boundaries
obey some algebraic relations (the so-called reflection equation).
However, although the integrability has been proven, the explicit
resolution
(eigenvalues and eigenvectors of the Hamiltonian)
of the models is not known in its full generality. In fact, for a
long
time, only the case of diagonal boundary matrices was solved, for
different kind of models, e.g. open XXX \cite{Gau} and its $su(N)$
\cite{byebye} or $su(N\vert M)$ \cite{RS,zaka}
generalizations, open XXZ \cite{alc} and its generalizations
\cite{mnsymm,ACDFR2,done2,sam},
using different versions of the
Bethe ansatz (analytical, algebraic or functional).
However, the classifications of boundary matrices (obeying a
reflection equation) \cite{dvgr2, gand, selene, nondiag-sol} clearly shows that
non-diagonal solutions do
exist, although explicit solutions for the eigenvalue problem were
not known. The problem laid
essentially in the construction of a reference state (a particular
Hamiltonian eigenvector) that allows to initiate the procedure.
Indeed, when the boundary matrices were not diagonal (or at least not
simultaneously diagonalizable), the existence of this reference state
was not ensured.
Recently, different approaches have been developed to overcome
these difficulties, such as gauge transformations that allow to go to a
diagonal basis \cite{Cao}, or fusion
relations for TQ relations that do not need the existence of a
reference state \cite{nepo,tq}. In all these
cases, the boundary matrices can be non-diagonal, but the
parameters entering their definition need to satisfy some constraints.
Note also the original approach \cite{BK} that uses another
presentation of the
reflection algebra, called $q$-Dolan-Grady relation, as well as the
`generalized' functional ansatz developed in \cite{galleas}. Both avoid the
use of constraints.
The Asymmetric Simple Exclusion Process (ASEP) is an
out-of-equilibrium statistical physics representation of the
Temperley-Lieb algebra \cite{simon09}, on which the XXZ Hamiltonian
is based. Many exact probabilistic results
\cite{dehp,schuetz1,schuetz2,jafarpour}, that are not necessarily
based on the integrability of the model, have been obtained in the
past fifteen years and shed some new light on integrable results. The
ASEP notations, as described in \cite{dGE}, are presented in this
paper and a specific section \ref{sec:matrixansatz} is dedicated to
the comparison of our results with old results on the ASEP, such as
the matrix ansatz \cite{dehp,mallicksandow,esr,corteel}.
In this paper, we present a construction based on the
coordinate Bethe ansatz \cite{bethe} for open XXZ and ASEP models, allowing the use of
non-diagonal boundary
matrices. As for the other approaches, the boundary matrices need to
obey some constraints, but the ones we find are more general than
those already known.
We present now the structure of the paper: section
\ref{sec:defandnotations} defines the models and introduces the
notations needed for the next sections; section \ref{sec:basis}
describes the different sets of reference states and the sets of
exceptional points they lead to. Section \ref{sec:CBA} gives the
structure of the coordinate Bethe ansatz as a combination of the
reference states and explains the role of the $BC_n$ Weyl group and gives in details the Bethe equations that correspond to the two sets of specific points. Finally, we conclude on two open problems: section \ref{sec:matrixansatz}
presents a short discussion of the relation between the specific points and
other previous results about the exclusion process such as the matrix ansatz; a discussion on the completeness of the spectrum and of the eigenvectors is tackled in section \ref{subsec:completeness}.
\section{XXZ and ASEP models with non-diagonal boundaries}
\label{sec:defandnotations}
The Markov transition matrix for the open ASEP model is given by
\begin{equation} \label{eq:hamasep}
W=\widehat K_1+K_L+\sum_{j=1}^{L-1}w_{j,j+1}\,,
\end{equation}
where the indices indicate the spaces in which the following matrices
act non trivially
\begin{eqnarray}
w=\left(
\begin{array}{c c c c}
0 & 0 & 0 & 0\\
0 &-q & p & 0\\
0 & q &-p & 0\\
0 & 0 & 0 & 0
\end{array}
\right)
\mb{,}
\widehat K=\left(
\begin{array}{c c}
-\alpha & \gamma e^{-s} \\
\alpha e^{s} & -\gamma
\end{array}
\right)
\mb{and}
K=\left(
\begin{array}{c c}
-\delta & \beta \\
\delta & -\beta
\end{array}
\right)\,.
\end{eqnarray}
It is well-established \cite{ssa,esr,dGE} that this ASEP model is
related by a
similarity transformation to
the following integrable open XXZ model
\begin{eqnarray} \label{eq:hamxxz}
H=\wh B_1 + B_L -\frac{1}{2}\sum_{j=1}^{L-1}\big(\sigma^x_j\sigma^x_{j+1}
+\sigma^y_j\sigma^y_{j+1}-\cos\eta~ \sigma^z_j\sigma^z_{j+1}\big)\,,
\end{eqnarray}
where $\sigma$ are the usual Pauli matrices and
\begin{eqnarray}
\widehat B&=&
\frac{\sin\eta}{\cos\omega_-+\cos\delta_-}
\left(
\begin{array}{c c}
\frac i2 (\cos\omega_--\cos\delta_-)-\sin\omega_- & e^{-i\theta_1}
\\
e^{i\theta_1} & -\frac i2 (\cos\omega_--\cos\delta_-)-\sin\omega_-
\end{array}
\right)
\,,
\\
B&=&
\frac{\sin\eta}{\cos\omega_++\cos\delta_+}
\left(
\begin{array}{c c}
-\frac i2 (\cos\omega_+-\cos\delta_+)-\sin\omega_+ & e^{-i\theta_2}
\\
e^{i\theta_2} & \frac i2 (\cos\omega_+-\cos\delta_+)-\sin\omega_+
\end{array}
\right)
\,.
\end{eqnarray}
The explicit form of the transformation can be found in e.g.
\cite{dGE}.
We reproduce it here for completeness:
\begin{eqnarray}
W &=& -\sqrt{pq}\,U^{-1}\,H\,U
\mb{with} U=\otimes_{j=1}^L\left(\begin{array}{cc} 1 & 0 \\
0 & \xi\left(\sqrt{\frac{q}{p}}\right)^{j-1}\end{array}\right)
\,,\\
\sqrt{\frac{\alpha}{\gamma}} &=&-ie^{i\omega_{-}} \mb{,}
\sqrt{\frac{\beta}{\delta}} =-ie^{i\omega_{+}}
\mb{,} \sqrt{\frac pq} = -e^{i\eta}\\
\xi\sqrt{\frac{\alpha}{\gamma}}\,e^s &=& e^{i\theta_{1}} \mb{and}
\xi\sqrt{\frac{\delta}{\beta}}\left(\sqrt{\frac{q}{p}}\right)^{L-1} =
e^{i\theta_{2}}\,,
\label{eq:asepXXZ}
\end{eqnarray}
where $\xi$ is an arbitrary (gauge) parameter that disappears in all
the following computations.
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=0.7]
\draw (0,0) -- (9,0) ;
\foreach \i in {0,1,...,9}
{\draw (\i,0) -- (\i,0.4) ;}
\draw[->,thick] (-0.4,0.9) arc (180:0:0.4) ; \node at (0.,1.5)
{$\alpha$};
\draw[->,thick] (0.4,-0.1) arc (0:-180:0.4) ; \node at (0.,-0.8)
{$\gamma$};
\draw (2.5,0.5) circle (0.3) [fill,circle] {};
\draw (4.5,0.5) circle (0.3) [fill,circle] {};
\draw (5.5,0.5) circle (0.3) [fill,circle] {};
\draw (8.5,0.5) circle (0.3) [fill,circle] {};
\draw[->,thick] (2.4,0.9) arc (0:180:0.4); \node at (2.,1.5) {$q$};
\draw[->,thick] (2.6,0.9) arc (180:0:0.4); \node at (3.,1.5){$p$};
\draw[->,thick] (4.4,0.9) arc (0:180:0.4); \node at (4.,1.5){$q$};
\draw[->,thick] (5.6,0.9) arc (180:0:0.4); \node at (6.,1.5) {$p$};
\draw[->,thick] (8.4,0.9) arc (0:180:0.4); \node at (8.,1.5){$q$};
\draw[->,thick] (8.6,0.9) arc (180:0:0.4) ; \node at (9.,1.5)
{$\beta$};
\draw[->,thick] (9.4,-0.1) arc (0:-180:0.4) ; \node at (9.,-0.8)
{$\delta$};
\node at (10.,0.4) [anchor=west] {ASEP};
\node at (4.5,-1.) {$\Updownarrow$};
\draw (0,-2.4) -- (9,-2.4) ;
\foreach \i in {0,1,...,9}
{\draw (\i,-2.4) -- (\i,-2.) ;}
\node at (2.5,-1.9) {$\downarrow$};
\node at (4.5,-1.9) {$\downarrow$};
\node at (5.5,-1.9) {$\downarrow$};
\node at (8.5,-1.9) {$\downarrow$};
\node at (0.5,-1.9) {$\uparrow$};
\node at (1.5,-1.9) {$\uparrow$};
\node at (3.5,-1.9) {$\uparrow$};
\node at (6.5,-1.9) {$\uparrow$};
\node at (7.5,-1.9) {$\uparrow$};
\node at (10.,-1.9) [anchor=west] {XXZ chain};
\end{tikzpicture}
\end{center}
\caption{Asymmetric exclusion process for a system of size $L$ with
two boundaries and mapping to the XXZ spin chain.}
\label{fig:schemaasep}
\end{figure}
Throughout the paper we will stick to this ASEP notation, keeping in
mind that by this similarity transformation we treat also the XXZ
model
with non-diagonal boundaries. However, in the ASEP model, all the
parameters must be
positive (transition rates) while there is not such constraint in the
XXZ spin chain.
To be as general as possible, we will not assume
this constraint in the paper.
The vectors of the canonical basis used for the previous definitions
are indexed by the spin value $\uparrow$ or $\downarrow$ for the
XXZ-spin chain and corresponds to the number of particle
$\tau_i\in\{0,1\}$ of each site in the ASEP (see figure
\ref{fig:schemaasep}).
\section{Basis vectors for the coordinate Bethe
ansatz\label{sec:basis}}
For a chain of $L$ sites with periodic or diagonal boundary
conditions, it is easy to find
one eigenstate, called pseudo-vacuum. It is usually chosen as $L$
spins up (resp. $L$ empty sites for the ASEP).
Then, excited states are constructed by adding some excitations
that flip a given number $n$ of spins (resp. add $n$ particles in
the ASEP).
The conventional Bethe methods (coordinate, algebraic or analytical)
allow us
to compute the linear combinations between the
$\left(\begin{array}{c}L\\n\end{array}\right)$
excited states that diagonalize the Hamiltonian.
For an open chain with non-diagonal boundary (which is the case we
want to
deal with, see equations
(\ref{eq:hamasep}) and (\ref{eq:hamxxz})),
the first problem consists in finding the pseudo-vacuum.
Different approaches have been elaborated to overcome this problem
as explained in the introduction. However, the coordinate Bethe
ansatz, which is the
historical method introduced by H. Bethe \cite{bethe}, has never been
successfully applied.
To treat this problem, we must generalize the ansatz. A first step in
this direction have been done in \cite{simon09}. The general
strategy we adopt here can be summarized as follows:
\begin{itemize}
\item we do not choose
anymore all spins up for the vacuum and spins down for the
excitations. We take general vectors (see their explicit
construction below),
\item Hamiltonian eigenfunctions will be constructed as linear
combinations
between states with $n$ excitations
\textit{together with} $m$ $(<n)$ excitations,
\item the explicit forms of the excitations and of the vacuum depend
on
the total number of excitations in the state we consider.
\end{itemize}
More physically, one allows that excitations may be destroyed
or created by the boundaries with the restriction that
one cannot create more than $n$ excitations.
Without any loss of generality, we can always choose that one of
the boundaries preserves the
number of excitations. In this paper, we choose the right-hand-side one.
Finally, we found two different ways to fulfill these conditions. In
both cases, some constraints between
the parameters of the model appear (as it is already the case in the
approaches \cite{Cao,dGE} to non-diagonal boundaries). We will show
that we
can solve the problem (within the framework of coordinate Bethe
ansatz) for all the sets of constraints that have already been
produced in the
literature as well as for some new sets.
The following subsection introduces some notations that are standard
for the ASEP and that will be used throughout the paper. Then, in the
next two subsections \ref{sec:first} and \ref{sec:second}, we will
present the two
different sets of vectors with the associated constraints.
Finally, in section \ref{sec:CBA}, we present their particular linear
combinations diagonalizing the Hamiltonian.
\subsection{Boundary operators and a duality}
\label{sec:boundarynotations}
Two particular sets of 2-dimensional vectors are relevant in the
study of the boudary dynamics. The first set corresponds to
vectors that diagonalize the boundary operators. We choose the
following arbitrary normalizations, that will become clearer later:
\begin{align}
\widehat{K} \begin{pmatrix} 1 \\ e^s/c_1 \end{pmatrix} &= \lambda_1
\begin{pmatrix} 1 \\ e^s /c_1 \end{pmatrix}\,,\qquad
K \begin{pmatrix} 1 \\ c_L \end{pmatrix} = \lambda_L
\begin{pmatrix} 1 \\ c_L \end{pmatrix} \,.
\end{align}
There are two solutions for the first equation (resp. the second)
given by $c_1=c_\pm(\alpha,\gamma)$ and $\lambda_1 =
\lambda_\pm(\alpha,\gamma)$ (resp. $c_L=c_\pm(\beta,\delta)$ and
$\lambda_L=\lambda_\pm(\beta,\delta)$) where
$c_\pm(u,v)$ and $\lambda_\pm(u,v)$ are the
roots of the two functions:
\begin{align}
P_{u,v}(X) &= uX +(u-v) -v/X \mb{i.e.} c_+(u,v) = v/u \mbox{ and }
c_-(u,v)=-1\,,\\
Q_{u,v}(X) &= X^2 + X(u+v) \mb{i.e.} \lambda_+(u,v)=0 \mbox{ and }
\lambda_-(u,v)=-u-v\,.
\end{align}
These notations allow for a
convenient way of parametrizing both boundaries at the same time.
The second set of relevant vectors satisfies the diagonal relations:
\begin{align}\label{eq:Kstar:def}
\left[ \widehat{K} - \begin{pmatrix} q & 0 \\ 0 & p
\end{pmatrix}\right]\begin{pmatrix} 1 \\ e^s/c^*_1 \end{pmatrix} =
\lambda^*_1 \begin{pmatrix} 1 \\ e^s/c^*_1 \end{pmatrix}
\,, \quad
\left[ K + \begin{pmatrix} q & 0 \\ 0 & p
\end{pmatrix}\right]\begin{pmatrix} 1 \\ c^*_L \end{pmatrix} =
(\lambda^*_L+p+q) \begin{pmatrix} 1 \\ c^*_L \end{pmatrix}
\,.
\end{align}
As previously, the coefficients $c^*_1$ and $\lambda^*_1$ (resp.
$c^*_L$ and $\lambda^*_L$) take the two possible values
$c^*_\pm(\alpha,\gamma)$ and $\lambda^*_{\pm}(\alpha,\gamma)$ (resp.
$c^*_\pm(\beta,\delta)$ and $\lambda^*_{\pm}(\beta,\delta)$), which
are the respective zeroes of
\begin{align}
P_{u,v}^*(X) &= uX + (u-v+q-p) - v/X= P_{u,v}(X)+(q-p) \,,\\
Q_{u,v}^*(X) &= X^2+ X(u+v+p+q) + (qp+pu+qv)\,.
\end{align}
The explicit values of $c^*_\pm(u,v)$
\footnote{These functions are sometimes called $\kappa_\pm(u,v)$.
We change this notation to be consistent throughout the paper.} and
$\lambda^*_\pm(u,v)$ are
given by:
\begin{align}
c^*_\pm(u,v) &= \frac{p-q+v-u \pm
\sqrt{(p-q+v-u)^2+4uv}}{2u}
\,,\\
\lambda^*_\pm(u,v) &= \frac{-p-q-v-u \pm \sqrt{(p-q+v-u)^2+4uv}}{2}
\,.
\end{align}
The definitions \eqref{eq:Kstar:def} suggest the definition of the
new operators:
\begin{subequations}
\label{eq:staroperator:def}
\begin{align}
\widehat{K}^* &= \widehat{K} - \begin{pmatrix} q & 0 \\ 0 & p
\end{pmatrix}
\,,\qquad
K^* = K + \begin{pmatrix} q & 0 \\ 0 & p \end{pmatrix}
\,, \\
w^* &= w + \begin{pmatrix} q & 0 \\ 0 & p
\end{pmatrix} \otimes I - I \otimes \begin{pmatrix} q & 0
\\ 0 & p \end{pmatrix} = \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 &
-p & p & 0 \\ 0 & q & -q & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}
\end{align}
\end{subequations}
where $I$ is the two by two identity matrix.
One verifies that the matrix
$W$ given in (\ref{eq:hamasep}) has the following second
representation:
\begin{equation}
\label{eq:secondrepres}
W =\widehat{K}_1^* + K_L^* + \sum_{i=1}^{L-1} w_{i,i+1}^*\,.
\end{equation}
This representation will be essential in the definitions of the new
exceptional points.
\subsection{First choice\label{sec:first}}
We first consider the matrix $W$ as written in eq.
(\ref{eq:hamasep}).
Let us define the family of vectors:
\begin{align}
\ket{\omega(u)}_i &= \begin{pmatrix} 1 \\ u (p/q)^{i-1} \end{pmatrix}
\,,
\\
\ket{V}_i &= \begin{pmatrix} q-p \\ 0 \end{pmatrix}
\,.
\end{align}
where $u$ is still an arbitrary parameter.
{From} these elementary vectors, we build the tensor product over the
sites $i$ to $j$ and write it as
\begin{equation}
\label{eq:def:Omega}
\ket{\Omega(u)}_i^j=
\ket{\omega(u)}_i\ket{\omega(u)}_{i+1}\ldots\ket{\omega(u)}_j
\,.
\end{equation}
We now fix an integer $n$ and introduce the state with $n-m$
excitations at the ordered positions $1\leq x_{m+1}< \ldots < x_n
\leq L$ defined as the $(\mathbb{C}^2)^{\otimes L}$-vector:
\begin{eqnarray}
&&\ket{x_{m+1},\ldots,x_n} \ =\
\left(\sqrt{\frac{q}{p}}\right)^{(x_{m+1}-1)+\ldots+(x_n-1)} \times
\label{eq:def:state}\\
&&\quad\times\
\ket{\Omega(u_{m+1})}_1^{x_{m+1}-1}\,\ket{\omega(v_{m+1})}_{x_{m+1}}\,
\ket{\Omega(u_{m+2})}_{x_{m+1}+1}^{x_{m+2}-1}\,\ket{\omega(v_{m+2})}_{x_{m+2}}
\ \ldots\ \ket{\omega(v_n)}_{x_n}\,\ket{\Omega(u_{n+1})}_{x_n+1}^L\,.
\nonumber
\end{eqnarray}
The overall factor $(\sqrt{\frac{q}{p}})$ is introduced only in order
to normalize the Bethe roots. The coefficients $u_m$ and $v_m$ are
related through the recursion relation:
\begin{equation}
\label{eq:recursion:ui:vi}
u_{m+1} = \frac{q}{p} u_m, \quad v_{m+1} = \frac{q}{p} v_m\,,
\end{equation}
and the initial coefficients $u_1$ and $v_1$ are still arbitrary.
These vectors correspond to states where $m$ excitations have left
the
system (through the left boundary). To clarify the notation, the state
with no excitation corresponds to $m=n$ and is given by
\begin{equation}
\ket{\emptyset}= \ket{\Omega(u_{n+1})}_{1}^{L}\,,
\end{equation}
while the state with one excitation ($m=n-1$) reads
\begin{equation}
\ket{x_{n}}= \left(\sqrt{\frac{q}{p}}\right)^{x_n-1}
\ket{\Omega(u_{n})}_1^{x_{n}-1}\,\ket{\omega(v_n)}_{x_n}\,\ket{\Omega(u_{n+1})}_{x_n+1}^L\,.
\end{equation}
The above states are product states and are linearly independent. We
denote by $\cB_n$ the set of these $\displaystyle \sum_{k=0}^n
\left(\begin{array}{c} L\\k\end{array}\right)$ independent vectors.
We also need the vectors $|x_{m+1},\dots,\overline
x_\alpha,\dots,x_n\rangle$
deduced from $|x_{m+1},\dots,x_n\rangle$ by replacing the vector
$\ket{\omega(u)}$ in
position $x_\alpha$ by $\ket{V}_{x_\alpha}$.
These vectors have been chosen such that we get for the Hamiltonian
bulk part:
\begin{equation}
w_{\beta,\beta+1}|x_{m+1},..,x_n\rangle=
\begin{cases}
0 &\text{if } \beta,\beta+1\neq x_{m+1},\dots,x_n
\,;\\
\sqrt{pq}|..,x_j-1,..\rangle
-q|..,x_j,..\rangle +|..,\overline x_j,..\rangle
&\text{if } \beta+1=x_j\ ,\,\beta\neq x_{j-1}
\,;\\
\sqrt{pq}|..,x_j+1,..\rangle
-p|..,x_j,..\rangle -|..,\overline x_j,..\rangle
&\text{if } \beta=x_j\ ,\,\beta+1\neq x_{j+1}
\,;\\
|..,x_j,\overline x_{j+1},..\rangle
-|..,\overline x_j,x_{j+1},..\rangle
&\text{if } \beta=x_j\ ,\,\beta+1=x_{j+1}\,.
\end{cases}
\end{equation}
Remark the additional states appearing on the right hand side
involving vector $|V\rangle$.
Because of the alternating signs, they cancel each other in the
Hamiltonian bulk part,
but the first and last terms (boundary terms). It is interesting to
remark that the same "telescopic" trick is also used in the proof of
the matrix ansatz in section \ref{sec:matrixansatz} and in the
definitions \eqref{eq:staroperator:def}.
The first and last terms $-\ket{V}_1$ and $\ket{V}_L$ of the
telescopic sum must be absorbed by the boundary operators and this
justifies the study of the operator \eqref{eq:Kstar:def}.
We now require that the left boundary diagonalizes
$\ket{\omega(u_1)}_1$ and that the right boundary diagonalizes
$\ket{\omega(u_{n+1})}_L$, hence excitations are created neither on
the right, nor on the left, when $n$
particles are already in the bulk. Thus, the action of the boundary
operators must be
\begin{align}\label{eq:K1}
\widehat K_1|x_{m+1},\dots,x_n\rangle &=
\begin{cases}
\Lambda_1^{(m)}|x_{m+1},\dots,x_n\rangle
+C_1^{(m)}|1,x_{m+1},\dots,x_n\rangle &\text{if } x_{m+1}>1\,,\\
\widetilde\Lambda_1^{(m)}|1,x_{m+2}\dots,x_n\rangle
+D_1^{(m)}|x_{m+2},\dots,x_n\rangle +
\ket{\overline{1},x_{m+2},..,x_n} &\text{if } x_{m+1}=1\,,
\end{cases} \\
K_L|x_{m+1},\dots,x_n\rangle &=
\begin{cases}
\Lambda_L|x_{m+1},\dots,x_n\rangle &\text{if } x_{n}<L\,,\\
\widetilde\Lambda_L|x_{m+1},\dots,x_{n-1}, L\rangle
-|x_{m+1},\dots,x_{n-1},\overline L\rangle&\text{if } x_{n}=L\,,
\end{cases} \label{eq:KL}
\end{align}
with the constraint
\begin{equation}
C_1^{(0)}=0 \label{eq:C1null}\,.
\end{equation}
We introduce also the more compact form $\Lambda_1=\Lambda_1^{(0)}$
since this value will appear in the energy.
The constraints \eqref{eq:C1null} and \eqref{eq:KL} lead to the
following trivial identifications with the parameters introduced in
section \ref{sec:boundarynotations}:
\begin{align}
u_1 &= e^s / c_\epsilon(\alpha,\gamma)\,, &\Lambda_1 &=
\lambda_\epsilon(\alpha,\gamma) \,,\\
(p/q)^{L-1} u_{n+1} &= c_{\epsilon'}(\beta,\delta)\,, &\Lambda_L
&= \lambda_{\epsilon'}(\beta,\delta) \,,\\
(p/q)^{L-1} v_n &= c^*_{\epsilon''}(\beta,\delta)\,,
&\widetilde{\Lambda}_L &= \lambda^*_{\epsilon''}(\beta,\delta)+q\,,
\end{align}
where $\epsilon$, $\epsilon'$, $\epsilon'' \in \{+,-\}$ are
arbitrary. The compatibility relation of the previous equalities with
the recursion relations \eqref{eq:recursion:ui:vi} lead to four
possible solutions summarized in Table~\ref{tab:cons1}. They
correspond to the different choices of the signs $\epsilon$,
$\epsilon'$ in the relation:
\begin{equation}
\label{eq:constraint1}
c_\epsilon(\alpha,\gamma) c_{\epsilon'}(\beta,\delta) = e^s
\left(\frac{p}{q}\right)^{L-1-n}\,.
\end{equation}
The third sign $\epsilon''$ does not appear in this condition, nor in
the energy, nor in the Bethe equations: it just corresponds to the
full
reflection of the excitations on the right boundary\footnote{Of
course,
we could have
also chosen a full reflection on the left boundary instead.}.
\begin{table}[h b t]
\begin{centering}
\begin{tabular}{|c | c || c | c || c |}
\hline
$\Lambda_1$ & $\Lambda_L$ & $c_\epsilon(\alpha,\gamma)$
&$c_{\epsilon'}(\beta,\delta)$
& Constraints
\\
\hline
$0$ & $0$ & $c_+(\alpha,\gamma) $ & $
c_+(\beta,\delta)$ &
$
\frac{\alpha\beta}{\gamma\delta}e^s\left(\frac{p}{q}\right)^{L-1-n}=1$\\
\hline
$-\alpha-\gamma$ & $-\beta-\delta$& $c_-(\alpha,\gamma) $ &
$ c_-(\beta,\delta)$
& $ e^s\left(\frac{p}{q}\right)^{L-1-n} = 1 $ \\
\hline
$-\alpha-\gamma$ & $0$ & $ c_-(\alpha,\gamma)$ & $
c_+(\beta,\delta)$
& $ -\frac{\beta}{\delta}e^s\left(\frac{p}{q}\right)^{L-1-n} =
1$ \\
\hline
$0$ & $-\beta-\delta$ & $ c_+(\alpha,\gamma)$ & $
c_-(\beta,\delta)$
& $ - \frac{\alpha}{\gamma}e^s\left(\frac{p}{q}\right)^{L-1-n} =
1$ \\
\hline
\end{tabular}
\caption{Different possible values for the parameters and the
constraints imposed by \eqref{eq:constraint1}.
\label{tab:cons1}}
\end{centering}
\end{table}
We then obtain the following values of the coefficients:
\begin{align}
C_1^{(m)} &= \frac{u_2}{v_1-u_2} P_{\alpha,\gamma}(e^s/u_{m+1}) \,,\\
D_1^{(m-1)} &= -\frac{v_1}{v_1-u_2}P_{\alpha,\gamma}^*(e^s/v_{m})\,,\\
\Lambda_1^{(m)} &=
\frac{(q/p)^m(\Lambda_1+\alpha)v_1-(p/q)^m(\Lambda_1+\gamma)u_2
+\gamma u_2-\alpha v_1}{v_1-u_2}\,, \\
\widetilde{\Lambda}_1^{(m-1)} &= \frac{(p/q)^m (\Lambda_1+\gamma)u_2
- (q/p)^m v_1 (\Lambda_1+\alpha) - \gamma v_1 +
(\alpha+q-p)u_2}{v_1-u_2}\,.
\end{align}
We also remind that for ASEP probabilistic models, all parameters
have to be
positive, so that only the two first lines of table \ref{tab:cons1}
have to be
considered in this case. These two constraints can be recasted into a
single one
\begin{equation}
\label{eq:firstcond}
\Big(\frac{\alpha\beta}{\gamma\delta}e^s
-\left(\frac{q}{p}\right)^{L-1-n}\Big)
\Big(e^s-\left(\frac{q}{p}\right)^{L-1+n}\Big)=0\,.
\end{equation}
Then, using the correspondence (\ref{eq:asepXXZ}), it takes the form
for XXZ model for even $L$:
\begin{eqnarray}
&&\cos(\theta_{1}-\theta_{2})=\cos(\omega_{+}+\omega_{-}-m\,\eta)\,,
\end{eqnarray}
that is just the constraint given in \cite{nepo,Cao,dGE} for some integer $m$.
Doing the same with the last two constraints, we get
\begin{equation}
\cos(\theta_{1}-\theta_{2})=\cos(\omega_{+}-\omega_{-}-n\,\eta)\,.
\end{equation}
\subsection{Second choice}
\label{sec:second}
Using the same technique as in the previous subsection, we introduce
a basis of states that is suitable to the diagonalization of the
alternative representation \eqref{eq:secondrepres} of the matrix $W$.
The vector
$\ket{V}_i$ has the same definition as before but we introduce the
family of vectors
\begin{equation}
\label{eq:def:omegastar}
\ket{\omega^*(u)}_i = \begin{pmatrix} 1 \\ u \end{pmatrix}\,.
\end{equation}
We define similarly the product state $\ket{\Omega^*(u)}_i^j$ as in
\eqref{eq:def:Omega} by replacing $\ket{\omega(u)}$ by
$\ket{\omega^*(u)}$ and we introduce the new state with the new
excitations in ordered positions $1\leq y_{m+1} < \ldots < y_n \leq
n$:
\begin{equation}
\label{eq:def:state:star}
\ket{y_{m+1},\ldots,y_n}^* =
\left(\sqrt{\frac{p}{q}}\right)^{(y_{m+1}-1)+\ldots+(y_n-1)}
\ket{\Omega^*(u_{m+1}^*)}_1^{y_{m+1}-1}\ket{\omega^*(v_{m+1}^*)}_{y_{m+1}}
\ldots \ket{\omega^*(v_n^*)}_{y_n}\ket{\Omega^*(u_{n+1}^*)}_{y_n+1}^L
\end{equation}
where the new coefficients $u_i^*$ and $v_i^*$ satisfy the recursion
relation
\begin{equation}
\label{eq:recursion:ui:vi:star}
u_{m+1}^* = (p/q) u_m^*\,, \quad v_{m+1}^* = (p/q) v_m^*\,.
\end{equation}
These vectors have been chosen such that we get for the Hamiltonian
bulk part
\begin{equation}
\begin{split}
w_{\beta,\beta+1}^*|y_{m+1},..,y_n\rangle^*=
\begin{cases}
0
&\text{if } \beta,\beta+1\neq y_{m+1},\dots,y_n\,;\\
\sqrt{pq}|..y_j-1..\rangle^*
-p|..y_j..\rangle^* -|..\overline{y_j}..\rangle^*
&\text{if } \beta+1=y_j\ ;\,\beta\neq y_{j-1}\,;\\
\sqrt{pq}|..y_j+1..\rangle^*
-q|..,y_j,..\rangle^* +|..,\overline{y_j},..\rangle^*
&\text{if } \beta=y_j\ ;\,\beta+1\neq y_{j+1}\,;\\
|..,\overline \beta,..\rangle^*
-|..,\overline{\beta+1},..\rangle^*
&\text{if } \beta=y_j\ ;\,\beta+1=y_{j+1}\,.
\end{cases}
\end{split}
\end{equation}
As for the previous choice, there are additional states $\ket{V}_1$
and $\ket{V}_L$ that survive the telescopic sum $\sum_{i=1}^{L-1}
w_{i,i+1}^*$ and that must be absorbed by the boundary operators. We
impose the dynamics:
\begin{align}
\label{eq:K1-2}
&\widehat{K}_1^*|x_{m+1},\ldots\rangle^* =
\begin{cases}
\Lambda_1^{*,(m)}|x_{m+1},\ldots\rangle^*
+C_1^{*,(m)}|1,x_{m+1},\ldots\rangle^*
&\text{if } x_{m+1}>1\,,\\
\widetilde{\Lambda}_1^{*,(m)}|1,x_{m+2}\dots\rangle^*
+D_1^{*,(m)}
|x_{m+2},\ldots\rangle^*-|\overline{1},x_{m+2},\dots\rangle^*
&\text{if } x_{m+1}=1\,,
\end{cases} \\
&K_L^*|x_{m+1},\dots,x_n\rangle^* =
\begin{cases}
\Lambda_L^*|x_{m+1},\dots,x_n\rangle^* &\text{if } x_{n}<L\,,\\
\widetilde\Lambda_L^*|x_{m+1},\dots,x_{n-1}, L\rangle^*
+|x_{m+1},\dots,x_{n-1},\overline
L\rangle^* &\text{if } x_{n}=L \,,
\end{cases}\label{eq:KL-2}
\end{align}
with the additional closure constraint
\begin{equation}
\label{eq:closurestar}
C_1^{*,(0)}=0\,.
\end{equation}
We introduce again the more compact form
$\Lambda_1^*=\Lambda_1^{*,(0)}$
since it is this value that will appear in the energy. {{From}} the
constraints \eqref{eq:KL-2} and \eqref{eq:closurestar} and the
notations introduced in section \ref{sec:boundarynotations}, we
identify directly the value of most parameters:
\begin{align}
u_1^* &= e^s/c^*_\epsilon(\alpha,\gamma) \,, &\Lambda_1^* &=
\lambda^*_\epsilon(\alpha,\gamma) \,,\\
u_{n+1}^* &= c^*_{\epsilon'}(\beta,\delta) , & \Lambda_L^* &=
\lambda^*_{\epsilon'}(\beta,\delta)+p+q \\
v_n^* &= c_{\epsilon''}(\beta,\delta) \,, &
\widetilde{\Lambda}_L^*&= p+\lambda_{\epsilon''}(\beta,\delta)
\,.
\end{align}
As in the previous subsection, these values are compatible with the
recursion relation \eqref{eq:recursion:ui:vi:star} if and only if the
parameters of the model satisfy the relation
\begin{equation}
\label{eq:cont-2}
c^*_\epsilon(\alpha,\gamma)c^*_{\epsilon'}(\beta,\delta) =
e^s\left(\frac{p}{q}\right)^{n}\,,
\end{equation}
for some signs $\epsilon$ and $\epsilon'$ and some integer $n$, which
then fixes the number of excitations. Similar exceptional points
already appeared in the literature in the context of the matrix
ansatz described in section \ref{sec:matrixansatz}. Table
\ref{tab:cons2} summarizes the different possibilities.
\begin{table}[h b t]
\begin{centering}
\begin{tabular}{|c || c | c || c |}
\hline
$\Lambda_1^*+\Lambda_L^*$ & $\epsilon$ &$\epsilon'$
& Constraints
\\
\hline
$ \frac{-\alpha-\gamma-\beta-\delta
+\sqrt{(p-q+\gamma-\alpha)^2+4\gamma\alpha} +
\sqrt{(p-q+\delta-\beta)^2+4\beta\delta}}{2}$ & $+$ &
$+$ &
$ c^*_+(\alpha,\gamma)c^*_+(\beta,\delta)
=e^s\left(\frac{p}{q}\right)^{n}$
\\
\hline
$ \frac{-\alpha-\gamma-\beta-\delta
-\sqrt{(p-q+\gamma-\alpha)^2+4\gamma\alpha} -
\sqrt{(p-q+\delta-\beta)^2+4\beta\delta}}{2}$ & $-$ &
$-$ &
$ c^*_-(\alpha,\gamma)c^*_-(\beta,\delta)
=e^s\left(\frac{p}{q}\right)^{n}$\\
\hline
$ \frac{-\alpha-\gamma-\beta-\delta
+\sqrt{(p-q+\gamma-\alpha)^2+4\gamma\alpha} -
\sqrt{(p-q+\delta-\beta)^2+4\beta\delta}}{2}$ & $+$ &
$-$ &
$c^*_+(\alpha,\gamma)c^*_-(\beta,\delta)
=e^s\left(\frac{p}{q}\right)^{n}$\\
\hline
$\frac{-\alpha-\gamma-\beta-\delta
-\sqrt{(p-q+\gamma-\alpha)^2+4\gamma\alpha} +
\sqrt{(p-q+\delta-\beta)^2+4\beta\delta}}{2}$ & $-$ &
$+$ &
$c^*_-(\alpha,\gamma)c^*_+(\beta,\delta)
=e^s\left(\frac{p}{q}\right)^{n}$\\
\hline
\end{tabular}
\caption{Different possible values for the parameters and the
constraints imposed by \eqref{eq:constraint1}.
\label{tab:cons2}}
\end{centering}
\end{table}
The values of all the coefficients are then given by:
\begin{align}
C_1^{*,(m)} &=
u^*_{2}\frac{P^*_{\alpha,\gamma}(e^s/u^*_{m+1})}{v^*_{1}-u^*_{2}}
\,,\\
D_1^{*,(m-1)} &=
-v^*_{1}\frac{P_{\alpha,\gamma}(e^s/v^*_{m})}{v^*_{1}-u^*_{2}}
\,,\\
\Lambda_1^{*,(m)} &=
\frac{(p/q)^m v_1^*
(\Lambda_1^*+\alpha+q)-(q/p)^mu_2^*(\Lambda_1^*+\gamma+p) +
(\gamma+p)u_2^* - (\alpha+q) v_1^*}{v_1^* -u_2^*}
\,, \\
\widetilde{\Lambda}_1^{*,(m-1)} &=
\frac{(q/p)^m u_2^� (\Lambda_1^*+\gamma+p) - (p/q)^m
(\Lambda_1^*+\alpha+q)v_1^* + (\alpha+p)u_2^* -
(\gamma+p)v_1^*}{v_1^*-u_2^*}
\,.
\end{align}
In the language of the XXZ spin chain, we should introduce the
additional angles $\Omega_\pm$ defined by
\begin{equation}
2 \cos \Omega_\pm = \cos \delta_\pm - \cos \omega_\pm \,,
\end{equation}
and the condition \eqref{eq:cont-2} takes the following simple form
for even $L$:
\begin{equation}
\cos(\theta_1-\theta_2) = \cos( \Omega_- \pm \Omega_+ -\eta m)\,,
\end{equation}
for some even integer $m$ between $-L$ and $L$. Up to our knowledge,
this condition is new. The associated Bethe equations are presented
in section \ref{subsec:betheeqs:secondpoints}.
\section{Coordinate Bethe Ansatz\label{sec:CBA}}
\subsection{The first set of specific points}
We are now in position to propose an ansatz for the eigenfunction
\begin{equation}
\label{eq:ansatz}
\Phi_n=\sum_{m=0}^n\ \sum_{x_{m+1}<\dots<x_n}\ \sum_{g\in G_m}\
A_g^{(m)}\ e^{i\boldsymbol{k}^{(m)}_g.\boldsymbol{x}^{(m)}}\
|x_{m+1},\dots,x_n\rangle\,,
\end{equation}
where $G_m$ is a full set of representatives of the following coset
$BC_n/BC_m$ (see appendix \ref{sec:BC}),
the vectors $|x_{m+1},\dots,x_n\rangle$ are either given by
(\ref{eq:def:state}) or by (\ref{eq:def:state:star})
and we introduce the notation $\boldsymbol{k}^{(m)}$ for the
following truncated vector
\begin{equation}
\boldsymbol{k}^{(m)}=(k_{m+1},\dots,k_n)\;.
\end{equation}
For this definition to be consistent, the coefficients $A_{g}^{(m)}$
do not have to depend on the choice of the representative i.e.
\begin{equation}
A_{gh}^{(m)}=A_{g}^{(m)}\mb{for any $h\in BC_m$.}
\end{equation}
The coefficients $A_{g}^{(m)}$ are complex numbers to be determined
such that $\Phi_n$ is
an eigenfunction of $H$ i.e. such that the following equation holds
\begin{equation}\label{eq:sch}
H \Phi_n = E \Phi_n\;.
\end{equation}
Due to the results of section \ref{sec:basis}, $H\cB_n\subset \cB_n$.
Then, we project equation (\ref{eq:sch}) on the different independent
vectors of $\cB_n$ to get constraints on the coefficients
$A^{(m)}_{g}$.
This projection depends on the two choices done for the vectors
$|x_{m+1},\dots,x_n\rangle$.
However, we chose the coefficients in such a way that one can be
deduced from the other easily by adding a star to most quantities.
Therefore,
we write only the projections for the choice of section
\ref{sec:first} and only the ones
leading to independent relations (one can check that the remaining
ones do not lead to new relations).
\paragraph{On $\boldsymbol{|x_{1},\dots,x_n\rangle}$ for
$\boldsymbol{(x_{1},\dots,x_n)}$ generic} (i.e. $1<x_1$, $x_n<L$ and
$1+x_j<x_{j+1}$)\\
As in the usual coordinate Bethe ansatz \cite{bethe}, this projection
provides the energy:
\begin{equation}
E=\Lambda_1+\Lambda_L+
\sum_{j=1}^n\lambda(e^{ik_j})
\mb{where}
\lambda(x)=\sqrt{pq}\big(x+\frac{1}{x}\big)-p-q
=\frac{\sqrt{pq}}{x}\left(x-\sqrt{\frac pq}\right)\left(x-\sqrt{\frac
qp}\right)\,.
\label{def:lambdax}
\end{equation}
Let us remark that, up to the boundary terms $\Lambda_1$ and
$\Lambda_L$, the energy takes
the same form as in the periodic case.
\paragraph{On $\boldsymbol{|x_{1},\dots,x_n\rangle}$ with
$\boldsymbol{x_{j+1}=1+x_{j}}$} (and
$x_1,\dots,x_{j-1},x_{j+2},\dots,x_n$ generic)\\
This projection is also a usual one and provides the scattering
matrix between excitations. It is given by a relation between
$A^{(0)}_{g}$ and $A^{(0)}_{g\sigma_j}$ where $\sigma_j$ is the
permutation of $j$ and $j+1$ (see appendix \ref{sec:BC}).
Namely, we get
\begin{equation}
\label{eq:S}
A^{(0)}_{g\sigma_j}
=S\left(e^{ik_{gj}},e^{ik_{g(j+1)}}\right)~A^{(0)}_{g}\,,
\end{equation}
with
\begin{equation}
S(u,v)=-\frac{a(u,v)}
{a(v,u)}
\mb{where}
a(x,y)=\frac{i}{xy-1}\left(\left(\sqrt{\frac{q}{p}}
+\sqrt{\frac{p}{q}}\right)y-xy-1\right)\,.
\label{def:alphax}
\end{equation}
The normalisation chosen for the function $a(x,y)$ is for
further simplifications.
As expected, this relation is similar to the periodic case since the
boundaries
are not involved in this process.
\paragraph{On $\boldsymbol{|1,x_{m+1}\dots,x_n\rangle}$}
($x_{m+1},\dots,x_n$ generic and $m\geq1$)\\
This relation is a new one and we must take into account that the
left
boundary can create one particle
at the site $1$. We finally get, for any $g\in G_m$,
\begin{eqnarray}\label{eq:1g}
&&\Big(\widetilde{\Lambda}_1^{(m-1)}-\Lambda_1-p
+\sum_{j=1}^{m}\big(p+q-\sqrt{pq}(e^{ik_{gj}}+e^{-ik_{gj}})\big)\Big)
\sum_{h\in H_m} A_{gh}^{(m-1)}e^{ik_{ghm}}
\nonumber \\
&&+\sqrt{pq}\sum_{h\in H_m}A_{gh}^{(m-1)}e^{2ik_{ghm}}
+C_1^{(m)}A_g^{(m)}=0\,,
\end{eqnarray}
where $H_m=BC_m/BC_{m-1}$.
To obtain this relation, we have used the following property
\begin{equation}
\sum_{g\in G_{m-1}}A_g
e^{ik_{g(m)}}e^{i\boldsymbol{k}_{g}^{(m)}\boldsymbol{x}^{(m)}}
=\sum_{g\in G_{m}}e^{i\boldsymbol{k}_{g}^{(m)}\boldsymbol{x}^{(m)}}
\sum_{h\in H_m}A_{gh}e^{ik_{gh(m)}}\;.
\end{equation}
Let us stress that equation (\ref{eq:1g}) is invariant by the choice
of representative of $G_m$.
\paragraph{On $\boldsymbol{|x_{m+1}\dots,x_n\rangle}$}
($x_{m+1},\dots,x_n$ generic and $m\geq 1$)\\
This projection provides a second relation between the coefficient
from the level $m-1$ and $m$
since we must take into account that the left boundary can destroy a
particle present on the site 1.
We obtain the following constraint, for any $g\in G_m$,
\begin{equation}\label{eq:2g}
D_1^{(m-1)}\sum_{h\in H_m}A_{gh}^{(m-1)}e^{ik_{gh(m)}}
+(\Lambda_1^{(m)}-\Lambda_1+
\sum_{j=1}^{m}(p+q-\sqrt{pq}(e^{ik_{gj}}+e^{-ik_{gj}})))A_g^{(m)}=0
\,.
\end{equation}
{From} (\ref{eq:1g}) and (\ref{eq:2g}), we may express all the
$A_g^{(m)}$ ($m\geq 1$) in terms of $A_g^{(0)}$
thanks to the following recursive relations defined for $m\geq 1$
\begin{equation}
\label{eq:recuT}
A_g^{(m)}=T^{(m)}(e^{ik_{g1}},\dots,e^{ik_{gm}})A_g^{(m-1)}\,,
\end{equation}
with the following definitions:
\begin{align}
T^{(m)}(x_{1},\dots,x_{m}) &=\frac{D_1^{(m-1)}}{p_1(x_{m})
V_1(x_{m})}
\frac{x_m^2-1}{\prod_{j=1}^{m-1}a(x_{m},x_j)a(x_{j},1/x_m)} \,,\\
V_1(Z) &=\lambda(Z)
+ (\Lambda_1+\gamma) \Big(1-\frac{1}{Z}\sqrt{\frac{p}{q}}\Big)
+ (\Lambda_1+\alpha)\Big(1-\frac{1}{Z}\sqrt{\frac{q}{p}}\Big)\,,
\label{def:V1x} \\
p_1(Z) &=Z+ r \label{def:p1x}\,, \\
r &= \frac{1}{\sqrt{pq}} \frac{pu_2-q v_1}{v_1-u_2} =
\frac{1}{\sqrt{pq}} \frac{pu_{m+1}-q v_m}{v_m-u_{m+1}} \,,
\label{def:coeff:r}
\end{align}
where $r$ describes the difference between $\ket{\omega(v_k)}$ and
the
vacuum states $\ket{\omega(u_k)}$ and $\ket{\omega(u_{k+1})}$. It is
independent from $k$.
The proof that (\ref{eq:recuT}) is a solution of
both equations (\ref{eq:1g}) and (\ref{eq:2g}) is postponed to
appendix \ref{sec:B} and relies on a residue computation.
The integrability of the model plays a role at this place, since there
are a priori too many constraints but not all of them are independent.
Let us emphasize that this recurrence relation (\ref{eq:recuT}) is
the main result of this article.
Indeed, firstly, it proves that the ansatz
(\ref{eq:ansatz}) and, in particular, the choice of cosets used to
define it, is the appropriate choice. Secondly, it gives a very
simple way to construct eigenfunctions, and we believe that it may be
used for further computations.
A consequence of (\ref{eq:recuT}) (for $m=1$) and
$A_{gr_1}^{(1)}=A_g^{(1)}$, is
\begin{equation}
A_{gr_1}^{(0)}
=\frac{T^{(1)}(e^{ik_{g1}})}{T^{(1)}(e^{-ik_{g1}})}A_g^{(0)}\,.
\end{equation}
This relation, together with (\ref{eq:S}), allow us to express $A_g^{(0)}$ for
any $g\in BC_n$ in terms of $A_{1}^{(0)}$
(where the subscript $1$ stands for the unit of $BC_n$ group).
Finally, using recursively (\ref{eq:recuT}), we can express all the
coefficients $A_g^{(m)}$ in terms of only $A_1^{(0)}$. This last
coefficient is
usually chosen such that the eigenfunction $\Phi_n$ be normed.
\paragraph{On $\boldsymbol{|x_{1}\dots,x_{n-1},L\rangle}$}
($x_{1}\dots,x_{n-1}$ generic)\\
This last constraint consists in the quantization of the excitations moments
since the
system is in a finite volume. In the context of the coordinate Bethe
ansatz,
this quantization leads to the so-called Bethe equations, explicitly
given by, for $1\leq j \leq n$,
\begin{equation}
\label{eq:bethe}
\prod_{\substack{\ell=1 \\ \ell\neq j}}^n
S(e^{ik_\ell},e^{ik_j})S(e^{-ik_j},e^{ik_\ell})
=e^{2iLk_j}
\frac{V_1(e^{ik_{j}})V_L(e^{ik_{j}})}
{V_1(e^{-ik_{j}})V_L(e^{-ik_{j}})}\,,
\end{equation}
\begin{equation}
V_{L}(x) = \lambda(x)+
\left(\Lambda_{L}+\beta\right)
\Big(1-\frac 1x\sqrt{\frac qp}\Big)
+(\Lambda_{L}+\delta)\Big(1-\frac 1x\sqrt{\frac
pq}\Big)\,.
\label{def:VLx}
\end{equation}
We remind that the parameters have to obey one of the relations given
in table \ref{tab:cons1}.
\subsection{The new second set of specific points}
\label{subsec:betheeqs:secondpoints}
When condition \eqref{eq:cont-2} is satisfied, the same construction
as in the previous section works, up to easy modifications in the
equations.
We introduce a new function $V_1^*(Z)$ and a new parameter $r^*$,
which replaces $r$, defined by:
\begin{align}
\label{def:V1star}
V_1^*(Z) &= \lambda(Z) +(\Lambda_1^*+\alpha+q) \left(
1-\frac{1}{Z}\sqrt{\frac{p}{q}}\right) + (\Lambda_1^*+\gamma+p)
\left(1-\frac{1}{Z}\sqrt{\frac{q}{p}}\right) \,,\\
V_L^*(Z) &= \lambda(Z) +(\Lambda_L^*+\alpha+q) \left(
1-\frac{1}{Z}\sqrt{\frac{p}{q}}\right) + (\Lambda_L^*+\gamma+p)
\left(1-\frac{1}{Z}\sqrt{\frac{q}{p}}\right) \,,\\
r^* &= \frac{1}{\sqrt{pq}}\frac{pv_1^*-qu_2^*}{u_2^*-v_1^*}\,.
\end{align}
The same proof as the one presented in appendix \ref{sec:BC} is valid
and we obtain the Bethe equations for $j\in\{1,\ldots,n\}$:
\begin{equation}
e^{2iL k_j} \frac{V_1^*(e^{ik_j})V_L^*(e^{ik_j}) }{V_1^*(e^{-ik_j})
V_L^*(e^{-ik_j})} = \prod_{\substack{l=1 \\ l\neq j}}^n
S(e^{ik_l},e^{ik_j})S(e^{-ik_j},e^{ik_l})\,,
\end{equation}
with the parameters obeying now one of the relations described in
table \ref{tab:cons2}.
\section{Conclusions}
We discuss in this section two open problems, which need further investigation.
\subsection{Connection with the matrix ansatz}
\label{sec:matrixansatz}
For $s=0$, the matrix $W$ is stochastic (Markov transition matrix of
the exclusion process) and hence has a known ground state eigenvalue
$E=0$ with a simple left eigenvector, whose components are all equal
to $1$ (conservation of total probability). Nevertheless, the
corresponding right eigenvector is non-trivial since the matrix is
not Hermitian and describes the stationary properties of the
asymmetric exclusion process. The structure of this specific
eigenvector was elucidated first in \cite{dehp} and then studied
algebraically in more details in \cite{esr,mallicksandow}.
As already explained, the state space is $2^L$ dimensional and the
canonical basis can be indexed by the values of the occupation
$\tau_i\in\{0,1\}$ (resp. spin $s_i\in\{-1,1\}$ in the XXZ language)
on each site $i$. The matrix ansatz states that the ground state of
the ASEP with $s=0$ has components given by:
\begin{equation}
\langle \tau_1\tau_2\ldots\tau_L | \Phi \rangle = \langle\langle V_1
| \prod_{1\leq i \leq n}^{\longrightarrow}\left( \tau_i D+
(1-\tau_i)E \right) | V_2 \rangle\rangle\,,
\end{equation}
where the arrow means that the product have to be build from left to
right when the index $i$ increases. One has for example $\langle
00\ldots00|\Phi\rangle= \langle\langle V_1 | E^L | V_2
\rangle\rangle$. The non-commuting matrices $D$ and $E$ act on an
abstract auxiliary vector space $\mathcal{V}$. The vector $| V_1
\rangle\rangle$ is in this space $\mathcal{V}$, whereas the vector
$\langle\langle V_2 |$ is in its dual. $\Phi$ is an eigenvector of
$W$ for $E=0$ when $s=0$ if the two matrices $D$ et $E$ and the two
boundary vectors satisfy the commutation rules \cite{dehp}:
\begin{subequations}
\label{matrixansatz}
\begin{align}
pDE-qED &= D+E \,,
\\
\langle\langle V_1 | (\gamma D - \alpha E) &= -\langle\langle V_1 |\,,
\\
(\beta D -\delta E) | V_2 \rangle\rangle &= | V_2 \rangle\rangle\,.
\end{align}
\end{subequations} with the condition $s=0$.
These three relations allow to determine recursively all the
components of the eigenvector $\Phi$ and do not need an explicit
representation of the algebra. In particular, the matrix ansatz gives
an easy access to classical correlation functions with standard
transfer matrix techniques.
The connection with the present Bethe ansatz approach comes from the
algebraic study of the algebra generated by $D$ and $E$, as performed
in \cite{esr,mallicksandow}, from computations that arise in the
combinatorics of some so-called \emph{staircase tableaux}
\cite{corteel}, as well as from the physical interpretation
\cite{schuetz1,jafarpour,schuetz2} of the matrix ansatz in terms of
so-called "shock" product state as introduced in section
\ref{sec:basis}.
The first remark is that the matrix ansatz described here
\emph{fails} for some values of the parameters
$p,q,\alpha,\beta,\gamma,\delta$. Failure happens when the recursion
relation
on the size $L$ induced by \eqref{matrixansatz} leads to
$\langle\langle V_2 | V_1 \rangle\rangle=0$ and thus more generally
to a null vector $\Phi$, as explained in \cite{esr}. This failure
condition for a system of size $L$ is the existence of an integer
$n\in\{0,1,\ldots, L-1\}$ such that
\begin{equation}
\frac{\alpha\beta}{\gamma\delta} \left(\frac{p}{q}\right)^n = 1\,.
\end{equation}
This relation is precisely one of the two cases of
\eqref{eq:firstcond} when $s=0$, for which the present Bethe ansatz
approach works.
The second remark is that the operators $D$ and $E$ do not have
generically finite dimensional representations. Detailed studies
shows that such finite-dimensional representations may exist at some
specific points \cite{mallicksandow,esr}. It appears that an
$n$-dimensional representation exists if one among the four cases of
the condition \eqref{eq:cont-2} is satisfied for some integer $n$ and
plus/minus signs, i.e.
\begin{equation}
\label{eq:ma:finiterep}
c^*_+(\alpha,\gamma) c^*_+(\beta,\delta) =
\left(\frac{p}{q}\right)^{n}.
\end{equation}
In \cite{mallicksandow}, the finite representation is used to study
how the ground state evolves through the phase transition that occurs
in the open ASEP along the manifold \eqref{eq:ma:finiterep}: the
present paper gives also the excited states on the same manifold and
the Bethe ansatz equations we obtain in this case now allow one to
study how the gap behaves near this phase transition. Once again,
atypical results appear in both the Bethe ansatz approach and the
matrix ansatz approach for the same specific parameters.
A third remark relies on the observation that the present
construction does not allow to express the ground state described by
the matrix ansatz in terms of a coordinate Bethe ansatz, precisely
because one fails when the second works. This mismatch leads us to
think that the matrix ansatz state may be used as a new reference
state to build the missing eigenvectors if one could manage to add
excitations on it. However, up to our knowledge, no deep
understanding of the relation between these two approaches exists and
has been exploited.
A last remark deals with the combinatorics work \cite{corteel}.
Besides the fact that quantities such as \eqref{eq:firstcond} and
\eqref{eq:cont-2} appear in numerators or denominators in their
computations and thus either simplifies or invalidates the matrix
ansatz, their rewriting of the matrix ansatz in terms of
\emph{staircase tableaux} sheds a new light on the precise structure
of this ansatz, and puts it in a form that is more suitable to
comparison with Bethe ansatz because of the relation of these
tableaux with permutation tableaux. In particular, the number of
so-called \emph{staircase tableaux} is $n!\,4^n$, whereas the cardinal
of $BC_n$ is precisely $n!\,2^n$. Understanding both the similarities
and the mismatch may lead to a better understanding of the Bethe
ansatz.
\subsection{Completeness of the spectrum}
\label{subsec:completeness}
As usual for Bethe Ansatz methods, the delicate point to check is the completeness of the spectrum. Indeed, in many cases, completeness is not proved but expected to be true: numerics for small size systems \cite{wup}, enumeration of the number of roots in the thermodynamic limit \cite{KIRI}.
In the present case, numerical checks of the completeness for small sizes have been considered in \cite{completeness}: the full spectrum is shown to be described by two sets of Bethe equations. Our approach for the right eigenvectors gives only one set of Bethe equations for each specific point and thus only one part of the spectrum and one part of the right eigenvectors. The same construction can be done for left eigenvectors (or right eigenvectors of the adjoint). In this case, one obtains the other part of the spectrum and one part of the left eigenvectors, as discussed in \cite{simon09}.
Let us stress that the operators we are considering may not be hermitian and thus left and right eigenvectors may not be simply related. Although the full spectrum is known, it would be interesting to build the full set of right and left eigenvectors.
\section*{Acknowledgements}
This work was partially supported by the PEPS-PTI
grant \textit{Applications des Mod\`eles Int\'egrables}. We also
thank Livia Ferro for a careful reading of the manuscript.
|
{
"timestamp": "2010-11-22T02:01:08",
"yymm": "1009",
"arxiv_id": "1009.4119",
"language": "en",
"url": "https://arxiv.org/abs/1009.4119"
}
|
\section*{Introduction}
The optimal stopping time problem has been wildely studied in case of {\em reward} given by a right continuous left limited (RCLL) positive adapted process $(\phi_t)$ defined on $[0, T]$ (see for example Shiryaev (1978), El Karoui (1981), Karatzas and Shreve (1998) or Peskir and Shiryaev (2006)).
If $T>0$ is the fixed time horizon and if $T_0$ denotes the set of stopping times $\theta$ smaller than $T$, the problem consists in computing the maximal reward given by
$$v(0)= \sup \{\,E[ \phi_\tau],\; \tau \in T_0\,\}\,,$$
in finding conditions for the existence of an optimal stopping time and giving a method to compute these optimal stopping times.
Classicaly, the {\em value function} at time $S\in T_0$ is defined by
$v(S)={\rm{ess}\;}\displaystyle{\sup}\{\, E[\phi_\tau\, |\,{\cal F}_S], \tau\in T_0\,{\rm and}\, \tau \geq S\,{\rm a.s.}\,\}$. The value function is given by a family of random variable $\{\,v(S), S\in T_0\,\}$. By using the right continuity of the reward $(\phi_t)$, it can be shown that there exists a RCLL adapted process $(v_t)$ which {\em aggregates} the family of random variable $\{\,v(S), S\in T_0\,\}$ that is such that $v_S=v(S)$ a.s. for each $S\in T_0$. This process is the {\em Snell envelope} of $(\phi_t)$, that is the smallest supermartingale process that dominates $\phi$.
Moreover, when the reward $(\phi_t)$ is continuous, the stopping time defined trajectorially by
\begin{equation*}\label{optun}
\overline \theta (S)=\inf\{\,t\geq S, \;v_t=\phi_t\,\}\,
\end{equation*}
is optimal.
Recall that El Karoui (1981) has introduced the more general notion of a reward given by a {\em family} $\{\,\phi(\theta), \theta\in T_0\,\}$ {\em of positive random variables} which satisfies some compatibility properties.
In the recent paper of Kobylanski et al. (2009), this notion appears to be the appropriate one to study
the $d$-multiple optimal stopping time problem. Moreover, in this work, Kobylanski et al. (2009) show that under quite weak assumptions (right and left continuity in expectation along stopping times of the reward), the minimal optimal stopping time for the value function at time $S$
\begin{equation}\label{casp}
v(S)= {\rm{ess}\;}\displaystyle{\sup}\{\,E[\phi(\theta)\, |\,{\cal F}_S], \;\theta \in T_0\,{\rm and}\, \theta \geq S\,{\rm a.s.} \,\}\,,
\end{equation}
is given by
\begin{equation}\label{te}\theta_{*}(S) :=\essinf \{\,\theta \in T_0,\,\, \theta \geq S\,{\rm a.s.} \,{\rm and} \, u(\theta) =\phi(\theta) \,\,\mbox {\rm a.s.} \,\}.
\end{equation}
Let us emphasize that the minimal optimal stopping time $\theta_*(S)$ is no longer defined as a hitting time of processes but as an essential infimum of random variables.
Also, this result allows to deal with the optimal stopping problem only in terms of admissible families of random variables. It presents the advantage that it does no longer require aggregation results. Indeed, the existence of optimal stopping times as well as the characterization of the minimal one can be done by using only the value function family and the reward family and no longer the aggregated processes. We stress on that in the multiple case, it avoids long and heavy proofs, due to some difficult aggregation problems. It allows to solve the problem under weaker assumptions than before in the unified framework of families of random variables.
In the present work, we consider the case of a one optimal stopping time problem with a discontinuous reward. More precisely, the reward is given by a family of random variables which satisfies some compatibility conditions and which is supposed to be upper-semicontinuous over stopping times in expectation. Note that these assumptions in terms of smoothness of the reward are optimal in order to ensure the existence of an optimal stopping time. Indeed, in the deterministic case, the upper-semicontinuity is the minimal assumption on a function
$\phi: [0,T] \rightarrow \mathbb{R}$; $t \mapsto \phi(t)$ which ensures that the supremum of $\phi$ is attained on any closed subset of $[0,T]$.
Under these assumptions, we show the existence of an optimal stopping time for the value function (\ref{casp}) which is given by the essential infimum $\theta_{*}(S)$ defined by (\ref{te}).
We stress on that the mathematical tools which are used in this proof are not sophisticated tools, as those of the general theory of processes, but just the use of well chosen supermartingale systems and an appropriate construction of penalized stopping times. We also show that $\theta_{*}(S)$ is the minimal optimal stopping time. Also, the stopping time given by
\begin{equation*}
\check{ \theta}(S) ={\rm{ess}\;}\displaystyle{\sup}\{\;\theta \in T_0,\,\, \theta \geq S\,\,{\rm a.s.}\,\,{\rm and}\,\, E[v(\theta)]= E[v(S)] \,\},
\end{equation*}
is proven to be the maximal optimal stopping time.
Note that an important tool in this work is the use of the family of random variables defined by
$v^+(S)={\rm{ess}\;}\displaystyle{\sup}\{\, E[\phi(\theta)\, |\,{\cal F}_S]$, $\theta \in T_0$ and $\theta > S$ a.s.\,for each stopping time $S$. Some properties and links between $v$, $v^+$ and $\phi$ are studied in this paper.
These new results allow to solve the case of a reward process $(\phi_t)$ which can be much less regular than in the previous works. For instance, this allows to solve the case of a reward process given by $\phi_t = f(X_t)$, where $f$ is upper-semicontinuous and $(X_t)$ is a RCLL process supposed to be left continuous along stopping times. This opens a way to a large range of applications, for instance in finance.
\vspace{0,2cm}
The paper is organised as follows. In section 1, we give some first properties on
$v$ and $v^+$. In particular, we have $v(S)= \phi(S)\vee v^+(S)$ a.s. for each $S$
$\in$ $T_0$ and the family $\{\, v^+(S), S \in T_0\,\}$ is right continuous along stopping times. In section 2, we show the existence of an optimal stopping time under some minimal assumptions. We begin by constructing $\varepsilon$-optimal stopping times which are appropriate to this case. Then,
these $\varepsilon$-optimal stopping times are shown to tend to $\theta_{*}(S)$
as $\varepsilon$ tends to $0$. Moreover, $\theta_{*}(S)$ is proven to be an optimal stopping time for $v(S)$ and even the minimal one. At last, the stopping time $\check{ \theta}(S)$ is proven to be the maximal optimal stopping time.
In section 3, we give some strict supermartingale conditions on $v$ which ensure the equality between $v$ and $\phi$ (locally). Secondly, we give some conditions on $v$ and $v^+$ which ensure the equality between
$v$ and $v^+$ (for some stopping times which are specified). At last, we give a few applications of the classical Doob-Meyer decomposition in particular when the reward is right continuous in expectation, which allows to use aggregation results. In section 4, we give some examples where the reward is given by an upper semicontinuous function of a RCLL adapted process $(X_t)$ which is left continuous in expectation. We stress on that, except in the last part, all the properties established in this work do not require any result of the general theory of processes.
\vspace{0,5cm}
Let ${\mathbb F}=(\Omega, {\cal F}, ({\cal F}_t)_{\; { 0\leq t\leq T}},P)$ be a probability space equipped with a filtration
$({\cal F}_t)_{\; { 0\leq t\leq T}}$ satisfying the usual conditions of right continuity and augmentation by the null sets ${\cal F}= {\cal F}_T$. We suppose that ${\cal F}_0$ contains only sets of probability $0$ or $1$. The time horizon is a fixed constant $T\in ]0,\infty[$.
We denote by $T_{0}$ the collection of stopping times of
${\mathbb{F} }$ with values in $[0 , T]$. More generally, for any
stopping times $S$, we denote by $T_{S}$ (resp. $T_{S^+}$) the class of stopping times
$\theta\in T_0$ with $\theta\geq S$ a.s.\, (resp. $\theta>S $ a.s. on $\{S<T\}$ and $\theta=T$ a.s. on $\{S=T\}$).
We also define $T_{[S, S^{'}]}$ the set of $\theta\in T_0$ with $S \leq \theta \leq S^{'}$ a.s. and $T_{]S, S^{'}]}$ the set of $\theta\in T_0$ with $S < \theta \leq S^{'}$ a.s..
Similarly, the set $T_{]S, S^{'}]}$ on $A$ will denote the set of $\theta\in T_0$ with $S < \theta \leq S^{'}$ a.s.\, on $A$.
We use the following notation: For $t\in\mathbb{R}$ and for real valued random variables $X$ and $X_n$, $n\in$ $\mathbb{N}$, ``$X_n\uparrow X$'' stands for ``the sequence $(X_n)$ is nondecreasing and converges to $X$ a.s.''.
|
{
"timestamp": "2010-09-28T02:03:39",
"yymm": "1009",
"arxiv_id": "1009.3862",
"language": "en",
"url": "https://arxiv.org/abs/1009.3862"
}
|
\section{Introduction}
The derivation of effective interactions from NN dynamics has been a
major task in Nuclear Physics ever since the pioneering works of
Moshinsky~\cite{Moshinsky195819} and Skyrme~\cite{Skyrme:1959zz}. The
use of those effective potentials, referred to as Skyrme forces, in
mean field calculations can hardly be exaggerated due to the enormous
simplifications that are implied as compared to the original many-body
problem~\cite{Vautherin:1971aw,Negele:1972zp,Chabanat:1997qh,Bender:2003jk}.
Similar ideas advanced by Moszkowski and
Scott~\cite{1960AnPhy..11...65M} have become rather useful in Shell
model calculations~\cite{Dean:2004ck,Coraggio:2008in}. The Skyrme
(pseudo)potential is usually written in coordinate space and
contains delta functions and its derivatives~\cite{Skyrme:1959zz}. In
momentum space it corresponds to a power expansion in the CM momenta
$({\bf p'}$ and $ {\bf p})$ corresponding to the initial and final
state respectively. To second order in momenta the potential reads
\begin{eqnarray}
&& V ({\bf
p}',{\bf p})
= \int d^3 x e^{-i {\bf x}\cdot ({\bf p'}-{\bf p})} V({\bf x} )
\nonumber \\ &=& t_0 (1 + x_0 P_\sigma ) + \frac{t_1}2(1 + x_1
P_\sigma ) ({\bf p}'^2 + {\bf p}^2) \nonumber \\ &+&
t_2 (1 + x_2
P_\sigma ) {\bf p}' \cdot {\bf p} + 2 i W_0 {\bf S} \cdot({\bf p}'
\wedge {\bf p}) \nonumber \\ &+&
\frac{t_T}2 \left[ \sigma_1 \cdot {\bf p}
\, \sigma_2 \cdot {\bf p}+ \sigma_1 \cdot {\bf p'} \, \sigma_2
\cdot {\bf p'} - \frac13 \sigma_1 \, \cdot
\sigma_2 ({\bf p'}^2+ {\bf p}^2)
\right] \nonumber \\ &+&
\frac{t_U}2 \left[ \sigma_1 \cdot {\bf p}
\, \sigma_2 \cdot {\bf p}'+ \sigma_1 \cdot {\bf p'} \, \sigma_2
\cdot {\bf p} - \frac23 \sigma_1 \, \cdot
\sigma_2 {\bf p'}\cdot {\bf p}
\right]
\label{eq:skyrme2}
\end{eqnarray}
where $P_\sigma = (1+ \sigma_1 \cdot \sigma_2)/2$ is the spin exchange
operator with $P_\sigma=-1$ for spin singlet $S=0$ and $P_\sigma=1$
for spin triplet $S=1$ states. In practice, these effective forces are
parameterized in terms of a few constants which encode the relevant
physical information and should be deduced directly from the
elementary and underlying NN interactions. Unfortunately, there is a
huge variety of Skyrme forces depending on the fitting strategy
employed (see e.g. \cite{Friedrich:1986zza,Klupfel:2008af}). This lack
of uniqueness may indicate that the systematic and/or statistical
uncertainties within the various schemes are not accounted for
completely. Interestingly, the natural units for those parameters have
been outlined in Ref.~\cite{Furnstahl:1997hq,Kortelainen:2010dt}
yielding the correct order of magnitude. A microscopic
basis~\cite{Baldo:2010nx,Stoitsov:2010ha} for the Density Functional
Theory (DFT) approach has also been set up, but still uncertainties
remain.
Although the pseudo-potential in Eq.~(\ref{eq:skyrme2}) may be taken
literally in mean field calculations, due to the finite extension of
the nucleus, its interpretation in the simplest two-body problem
requires some regularization to give a precise meaning to the Dirac
delta interactions. The standard view of a pseudo-potential (in the
sense of Fermi) is that it corresponds to the potential which in the
Born approximation yields the real part of the full scattering
amplitude. This is a prescription which implements unitarity, but
necessarily fails when the scattering length is unnaturally large as
it is the case for NN interactions. On the contrary, the Wilsonian
viewpoint corresponds to a coarse graining of the NN interaction to a
certain energy scale. There are several schemes to coarse grain interactions in
Nuclear Physics. The traditional way has been by using the oscillator
shell model, where matrix elements of NN interactions are evaluated
with oscillator constants of about $b=1.4-2 \,{\rm
fm}$~\cite{Coraggio:2008in}. A modern way of coarse graining nuclear
interactions is represented by the $V_{\rm low k}$
method~\cite{Bogner:2001gq} (for a review see \cite{Bogner:2009bt})
where all momentum scales above $2 {\rm fm}^{-1}$ are integrated
out. The recent Euclidean Lattice Effective Field Theory (EFT)
calculations~(for a review see e.g. \cite{Lee:2008fa}), although
breaking rotational symmetry explicitly, provide a competitive scheme
where coarse grained interactions allow ab initio calculations
combining the insight of EFT and Monte-Carlo lattice experience, with
lattice spacings as large as $a=2 {\rm fm}$. These length scales match
the typical inter-particle distance of nuclear matter $d = 1
/\rho^{\frac13} \sim 2 {\rm fm}$. Actually, the three approaches
feature energy-, momentum- and configuration space coarse graining
respectively and ignore explicit dynamical effects below distances
$\sim b \sim 1/\Lambda \sim a $ which advantageously sidestep the
problems related to the hard core and confirm the modern view that
{\it ab initio} calculations are subjected to larger systematic
uncertainties than assumed hitherto. Clearly, any computational set up
implementing the coarse graining philosophy yields by itself a {\it
unique} definition of the effective interaction. However, there is
no universal effective interaction definition. For definiteness, we will follow here the $V_{\rm low k}$ scheme to
determine the effective parameters because within this framework some
underlying old nuclear symmetries, namely those implied by
Wigner and Serber forces, are vividly
displayed~\cite{CalleCordon:2008cz,CalleCordon:2009ps,RuizArriola:2009bg,Arriola:2010qk}.
In the present paper we want to show that in fact these parameters can
uniquely be determined from known NN scattering threshold parameters
by rather simple calculations by just coarse grain the interaction
over all wavelengths larger than the typical ones occurring in finite
nuclei. As we will show, this
introduces a momentum scale $\Lambda$ in the 9 effective parameters
$t_{0,1,2}$, $x_{0,1,2}$ and $t_{U,T,V}$ which allow to connect the
two body problem to the many body problem.
Going beyond Eq.~(\ref{eq:skyrme2}) requires further information than
just two-body low energy scattering, in particular knowledge about
three and four body forces and their scale dependence consistently
inherited from their NN counterpart. The finite $k_F$ situation
relevant for heavy nuclei and nuclear matter involves mixing between
operators with different particle number and, in principle, could be
conveniently tackled with the method outlined in
Ref.~\cite{delaPlata:1996fe} where the lack of genuine medium effects
is manifestly built in.
For completeness, we review here the $V_{\rm low k}$
approach~\cite{Bogner:2003wn} in a way that our points can be easily
stated. The starting point is a {\it given} phenomenological NN
potential, $V$, and usually denominated {\it bare potential}, whence
the scattering amplitude or $T$ matrix is obtained as the solution of
the half-off shell Lippmann-Schwinger (LS) coupled channel equation in
the CM system
\begin{eqnarray}
&& T_{l',l}^J (k',k; k^2) = V_{l',l}^J (k',k)
\nonumber \\
&&+ \sum_{l''} \int_0^\infty \,
\frac{M_N}{(2\pi)^3}
\frac{dq \, q^2}{k^2-q^2} V_{l',l''}^J ( k' , q) T_{l'',l}^J ( q , k;k^2) \, ,
\end{eqnarray}
where $J$ is the total angular momentum and $l,l'$ are orbital angular
momentum quantum numbers $p,p',q$ are CM momenta and $M_N$ is the
Nucleon mass. The unitary (coupled channel) S-matrix is obtained as
usual
\begin{eqnarray}
S_{l',l}^J (p) = \delta_{l',l} - {\rm i} \frac{p M_N}{8 \pi^2} T_{l',l}^J (p,p) \, .
\end{eqnarray}
Using the matrix representation ${\bf S}^J = ({\bf M}^J - i {\bf 1}
)({\bf M}^J + i {\bf 1})^{-1} $ with $({\bf M}^J)^\dagger = {\bf M}^J$
a hermitian coupled channel matrix, at low energies the effective
range theory for coupled channels reads
\begin{eqnarray}
p^{l+l'+1} M_{l',l}^J (p) = -(\alpha^{-1})^{J}_{l,l'} + \frac12 (r)^{J}_{l,l'}p^2+ (v)^{J}_{l,l'}p^4 + \dots
\label{eq:ERE-coup}
\end{eqnarray}
which in the absence of mixing and using $S_l(p) = e^{2 i
\delta_l (p)}$ reduces to the well-known expression
\begin{eqnarray}
p^{2 l +1} \cot \delta_l (p) = - \frac{1}{\alpha_l} + \frac12 r_l p^2
+ v_l p^4 + \dots
\label{eq:ERE}
\end{eqnarray}
An extensive study and determination of the low energy parameters for
all partial waves has been carried out in
Ref.~\cite{PavonValderrama:2005ku} for both the NijmII and the Reid93
potentials~\cite{Stoks:1994wp} yielding similar numerical results.
Dropping these coupled channel indices for simplicity the $V_{\rm low
k}$ potential is then defined by the equation
\begin{eqnarray}
&& T(k',k;k^2) = V_{\rm low k} ( k' , k) \nonumber \\
&& + \int_0^\Lambda \, \frac{M_N}{(2\pi)^3} \frac{dq \, q^2}{k^2-q^2}
V_{\rm low k} ( k' , q) T ( q , k;k^2) \, ,
\label{eq:Tlowk}
\end{eqnarray}
where $ (k,k') \le \Lambda $. We use here a sharp three-dimensional
cut-off $\Lambda$ to separate between low and high momenta since our
results are not sensitive to the specific form of the regularization.
Thus, eliminating the $T$ matrix we get the equation for the effective
potential which evidently depends on the cut-off scale $\Lambda$ and
corresponds to the effective interaction which nucleons see when all
momenta higher then the momentum scale $\Lambda$ are integrated out.
It has been found~\cite{Bogner:2003wn} that high precision potential
models, i.e. fitting the NN data to high accuracy incorporating One
Pion Exchange (OPE) at large distances and describing the deuteron
form factors, collapse into a unique self-adjoint nonlocal potential
for $\Lambda \sim 400-450 {\rm MeV}$. This is a not a unreasonable
result since all the potentials provide a rather satisfactory
description of elastic NN scattering data up to $p \sim 400 {\rm
MeV}$. Note that this universality requires a marginal effect of
off-shell ambiguities (beyond OPE off-shellness), which is a great
advantage as this is a traditional source for uncertainties in nuclear
structure. Actually, in the extreme limit when $\Lambda \to 0$ one is
left with zero energy {\it on shell} scattering yielding $T(k,k) \to
-(2\pi)^3 \alpha_0 /M_N$.
Moreover, for sufficiently small $\Lambda$, the potential which comes
out from eliminating high energy modes can be accurately represented
as the sum of the truncated original potential and a polynomial in the
momentum~\cite{Holt:2003rj}. However, as discussed in
\cite{CalleCordon:2009ps} a more convenient representation is to
separate off all polynomial dependence explicitly from the original
potential
\begin{eqnarray}
V_{\rm low k} ( k' , k) = \bar V_{\rm NN} ( k' , k) + \bar V_{\rm CT}^\Lambda ( k' , k)\, ,
\label{eq:vlowk2}
\end{eqnarray}
with $ (k,k') \le \Lambda $, so that if $\bar V_{\rm CT}^\Lambda ( k'
, k)$ contains up to ${\cal O} (p^n)$ then $\bar V_{\rm NN} ( k' , k)$
starts off at ${\cal O} (p^{n+1})$, i.e. the next higher order. This
way the departures from a pure polynomial may be viewed as true and
explicit effects due to the potential and more precisely from the
logarithmic left cut located at CM momentum $p= i m/2$ at the partial
wave amplitude level due to particle exchange with mass
$m$. Specifically,
\begin{eqnarray}
V_{\rm CT}^\Lambda ( k' , k) = k^l k'^{l'} \left[ C_J^{l l'}
(\Lambda)+ D_J^{l l'}(\Lambda) ( k^2+ k'^2 ) + \dots \right] \, ,
\label{eq:Vlowk}
\end{eqnarray}
where the coefficients $C_J^{l l'}(\Lambda) $ and $D_J^{l l'}(\Lambda)
$ include all contributions to the effective interaction at low
energies. Although we cannot calculate them {\it ab initio} we may
relate them to low energy scattering data, in harmony with the
expectation that off-shell effects are marginal. Not surprisingly the
physics encoding the effective interaction in Eq.~(\ref{eq:Vlowk})
will be related to the threshold parameters defined by
Eq.~(\ref{eq:ERE-coup}). Thus, the relevance of specific microscopic
nuclear forces to the effective (coarse grained) forces has to do with
the extent to which these threshold parameters are described by the
underlying forces and not so much with their detailed structure. We
will discuss below the limitations to this universal pattern.
Using the partial wave projection~\cite{1971NuPhA.176..413E} we get
the potentials in different angular momentum channels. These
parameters can be related to the spectroscopic notation used in
Ref.~\cite{Epelbaum:1999dj}. The S- and P-wave potentials are
\begin{eqnarray}
V_{^1S_0} (p',p) &=& C_{^1S_0} + D_{^1S_0} (p'^2+p^2) \, ,\nonumber \\
V_{^3S_1} (p',p) &=& C_{^3S_1} + D_{^3S_1} (p'^2+p^2) \, ,\nonumber \\
V_{E_1} (p',p) &=& D_{E_1} p^2 \, , \nonumber \\
V_{^3P_0} (p',p) &=& C_{^3P_0} p' p \, , \nonumber \\
V_{^3P_1} (p',p) &=& C_{^3P_1} p' p \, , \nonumber \\
V_{^3P_2} (p',p) &=& C_{^3P_2} p' p \, , \nonumber \\
V_{^1P_1} (p',p) &=& C_{^1P_1} p' p \, .
\label{eq:Vlowkp2}
\end{eqnarray}
The 9 effective parameters depend on the scale $\Lambda$ and can
be related to the effective force representation $t_{0,1,2}$,
$x_{0,1,2}$ and $t_{V,U,S}$ of Eq.~(\ref{eq:skyrme2}) by the following
explicit relations,
\begin{eqnarray}
t_0 &=& \frac1{8\pi}\left( C_{^3S_1}+C_{^1S_0} \right) \, ,\nonumber
\\
x_0 &=& \frac{C_{^3S_1}-C_{^1S_0}}{C_{^3S_1}+C_{^1S_0}} \, , \nonumber
\\
t_1 &=& \frac1{8\pi}\left( D_{^3S_1}+D_{^1S_0} \right) \, , \nonumber
\\
x_1 &=& \frac{D_{^3S_1}-D_{^1S_0}}{D_{^3S_1}+D_{^1S_0}} \, , \nonumber
\\
t_2 &=& \frac1{32\pi}\left( 9 C_{^1P_1}+C_{^3P_0}+ 3 C_{^3P_1}+5
C_{^3P_2} \right) \, , \nonumber
\\
x_2 &=& \frac{ -9 C_{^1P_1}+C_{^3P_0}+ 3 C_{^3P_1}+5
C_{^3P_2}}{ 9 C_{^1P_1}+C_{^3P_0}+ 3 C_{^3P_1}+5
C_{^3P_2}} \, , \nonumber \\
t_T &=& -\frac3{4 \sqrt{2}\pi} D_{E_1} \, , \nonumber \\
t_V &=& \frac1{32\pi}
\left( 2 C_{^3P_0}+ 3 C_{^3P_1}-5 C_{^3P_2} \right) \, ,
\nonumber
\\
t_U &=& \frac1{16\pi}
\left( -2 C_{^3P_0}+ 3 C_{^3P_1}- C_{^3P_2} \right) \, .
\end{eqnarray}
\begin{figure}[ttt]
\includegraphics[height=5.5cm,width=7cm,angle=0]{VSlowk-mix.eps}
\hskip.5cm
\includegraphics[height=5.5cm,width=7cm,angle=0]{VPlowk.eps}
\caption{(Color on-line) Counterterms for the S- (in ${\rm MeV} {\rm
fm}^3$, upper panel) and P-waves (${\rm MeV} {\rm fm}^5$, lower
panel) as a function of the momentum scale $\Lambda$ (in MeV). C's
from Eqs.~(\ref{eq:Vlowkp2}) solving Eqs.~(\ref{eq:Tlowk}) including
the D's using just the low energy threshold parameters from
Ref.~\cite{PavonValderrama:2005ku} (Thick Solid). C's extracted
from the diagonal $V_{\rm low k} (p,p)$
potentials~\cite{Bogner:2003wn} at fixed $\Lambda= 420 {\rm MeV}$
for the Argonne-V18~\cite{Wiringa:1994wb} (dashed). C's for P-waves
including D-terms without mixings (thick dotted).}
\label{fig:vlowk}
\end{figure}
\begin{figure}[ttt]
\includegraphics[height=5.5cm,width=7cm,angle=0]{VSlowk-nomix-mix.eps}
\caption{(Color on-line) $S-D$ waves mixing on $C_{^3S_1}$ (in ${\rm
MeV}{\rm fm}^3$) as a function of the momentum space cut-off
$\Lambda$ (in MeV) in the $^3S_1$ channel. We compare when including
a) only $C_{^3S_1}$ (dashed), b) $C_{^3S_1}$ and $D_{^3S_1}$
(dotted), c) $C_{^3S_1}$, $D_{^3S_1}$ and $D_{E_1}$ (solid). Wigner
symmetry is displayed when the mixing is included. See also
Fig.~\ref{fig:vlowk} and Eq.~(\ref{eq:vlowk2}) in the main text.}
\label{fig:nomix-mix}
\end{figure}
\begin{figure*}[tbc]
\includegraphics[height=3.5cm,width=4.4cm,angle=0]{Skyrme-Vlowk-t0.eps}
\hskip.3cm
\includegraphics[height=3.5cm,width=4.4cm,angle=0]{Skyrme-Vlowk-x0.eps}
\hskip.3cm
\includegraphics[height=3.5cm,width=4.4cm,angle=0]{Skyrme-Vlowk-t2.eps}\\
\includegraphics[height=3.5cm,width=4.4cm,angle=0]{Skyrme-Vlowk-x2.eps}
\hskip.3cm
\includegraphics[height=3.5cm,width=4.4cm,angle=0]{Skyrme-Vlowk-W0.eps}
\hskip.3cm
\includegraphics[height=3.5cm,width=4.4cm,angle=0]{Skyrme-Vlowk-U.eps}
\caption{(Color on line) Skyrme force parameters as a function of the
scale $\Lambda$ (in MeV). We compare with the
Argonne-V18~\cite{Wiringa:1994wb} exact $V_{\rm low k}$ values
evaluated at $\Lambda=420 {\rm MeV}$~\cite{Bogner:2003wn}. See also
Fig.~\ref{fig:vlowk} and main text. }
\label{fig:skyrme}
\end{figure*}
The corresponding T-matrices are conveniently solved by factoring out
the centrifugal terms
\begin{eqnarray}
T_{l',l}(k',k) = k^l k'^{l'} \left[ t_{l',l}^J (p) + u_{l',l}(p) (k^2+k'^2) + \dots
\right]
\end{eqnarray}
which reduce the LS equation to a finite set of algebraic
equations which are analytically solvable (see
e.g. Ref.~\cite{Entem:2007jg} and references therein). In the simplest
case where only the $C's$ are taken into account the explicit
solutions for S- and P-waves are,
\begin{eqnarray}
C_{S} (\Lambda) &=& \frac{16 \pi^2 \alpha_{0}}{M_N (1 - 2 \alpha_{0}
\Lambda/\pi)} \, , \nonumber \\ C_P (\Lambda) &=& \frac{16 \pi^2
\alpha_1}{M_N (1 - 2 \alpha_{1} \Lambda^3/ 3 \pi) } \, ,
\label{eq:C's}
\end{eqnarray}
where $\alpha_0$($\alpha_1$) is the scattering length (volume) defined by
Eq.~(\ref{eq:ERE}). The Eq.~(\ref{eq:C's}) illustrates the difference
between a Fermi pseudo-potential and a coarse grained potential as the
former corresponds to $\alpha_0 \Lambda \ll 1 $ where $ C_{S}
(\Lambda) \sim 16 \pi^2 \alpha_{0}/M_N $. In the case $\alpha_0 \Lambda
\gg 1 $ one has instead $ C_{S} (\Lambda) \sim - 8 \pi / ( M
\Lambda)$. Full solutions including the $D$'s are also analytical
although a bit messier, so we do not display them explicitly.
They rely on Eq.~(\ref{eq:ERE-coup}) with $\alpha_{^1S_0}$,
$\alpha_{^3S_1}$, $\alpha_{^3P_0}$, $\alpha_{^3P_1}$,
$\alpha_{^3P_2}$, $\alpha_{^1P_1}$, $\alpha_{E_1}$, $r_{^3S_1}$ and
$r_{^1S_0}$ (see Ref.~\cite{PavonValderrama:2005ku} for numerical
values for NijmII and Reid93 potentials). At the order considered
here we just mention that while all P-waves constants run
independently of each other with $\Lambda$ the spin-singlet parameters
$C_{^1S_0} $, $D_{^1S_0} $ on the one hand and the spin-triplet
parameters $C_{^3S_1} $, $D_{^3S_1} $ and $C_{E_1} $ on the other are
intertwined.
We now turn to our numerical results. As can be seen from
Fig.~\ref{fig:vlowk} the comparison of contact interactions using
threshold parameters with $V_{\rm low k}$ results evolved to
$\Lambda=420 {\rm MeV}$~\cite{Bogner:2003wn} (note the different
normalization as ours) from the Argonne-V18 bare
potential~\cite{Wiringa:1994wb} are saturated for $\Lambda=250 {\rm
MeV}$ for S-waves and for much lower cut-offs for P-waves. Note that
this holds regardless on the details of the potential as we only need
the low energy threshold parameters as determined e.g. in
Ref.~\cite{PavonValderrama:2005ku}. The strong dependence observed at
larger $\Lambda$ values just reflects the inadequacy of the second
order truncation in Eqs.~(\ref{eq:Vlowkp2}). This also reflects in the
$25\%-50\%$ inaccuracy are off the exact $V_{\rm low k}$ of the D's
themselves despite showing plateaus, and thus will not be discussed
any further.
The identity $C_{^1S_0}(\Lambda)=C_{^3S_1}(\Lambda)$ for $\Lambda \ge
250 {\rm MeV}$ features the appearance of Wigner symmetry as pointed
out in Ref.~\cite{CalleCordon:2008cz}, but now we see that this does
not depend on details of the force. Actually, the effect of the
$^3S_1-^3D_1$wave mixing represented by a non-vanishing off diagonal
potential $V_{E_1} (p',p)$ becomes essential to achieve this identity
(a fact disregarded in Ref.~\cite{Mehen:1999qs}). As can be seen from
Fig.~\ref{fig:nomix-mix} there is a large mismatch at values of
$\Lambda \sim 200-300 {\rm MeV}$ when $D_{E_1}$ is set to zero (and
hence $\alpha_{E_1}=0$) as compared with the case $D_{E_1} \neq 0$.
The scale dependence of the Skyrme interaction parameters (not
involving the D's) can be seen in Fig.~\ref{fig:skyrme} in comparison
with the $V_{\rm low k} $ potentials~\cite{Bogner:2003wn} deduced from
the Argonne-V18 bare potentials~\cite{Wiringa:1994wb}. The plateaus
observed in the different partial waves are corroborated here as well
as a remarkable accuracy in reproducing the exact $V_{\rm low k} $
numbers. Moreover,
the weak cut-off dependence of the spin orbit interaction
observed in Fig.~\ref{fig:skyrme} suggests taking $\Lambda \to 0$ in
which case
\begin{eqnarray}
W_0 = \frac{\pi}{ 2 M_N} \left( 2 \alpha_{^3P_0} + 3 \alpha_{^3P_1} -5
\alpha_{^3P_2} \right) \, ,
\end{eqnarray}
which upon using Ref.~\cite{PavonValderrama:2005ku} yields $ W_0 = 72
{\rm MeV fm}^4$. This numerical value reproduces within less than
$10\%$ the exact $V_{\rm low k}$ value. As can be seen from
Fig.~\ref{fig:skyrme} the effective range correction $r_1$ provides
via additional $D$ coefficients the missing
contribution. This is a bit lower than what it is found in
phenomenological approaches from the $p_{3/2}-p_{1/2}$ level splitting
in $^{16}$O~\cite{Bender:2003jk}. In any case, the comparison with
phenomenological approaches based on mean field calculations may be
tricky since as already mentioned not all the terms are always kept,
and selective fits to finite nuclear properties may overemphasize the
role played by specific terms.
It has recently been argued that counterterms are fingerprints of long
distance
symmetries~\cite{CalleCordon:2008cz,CalleCordon:2009ps,RuizArriola:2009bg}.
This remarkable result holds regardless on the nature of the forces
and applies in particular to both Wigner and Serber symmetries. We
confirm that to great accuracy, $x_0=0$ (Wigner symmetry) and
$x_2=-1$ (Serber symmetry). The astonishing large-$N_c$ ($N_c$ is the
number of colours in QCD) relations discussed in
Refs.~\cite{CalleCordon:2008cz,CalleCordon:2009ps,RuizArriola:2009bg,Arriola:2010qk}
provide a direct link to the underlying quark and gluon dynamics and
after~\cite{Kaplan:1996rk} suggests a $1/N_c^2$ accuracy of the Wigner
symmetry in even-L partial waves. Wigner symmetry has proven crucial
in Nuclear coarse lattice ($a \sim 2{\rm fm}$)
calculations~\cite{Lee:2008fa} in sidestepping the sign problem for
fermions. As we see for the scales typically involved there this works
with great accuracy already at $\Lambda \sim 250 {\rm MeV}$. Taking
into account that we deal with low energies, it is thus puzzling that
Chiral interactions to N$^3$LO~\cite{Entem:2003ft} having chiral
cut-offs $\Lambda_\chi \sim 600 {\rm MeV} $ tend to violate Wigner
symmetry in the $V_{\rm low k}$ sense, i.e. $C_{^1S_0}^\chi \neq
C_{^3S_1}^\chi $, whereas smaller values $\Lambda_\chi \sim 450 {\rm
MeV} $~\cite{CalleCordon:2009ps} are preferred.
Within the low energy expansion we have neglected terms ${\cal O}
({\bf p'}^4 , {\bf p}^4 , {\bf p'}^2 {\bf p}^2 )$ which correspond to
P-waves and S-wave range corrections. In configuration space this
corresponds to a dimensional expansion, since $\delta (\vec r_{12}) =
{\cal O} (\Lambda^3) $ and $ \{P^2, \delta (\vec r_{12}) \} = {\cal O}
(\Lambda^5) $, $ \{P^4, \delta (\vec r_{12}) \} = {\cal O} (\Lambda^7)
$. Within such a scheme going to higher orders requires also to
include three-body interactions, $\sim \delta (r_{12}) \delta (r_{13})
= {\cal O} (\Lambda^6) $. Actually, at the two body level there are
more potential parameters than low energy threshold parameters. For
instance, in the $^1S_0$ channel one has two independent hermitean
operators, ${\bf p'}^4 + {\bf p}^4 $ and $ 2 {\bf p'}^2 {\bf p}^2 $
(which are on-shell equivalent), but only one $v_{^1S_0}$ threshold
parameter in the low energy expansion (see
Eq.~(\ref{eq:ERE-coup})). As it was shown in Ref.~\cite{Amghar:1995av}
(see also Ref.~\cite{Furnstahl:2000we}) these two features are
interrelated since this two body off-shell ambiguity is cancelled when
a three body observable, like e.g. the triton binding energy, is
fixed . An intriguing aspect of the present investigation is the
modification induced by potential tails due to e.g. pion exchange
which cannot be represented by a polynomial since particle exchange
generates a cut in the complex energy plane. The important issue,
however, is that the low scale saturation unveiled in the present
paper works accurately just to second order as long as the low energy
parameters determined from on-shell scattering are properly
reproduced.
{\it I thank M. Pav\'on Valderrama and L.L. Salcedo
for a critical reading of the ms
and Jes\'us Navarro, A. Calle Cord\'on, T. Frederico
and V.S. Timoteo for discussions. Work supported by the Spanish DGI
and FEDER funds with grant FIS2008-01143/FIS, Junta de
Andaluc{\'\i}a grant FQM225-05.}
|
{
"timestamp": "2010-09-30T02:02:09",
"yymm": "1009",
"arxiv_id": "1009.4161",
"language": "en",
"url": "https://arxiv.org/abs/1009.4161"
}
|
\section*{Introduction}
We discuss in this paper the computation of certain advanced representation theory invariants, for the main examples of ``easy quantum groups''. These are the groups $S_n,O_n,B_n,H_n$, their free versions $S_n^+,O_n^+,B_n^+,H_n^+$, and the half-liberated versions $O_n^*,H_n^*$. Here $S_n,O_n$ are the permutation and orthogonal groups, $B_n$ is the bistochastic group consisting of orthogonal matrices whose rows and columns sum to 1, and $H_n = \mathbb Z_2 \wr S_n$ is the hyperoctahedral group. For a global introduction to these groups and quantum groups, we refer to our previous papers \cite{ez1}, \cite{ez2}, \cite{ez3}.
According to a paper of Weingarten \cite{wei}, further processed and generalized by Collins \cite{col}, then Collins and \'Sniady \cite{csn}, a number of advanced representation theory invariants of the quantum group are encoded in a certain associated matrix $G_{kn}$, called Gram matrix. For instance the inverse of the Gram matrix $G_{kn}$ is the Weingarten matrix $W_{kn}$, whose knowledge allows the full computation of the Haar functional. See \cite{wei}, \cite{col}, \cite{csn}.
Among these invariants, the central object is the Gram matrix determinant, $\det(G_{kn})$. For instance the roots of $\det(G_{kn})$ are the poles of the Weingarten function $W_{kn}$, and the knowledge of these numbers clarifies the invertibility assumptions in \cite{ez1}, \cite{ez2}, \cite{ez3}.
The quantity $\det(G_{kn})$ appears in fact naturally in relation with many other questions, and its exact computation a well-known problem. A first purpose of the present work is to collect all the available formulae from the literature, and to write them down by using our unified ``easy quantum group'' formalism, along with complete, simplified proofs.
The basic example of such a formula is that for $S_n,H_n,H_n^*$. Here the Gram matrix is upper triangular, up to a simple determinant-preserving operation. The determinant, computed by Jackson \cite{jac} and Lindstr\"om \cite{lin}, is as follows:
$$\det(G_{kn})=\prod_{\pi\in \mathcal P(k)}\frac{n!}{(n-|\pi|)!}$$
Here $\mathcal P(k)$ is the set of partitions associated to the quantum group, namely all partitions for $S_n$, all partitions with even blocks for $H_n$, and all partitions with blocks having the same number of odd and even legs for $H_n^*$, and $|.|$ is the number of blocks.
In the general case the situation is much more complicated. However, one may still wonder for a general decomposition result, of the following type:
$$\det(G_{kn})=\prod_{\pi\in \mathcal P(k)}\varphi(\pi)$$
This question is of course quite vague, depending on how explicit we would like our functions $\varphi$ to be. For instance a natural requirement would be that in the case of a liberation $G_n\to G_n^+$, the functions $\varphi$ are related by an induction/restriction procedure.
This kind of specialized question appears to be quite difficult. In this paper we will obtain some preliminary decomposition results of the above type, as follows:
\begin{enumerate}
\item For $O_n,B_n,O_n^*$ a formula comes from the work of Collins-Matsumoto \cite{cma} and Zinn-Justin \cite{zin}. The natural decomposition here is over Young diagrams, and in principle one can pass to partitions by applying a certain surjective map.
\item For $O_n^+,B_n^+,S_n^+,H_n^+$ we use the work of Di Francesco, Golinelli and Guitter \cite{df1}, \cite{df2}, \cite{dg1}, \cite{dg2}, Tutte \cite{tut} and Dahab \cite{dah}. We will obtain some evidence towards the existence of contributions $\varphi(\pi)$, of ``trigonometric'' nature.
\end{enumerate}
We will make as well a number of speculations in relation with quantum group/planar algebra methods, and with spectral measure/orthogonal polynomial interpretations.
As a conclusion, there is still a lot of work to be done, mostly towards the conceptual understanding, at the level of Gram determinants, of the operation $G_n\to G_n^+$.
The paper is organized as follows: 1-2 are preliminary sections, in 3-4 we discuss the classical and half-liberated cases, and in 5-6 we discuss the free case. The final sections, 7-9, contain a number of speculations on the formulae, and a few concluding remarks.
\subsection*{Acknowledgements}
This work was started at the Bedlewo 2009 workshop ``Noncommutative harmonic analysis'', and we are highly grateful to M. Bo\.zejko for the invitation, and for several stimulating discussions. S.C. would like to thank the Paul Sabatier University and the Cergy-Pontoise University, where another part of this work was done. The work of T.B. was supported by the ANR grants ``Galoisint'' and ``Granma'', and the work of S.C. was partially supported by an NSF Postdoctoral Fellowship.
\section{Easy quantum groups}
Let $\mathcal P_s$ be the category of all partitions. That is, $\mathcal P_s(k,l)$ is the set of partitions between an upper row of $k$ points and a lower row of $l$ points, and the categorical operations are the horizontal and vertical concatenation, and the upside-down turning.
A category of partitions $\mathcal P \subset \mathcal P_s$ is by definition a collection of sets $\mathcal P(k,l)\subset \mathcal P_s(k,l)$, which is stable under the categorical operations. We have the following examples.
\begin{proposition}
The following are categories of partitions:
\begin{enumerate}
\item $\mathcal P_o/\mathcal P_o^+$: all pairings/all noncrossing pairings.
\item $\mathcal P_o^*$: pairings with each string having an odd leg and an even leg.
\item $\mathcal P_b/\mathcal P_b^+$: singletons plus pairings/noncrossing pairings.
\item $\mathcal P_s/\mathcal P_s^+$: all partitions/all noncrossing partitions.
\item $\mathcal P_h/\mathcal P_h^+$: partitions/noncrossing partitions with blocks of even size.
\item $\mathcal P_h^*$: partitions with blocks having the same number of odd and even legs.
\end{enumerate}
\end{proposition}
\begin{proof}
This is clear from definitions. Note that $\mathcal P_g^\times$ corresponds via Tannakian duality \cite{wo1}, \cite{wo2} to the easy quantum group $G^\times=(G_n^\times)$, with the notations in \cite{ez1}, \cite{bsp}.
\end{proof}
We use the notation $\mathcal P(k)=\mathcal P(0,k)$. We denote by $\vee$ and $\wedge$ the set-theoretic sup and inf of partitions, always taken with respect to $\mathcal P_s$, and by $|.|$ the number of blocks.
\begin{definition}
Associated to any category of partitions $\mathcal P$ and to any numbers $k,n\geq 0$ are the following matrices, with entries indexed by $\pi,\sigma\in \mathcal P(k)$:
\begin{enumerate}
\item Gram matrix: $G_{kn}(\pi,\sigma)=n^{|\pi\vee\sigma|}$.
\item Weingarten matrix: $W_{kn}=G_{kn}^{-1}$.
\end{enumerate}
\end{definition}
In order for $G_{kn}$ to be invertible, $n$ must be big enough, and $n\geq k$ is known to be sufficient. The precise bounds depend on the category of partitions, and can be deduced from the various explicit formulae of $\det(G_{kn})$, to be given later on in this paper.
The interest in the above matrices comes from the fact that in the case $\mathcal P= \mathcal P_g^\times$, they describe the integration over the corresponding easy quantum group $G_n^\times$.
\begin{theorem}
We have the Weingarten formula
$$\int_{G_n^\times}u_{i_1j_1}\ldots u_{i_kj_k}\,du=\sum_{\pi,\sigma\in \mathcal P_g^\times(k)}\delta_\pi(i)\delta_\sigma(j)W_{kn}(\pi,\sigma)$$
where the $\delta$ symbols are $0$ or $1$, depending on whether the indices fit or not.
\end{theorem}
\begin{proof}
This follows by using a classical argument from \cite{wei}, \cite{csn}. See \cite{ez1}, \cite{bsp}.
\end{proof}
The exact computation of the Weingarten matrix is a quite subtle problem. A precise result is available only in the finite group case, where the formula is given in terms of the M\"obius function $\mu$ on $\mathcal P$ as follows.
\begin{proposition}
For $S_n,H_n$ the Weingarten function is given by
$$W_{kn}(\pi,\sigma)=\sum_{\tau\leq\pi\wedge\sigma}\mu(\tau,\pi)\mu(\tau,\sigma)\frac{(n-|\tau|)!}{n!}$$
and satisfies $W_{kn}(\pi,\sigma)=n^{-|\pi\wedge\sigma|}(
\mu(\pi\wedge\sigma,\pi)\mu(\pi\wedge\sigma,\sigma)+O(n^{-1}))$.
\end{proposition}
\begin{proof}
The first assertion follows from the Weingarten formula: in that formula the integrals on the left are known, and this allows the computation of the right term, via the M\"obius inversion formula. The second assertion follows from the first one.
\end{proof}
In the general case we have the following result, which is useful for applications.
\begin{proposition}
For $\pi\leq\sigma$ we have the estimate
$$W_{kn}(\pi,\sigma)=n^{-|\pi|}(\mu(\pi,\sigma)+O(n^{-1}))$$
and for $\pi,\sigma$ arbitrary we have $W_{kn}(\pi,\sigma)=O(n^{|\pi\vee\sigma|-|\pi|-|\sigma|})$.
\end{proposition}
\begin{proof}
Once again this follows by using a classical argument, see \cite{ez1}.
\end{proof}
\section{Gram determinants}
In this paper, we will be mainly interested in the computation of $\det(G_{kn})$. Let us being with some simple observations, coming from definitions.
\begin{proposition}
Let $D_k(n)=\det(G_{kn})$, viewed as element of $\mathbb Z[n]$.
\begin{enumerate}
\item $D_k$ is monic, of degree $s_k=\sum_{\pi\in \mathcal P(k)}|\pi|$.
\item We have $n^{b_k}|D_k$, where $b_k=\# \mathcal P(k)$.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) This follows from $|\pi\vee\sigma|\leq|\pi|$, with equality if and only if $\sigma\leq\pi$. Indeed, from the inequality we get $\deg(D_k)\leq s_k$. Now the coefficient of $n^{s_k}$ is the signed number of permutations $f:\mathcal P(k)\to \mathcal P(k)$ satisfying $f(\pi)\leq\pi$ for any $\pi$, and since there is only one such permutation, namely the identity, we obtain that this coefficient is 1.
(2) This is clear from the definition of $D_k$, and from $|\pi\vee\sigma|\geq 1$.
\end{proof}
The above result raises the question of computing the numbers $b_k=\#\mathcal P(k)$ and $s_k=\sum_{\pi\in \mathcal P(k)}|\pi|$. It is convenient here to introduce as well the related numbers $m_k=s_k/b_k$ and $a_k=2s_k-kb_k=(2m_k-k)b_k$, which will appear several times in what follows.
\begin{proposition}
The numbers $b_k,s_k,m_k,a_k$ are as follows:
\begin{enumerate}
\item $O_n,O_n^*,O_n^+$: $b_{2l}=(2l)!!,l!,\frac{1}{l+1}\binom{2l}{l}$, $s_{2l}=lb_{2l}$, $m_{2l}=l$, $a_{2l}=0$.
\item $S_n$: $b_k=$ Bell, $s_k=b_{k+1}-b_k$, $m_k=\frac{b_{k+1}}{b_k}-1$, $a_k=2b_{k+1}-(k+2)b_k$.
\item $S_n^+$: $b_k=\frac{1}{k+1}\binom{2k}{k}$, $s_k=\frac{1}{2}\binom{2k}{k}$, $m_k=\frac{k+1}{2}$, $a_k=b_k$.
\item $H_n^+$: $b_{2l}=\frac{1}{2l+1}\binom{3l}{l}$, $s_{2l}=\binom{3l-1}{l-1}$, $m_{2l}=\frac{2l+1}{3}$, $a_{2l}=-2\binom{3l-1}{l-2}$.
\end{enumerate}
\end{proposition}
\begin{proof}
All these results are well-known.
\end{proof}
For the remaining quantum groups, namely $B_n,B_n^+,H_n,H_n^*$, the numbers $b_k,s_k,m_k,a_k$ are given by quite complicated formulae. The best approach to their computation is via the trace of the Gram matrix, and its analytic interpretations.
So, let us first reformulate Proposition 2.1, in the following way.
\begin{proposition}
With $D_k(n)=\det(G_{kn})$ and $T_k(t)=Tr(G_{kt})$, we have:
\begin{enumerate}
\item $D_k(n)=n^{s_k}(1+O(n^{-1}))$ as $n \to \infty$, where $s_k=T_k'(1)$.
\item $D_k(n)=O(n^{b_k})$ as $n \to 0$, where $b_k=T_k(1)$.
\end{enumerate}
\end{proposition}
\begin{proof}
This is indeed just a reformulation of Proposition 2.1, using a variable $t$ around 1. Note that in (2) we regard the variable $n$ as a formal parameter, going to $0$.
\end{proof}
The trace can be understood in terms of the associated Stirling numbers.
\begin{proposition}
We have the formula
$$T_k(t)=\sum_{r=1}^kS_{kr}t^r$$
where $S_{kr}=\#\{\pi\in \mathcal P(k):|\pi|=r\}$ are the Stirling numbers.
\end{proposition}
\begin{proof}
This is clear from definitions.
\end{proof}
Another interpretation of the trace, analytic this time, is as follows.
\begin{proposition}
For any $t\in(0,1]$ we have the formula
$$T_k(t)=\lim_{n\to\infty}\int_{G_n^\times}\chi_t^k$$
where $\chi_t=\sum_{i=1}^{[tn]}u_{ii}$ are the truncated characters of the quantum group.
\end{proposition}
\begin{proof}
As explained in \cite{bsp}, \cite{ez1}, this follows from the Weingarten formula.
\end{proof}
In general, the Stirling numbers $S_{kr}$ and the trace $T_k(t)$ are given by quite complicated formulae, unless we are in the situation of one of the quantum groups in Proposition 2.2. Here these invariants are well-known in the $O,S$ cases, and for $H^+$ we have:
$$T_{2l}(t)=\sum_{r=1}^l\frac{1}{r}\binom{l-1}{r-1}\binom{2l}{r-1}t^r$$
See \cite{bb+}. In general now, the conceptual result concerns the asymptotic measures of truncated characters, i.e. the probability measures $\mu_t$ satisfying $T_k(t)=\int x^kd\mu_t(x)$.
\begin{theorem}
The asymptotic measures of truncated characters are as follows:
\begin{enumerate}
\item $S_n/S_n^+$: Poisson/free Poisson.
\item $O_n/O_n^+$: Gaussian/semicircular.
\item $H_n/H_n^+$: Bessel/free Bessel.
\item $B_n/B_n^+$: shifted Gaussian/shifted semicircular.
\item $O_n^*/H_n^*$: symmetrized Rayleigh/squeezed $\infty$-Bessel.
\end{enumerate}
\end{theorem}
\begin{proof}
The one-parameter measures in the statement are best found via a direct computation, by using classical and free cumulants. See \cite{bsp}, \cite{ez1}, \cite{ez2}.
\end{proof}
\section{The basic formula}
We discuss now the explicit computation of the Gram determinants. The basic formula here, coming from the work of Jackson \cite{jac} and Lindstr\"om \cite{lin}, is as follows.
\begin{theorem}
For $S_n,H_n,H_n^*$ we have
$$\det(G_{kn})=\prod_{\pi\in \mathcal P(k)}\frac{n!}{(n-|\pi|)!}$$
where $|.|$ is the number of blocks.
\end{theorem}
\begin{proof}
We use the fact that the partitions have the property of forming semilattices under $\vee$. The proof uses the upper triangularization procedure in \cite{lin} together with the explicit knowledge of the M\"obius function on $\mathcal P(k)$ as in \cite{jac}. Consider the following matrix, obtained by making determinant-preserving operations:
$$G_{kn}'(\pi,\sigma)=\sum_{\pi\leq\tau}\mu(\pi,\tau)n^{|\tau\vee\sigma|}$$
It follows from the M\"obius inversion formula that we have:
$$G_{kn}'(\pi,\sigma)=
\begin{cases}
n(n-1)\ldots(n-|\sigma|+1)&{\rm if}\ \pi\leq\sigma\\
0&{\rm if\ not}
\end{cases}$$
Thus the matrix is upper triangular, and by computing the product on the diagonal we obtain the formula in the statement.
\end{proof}
A first remarkable feature of the above result is that the determinant for $S_n,H_n,H_n^*$ can be computed from the trace: indeed, the Gram trace gives the Stirling numbers, which in turn give the Gram determinant. However, the connecting formula is quite complicated, so let us just record here an improvement of the first estimate in Proposition 2.3.
\begin{proposition}
With $D_k(n)=\det(G_{kn})$ and $T_k(t)=Tr(G_{kt})$ we have
$$D_k(n)=n^{s_k}\left(1-\frac{z_k}{2}\,n^{-1}+O(n^{-2})\right)$$
where $s_k=T_k'(1)$ and $z_k=T_k''(1)$.
\end{proposition}
\begin{proof}
In terms of Stirling numbers, the formula in Theorem 3.1 reads:
$$D_k(n)=\prod_{r=1}^k\left(\frac{n!}{(n-r)!}\right)^{S_{kr}}$$
We use now the following basic estimate:
$$\frac{n!}{(n-r)!}=n^r\prod_{s=1}^{r-1}\left(1-\frac{s}{n}\right)=n^r\left(1-\frac{r(r-1)}{2}\,n^{-1}+O(n^{-2})\right)$$
Together with $T_k(t)=\sum_{r=1}^kS_{kr}t^r$, this gives the result.
\end{proof}
The above discussion raises the general question on whether the Gram determinant can be computed or not from the Gram trace, or from the measures in Theorem 2.6.
Since the connecting formula for $S_n,H_n,H_n^*$ is already quite complicated, let us formulate for the moment a more modest conjecture, as follows.
\begin{conjecture}
For any easy quantum group we have a formula of type
$$\det(G_{kn})=\prod_{\pi\in \mathcal P(k)}\varphi(\pi)$$
with the ``contributions'' being given by an explicit function $\varphi:\mathcal P(k)\to\mathbb Q(n)$.
\end{conjecture}
This statement is of course quite vague, depending of the meaning of the above word ``explicit''. As already mentioned, one would expect $\varphi$ to come from the Gram trace, or from the Stirling numbers, or, even better, from the measures in Theorem 2.6. Such a decomposition could potentially clarify the behavior of the Gram determinants under the ``liberation'' procedure $G \to G^+$.
This kind of general question appears to be quite difficult. In what follows we will obtain some evidence towards such general decomposition results.
\section{The orthogonal case}
We discuss now the cases $O,B,O^*$. Here the combinatorics is that of the Young diagrams. We denote by $|.|$ the number of boxes, and we use quantity $f^\lambda$, which gives the number of standard Young tableaux of shape $\lambda$.
\begin{theorem}
For $O_n$ we have
$$\det(G_{kn})=\prod_{|\lambda|=k/2}f_n(\lambda)^{f^{2\lambda}}$$
where $f_n(\lambda)=\prod_{(i,j)\in\lambda}(n+2j-i-1)$.
\end{theorem}
\begin{proof}
This follows from the results of Collins and Matsumoto \cite{cma} and Zinn-Justin \cite{zin}. Indeed, it is known from \cite{zin} that the Gram matrix is diagonalizable, as follows:
$$G_{kn}=\sum_{|\lambda|=k/2}f_n(\lambda)P_{2\lambda}$$
Here $1=\Sigma P_{2\lambda}$ is the standard partition of unity associated to the Young diagrams having $k/2$ boxes, and the coefficients $f_n(\lambda)$ are those in the statement. Now since we have $Tr(P_{2\lambda})=f^{2\lambda}$, this gives the result.
\end{proof}
\begin{theorem}
For $B_n$ we have
$$\det(G_{kn})=n^{a_k}\prod_{|\lambda|\leq k/2}f_n(\lambda)^{\binom{k}{2|\lambda|}f^{2\lambda}}$$
where $a_k=\sum_{\pi\in \mathcal P(k)}(2|\pi|-k)$, and $f_n(\lambda)=\prod_{(i,j)\in\lambda}(n+2j-i-2)$.
\end{theorem}
\begin{proof}
We recall from \cite{bsp} that we have an isomorphism $B_n\simeq O_{n-1}$, given by $u=v+1$, where $u,v$ are the fundamental representations of $B_n,O_{n-1}$. We get:
$$Fix(u^{\otimes k})
=Fix\left((v+1)^{\otimes k}\right)
=Fix\left(\sum_{r=0}^k\binom{k}{r}v^{\otimes r}\right)$$
Now if we denote by ${\rm det}',f'$ the objects in Theorem 4.1, we obtain:
$$\det(G_{kn})
=n^{a_k}\prod_{r=1}^k{\rm det}'(G_{r,n-1})^{\binom{k}{r}}
=n^{a_k}\prod_{r=1}^k\left(\prod_{|\lambda|=r/2}f'_{n-1}(\lambda)^{f^{2\lambda}}\right)^{\binom{k}{r}}$$
This gives the formula in the statement.
\end{proof}
\begin{theorem}
For $O_n^*$ we have
$$\det(G_{kn})=\prod_{|\lambda|=k/2}f_n(\lambda)^{{f^\lambda}^2}$$
where $f_n(\lambda)=\prod_{(i,j)\in\lambda}(n+j-i)$.
\end{theorem}
\begin{proof}
We use the isomorphism of projective versions $PO_n^*=PU_n$, established in \cite{bve}. This isomorphism shows that the Gram matrices for $O_n^*$ are the same as those for $U_n$. But for $U_n$ it is known from \cite{zin} that the Gram matrix is diagonalizable, as follows:
$$G_{kn}=\sum_{|\lambda|=k/2}f_n(\lambda)P_\lambda$$
Here $1=\Sigma P_\lambda$ is the standard partition of unity associated to the Young diagrams having $k/2$ boxes, and the coefficients $f_n(\lambda)$ are those in the statement. Now since we have $Tr(P_\lambda)={f^\lambda}^2$, this gives the result.
\end{proof}
Observe that the above results provide a kind of answer to Conjecture 3.3, but with the Young diagrams contributing to the determinant, instead of the partitions. The remaining problems are to find the relevant surjective map from diagrams to partitions, and to see if the above formulae further simplify by using this surjective map.
\section{Meander determinants}
In this section we discuss the computation of the Gram matrix determinant, in the free cases $O_n^+,B_n^+,S_n^+,H_n^+$. Let $P_r$ be the Chebycheff polynomials, given by $P_0=1,P_1=n$ and $P_{r+1}=nP_r-P_{r-1}$. Consider also the following numbers, depending on $k,r\in\mathbb Z$:
$$f_{kr}=\binom{2k}{k-r}-\binom{2k}{k-r-1}$$
We set $f_{kr}=0$ for $k\notin\mathbb Z$. The following key result was proved in \cite{dg1}.
\begin{theorem}
For $O_n^+$ we have
$$\det(G_{kn})=\prod_{r=1}^{[k/2]}P_r(n)^{d_{k/2,r}}$$
where $d_{kr}=f_{kr}-f_{k+1,r}$.
\end{theorem}
\begin{proof}
As already mentioned, the result is from \cite{dg1}. We present below a short proof. The result holds when $k$ is odd, all the exponents being 0, so we assume that $k$ is even.
{\bf Step 1.} We use a general formula of type $G_{kn}(\pi,\sigma)=<f_\pi,f_\sigma>$.
Let $\Gamma$ be a locally finite bipartite graph, with distinguished vertex 0 and adjacency matrix $A$, and let $\mu$ be an eigenvector of $A$, with eigenvalue $n$. Let $L_k$ be the set of length $k$ loops $l=l_1\ldots l_k$ based at $0$, and $H_k=span(L_k)$. For $\pi\in \mathcal P_{o^+}(k)$ define $f_\pi\in H_k$ by:
$$f_\pi=\sum_{l\in L_k}\left(\prod_{i\sim_\pi j}\delta(l_i,l_j^o)\gamma(l_i)\right)l$$
Here $e\to e^o$ is the edge reversing, and the ``spin factor'' is $\gamma=\sqrt{\mu(t)/\mu(s)}$, where $s,t$ are the source and target of the edges. The point is that we have $G_{kn}(\pi,\sigma)=<f_\pi,f_\sigma>$. We refer to \cite{mar}, \cite{jo1}, \cite{gjs} for full details regarding this formula.
{\bf Step 2.} With a suitable choice of $(\Gamma,\mu)$, we obtain a fomula of type $G_{kn}=T_{kn}T_{kn}^t$.
Indeed, let us choose $\Gamma=\mathbb N$ to be the Cayley graph of $O_n^+$, and the eigenvector entries $\mu(r)$ to be the Chebycheff polynomials $P_r(n)$, i.e. the orthogonal polynomials for $O_n^+$.
In this case, we have a bijection $\mathcal P_{o^+}(k)\to L_k$, constructed as follows. For $\pi \in \mathcal P_{o^+}(k)$ and $0\leq i\leq k$ we define $h_\pi(i)$ to be the number of $1\leq j \leq i$ which are joined by $\pi$ to a number strictly larger than $i$. We then define a loop $l(\pi)=l(\pi)_1\ldots l(\pi)_k$, where $l(\pi)_i$ is the edge from $h_{\pi}(i-1)$ to $h_{\pi}(i)$. Consider now the following matrix:
$$T_{kn}(\pi,\sigma)=\prod_{i\sim_\pi j}\delta(l(\sigma)_i,l(\sigma)_j^o)\gamma(l(\sigma)_i)$$
We have $f_\pi=\sum_\sigma T_{kn}(\pi,\sigma)\cdot l(\sigma)$, so we obtain as desired $G_{kn}=T_{kn}T_{kn}^t$.
{\bf Step 3.} We show that, with suitable conventions, $T_{kn}$ is lower triangular.
Indeed, consider the partial order on $\mathcal P_{o^+}(k)$ given by $\pi\leq\sigma$ if $h_\pi(i)\leq h_\sigma(i)$ for $i=1,\ldots,k$. Our claim is that $\sigma\not\leq\pi$ implies $T_{kn}(\pi,\sigma)=0$.
Indeed, suppose that $\sigma\not\leq\pi$, and let $j$ be the least number with $h_\sigma(j)>h_\pi(j)$. Note that we must have $h_\sigma(j-1)=h_\pi(j-1)$ and $h_\sigma(j)=h_\pi(j)+2$. It follows that we have $i\sim_\pi j$ for some $i<j$. From the definitions of $T_{kn}$ and $l(\sigma)$, if $T_{kn}(\pi,\sigma)\neq 0$ then we must have $h_\sigma(i-1)=h_\sigma(j)=h_\pi(j)+2$. But we also have $h_\pi(i-1)=h_\pi(j)$, so that $h_\sigma(i-1)=h_\pi(i-1)+2$, which contradicts the minimality of $j$.
{\bf Step 4.} End of the proof, by computing the determinant of $T_{kn}$.
Since $T_{kn}$ is lower triangular we have:
$$\det(T_{kn})
=\prod_\pi T_{kn}(\pi,\pi)
=\prod_\pi\prod_{i\sim_\pi j}
\sqrt{\frac{P_{h_{\pi(i)}}}{P_{h_{\pi(i)-1}}}}
=\prod_{r=1}^{k/2}P_r^{e_{kr}/2}$$
Here the exponents appearing on the right are by definition as follows:
$$e_{kr}=\sum_\pi\sum_{i\sim_\pi j}\delta_{h_\pi(i),r}-\delta_{h_\pi(i),r+1}$$
Our claim now, which finishes the proof, is that for $1\leq r\leq k/2$ we have:
$$\sum_\pi\sum_{i\sim_\pi j}\delta_{h_\pi(i)r}=f_{k/2,r}$$
Indeed, note that the left term counts the number of times that the edge $(r,r+1)$ appears in all loops in $L_k$. Define a shift operator $S$ on the edges of $\Gamma$ by $S(s,t)=(s+1,t+1)$. Given a loop $l=l_1\ldots l_k$ and $1\leq s\leq k$ with $l_s=(r,r+1)$, define a path $S^r(l_s)\ldots S^r(l_k)l_{s-1}^o\ldots l_1^o$. Observe that this is a path in $\Gamma$ from $2r$ to $0$ whose first edge is $(2r,2r+1)$ and first reaches $r-1$ after $k-s+1$ steps.
Conversely, given a path $f_1\ldots f_k$ in $\Gamma$ from $2r$ to $0$ whose first edge is $(2r,2r+1)$ and first reaches $r-1$ after $s$ steps, define a loop $f_k^o\ldots f_s^oS^{-r}(f_1)\ldots S^{-r}(f_{s-1})$. Observe that this is a loop in $\Gamma$ based at 0 whose $k-s+1$ edge is $(r,r+1)$.
These two operations are inverse to each other, so we have established a bijection between $k$-loops in $\Gamma$ based at $0$ whose $s$-th edge is $(r,r+1)$ and $k$-paths in $\Gamma$ from $2r$ to $0$ whose first edge is $(2r,2r+1)$ and which first reach $r-1$ after $k-s+1$ steps.
It follows that the left hand side is equal to the number of paths in $\Gamma=\mathbb N$ from $2r$ to $0$ whose first edge is $(2r, 2r+1)$. By the usual reflection trick, this is the difference of binomials defining $f_{k/2,r}$, and we are done.
\end{proof}
We use the notation $a_k=\sum_{\pi\in \mathcal P(k)}(2|\pi|-k)$, which already appeared in section 2.
\begin{theorem}
For $B_n^+$ we have:
$$\det(G_{kn})=n^{a_k}\prod_{r=1}^{[k/2]}P_r(n-1)^{\sum_{l=1}^{[k/2]}\binom{k}{2l}d_{lr}}$$
\end{theorem}
\begin{proof}
We have $B_n^+\simeq O_{n-1}^+$, see e.g. \cite{rau}, so we can use the same method as in the proof of Theorem 4.2. By using prime exponents for the various $O_n^+$-related objects, we get:
$$\det(G_{kn})
=n^{a_k}\prod_{l=1}^{[k/2]}{\rm det}'(G_{2l,n-1})^{\binom{k}{2l}}
=n^{a_k}\prod_{l=1}^{[k/2]}\left(\prod_{r=1}^lP_r(n-1)^{d_{lr}}\right)^{\binom{k}{2l}}$$
Together with Theorem 5.1, this gives the formula in the statement.
\end{proof}
\begin{theorem}
For $S_n^+$ we have:
$$\det(G_{kn})=(\sqrt{n})^{a_k}\prod_{r=1}^kP_r(\sqrt{n})^{d_{kr}}$$
\end{theorem}
\begin{proof}
Let $\pi\to\widetilde\pi$ be the ``cabling'' operation, obtained by collapsing neighbors. According to the results of Kodiyalam-Sunder \cite{ksu} and Chen-Przytycki \cite{cpr}, we have:
$$|\pi\vee\sigma|=k/2+2|\widetilde\pi\vee\widetilde\sigma|-|\widetilde\pi|-|\widetilde\sigma|$$
In terms of Gram matrices we get $G_{kn}=D_{kn}G'_{2k,\sqrt{n}}D_{kn}$, where $D_{kn}=diag(n^{|\widetilde\pi|/2-k/4})$, and where $G'$ is the Gram matrix for $O_n^+$, so the result follows from Theorem 5.1.
\end{proof}
\begin{theorem}
For $H_n^+$ we have the formula
$$\det(G_{kn})=(\sqrt{n})^{a_k}\prod_{r=1}^{[k/2]}P_r(\sqrt{n})^{2d_{k/2,r}'}$$
with $d_{sr}'=f_{sr}'-f_{s,r+1}'$, where
$f_{sr}'=\binom{3s}{s-r}-\binom{3s}{s-r-1}$ for $s\in\mathbb Z$, $f_{sr}'=0$ for $s\notin\mathbb Z$.
\end{theorem}
\begin{proof}
According to \cite{bbc}, the diagrams for $H_n^+$ are the ``cablings'' of the Fuss-Catalan diagrams \cite{bjo}, so we can use the same method as in the previous proof. So, by using the above formula from \cite{ksu}, \cite{cpr}, we have $G_{kn}=D_{kn}G'_{2k,\sqrt{n}}D_{kn}$, where $D_{kn}=diag(n^{|\widetilde\pi|/2-k/4})$, and where $G'$ is the Gram determinant for the Fuss-Catalan algebra. But this latter determinant was computed by Di Francesco in \cite{df2}, and this gives the result.
\end{proof}
\section{Algebraic manipulations}
In this section we perform some algebraic manipulations on the formulae found in the previous sections. Consider the quantity $a_k=\sum_{\pi\in \mathcal P(k)}(2|\pi|-k)$, which already appeared, several times. Then $n^{a_k}$ is a true ``contribution'', in the sense of Conjecture 3.3.
We will prove here that a $n^{a_k}$ factor appears naturally, in all the 10 formulae of Gram determinants. This can be regarded as a piece of evidence towards Conjecture 3.3.
In the classical and half-liberated cases there is no need for supplementary work in order to isolate this $n^{a_k}$ factor, and the unified result is as follows.
\begin{theorem}
In the classical and half-liberated cases, we have
\begin{eqnarray*}
S_n,H_n,H_n^*:\quad\det(G_{kn})&=&n^{a_k}\prod_{\pi\in \mathcal P(k)}\frac{n^{k-2|\pi|}n!}{(n-|\pi|)!}\\
O_n:\quad\det(G_{kn})&=&n^{a_k}\prod_{|\lambda|=k/2}f_n(\lambda)^{f^{2\lambda}}\\
B_n:\quad\det(G_{kn})&=&n^{a_k}\prod_{|\lambda|\leq k/2}f_n'(\lambda)^{\binom{k}{2|\lambda|}f^{2\lambda}}\\
O_n^*:\quad\det(G_{kn})&=&n^{a_k}\prod_{|\lambda|=k/2}f_n''(\lambda)^{{f^\lambda}^2}
\end{eqnarray*}
where $f_n^\circ(\lambda)=\prod_{(i,j)\in\lambda}(n+j-i+\varphi^\circ)$, with $\varphi=j-1,\varphi'=j-2,\varphi''=0$.
\end{theorem}
\begin{proof}
This is a reformulation of the results in section 4, by using $a_k=0$ for $O_n,O_n^*$.
\end{proof}
In order to process the formulae in section 5, we need the following technical result.
\begin{proposition}
The Chebycheff polynomials $P_r$ have the following properties:
\begin{enumerate}
\item $P_r(n-1)=Q_r(n)$, with $Q_0=1,Q_1=n-1$ and $Q_{r+1}=(n-1)Q_r-Q_{r-1}$.
\item $P_{2l}(n)=R_{2l}(n^2)$, with $R_0=1,R_2=n-1$ and $R_{2l+2}=(n-2)R_{2l}-R_{2l-2}$.
\item $P_{2l+1}(n)=nR_{2l+1}(n^2)$, with $R_1=1,R_3=n-2$ and $R_{2l+3}=(n-2)R_{2l+1}-R_{2l-1}$.
\item $P_{2l}(n)=n^{-l}S_{2l}(n^2)^{1/2}$, with $S_0=1,S_2=n(n-1)^2$ and so on.
\item $P_{2l+1}(n)=n^{-l}S_{2l+1}(n^2)^{1/2}$, with $S_1=n,S_3=n^2(n-2)^2$ and so on.
\end{enumerate}
\end{proposition}
\begin{proof}
This is routine. As pointed out in section 7 below, $Q_r$ are the orthogonal polynomials for $B_n^+$, and $R_{2l}$ are the orthogonal polynomials for $S_n^+$. As for the polynomials $S_r$, these are some technical objects, introduced in relation with the $H_n^+$ computation.
\end{proof}
\begin{theorem}
In the free cases, we have
\begin{eqnarray*}
O_n^+:\quad\det(G_{kn})&=&n^{a_k}\prod_{r=1}^{[k/2]}P_r(n)^{d_{k/2,r}^1}\\
B_n^+:\quad\det(G_{kn})&=&n^{a_k}\prod_{r=1}^{[k/2]}Q_r(n)^{\sum_{l=1}^{[k/2]}\binom{k}{2l}d_{lr}^1}\\
S_n^+:\quad\det(G_{kn})&=&n^{a_k}\prod_{r=1}^kR_r(n)^{d_{kr}^1}\\
H_n^+:\quad\det(G_{kn})&=&n^{a_k}\prod_{r=1}^{[k/2]}S_r(n)^{d_{k/2,r}^2}
\end{eqnarray*}
where $d_{kr}^i=f_{kr}^i-f_{k,r+1}^i$, with $f_{kr}^i=\binom{(i+1)k}{k-r}-\binom{(i+1)k}{k-r-1}$ for $k\in\mathbb Z$, and $f_{kr}^i=0$ for $k\notin\mathbb Z$.
\end{theorem}
\begin{proof}
The $O_n^+$ formula is the one in Theorem 5.1, with a $n^{a_k}=1$ factor inserted.
The $B_n^+$ formula is the one in Theorem 5.2, with $P_r(n-1)$ replaced by $Q_r(n)$.
For the $S_n^+$ formula, we use Theorem 5.3. By replacing the Chebycheff polynomials $P_{2l},P_{2l+1}$ by the polynomials $R_{2l},R_{2l+1}$ from Proposition 6.2, we get:
$$\det(G_{kn})
=(\sqrt{n})^{a_k}\prod_{r=1}^kP_r(\sqrt{n})^{d_{kr}^1}
=(\sqrt{n})^{a_k}\sqrt{n}^{\sum_{l=1}^{[(k+1)/2]}d_{k,2l-1}^1}\prod_{r=1}^kR_r(n)^{d_{kr}}$$
Now recall from Proposition 2.2 that $a_k=\frac{1}{k+1}\binom{2k}{k}$. On the other hand a direct computation gives $\sum_{l=1}^{[(k+1)/2]}d_{k,2l-1}^1=\frac{1}{k+1}\binom{2k}{k}$, so we get the formula in the statement.
For the $H_n^+$ formula we use a similar method. With $k=2l$, Theorem 5.4 gives:
$$\det(G_{2l,n})=(\sqrt{n})^{a_{2l}}\prod_{r=1}^{l}P_r(\sqrt{n})^{2d_{lr}^2}
=(\sqrt{n})^{a_{2l}}(\sqrt{n})^{-2\sum_{s=2}^l[s/2]d_{ls}^2}\prod_{r=1}^{l}S_r(n)^{d_{lr}^2}$$
Now recall from Proposition 2.2 that $a_{2l}=-2\binom{3l-1}{l-2}$. On the other hand a direct computation gives $\sum_{s=2}^l[s/2]d_{ls}^2=\binom{3l-1}{l-2}$, so we get the formula in the statement.
\end{proof}
As a conclusion, the formulae in Theorem 6.1 and Theorem 6.3 are an intermediate step towards a general decomposition result of type $\det(G_{kn})=\prod_{\pi\in \mathcal P(k)}\varphi(\pi)$. We will come back to the question of finding such a general decomposition result in section 8 below.
\section{Orthogonal polynomials}
We present here a speculation in the free case, in relation with orthogonal polynomials. As we will see, this speculation works for $S_n^+,O_n^+,B_n^+$, but doesn't work for $H_n^+$.
\begin{definition}
The orthogonal polynomials for a real probability measure $\mu$ are the polynomials $Q_0,Q_1,Q_2,\ldots$ satisfying the following conditions:
\begin{enumerate}
\item $Q_k(n)=n^k+a_1n^{k-1}+\ldots+a_{k-1}n+a_k$, with $a_i\in\mathbb R$.
\item For any $k\neq l$ we have $\int Q_k(n)Q_l(n)\,d\mu(n)=0$.
\end{enumerate}
\end{definition}
The orthogonal polynomials can be constructed by using a recursive formula, of type $Q_{k+1}=(n-\alpha_k)Q_k-\beta_kQ_{k-1}$. Here the parameters $\alpha_k,\beta_k\in\mathbb R$ are uniquely determined by the linear equations coming from the fact that $Q_{k+1}$ must be orthogonal to $n^{k-1},n^k$.
More precisely, by solving these two equations we obtain the following formulae, where the integral sign denotes the integration with respect to $\mu$:
$$\alpha_k=\frac{\int n^{k+1}Q_k}{\int n^kQ_k}-\frac{\int n^kQ_{k-1}}{\int n^{k-1}Q_{k-1}},\quad\beta_k=\frac{\int n^kQ_k}{\int n^{k-1}Q_{k-1}}$$
The numbers $\alpha_k,\beta_k$ are called Jacobi parameters of the sequence $\{Q_k\}$. Since $Q_0=1$, in order to describe $\{Q_k\}$ we just need to specify $Q_1$, and the Jacobi parameters.
The orthogonal polynomials for an easy quantum group are by definition those for the asymptotic measure of the main character, given in Theorem 2.6.
\begin{proposition}
The basic orthogonal polynomials are as follows:
\begin{enumerate}
\item $O_n$: here $Q_1=n$ and $Q_{k+1}=nQ_k-kQ_{k-1}$.
\item $B_n$: here $Q_1=n-1$ and $Q_{k+1}=(n-1)P_k-kQ_{k-1}$.
\item $O_n^*$: here $Q_1=n$ and $Q_{k+1}=nQ_k-[(k+1)/2]Q_{k-1}$.
\item $S_n$: here $Q_1=n-1$ and $Q_{k+1}=(n-k-1)Q_k-kQ_{k-1}$.
\item $O_n^+$: here $Q_1=n$ and $Q_{k+1}=nQ_k-Q_{k-1}$.
\item $B_n^+$: here $Q_1=n-1$ and $Q_{k+1}=(n-1)Q_k-Q_{k-1}$.
\item $S_n^+$: here $Q_1=n-1$ and $Q_{k+1}=(n-2)Q_k-Q_{k-1}$.
\end{enumerate}
\end{proposition}
\begin{proof}
This result is well-known, and easy to deduce from definitions. Note that all the polynomials in the above statement are versions of the polynomials appearing in (1,3,4,5), which are respectively the Hermite, Charlier and Chebycheff polynomials.
\end{proof}
Let us go back now to the considerations in section 6. The polynomials $R_{2l}$ appearing in Proposition 6.2 are the orthogonal polynomials for $S_n^+$, and it is natural to call $\{R_n|n\in\mathbb N\}$ the family of ``extended orthogonal polynomials'' for $S_n^+$.
\begin{theorem}
In the $O_n^+,B_n^+,S_n^+$ cases we have a formula of type
$$\det(G_{kn})=n^{a_k}\prod_{r=1}^kQ_r(n)^{d_{kr}}$$
with $d_{kr}\in\mathbb N$, where $Q_r(n)$ are the corresponding extended orthogonal polynomials.
\end{theorem}
\begin{proof}
This follows from Theorem 6.3 and Proposition 7.2.
\end{proof}
Regarding now $H_n^+$, the combinatorics here is that of the Fuss-Catalan algebra \cite{bjo}, see \cite{bb+}, \cite{bbc}. Since $\mu$ is symmetric, the orthogonal polynomials are given by $Q_1=n$ and $Q_{k+1}=nQ_k-\beta_kQ_{k-1}$, where $\beta_k=\gamma_k/\gamma_{k-1}$, with $\gamma_k=\int n^kP_k$. The data is as follows:
\begin{center}
\begin{tabular}[t]{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l}
\hline &1&2&3&4&5&6&7&8\\
\hline $c_k$&1&3&12&55&273&1428&7752&43263\\
\hline $\gamma_k$&1&2&3&11/2&26/3&170/11&17.19/13&19.23/10\\
\hline $\beta_k$&1&2&3/2&11/6&52/33&15.17/11.13&11.19/130&13.23/170\\
\hline
\end{tabular}
\end{center}
\medskip
This suggests the following general formula:
$$\beta_k=
\begin{cases}
\displaystyle{\frac{3(3k-1)(3k+2)}{4(2k-1)(2k+1)}}&(k\ {\rm even})\\
\\
\displaystyle{\frac{3(3k-2)(3k+1)}{4(2k-1)(2k+1)}}&(k\ {\rm odd})
\end{cases}$$
The problem can be probably investigated by using techniques from \cite{hml}, \cite{leh}, \cite{mlo}. Our main problem is of course: what is the analogue of Theorem 7.3 for $H_n^+$?
Let us also mention that the computation of the orthogonal polynomials for $H_n,H_n^*$ looks like a quite difficult problem. Probably the good framework here is that of the quantum groups $H_n^{(s)}$ from \cite{ez1}, because at $s=2,\infty$ we have $H_n,H_n^*$.
We have as well the following question: is there a quantum group/planar algebra proof of Theorem 7.3, in the cases $B_n^+,S_n^+$? For $O_n^+$ this was done in Theorem 5.1.
\section{More manipulations}
We have seen in the previous section that the quantum group $H_n^+$ is somehow of a more complicated nature than the other quantum groups under consideration.
In this section we restrict attention to $O_n^+,B_n^+,S_n^+$, and we further rearrange the formulae in Theorem 6.3. The idea comes from the formula of $O_n^+$. Indeed, the numbers $f_{kr}$ for $O_n^+$ count the $\mathcal P_{o^+}$ diagrams with $2r$ upper points and $2k$ lower points, with the property that each upper point is paired with a lower point. This kind of diagrams, called ``epi" in the paper of Jones, Shlyakhtenko and Walker \cite{jsw}, have the following generalization.
\begin{definition}
Let $\mathcal P$ be a category of partitions, and let $0\leq r\leq k$.
\begin{enumerate}
\item We let $\mathcal P^r(k)$ be the set of partitions $\sigma\in \mathcal P(r,k)$, with $0\leq r\leq k$, such that each upper point is connected to lower points only, and to at least one of them.
\item The elements of $\mathcal P^r(k)$ are called ``epi''. We let $\mathcal P^+(k)=\cup_{r=0}^k\mathcal P^r(k)$. For an epi $\sigma\in \mathcal P^r(k)$, we denote by $r(\sigma)=r$ the number of its upper legs.
\end{enumerate}
\end{definition}
With these notations, we can now state and prove our main result. This is a global formula for the Gram determinants associated to the quantum groups $O_n^+,B_n^+,S_n^+$.
\begin{theorem}
For $O_n^+,B_n^+,S_n^+$ we have the formula
$$\det(G_{kn})=n^{a_k}\prod_{\sigma\in \mathcal P^+(k)}\frac{F_{r(\sigma)}}{F_{r(\sigma)-1}}$$
where $F_r=P_{r/2},Q_{r/2},R_r$ are the corresponding extended orthogonal polynomials.
\end{theorem}
\begin{proof}
Observe first that the $F_r$ quantities in the statement make indeed sense. This is because the epi for $O_n^+,B_n^+$ must have an even number of upper legs.
(1) For $O_n^+$ we have $f_{sr}^1=\#\mathcal P^{2r}(2s)$, so the formula in Theorem 6.3 becomes:
$$\det(G_{kn})=n^{a_k}\prod_{r=0}^{[k/2]}P_r(n)^{f_{k/2,r}-f_{k/2,r+1}}
=n^{a_k}\prod_{r=0}^{[k/2]}P_r(n)^{\#\mathcal P^{2r}(k)-\#\mathcal P^{2r+2}(k)}$$
Now since we have $\mathcal P^+(k)=\cup_{r=0}^{[k/2]}\mathcal P^{2r}(k)$, we obtain the formula in the statement:
$$\det(G_{kn})=n^{a_k}\prod_{r=0}^{[k/2]}\left(\prod_{\sigma\in \mathcal P^{2r}(k)}\frac{Q_r(n)}{P_{r-1}(n)}\right)=n^{a_k}\prod_{\sigma\in \mathcal P^+(k)}\frac{P_{r(\sigma)/2}(n)}{P_{r(\sigma)/2-1}(n)}$$
(2) For $B_n^+$ the epi have, according to our definitions, singletons only in the lower row. Thus these epi can be counted as function of those for $O_n^+$, and we get:
$$\det(G_{kn})=n^{a_k}\prod_{r=0}^{[k/2]}Q_r(n)^{\sum_{l=1}^{[k/2]}\binom{k}{2l}d_{lr}^1}
=n^{a_k}\prod_{r=0}^{[k/2]}Q_r(n)^{\#\mathcal P^{2r}(k)-\#\mathcal P^{2r+2}(k)}$$
A similar manipulation as in (1) gives now the formula in the statement.
(3) For $S_n^+$ the epi are in standard bijection (via fatenning/collapsing of neighbors) with the epi for $O_n^+$. Thus the formula in Theorem 6.3 becomes:
$$\det(G_{kn})=n^{a_k}\prod_{r=0}^kR_r(n)^{d_{kr}^1}=n^{a_k}\prod_{r=0}^kR_r(n)^{\#\mathcal P^r(k)-\#\mathcal P^{2r+1}(k)}$$
Once again, a similar manipulation as in (1) gives the formula in the statement.
\end{proof}
Observe that the quantum group $H_n^+$ cannot be included into the above general theorem, and this for 2 reasons: first, because the orthogonal polynomial interpretation of the polynomials appearing in Theorem 6.3. fails, cf. the previous section, and second, because the epi interpretation of the exponents appearing in Theorem 6.3 seems to fail as well.
\section{Concluding remarks}
We have seen in this paper that the Gram matrix determinants have a natural interpretation in the easy quantum group framework, developed in \cite{bsp}, \cite{ez1}, \cite{ez2}, \cite{ez3}. The known computations, that we partly extended, simplified, or rearranged in this paper, provide a complete set of formulae for the main examples of easy quantum groups.
Our conjecture is that these Gram determinants should have general decompositions of type $\det(G_{kn})=\prod_{\pi\in \mathcal P(k)}\varphi(\pi)$. More precisely, the situation here is as follows:
\begin{enumerate}
\item For $S_n,H_n,H_n^*$ the conjecture holds, with $\varphi(\pi)=n!/(n-|\pi|)!$.
\item For $O_n,B_n,O_n^*$ we have a decomposition result, but over Young diagrams.
\item For $O_n^+,B_n^+,S_n^+$ we have a decomposition result, but over the associated epi.
\end{enumerate}
The remaining problem is to find the correct surjective maps for (2,3), i.e. the correct surjections from diagrams/epi to partitions. Of course, this question is not very clearly formulated. The main problem is probably to understand the behavior of the Gram matrix determinants in relation with the liberation operation $G_n\to G_n^+$. Indeed, we expect in this situation the contributions $\varphi$ to be related by a kind of induction/restriction procedure.
In addition to the concrete computations performed in this paper, let us mention that there are as well some quite heavy, abstract methods, that we haven't really tried yet. First, the inclusion $G_n\subset G_n^+$ gives rise to a planar algebra module in the sense of Jones \cite{jo2}, and our above ``liberation conjecture'' can be understood as saying that the Gram matrix combinatorics behaves well with respect to this planar module structure. And second, modulo the orthogonal polynomial issues discussed in the previous section, some useful tools should come from the analytic theory of the Bercovici-Pata bijection \cite{bpa}.
|
{
"timestamp": "2010-10-04T02:00:37",
"yymm": "1009",
"arxiv_id": "1009.4036",
"language": "en",
"url": "https://arxiv.org/abs/1009.4036"
}
|
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\title{Covariant Dirac Operators on Quantum Groups}
\author{Antti J. Harju\footnote{ Department of Mathematics and Statistics, University of Helsinki, antti.harju@helsinki.fi}}
\date{}
\begin{document}
\maketitle
\begin{abstract}
We give a construction of a Dirac operator on a quantum group based on any simple Lie algebra of classical type. The Dirac operator is an element in the vector space $U_q(\mathfrak{g}) \otimes \mathrm{cl}_q(\mathfrak{g})$ where the second tensor factor is a $q$-deformation of the classical Clifford algebra. The tensor space $ U_q(\mathfrak{g}) \otimes \mathrm{cl}_q(\mathfrak{g})$ is given a structure of the adjoint module of the quantum group and the Dirac operator is invariant under this action. The purpose of this approach is to construct equivariant Fredholm modules and $K$-homology cycles. This work generalizes the operator introduced by Bibikov and Kulish in \cite{BK}. \\
\noindent MSC: 17B37, 17B10\\
\noindent Keywords: Quantum group; Dirac operator; Clifford algebra
\end{abstract}
\textbf{1.} The Dirac operator on a simple Lie group $G$ is an element in the noncommutative Weyl algebra $U(\mathfrak{g}) \otimes \mathrm{cl}(\mathfrak{g})$ where $U(\mathfrak{g})$ is the enveloping algebra for the Lie algebra of $G$. The vector space $\mathfrak{g}$ generates a Clifford algebra $\mathrm{cl}(\mathfrak{g})$ whose structure is determined by the Killing form of $\mathfrak{g}$. Since $\mathfrak{g}$ acts on itself by the adjoint action, $ U(\mathfrak{g}) \otimes \mathrm{cl}(\mathfrak{g})$ is a $\mathfrak{g}$-module. The Dirac operator on $G$ spans a one dimensional invariant submodule of $ U(\mathfrak{g}) \otimes \mathrm{cl}(\mathfrak{g})$ which is of the first order in $\mathrm{cl}(\mathfrak{g})$ and in $U(\mathfrak{g})$. Kostant's Dirac operator \cite{Kos} has an additional cubical term in $\mathrm{cl}(\mathfrak{g})$ which is constant in $U(\mathfrak{g})$.
The noncommutative Weyl algebra is given an action on a Hilbert space which is also a $\mathfrak{g}$-module. Since the Dirac operator is the invariant subspace it follows that it commutes with the action of $\mathfrak{g}$ and hence the Dirac operator acts as a constant on each irreducible component in the representation of $\mathfrak{g}$ on the Hilbert space. The spectrum of the Dirac operator captures the metric properties of the the Riemannian manifold $G$ \cite{Con}.
In \cite{BK} Bibikov and Kulish considered the quantum group deformation of $\mathfrak{su(2)}$ and constructed a Dirac operator which is invariant under the adjoint action of the quantum group. In this approach the spectrum of the operator grows exponentially as a function of the highest weight of the representation of the quantum group. The sign operators of this type of Dirac operators can be used to define equivariant Fredholm modules which can be applied in the study of $D$-branes, $K$-homology, index theory and cyclic cohomology.
The purpose of this paper is to study further the approach of the reference \cite{BK}. We are looking for an operator in the vector space $U_q(\mathfrak{g}) \otimes \mathrm{cl}_q(\mathfrak{g})$, where $\mathrm{cl}_q(\mathfrak{g})$ is a $q$-deformation of the Clifford algebra which transforms covariantly under the action of the quantum group. Furthermore, we postulate the following defining principles for the covariant Dirac operator $D$
\begin{enumerate}
\item Considered as a submodule of $U_q(\mathfrak{g}) \otimes \mathrm{cl}_q(\mathfrak{g})$, $D$ spans a one dimensional trivial module under the adjoint action of $U_q(\mathfrak{g})$.
\item Considered as an operator on a Hilbert space, $D$ commutes with the representation of $U_q(\mathfrak{g})$.
\end{enumerate}
For the Lie algebras the property 2. is a consequence of 1. However, if we let a quantum group act on the tensor product $U_q(\mathfrak{g}) \otimes \mathrm{cl}_q(\mathfrak{g})$ with its coproduct, we need to choose the module structures in $\mathrm{cl}_q(\mathfrak{g})$ and $U_q(\mathfrak{g})$ carefully to make an operator with property 1. verify 2. This is not a general fact and will be explained in 8.
We consider the adjoint representation of the quantum group and define a bilinear form in this module which is invariant under the action. The braiding operator $\check{R}$ commutes with the coproduct of $U_q(\mathfrak{g})$ and it can be considered as a $q$-analogue of the permutation of a tensor product. We let $\check{R}$ act on a tensor product of adjoint representations and use the spectral decomposition of this action to define the $q$-Clifford algebra. The eigenvectors of $\check{R}$ split into two parts which can be considered as $q$-deformations of symmetric and antisymmetric tensor products. We identify the '$q$-symmetric' tensors with their image in the bilinear form. The practical difficulty in this approach is that there are no general formulas for the spectral decompositions. However, the explicit form of the $\check{R}$-operator is well known and so one can solve the eigenvalue problem with some mathematical software for any chosen quantum group and apply our results.
We give a constructive proof for the existence of the covariant Dirac operator with the properties 1. and 2. on any quantum group based on any complex simple Lie algebra of classical type. The deformation parameter $q$ is supposed to be strictly positive real number. We give an explicit construction in the case of $SU_q(2)$ and build a Fredholm module from the sign operator. \\
\textbf{2. Conventions.}\ Let $\mathfrak{g}$ be a simple finite dimensional Lie algebra with a set of simple roots $\Delta = \{\alpha_i: 1 \leq i \leq n\}$ and Cartan matrix $a_{ij}$. Let $q\neq 1$ be a complex number. The quantum group $U_q(\mathfrak{g})$ is the unital associative algebra with generators $k_{i}, k_{i}^{-1}, e_{i}, f_{i}$ ($1 \leq i \leq n$) subject to \cite{Dr,Ji}
\begin{eqnarray*}
& &[k_{i}, k_{j}] = 0,\hspace{0,5cm} k_i k_i^{-1} = 1 \hspace{0,5cm} k_{i} e_j k_{i}^{-1} = q_i^{a_{ij}/2} e_j,\hspace{0,5cm} k_{i} f_j k_{i}^{-1} = q_i^{-a_{ij}/2} f_j, \\
& &[e_i, f_j] = \delta_{ij}\frac{k_{i}^2 - k_{i}^{-2}}{q_i -q_i^{-1}}, \\
& &\sum_{s=0}^{1-{a_{ij}}}(-1)^s \left[ \begin{array}{c}
1- a_{ij} \\
s
\end{array} \right]_{q_i} e_i^{s} e_j e_i^{1- a_{ij} - s} = 0 = \sum_{s=0}^{1-{a_{ij}}}(-1)^s \left[ \begin{array}{c}
1- a_{ij} \\
s
\end{array} \right]_{q_i} f_i^{s}f_j f_i^{1- a_{ij} - s}\hspace{0,5cm} (i \neq j),
\end{eqnarray*}
where $q_i = q^{d_i}$, $d_i$'s being the coprime integers such that $d_i a_{ij}$ is a symmetric matrix and the $q$-binomial coefficients are defined by
\begin{eqnarray*}
& &(m)_{q_i} = (q_i-q_i^{-1})(q_i^2-q_i^{-2}) \cdots (q_i^m -q_i^{-m}),\hspace{0,5cm} [0]_{q_i} = 1 \\
& &\left[ \begin{array}{c}
m \\
n
\end{array} \right]_{q_i} = \frac{(m)_{q_i}}{(n)_{q_i}(m-n)_{q_i}}.
\end{eqnarray*}
In the limit $q \rightarrow 1$ the algebra $U_q(\mathfrak{g})$ reduces to $U(\mathfrak{g})$. $U_q(\mathfrak{g})$ is a Hopf algebra with a coproduct $\triangle: U_q(\mathfrak{g}) \rightarrow U_q(\mathfrak{g}) \otimes U_q(\mathfrak{g})$, an antipode $S: U_q(\mathfrak{g}) \rightarrow U_q(\mathfrak{g})$ and a counit $\epsilon: U_q(\mathfrak{g}) \rightarrow \mathbb{C}$. The vectors $k_i$ and $k_i^{-1}$ are grouplike so that $\epsilon(k_i)= \epsilon(k_i^{-1}) = 1$ whereas $\epsilon(e_i) = \epsilon(f_i) = 0$. The coproduct $\triangle$ is noncocommutative and there exists a universal $R$-matrix such that
\begin{eqnarray*}
R \triangle(x) R^{-1} = \sigma \triangle(x),
\end{eqnarray*}
where $\sigma$ permutes the tensor product. The $R$-matrix is an infinite sum defined in some completion of the tensor product $U_q(\mathfrak{g})\otimes U_q(\mathfrak{g})$ but only a finite number of terms are nonzero in any finite dimensional representation. Here we always assume that the parameter $q$ is a strictly positive real number. In this case the braiding operator $\check{R} = \sigma R$ is a selfadjoint operator in any finite dimensional representation. In the limit $q \rightarrow 1$, $\check{R}$ becomes the permutation operator.
Let $\mathfrak{h}$ be the Cartan subalgebra of $\mathfrak{g}$ with a basis $h_i$ satisfying $\alpha_j(h_i) = a_{ij}$. The representation $(V_{\lambda}, \pi_{\lambda,q})$ of $U_q(\mathfrak{g})$ is of the highest weight $\lambda \in \mathfrak{h}^*$ if there exists a vector $\xi$ so that $\pi_{\lambda,q} (e_i) \xi = 0$ and $\pi_{\lambda,q}(k_i)\xi = q^{\lambda(h_i)/2}\xi$ for all $i$ and the action of the operators $\pi_{\lambda,q}(f_i)$ on $\xi$ generate $V_{\lambda}$.
Denote by $P_+$ the set of integral dominant weights of a simple Lie algebra $\mathfrak{g}$. As was shown in \cite{Lus,ros} the theory of finite dimensional representations of $\mathfrak{g}$ and quantum group $U_q(\mathfrak{g})$ are identical in the case $q$ is not a root of unity. A highest weight module $(V_{\lambda},\pi_{\lambda,q})$ of $U_q(\mathfrak{g})$ is finite dimensional if and only if $\lambda \in P_+$. If $\lambda \in P_+$ then the dimension of each weight space is equal to the dimension of the corresponding weight space in the highest weight module $( V_{\lambda}, \pi_{\lambda})$ of $U(\mathfrak{g})$. The category of representations is semisimple with simple objects $(V_{\lambda}, \pi_{\lambda,q})$, $\lambda \in P_+$. For each morphism in the category of $U(\mathfrak{g})$ representations there exists a corresponding morphism in the category of $U_q(\mathfrak{g})$ representations.
The matrix elements of an irreducible module $(V_{\lambda},\pi_{\lambda,q})$ of $U_q(\mathfrak{g})$ depend continuously on the parameter $q$ so that in the limit $q \rightarrow 1$, $\pi_{\lambda,q}(e_i)$ and $\pi_{\lambda,q}(f_i)$ define representation matrices for generators of $U(\mathfrak{g})$. These operators fix the representation $(V_{\lambda},\pi_{\lambda})$ of $U(\mathfrak{g})$ completely.
We shall drop the $q$-subscript from the representations most often used in this work and denote by $(U, \pi)$, $(U^*, \pi^*)$ and $(V, \rho)$ the defining representation, its dual and the adjoint representation of $U_q(\mathfrak{g})$.
The $q$-integers are defined by
\begin{eqnarray*}
[n] = \frac{q^n - q^{-n}}{q-q^{-1}}.
\end{eqnarray*}
\textbf{3.} The adjoint action of the quantum group on itself is an algebra homomorphism $U_q(\mathfrak{g}) \times U_q(\mathfrak{g}) \rightarrow U_q(\mathfrak{g})$ defined by
\begin{eqnarray*}
x \stackrel{\mathrm{ad}}{\blacktriangleright} y = x'y S(x''),
\end{eqnarray*}
where $\triangle(x) = x' \otimes x''$. One can find a finite dimensional submodule in $U_q(\mathfrak{g})$ which is isomorphic to the adjoint representation of the quantum group \cite{Del}. Let $q = e^h$, $X = h^{-1} (R^{t}R-1)$ where $R^t = \sigma R \sigma$ and
\begin{eqnarray*}
X_{lk} = (\pi_{lk} \otimes \mathrm{id})X \in \mathbb{C} \otimes U_q(\mathfrak{g}) \simeq U_q(\mathfrak{g}),
\end{eqnarray*}
where $\pi_{lk}$ are the matrix elements of the defining representation of $U_q(\mathfrak{g})$. Denote by $\pi^*$ the dual representation
\begin{eqnarray*}
\pi_{il}^*(x) = \pi_{li}(S(x)).
\end{eqnarray*}
The vectors $X_{lk}$ transform covariantly under the adjoint action
\begin{eqnarray*}
x \stackrel{\mathrm{ad}}{\blacktriangleright} X_{lk} = \sum_{i,j} X_{ij} \pi^*_{il}(x') \pi_{jk}(x''),\hspace{0,5cm} \hbox{for all}\hspace{0,5cm} x \in U_q(\mathfrak{g}).
\end{eqnarray*}
According to the termionology of \cite{Del}, a (weak) quantum Lie algebra is an invariant submodule in $U_q(\mathfrak{g})$ which is a deformation of $\mathfrak{g}$ and transforms covariantly under the adjoint action. Denote by $\{u_i\}$ and $\{u_i^*\}$ the basis vectors of the defining representation and its dual and by $\{v_i\}$ the basis of the adjoint representation. Using the matrix coefficients of the module isomorphism $v_a \mapsto \sum_{i,j} K_a^{ij}(u_i^* \otimes u_j)$ we define
\begin{eqnarray*}
X_a = \sum_{i,j} K_{a}^{ij}(\pi_{ij} \otimes \mathrm{id}) X.
\end{eqnarray*}
$X_a$'s span a quantum Lie algebra $\mathfrak{L}_q(\mathfrak{g})$ inside $U_q(\mathfrak{g})$ which is isomorphic to the adjoint representation of $U_q(\mathfrak{g})$. \\
\textbf{4.} Let $\mathfrak{g}$ be a simple Lie algebra of classical type. The adjoint representation can be considered as an invariant submodule $V \subset U^* \otimes U$ for the action
\begin{eqnarray*}
x \stackrel{\mathrm{ad}}{\triangleright} (u_l^* \otimes u_k) = \sum_{i,j} \pi_{li}(S(x')) u_i^* \otimes \pi_{jk}(x'') u_j
\end{eqnarray*}
for all $x \in U_q(\mathfrak{g})$ and $u_l^* \otimes u_k \in V$. \\
\noindent \textbf{Proposition.} There exists a nondegenerate bilinear form $B_q: V \otimes V \rightarrow \mathbb{C}$ which is invariant under the adjoint action of $U_q(\mathfrak{g})$, i.e.
\begin{eqnarray*}
B_q(x \stackrel{\mathrm{ad}}{\triangleright} (v \otimes w)):= B_q(x'\stackrel{\mathrm{ad}}{\triangleright} v \otimes x''\stackrel{\mathrm{ad}}{\triangleright} w) = \epsilon(x) B_q(v \otimes w).
\end{eqnarray*}
$B_q$ is unique up to a multiplicative constant. \\
\noindent Proof. The adjoint representation $(V, \rho)$ and the dual representation $(V^*, \rho^*)$ are isomorphic. Let $\tau: V \rightarrow V^*$ denote a module isomorphism. Choose a basis $\{v_i\}$ of $V$ and let $\{v_i^*\}$ denote the dual basis. The canonical pairing defined on the generators by $\mathrm{eval}(v_j^* \otimes v_k) = v_j^* (v_k) = \delta_{jk}$ is nondegenerate. Thus, the composition
\begin{eqnarray*}
B_q:& & V \otimes V \stackrel{\sim}{\rightarrow} V^* \otimes V \stackrel{\mathrm{eval}}{\rightarrow} \mathbb{C}, \\
& &v \otimes w \mapsto \mathrm{eval}(\tau(v) \otimes w)
\end{eqnarray*}
is nondegenerate. $B_q$ is invariant because
\begin{eqnarray*}
& &B_q(x'\stackrel{\mathrm{ad}}{\triangleright} v \otimes x''\stackrel{\mathrm{ad}}{\triangleright} w) = \mathrm{eval}(\tau( x'\stackrel{\mathrm{ad}}{\triangleright} v) \otimes x''\stackrel{\mathrm{ad}}{\triangleright} w) = \tau(v)(S(x')x''\stackrel{\mathrm{ad}}{\triangleright} (w)) = \epsilon(x) B_q(v \otimes w),
\end{eqnarray*}
for all $x \in U_q(\mathfrak{g})$ and $v,w \in V$.
Let $\phi$ be the map $V \rightarrow V^*$ which sends $v \in V$ to the functional $\phi(v) (w) = B_q(v \otimes w) \in V^*$. Using the Hopf algebra axioms we see that $\phi$ is a module homomorphism:
\begin{eqnarray*}
\phi(x\stackrel{\mathrm{ad}}{\triangleright} v)(w) &=& \phi(x'\stackrel{\mathrm{ad}}{\triangleright} v)(\epsilon(x'')w) = \phi(x'\stackrel{\mathrm{ad}}{\triangleright} v) ((x''S(x'''))\stackrel{\mathrm{ad}}{\triangleright} w) \\
&=& B_q(x'\stackrel{\mathrm{ad}}{\triangleright} v \otimes x''\stackrel{\mathrm{ad}}{\triangleright} (S(x''')\stackrel{\mathrm{ad}}{\triangleright} w)) = \epsilon(x') B_q(v \otimes S(x'')\stackrel{\mathrm{ad}}{\triangleright} w) \\
&=& \phi(v)(S(x)\stackrel{\mathrm{ad}}{\triangleright} w)
\end{eqnarray*}
for all $x \in U_q(\mathfrak{g})$ and $v, w \in V$. Furthermore, $\phi$ is a module isomorphism because $B_q$ is nondegenerate. If $\theta$ is another module isomorphism $\theta : V \rightarrow V^*$ we can define a module isomorphism $\theta^{-1} \circ \phi: V \rightarrow V$ which commutes with the action of the quantum group. $\mathfrak{g}$ is simple and so the adjoint module $V$ is irreducible and the uniqueness follows from Schur's lemma. $\square$ \\
In practical calculations the form $B_q$ is easiest to find by fixing the costants directly from the invariance condition. \\
\textbf{5.} Let $\check{R}_i=\sigma_i R_i$ ( $1 \leq i \leq N-1$) be the linear operator where $R_i$ is the $R$ matrix acting on the $i$'th and $(i+1)$'th component in the tensor product space $V^{\otimes N}$ and $\sigma_i$ permutes the tensor components. The braiding operator $\check{R}_i$ commutes with the action of $U_q(\mathfrak{g})$ on the tensor product and thus the eigenspaces of $\check{R}_i$ are invariant subspaces of $U_q(\mathfrak{g})$. $\check{R}_i$ is a selfadjoint operator and its eigenvalues are real. Furthermore, the eigenvalue of a nonzero eigenspace is not equal to zero for any $q > 0$ because $\check{R}_i$ is an isomorphism. Thus, the tensor product splits into parts consisting of the vectors with strictly positive eigenvalues and strictly negative eigenvalues for any allowed value of $q$. In the classical limit $q \rightarrow 1$ these eigenspaces become the symmetric and antisymmetric tensor products in the $i$'th and $(i+1)$'th component.
Given a spectral resolution of $\check{R}_i$ denote by $\{ a_{i,k}: k \in I\}$ the negative eigenvalues and by $\{ b_{i,k}: k \in J\}$ the positive eigenvalues of $\check{R}_i$. The braiding operators form a generalized Hecke-algebra with relations
\begin{eqnarray*}
& &\check{R}_{i} \check{R}_{i+1} \check{R}_{i} = \check{R}_{i+1} \check{R}_{i} \check{R}_{i+1}\\
& &\check{R}_{i} \check{R}_{j} = \check{R}_{j} \check{R}_{i}, \\
& & \prod_{k \in I}(\check{R}_{i}- a_{i,k})\prod_{l \in J}(\check{R}_{i}-b_{i,l}) = 0
\end{eqnarray*}
for all $1 \leq i,j \leq N-1$ and $|i-j|>1$.
Let $T(V)$ be the tensor algebra of $V$. We define the covariant Clifford algebra as a projection of $T(V)$ on the $q$-analogue of the antisymmetric tensor products by
\begin{eqnarray*}
\mathrm{cl}_q(\mathfrak{g}) = T(V)/\mathfrak{I}
\end{eqnarray*}
where the ideal $\mathfrak{I}$ is defined by
\begin{eqnarray*}
\mathfrak{I} = \{ (\mathrm{id} - B_q^{i})v : v \in \mathrm{Ker}(\check{R}_i - b_{i,k})\hspace{0,5cm} \hbox{for some}\hspace{0,5cm} i \in \mathbb{N}, k \in J \}.
\end{eqnarray*}
$B_q^{i}$ is the invariant pairing of $i$'th and $(i+1)$'th tensor component. $\mathrm{cl}_q(\mathfrak{g})$ transforms covariantly under the adjoint action of $U_q(\mathfrak{g})$ because $B^i_q$ is invariant and the operators $\check{R}_i - b_{i,k}$ commute with the action of $U_q(\mathfrak{g})$. Therefore, it is a $U_q(\mathfrak{g})$-module algebra. Let us denote by $\gamma_q: V \rightarrow \mathrm{cl}_q(\mathfrak{g})$ the canonical embeddings. \\
\textbf{6.} There exists a Lie algebra homomorphism $\widetilde{\mathrm{ad}}: \mathfrak{g} \rightarrow \mathrm{cl}(\mathfrak{g})$ which satisfies the equivariance condition
\begin{eqnarray*}
\gamma([x,y]) = [\widetilde{\mathrm{ad}}(x),\gamma(y)],
\end{eqnarray*}
for all $x,y \in \mathfrak{g}$ where $\gamma$ is the canonical embedding $\mathfrak{g} \rightarrow \mathrm{cl}(\mathfrak{g})$. Once the irreducible representation $(\Sigma,s)$ of $\mathrm{cl}(\mathfrak{g})$ is given this fixes the representation of $\mathfrak{g}$ on $\Sigma$. \\
\noindent \textbf{Proposition.} The algebras $\mathrm{cl}_q(\mathfrak{g})$ and $U_q(\mathfrak{g})$ have representations $s_q$ and $\sigma_q$ on $\Sigma$ which satisfy the $q$-deformed equivariance condition
\begin{eqnarray}\label{comp}
s_q (\gamma_q(x \stackrel{\mathrm{ad}}{\triangleright} v)) = \sigma_q(x') s_q(\gamma_q(v)) \sigma_q(S(x''))
\end{eqnarray}
for any $x \in U_q(\mathfrak{g})$ and $v \in V$. The representation $s_q$ of $\mathrm{cl}_q(\mathfrak{g})$ is irreducible. \\
\noindent Proof. Denote by $(\Sigma, s)$ an irreducible $\mathrm{cl}(\mathfrak{g})$ representation. The vector space $B(\Sigma)$ of all endomorphisms of $\Sigma$ is a $\mathfrak{g}$-module equipped with the action given by commutator
\begin{eqnarray*}
x \triangleright T = s( \widetilde{\mathrm{ad}}(x')) T s (\widetilde{\mathrm{ad}}(S(x''))).
\end{eqnarray*}
Let us denote by $\sigma_q$ the representation of $U_q(\mathfrak{g})$ on $\Sigma$ which corresponds to the representation $s(\widetilde{\mathrm{ad}})$ of $U(\mathfrak{g})$ in the classical limit. The action
\begin{eqnarray*}
x \triangleright_q T = \sigma_q(x') T \sigma_q(S(x''))
\end{eqnarray*}
defines a representation of $U_q(\mathfrak{g})$ on $B(\Sigma)$. The module $B(\Sigma)$ is reducible and the decomposition to irreducible subspaces is the same as in the classical case.
The proof is based on the following observation. If $X$ and $Y$ are $U(\mathfrak{g})$ or $U_q(\mathfrak{g})$ submodules of $B(\Sigma)$ then the space $XY$ spanned by the matrix products is also a $U(\mathfrak{g})$ or $U_q(\mathfrak{g})$ submodule of $B(\Sigma)$. This is true because by using the Hopf algebra properties we get
\begin{eqnarray*}
x \triangleright MN = (x' \triangleright M)(x'' \triangleright N) \in XY
\end{eqnarray*}
for all $x \in U(\mathfrak{g})$, $M \in X$ and $N \in Y$. The corresponding formula holds also for the quantum group action $\triangleright_q$. In both cases the matrix multiplication defines a module homomorphism $m: X \otimes Y \rightarrow XY$ where the action on the tensor product is given by composing the coproduct with $\triangleright$ or $\triangleright_q$.
By equivariance, the submodule $B(\Sigma)' = s(\gamma(\mathfrak{g}))$ is isomorphic to the adjoint representation. The module $B(\Sigma)' \otimes B(\Sigma)'$ reduces to invariant symmetric and antisymmetric components. The multiplication restricted to the symmetric submodule is a module homomorphism getting values in the trivial module $\mathbb{C} \textbf{1}$
\begin{eqnarray*}
m: s(\widetilde{\mathrm{ad}}(x)) \otimes s(\widetilde{\mathrm{ad}}(y)) + s(\widetilde{\mathrm{ad}}(y)) \otimes s(\widetilde{\mathrm{ad}}(x)) \mapsto B(x,y)\textbf{1},
\end{eqnarray*}
for all $x,y \in \mathfrak{g}$, where $B$ is proportional to the Killing form.
Denote by $B(\Sigma)_q'$ the submodule of $B(\Sigma)$ which is isomorphic to the adjoint representation of $U_q(\mathfrak{g})$ and reduces to the representation $s(\gamma(\mathfrak{g}))$ in the classical limit. The tensor product $B(\Sigma)_q' \otimes B(\Sigma)_q'$ decomposes again into invariant components which are symmetric and antisymmetric in the $q$-deformed sense. The multiplication restricted to the $q$-symmetric part is a module homomorphism and must get values in the trivial module $\mathbb{C} \textbf{1}$ because the representation theory corresponds to the classical one. Especially the homomorphism must be determined by the invariant bilinear form which is unique. Therefore we find a representation $s_q$ of $\mathrm{cl}_q(\mathfrak{g})$ on $\Sigma$ which satisfies \eqref{comp} with $\sigma_q$.
It remains to show irreducibility. If the dimension of $\mathfrak{g}$ is $2l$ the irreducible representation $\mathrm{cl}(\mathfrak{g}) \rightarrow B(\Sigma)$ is an algebra isomorphism. If the dimension is $2l+1$ there exists two nonisomorphic irreducible representations which give an algebra isomorphism $\mathrm{cl}(\mathfrak{g}) \rightarrow B(\Sigma) \oplus B(\Sigma)$. Since $\gamma(\mathfrak{g})$ generates $\mathrm{cl}(\mathfrak{g})$ as an algebra, each endomorphism algebra $B(\Sigma)$ must be generated by the subspace $s(\gamma(\mathfrak{g})) = B(\Sigma)'$. In terms of representation theory this has the following interpretation: Starting with $B(\Sigma)' \otimes B(\Sigma)'$ one can apply the multiplication homomorphism to construct more irreducible $U(\mathfrak{g})$ representations and after taking finite number of steps each irreducible component of $B(\Sigma)$ is found. Similarly, we can consider $B(\Sigma)$ as $U_q(\mathfrak{g})$-module and apply the multiplication process to $B(\Sigma)_q'$ which gives all the irreducible components of irreducible $U_q(\mathfrak{g})$ representations in $B(\Sigma)$, but then the module algebra generated by $B(\Sigma)'_q$ is the whole space $B(\Sigma)$ and therefore no nontrivial invariant subspaces exist. $\hspace{0,5cm} \square$ \\
\noindent \textbf{Corollary} The algebras $\mathrm{cl}_q(\mathfrak{g})$ and $\mathrm{cl}(\mathfrak{g})$ are isomorphic as associative algebras. \\
\noindent Proof. It follows from the above proof that the representations $\mathrm{cl}_q(\mathfrak{g}) \rightarrow B(\Sigma)$ in the $2l$ dimensional case and $\mathrm{cl}_q(\mathfrak{g}) \rightarrow B(\Sigma) \otimes B(\Sigma)$ in the $2l+1$ dimensioal case are onto. Since the space of symmetric tensor products preserve the dimensionality in the $q$-deformation we get $\mathrm{dim}(\mathrm{cl}_q(\mathfrak{g}))=\mathrm{dim}(\mathrm{cl}(\mathfrak{g}))$. Then $\mathrm{dim}(\mathrm{cl}_q(\mathfrak{g})) = \mathrm{dim}(B(\Sigma))$ or $\mathrm{dim}(\mathrm{cl}_q(\mathfrak{g})) = 2\mathrm{dim}(B(\Sigma))$ in $2l$ and $2l+1$ dimensions. Thus, the irreducible representations define algebra isomorphisms
\begin{eqnarray*}
& &s_q: \mathrm{cl}_q(\mathfrak{g}) \rightarrow B(\Sigma),\hspace{0,5cm} \hbox{dim$(\mathfrak{g}) = 2l$} \\
& &s_q^1 \oplus s_q^2 : \mathrm{cl}_q(\mathfrak{g}) \rightarrow B(\Sigma) \oplus B(\Sigma),\hspace{0,5cm} \hbox{dim$(\mathfrak{g}) = 2l+1$} .
\end{eqnarray*}
Especially, the algebras $\mathrm{cl}_q(\mathfrak{g})$ and $\mathrm{cl}(\mathfrak{g})$ are isomorphic. $\hspace{0,5cm} \square$ \\
\textbf{7.} We first define an invariant one dimensional subspace in the vector space $V \otimes V^*$ and then define $D$ in the image of this subspace in a module isomorphism from $V \otimes V^*$ to a submodule of $U_q(\mathfrak{g}) \otimes \mathrm{cl}_q(\mathfrak{g})$. The module structure of $ U_q(\mathfrak{g}) \otimes \mathrm{cl}_q(\mathfrak{g})$ is chosen so that $D$ commutes with the representation. \\
\noindent \textbf{Proposition.} Let $\{v_i \}$ denote the basis of $V$ and $\{v_i^*\}$ the dual basis. The vector $\Omega \in V \otimes V^*$ defined by
\begin{eqnarray*}
\Omega = \sum_{i} v_i \otimes v_i^*
\end{eqnarray*}
is invariant under the action of $U_q(\mathfrak{g})$. \\
\noindent Proof. For all $x \in U_q(\mathfrak{g})$
\begin{eqnarray*}
x \stackrel{\mathrm{ad}}{\triangleright} \Omega &=& \sum_{i,k,l} \rho_{ki}(x') v_k \otimes \rho^*_{li}(x'') v^*_l \\
&=& \sum_{i,k,l} \rho_{ki}(x')\rho_{il}(S(x''))v_k \otimes v^*_l \\
&=& \sum_{k,l} \rho_{kl}(x' S(x''))v_k \otimes v^*_l \\
&=& \sum_{k,l} \epsilon(x) \delta_{kl} v_k \otimes v^*_l = \epsilon(x)\Omega.\hspace{0,5cm} \square
\end{eqnarray*}
\textbf{8.} Representations on tensor products are dependent on the choice of the Hopf algebra structure for the quantum group. If we fix a Hopf structure, then by definition, the adjoint action on a vector $Z \otimes \psi \in U_q(\mathfrak{\mathfrak{g}}) \otimes \mathrm{cl}_q(\mathfrak{g})$ is
\begin{eqnarray}\label{naive}
x \stackrel{\mathrm{ad}}{\triangleright} (Z \otimes \psi) := x' \stackrel{\mathrm{ad}}{\blacktriangleright} Z \otimes x'' \stackrel{\mathrm{ad}}{\triangleright} \psi.
\end{eqnarray}
Let $\phi': V \rightarrow \mathfrak{L}_q(\mathfrak{g})$ and $\tau: V^* \rightarrow V$ be module isomorphisms. We could try to define the Dirac operator by $A = (\phi' \otimes \gamma_q \circ \tau)(\Omega)$. This is certainly invariant under the action \eqref{naive} of the quantum group. Let us write $A = \sum_{i,j} \alpha_{ij} X_i \otimes \psi_j$ for some complex numbers $\alpha_{ij}$.
Denote by $s_q$ the irreducible representation of $\mathrm{cl}_q(\mathfrak{g})$ and $\sigma_q$ the representation of $U_q(\mathfrak{g})$ on $\Sigma$ which satisfy \eqref{comp}. Let $U_q(\mathfrak{g})$ act on $V_{\lambda} \otimes \Sigma$ by $x \mapsto \pi_{\lambda,q}(x') \otimes \sigma_q(x'')$. Even though $A$ is invariant it fails to commute with the representation of $U_q(\mathfrak{g})$. This can be seen using $x = x'\epsilon(x'')$ and $\epsilon(x) = S(x')x''$ twice
\begin{eqnarray*}
xA.| \omega \rangle &=& (\pi_{\lambda,q}(x') \otimes \sigma_q(x''))(\sum_{i,j} \alpha_{ij} \pi_{\lambda,q}(X_i) \otimes s_q(\psi_j))| \omega \rangle \\
&=& (\sum_{i,j} \alpha_{ij} \pi_{\lambda,q}(x' \stackrel{\mathrm{ad}}{\blacktriangleright} X_i) \otimes s_q(x''' \stackrel{\mathrm{ad}}{\triangleright} \psi_j))(\pi_{\lambda,q}(x'') \otimes \sigma_q(x''''))| \omega \rangle \\
&\neq& \epsilon(x')(\sum_{i,j} \alpha_{ij} \pi_{\lambda,q}(X_i) \otimes s_q(\psi_j))(\pi_{\lambda,q}(x'') \otimes \sigma_q(x'''))| \omega \rangle = Ax.| \omega \rangle
\end{eqnarray*}
for some $x\in U_q(\mathfrak{g})$ and $| \omega \rangle \in V_{\lambda} \otimes \Sigma$ because of the noncocommutativity of the coproduct. \\
\textbf{9.} The problem above can be cured by modifying the structure of the module $\mathfrak{L}_q(\mathfrak{g})$. Let us choose the primary Hopf algebra structure for $U_q(\mathfrak{g})$ by
\begin{eqnarray*}
& &\triangle(e_i) = e_i \otimes k_i + k_i^{-1} \otimes e_i,\hspace{0,5cm} \triangle(f_i) = f_i \otimes k_i + k_i^{-1} \otimes f_i,\hspace{0,5cm} \triangle(k_i) = k_i \otimes k_i \\
& & S(k_i) = k_i^{-1},\hspace{0,5cm} S(e_i) = - q e_i,\hspace{0,5cm} S(f_i) = - q^{-1} f_i.
\end{eqnarray*}
The Sweedler's notation $\triangle(x) = x' \otimes x''$ is always applied to this one. The opposite Hopf algebra structure is defined by
\begin{eqnarray*}
\triangle^{\mathrm{op}}(x) = x'' \otimes x',\hspace{0,5cm} S^{\mathrm{op}}(k_i) = k_i^{-1},\hspace{0,5cm} S^{\mathrm{op}}(e_i) = - q^{-1} e_i ,\hspace{0,5cm} S^{\mathrm{op}}(f_i) = - q f_i.\
\end{eqnarray*}
We apply the method of 3. to construct a quantum Lie algebra for the Hopf algebra $U_q(\mathfrak{g})$ with the opposite Hopf structure. This gives us a module $\mathfrak{L}_q^{\mathrm{op}}(\mathfrak{g})$ with a basis $\{Z_i\}$ transforming as
\begin{eqnarray*}
x \stackrel{\mathrm{oad}}{\blacktriangleright} Z_i = x''Z_i S^{\mathrm{op}}(x') = \sum_j \rho_{ji}(x) Z_j
\end{eqnarray*}
for each $x \in U_q(\mathfrak{g})$ and $\rho_{ji}$ are the matrix elements of the fixed adjoint representation $V$. Denote by $\phi$ the module isomorphism $V \rightarrow \mathfrak{L}^{\mathrm{op}}_q(\mathfrak{g})$. \\
\noindent \textbf{Theorem.} Consider the representation theory of $\mathrm{cl}_q(\mathfrak{g})$ and $U_q(\mathfrak{g})$ given by $(\Sigma,s_q)$ and $\sigma_q$ as in the proposition of 6. The operator
\begin{eqnarray*}
D = (\phi \otimes \gamma_q \circ \tau)(\Omega) \in \mathfrak{L}^{\mathrm{op}}_q(\mathfrak{g}) \otimes \mathrm{cl}_q(\mathfrak{g}) \subset U_q(\mathfrak{g}) \otimes \mathrm{cl}_q(\mathfrak{g})
\end{eqnarray*}
is invariant under the action of $U_q(\mathfrak{g})$: $\triangle(x)( \stackrel{\mathrm{oad}}{\blacktriangleright} \otimes \stackrel{\mathrm{ad}}{\triangleright})D = \epsilon(x)D$. $D$ commutes with the action of $U_q(\mathfrak{g})$ on $V_{\lambda} \otimes \Sigma$ for any $\lambda \in P_+$. \\
\noindent Proof. $D$ clearly spans a singlet in $U_q(\mathfrak{g}) \otimes \mathrm{cl}_q(\mathfrak{g})$ because $\Omega$ does. Let us write $D = \sum_{i,j} \alpha_{ij} Z_i \otimes \psi_j$. Let $| \omega \rangle \in V_{\lambda} \otimes \Sigma$ and $x\in U_q(\mathfrak{g})$. Using the Hopf algebra properties
\begin{eqnarray*}
\epsilon(x) = S(x')x'' = S^{\mathrm{op}}(x'')x'
\end{eqnarray*}
we find that
\begin{eqnarray*}
xD.| \omega \rangle &=& (\pi_{\lambda,q}(x') \otimes \sigma_q(x''))(\sum_{i,j} \alpha_{ij} \pi_{\lambda,q}(Z_i) \otimes s_q(\psi_j))| \omega \rangle \\
&=& (\sum_{i,j} \alpha_{ij} \pi_{\lambda,q}(\epsilon(x')x'' Z_i)) \otimes (\sigma_q(x''' \epsilon(x'''') ) s_q(Z_j))| \omega \rangle \\
&=& (\sum_{i,j} \alpha_{ij} \pi_{\lambda,q}(x^{(3)} Z_{i} S^{\mathrm{op}}(x^{(2)}))\pi_{\lambda,q}(x^{(1)})) \otimes (\sigma_q(x^{(4)}) s_q( \psi_{j}) \sigma_q(S(x^{(5)}))) \sigma_q(x^{(6)})| \omega \rangle \\
&=& (\sum_{i,j} \alpha_{ij} \pi_{\lambda,q} (x'' \stackrel{\mathrm{oad}}{\blacktriangleright} Z_{i}) \otimes s_q(x''' \stackrel{\mathrm{ad}}{\triangleright} \psi_{j}))(\pi_{\lambda,q}(x') \otimes \sigma_q(x''''))| \omega \rangle \\
&=& \epsilon(x'')D(\pi_{\lambda,q}(x') \otimes \sigma_q(x''')) | \omega \rangle = Dx.| \omega \rangle. \hspace{0,5cm} \square
\end{eqnarray*}
The limit $q \rightarrow 1$ of this operator is the classical Dirac operator with a reduced connection. We can also add the cubical in $\mathrm{cl}_q(\mathfrak{g})$ to $D$ so that it satisfies the required properties. The tensor product $V \otimes V$ contains an invariant subspace isomorphic to the adjoint representation in the negative spectral subspace of the braid operator. Thus, it does not vanish when embedded into the algebra $\mathrm{cl}_q(\mathfrak{g})$. Let $\theta : V \rightarrow \mathrm{cl}_q(\mathfrak{g})$ denote the corresponding module isomorphism. The cubical part of the Dirac operator is defined by
\begin{eqnarray*}
\Gamma = 1 \otimes (m(\theta \otimes \gamma_q \circ \tau) (\Omega)) \in U_q(\mathfrak{g}) \otimes \mathrm{cl}_q(\mathfrak{g})
\end{eqnarray*}
where $m$ is the product of $\mathrm{cl}_q(\mathfrak{g})$. $\Gamma$ is invariant under the action of $U_q(\mathfrak{g})$ and commutes with the representation. For a suitable choice of constant $N$ the operator $D + N \Gamma$ reduces to the geometric Dirac operator equipped with a Levi-Civita connection or to Kostant's Dirac operator \cite{Kos} in the limit $q \rightarrow 1$.
\\
\textbf{10. Example: $SU_q(2)$.} For each $l \in \frac{1}{2} \mathbb{N}_0$ choose a basis of $2n+1$ dimensional vector space $V_l$ by $\{ |l,m \rangle: -l \leq m \leq l\}$. The irreducible finite dimensional representations $(V_{l}, \pi_{l,q}$) of $U_q(\mathfrak{su_2})$ are
\begin{eqnarray*}
\pi_{l,q}(k)|l,m \rangle &=& q^m |l,m \rangle \\ \nonumber
\pi_{l,q}(e)|l,m \rangle &=& \sqrt{[l-m][l+m+1]} |l,m+1 \rangle \\ \nonumber
\pi_{l,q}(f)|l,m \rangle &=& \sqrt{[l-m+1][l+m]} |l,m-1 \rangle.
\end{eqnarray*}
$U = V_{1/2}$ is the defining representation and $V = V_1$ is the adjoint representation.
The module isomorphism $\phi: V \rightarrow \mathfrak{L}^{\mathrm{op}}_q(\mathfrak{su_2})$ is defined by
\begin{eqnarray*}
\phi(|1,1 \rangle) = t^{-1}e, \hspace{0,5cm} \phi(|1,0 \rangle) = \frac{1}{\sqrt{[2]}}(q^{-1}fe - qef),\hspace{0,5cm} \phi(|1,-1 \rangle) = - t^{-1}f.
\end{eqnarray*}
Let us write $\phi(|1,m \rangle) = Z_m$.
Using the Clebsch-Gordan rule we find
\begin{eqnarray*}
V \otimes V \simeq V_0 \oplus V \oplus V_2
\end{eqnarray*}
The subspace $V_0 \oplus V_2$ is in the positive spectral subspace of $\check{R}$ and the adjoint module $V$ is in the negative spectral subspace. The following Clifford algebra relations can be written down immediately by identifying the basis vectors of $V_0 \oplus V_2$ with their image in $B_q$
\begin{eqnarray*}
& &\psi_1 \psi_1 = \psi_{-1} \psi_{-1} = 0 \\
& & q^{-1} \psi_1 \psi_0 + q \psi_0 \psi_1 = 0 \\
& &q^{-2} \psi_1 \psi_{-1} + [2] \psi_0 \psi_0 + q^2 \psi_{-1} \psi_1 = 0 \\
& &\psi_0 \psi_{-1} + q^2 \psi_{-1} \psi_0 = 0 \\
& &\psi_1 \psi_{-1} + \psi_{-1} \psi_1 = b,
\end{eqnarray*}
where $\psi_i = \gamma_q(|1,i \rangle)$ and $b$ is some constant fixed from the normalization of the form $B_q$.
The irreducible representation space for $\mathrm{cl}_q(\mathfrak{g})$ is $2$-dimensional, $\Sigma = V_{1/2}$. Let us choose $\sigma_q = \pi_{1/2,q}$. The vector space $B(\Sigma)$ becomes a $U_q(\mathfrak{g})$-module and the submodule isomorphic to the adjoint representation under the action \eqref{comp} has a basis
\begin{eqnarray*}
& &s_q(\psi_1) = \begin{pmatrix}
0 & \sqrt{q} \\
0 & 0 \end{pmatrix}, \hspace{0,5cm} s_q(\psi_0) = - \frac{1}{\sqrt{[2]}}\begin{pmatrix}
q^{-1} & 0 \\
0 & -q \end{pmatrix}, \hspace{0,5cm} s_q(\psi_{-1}) = \begin{pmatrix}
0 & 0 \\
-\sqrt{q^{-1}} & 0 \end{pmatrix}
\end{eqnarray*}
These matrices satisfy the $q$-Clifford algebra relations with $b = -1$.
The module isomorphism $\tau: V^* \rightarrow V$ is given by
\begin{eqnarray*}
\tau(|1,1 \rangle^*) = -q [2]|1,-1 \rangle,\hspace{0,5cm} \tau(|1,0 \rangle^*) = [2]|1,0 \rangle ,\hspace{0,5cm} \tau(|1,-1 \rangle^*) = -q^{-1} [2]|1, 1 \rangle.
\end{eqnarray*}
Now we can write
\begin{eqnarray*}
D = \sum_l \phi(|1,l \rangle ) \otimes \gamma_q \circ \tau( |1, l \rangle^*) = -q[2] Z_1 \otimes \psi_{-1} +[2] Z_0 \otimes \psi_0 - q^{-1}[2] Z_{-1} \otimes \psi_{1}.
\end{eqnarray*}
For any $l \in \frac{1}{2}\mathbb{N}_0$ we define $H_l = V_{l} \otimes \Sigma$. As a $U_q(\mathfrak{su(2)})$-module this decomposes as
\begin{eqnarray*}
H_l \simeq V_{l+1/2} \oplus V_{l-1/2},
\end{eqnarray*}
if $l > 0$ and $H_0 \simeq V_{1/2}$. Let us write $j\pm1/2 = j^{\pm}$. Denote by $|j^+,\mu \rangle$ $(-j^+ \leq \mu \leq j^+)$ and $|j^-,\mu \rangle$ $(-j^- \leq \mu \leq j^-)$ the basis vectors of the irreducible components. The Dirac operator acts on $H_l$ by
\begin{eqnarray*}
D.|j^+ \mu \rangle = [2j] |j^+ \mu \rangle,\hspace{0,5cm} D.|j^- \mu \rangle = -[2j+2] |j^- \mu \rangle
\end{eqnarray*}
if $l > 0$ and $H_0 \in \mathrm{Ker}(D)$. \\
\textbf{11.} The matrix elements of the irreducible finite dimensional representations of $U_q(\mathfrak{su_2})$ span the space of polynomial functions on the quantum group
\begin{eqnarray*}
\mathbb{C}[SU_q(2)] = \bigoplus_{l \in \frac{1}{2}\mathbb{N}_0} V_l^* \otimes V_l \simeq \bigoplus_{l \in \frac{1}{2}\mathbb{N}_0} V_l \otimes V_l.
\end{eqnarray*}
The multiplication is derived from the Clebsch-Gordan coefficients. We use the Haar state of $\mathbb{C}[SU_q(2)]$ to complete $\mathbb{C}[SU_q(2)] \otimes \Sigma$ to a Hilbert space $\mathbf{H}$. The algebra $U_q(\mathfrak{g})$ acts from left by
\begin{eqnarray*}
x.(|l,m \rangle \otimes |l,n \rangle \otimes |\frac{1}{2},s \rangle) = (\pi_{l,q}(x')|l,m \rangle) \otimes |l,n \rangle \otimes (\pi_{1/2,q}(x'') |\frac{1}{2},s \rangle),
\end{eqnarray*}
for all $|l,m \rangle \otimes |l,n \rangle \otimes |\frac{1}{2}, s \rangle \in \mathbf{H}$. The explicite decomposition of the prehilbert space into irreducible components under the left action is given in \cite{DB}
\begin{eqnarray*}
(\bigoplus_{l \in \frac{1}{2} \mathbb{N}_0} V_{l} \otimes V_{l}) \otimes \Sigma \simeq V_{1/2} \oplus \bigoplus_{l \in \frac{1}{2} \mathbb{N}} (V_{j+1/2} \otimes V_{j}) \oplus (V_{j-1/2} \otimes V_{j}) := W^{\uparrow}_0 \oplus \bigoplus_{l \in \frac{1}{2} \mathbb{N}} W^{\uparrow}_j \oplus W^{\downarrow}_j.
\end{eqnarray*}
The components $W^{\uparrow}_j$ and $W^{\downarrow}_j$ have multiplicities $(2j+2)(2j+1)$ and $2j(2j+1)$. The orthonormal basis of $\mathbf{H}$ is chosen by
\begin{eqnarray*}
|j \mu n \uparrow \rangle \in W^{\uparrow}_j, \hspace{0,5cm} |j' \mu' n \downarrow \rangle \in W^{\downarrow}_j
\end{eqnarray*}
for $j \in \frac{1}{2} \mathbb{N}_0$, $ j' \in \frac{1}{2} \mathbb{N}$, $ -j^+ \leq \mu \leq j^+$, $ -j^- \leq \mu' \leq j^-$ and $ -j \leq n \leq j$.
Let us adopt the column vector notation of \cite{DB}
\begin{eqnarray*}
|j \mu n \rangle \rangle = \begin{pmatrix}
|j \mu n \uparrow \rangle \\
|j \mu n \downarrow \rangle
\end{pmatrix}.
\end{eqnarray*}
A faithful $*$-representation for the algebra $\mathbb{C}[SU_q(2)]$ on $\mathbf{H}$ is developed in \cite{DB} (Proposition 4.4), where the notation matches with ours. This representation coincides with the GNS representation. Since we know how $D$ acts on each irreducible piece we get
\begin{eqnarray*}
D.|j \mu n \rangle \rangle = \begin{pmatrix}
[2j] & 0 \\
0 & -[2j+2] \end{pmatrix} |j \mu n \rangle \rangle,
\end{eqnarray*}
when $j > 0$ and $W^{\uparrow}_0 \in \mathrm{Ker}(D)$. The kernel is nontrivial and we define an approximated sign operator in the usual way: $F = D(1+D^2)^{-1/2}$. \\
\noindent \textbf{Proposition.} Triple $(\mathbb{C}[SU_q(2)], F, \mathbf{H})$ is a $1$-summable Fredholm module. \\
\noindent Proof. The spectrum of $D$ grows exponentially as a function of $j$ and therefore $F^2-1$ is a trace class operator
\begin{eqnarray*}
F^2-1 = -(1+D^2)^{-1} \in L^1(\mathbf{H}).
\end{eqnarray*}
The operators of the irreducible $*$-representation \cite{DB} of $\mathbb{C}[SU_q(2)]$ on $\mathbf{H}$ are of the form
\begin{eqnarray*}
\eta(x) = \sum_{i} X(i),\hspace{0,5cm} X(k) = \begin{pmatrix}
X(k)_{\uparrow \uparrow} & X(k)_{\uparrow \downarrow} \\
X(k)_{\downarrow \uparrow} & X(k)_{\downarrow \downarrow} \end{pmatrix},
\end{eqnarray*}
where $X(k)_{\uparrow \uparrow}, X(k)_{\downarrow \downarrow} \in B(\mathbf{H})$ and $X(k)_{\uparrow \downarrow}, X(k)_{\downarrow \uparrow} \in L^1(\mathbf{H})$ and each $X(k)$ shifts the index $j$ of the basis vector by $k \in \frac{1}{2} \mathbb{Z}$. The sum is always finite. We have
\begin{eqnarray*}
[F, X(k)_{\uparrow \uparrow}]|j \mu n \rangle = \Big( \frac{[j+k]}{(1 + [j+k]^2)^{1/2}} - \frac{[j]}{(1 + [j]^2)^{1/2}} \Big) X(k)_{\uparrow \uparrow}|j \mu n \rangle.
\end{eqnarray*}
The first term can be easily seen to give a sequence which decays rapidly as a function of $j$. Therefore $[F, X(k)_{\uparrow \uparrow}] \in L^1(\mathbf{H})$. Similarly we see that $[F, x_{\downarrow \downarrow}] \in L^1(\mathbf{H})$. The commutators $[F, X(k)_{\uparrow \downarrow}]$ and $[F, X(k)_{\downarrow \uparrow}]$ are in trace class because the off diagonal blocks are trace class operators and $F$ is bounded. Thus, $[F,X(k)] \in L^1(\mathbf{H})$ for all $k$ and thus $[F, \eta(x)] \in L^1(\mathbf{H})$. $\hspace{0,5cm} \square$ \\
\textbf{12. Acknowledgements.} The author wishes to thank his supervisor Jouko Mickelsson for several helpful discussions. This project was supported by the V$\ddot{a}$is$\ddot{a}$l$\ddot{a}$ foundation of the Finnish Academy of Science and Letters and The Finnish National Graduate School in Mathematics and its Applications.
|
{
"timestamp": "2011-06-13T02:00:19",
"yymm": "1009",
"arxiv_id": "1009.3913",
"language": "en",
"url": "https://arxiv.org/abs/1009.3913"
}
|
\section{Real-Time Maude and Sockets}
\label{sec:maude-background}
In this section we briefly cover the important constructs we used from
Real-Time Maude and Maude Sockets. We assume the reader is familiar with basic
Maude constructs including modules ({\tt mod}), sorts ({\tt sort}), operators
({\tt op}), unconditional and conditional equations ({\tt eq} and {\tt ceq})
and unconditional and conditional rules ({\tt rl} and {\tt crl}).
\subsection{Full Maude and Real-Time Maude}
Full Maude \cite{fullmaude} is a Maude interpreter written in Maude, which in
addition to the Core Maude constructs provides syntactic constructs such as
object oriented modules. Object oriented (OO) modules implicitly add sorts
{\tt Object} and {\tt Msg}. Furthermore, OO-modules add a sort called {\tt
Configuration} which consists of a multiset of terms of sort {\tt Object} or {\tt
Msg}.
Objects are represented as records:
\begin{footnotesize}
\begin{alltt}
< {\em objectID} : {\em classID} | {\em AttributeName} : {\em Attribute}, ... >
\end{alltt}
\end{footnotesize}
Rewrite rules are then used to describe state transitions of objects
based on consumption of messages. For example, the following rule expresses
the fact that a pacemaker object consumes a message to set the pacing
period to T:
\begin{footnotesize}
\begin{alltt}
rl setPeriod(pm, T)
< pm : Pacing-Module | pacing-period : PERIOD >
=> < pm : Pacing-Module | pacing-period : T > .
\end{alltt}
\end{footnotesize}
Real-Time Maude \cite{rt-maude} is a real-time extension of
Maude in Full Maude. It adds syntactic constructs for defining timed
modules. Timed modules automatically import the {\tt TIME} module, which defines
the sort {\tt Time} (which can be chosen to be discrete or continuous) along
with various arithmetic and comparison operations on {\tt Time}. Timed modules
also provide a sort {\tt System} which encapsulates a {\tt Configuration} and
implicitly associates with it a time stamp of sort {\tt Time}. After defining a
time-advancing strategy, Real-Time Maude provides timed execution ({\tt
trew}), timed search ({\tt tsearch}), which performs search on
a term of sort {\tt System} based on the time advancement strategy, and
timed and untimed LTL model checking commands.
\subsection{Deterministic Timed Rewriting in Real-Time Maude}
We are interested in emulations of real-time systems specified in Real-Time
Maude. For useful real execution, a self-evident condition is that the
time-advancing rewrite rules in the specification should be deterministic. This
can be achieved by defining only one time-advancing rewrite rule on the system
with two auxiliary operators {\em tick} and {\em mte} \cite{rtmaude-manual}. In
our specification this is captured in {\tt TIME-ADV-SEMANTICS}, which is
included by all other timed modules:
\begin{footnotesize}
\begin{verbatim}
(mod TIME-ADV-SEMANTICS is ...
op mte : Configuration ~> TimeInf . ...
eq mte(none) = INF .
eq mte(C C') = minimum(mte(C), mte(C')) .
op tick : Configuration Time ~> Configuration . ...
eq tick(none, T) = none .
eq tick(C C', T) = tick(C, T) tick(C', T) .
op def-te : -> Time .
op max-te : -> TimeInf .
crl {GC} => {tick(GC, T)} in time T if T <= mte(GC) [nonexec] .
endm)
\end{verbatim}
\end{footnotesize}
The rewrite rule at the end is assumed to be the only timed rewrite rule (a
rewrite rule that advances the time stamp of the system) in the system
specification. The {\tt mte} operator defines the maximum time elapse before
any 0-time rewrite rule can be applied. The {\tt tick} operator defines how
the system state changes due to time advancement between applications of
0-time rewrite rules. We also define a default time elapse, {\tt def-te},
and maximum time elapse, {\tt max-te}, inside the module to be used as
parameters during real execution.
\subsection{Socket Programming in Maude}
Maude supports the Berkley sockets API for TCP communication. This is done by
having a special gateway object, denoted {\tt <>}, to consume all the messages
responsible for setting up sockets and communicating to an external environment
(e.g. {\tt createClientTcpSocket}, {\tt send}, {\tt receive}). The gateway
object will also generate messages upon status updates from the socket (e.g.
{\tt sent}, {\tt received}, {\tt closedSocket}). Consuming and generating
messages from the gateway object is captured by external rewrite rules which
can be executed using the {\tt erew} command in Core Maude. An important thing
worth pointing out about external rewrite rules is that {\em external rewrite
rules are only applied when no internal rewrite rules can be applied}. Also,
using external rewrite rules with Real-Time Maude specifications (built on top
of Full Maude) requires reflecting the specification down to a Core Maude
module before executing.
\section{Case Studies}
\label{sec:case}
\subsection{A Pattern for Medical Device Execution}
We briefly described in the introduction at a high level that the model
execution framework is to support rapid prototyping of instantiated medical
device safety patterns. In \cite{sun-meseguer-sha-wrla10} and \cite{tech-rep},
we have described in detail the command shaper pattern for medical device
safety. In essence, the command shaper pattern can modify commands to an
existing medical device to guarantee specific safety properties in terms of
limiting durations of stressful states and limiting the rate of change (Figure
\ref{fig:commandshaper}).
\begin{figure}
\centering \includegraphics[width=10cm, angle=0]{figures/pacemaker_wrapper}
\caption{Command Shaper Pattern for a Pacemaker}
\label{fig:commandshaper}
\end{figure}
\subsection{Pacemaker Simulation Case Study}
One of the applications for the command shaper pattern is a pacemaker system
\cite{sun-meseguer-sha-wrla10}. At a high level the safety properties
guaranteed by the command shaper pattern is that the pacemaker will not pace at
fast heart rates too frequently or for too long, and the pacing rate will
change gradually. We omit the details of instantiating the medical device
pattern, but the final wrapper object provided by the pattern is:
\begin{footnotesize}
\begin{verbatim}
(tomod PARAM-PACEMAKER is pr EPR-WRAPPER-EXEC{Safe-Pacer} ...
eq wrapper-init =
< pacing-module : EPR-Wrapper{Safe-Pacer} |
inside :
< pacing-module : Pacing-Module |
nextPace : t(0),
period : safe-dur >,
val : safe-dur,
next-val : safe-dur,
disp : t(period),
stress-intervals : (nil).Event-Log{Stress-Relax} > .
... endtom)
\end{verbatim}
\end{footnotesize}
This says that a wrapper is placed around a pacing module, and the initial
pacing rate is set as the default safe-duration ({\tt safe-dur} is 750 ms or 80
heart beats per minute). Verifying this instantiation (with a simple pacemaker
lead model \cite{tech-rep}) indicates that the safety properties are met by the
pattern. However, with the power of the model emulation framework, we can
immediately use this specification to run with an actual pacemaker. In this
paper we demonstrate this emulation capability not on an actual pacemaker but
on a pacemaker simulator (a Java widget that receives messages about when to
pace and draws a simple line graph resembling an ECG trace). Before the system
can be emulated with the pacemaker simulator, some interface information must
be provided. The entire module providing all the necessary interface
information is shown below:
\begin{footnotesize}
\begin{verbatim}
(mod CREATE-TICKER is
inc PARAM-PACEMAKER .
inc TIME-CLIENT .
inc SEND-RECEIVE-CLIENT .
eq addr = "localhost" .
eq port = 4444 .
eq def-te = 1 .
eq max-te = INF .
eq time-grain = 10 . --- milliseconds
op pacer-client : -> Oid .
eq internal = wrapper-init .
eq out-adapter(shock)
= createSendReceiveClient(pacer-client, "localhost", 4451, "shock") .
eq in-adapter(msg-received(pacer-client, "shocked\n"))
= set-period(pacing-module, 50) .
endm)
\end{verbatim}
\end{footnotesize}
The module first indicates that the TCP
socket interface to the pacemaker simulator is {\em localhost} on {\em port}
4444. The default time elapse for one tick is 1 time unit. The maximum time elapse
for one tick step is infinity (i.e. there is no maximum). The duration of one time
unit is 10 milliseconds. The time units are in terms of milliseconds since the
minimum time granularity provided by the Java time interfaces is 1 millisecond.
The equation for {\tt internal} specifies that the internal configuration to be
executed is the configuration defined by {\tt wrapper-init} (as defined in {\tt
PARAM-PACEMAKER}). Also, the last two equations specify that the output message
shock should be mapped to a string ``shock'' sent over the socket, and upon
receiving the acknowledgment message ``shocked'' set the pacing period to 500
ms (120 bpm - a really fast heart rate). The last equation creates the scenario
where a stressful heart rate is always being sent to the pacing module. Since
the command shaper pattern should prevent this unsafe behavior, we should see
the pacing automatically slow down from 120 bpm after some time interval.
The module is executed by first reflecting the {\tt CREATE-TICKER} module down
to Core-Maude (with the command {\tt show all CREATE-TICKER}), and executing
with the {\tt erew} command. A snapshot of the ``ECG'' trace of the pacemaker
simulator is show in Figure \ref{fig:pacing}. For validation, we measured the
jitter for executing such a system -- the physical time required to completely
execute 0-time rules and finish communication (Figure \ref{fig:jitter}). The
results were obtained from a 1.67 GHz Dual-Core Intel Centrino with Maude
running in Windows through Cygwin (tracing was turned off). The main thing to
notice is that the jitter is mostly below 0.1 seconds and almost never exceeds
0.2 seconds. This amount of jitter is tolerable since most medical devices need
to respond in the order of seconds. The pacemaker is a bit more strict in terms
of its timing requirements. To evaluate suitability for the pacemaker, we
plotted the recorded the physical time duration between pacing events (Figure
\ref{fig:pacingintervals}). Notice in this example the heart rate increases
(duration decreases) up to a limit and then the heart rate starts to decrease
(duration increases) and the cycle repeats. It is clear that the jitter in
control seems tolerable since there are no sharp spikes in the graph of the
pacing durations.
\begin{figure}
\centering \includegraphics[width=6cm, angle=0]{figures/pacing}
\caption{Trace from Pacemaker Simulator}
\label{fig:pacing}
\end{figure}
\begin{figure}
\centering \includegraphics[width=12cm, angle=0]{figures/execution_jitter}
\caption{Model Execution Jitter Distribution}
\label{fig:jitter}
\end{figure}
\begin{figure}
\centering \includegraphics[width=12cm, angle=0]{figures/pacing_intervals}
\caption{Pacing periods recorded by the pacemaker simulator (jitter effects
are reflected by noise on the curve)}
\label{fig:pacingintervals}
\end{figure}
\subsection{Syringe Pump Case Study}
The pacemaker emulation example was demonstrated through a simulated
pacemaker mostly because current pacemakers do not have external interfaces for
setting when to pace (and rightly so). However, for devices such as electronic
syringe pumps these interfaces are available. Syringe pumps and infusion pumps
in general deliver intravenous injections into a patient. For this scenario, we
assume that the syringe pump is delivering an analgesic (e.g.\ morphine) to the
patient, and we would like to prevent overdose. We assume that for a normal
patient overdoses do not occur at the base rate of infusion and can only result
if a bolus dose is administered too often for the patient. We again use the
command shaper pattern to limit the frequency and duration of bolus doses. The
instantiated pump is as follows:
\begin{footnotesize}
\begin{verbatim}
(tomod PARAM-PUMP is
pr EPR-WRAPPER-EXEC{Safe-Pump} .
pr DELAY-MSG .
...
eq msgs-init =
delay(set-mode(pump-module, bolus), t(9))
delay(set-mode(pump-module, bolus), t(11))
delay(set-mode(pump-module, bolus), t(12)) ... .
eq wrapper-init =
< pump-module : EPR-Wrapper{Safe-Pump} |
inside :
< pump-module : Pump-Module |
mode : base
> base,
val : base,
next-val : base,
disp : t(period),
stress-intervals : (nil).Stress-Relax-Log > .
... endtom)
\end{verbatim}
\end{footnotesize}
This module shows the initialized wrapper object for the pump, with the initial
state being the base rate of infusion. Furthermore, there is also a set of
delayed messages that will be sent to the pump. In the term {\tt msgs-init}, the
model will send bolus requests at 9 time units, 11 time units, 12 time units,
\ldots after the start of execution for the system. Again, creating a simulated
patient model, we can verify the safety of the instantiated pattern
\cite{tech-rep}. Instantiating the pump is similar to instantiating the
pacemaker, except that there are a few more types of output messages.
\begin{footnotesize}
\begin{verbatim}
(mod CREATE-TICKER is
inc PARAM-PUMP .
inc TIME-CLIENT .
inc SEND-RECEIVE-CLIENT .
eq addr = "localhost" .
eq port = 4444 .
eq def-te = 1 .
eq max-te = INF .
eq time-grain = 1000 . --- milliseconds
ops pump-client pump-client' : -> Oid .
eq internal = wrapper-init msgs-init .
eq out-adapter(stop)
= createSendReceiveClient(pump-client, "localhost", 1234, "STP") .
eq out-adapter(base)
= createSendReceiveClient(pump-client, "localhost", 1234, "RAT1")
createSendReceiveClient(pump-client', "localhost", 1234, "RUN") .
eq out-adapter(bolus)
= createSendReceiveClient(pump-client, "localhost", 1234, "RAT2")
createSendReceiveClient(pump-client', "localhost", 1234, "RUN") .
var S : String .
eq in-adapter(msg-received(pump-client, S))
= none .
eq in-adapter(msg-received(pump-client', S))
= none .
endm)
\end{verbatim}
\end{footnotesize}
The model is communicating with {\em localhost} on {\em port} 4444. The time
granularity is 1 second. The internal configuration being executed is the
wrapped pump as well as the set of messages that will deliver bolus requests.
The output requests are handled by a Java thread listening on port 1234 and
forwarding the request string to the actual {\em Multi-Phaser NE-500} Syringe
Pump (Figure \ref{fig:pump}). A few important requests to the pump are: {\tt
STP} stop the pump, {\tt RAT <n>} set infusion rate to {\tt n} ml/hr, {\tt RUN}
start the infusion. Reflecting down the {\tt CREATE-TICKER} module and
executing with {\tt erew} will now control the physical pump motor!
\begin{figure}
\centering \includegraphics[width=8cm, angle=0]{figures/pump}
\caption{Multi-Phaser NE-500 Syringe Pump}
\label{fig:pump}
\end{figure}
As a validation for correct pump control, we used a Salter Brecknell 7010SB
scale to weigh the amount of liquid infused from the syringe pump over time
(Figure \ref{fig:pumpdata}). The data granularity is a bit rough since the
scale can only measure within a precision of 0.1 oz. For this example, to
clearly distinguish between two pump states, we let the base rate of infusion
be zero (horizontal parts of the graph) and the bolus rate be the maximum
infusion rate provided by the pump (positive sloped parts of the graph). Bolus
requests are continuously sent to the pump. The safety properties require that
bolus doses last no longer than 30 seconds, and there must be 10 seconds
between bolus doses, and at most 3 bolus doses for a window size of 3 minutes.
The graph validates that these properties are indeed satisfied for this
particular execution of the pump.
\begin{figure}
\centering \includegraphics[width=8cm, angle=0]{figures/pump_data}
\caption{Infusion Volume over Time}
\label{fig:pumpdata}
\end{figure}
\section{Conclusion}
\label{sec:conc}
Safety of medical devices and of their interoperation is an unresolved issue
causing severe and sometimes deadly accidents for patients. Formal methods,
particularly in support of highly generic and reusable formal patterns whose
safety properties have been verified can help in ensuring the safety of
specific components, but this still leaves several open problems including: (i)
how to pass from specifications to code and from logical time to physical time
in a correctness-preserving ways; and (ii) how to experimentally validate
medical safety architectures in realistic scenarios in which actual devices and
models of patients and doctors can interact with formally specified and
provably safe designs of device components.
By developing virtual emulation environments in which highly generic and
reusable formally verified patterns in Real-Time Maude can be easily
transformed into emulations in physical time which can interact with other real
devices and with simulations of patient and/or doctor behaviors, we have taken
some first steps towards a seamless integration of formal specification and
verification with emulation and testing, and ultimately with deployment of
medical DES systems that offer much stronger safety guarantees than what is
currently available. Much work remains ahead. As we explain in
\cite{sun-meseguer-sha-wrla10}, the provably safe formal pattern used in the
experiments of this paper is just one such pattern: it covers a useful class,
but does not cover other kinds of safety needed in other medical devices. Also,
other safety concerns, such as so-called open-loop safety, ensuring that
medical devices will always be in states safe for the patient even under key
infrastructure failures, such as network disconnection, have not been addressed
in this work. However, we believe that the general methodology presented here
to pass from formal specifications to virtual emulation environments and
eventually to deployed systems should also be applicable to those new formally
verified patterns that have yet to be developed.
\section{Introduction}
\label{sec:intro}
Each year, just in the US hospitals, a shocking and almost unacceptable number
of medical accidents occur. In a 2009 study, reports estimate 40,000 instances
of medical harm occur daily, and from the 2005 through 2007 period, at least
92,882 deaths were potentially preventable \cite{HealthGrades2009}. Many of
these accidents happen due to mistakes and failures in the
\emph{interoperation} of medical devices. A modern hospital's operating room is
in fact a quite complex distributed embedded system (DES) with many devices
involved in either passively monitoring the patient state or actively
performing different parts of a procedure. Both the safety of the individual
devices and the safety of interoperation between devices (and between the
patients and doctors) are of paramount importance. Presently, this safety is
not adequately guaranteed.
When the reported accidents are analyzed, it becomes clear that many of them
could and should have been avoided if the DES formed by the devices, the
patient, and the doctors had been properly designed and analyzed, so that many
unsafe interactions become \emph{impossible} by design. The use of formal
methods can clearly help in this respect, and promising research advances have
already been made in this direction (see, e.g.,
\cite{alur04,arney07,ray04,jetley06}).
In our recent work \cite{tech-rep}\cite{sun-meseguer-sha-wrla10}, we have
developed a scalable and highly reusable approach to the safety of medical
devices by means of \emph{formally verified patterns} that: (i) are formally
specified as real-time rewrite theories in Real-Time Maude; (ii) are generic,
so that they apply not to a single device but to a wide range of devices, and
are therefore specified as \emph{parameterized} modules; (iii) come with
explicit \emph{formal requirements} (specified in their parameter theories)
that must be met by any pattern instantiation to be correct; and (iv) come with
\emph{formal safety guarantees} that will be satisfied by any correct
instantiation of the pattern. For example, in \cite{tech-rep} we present one
such pattern and show how it can be instantiated to obtain safe controllers for
quite different devices, such as a pacemaker, an infusion pump for analgesia,
and the interoperation of a ventilator with an X-ray machine.
However, there is still a substantial gap between the verified safety of
designs in formal specifications and the actual safety of real medical devices,
patients and doctors in an operating room for at least two reasons. First, the
formal specifications are somewhat idealized abstractions in which, for
example, time is not the actual physical time that devices need to operate in,
but logical time, and the code of the actual devices and controllers is not
used but instead some formal specifications are used. Second, it is important
to consider not just the safety of a single device or small collection of
devices, but also that of their \emph{interoperation} with other devices and
with the patient and the doctors. This work takes some first steps towards the
goal of bridging the gap between formal specifications and actual devices in a
hospital to help ensure that safety properties are preserved in the passage
from specification to actual code and physical devices. To achieve this goal
we propose the use of \emph{virtual emulation environments} in which:
\begin{enumerate}
\item formally verified patterns \cite{sun-meseguer-sha-wrla10} can be
instantiated to obtain various concrete specifications of desired devices and
controllers;
\item the so-obtained formal executable specifications of devices and
controllers are used \emph{directly} to generate emulators that perform the
same specified behavior in physical time;
\item actual devices, as well as actual executable models of patient and doctor
behavior, can be seamlessly integrated with specification-based emulators to
validate the safety not just of individual devices but also of various
DESs that are needed in practice in actual operating room conditions and
scenarios.
\end{enumerate}
The advantage of point (1) is a great degree of reusability, and amortizing the
formal verification effort across potentially many devices. The advantage of
(2) is that, since each specification-based emulator executes the exact same
formal specification that has been proved safe for the given device, the safety
of such an emulator is automatically guaranteed, and a path remains open to
correctly generate actual code for it preserving such safety in an actual
implementation. The advantage of point (3) is that system experimentation with
physical time and actual devices becomes available from very early in the
design process and are available afterwards along the entire development
process: initially, only specifications may be emulated; at intermediate
stages, both specifications and actual devices form the virtual emulation
environment; and in the end the emulating environment seamlessly becomes an
actual implementation.
Technically, the way such virtual emulation environments are obtained from
formal specifications is by using a key idea first demonstrated by Musab
Al-Turki to semi-automatically pass from a Real-Time Maude formal executable
specification operating in \emph{logical time} to a corresponding
\emph{physical emulation} of the same specification operating in \emph{physical
time} and possibly interacting with other devices in a distributed way. The
key observations are: (i) in Real-Time Maude rewrite rules are either 0-time
rules requiring no time, or time-advancing rules moving the entire system
forward in logical time; (ii) time advancing rules (typically a single such
rule) can be physically implemented by an external object that sends time ticks
according to physical time; (iii) although so-called 0-time rules do take some
physical time to be executed, if this time is small enough in comparison with
the time granularity of the physical time period chosen, for all practical
purposes they can be considered to take 0 time units to execute; and (iv) the
Maude infrastructure for Maude computations to interact with external objects
via sockets can be used to interface the Maude objects in the formal
specification with the external ticker object and also to other external
devices.
\subsection{Our Contribution.} This is the first work we are aware of in which
formal specifications of real-time components are directly used in the area of
distributed embedded systems for medical applications to obtain a virtual
emulation environment in which specifications, patients, doctors, and actual
devices can be emulated in physical time, and such that the correctness of
verified specifications is preserved, provided adequate timing
constraints are obeyed (see Section \ref{sec:issues}). This work is a first
step towards a seamless integration of formal specification, verification, and
system development and testing for safe medical systems. The associated notion
of a virtual emulation environment plays a crucial role in passing from
specifications to code and devices, and from logical time to physical time. We
have demonstrated both the feasibility and the usefulness of these methods in
two concrete scenarios: one in which a pacemaker interacts adaptively in
physical time with a simulated model of a patient heart and keeps heart rates
within a safe envelope; and another in which a safety controller for
patient-controlled analgesia interacts in physical time with an actual drug
infusion device and with a simulation of patient behavior.
In the passage from formal real-time specifications to their corresponding
emulators there are additional novel contributions that were required for this
work, including: (i) advancing time by the maximum time elapsable as opposed to
by a fixed ticking period; (ii) handling asynchronous interrupts in addition to
synchronous communication; (iii) emulating the interaction of real medical
devices, patient models, and formal specifications; and (iv) generating a timed
wrapper for each component specification almost for free with deterministic
Real-Time Maude specifications using both a time ticker and the computation of
the maximum time elapsable for each time advance.
The paper is organized as follows: Section \ref{sec:overview} provides the high
level ideas of the framework from formal patterns to real-time execution.
Section \ref{sec:maude-background} covers the basics of Real-Time Maude and
Maude's support for socket programming. Section \ref{sec:io} and
\ref{sec:emulation} describes the core of the execution framework which allows
seamless passage from Real-Time Maude specifications to execution with physical
time and physical devices. Section \ref{sec:case} covers case studies for a
pacemaker and a syringe pump to evaluate the feasibility of using formal models
to execute medical devices. Finally, we describe some fundamental assumptions
required for formal model execution to work for medical devices in Section
\ref{sec:issues}, and we conclude in Section \ref{sec:conc}.
\section{Mapping Internal Messages to External I/O}
\label{sec:io}
Validating the design of a device in an execution environment requires handling
its outputs. After all, the end validation of a system's behavior is based on
its outputs. Thus, it seems reasonable to talk about how internal messages in
the model can be converted into messages for communicating with the external
world. This section also serves as an explanation for unfamiliar readers of how
Maude sockets are used.
In order to talk about external communication, we must first define in the
model what is external. The model will have an internal distributed actor
configuration with internal messages as well as messages to be output to the
external world. Thus, the first definition is {\tt EXTERNAL-CONFIGURATION}
which defines external messages {\tt ExtMsg} as subsort of {\tt Msg}.
Furthermore, external messages are classified in terms of incoming external
messages {\tt InExtMsg} and outgoing external messages {\tt OutExtMsg}. A
configuration is called {\em open} if there are external messages present in
the configuration: either an incoming external message has not been delivered,
or an outgoing external message has not been sent. The predicate {\tt open?} is
defined accordingly.
\begin{footnotesize}
\begin{verbatim}
subsorts InExtMsg OutExtMsg < ExtMsg < Msg .
op open? : Configuration -> Bool .
eq open?(C C') = open?(C) or open?(C') .
eq open?(O) = false .
eq open?(M) = M :: ExtMsg .
\end{verbatim}
\end{footnotesize}
Actually sending an external message may be more complex than just forwarding
the message through the gateway object. External messages may not be the same
in the internal configuration and in the external configuration. For example, a
simple output message in the internal configuration may need to be mapped to a
client object that initiates the communication to deliver the message.
Operators {\tt in-adapter} and {\tt out-adapter} are defined to perform these
mappings from external message client configurations to internal messages.
An example of an output adapter for a pacemaker message to beat the heart
may be:
\begin{footnotesize}
\begin{verbatim}
eq out-adapter(shock)
= createSendReceiveClient(pacer-client, "localhost", 4451, "SetLeadVoltage 5V")
\end{verbatim}
\end{footnotesize}
In this example, the message {\tt shock} is transformed into a client object
which sends a message on port 4451 with the string {\tt "SetLeadVoltage 5V"}
indicating that the proxy server will then proceed to set a 5V voltage on the
pacemaker lead.
\subsection{One-Round Communication Clients}
Once the external message is mapped into a client configuration, we must define
the rewrite rules to specify how the communication protocol works with the
external device. Here we describe a simple {\tt SEND-RECEIVE-CLIENT} which is
responsible for establishing communication, sending a message, receiving a
reply, and then closing the communication. Although simple, this type of
protocol is sufficient for most of the communication for medical devices we have
used in our case studies.
\begin{footnotesize}
\begin{verbatim}
(mod SEND-RECEIVE-CLIENT is ...
op createSendReceiveClient : Oid String Nat String -> Configuration .
eq createSendReceiveClient(CLIENT, ADDRESS, PORT, SEND-CONTENTS)
= < CLIENT : SendReceiveClient | ... >
createClientTcpSocket(socketManager, CLIENT, ADDRESS, PORT) .
op msg-received : Oid String -> InExtMsg .
...endm)
\end{verbatim}
\end{footnotesize}
After creating the client and establishing communication, the client goes into
one round of send and receive before the socket is closed. Once the socket is
closed, the entire client object is converted into one reply message to be
delivered to the internal configuration using the operator {\tt msg-received}.
\begin{footnotesize}
\begin{verbatim}
--- send contents
rl createdSocket(CLIENT, socketManager, SOCKET-DST)
< CLIENT : SendReceiveClient | ... send-contents : SEND-CONTENTS >
=> < CLIENT : SendReceiveClient | ... > send(SOCKET-DST, CLIENT, SEND-CONTENTS) .
--- receive contents
rl sent(CLIENT, SOCKET-DST) < CLIENT : SendReceiveClient | ... >
=> < CLIENT : SendReceiveClient | ... > receive(SOCKET-DST, CLIENT) .
--- close socket
rl received(CLIENT, SOCKET-DST, RECEIVE-CONTENTS) < CLIENT : SendReceiveClient | ... >
=> < CLIENT : SendReceiveClient | ... recv-contents : RECEIVE-CONTENTS >
closeSocket(SOCKET-DST, CLIENT) .
--- done
rl closedSocket(CLIENT, SOCKET-DST, "")
< CLIENT : SendReceiveClient | ... recv-contents : RECEIVE-CONTENTS >
=> msg-received(CLIENT, RECEIVE-CONTENTS) .
\end{verbatim}
\end{footnotesize}
\section{Assumptions and Issues}
\label{sec:issues}
In this section we discuss the timing assumptions that need to be taken into
account to ensure that the emulation of a Real-Time Maude specification
correctly implements the logical time behavior.
Figure \ref{fig:timing} shows one round of communication between the time server
and the formal model.
$t_{comm,i}$ denotes the delays due to each stage of communication.
$t_{rew,i}$ denotes the delays incurred by each stage of rewriting. $t_{proc}$
denotes the time needed at the physical device interface to process
the commands. Thus, the entire time to finish a round is $t_{round} =
t_{comm1} + t_{rew1} + t_{comm2} + t_{proc} + t_{comm3} +
t_{rew2} + t_{comm4}$.
\begin{figure}
\centering \includegraphics[width=10cm, angle=0]{figures/timing}
\caption{Timing Considerations}
\label{fig:timing}
\end{figure}
In logical time, no time advancement should actually take place in a
communication round. All computation and message communication is assumed to
take zero time. Of course, for proper timed operation we can relax these
constraints to first allow the model to have a non-zero (but bounded) delay for
these computations and communications. The maximum bound on these
communications is $t_{round} \leq mte(C_{next})$. Otherwise, by the time the
round has completed, the execution is already delayed passed the time for the
maximum time elapse for the next 0-timed rewrite rule; i.e., the maximum speed
of execution of the formal model and communication is slower than the real time
requirements. Now, assuming that the constraint $t_{round} \leq mte(C_{next})$
is satisfied, there is still another problem we must deal with. The actual
commands sent to the device are not received until $t_{jitter} = t_{comm1} +
t_{rew1} + t_{comm2}$ time after they are actually supposed to be executed. This
could be very problematic. Even if the model can keep up with real-time, the
time in which it issues commands will be delayed. For example, the shocks from
a pacemaker may be issued at the correct time by the model, but the real shock
is not delivered until 0.1 seconds later. To meet this requirement, we need to
look at the finer requirements of medical devices and patient parameters. How
much jitter in control can a patient tolerate? As we have seen in the Section
\ref{sec:case}, the jitter seems to be tolerable for the applications we
considered, and furthermore, the end-to-end round communication timing
constraints are also satisfied by our case studies.
\section{Overview of the Model Execution Framework}
\label{sec:overview}
An envisioned design framework from generic design patterns to executable
specifications is shown in Figure \ref{fig:pattern-to-execution}. We start with
a safety pattern, which is a parameteric module with well-defined parameters
with formal requirements provided by an input theory. The next stage is to
instantiate the pattern to a concrete instance, which will of course still
satisfy all the safety properties ensured by the pattern. Finally, in the last
stage (the focus of this paper), the entire executable specification of a
system can really be executed in the real world by encapsulating the
specification in an external wrapper for model execution. Figure
\ref{fig:pattern-to-execution} illustrates these various stages for the design
of a cardiac pacemaker system.
\begin{figure}
\centering \includegraphics[width=9cm, angle=0]{figures/mde_process}
\caption{From Formal Patterns to Real-Time Execution}
\label{fig:pattern-to-execution}
\end{figure}
The first step formally defines a safety pattern as a parameterized module. In
our pacemaker example, the pattern is a generic safety wrapper for medical
devices. We briefly describe the pattern in this paper, summarizing the details
presented in \cite{tech-rep}\cite{sun-meseguer-sha-wrla10}. The safety wrapper
filters the input commands, so that state changes in a medical device fall
within safe physiological ranges and constraints. The white boxes in the
diagram denote pattern parameters that must be instantiated. For the device
safety wrapper these parameters include the type of device that is being
considered, the period for updating device states, states of the device
considered stressful for the patient, etc. Aside from the parameters, there are
also formal constraints that these parameters must conform to in order for a
parameter instance to be acceptable. For example, stress states and relaxed
states must be disjoint for a device.
The next stage is to instantiate the parameters of the pattern. In our
example, the medical device safety wrapper is instantiated to filter the input
commands to a pacemaker. The state of the pacemaker is assumed to be its pacing
rate. The necessary parameters are then filled in. The period for updating the
pacing rate is every 10 ms; pacing rates between 90 bpm to 120 bpm are
considered stressful for the patient; etc. This instantiated model will satisfy
the safety properties guaranteed by the safety pattern, provided all the
parameters satisfy the necessary constraints and formal parameter requirements.
Furthermore, once instantiated, the wrapped pacemaker is just another
executable Real-Time Maude model. Thus, we can use this model for simulation and
model checking purposes in Real-Time Maude.
In the final stage, which is the main focus of this paper, we transform the
model to execute in {\em real world} time with physical devices in a medical
device emulation environment. For this purpose, we take the model of the
wrapped pacemaker and wrap it again in an external execution wrapper (Figure
\ref{fig:mdewrapper}). The execution wrapper is responsible for conveying to
the model the notion of real world time as well as providing a communication
interface to the external world. A dedicated timer thread is responsible for
``ticking'' the model by sending a minimal number of messages to advance the
model's logical time. The timer thread also intercepts all asynchronous
(interrupt) messages and relays them to the model. Another aspect of the
execution wrapper is the ability to map external I/O messages to communicate
with the external devices. For example, in the pacemaker specification, an
internal message called {\em paceVentricle} may be mapped into an entire client
configuration to send a message for setting the final voltage on a pacing lead.
\begin{figure}
\centering \includegraphics[width=6cm, angle=-90]{figures/mde_wrapper}
\caption{Real-Time Model Execution Wrapper}
\label{fig:mdewrapper}
\end{figure}
For the design of a medical device or, more generally, of any safety-critical
system, all the stages can work together to achieve a modular component design,
and to support experimentation and testing in the context of other real devices
that the safe component being designed has to interact with. In the first
stages of safety pattern specification, we create a parameteric formal
specification that essentially isolates important safety properties of a device
from the rest of the system. We then can use theorem proving techniques to
provide provable properties of the safety pattern. Although theorem proving
may be time-consuming, as we show with our case studies one safety pattern can
be applied to many different applications, so the time spent proving properties
of the pattern is well worth the effort. The second stage with pattern
instantiation is necessary to obtain a fully specified and executable model.
The last stage of course takes the existing models, with minimal auxiliary
information for external interfaces, and provides an executable prototype
essentially for free. In this way, it becomes possible to emulate the behavior
of safe medical components in an experimental environment involving interactions
with real medical devices.
\section{Distributed Emulation of Safe Medical Devices}
\label{sec:emulation}
The external execution wrapper is an object that encapsulates the original
formal model. It is primarily responsible for interfacing constructs between
the physical world (the real interfaces to devices) and the logical world (the
world as seen by the formal model). In particular, the execution wrapper is
responsible for conveying the measurement of real time elapsed to the model and
also for mapping logical communication messages to communication configurations
that can deliver the message to real devices. The most important feature of the
external execution wrapper is its modularity. Aside from adding the minimal
information about how to map external I/O to messages in the model, no further
specifications are required to execute the logical model within an external
environment.
\subsection{Mapping Logical Time to Physical Time}
As mentioned earlier, time advancement of the system is achieved by defining
the {\em tick} and {\em mte} operators. Ideally the system continuously evolves
over time (possibly nondeterministically). Of course, we cannot capture the
notion of continuous time without abstractions in the model, so to advance time
discretely, an {\em mte} (maximum time elapsable) operator is introduced. A
correctly defined {\em mte} operator ensures that if a system is in state $S$,
then for any time $T < mte(S)$, no 0-time rewrite rules (state transitions)
can apply to $tick(S, T)$. That is, if a system is in state $S$, and $T \leq
mte(S)$, then $tick(S, T)$ will be equivalent to the state $S$ advancing in
continuous time for $T$ time units. This ideal semantics of time is shown on the
left side of Figure \ref{fig:time}. The figure shows that 0-time rewrite
rules are assumed to take zero time, and ideally, the system continuously
evolves over time between the 0-time rewrite rules.
\begin{figure}
\centering \includegraphics[width=14cm, angle=0]{figures/logical_time_to_physical_time}
\caption{From Ideal Time Advancement Semantics to Physical Time Advancement}
\label{fig:time}
\end{figure}
Of course, in a real execution of the model, the ideal notion of time with
0-time rewrite rules and time-advancing rules is only an idealized
abstraction. Performing rewrites cannot take zero time, and we cannot
continuously rewrite states of the system over time. We could of course create
a model in discrete time with very fine time granularity and drive it by a high
frequency clock like in hardware. However, this would introduce a lot of
unnecessary overhead in terms of communication of timing messages and
performing rewrite rules to change the model for every clock tick. We resolve
this problem by observing that the actual internal state of the model is not
important at most instants in time unless it is communicating with the external
world. The model states only generate output messages with 0-time rewrite
rules, so we can essentially let the model in state $S$ remain unaffected by
the passage of time until the next time instant in which a 0-time rewrite
rule can be applied; this is exactly $mte(S)$ time units later. This method of
driving execution is shown in the right part of Figure \ref{fig:time}. We have
created a dedicated timer server thread (in Java) that has access to the system
time. When the execution of the wrapped model starts, it will send a start
request which includes the time units of the model or the minimum granularity
of time for model execution in milliseconds. Once the timer thread processes
all the initial information, it will send a {\em Go!} message to signal the
model to start executing. The model then calculates the maximum time elapsable
(which is 10 seconds in the example) and sends this information to the timer
thread. The model then proceeds to sleep until the timer thread wakes it in
time for the next 0-time rewrite rule. The process then continues. There are
two key points to notice about this example. The input and output messages from
the model may be delayed by an amount of time equal to the communication jitter
plus the time to complete rewriting. Normally this delay is on the order of 10
ms, but this is still suitable for medical devices which normally receive
commands on the order of seconds or more. Also, the timer thread sets the
timeout from the last time it sent a time advancement message to the model and
not from the time it receives the $mte$ message from the model. This ensures
that clock skew and jitter are bounded over time.
\subsection{Synchronous Timed Execution}
The {\em communication wrapper} ({\tt commwrap}) is represented as an object
with the attributes for the communication state, the socket information for
communication, and the internal wrapped (Real-Time Maude) system model being
executed. The top level system is of sort {\tt CommWrapConfiguration} for any
communicating model.
\begin{footnotesize}
\begin{verbatim}
op commwrap : Configuration -> CommWrapConfiguration .
op wrap-client : Configuration -> Configuration .
eq wrap-client(C) = < client : TickClient |
state : start, internal : [ {C} in time 0 ], socket-name : no-oid > .
op init-client : -> CommWrapConfiguration .
eq init-client = commwrap( <> wrap-client(internal)
createClientTcpSocket(socketManager, client, addr, port) ) .
\end{verbatim}
\end{footnotesize}
The communication wrapper initializes a wrapped communication client that
receives messages from the tick server (a Java thread executing in real-time
that sends it messages for time advancement). After creating the TCP socket,
the first message sent from the client to the tick server is the
time-granularity ({\tt time-grain}), which is a rational number specifying the
number of milliseconds in one time unit. Then, the actual execution starts when
the communication wrapper receives a {\tt GO} message from the tick server. The
time when the tick server sends the {\tt GO} message is the starting point from
which time elapses are being measured. Upon receiving the {\tt GO} message, the
formal model will immediately start to execute ({\tt state : run}).
\begin{footnotesize}
\begin{verbatim}
rl [send-init] :
commwrap( <> createdSocket(...) < client : TickClient | ... > )
=> commwrap( <> < client : TickClient | ... > send(..., string(time-grain)) ) .
rl [wait-for-go] :
commwrap( <> sent(...) < client : TickClient | ... > )
=> commwrap( <> < client : TickClient | ... > receive(...) ) .
rl [start-running] :
commwrap( <> received(..., "GO\r\n") < client : TickClient | ... > )
=> commwrap( <> < client : TickClient | state : run, ... > .
\end{verbatim}
\end{footnotesize}
The formal model executes until {\tt mte} becomes non-zero (no other 0-time
rewrite rules can be applied), and the model sends a message to request the
next time advancement message after the maximum time elapse and blocks. After
sending this waiting duration, the tick server will sleep for this time
duration and then send a time advancement message when the time has expired.
The model will then advance time (tick) the model for the time duration expired
and perform 0-time rewrite rules. The model now blocks again for the next {\tt
mte}, and the cycle repeats.
\begin{footnotesize}
\begin{verbatim}
crl [request-wait-timer] :
commwrap( <>
< client : TickClient |
state : run,
internal : [ {C} in time T ], ... > )
=> commwrap( <>
< client : TickClient |
state : request, ... >
send(..., string(mte(C, T))) )
if mte(C,T) :: TimeInf /\ mte(C,T) > 0 /\ not open?(C) . ...
rl [block] :
commwrap( <> sent(...)
< client : TickClient |
state : request, ... > )
=> commwrap( <>
< client : TickClient |
state : wait, ... >
receive(SOCKET-NAME, client) ) .
rl [wake-up] :
commwrap( <> received(..., ADV-STR)
< client : TickClient |
state : wait, ... > )
=> commwrap( <>
< client : TickClient |
state : run,
internal : [ {tick(C, rat(ADV-STR))} in time rat(ADV-STR) in time T ], ... > ) .
\end{verbatim}
\end{footnotesize}
\subsection{Handling Asynchronous External Events}
So far, the model can only handle synchronous events (polling and blocking
communication). However, in general a useful design must be able to react to
external events from the environment. For example, an EKG sensor detects a QRS
waveform, and sends this information to the pacemaker. This points to the fact
that our model needs to be able to handle external events asynchronously.
\begin{figure}
\centering \includegraphics[width=8cm, angle=0]{figures/interrupts}
\caption{Handling Interrupts and Asynchronous Communication Semantics}
\label{fig:async}
\end{figure}
An external message would trigger a 0-time rewrite rule to receive the message
by some object and process it. More precisely, if we have $C_M$ and $C_{Ext}$
as the model configuration and the external (environment) configuration
respectively, the maximal time elapse for the system $C_M C_{Ext}$ should be
$min(mte(C_M), mte(C_{Ext}))$, where $mte(C_{Ext})$ denotes time duration
before the next interrupt message. This semantics is captured by having
interrupt messages forwarded by the timer thread, as shown in Figure
\ref{fig:async}. The timer thread will only check for interrupts when it is
waiting for the next timeout, so when the interrupt message arrives, it will
wake up and immediately forward the interrupt message to the model with the
amount of time that has elapsed. Any future timeouts are canceled. Introducing
the notion of interrupts requires us to modify the wake-up rule for the model
to not only advance time, but also check for potential interrupt messages as
well.
\begin{footnotesize}
\begin{verbatim}
rl [wake-up] :
commwrap( <> received(client, SOCKET-NAME, INTR-STR)
< client : TickClient |
state : wait,
internal : [ {C} in time T ],
socket-name : SOCKET-NAME > )
=>
commwrap( <> < client : TickClient |
state : run,
internal : [
{tick(C, recv->rat(INTR-STR)) recv->conf(INTR-STR)}
in time recv->rat(INTR-STR) in time T
],
socket-name : SOCKET-NAME > ) .
\end{verbatim}
\end{footnotesize}
|
{
"timestamp": "2010-09-23T02:01:00",
"yymm": "1009",
"arxiv_id": "1009.4266",
"language": "en",
"url": "https://arxiv.org/abs/1009.4266"
}
|
\section{Introduction}
Superstring theory is currently the most promising candidate for
a theory of everything. Yet, it is not clear what superstring theory is.
Moreover, even the perturbation theory of the standard (RNS)
formulation of string theory is not yet completely established beyond some
loop level, due to complications related to supermoduli
spaces~\cite{D'Hoker:2002gw,Grushevsky:2008zm,DuninBarkowski:2009ej}.
A possible way to define superstring theory that might also resolve the
problems with supermoduli spaces, is as a superstring field theory, i.e.,
as a field theory of strings (see~\cite{Fuchs:2008cc} for a recent review).
It is natural to expect from a reliable formulation of superstring field
theory that it respects the underlying symmetries
of string theory, i.e., it should be covariant and, moreover,
universal~\cite{Sen:1999xm}.
There are several variants of superstring field theories of this kind, e.g.,
the heterotic~\cite{Berkovits:2004xh} and
open~\cite{Arefeva:1989cp,Preitschopf:1989fc,Berkovits:1995ab,Kroyter:2009rn}
theories. However, it was never checked if any of those is really well
defined at the quantum level. It is our intention to address this question
using lattice techniques.
Some of the formulations of superstring field theory cannot be expected, at
this stage, to be consistent at the quantum level, since, e.g., they don't
include a consistent Ramond sector. The fermions of the Ramond sector, as
well as the notion of ``picture''~\cite{Friedan:1985ge} introduce, in any
case, some complications for the theory.
Hence, it might be advisable to start with a simpler model,
such as that of the bosonic string. The bosonic closed string field
theory~\cite{Zwiebach:1992ie} is much more complicated than the open
one~\cite{Witten:1986cc}. Hence, we concentrate on the later.
It is known that the bosonic string theory is consistent in flat space only
in 26 dimensions. Simulating any theory on a 26 dimensional lattice is almost
a hopeless task. Moreover, the theory has tachyons, both in the open and in
the closed string spectrum. It is by now understood that the open string
tachyon is related to a condensation of an unstable
D-brane~\cite{Sen:1999xm,Sen:1999mh,Sen:1999nx,Schnabl:2005gv}. However,
no analogous understanding exists regarding the fate of the closed string
tachyon~\cite{Yang:2005rw,Yang:2005rx,Moeller:2006cv}.
Both problems can be avoided by considering ``non-critical'' string theory.
The non-critical theory lives at lower dimensions and for $d\leq 2$ the
tachyon is absent. On the other hand, a new complication is introduced,
namely the theory includes a linear dilaton, which breaks Poincar\'e
invariance. Moreover, all the coupling constants are proportional to the
dilaton vacuum expectation value, which runs to infinity in one direction. These issues pose
a challenge to a lattice simulation.
A simple consideration of world-sheet gauge symmetry and degrees of freedom
reveals that two of the $d$ dimensions in which the string lives are
unphysical. Indeed, the two-dimensional string theory is already not a
string theory, but a field theory, in the sense that only one physical field
out of the infinitely many ones, remains. This field is the ``tachyon'',
which is now a massless field. Going on to ``lower dimensions'' is possible.
The dimension is replaced by the central charge of the conformal field
theory. This is natural, since $d$ flat dimensions correspond to $c=d$,
where $c$ is the central charge. The two-dimensional theory includes a single
flat scalar field, i.e., a $c=1$ system, coupled to a single linear dilaton
direction. Theories with $0\leq c<1$ exist and are well studied. They go
under the name of ``minimal models''~\cite{Belavin:1984vu,9304011}. For
simplicity we are starting this study with the simplest model of all,
the $c=0$ theory that includes only the one dimensional linear dilaton
direction with $c=26$ and the canonical ghost system with $c=-26$.
While minimal models have only a finite number of degrees of freedom,
the string field theory that describe them still contains an infinite
number of fields. Almost all of the degrees of freedom of the theory should
then be removed by the very large gauge symmetry present. This fact raises
two further problem that one should address. First, we have to truncate the
infinite number of fields to a finite number while taking this number to
infinity eventually. Second, we have to address the existence of the gauge
symmetry. The first issue of ``level truncation'' was much employed in the
string field theory
literature~\cite{Kostelecky:1990nt,Kostelecky:1988ta,Gaiotto:2002wy},
albeit only at the classical level. It is
not even a-priori clear that it would be a consistent regularization at
the quantum level. Here, we apply an ``experimental approach'' towards this
question. The issue of gauge symmetry arises only at the next level.
Since in this report we only concentrate on preliminary results from the
lowest level, we ignore this issue for now.
\section{Methods}
The ``level'' of level truncation is, up to an additive constant, the
eigenvalue of the ``Hamiltonian'' $L_0$ (the zeroth Virasoro generator).
As such, the level includes two contributions,
that of the field itself, which is different for any of the infinitely many
fields of which the string field is composed and a momentum contribution for
each possible mode of the field. The former contribution is denoted $l_0$ and
the total level is given by,
\begin{equation}
l=l_0+\al' p^2\,,
\end{equation}
where $p$ is the momentum and $\al'$ is a dimensional constant setting the
string scale. We also assumed that the fields were properly redefined,
e.g., instead of the canonical ``tachyon field'' $T(x)$ we consider
$\tau(x)=e^{-\frac{Vx}{2}}T(x)$. This is the only field with $l_0=0$.
The $l_0=0$ level action is
\begin{equation}
S=-\frac{1}{2}\int d x\, \big(m_0^2\tau^2+(\nabla \tau)^2\big)
-\frac{g_o K^{3\big(1-\frac{\al' V^2}{4}\big)}}{3}\int d x\,
e^{-\frac{V \cdot x}{2}}\tilde \tau (x)^3 \,,
\end{equation}
where $K=\frac{3\sqrt{3}}{4}$, $g_o$ is the open string coupling constant,
$V$ is the dilaton gradient
$V=-\sqrt{\frac{25}{6\al'}}$ and $m_0^2$ is the mass squared of the
``tachyon field'', $m_0^2=\frac{V^2}{4}-\frac{1}{\al'}=\frac{1}{24\al'}$.
The second term depends on a non-local variant of $\tau$, namely
$\tilde \tau(x)=K^{\al' \nabla^2} \tau(x)$.
This action is both non-local and space-dependent. Furthermore, we
cannot use periodic boundary conditions, since this would unphysically
glue together the strong-coupling region at large $x$ with the
weak-coupling region at small $x$. Instead we choose Dirichlet
boundary conditions and expand $\tau(x)$ on an interval $x_{min}< x <
x_{max}=x_{min}+L$ in sine waves:
\begin{equation}
\tau(x)=\sqrt{\frac{2}{L}}\sum_{n=1}^N
\tau_n \sin\Big(\frac{\pi n x}{L}\Big)\,.
\end{equation}
Here the level of each mode is given by $l(\tau_n)=\al' (\frac{\pi
n}{L})^2$. We choose $N$ so that all modes have $l < 1$ since we are
working at zero $l_0$ level. We also set $g_o=1$, which amounts to a shift
in $x$.
In terms of the $\tau_n$, the action is
\begin{equation}
S=-\frac{1}{2}\sum_{n=1}^N \Big(\frac{1}{24\al'}
+\big(\frac{\pi n}{L}\big)^2\Big) \tau_n^2
-\frac{g_o K^{3\big(1-\frac{\al' V^2}{4}\big)}}{3} \sum_{n_{1,2,3}=1}^N
K^{-\al'\big(\frac{\pi}{L}\big)^2(n_1^2+n_2^2+n_3^2)}
\tau_{n_1}\tau_{n_2}\tau_{n_3}f_{n_1,n_2,n_3}\,,
\label{tau_n action}
\end{equation}
where
\begin{equation}
f_{n_1,n_2,n_3} = \Big(\frac{2}{L}\Big)^\frac{3}{2}\int_{x_{min}}^{x_{max}} dx
\,e^{-\frac{V x}{2}}\sin\Big(\frac{\pi n_1 x}{L}\Big)
\sin\Big(\frac{\pi n_2 x}{L}\Big)\sin\Big(\frac{\pi n_3
x}{L}\Big).
\label{f_n1n2n3}
\end{equation}
This is the action we want to consider. Note that the weight of a
configuration in the path integral is $e^S$ rather than $e^{-S}$ due
to the way we Wick-rotated.
We see an immediate problem: the action~(\ref{tau_n action}) has a
cubic instability. To proceed, we consider the integral $\int d
\tau_n$ over each mode as a complex integral, and deform the
integration contour to be a straight line at an angle $\gamma$ to the
real axis.
If we choose $\gamma=\pi/6$, the cubic part of the action becomes pure
imaginary and so the action is no longer unstable. This is similar to
the contour deformation used to consider an analytic continuation of
Chern-Simons theory~\cite{Witten:2010cx}.
However, taking the $\tau_n$ to be complex introduces another problem;
the action also becomes complex and so cannot be interpreted as a
weight for a Markov chain. Instead we simulate in the phase-quenched
ensemble and reweight. That is, we split $e^{S}$ into an amplitude and
a phase:
\begin{equation}
e^{S}=|e^{S}| e^{i\theta},
\end{equation}
and calculate the expectation value of an observable $\mathcal{O}$
using the identity
\begin{eqnarray}
\langle \mathcal{O} \rangle & = & \frac{\int \mathcal{O}|e^{S}|e^{i \theta}}{\int
|e^{S}|e^{i \theta}} \\
& = & \frac{\langle \mathcal{O}e^{i \theta} \rangle_\mathrm{PQ}}{\langle e^{i
\theta} \rangle_\mathrm{PQ}},
\end{eqnarray}
where the label $\mathrm{PQ}$ means the expectation value is evaluated in
the phase-quenched ensemble, i.e. with the weight $|e^{S}|$. This
is a real, positive weight, so can be used in a Monte Carlo simulation.
We generate configurations in the phase-quenched ensemble using a
Metropolis algorithm. The observables we measure are the action $S$
and the fields $\tau_n$. We estimate errors with the jackknife method.
\section{Results}
In principle, we expect that the theory will be unstable for all
values of the parameters, since there is always a cubic term in the
action. However, due to the factor $e^{-\frac{V x}{2}}$ in
$f_{n_1,n_2,n_3}$, this term can be exponentially small, in which case
the theory will be stable for all practical purposes. We have
performed scans in parameter space to search for the onset of instability.
The instability can be seen by looking at the imaginary parts of
$\langle S \rangle$ and $\langle \tau_n \rangle$, which will be zero
for a stable set of parameters and will become non-zero as the
instability increases. In practice we find that the errors are smaller
for the $\langle \tau_n \rangle$ than for the action, so we will
concentrate on the former from now on; however the behaviour of the
action is very similar.
We find that the onset of the instability is rather rapid. We show an
example in Fig.~\ref{complx_fig}. Here we have $N=6$, which is the
maximum allowed for $L=20$. In each case we observe that the $\langle
\tau_n \rangle$ oscillate, but at $x_{min}=-20.5$ they have negligible
imaginary parts, whereas
by $x_{min}=-19.5$ the imaginary parts are as large as the real parts.
Hence the instability appears roughly when $x_{max}=0$, that is when
we start to include the region $x>0$ where the cubic terms become
large. The behaviour for other values of $L$ is very similar, with the
instability first appearing around $x_{max}=0$ in each case.
\begin{figure}
\centering
\psfrag{Re<tau>}{$\mathrm{Re}\langle \tau_n \rangle$}
\psfrag{Im<tau>}{$\mathrm{Im}\langle \tau_n \rangle$}
\epsfig{file=figs/tau_6b.eps,scale=0.92,clip}
\caption{$\langle \tau_n \rangle$ in the complex plane for
$x_{min}=-20.5$ (red),
$-20$ (green), and $-19.5$ (blue). All for $L=20$, $N=6$, $\al'=1$,
$V=-\sqrt{\frac{25}{6\al'}}$. In each case $\langle \tau_1
\rangle$ is the point furthest to the left.}
\label{complx_fig}
\end{figure}
Since the action is space-dependent, the meaning of the
infinite-volume limit is unclear. It is straightforward to decrease $x_{min}$,
since the cubic term becomes exponentially small at small
$x$. However, it is not clear if the limit $x_{max} \rightarrow
\infty$ is well-defined, since the cubic term continues to get
stronger in this direction. Indeed, we find that the imaginary parts
of the $\tau_n$ continue to increase rapidly when we increase $x_{max}$.
\subsection{Continuum limit}
In momentum space, the continuum limit is approached by increasing the
number of modes~$N$. We can only increase $N$ up to a
maximum value of $L/\pi\sqrt{\al'}$ since we require $l<1$. In this
range we find that the instability becomes stronger as $N$ is
increased, presumably because the number of unstable cubic terms
increases rapidly with $N$. We show an example in
Fig.~\ref{contlimit_fig},
where the maximum level increases from 0.05 (where $N=1$)
to 0.95 ($N=6$).
To approach closer to the continuum limit,
we will have to include more fields at higher
$l_0$-levels. Calculations at level-1 are in progress.
\begin{figure}
\centering
\psfrag{l}{$l$}
\psfrag{Im<tau1>}{$\mathrm{Im}\langle \tau_1 \rangle$}
\epsfig{file=figs/contlimit.eps,scale=0.46,angle=-90,clip}
\caption{$Im \langle \tau_1 \rangle$ as a function of level $l$, for
$x_{min}=-20$, $L=20$, $\al'=1$,
$V=-\sqrt{\frac{25}{6\al'}}$.}
\label{contlimit_fig}
\end{figure}
\subsection{Dilaton}
Finally, we have considered what happens when we vary the dilaton
$V$. In the full string field theory we require
$V=-\sqrt{\frac{25}{6\al'}}$. We might expect that varying $V$ away
from this value would increase the instability. However, this is not
the case, at least for the level-0 theory. Increasing $V$ above
$-\sqrt{\frac{24}{6\al'}}$ changes the sign of $m_0^2$, making the
theory tachyonic and hence more unstable; on the other hand decreasing
$V$ makes $m_0^2$ larger and the cubic terms smaller and hence
decreases the instability. Our results when we vary $V$ are
consistent with this picture. However, it should be noted that
decreasing $V$ does not remove the instability entirely, and it will
always become strong at some sufficiently large value of $x$.
\section{Conclusions}
We have implemented a Monte Carlo simulation of the 1-d linear
dilation truncated to zero $l_0$-level. We observe non-trivial
quantum effects: the classical solution to the equations of motion is
$\tau_n=0$, but we observe non-zero $\tau_n$. We also find that as
expected, the theory is unstable at large $x$, as shown by the large
imaginary parts the expectation values of the field develop.
There are several possible explanations for our result. Firstly, it
may be that the instability is a real feature of the full,
non-truncated theory. This should not be the case for the theory at hand.
Another possibility is that the instability is just an artifact of the
level-truncation. In this scenario the higher-level fields, which we
have not included, would stabilise the theory. Alternatively, it might
also be the case that level-truncation is not a consistent regularization
of the quantum theory.
Finally it is also possible that the instability represents some
fundamental problem with open string field theory as a method for
quantising string theory. This could be attributed, e.g., to the lack of
control over closed string degrees of freedom or to the somewhat
singular nature of the star product.
Calculations including level-1 fields are underway. At this level it
is not obvious how to deal with the gauge and ghosts degrees of
freedom, and there are several possible choices. It will be
interesting to see how these compare. A practical issue is that
Grassmann-odd fields will appear and will have to be dealt with.
Looking further ahead, it would be interesting to increase the number
of dimensions. Ultimately the target would be to work in ten dimensions
and to include fermionic degrees of freedom, with the aim of reaching a full
quantum, non-perturbative definition of superstring theory.
\section*{Acknowledgements}
We would like to thank Y.~Oz, L.~Rastelli and B.~Zwiebach for discussions.
The research of M.~K is supported by a Marie Curie OIF. The views presented
are those of the authors and do not necessarily reflect those of the European
Community.
|
{
"timestamp": "2010-09-23T02:02:32",
"yymm": "1009",
"arxiv_id": "1009.4414",
"language": "en",
"url": "https://arxiv.org/abs/1009.4414"
}
|
\section{Introduction}
The application of chaotic dynamics concepts to asteroidal dynamics
led to the understanding of the main structural characteristics of
asteroids distribution within the solar system. It was verified that
chaotic region are generally devoid of larger asteroids while, in
contrast, regular regions exhibit a great number of them (see
for instance, Berry 1978, Wisdom 1982, Dermott and Murray 1983,
Hadjidemetriou and Ichtiaroglou 1984, Ferraz-Mello et al. 1997,
Tsiganis et al. 2002b, Kne\v{z}evi\'{c} 2004, Varvoglis 2004)
It was soon
accepted that chaos was related inevitably to instability, which may
be local or global. Subsequent
investigations searched for initial conditions leading to instabilities
in relatively short time. In many applications, the determination
of Lyapunov exponent on a grid of initial conditions was used to get
quantitative informations on stability. The inverse of the largest
Lyapunov exponent, called Lyapunov time, should be in some way linked
to the characteristic time for the onset of chaos (Morbidelli and
Froeschl\'{e}, 1996).
However, some investigations have shown that many asteroids exhibit
intermediary behavior between chaos and regularity. The first registered
case was the asteroid (522) Helga (Milani and Nobili, 1992). This
asteroid was in chaotic orbit with a Lyapunov time inch shorter than the age of the solar system, but
it exhibited a long period stability. No significant evolution was
observed in the orbital elements of (522) Helga for times up to one
thousand times its Lyapunov time. Since then, other asteroids have
been shown to have Lyapunov times much shorter than the stability
times unraveled by simulations (e.g., Trojans, cf. Milani 1993). Currently,
this behavior is known in literature as \textit{stable chaos} (Milani
et al. 1997, Tsiganis et al. 2002a, Tsiganis et al. 2002b). Indeed, there is
strong evidence that local instability does not mean chaotic diffusion, in the sense that nothing can be
said about how much global or local integrals (or orbital elements)
could change in a chaotic domain, even when a linear stability analysis
shows rather short Lyapunov times (see Giordano and Cincotta, 2004, Cincotta and
Giordano 2008).
Nesvorn\'{y} and Morbidelli (1998, 1999) demonstrated that one source
of stable chaos is related with three-body (three-orbit) mean-motion
resonances (Jupiter-Saturn-asteroid system). They observed that asteroids in these
resonances exhibit a slow diffusion in eccentricity and inclination, but no diffusion in the semi-major axis.
According to the estimates of Nesvorn\'{y} and Morbidelli (1998), about
1500 among the first numbered asteroids are affected by three-body
mean-motion resonances.
The three-body mean-motion resonances are very narrow since they appear
at second order in planetary masses, their typical width being $\sim10^{-3}$
AU, but they are much more dense (in phase space) than standard two-body
mean-motion resonances of similar size. Nesvorn\'{y} and Morbidelli (1999)
developed a detailed model for the three-body mean-motion resonance
and presented analytical and numerical evidence that most of them
exhibit a highly chaotic dynamics (at moderate-to-low-eccentricities)
which may be explained in terms of an overlap of their associated
multiplets. By multiplet, we refer to all resonances for which the
time-derivative of the resonant angle, $\sigma_{p,p_{J},p_{S}}$,
satisfies\begin{equation}
\dot{\sigma}_{p,p_{J},p_{S}}=m_{J}\dot{\lambda}_{J}+m_{S}\dot{\lambda}_{S}+m\dot{\lambda}+p\dot{\varpi}+p_{J}\dot{\varpi}_{J}+p_{S}\dot{\varpi}_{S}\simeq0,\label{resonant_angle}\end{equation}
for given $\left(m_{J},m_{S},m\right)\in\mathbb{Z}^{3}/\{0\}$. In
(\ref{resonant_angle}) the $\lambda$'s and $\varpi$'s denote, as
usual, the mean longitudes and perihelion longitudes, respectively;
$\left(p,p_{J},p_{S}\right)\in\mathbb{Z}^{3}$ are integers such that
$\sum_{i}\left(m_{i}+p_{i}\right)=0$ for \textit{i} ranging over
three bodies (Jupiter, Saturn and the asteroid).
We will be dealing in this paper with the case $\left(m_{J},m_{S},m\right)=\left(5,-2,-2\right)$.
This three-body resonance seems to dominate the dynamics of, for instance,
the asteroids (3460) Ashkova, (2039) Payne-Gaposchkin and (490) Veritas
(see Nesvorn\'{y} and Morbidelli 1999). In the case of the first two of
those asteroids (with relatively large eccentricity, $\sim0.15-0.20$),
their behavior looks regular over comparatively long time-scales (typically
$\sim1-10\times10^{3}$ years) while in case of (490) Veritas (with
eccentricity, $\sim0.06$) its dynamics looks rather chaotic over
similar time-scales. The determination of the age of (490) Veritas
family has been the concern of some authors who studied stable chaos
(Milani and Farinella 1994, Kne\v{z}evi\'{c} 1999, Kne\v{z}evi\'{c} et al. 2002, Kne\v{z}evi\'{c} 2003, Kne\v{z}evi\'{c} et al. 2004, Tsiganis et al. 2007, Kne\v{z}evi\'{c} 2007, Novakovi\'{c} et al. 2009).
Herein, we investigate chaotic diffusion \textit{along} (and also
\textit{across}) the above mentioned three-body mean-motion resonance
by means of a classical diffusion approach. We use (partially)
the formulations given by Chirikov (1979). That formulation are developed
to study specifically Arnold diffusion or some kind of diffusion that
geometrically resembles it initially called Fast-Arnold diffusion by
Chirikov and Vecheslavov (1989, 1993), as well as the so called
modulational diffusion (Chirikov et. al, 1985).
However, although from the purely mathematical point of view several
restrictions should be imposed, there are many unsolved aspects
regarding general phase space diffusion (see for instance Lochak 1999,
Cincotta 2002, Cincotta and Giordano 2008).
The structure of the Hamiltonian used by Chirikov in his first formulation
is similar to the
Hamiltonians obtained with the perturbations theories of Celestial
Mechanics. In particular, the Hamiltonians of analytic models of the
three-body mean-motion resonances are directly adaptable,
with some restrictions, to Chirikov's formulations.
Let us mention that some progress has been done
in the study of Arnold diffusion, particularly when applied to simple
dynamical systems, like maps, the latest ones are for instance, the
works of Guzzo et. al (2009a),(2009b), Lega (2009). However
the link between strictly Arnold diffusion and general diffusion
in phase space is still an open matter. Indeed, Arnold diffusion requires
a rather small perturbation, when the measure of the regular component
of phase space is close to one. Thus, as far as we know, almost
all investigations regarding Arnold diffusion involves relatively
simple dynamical systems like quasi--integrable maps.
In more real systems, like the one investigated in this paper,
the scenario is much more complex in the sense that the domain of
the three body resonance is almost completely chaotic.
Finally, this work is justified by the fact that an application
of all those theories to real astronomical models is still needed.
In Sect. 2, we summarize the general problem of computation of the
diffusion rate \textit{along} the resonance and we discuss the limitations
and difficulties to follow Chirikov approach in case of this particular
three-body mean-motion resonance. Section 3 is devoted to the resonant
Hamiltonian (given in Nesvorn\'{y} and Morbidelli (1999)) and its application
to the $\left(5,-2,-2\right)$ resonance. In Sect. 4, we construct
the simplified (or two-resonance) and complete (or three-resonance)
numerical models used in our investigations. Moreover, informations
about the algorithms and initial conditions used for numerical integrations
and the procedure for estimation of the diffusion are also considered
in this section. In Sect. 5, we discuss the numerical results on
diffusion in the $\left(5,-2,-2\right)$ resonance. In this application
of the Chirikov theory, we are concerned with the role of the perturbing
resonances in the diffusion \textit{across} and \textit{along} the
$\left(5,-2,-2\right)$ resonance and their relationship to diffusion
in semi-major axis and eccentricity. Finally, in Sect. 6, we investigate
the behavior of the asymptotic diffusion decreasing the intensity
of the perturbations in the $\left(5,-2,-2\right)$ resonance. In
this case, we are interested in the study of the diffusion under the
action of an arbitrarily weak perturbation considering scenarios close
to that of the Arnold diffusion.
\section{Chirikov's Diffusion Theory}
In this section we give Chirikov's (1979) as well as Cincotta's (2002)
description of diffusion theory in phase space in order to provide a
self-consistent presentation of the subject. Since most of the results and
discussions given here are included in at least these two reviews, we just
address the basic theoretical aspects.
Let us consider a Hamiltonian system having several periodic perturbations
that can create resonances. The initial conditions are chosen such that
the system is in the domain of a main resonance, called \textit{guiding
resonance}. The term of perturbation corresponding to the guiding
resonance is separated from the others, which will be called of \textit{perturbing
resonances}. The Hamiltonian has the following form
\begin{equation}
H=H_{0}\left(\mathbf{I}\right)+\epsilon V_{G}\left(\mathbf{I}\right)
\cos\left(\mathbf{m}_{G}\cdot\boldsymbol{\theta}\right)+
\epsilon V\left(\mathbf{I},\boldsymbol{\theta}\right),\label{Hamilton_original}
\end{equation}
with
\begin{equation}
\epsilon V=\epsilon\sum_{\mathbf{m}\neq\mathbf{m}_{G}}
V_{\mathbf{m}}\left(\mathbf{I}\right)\cos\left(\mathbf{m}\cdot\boldsymbol{\theta}\right),
\label{perturbation_original}
\end{equation}
where $V_{G}$ and $\mathbf{\mathbf{m}}_{G}$ are, respectively, the
amplitude and resonant vector of the \textit{guiding resonance}, $V_{\mathbf{m}}$
and $\mathbf{m}$ are, respectively, the amplitude and resonant vectors
of perturbing resonances. Here $\left(\mathbf{I},\boldsymbol{\theta}\right)$
are the usual \textit{N}-dimensional action-angle coordinates for
the unperturbed Hamiltonian $H_{0}\left(N\geq3\right)$, the vectors
$\mathbf{m}_{G}$, $\mathbf{m}\in\mathbb{Z}^{N}/\{0\}$ and $V_{G}$,
$V_{\mathbf{m}}$ are real functions. The small parameter perturbation,
$\epsilon$, is a real number such that $\epsilon\ll 1$. The resonance
condition is fixed by
\begin{equation}
S\left(\mathbf{I}^{r}\right)\mathbf{=}
\mathbf{m}_{G}\cdot\mathbf{\boldsymbol{\omega}}\left(\mathbf{I}^{r}\right)=0.
\label{eq:resonance_condition}
\end{equation}
The surface \textbf{$S\left(\mathbf{I}^{r}\right)=0$} in the action
space, is called \textit{resonant surface}.
\subsection{Dynamics of the guiding resonance in the actions space}
Let us first consider the simple case of one single resonance, that
is, let us assume that all $V_{\mathbf{m}}=0$ for $\mathbf{m}\neq\mathbf{m}_{G}$,
and we chose initial conditions close to the separatrix of the guiding
resonance. In the $\boldsymbol{\omega}$-space, the resonance condition
$\mathbf{m}_{G}\cdot\boldsymbol{\omega}^{r}=0$ has a very simple
structure, just a $\left(N-1\right)$-dimensional plane the
normal of which is the resonant vector $\mathbf{m}_{G}$. In the \textbf{$\mathbf{I}$}-space,
$\mathbf{m}_{G}\cdot\boldsymbol{\omega}^{r}=0$ leads to the $\left(N-1\right)$-dimensional
resonant surface $S\left(\mathbf{I}^{r}\right)=0$, whose local normal
at the point $\mathbf{I}=\mathbf{I}^{r}$ is
\begin{equation}
\mathbf{n}^{r}=
\left(\frac{\partial}{\partial\mathbf{I}}
\left[\mathbf{m}_{G}\cdot
\boldsymbol{\omega}\left(\mathbf{I}\right)\right]\right)_{\mathbf{I=}\mathbf{I}^{r}}
\label{eq:normalvector}
\end{equation}
In addition, we consider the $\left(N-1\right)$-dimensional surface
$H_{0}\left(\mathbf{I}\right)=E$ (in $\mathbf{I}$-space) and, if
we suppose that $\boldsymbol{\omega}\left(\mathbf{I}^{r}\right)$
is an one-to-one application, we can also write
$\widetilde{H}_{0}(\boldsymbol{\omega})=
H_{0}\left(\mathbf{I}\left(\boldsymbol{\omega}\right)\right)=E$
(in $\boldsymbol{\omega}$-space).
The manifolds
defined by the intersection of both resonant and energy surfaces has, in general,
dimension $N-2.$ By definition, the frequency vector $\boldsymbol{\omega}$
is normal to the energy surface in $\mathbf{I}$-space, since it is
the $\mathbf{I}$-gradient of $H_{0}.$ The latter condition, together
with the resonance condition Eqn. (\ref{eq:resonance_condition}),
shows that the resonant vector $\mathbf{m}_{G}$ lies on a plane tangent
to the energy surface at $\mathbf{I}=\mathbf{I}^{r}$. Furthermore,
the equations of motion (only with $H_{0}$
and the guiding resonant term) show that $\mathbf{\dot{\mathbf{I}}}$ is
parallel to the constant vector $\mathbf{m}_{G}$. Thus the motion under a single
resonant perturbation lies on the tangent plane to the energy surface
at the point $\mathbf{I}=\mathbf{I}^{r}$ in the direction of the
resonant vector.
\subsection{Local change of basis}
Now, let us introduce a canonical transformation
$\left(\mathbf{I},\boldsymbol{\theta}\right)\rightarrow\left(\mathbf{p},\boldsymbol{\psi}\right)$
by means of a generating function
\begin{equation}
F\left(\mathbf{p},\boldsymbol{\theta}\right)=\sum\limits _{i=1}^{N}\left(I_{i}^{r}+
\sum\limits _{k=1}^{N}p_{k}\mu_{ki}\right)\theta_{i},\label{geratrix_function}
\end{equation}
where $\mu_{ik}$ is a $N\times N$ matrix with $\mu_{1i}=\left(\mathbf{m}_{G}\right)_{i}$.
The transform action equations are
\begin{equation}
I_{i}=I_{i}^{r}+\sum\limits _{k=1}^{N}p_{k}\mu_{ki};\qquad\psi_{k}=
\sum\limits _{\ell=1}^{N}\mu_{k\ell}\theta_{\ell}.
\label{transform_equations}
\end{equation}
The phases $\psi_{k},k=1,\ldots,N$ are supposed to be non degenerate,
i.e., $\tfrac{\partial H_{0}}{\partial I_{k}}\neq0$. As Cincotta (2002) has shown,
this transformation
should better be thought as a local change of basis rather than as
a local change of coordinates. The action vector whose components
are $\left(I_{j}-I_{j}^{r}\right)$ in the original basis $\left\{ u_{j},j=1,\ldots,N\right\} $,
has components $p_{j}$ in the new basis $\left\{ \mu_{j},j=1,\ldots,N\right\} $
constructed taking advantage of the particular geometry of resonances
in action space.
We choose, $\mathbf{\boldsymbol{\mu}}_{1}=\mathbf{m}_{1}\equiv\mathbf{m}_{G}$
and since the vector $\mathbf{m}_{G}$ is orthogonal to the frequency
vector $\boldsymbol{\mathbf{\omega}}^{r}$ (due to the resonance condition),
it seems natural to take $\mathbf{\boldsymbol{\mu}}_{2}=
\boldsymbol{\mathbf{\omega}}^{r}/\left|\boldsymbol{\mathbf{\omega}}^{r}\right|.$
The remaining vectors of the basis are $\boldsymbol{\mathbf{\mu}}_{k}=
\mathbf{e}_{k},k=3,\ldots,N,$
the vectors $\mathbf{e}_{k}$ are orthonormal to each other and to
$\boldsymbol{\mathbf{\mu}}_{2}.$ Let us define one of the $\mathbf{e}_{k}$,
say $\mathbf{e}_{s}$, orthogonal to the normal $\mathbf{n}^{r}$
to the guiding resonance surface. In general, all the vectors $\mathbf{e}_{k}$
will be orthogonal also to $\boldsymbol{\mathbf{\mu}}_{1}$, except
$\mathbf{e}_{s}$. In general, $\mathbf{e}_{s}$ will not be orthogonal
to $\mathbf{m}_{G}$.
Then, considering $N=3$ and since $\mathbf{p}=p_{i}\boldsymbol{\mathbf{\mu}}_{i},
i=1,\ldots,3,$
we can say that $p_{1}$ measures the deviations of the
actual motion from the resonant point across the guiding resonance
surface, $p_{3}$ measures the deviation from the resonant value
along the guiding resonance, while $p_{2}$ measures the
variations in the unperturbed energy.
For $N\ge3$ degrees of freedom the subspace of intersection of the
two surfaces leads to a manifold of $N-2$ dimensions. Following Chirikov (1979),
this subspace is called \textit{diffusion manifold}. The $N-2$ vectors
$\mathbf{e}_{k}$ locally span (at the resonant value) a tangent plane
to the diffusion manifold called \textit{the diffusion plane.} Then,
in the new basis, the action vector may be written as:
$\mathbf{p}=p_{1}\mathbf{m}_{G}+p_{2}\boldsymbol{\mathbf{\omega}}^{r}/
\left|\boldsymbol{\mathbf{\omega}}^{r}\right|+\mathbf{q}$,
where $\mathbf{q}$ is confined to the diffusion plane
$\mathbf{q}=\sum_{k}q_{k}\mathbf{e}_{k}$
with $q_{k}=p_{k}$ for $k=3,\ldots,N$.
We write now the Hamiltonian (\ref{Hamilton_original}) in terms of
the new components of the action. Expanding up to second order in
$p_{k}$, using the orthogonal properties of the new basis, recalling
that $\psi_{1}$ is the resonant phase and neglecting the constant
terms, we obtain for $\left(k,\ell\right)\neq\left(1,1\right)$
\begin{equation}
H\left(\mathbf{p},\boldsymbol{\mathbf{\psi}}\right)\approx
\frac{p_{1}^{2}}{2M_{G}}+\epsilon V_{G}\cos\psi_{1}+\left|\omega^{r}\right|p_{2}+
\sum\limits _{k=1}^{N}\sum\limits _{\ell=1}^{N}\frac{p_{k}p_{\ell}}{2M_{k\ell}}+
\epsilon V\left(\boldsymbol{\psi}\right),
\label{Hamilton_transformed}
\end{equation}
with
\begin{equation}
\frac{1}{M_{k\ell}}=
\sum\limits _{i=1}^{N}\sum\limits _{j=1}^{N}\mu_{ki}\frac{\partial\omega_{i}^{r}}{\partial I_{j}}
\mu_{\ell j},
\label{tensor_massa}
\end{equation}
\begin{equation}
\frac{1}{M_{G}}=\frac{1}{M_{11}}=m_{Gi}\frac{\partial\omega_{i}^{r}}{\partial I_{j}}m_{Gj};
\label{mass_tensor_guiding}
\end{equation}
where we have written $V_{G},V\left(\boldsymbol{\psi}\right)$ instead
of $V_{G}\left(\mathbf{p}\right),V\left(\mathbf{p},\mathbf{\boldsymbol{\psi}}\right)$.
These functions are evaluated at the point $\mathbf{I}=
\mathbf{I}^{\mathit{r}}\textrm{ or }\mathbf{p=0}$.
In absence of perturbation $\left(V=0\right)$, the components $p_{k},k=2,\ldots,N$
are integrals of motion, which we set equal to zero so that $\mathbf{I}^{r}$
is a point of the orbit. Then the Hamiltonian (\ref{Hamilton_transformed})
reduces to
\begin{equation}
H\left(\mathbf{p,}\boldsymbol{\mathbf{\psi}}\right)\approx
H_{1}\left(p_{1},\psi_{1}\right)+
\epsilon V\left(\boldsymbol{\psi}\right),
\label{Hamilton_reduced}
\end{equation}
where
\begin{equation}
H_{1}=\frac{p_{1}^{2}}{2M_{G}}+\epsilon V_{G}\cos\psi_{1}
\label{Hamilton_pendulum}
\end{equation}
is the resonant Hamiltonian associated to the guiding resonance. It
is a simple pendulum. Note that the stable equilibrium point of the
pendulum is $\psi_{1}=\pi$ if $M_{G}V_{G}>0$, or $\psi_{1}=0$ if
$M_{G}V_{G}<0$.
To transform the phase variables, we take into account that the dot
product is invariant under a change of basis. Recalling
that $\psi_{k}=\sum_{\ell}\mu_{k\ell}\theta_{\ell}$,
then if $\mathbf{\boldsymbol{\nu}}$ denotes the vector $\mathbf{m}$
in the new basis, we have: $\varphi_{\mathbf{m}}\equiv
\mathbf{m}\boldsymbol{\cdot\mathbf{\theta}}=\boldsymbol{\mathbf{\nu}\cdot\mathbf{\psi}}$,
where $m_{k}=\sum_{\ell}\nu_{\ell}\mu_{\ell k}$. As we can see, while
the $m_{k}$ are integers, the quantities $\nu_{k}$ are, in general,
non-integer numbers, due to the scaling of the phase variables.
\subsection{Changes due to perturbation}
As mentioned above, for $V=0$ the $p_{k}$ are integrals of motion
and since $H_{1}$ is also an integral, we have the full set of $N$
unperturbed integrals: $H_{1},p_{2},q_{k},k=3,\ldots,N.$ But if we
switch on the perturbation, these quantities will change with time.
This can be seen using the equations of motion for the Hamiltonian
(\ref{Hamilton_transformed}), where $\dot{\psi}_{j}=\partial H/\partial p_{j},j=1,\ldots,N$.
Performing derivatives and integrating, considering
that for $V=0$, $p_{\ell}\left(\ell\neq2\right)$ are constants and
$p_{1}=M_{G}\dot{\psi}_{1}-\sum_{\ell=2}^{N}\frac{M_{G}}{M_{1\ell}}p_{\ell}$,
we obtain\begin{equation}
\begin{array}{ccc}
\psi_{k}\left(t\right)=\left|\boldsymbol{\omega}^{r}\right|t\delta_{2k}+\sum\limits _{\ell=2}^{N}\left(\frac{1}{M_{k\ell}}-\frac{M_{G}}{M_{k\ell}M_{1\ell}}\right)p_{\ell}t+\frac{M_{G}}{M_{k1}}\psi_{1}\left(t\right)+\psi_{k0}, & & k>1\end{array}\label{new_phase}\end{equation}
where $\delta_{ij}$ is the Kronecker's delta and $\psi_{j0}$ is
a constant. To get $\varphi_{\mathbf{m}}\left(t\right)$, we evaluate
the dot product $\sum_{i}\nu_{i}\psi_{i}$\begin{equation}
\varphi_{\mathbf{m}}=\mathbf{m}\cdot\boldsymbol{\mathbf{\theta}}=\boldsymbol{\mathbf{\nu}}\cdot\boldsymbol{\mathbf{\psi}}=\xi_{\mathbf{m}}\psi_{1}\left(t\right)+\omega_{\mathbf{m}}t+\beta_{\mathbf{m}}+K_{\mathbf{m}},\label{new_argument}\end{equation}
where\begin{equation}
\xi_{\mathbf{m}}=\sum\limits _{k=1}^{N}\nu_{k}\left(\mathbf{m}\right)\frac{M_{G}}{M_{k1}},\qquad\omega_{\mathbf{m}}=\mathbf{\mathbf{m}}\cdot\mathbf{\boldsymbol{\omega}}^{r},\label{constant_argument}\end{equation}
and $\beta_{\mathbf{m}}$ is a constant and\begin{equation}
K_{\mathbf{m}}=\sum_{\ell=2}^{N}\nu_{k}\left(\mathbf{m}\right)\left(\frac{1}{M_{k\ell}}-\frac{M_{G}}{M_{k\ell}M_{1\ell}}\right)p_{\ell}t.\label{eq:constphase}\end{equation}
The second relation of (\ref{Hamilton_pendulum}) is obtained taking
into account that \begin{equation}
\mathbf{m}\cdot\boldsymbol{\mathbf{\omega}}^{r}=\left(\sum_{i}\nu_{i}\left(\mathbf{m}\right)\boldsymbol{\mathbf{\mu}}_{i}\right)\cdot\left(\boldsymbol{\mathbf{\mu}}_{2}\left|\boldsymbol{\mathbf{\omega}}^{r}\right|\right)\label{eq:prodmomegar}\end{equation}
and the fact that, since $\mathbf{\boldsymbol{\mu}}_{2}$ is orthogonal
to all $\boldsymbol{\mathbf{\mu}}_{i},i\neq2$ and $\mu_{2}\cdot\mu_{2}=1$,
the dot product only contributes to $i=2$. Then,\begin{equation}
\mathbf{\omega_{\mathbf{m}}=m}\cdot\boldsymbol{\mathbf{\omega}}^{r}=\nu_{2}\left(\mathbf{m}\right)\left|\mathbf{\boldsymbol{\omega}}^{r}\right|.\label{eq:freq_m}\end{equation}
We are now ready to compute the time variation of the unperturbed
integrals. From (\ref{Hamilton_reduced}) and (\ref{perturbation_original}),
for $\dot{p}_{k}=-\partial H/\partial\psi_{k},k\neq1$, we easily
find\begin{equation}
\dot{p}_{k}\left(t\right)\approx\epsilon\sum\limits _{\mathbf{m\neq\mathbf{m}}_{G}}\nu_{k}\left(\mathbf{m}\right)V_{\mathbf{m}}^{r}\sin\varphi_{\mathbf{m}}\left(t\right).\label{time_variation_pk}\end{equation}
where $V_{\mathbf{m}}^{r}=V_{\mathbf{m}}\left(\mathbf{I}^{r}\right).$
This equation holds for every component of the momentum $\mathbf{p}$,
except for $p_{1}$. Since $p_{1}$ is not an integral, we use $H_{1}$,
instead of $p_{1}$.
Chirikov (1979) calculated the total variation of $H_{1}$ with
the aim of constructing a whisker map to describe the Arnold diffusion.
However, instead of it, we prefer, in the study of three-body resonances,
to compute the evolution of the components of the momentum
$\mathbf{p}$ by means of numerical integrations or, alternatively,
by mean of a Hadjidemetriou-type sympletic map (see Sect. 4). However,
we use a variation of Chirikov's construction to obtain a theoretical
estimate of the slow diffusion. We then proceed and compute the
total variation of $p_{k}$. For details about the construction of
the whisker map we refer to Chirikov (1979, Sect. 7.3) (see also
Cincotta 2002 and the Appendix B of Ferraz-Mello 2007).
If $\epsilon$ is small enough, the phase space domains associated
with all resonances present in (\ref{perturbation_original}) do not
overlap. Then a standard procedure is to replace $\psi_{1}\left(t\right)$
and $\dot{\psi}_{1}\left(t\right)$ by the values on the unperturbed
separatrix and to solve analytically (\ref{time_variation_pk}). We
make first the integration of (\ref{time_variation_pk}) over a complete
trajectory inside the stochastic layer assuming that $\psi_{1}=\psi_{1}^{sx}$
and $K_{m}=0$. Indeed, as mentioned previously for $V=0$ the $p_{\ell}\left(\ell\neq1\right)$
are integrals of motion and the phases $\varphi_{\mathbf{m}}$ can
be estimated considering $p_{\ell}\left(\ell\neq1\right)=0$, such
that $K_{\mathbf{m}}=0$. Then, the total variations of the $p_{k}$'s
are given by\begin{equation}
\Delta p_{k}\left(t\right)\approx\epsilon\sum\limits _{\mathbf{m\neq\mathbf{m}}_{G}}\nu_{k}\left(\mathbf{m}\right)V_{\mathbf{m}}^{r}\int_{_{-\infty}}^{^{+\infty}}\sin\varphi_{\mathbf{m}}^{sx}\left(t\right)dt,\label{eq:varpkint}\end{equation}
where $\varphi_{\mathbf{m}}^{sx}\left(t\right)=\xi_{\mathbf{m}}\psi_{1}^{sx}\left(t\right)+\omega_{\mathbf{m}}t+\beta_{\mathbf{m}}$. The estimate of integral into (\ref{eq:varpkint}) is done considering the known solutions for the phase $\psi_{1}^{sx}\left(t\right)$ obtained near both branches of unperturbed separatrix of the pendulum $H_{1}$. More details about these calculations are given in the appendix of this paper. Here we only described the main steps and the final result for $\Delta p_{k}\left(t\right)$.
Chirikov shown that the contributions of integral in (\ref{eq:varpkint}) in both branches of separatrix are described in terms of the Melnikov integral with arguments \begin{equation}
\pm\left|\lambda_{\mathbf{m}}\right|=\pm\left|\frac{\omega_{\mathbf{m}}}{\Omega_{G}}\right|,\label{eq:lambda}
\end{equation}
where the double sign indicates the both separatrix branches and $\Omega_{G}$ is the proper frequency of the pendulum Hamiltonian $H_{1}$. In order to simplify the calculations, Chirikov considered only even perturbing resonances and the contribution of Melnikov integral with negative argument was neglected under the condition $\left|\lambda_{\mathbf{m}}\right|\gg1$. In contrast, the perturbations in the three-body mean-motion resonance model are non even and the arguments are small. Moreover, the asymmetry in the Nesvorn\'{y}-Morbidelli model implies that the time of permanence of the motion near each separatrix is different. Thus, we introduce the factor $R_{T}$ which takes into account the difference in the time of permanence of the motion in each separatrix branch. Hence, after some algebraic manipulations the Eqn. (\ref{eq:varpkint}) is rewritten as
\begin{equation}
\Delta p_{k}\approx\frac{\epsilon}{\Omega_{G}}\sum\limits _{\mathbf{m}\neq\mathbf{m}_{G}}\nu_{k}\left(\mathbf{m}\right)Q_{\mathbf{m}}\sin\varphi_{\mathbf{m}}^{0},\label{eq:totalvarpk}\end{equation}
with\begin{equation}
Q_{\mathbf{m}}=V_{\mathbf{m}}^{r}\left[R_{T}A_{2\left|\xi_{\mathbf{m}}\right|}\left(\left|\lambda_{\mathbf{m}}\right|\right)+\left(1-R_{T}\right)A_{2\left|\xi_{\mathbf{m}}\right|}\left(-\left|\lambda_{\mathbf{m}}\right|\right)\right],\label{Melnikov_integral1}\end{equation}
where $\varphi_{\mathbf{m}}^{0}=\varphi_{\mathbf{m}}^{sx}\left(t=t^{0}\right)$
with $\psi_{1}^{sx}\left(t=t^{0}\right)=\pi$. Equation (\ref{eq:totalvarpk}) is a theoretical
estimate for the total variation of the momenta $p_{k}$'s inside the stochastic layer around
of separatrix of the pendulum Hamiltonian $H_{1}$, and it is valid for non-even perturbation
and for small $\lambda_{\mathbf{m}}$. Estimations of the Melnikov integral,
$A_{2\left|\xi_{\mathbf{m}}\right|}\left(\left|\lambda_{\mathbf{m}}\right|\right)$,
in terms of ordinary function can be obtained from the values of
$\left|\lambda_{\mathbf{m}}\right|$ and $\left|\xi_{\mathbf{m}}\right|$.
On the other hand, the factor $R_{T}$ can be estimated from numerical experiments.
\subsection{\label{sub:thediffusionrate}The diffusion rate}
In Chirikov's theory of slow diffusion, each resonance has a role
in the dynamics of system. The main resonance, that is the guiding resonance,
defines the domain where diffusion occurs. The stronger perturbing resonance
is called \textit{layer
resonance}. That resonance perturbs the guiding resonance separatrix
and it generates the stochastic layer and its
properties (width, KS-entropy, etc.). Thus, the layer resonance controls
the dynamics $\textrm{\underline{across}}$ the stochastic layer.
The weaker perturbing resonances are called \textit{driving resonances}.
They perturb the stochastic layer and control the dynamics $\textrm{\underline{along}}$
the stochastic layer. Then, the driving resonances are responsible
for the drift along the stochastic layer, i.e., the slow diffusion.
We are interested in obtaining an analytical estimate for the slow
diffusion. To fulfill this task, we will estimate the diffusion in
the actions whose direction is given along the stochastic layer.
We introduced the slow diffusion tensor\begin{equation}
D_{ij}=\frac{\overline{\Delta p_{i}\left(t\right)\Delta p_{j}\left(t\right)}}{T_{a}}\qquad i,j=3,\ldots N,\label{Diffusion_tensor}\end{equation}
where $T_{a}=\ln\left(32e/w_{s}\right)/\Omega_{G}$ is the characteristic
time of the motion within the stochastic layer of the guiding resonance
(equal to half the period of libration or to one period of circulation
of $\psi_{1}$ near the separatrix) and the average in the numerator
is done over successive values of $\varphi_{\mathbf{m}}^{0}$. Here $w_{s}$
is the width of the stochastic layer given by\begin{equation}
w_{s}=-\frac{\left|\boldsymbol{\omega}^{r}\right|}{\Omega_{G}^{2}}\omega_{\mathbf{\mathbf{m}}_{L}}\frac{\nu_{\mathnormal{1}}\left(\mathbf{\mathbf{m}}_{L}\right)\nu_{\boldsymbol{2}}\left(\mathbf{\mathbf{m}}_{L}\right)}{\xi_{\mathbf{\mathbf{m}}_{L}}}Q_{\mathbf{m}_{L}}>0.\label{width_layer}\end{equation}
(see Sects. 6.2 and 7.3 of Chirikov 1979). In the last equation,
the subscript $L$ indicate the layer resonance.
The components of the diffusion tensor (\ref{Diffusion_tensor}) are estimated using the Eqn. (\ref{eq:totalvarpk}). Hence, because of dependence with the phase $\varphi_{\mathbf{m}_{{\it D}}}^{0}$, the average in (\ref{Diffusion_tensor}) depends: (1) of the correlation between successive values $\varphi_{\mathbf{m}_{{\it D}}}^{0}$
when the system approaches the edges of the layer; (2) of the possible
interferences of several driving resonances. However, the analysis done by Chirikov shown that the terms that contribute to the diffusion must have the same phase $\varphi_{\mathbf{m}_{{\it D}}}^{0}$ (see Sect. 7.5 of Chirikov 1979 and Cincotta 2002 for more details). Hence, using (\ref{eq:totalvarpk}) the diffusion tensor components in (\ref{Diffusion_tensor}) are described as
\begin{equation}
D_{ij}=\frac{\epsilon^{2}}{T_{a}\Omega_{G}^{2}}\sum\limits _{\mathbf{m}_{D}}\nu_{i}\left(\mathbf{m}_{D}\right)\nu_{j}\left(\mathbf{m}_{D}\right)Q_{\mathbf{m}_{D}}^{2}\overline{\sin^{2}\varphi_{\mathbf{m}_{D}}^{0}}.\label{eq:diffusion_mean}\end{equation}
Terms with different $\mathbf{m}_{D}$ are averaged out.
Now, there still remains the problem of estimating $\overline{\sin^{2}\varphi_{\mathbf{m}_{D}}^{0}}$.
To solve this problem we need to consider that the structure of the stochastic layer
affects the motion of the system. In fact, studies of the slow diffusion theories have shown
that the stochastic layer is formed by two different regions.
The first, more central, near the unperturbed
separatrix, is totally chaotic. The second, more external, near the
edge of the stochastic layer, includes domains of regular motion forming
stability islands. When the solution approaches the edge of the stochastic
layer, it could remain rather close to the neighborhood of those stability islands
for long times. This phenomenon, called stickiness, leads to
a reduction in the diffusion rate (for more details about
the stickiness phenomenon see the recent work of Sun and Zhou 2009 and references
therein). Thus, near stability islands some
correlations in the phases arise, which dominate
the motion across and along the stochastic layer. In this case, the
evolution of phases $\varphi_{\mathbf{m}_{L}}^{0}$ and $\varphi_{\mathbf{m}_{D}}^{0}$
cannot be random simultaneously, and their correlation decreases the
diffusion rate (see Chirikov 1979, Cincotta 2002).
In order to estimate the correlation between $\overline{\sin^{2}\varphi_{\mathbf{m}_{D}}^{0}}$
and $\overline{\sin^{2}\varphi_{\mathbf{m}_{L}}^{0}}$, we use the so called \textit{reduced stochasticity approximation}, introduced by Chirikov (1979) like an additional hypothesis. Hence, the theoretical rate of diffusion given by (\ref{eq:diffusion_mean}) may be now evaluated and has the form
\begin{equation}
D_{ij}=\frac{\epsilon^{2}}{2\Omega_{G}^{2}T_{a}}\sum\limits _{\mathbf{m}_{D}}R_{\mathbf{m}_{D}}\nu_{i}\left(\mathbf{m}_{D}\right)\nu_{j}\left(\mathbf{m}_{D}\right)Q_{\mathbf{m}_{D}}^{2}\qquad i,j=3,\ldots,N.\label{Chirikov_diffusion_tensor}
\end{equation}
The Eqn. (\ref{Chirikov_diffusion_tensor}) is an estimate for the theoretical diffusion inside the stochastic layer. The diffusion coefficient includes two parameters that reduce the diffusion rate: $R_{T}$due to non-even perturbations and $R_{\mathbf{m}_{D}}$ due to the reduced stochasticity approximation.
The expression given here for the diffusion tensor is different of that given by Chirikov because of
the introduction of the parameter $R_{T}$ and by the possibility of having a small argument in the
Melnikov integral. Moreover, we have considered that the reduction factor due the reduced
stochasticity approximation is different for each driving resonance,
while Chirikov considers the same value for all of them.
\section{Application to 3-body mean-motion resonance}
The Hamiltonian, in the extended phase space, associated to a given $\left(m_{J},m_{S},m\right)$
resonance, in Delaunay action-angles variables, is
\begin{equation}
H=-\frac{1}{2L^{2}}+n_{J}\Lambda_{J}+n_{S}\Lambda_{S}+v_{J}\Pi_{J}+v_{S}\Pi_{S}+\mathcal{P}_{\text{sec}}+\mathcal{P}_{\text{res}},\label{Hamilton_Nesvorny}\end{equation}
where
\begin{equation}
\lambda,\varpi,\lambda_{J},\varpi_{J},\lambda_{S},\varpi_{S}\label{Delaunay_angles}\end{equation}
are the mean longitudes and longitudes of the perihelions of the asteroid,
Jupiter and Saturn, respectively, and \begin{equation}
L=\sqrt{a};\Pi=\sqrt{a}\left(\sqrt{1-e^{2}}-1\right),\Lambda_{J},\Pi_{J},\Lambda_{S},\Pi_{S}\label{Delaunay_actions}\end{equation}
are the actions conjugated to them. The frequencies $n_{J},v_{J},n_{S},v_{S}$
are the mean-motion and perihelion motions of Jupiter and Saturn,
respectively.
The first term in (\ref{Hamilton_Nesvorny}) describes the Keplerian
motion of the asteroid and the terms proportional to the planetary
actions extend the phase space to incorporate the motion of the angles
$\lambda,\varpi,\lambda_{J},\varpi_{J},\lambda_{S},\varpi_{S}$ in
the unperturbed Hamiltonian. Details concerning the derivation of
this Hamiltonian are given by Nesvorn\'{y} and Morbidelli (1999), whose
main results and formula are used in this paper. Note that this Hamiltonian
does not satisfy the convexity condition, however, this fact should not be
a restriction for the application of Chirikov's diffusion theory.
The perturbing function, following Nesvorn\'{y} and Morbidelli (1999),
has been splitted into its secular and resonant parts
\begin{equation}
\mathcal{P}_{\text{sec}}=\frac{\mu_{J}}{a_{J}}\sum\limits _{k_{J},k_{S},k,i_{J},i_{S},i}P_{\text{sec}}\left(\alpha_{\text{res}}\right)e^{k}e_{J}^{k_{J}}e_{S}^{k_{S}}\cos\left(i_{J}\varpi_{J}+i_{S}\varpi_{S}+i\varpi\right)\label{Secular_perturbation}\end{equation}
\begin{equation}
\mathcal{P}_{\text{res}}=\frac{\mu_{J}}{a_{J}}\sum\limits _{k_{J},k_{S},k,p_{J},p_{S},p}P_{\text{res}}\left(\alpha_{\text{res}}\right)e^{k}e_{J}^{k_{J}}e_{S}^{k_{S}}\cos\left(\sigma_{p,p_{J},p_{S}}\right)\label{Resonant_perturbation}\end{equation}
where, $\alpha_{\text{res}}=a_{\text{res}}/a_{J}$ is the semi-major
axis corresponding to the exact resonance, $\sigma_{p,p_{J},p_{S}}=m_{J}\lambda_{J}+m_{S}\lambda_{S}+m\lambda+p\varpi+p_{J}\varpi_{J}+p_{S}\varpi_{S},$
$\mu_{J}$ is Jupiter's mass, $e$, $e_{J}$, $e_{S}$ are the asteroid,
Jupiter and Saturn's eccentricities, respectively, and $P_{\text{sec}}\left(\alpha_{\text{res}}\right)$,
$P_{\text{res}}\left(\alpha_{\text{res}}\right)$ are given functions that are linear in Saturn's mass
(see bellow). The harmonic coefficients satisfy d'Alembert rules,
$i_{J}+i_{S}+i=0$, $m_{J}+m_{S}+m+p+p_{J}+p_{S}=0$ and the series
are truncated at some order in $\left|k_{J}\right|+\left|k_{S}\right|+\left|k\right|$,
$\left|i_{J}\right|+\left|i_{S}\right|+\left|i\right|$ and $\left|m_{J}\right|+\left|m_{S}\right|+\left|m\right|+\left|p\right|+\left|p_{J}\right|+\left|p_{S}\right|$.
Next, we reduce the secular part (\ref{Secular_perturbation}) to
the quadratic term in asteroid's eccentricity in order to break the degeneracy
of the unperturbed Hamiltonian, and introduce in (\ref{Hamilton_Nesvorny})
the new action-angle variables, $\left(\mathbf{I'},\boldsymbol{\mathbf{\theta}'}\right)$:
\begin{equation}
\mathbf{I'}\boldsymbol{=}\left(N,N_{J},N_{S},\Pi,\Pi_{J},\Pi_{S}\right)\qquad\textrm{(actions)}\label{eq:new_action}\end{equation}
\begin{equation}
\boldsymbol{\mathbf{\theta}'=}\left(\nu,\tilde{\nu}_{J},\tilde{\nu}_{S},\varpi,\varpi_{J},\varpi_{S}\right)\qquad\textrm{(angles)}\label{eq:new_angle}\end{equation}
defined by\begin{equation}
\nu=m_{J}\lambda_{J}+m_{S}\lambda_{S}+m\lambda,\qquad\tilde{\nu}_{J}=\lambda_{J},\qquad\tilde{\nu}_{S}=\lambda_{S},\label{new_old_angles}\end{equation}
and\begin{equation}
L=mN,\qquad\Lambda_{J}=m_{J}N+N_{J},\qquad\Lambda_{S}=m_{S}N+N_{S}.\label{new_old_actions}\end{equation}
The variables ($\Pi_{J},\varpi_{J},\Pi_{S},\varpi_{S}$) remain unchanged.
We recall that the resonant perturbation (\ref{Resonant_perturbation})
does not depend on $\tilde{\nu}_{J}$ and $\tilde{\nu}_{S}$ (so that
$N_{J},N_{S}$ are constant that we can take as equal to zero). Let
us write\begin{equation}
\mathbf{I}\boldsymbol{\equiv}\left(N,\Pi,\Pi_{J},\Pi_{S}\right),\qquad\boldsymbol{\mathbf{\theta}\equiv}\left(\nu,\varpi,\varpi_{J},\varpi_{S}\right).\label{variables_transformed}\end{equation}
Eliminating the constant terms, the Hamiltonian (\ref{Hamilton_Nesvorny})
may be written\begin{equation}
H\left(\mathbf{I,}\boldsymbol{\mathbf{\theta}}\right)=H_{0}\left(\mathbf{I}\right)+\tilde{V}\left(\mathbf{I,}\boldsymbol{\mathbf{\theta}}\right),\label{Hamilton_reduced_Nesvorny}\end{equation}
where\begin{equation}
H_{0}\left(\mathbf{I}\right)=-\frac{1}{2m^{2}N^{2}}-\beta_{0}\left(1+\frac{\Pi}{mN}\right)^{2}+\left(m_{J}n_{J}+m_{S}n_{S}\right)N+\nu_{J}\Pi_{J}+\nu_{S}\Pi_{S},\label{Hamilton_Nesvorny_transformed}\end{equation}
is the unperturbed Hamiltonian and the perturbation is described
by \begin{equation}
\tilde{V}\left(\mathbf{I,}\boldsymbol{\mathbf{\theta}}\right)=\sum\limits _{\mathbf{m}}\beta_{\mathbf{m}}\left(\mathbf{I}\right)\cos\left(\mathbf{m}\boldsymbol{\cdot\mathbf{\theta}}\right),\label{Perturbation_transformed}\end{equation}
with\begin{equation}
\mathbf{m}=\left(1,p,p_{J},p_{S}\right),\label{eq:resonantvector}\end{equation}
and\begin{equation}
\beta_{\mathbf{m}}\left(\mathbf{I}\right)=\frac{\mu_{J}}{a_{J}}\sum\limits _{k_{J},k_{S},k}P_{\text{res}}\left(\alpha_{\text{res}}\right)e^{k}e_{J}^{k_{J}}e_{S}^{k_{S}}.\label{Coefficient_perturbation}\end{equation}
Nesvorn\'{y} developed a procedure allowing to obtain the coefficients
(\ref{Coefficient_perturbation}) in terms of power series of the
asteroid eccentricity only. In the last column of Table \ref{tabvectors}
are the coefficients calculated by Nesvorn\'{y} for the guiding (\textit{G}),
layer (\textit{L}) and driving (\textit{D}) resonances used in our
numerical experiments.
We have considered the guiding resonance, defined by the vector $\mathbf{m}_{G}=\left(1,-1,0,0\right)$,
the layer resonance, defined by the vector $\mathbf{m}_{L}=\left(1,0,-1,0\right)$
and the driving resonance defined by vector $\mathbf{m}_{D}=\left(1,0,0,-1\right)$.
The unperturbed separatrices of those resonances in the plane $a-e$
are shown in Fig. (\ref{fig:figsep}).
The next step is to introduce the Chirikov variables $\left(\mathbf{p,}\boldsymbol{\psi}\right)$
allowing to have a separate representation of the actions \textit{across}
and \textit{along} the resonance within the stochastic domain of the
guiding resonance. The canonical transformation, is performed by the generating
function (\ref{geratrix_function}), with a transformation matrix,
$\mathbf{\boldsymbol{\mu}}$, given by
\begin{equation}
\boldsymbol{\mathbf{\mu}} = \left(\begin{array}{cccc}
1 & -1 & 0 & 0\\
\displaystyle{\frac{\omega_{2}^{r}}{\left|\boldsymbol{\omega}^{r}\right|}} & \displaystyle{\frac{\omega_{2}^{r}}{\left|\boldsymbol{\omega}^{r}\right|}} & \displaystyle{\frac{\nu_{J}}{\left|\boldsymbol{\omega}^{r}\right|}} & \displaystyle{\frac{\nu_{S}}{\left|\boldsymbol{\omega}^{r}\right|}}\\
\displaystyle{-\frac{2v_{S}n_{2}^{r}\omega_{2}^{r}}{\left|\mathbf{q}^{r}\right|}} & \displaystyle{\frac{2v_{S}n_{1}^{r}\omega_{2}^{r}}{\left|\mathbf{q}^{r}\right|}} & \displaystyle{-\frac{\nu_{J}v_{S}\left(n_{2}^{r}-n_{1}^{r}\right)}{\left|\mathbf{q}^{r}\right|}} & \displaystyle{\frac{\left|\mathbf{v}\right|^{2}\left(n_{2}^{r}-n_{1}^{r}\right)}{\left|\mathbf{q}^{r}\right|}}\\
\displaystyle{\frac{\sqrt{2}v_{J}}{2\left|\mathbf{v}\right|}} & \displaystyle{\frac{\sqrt{2}v_{J}}{2\left|\mathbf{v}\right|}} & \displaystyle{-\frac{\sqrt{2}\omega_{2}^{r}}{\left|\mathbf{v}\right|}} & 0\end{array}\right)\label{transformation_matrix}\end{equation}
where\[
\left|\mathbf{q}^{r}\right|=\sqrt{\left(v_{J}^{2}v_{S}^{2}+\left|\mathbf{v}\right|^{4}\right)\left(n_{2}^{r}-n_{1}^{r}\right)^{2}+4\nu_{S}^{2}\omega_{2}^{r2}\left|\mathbf{n}^{r}\right|^{2}},\]
\[
\left|\mathbf{\mathbf{n}}^{r}\right|=\sqrt{\left(n_{1}^{r}\right)^{2}+\left(n_{2}^{r}\right)^{2}},\qquad\left|\mathbf{v}\right|=\sqrt{\nu_{J}^{2}+2\left(\omega_{2}^{r}\right)^{2}}.\]
\begin{table}
\caption{Old and new resonant vectors, and coefficients of the guiding (\textit{G}),
layer (\textit{L}) and driving (\textit{D}) resonances (The coefficients
were taken from Nesvorn\'{y} and Morbidelli, 1999).}
\label{tabvectors}
\centering
\begin{tabular}{llll}
\hline\noalign{\smallskip}
& vectors $\mathbf{m}$ & vectors $\mathbf{\boldsymbol{\nu}}$ & coefficients $\widetilde{\beta}_{\mathbf{m}}\left(\times10^{-8}\right)$\tabularnewline
\noalign{\smallskip}\hline\noalign{\smallskip}
\textit{G } & $\left(1,-1,0,0\right)$ & $(1,0,0,0)$ & $45.59e-32.24e^{3}$\tabularnewline
\textit{L } & $\left(1,0,-1,0\right)$ & $(0.55,0.66,0.76,0.70)$ & $-2.76+0.93e^{2}$\tabularnewline
\textit{D} & $\left(1,0,0,-1\right)$ & $(0.68,0.47,0.92,-0.70)$ & $1.18-0.38e^{2}$\tabularnewline
\noalign{\smallskip}\hline
\end{tabular}
\end{table}
\begin{figure}
\centering
\includegraphics[scale=0.4]{fig2.eps}
\caption{Unperturbed separatrices of guiding, layer and driving
resonances in the plane $(a,e)$. (Nesvorn\'{y} and Morbidelli, 1999)}
\label{fig:figsep}
\end{figure}
Once the matrix of the transformation is defined, the new variables
$\left(\mathbf{p,}\boldsymbol{\mathbf{\psi}}\right)$ can be rapidly
obtained using the relations (\ref{transform_equations}). In the
new basis the arguments of the periodic terms change. The new vectors
$\mathbf{\boldsymbol{\nu}}$ defined by $\mathbf{m}\cdot\mathbf{\boldsymbol{\theta}}=\boldsymbol{\mathbf{\nu}}\cdot\boldsymbol{\mathbf{\psi}}$
are shown in Table \ref{tabvectors}, in addition to the resonant
vectors $\mathbf{m}$ and their respective coefficients.
The procedure of previous section was applied in the Nesvorn\'{y}-Morbidelli
model, and leads to the Hamiltonian (\ref{Hamilton_transformed})
with $N=4$. The three perturbation coefficients $\beta_{G}$, $\beta_{L}$
and $\beta_{D}$ of the guiding, layer and driving resonances, respectively,
are calculated at the resonant values $\mathbf{I}^{r}$, which satisfies
(\ref{eq:resonance_condition}). In the plane $N\Pi$ the resonant
condition (\ref{eq:resonance_condition}) leads to a curve satisfying
to\begin{equation}
\Pi^{r^{2}}+C_{1}\Pi^{r}+C_{2}=0,\label{eq:curvasresonantes}\end{equation}
where $C_{1}\left(N^{r}\right)$ and $C_{2}\left(N^{r}\right)$ are
given in terms of $N^{r}$. Then, the solutions of (\ref{eq:curvasresonantes})
can be obtained analytically for a fixed value of $N^{r}$. However,
in Nesvorn\'{y}-Morbidelli model the coefficients $\tilde{\beta}_{\mathbf{m}}$'s
are given as functions of the asteroid eccentricity (see Table \ref{tabvectors}).
Therefore, we must use the definitions of Delaunay variables (\ref{Delaunay_actions})
to determinate $(a^{r},e^{r})$. The resonant eccentricity is determined
through $e^{r}=\sqrt{1-\left(1+\Pi^{r}/N^{r}\right)^{2}},$ where
$\left(N^{r},\Pi^{r}\right)$ satisfies (\ref{eq:resonance_condition}).
The resonant semi-major axis is determinate using $a^{r}=\left(N^{r}/2\right)^{2}.$
\section{Numerical Experiments}
In this section, we describe the numerical experiments done
to investigate the diffusion \textit{across} and \textit{along} the
stochastic layer of the three-body mean-motion resonance $\left(m_{J},m_{S},m\right)=(5,-2,-2)$ and its relations with the diffusion in semi-major axis and eccentricity.
In these investigations the diffusion \textit{across} will be described
by the actions $(p_{1},p_{2})$ and the diffusion \textit{along} by
the actions $(p_{3},p_{4})$. In order to determine the time evolution
of each action $\mathbf{p}$, we use the equations of motion obtained
from Hamiltonian (\ref{Hamilton_transformed}) with $N=4$. Then,
for each value $p_{k}(t)$, we use the equations of transformation
(\ref{transform_equations}) to obtain the respective values of $N(t)$
and $\Pi(t)$ and the definition of the Delaunay actions in (\ref{Delaunay_actions})
to obtain $a(t)$ and $e(t)$.
Two main models were considered in the numerical experiments: (i)
simplified (or two-resonance model) and (ii) complete (or three-resonance
model). In the first one, only one term of the perturbation - the
layer resonance - is considered. In the complete model, two terms
are considered: the layer and one driving resonance. In both cases
the guiding resonance is given by $\mathbf{m}_{G}=\left(1,-1,0,0\right)$.
Two different techniques were used to construct the solutions. In
a first set of experiments, the equations of motion of the Hamiltonian
(\ref{Hamilton_transformed}) were numerically integrated using the
Burlish-St\"{o}er method, for times in the interval $10^{2}\leq t_{int}\leq10^{8}$
years. The results of these simulations were sampled with an output
time step of 10 years. The simulations were done for the eccentricities
0.05 and 0.25 with the initial conditions given on the separatrix
of the guiding resonance $(p_{10}=2\sqrt{\left|M_{G}\beta_{G}^{r}\right|},p_{20}=0,p_{30}=0,p_{40}=0,\boldsymbol{\psi}=0)$.
The main goal in these experiments was the study of the variation
of the rate of diffusion \textit{across} and \textit{along} as a function
of the total time of the simulations. Moreover, we investigate the
correlations between the diffusion in the Chirikov actions $\mathbf{p}$
and the diffusion in semi-axis major and eccentricity.
In the other set of experiments, the simulations were done using an
Hadjidemetriou-type sympletic mapping (Hadjidemetriou 1986, 1988, 1991, 1993; Ferraz-Mello 1997; Roig and Ferraz-Mello 1999, Lhotka 2009) defined
by the canonical transformation $\left(\mathbf{p}^{n},\mathbf{\boldsymbol{\psi}}^{n}\right)\rightarrow\left(\mathbf{p}^{n+1},\mathbf{\boldsymbol{\psi}}^{n+1}\right),$
whose generating function is given by \begin{equation}
S\left(\mathbf{p}^{n+1},\mathbf{\boldsymbol{\psi}}^{n}\right)=\sum\limits _{i=1}^{3}\psi_{i}^{n}p_{i}^{n+1}+\eta H\left(\mathbf{p}^{n+1},\mathbf{\boldsymbol{\psi}}^{n}\right),\label{function_geratrix_map}\end{equation}
where $\eta$ is the mapping step and the Hamiltonian is given by
(\ref{Hamilton_transformed}). The mapping equations are
\begin{equation}
p_{i}^{n+1} = p_{i}^{n}-\eta\frac{\partial H\left(\mathbf{p}^{n+1},\mathbf{\boldsymbol{\psi}}^{n}\right)}{\partial\psi_{i}^{n}}\label{equations_map}\end{equation}
\begin{equation}
\psi_{i}^{n+1} = \psi_{i}^{n}+\eta\frac{\partial H\left(\mathbf{p}^{n+1},\mathbf{\boldsymbol{\psi}}^{n}\right)}{\partial p_{i}^{n+1}}\qquad i=1,2,3.\label{eq:psi_map}
\end{equation}
The procedure to determinate the semi-major axis and eccentricity
for each point $(p_{i}^{n+1},\psi_{i}^{n+1})$ of the trajectory is
analogous to that discussed above. The goal of these experiments is
to obtain the diffusion contour plots in the region of the 5,-2,-2
resonance in the plane $(a,e)$ (that plane is shown in Fig. \ref{fig:figsep})
for the two models considered. In this case, the total time of integration
used is the $10^{8}$ years with $\eta=10$ years. The initial conditions
are defined by the knots of a grid in the plane $(a,e)$, on the rectangle $\left(3.17\leq a_{0}\leq3.18\right)$U.A.,
$\left(0.01\leq e_{0}\leq0.30\right)$. The initial condition of the
state vector $\mathbf{p}_{0}$, for each point of the grid, was obtained
using the transformation equations (\ref{transform_equations}) and
the definitions of Delaunay variables. The initial condition for the
phases is $\psi_{k0}=0,k=1,\ldots4$. The use of the Hadjidemetriou
map was instrumental allowing the computation of the solutions starting on each point of the grid which, otherwise, would demand an excessively
large amount of CPU-time. The comparison of results provided by the map with those
obtained by integrating the Hamiltonian flow, do not show significant differences
in the numerical computation of the diffusion coefficient (see below), at least for the two values of the eccentricity used (0.05 and 0.25).
Finally, we need a numerical procedure to estimate the diffusion coefficient
of each element of the set $(p_{1},p_{2},p_{3},p_{4},a,e)$. In his
investigations, Chirikov (1979, et al. 1979, 1985) used a particular
method to determine the diffusion coefficient of the total energy
$H$ of the system. After Chirikov (1979), this procedure allows the
processes that are really stochastic to be separated from those associated
to bounded oscillations of periodic nature. Chirikov's procedure for
experimental determination of the diffusion coefficient consist in
dividing the total time of simulation $t_{int}$ in $N_{k}$ sub-intervals
of length $\left(\Delta t\right)_{k}$ and the calculation of the
mean value, $\bar{p}_{i}$, for every sub-interval. The contribution
to the diffusion rate for a given pair $\bar{p}_{i_{m}}$, separated
by interval of time $\left(m-\ell\right)\left(\Delta t\right)_{k}$,
is given by $\left(\bar{p}_{i_{m}}-\bar{p}_{i_{\ell}}\right)^{2}/\left|m-\ell\right|\left(\Delta t\right)_{k}$.
To obtain the rate of diffusion, the contributions of the considered
pairs are averaged over all the combinations $m\neq\ell$. That is,\begin{equation}
D_{i}^{k}=\frac{2}{N_{k}\left(N_{k}-1\right)}\sum_{m>\ell}\frac{\left(\bar{p}_{i_{m}}-\bar{p}_{i_{\ell}}\right)^{2}}{\left(\Delta t\right)_{k}\left(m-\ell\right)}.\label{diffusion_coef_exp}\end{equation}
The sub-intervals, used to estimate the mean values of quantities
$\bar{p}_{i}$, were obtained with $k=10$ and the length $(\Delta t)_{10}=t_{int}/10$.
The same procedure was used to determine the diffusion of the semi-major
axis, $D_{a}^{k}$, and of the eccentricity, $D_{e}^{k}$. We have
also estimated the eccentricity variation in these experiments using
a definition of diffusion rate of the random walking type (see for
example Eqn. (\ref{Diffusion_tensor})): \begin{equation}
\delta e\sim\sqrt{D_{e}^{k}t_{int}}.\label{variation_exp}\end{equation}
\section{Results and discussion}
In this sections, we discuss the results obtained in the numerical
experiments described above. In the discussion we will call action
\textit{across} to $(p_{1},p_{2})$, and \textit{across} diffusion
to $(D_{1},D_{2})$, where we suppressed the superscript $k$. In
the same way we call action \textit{along} to $(p_{3},p_{4})$ and
\textit{along} diffusion to $(D_{3},D_{4})$.
\subsection{\label{sub:role}The role of number of the perturbing resonances
in the diffusion}
In his theory, Chirikov showed that the number of perturbing resonances
is important for the dynamics of systems with many-dimensional Hamiltonians.
The results, in this case, repeat what is know from the general theory
of Hamiltonian systems. In a system with two degrees of freedom, the
resonances may be isolated by KAM tori, but for $(N>3)$ the dimensionality
may allows, in principle, a solution to visit the whole phase space when
$t\to\infty$.
Several experiments, using the Burlish-St\"{o}er integrator, were done
see the way in which the number of perturbing resonances in the diffusion
behavior. Figure \ref{fig:resonumber} shows the results for the diffusion
coefficients $D_{i},i=1,2,3,4$ in the simplified and complete models
as function of total integration time for eccentricities equal to
0.05 and 0.25, . In the plots of Fig. \ref{fig:resonumber}, we
see that the estimated diffusion increases in the low eccentricities
up to a maximum reached for $10^{3}-10^{4}$ years. This behavior
is explained by the fact that the solution needs to fill the stochastic
domain in the direction \textit{across} to it. After that maximum, in
the simplified model the diffusion coefficients for all actions decrease
continuously. This decrease indicates that the variation of the momenta
in both directions, \textit{across} and \textit{along} the stochastic
layer are bounded (as the total time increase, only the denominator
of (\ref{diffusion_coef_exp}) grows making the result to decrease).
As predicted by Chirikov's theory of slow diffusion, the actions $p_{3}$
and $p_{4}$ \textit{along} the resonance do not evolve, notwithstanding
the absence of topological barriers for its evolution. Without a driving
resonance, there is no long-period evolution of the solution \textit{along}
the stochastic domain. In our experiments, a very distinctive reduction
in the diffusion is observed in the case $e=0.25$ after $t_{int}\sim10^{7}$
years. This behavior is likely due to a sticking of the solution
to some regular domain.
The behavior of the diffusion in the complete model is more complicated.
In the experiment with $e=0.05$, the diffusion coefficients for the
actions \textit{across} the stochastic domain after $10^{8}$ years are
smaller than for the actions \textit{along} it. This difference reaches
approximately four orders of magnitude in this case and is due, probably,
to the limitation of the motion \textit{across} the stochastic layer
imposed by its width. For $e=0.25$ (right plot of Fig. \ref{fig:resonumber}),
the diffusion coefficients in the two models present almost the same
characteristics observed for $e=0.05$, except by the fact that, now,
the diffusion in the actions \textit{along} the stochastic layer,
present a slow reduction with the integration time after $10^{4}$
years. This behavior is likely due to the absence of overlapping of
resonance at high eccentricities, in contrast with the case of low
eccentricities, where the three resonances overlap (see Fig. \ref{fig:figsep}).
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{fig3.eps}
\caption{Diffusion coefficients of the actions associated
with motion \textit{across} and \textit{along} the guiding resonance,
in experiments over times from $10^{2}-10^{8}$\textit{ }years for
two different initial eccentricities.}
\label{fig:resonumber}
\end{figure*}
\subsection{Diffusion in semi-major axis and eccentricity in the complete model }
The study of the previous section was completed
with the computation of the diffusion coefficients for the orbital
elements: semi-major axis and eccentricity in the complete model. Figure \ref{fig:orbitalelements} presents the results. The results for the actions shown in this figure are the same shown in Fig. \ref{fig:resonumber}, but with a magnified
scale. We see that, for large total times, there exist a correspondence
between the diffusion coefficients of the actions \textit{across}
$(p_{1},p_{2})$ and of the semi-major axis, and between the diffusion
coefficients of the actions \textit{along} the resonance $(p_{3},p_{4})$
and of the eccentricity. This behavior can be understood observing
the geometry of resonance $\left(5,-2,-2\right)$ shown in the Fig. \ref{fig:figsep}.
The separatrices of resonances are straight lines and the motion,
\textit{along} one of these separatrices, has constant semi-major
axis and variable eccentricity. Following the discussion presented in
Sect. \ref{sub:role}, and the comparison done in the previous section
for the simplified and completed models, we know that the drift \textit{along}
the separatrices only occurs if there is at least one driving resonance.
Hence, the eccentricity diffusion is due to the driving resonance.
As a complement to the previous discussion, we note that the variations
in semi-major axis occurs in the horizontal direction, the same direction
of the actions $(p_{1},p_{2})$. The behavior of the diffusion in
semi-major axis is similar to the diffusion of the actions \textit{across}
the resonance $(p_{1},p_{2})$ and is bounded by the width of the
stochastic domain. A consequence of this fact is that the diffusion
coefficient in the semi-major axis is smaller than that for the eccentricity
(in the complete model, the diffusion \textit{along} is not bounded).
The Fig. \ref{fig:eccentricityvariation} shows the variation of
the eccentricity calculated using initial conditions forming a grid
in the plane $(a,e)$ (the same grid of Fig. \ref{fig:contourn})
in $t_{int}=10^{8}$ years. The results for the simplified model are
shown in Fig. \ref{fig:eccentricityvariation}(a). In this case,
the larger variations in eccentricity occur for small values of the
eccentricity. Two shallow maximums are formed, which are likely related
with the eccentricity value at the intersection of the separatrices
of the guiding and layer resonances (the only secondary resonance
considered in the simplified model). The results for the complete
model are shown in Fig. \ref{fig:eccentricityvariation}(b). In
this case, the eccentricity variation reaches high values in the domain
of low eccentricities - between $0.01$and $0.125$ - with a maximum
for $<e>\sim0.05$. This maximum is certainly a result of the overlapping
of the resonances in low eccentricities, forcing the actions \textit{along}
the resonance.
In this model for mean eccentricities between 0.125 and 0.20 the variations
are of the same order. The distributions observed in the Fig. \ref{fig:eccentricityvariation}(b)
is in agreement with Nesvorn\'{y}'s unpublished
data for 45 numbered asteroids of the $(5,-2,-2)$ resonance (see
the Table 2 in Nesvorn\'{y} and Morbidelli 1998). The use of the models with only one perturbing resonance does not allow to get the distribution of the eccentricity variation observed in Fig. \ref{fig:eccentricityvariation}(b).
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{fig4.eps}
\caption{Diffusion Coefficient for actions across
and along, semi-major axis and eccentricity in experiments, obtained
for complete model, for times from $10^{2}$ up to $10^{8}$ years.
Each point is one experiment with initial conditions upon the unperturbed
separatrix of the guiding resonance.}
\label{fig:orbitalelements}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{fig5.eps}
\vspace{2.5mm}
\caption{Variation of the eccentricity versus
mean eccentricity for (a) simplified model and (b) complete model
on a net of points in the plane $(a,e)$.}
\label{fig:eccentricityvariation}
\end{figure*}
\subsection{The stochastic domain in the plane (\textit{a},\textit{e}). Dependence
on the initial conditions}
The diffusion coefficients were calculated on a large set of initial
conditions to assess the domain where the solutions present stochastic
behavior. The analysis was done using simulations over $t_{int}\sim10^{8}$
years, on the points of a grid of initial conditions in plane $(a,e)$.
A Hadjidemetriou-type sympletic mapping was used instead of expensive
numerical integration to allow a large number of simulations. Figure
\ref{fig:contourn} shows the contour plots of the diffusion coefficients
of $p_{4}$. It shows the stochastic domain of the guiding resonance
(the light gray areas in Fig. \ref{fig:contourn}). Note that the
stochastic domain follows the geometry of the unperturbed separatrix
of Fig. \ref{fig:figsep}. Also note that the results for the complete
model show a stochastic domain $(a,e)$ larger than that observed for
the simplified model.
These differences are easily understood if we note the overlapping
of the three resonances in the considered range of eccentricities.
Figure \ref{fig:figsep} shows that the separatrices of the layer and
driving resonances are, for almost all eccentricities, interior to
the domain of the guiding resonance. At low eccentricities, however,
the separatrices cross one another. Thus, in low eccentricities, one
solution crossing the chaotic neighborhood of the separatrix of the
guiding resonance, also cross the separatrices of the layer and driving
resonances. The driving resonance acts pushing the actions \textit{along}
the guiding resonance. The magnitude of the push is determined by
the phase $\varphi_{\mathbf{m}_{D}}$ and amplitude $\beta_{\mathbf{m}_{D}}$.
At variance with the complete model, the simplified model presents
very low diffusion, in low eccentricities, as seen in Fig. \ref{fig:resonumber}.
In this case, the absence of the driving resonance (only the guiding
and layer are considered in the simplified model) implies in the
absence of evolution along the guiding resonance.
A remarkable feature in both results is the formation of a wide region,
in the central part of the domain of the guiding resonance, where
the diffusion is negligible. The motion appears regular for initial
conditions inside that region even when considering very long time
spans. This result confirms what Nesvorn\'{y} and Morbidelli (1999) observed
in surface of sections for eccentricity $0.20$ using this same analytic
model reduced to two degrees of freedom and two resonances. This is
different from the situation observed in low eccentricities, where
the separatrices of the resonances overlap.
\begin{figure}
\centering
\includegraphics[width=1.0\textwidth]{fig6.eps}
\caption{Diffusion coefficients for action \textit{along}
$p_{4}$ for initial conditions in the interval $3.17<a_{0}<3.18$
U.A. for semi-major axis and $0.01<e_{0}<0.30$ for eccentricity for
both simplified and complete models. The results were obtained for
total integration time equal to $10^{8}$ years.}
\label{fig:contourn}
\end{figure}
\section{Asymptotic behavior}
Chirikov theory of slow diffusion was constructed to study the diffusion
under the action of an arbitrarily weak perturbations, and the diffusion
coefficient was computed there using the asymptotic estimate of Melnikov's
Integral. The asymptotic behavior of the three-body resonance model
of Nesvorn\'{y} and Morbidelli was studied using the same technique devised
by Chirikov. Figure \ref{fig:asymptoticbehavior} shows the variation
of the diffusion coefficients \textit{along} the resonance, for two
different initial eccentricities (0.05 and 0.2), as functions of the
parameter $\lambda_{\mathbf{m}_{D}}=\omega_{\mathbf{m}_{D}}/\Omega_{G}$
appearing as argument of the Melnikov integrals in Sect. 2.3 in
the case the driving resonance $\mathbf{m}=\left(1,0,0,-1\right)$.
Small values of $\lambda_{\mathbf{m}_{D}}$ are obtained decreasing
the intensity of the guiding and perturbing resonances.
The Hadjidemetriou-like mapping was used to allow us to compute the solutions over $10^{10}$ years for a great deal of different conditions. The diffusion coefficient was calculated for initial conditions over
the separatrix of the guiding resonance. A background value $D_{b}$,
to be used as reference, was also obtained with initial conditions
in the central part of the guiding resonance, far from the separatrices.
For small values of the perturbation, the motion in the central part
of the resonance domain is regular and the background diffusion appear
as smaller than the diffusion shown by solution starting on the separatrices.
For high values of the perturbation intensity, the motion is chaotic
over the whole domain and the diffusion coefficients in the central
part are not different form those of solutions starting on the separatrices.
Figure \ref{fig:asymptoticbehavior} shows the diffusion coefficient
$D_{3}$ for $e=0.2$ and for the low eccentricity case $e=0.05$.
The figures for the coefficient $D_{4}$ are not shown since they
are almost identical to those shown for $D_{3}$. Figure \ref{fig:asymptoticbehavior}
shows the 3 different possibilities.
\begin{enumerate}
\item The first section of the figures, corresponding roughly to $\lambda_{\mathbf{m}_{D}}\lesssim2$,
is characterized by complete chaos. For the smallest $\lambda_{\mathbf{m}_{D}}$,
one sees the same phenomenon discussed in Sect. \ref{sub:role}:
the background diffusion appear small for some shorter runs because
they do not cover the time necessary to allow the solution to fill
the chaotic layer; but when time span grows, the diffusion values
increase as expected. In this section, in general the diffusion coefficients
for solutions starting in the central part or on the separatrices
are equal showing that the whole resonance domain is chaotic. A few
exceptions appear, as shown in Fig. \ref{fig:asymptoticbehavior}(b).
In addition we may see in this figure, for $\lambda_{\mathbf{m}_{D}}\sim1$,
a sudden decrease of the background diffusion indicating that the
corresponding solution stuck to some regularity island during its
evolution. However this sticking is not permanent and the background
diffusion grows when longer time spans are considered. The background
values are shown in Fig. \ref{fig:asymptoticbehavior}(a) only for
the time span $10^{10}$ years to allow a better comparison of the
numerical results with the dashed lines representing results from
Chirikov's model,
\item For $\lambda_{\mathbf{m}_{D}}\sim2$, the background diffusion shows
a discontinuity which, for the longest runs, reaches up to 14 orders
of magnitude. This means that the center of the resonance domain becomes
regular and the stochasticity remains confined to layers around the
separatrix. This is the domain where Chirikov's slow diffusion theories
are valid and where the results may be compared to the theoretical
results obtained in Sect. \ref{sub:thediffusionrate}. The
integration time is a crucial factor in the detection of the slow
diffusion. For instance, one may see that for simulations over only $10^{5}$ years, the diffusion near separatrix is equal to the background diffusion for values of $\lambda_{\mathbf{m}_{D}}$ close to 1, while for simulations over $10^{10}$ years, the equality is reached only for $\lambda_{\mathbf{m}_{D}}=9$.
\item In the last section of the Figs. \ref{fig:asymptoticbehavior} the
solutions starting close to the separatrices show a diffusion equal
to the background diffusion. The interpretations is that the stochastic
layer is this case is so thin that the used initial conditions are
no longer within them. (For that sake, the locus of the separatrices
should be computed with very large precision. See e.g. Froeschl\'{e} et
al. 2006). One striking feature in this section is that an increase
in the time span by a factor 10 means a decrease in the background
diffusion by a factor $10^{3}$. This is a clue for the fact that
the solutions are dominated by periodic terms. Indeed, if we consider
one periodic term with amplitude proportional to $\epsilon$ and frequency
$\omega$, its contribution to the average momentum in an interval
$\left[a,b\right]$ is proportional to \[
\frac{1}{\Delta t}\intop_{a}^{b}\epsilon\mbox{cos}\omega tdt\]
where $\Delta t=b-a$. This integral is elementary and the integration
of the result over all frequencies below a upper limit $\omega_{\text{lim}}$,
gives \[
\frac{\epsilon}{\Delta t}\left[\mbox{si}\left(b\omega_{\text{lim}}\right)-\mbox{si}\left(a\omega_{\text{lim}}\right)\right]\]
where $\mbox{si}$ is the sine-integral function. The diffusion coefficients
are given by the square of the average variation of the momentum divided
by the total time (see Eqn. \ref{Diffusion_tensor}) and then $D\sim\Delta t^{-3}$.
We also have $D\varpropto\epsilon^{2}\sim\Omega_{G}^{-2}\sim\lambda_{D}^{-4}$.
The inclination -4 of the straight lines in the log-log plots can
be easily checked.
\end{enumerate}
The diffusion of the solutions in the neighborhood of the separatrix
may be determined from Eqn. \ref{Chirikov_diffusion_tensor}. This
equation involves the intensity of the perturbation (related to $\lambda_{\mathbf{m}_{D}}$)
and two unknown parameters: the factor of reduction $R_{\mathbf{m}_{D}}$
and the factor of odd perturbations $R_{T}$. The factor of reduction
corresponds to Chirikov's hypothesis of reduced stochasticity (due
to holes, the solution does not fill the strip around the separatrix);
the other factor comes from the fact that the perturbation is not
even and thus the values of the diffusion coefficient are not the
same for solutions in both separatrices (the solution may remain circulating
near one of the separatrices at time different of the time it remain
near the other).
The results obtained with Chirikov are shown in Figs. \ref{fig:asymptoticbehavior}
by dashed lines. In Figs. \ref{fig:asymptoticbehavior}(a) tree
different solutions are shown (calculated with the reduction factors
indicated in the figure). The better agreement is obtained with $R_{\mathbf{m}_{D}}=0.25$.
The two values used for $R_{T}$ (0.6 and 0.9) give almost the same
result, showing that the motion near deviation for the weakest perturbation
(larger $\lambda_{\mathbf{m}_{D}}$). In the other two figures, only
the two solutions with $R_{\mathbf{m}_{D}}=0.25$ are shown.
\begin{figure*}
\centering
\includegraphics[width=1.0\textwidth]{fig7.eps}
\caption{Asymptotic behavior of the diffusion coefficient $D_{3}$ for initial conditions over the separatrix and on the central part of the guiding resonance for (a) $e=0.05$ and (b) $e=0.20$. The dashed lines show the behavior predicted with Chirikov's theory.}
\label{fig:asymptoticbehavior}
\end{figure*}
\section{Conclusion}
Chirikov's theories provide heuristic tools to understand the diffusion
observed in both eccentricity and semi-major axis of asteroids inside
the $\left(5,-2,-2\right)$ resonance. The multi-dimensional Hamiltonians of the
three-body (three orbit) mean-motion resonances may be studied with the
theories developed by Chirikov and collaborators,
mainly because of the particular geometry of those resonances in the
plane $\left(a,e\right)$. The results obtained
in this paper for the $\left(5,-2,-2\right)$ three-body mean-motion resonance
confirms the role of the resonances in the raising of diffusion \textit{across}
and \textit{along} the main resonance as foreseen in Chirikov's theories.
The diffusion calculations presented in this paper show that diffusion
in semi-major axis is related with the diffusion in the \textit{across}
actions $(p_{1},p_{2})$ while the diffusion in eccentricity is related
with the diffusion in the \textit{along} actions $(p_{3},p_{4})$.
The diffusion coefficient for the semi-major axis tends to small values
showing that the variation of the semi-major axis remains small. It
indicates the existence of barriers on both sides of the
stochastic layer limiting the motion \textit{across} the resonance.
The comparison between simplified and complete model results shown that the diffusion
in eccentricity is presumably due to the presence of at least one resonance driving
the motion along the guiding resonance. This behavior is similar to the expected behavior
of the Arnold diffusion, but, differently of it, the diffusion here is well apparent and
the diffusion coefficients remain high. For this reason it was sometimes
called Fast Arnold Diffusion (Chirikov and Vecheslavov 1989, 1993).
The structure of the $\left(5,-2,-2\right)$ resonance is formed by several overlapping
resonances, particularly at low eccentricities. Thus, diffusion across
$\left(5,-2,-2\right)$ resonance may be no longer limited to the thin chaotic
layers (stochastic layers), but it fills the whole resonance zone.
The diffusion \textit{along} the resonance could be due to a multiplet.
In this scenario, we have a random motion \textit{across} the resonance,
due to the overlap of several resonances belonging to a multiplet,
and another, likely due to weaker resonances, which drive the diffusion
\textit{along} the guiding resonance. Arnold diffusion
might occur inside the stochastic layer formed around the separatrix
of the guiding resonance under the action of sufficiently weak perturbations.
At variance, thick layer diffusion can appear for perturbation parameters
in a broad interval. Although this mechanism show a similar exponential
dependence of diffusion rate as a function of some system parameters,
the mean rate of thick layer diffusion is generally larger than any
theoretical estimation of Arnold diffusion. Therefore, it seems that for the
real problem would be more appropriate to call the diffusion with another name -
asymptotic diffusion of Chirikov-Arnold - due to the fact that this keeps some
features of the Arnold diffusion, but late very well characterized by
Chirikov like some distinct.
As we mentioned before, we believe that the connection between rigorous
investigations concerning strictly Arnold diffusion and that observed
in real physical systems like this, is still an open subject.
As Lochak (1999) pointed out, the global instability properties of near--integrable
Hamiltonian systems are far from well--understood. It could almost be said
that little progress has been made after pioneering work Arnold,
and new ideas are definitely called for.
Finally, the good results showed that the Chirikov slow diffusion theory can be used
in broader investigations considering more resonances for $\left(5,-2,-2\right)$, as well applied for the
others three-body (three orbit) mean motion resonances and also can include the inclination
of the asteroid orbit.
|
{
"timestamp": "2010-09-21T02:00:55",
"yymm": "1009",
"arxiv_id": "1009.3558",
"language": "en",
"url": "https://arxiv.org/abs/1009.3558"
}
|
\section{Background}
A classical result due to Furstenberg, Katznelson, and Weiss (\cite{FKW90}; see also \cite{B86}) says that if $E \subset {\mathbb R}^2$ has positive upper Lebesgue density, then for any $\delta>0$, the $\delta$-neighborhood of $E$ contains a congruent copy of a sufficiently large dilate of every three point configuration. An example due to Bourgain shows that if the three point configuration in question is an arithmetic progression, then taking a $\delta$-neighborhood is in fact necessary and the result is not otherwise true. However, it seems reasonable to conjecture that if the three point configuration is non-degenerate in the sense that the three points do not lie on the same line, then a set of positive density contains a sufficiently large dilate of this configuration.
When the size of the point set is smaller than the dimension of the ambient Euclidean space, taking a $\delta$-neighborhood is not necessary, as shown by Bourgain (\cite{B86}). He proves that if $E \subset \mathbb{R}^d$ has positive upper density and $\Delta_k$ is a $k$-simplex (a set of $k+1$ points which spans a $k$-dimensional subspace) with $k<d$, then $E$ contains a rotated and translated image of every large dilate of $\Delta_k$. The cases $k=d$ and $k=d+1$ remain open. See also, for example, \cite{Berg96}, \cite{F81}, \cite{K07}, \cite{TV06}, and \cite{Z99} on related problems and their connections with discrete analogs.
In the geometry of the integer lattice $\mathbb{Z}^d$, related problems have been recently investigated by Akos Magyar in \cite{M06} and \cite{M07}. In particular, he proves (\cite{M07}) that if $d>2k+4$ and $E \subset \mathbb{Z}^d$ has positive upper density, then all large (depending on the density of $E$) dilates of a $k$-simplex in $\mathbb{Z}^d$ can be embedded in $E$. Once again, serious difficulties arise when the size of the simplex is sufficiently large with respect to the ambient dimension.
We aim to investigate an analog of this question in finite field geometries. A step in this direction was taken (\cite{HI07}) by the second and third listed authors. They prove that if $E \subset \mathbb{F}_q^d$, the $d$-dimensional vector space over the finite field with $q$ elements, has $|E| \gtrsim q^{d\frac{k}{k+1}+\frac{k}{2}}$ and $\Delta$ is a $k$-simplex determined by (with vertices lying in) $E$, then there exists $\tau \in \mathbb{F}_q^d$ and $O \in O_d(\mathbb{F}_q)$ such that $\tau+O(\Delta) \subset E$. The result is only non-trivial in the range $d \geq {k + 1 \choose 2}$ as larger simplices are out of range of the methods used.
Le Anh Vinh has also investigated $k$-point configurations in $\mathbb{F}_q^d$. He showed in \cite{Vinh2} that if $E \subset \mathbb{F}_q^d$, $|E| \gtrsim q^{\frac{d-1}{2} + k}$, and $d \geq 2k$ then $E$ contains an isometric copy of every $k$-simplex. Also, he showed (\cite{Vinh3}) that if an arbitrary set $E \subset \mathbb{F}_q^d$ has size $E \gtrsim q^{\frac{d+2}{2}}$ (for $d \geq 3$), then it determines a positive proportion of all triangles. Based on an earlier draft of this paper, Vinh proved (\cite{Vinh1}) the $2$-dimensional version of our main theorem (see Theorem \ref{main} below) using graph-theoretic methods. Namely, if $E \subset \mathbb{F}_q^2$ has size $|E| \geq \rho q^2$ for some $ q^{- \frac{1}{2}} \ll \rho \leq 1$, then the set of triangles determined by $E$ has size $\geq c \rho q^3$.
The purpose of this paper is to address the case of $d$-simplices. As before we let $\Delta_k$ denote a $k$-simplex, i.e.~a set of $k+1$ points which span a $k$-dimensional subspace. Given $E \subset {\mathbb F}_q^d$, let the set of $k$-simplices determined by $E$ up to congruence be denoted by
\[
T_{k}(E)=\{\Delta_k \in E^{k+1}\} \ / \sim
\]
where two $k$-simplices are equivalent if one is a rotated, shifted, reflected copy of the other.
Note that $T_k(E)$ is a natural subset of $\mathbb{F}_q^{k +1 \choose 2}$ (see Lemma \ref{linalglemma} below). Our main result is the following.
\begin{theorem} \label{main}
Let $E \subset \mathbb{F}_q^d$ with $|E| \geq \rho q^d$ for $q^{-1/2} \ll \rho \leq 1$. Then, there exists $c > 0$ so that
\[
|T_{d}(E)| \geq c \rho^{d-1} q^{d+1 \choose 2}.
\]
\end{theorem}
\begin{remark} The viable range for $\rho$ in Theorem \ref{main} is $q^{- (d-\alpha)} \ll \rho \leq 1$, where $\alpha$ is threshold so that
\[
\sum_{x,x^1, \dots , x^d}E(x)E(x^1) \dots E(x^d)S(x-x^1)\dots S(x - x^d) = \frac{|E|^{d+1}}{q^d}(1 + o(1)),
\]
whenever $|E| \gg q^{\alpha}$. Theorem \ref{dhinges} gives $\alpha = q^{d - \frac{1}{2}}$, although it is reasonable to expect $\alpha = q^\frac{d+1}{2}$
\end{remark}
\begin{remark}
We deal only with finite fields $\mathbb{F}_q$ with characteristic $p > 2$. We also assume $q$ is much larger than the dimension $d$. Also, note that the error terms appearing in Theorems \ref{sphere} and \ref{gen} are always of lower order in the effective range of Theorem \ref{main} for $d \geq 2$.
\end{remark}
\begin{remark}
The assumption that $|E| \geq \rho q^d$ implies that the number of $(d+1)$-point configurations determined by $E$ (up to congruence) is at least
\[
\frac{|E|^{d+1}}{\rho q^d \cdot q^{d \choose 2}} \geq \rho^d q^{d+1 \choose 2},
\]
since the size of the subset of the translation group that maps points in $E$ to a set of size $|E|$ is no larger than $ \rho q^d$ and the rotation group is of size $\approx q^{d \choose 2}$. Our result shaves off a power of $\rho$ from this trivial estimate.
\end{remark}
\section{Proof of the main result (Theorem \ref{main})}
Here, we roughly state the argument. We prove Theorem \ref{main} by first making a reduction to a statistical statement about hinges (defined below). Having made this reduction, we next show, using a pigeon-holing argument that for some $x \in E$, the hinge is large. To finish the argument, we realize a dichotomy. If the number of transformations mapping the hinge to itself is small, then a purely probabilistic argument gives the number of distinct (incongruent) $(d+1)$-point configurations is what we claim. If the number of transformations mapping the hinge to itself is large, then a purely combinatorial argument gives the result.
We start with the statistical reduction. We observe that if $|E| \geq \rho q^d$, for $\rho$ as above, then it suffices to show that this implies that
\begin{equation}
\left|\left\{ \left(a_{i,j} \right)_{1 \leq i < j \leq d+1} \in \mathbb{F}_q^{d+1 \choose 2} : |R_a(E) | > 0\right\}\right| \geq c \rho^{d-1} q^{d +1 \choose 2},
\end{equation}
where
\[
R_a(E) = \{(y^1, \dots , y^{d+1}) \in E\times \dots \times E : \| y^i - y^j \| = a_{i,j}\},
\]
and
\[
\| x \| = \sum_{j=1}^d x_j^2.
\]
This follows immediately from the following simple linear algebra lemma. The proof of this lemma will appear in Section \ref{linalgproof} for completeness.
\begin{lemma} \label{linalglemma}
Let $V$ be a simplex with vertices $V_i \in \mathbb{F}_q^d$, where $i = 0, \dots , k$. Let $W$ be another simplex with vertices $W_i \in \mathbb{F}_q^d$ for $i = 0 , \dots , k$. Suppose that
\begin{equation}
\| V_ i - V_j \| = \| W_i - W_j\|
\end{equation}
for all $i , j$. Then $ V \sim W$ in the sense of $T_k(E)$.
\end{lemma}
Our main estimate is the following:
\begin{theorem} \label{dhinges}
Suppose that $\alpha_i \in \mathbb{F}_q \backslash \{0\}$ for $i = 1, \dots, d$, and let $E \subset \mathbb{F}_q^d$. Then,
\[
|\{(x,x^1, \dots , x^d) \in E \times \dots \times E : \| x - x^i \| = \alpha_i \}| = \frac{|E|^{d+1}}{q^d}(1 +o(1))
\]
whenever $|E| \gg q^{d - \frac{1}{2}}$.
\end{theorem}
This implies that there exists $x \in E$ so that
\begin{equation} \label{d-star}
|\{(x^1, \dots , x^d)\in E \times \dots \times E : \| x - x^i \| = \alpha_i \}| \geq \frac{|E|^d}{q^d}(1 + o(1)).
\end{equation}
Fix a $d$-tuple $\alpha = (\alpha_i)_{i = 1}^d$, with $\alpha_i \in \mathbb{F}_q \backslash \{ 0 \}$, for $i = 1, \dots , d$. Define a {\it hinge} $h_{x,\alpha}$ to be the set $\{(x^1, \dots , x^d) \in E \times \dots \times E : \| x - x^i \| = \alpha_i\}$.
Let $M_{x, \alpha} \subset O_d(\mathbb{F}_q)$ denote the set of orthogonal matrices which maps the hinge $h_{x,\alpha}$ to itself. We next turn our attention to the following dichotomy:
Suppose that $|M_{x, \alpha}| \leq \rho q^{d \choose 2}$. By \eqref{d-star}, the number of distinct $d$-point configurations between the $d$ sets $\{x^i \in E : \| x - x^i \| = \alpha_i\}$ is at least
\begin{equation} \label{smallhinge}
\frac{|h_{x, \alpha}|}{|M_{x, \alpha}|}\geq \frac{|E|^d q^{-d} (1 + o(1))}{\rho q^{d \choose 2}}\geq c \rho^{d-1} q^{d \choose 2}.
\end{equation}
We are left only to deal with the case when $|M_{x, \alpha}| > \rho q^{d \choose 2}$. We put $A_i = \{x^i \in E : \| x - x^i \| = \alpha_i\}.$ It is worthwhile to point out the possibility that $A_i = A_j$. Also, although the sets $A_i$ are not themselves spheres, they are subsets of spheres and therefore inherit some of their intersection properties. When dealing with the case $|M_{x, \alpha}| > \rho q^{d \choose 2}$ we are faced with two possibilities. First, suppose that for some $i \in \{ 1, \dots , d \}$ we have that $|A_i| \leq \rho q^{d-1}$. In this case we utilize the orbit-stabilizer theorem from elementary group theory:
\begin{proposition} \label{ost} $($\cite{Lang}$)$
Let a group $G$ act on a set $S$. Let $Gs = \{gs : g \in G\}$ be the orbit of $s \in S$, and $G_s = \{g : gs = s\}$ the isotropy group of $s \in S$. Then there is a bijection between $Gs$ and $G/G_s$. Consequently,
\[
|Gs| = (G : G_s) = |G| / |G_s|.
\]
\end{proposition}
We let the group $O_d(\mathbb{F}_q)$ act on $\mathbb{F}_q^d$. Recalling that $|O_d(\mathbb{F}_q)| \approx q ^{d \choose 2}$, and since orthogonal maps preserve the length of a certain vector, we get that the size of the orbit of any point is exactly $q^{d-1}$. Hence, picking some $z$ from the previously mentioned set $A_i$, we get that the size of the stabilizer group of this element $z$ is
\[
|G_z| = \frac{|G|}{|Gz|} \approx \frac{q^{d \choose 2}}{q^{d-1}}.
\]
The final element here is to notice that
\[
|M_{x, \alpha}| \leq |G_z| |A_i| \leq \frac{q^{d \choose 2}}{q^{d-1}} \cdot \rho q^{d-1} = \rho q^{d \choose 2},
\]
since the number of hinge-preserving orthogonal matrices is no more than the number of orthogonal transformations which fix a given vector $z \in A_i$, times the number of choices for that vector $z$, which is a contradiction. We may therefore assume $|A_i| > \rho q^{d-1}$ for all $i = 1, \dots , d$. Recall that we are working with the hinge $h_{x, \alpha} = \{(x^1, \dots , x^d) \in E \times \dots \times E : \| x - x^i \| = \alpha_i\}$, and we aim to show that the number of incongruent $d$-point configurations is bounded below by $c \rho^{d-1} q^{d \choose 2}$.
We start by picking a point $a_1 \in A_1$. We want to know how many distinct distances occur between $a_1$ and points in the set $A_2$. To achieve this, we count how often a given distance may occur between $a_1$ and the points on $A_2$. This amounts to intersecting $E$ with two spheres: one sphere of a given radius, centered at $a_1$, and the set $A_2$, which is, itself, a sphere intersected with $E$. The intersection must contain fewer than $q^{d-2}$ possible points on the set $A_2$ which are a given distance from $a_1$. Since $|A_2| > \rho q^{d-1}$, there must be at least $\rho q^{d-1} / q^{d-2} = \rho q$ different distances between $a_1$ and points on $A_2$, by pigeonholing.
For each of the $\rho q$ choices of $a_2$ which are different distances from $a_1$, we need to find the number of 3-point configurations that $a_1$ and $a_2$ can make with points on $A_3$. Now we are intersecting $E$ with spheres of two (possibly the same) radii about $a_1$ and $a_2$ with the sphere containing $S_3$. There can be no more than $q^{d-3}$ points in this intersection, which would each correspond to the same 3-point configuration. So there must be $\rho q^{d-1} / q^{d-3} = \rho q^2$ distinct 3-point configurations for each of the $\rho q$ different pairs we found before, which gives us a total of $\rho q \cdot \rho q^2 = \rho^2 q^3$ different 3-point configurations. Repeating this process, we see that we will pick up $\rho q^p$ different $(p-1)$-point configurations at each step. If we multiply all of these together, we will get a grand total of
\begin{equation} \label{largehinge}
\rho q \cdot \rho q^2 \cdot \dots \cdot \rho q^{d-1} = \rho^{d-1} q^{d \choose 2}
\end{equation} distinct $d$-point configurations.
From \eqref{smallhinge} and \eqref{largehinge}, we see that in any case, there exist no less than $c \rho^{d-1}q^{d \choose 2}$ many distinct $d$-point configurations. Since this holds for any fixed vector $\alpha = (\alpha_i)_{i = 1}^d$, and since there are $q - 1$ choices for each $\alpha_i \in \mathbb{F}_q \backslash \{ 0 \}$, then there are at least
\[
c \rho^{d-1} q^{d \choose 2} (q -1)^d \geq c \rho^{d-1} q^{d + 1 \choose 2}
\]
many distinct $(d+1)$-point configurations determined by $E$.
\subsection{Fourier analysis}
\vskip.125in
The Fourier transform of a function $f : \mathbb{F}_q^d \to \mathbb{C}$ is given by
\[
\widehat{f}(m) = q^{-d} \sum_{x \in \mathbb{F}_q^d} f(x) \chi(- x \cdot m)
\]
where $\chi$ is a non-trivial additive character on $\mathbb{F}_q$. By orthogonality,
\[
\sum_{x \in \mathbb{F}_q^d} \chi(- x \cdot m) =
\left\{
\begin{array}{lcc}
q^d & & m = (0,\dots,0) \\
0 & & m \neq (0,\dots , 0)
\end{array}
\right.
\]
\begin{lemma} \label{fourierproperties}
Let $f,g : \mathbb{F}_q^d \to \mathbb{C}$. Then,
\[
\widehat{f}(0,\dots,0) = q^{-d} \sum_{x \in \mathbb{F}_q^d} f(x),
\]
\[
q^{-d} \sum_{x \in \mathbb{F}_q^d} f(x) \overline{g(x)} = \sum_{m \in \mathbb{F}_q^d} \widehat{f}(m) \overline{\widehat{g}(m)},
\]
\[
f(x) = \sum_{m \in \mathbb{F}_q^d} \widehat{f}(m) \chi(x \cdot m).
\]
\end{lemma}
\section{Proof of Theorem \ref{dhinges}}
In order to prove Theorem \ref{dhinges} we will actually prove the more general following theorem. \begin{theorem} \label{gen}
Let $r>2$ be an integer, and let $H_{r,\alpha}$ represent the set of $r-$hinges, with distances $\alpha= \{\alpha_i\}_{i=1}^{r-1}$, which are present in $E$. That is,
\[
H_{r,\alpha} = \{(x,x^1, \dots x^{r-1}) \in E \times \dots \times E : \| x - x^i \| = \alpha_i\},
\]
where $\alpha_i \neq 0$ for $i = 1 , \dots , r-1$. Then,
\[
|H_{r,\alpha}| = \frac{|E|^{r}}{q^{r-1}}(1 + o(1)),\] whenever
$|E|\gg q^{\frac{2r-5}{2r-4}d+\frac{1}{2r-4}}$
\end{theorem}
Setting $r=d+1$ in Theorem \ref{gen} gives Theorem \ref{dhinges}.
We will need the following estimates whose proof we delay to the end of the paper.
\begin{theorem} \label{sphere}
Let $S_t = \{x \in \mathbb{F}_q^d : \| x \| = t\}$. Identify $S_t$ with its characteristic function. For $t \neq 0$,
\begin{equation} \label{sizeofsphere}
|S_t| = q^{d-1}(1 + o(1))
\end{equation}
and if also $m \neq (0, \dots , 0)$,
\begin{equation} \label{decaysphere}
| \widehat{S}_t(m) | \lesssim q^{- \frac{d+1}{2}}.
\end{equation}
\end{theorem}
We will proceed by induction. Before we handle the case $r=3$ we first observe the following estimate which originally appeared in \cite{IR}.
\begin{lemma} \label{2hinge}
Using the notation as above, we have $|H_{2,\alpha}| = \frac{|E|^2}{q} + O(q^{\frac{d-1}{2}}|E|)$.
\end{lemma}
To see this, write
\begin{align*}
|H_{2,\alpha}| &= \sum_{x,y} E(x) E(y) S(x - y)
\\
&= q^{2d} \sum_m \left| \widehat{E}(m) \right|^2 \widehat{S}(m)
\\
&= q^{-d} |E|^2 |S| + q^{2d} \sum_{m \neq 0} \left| \widehat{E}(m) \right|^2 \widehat{S}(m)
\end{align*}
and
\begin{align*}
q^{2d} \left| \sum_{m \neq 0} \left| \widehat{E}(m) \right|^2 \widehat{S}(m)\right| &\leq 2 q^{2d} q^{- \frac{d+1}{2}} q^{-d} |E|
\\
&= 2q^{\frac{d-1}{2}}|E|.
\end{align*}
We now illustrate the base step. First we write
\[
|H_{3,\alpha}|=\sum_{x\in E} |E \cap (x - S)| ^2.
\]
Now,
\[
|E \cap (x - S)|=\sum_y E(y)S(x-y)=q^d\sum_m \widehat{E}(m)\widehat{S}(m)\chi(m\cdot x)
\]
\[
=|E||S|q^{-d}+q^d\sum_{m \neq 0} \widehat{E}(m)\widehat{S}(m)\chi(m\cdot x),
\]
which gives
\begin{align*}
|H_{3,\alpha}| &= \sum_{x\in E} |E \cap (x - S)| ^2
\\
&= |E|^3|S|^2q^{-2d}+2|E||S|q^d\sum_{m\neq 0} |\widehat{E}(m)|^2|\widehat{S}(m)|
+q^{2d}\sum_{x}\left|\sum_{m\neq 0} \widehat{E}(m)\widehat{S}(m)\chi(m\cdot x)\right|^2
\\
&=|E|^3|S|^2q^{-2d}+O\left(|E|^2|S|q^{-d}q^{(d-1)/2}+q^{3d}\sum_{m \neq 0} |\widehat{E}(m)|^2|\widehat{S}(m)|^2\right)
\\
&= |E|^3|S|^2q^{-2d}+O\left(|E|^2|S|q^{-d}q^{(d-1)/2}+q^{d-1}|E|\right).
\end{align*}
If $|E|\gg q^{\frac{d+1}{2}}$ then
\[
|H_{3,\alpha}|= |E|^3 q^{-2}(1+o(1)),
\]
For the inductive step, assume that we are in the case $|H_{r,\alpha}| = \frac{|E|^r}{q^{r-1}}(1 + o(1))$ for $|E|\gg q^{\frac{2r-5}{2r-4}d+\frac{1}{2r-4}}.$
We begin by writing
\begin{align*}
|H_{r+1,\alpha}| &= \sum_{x, x^1 , \dots, x^r} H_{r,\alpha}(x , x^1 , \dots , x^{r-1}) E(x^r) S(x - x^r)
\\
&= q^{(r+1)d} \sum_{m} \widehat{H}_{r,\alpha}(m, 0 , \dots , 0) \widehat{S}(m) \widehat{E}(m)
\\
&= q^{-d} |E| |S| |H_{r,\alpha}| + q^{(r+1)d} \sum_{m \neq 0} \widehat{H}_{r,\alpha}(m, 0 , \dots , 0) \widehat{S}(m) \widehat{E}(m)
\\
&= q^{-d} |E| |S| |H_{r,\alpha}| + R.
\end{align*}
Applying Cauchy-Schwarz gives
\begin{align*}
R^2 &\leq q^{2d(r+1)} \sum_{m \neq 0} |\widehat{S}(m)|^2 |\widehat{E}(m)|^2 \sum_{m \neq 0} |\widehat{H}_{r,\alpha}(m, 0 , \dots, 0)|^2
\\
&\lesssim q^{2d(r+1)}q^{-d-1}q^{-d}|E| \sum_{m \neq 0} |\widehat{H}_{r,\alpha}(m, 0 , \dots , 0)|^2
\\
&\leq q^{2dr - 1}|E| \sum_{m} |\widehat{H}_{r,\alpha}(m, 0, \dots , 0)|^2
\end{align*}
Also, we have that
\begin{align*}
\widehat{H}_{r,\alpha}(m, 0 , \dots , 0) &= q^{-rd} \sum_{x, x^1, \dots , x^{r-1}} \chi(x \cdot m) E(x) E(x^1) \dots E(x^{r-1}) S(x - x^1) \dots S(x - x^{r-1})
\\
&= q^{-rd + d} \widehat{f}(m)
\end{align*}
where
\[
f(x) = E(x) \sum E(x^1) \dots , E(x^{r-1}) S(x - x^1) \dots S(x - x^{r-1}) = E(x)|E \cap (x - S)|^{r-1}.
\]
Since $|E \cap (x - S)| \leq q^{d-1}$, it follows that
\begin{align*}
A = \sum_{m} |\widehat{H}_{r,\alpha}(m, 0 , \dots , 0 ) |^2 &= q^{-2rd + 2d} \sum_{m} |\widehat{f}(m)|^2
\\
&= q^{-2rd + d} \sum_{x} |f(x)|^2
\\
&\leq q^{-2rd + d} \left( q^{d-1}\right)^{2(r-2)} |H_{3,\alpha}|
\end{align*}
and
\[
A \lesssim q^{-2rd + d} \left( q^{d-1}\right)^{2(r-2)} |E|^3 q^{-2}(1+o(1)).
\]
Finally,
\[
R^2 \lesssim q^{ - 3}q^{ d} \left( q^{d-1}\right)^{2(r-2)} |E|^4 (1+o(1))\leq q^{(2r-3)d-2r+1}|E|^4 (1+o(1)).
\]
Therefore,
\[
|H_{r+1,\alpha}| = q^{-d} |E| |S| |H_{r,\alpha}| + O(q^{d \frac{2r-3}{2} - r + \frac{1}{2}}|E|^2),
\]
and we have that
\[
|H_{r+1,\alpha}| = \frac{|E|^{r+1}}{q^r}(1 + o(1)),
\]
whenever
\[
|E|\gg q^{\frac{2r-3}{2r-2}d+\frac{1}{2r-2}}.
\]
\section{Proof of Lemma \ref{linalglemma}} \label{linalgproof}
Let $\pi_r(x)$ denote the $r$-th coordinate of $x$. By translating, we may assume that $V_0 = \vec{0}$. We may also assume that $V_1 , \dots , V_k$ are contained in $\mathbb{F}_q^k$. The condition that $\| V_i - V_j \| = \| W_i - W_j\|$ for all $ i, j$ implies that
\begin{equation} \label{fixeddistance}
\sum_{r = 1}^k \pi_r(V_i)\pi_r(V_j) = \sum_{r=1}^k \pi_r(W_i) \pi_r (W_j).
\end{equation}
Let $T$ be the transformation uniquely defined by $T(V_i) = W_i$. To show that $T$ is orthogonal it suffices to show that $\| Tx \| = \| x \|$ for all $x$. By assumption, the $V_i$'s form a basis, so we have
\[
x = \sum_{i} t_i V_i.
\]
Thus, by \eqref{fixeddistance}, we have that
\[
\| Tx \| = \sum_{r} \sum_{i , j} t_i t_j \pi_r(W_i) \pi_r(W_j) = \sum_r \sum_{i,j} t_i t_j \pi_r(V_i) \pi_r(V_j) = \| x \|,
\]
giving the result.
\section{Proof of Theorem \ref{sphere}} \label{sphereproof}
For any $l\in{\mathbb F}^d_q$, we have
\begin{equation} \label{sphereparade}
\begin{array}{llllll} \widehat{S}_t(l)&=&
q^{-d} \displaystyle\sum_{x \in {\mathbb F}^d_q} q^{-1} \sum_{j \in {\mathbb F}_q} \chi(
j(\|x\|-t)) \chi( - x \cdot l)\\ \hfill \\&=&q^{-1}\delta(l) +
q^{-d-1} \displaystyle\sum_{j \in {\mathbb F}^{*}_q} \chi(-jt) \sum_{x}
\chi( j\|x\|) \chi(- x \cdot l),\\
\end{array}\end{equation}where the notation $\delta(l)=1$ if $l=(0\ldots,0)$ and $\delta(l)=0$ otherwise.
Now
\[
\widehat{S}_t(l)=q^{-1}\delta(l)+ Q^d q^{-\frac{d+2}{2}} \sum_{j \in {\mathbb F}^{*}_q}
\chi\left(\frac{\|l\|}{4j}+jt\right)\eta^d(-j).
\]
In the last line we have completed the square, changed $j$ to
$-j$, and used $d$ times the Gauss sum equality
\begin{equation} \sum_{c\in {\mathbb F}_q} \chi(jc^2) = \eta(j)\sum_{c\in{\mathbb F}_q}\eta(c)\chi(c)=\eta(j)\sum_{c\in{\mathbb F}_q^*}\eta(c)\chi(c) =Q\sqrt{q}\,\eta(j), \label{gauss}\end{equation} where the constant $Q$ equals $\pm1$ or $\pm i$, depending on $q$, and $\eta$ is the quadratic multiplicative character (or the Legendre symbol) of ${\mathbb F}_q^*$. (see, e.g. \cite{LN97}, for more information).
The conclusion to both parts of Theorem \ref{sphere} now follows from the following classical estimate due to A. Weil (\cite{Weil}).
\begin{theorem} \label{kloosterman} Let
\[
K(a)=\sum_{s \not=0} \chi(as+s^{-1}) \psi(s),
\]
where $\psi$ is a multiplicative character on ${\mathbb F}_q\backslash\{0\}$. Then if $a \neq 0$,
\[
|K(a)| \leq 2 \sqrt{q}.
\]
\end{theorem}
|
{
"timestamp": "2010-09-22T02:00:58",
"yymm": "1009",
"arxiv_id": "1009.3991",
"language": "en",
"url": "https://arxiv.org/abs/1009.3991"
}
|
\section[#1]{#2}}
\def\pr {\noindent {\it Proof.} }
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\textup{2000} Mathematics Subject Classification}
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\begin{document}
\title
[The Gauss image and Bernstein type theorems] {The Gauss image of
entire graphs of higher codimension and Bernstein type theorems}
\author
[J. Jost, Y. L. Xin and Ling Yang]{J. Jost, Y. L. Xin and Ling
Yang}
\address{Max Planck Institute for Mathematics in the
Sciences, Inselstr. 22, 04103 Leipzig, Germany.}
\email{jost@mis.mpg.de}
\address {Institute of Mathematics, Fudan University,
Shanghai 200433, China.} \email{ylxin@fudan.edu.cn}
\address{Max Planck Institute for Mathematics in the
Sciences, Inselstr. 22, 04103 Leipzig, Germany.}
\email{lingyang@mis.mpg.de}
\thanks{The second named author is grateful to the Max Planck
Institute for Mathematics in the Sciences in Leipzig for its
hospitality and continuous support. He is also partially
supported by NSFC and SFMEC }
\begin{abstract}
Under suitable conditions on the range of the Gauss map of a
complete submanifold of Euclidean space with parallel mean
curvature, we construct a strongly subharmonic function and
derive a-priori estimates for the harmonic Gauss map. The
required conditions here are more general than in previous work
and they therefore enable us to improve substantially previous
results for the Lawson-Osseman problem concerning the regularity
of minimal submanifolds in higher codimension and to derive
Bernstein type results.
\end{abstract}
\maketitle
\Section{Introduction}{Introduction}
We consider an oriented $n$-dimensional submanifold $M$ in
$\ir{n+m}$ with $n\ge 3,\; m\ge 2.$ The
Gauss map $\g:M\to \grs{n}{m}$ maps $M$ into a Grassmann
manifold. In fact, for codimension $m=1$, this Grassmann
manifold $\grs{n}{1}$ is the unit sphere $S^n$. In this paper,
however, we are interested in the case $m\ge 2$ where the geometry of
this Grassmann manifold is more complicated. By the theorem of Ruh-Vilms \cite{r-v}, $\g$ is
harmonic if and only if $M$ has parallel mean curvature. This result
applies thus in particular to the case where $M$ is a {\it minimal}
submanifold of Euclidean space.
Now, the Bernstein problem for entire minimal graphs is one of
the central problems in geometric analysis. Let us summarize the
status of this problem, first for the case of codimension 1. The
central result is that an entire minimal graph $M$ of dimension
$n\le 7 $ and codimension 1 has to be planar, but there are
counterexamples to such a Bernstein type theorem in dimension $8$
or higher. However, when the additional condition is imposed that
the slope of the graph be uniformly bounded, then a theorem of
Moser \cite{m}, called a weak Bernstein theorem, asserts that
such an $M$ in arbitrary dimension has to be planar. Thus, the
counterexamples arise from a non-uniform behavior at infinity. In
fact, by a general scaling argument, the Bernstein theorems are
intimately related to the regularity question for the minimal
hypersurface equation.
A natural and important question then is to what extent such
Bernstein type theorems generalize to entire minimal graphs of
codimension $m\ge 2$. Moser's result has been extended to higher
codimension by Chern-Osserman for dimension $n=2$ \cite{c-o} and
Barbosa \cite{b} and Fisher-Colbrie \cite{fc} for dimension
$n=3$. For dimension $n=4$ and codimension $m=3$, however, there
is a counterexample given by Lawson-Osserman \cite{l-o}. In fact,
their paper emphasizes the stark contrast between the cases of
codimension 1 and greater than 1 for the minimal submanifold
system, concerning regularity, uniqueness, and existence. The
Lawson-Osserman problem then is concerned with a systematic
understanding of the analytic aspects of the minimal submanifold
system in higher codimension. As in the case of codimension 1,
the Bernstein problem provides a key towards this aim.
While the work of Lawson-Osserman produced a counterexample for a
general Bernstein theorem, there are also some positive results
in this direction which we shall now summarize.
Hildebrandt-Jost-Widman \cite{h-j-w} started a systematic
approach on the basis of the aforementioned Ruh-Vilms theorem.
That is, they developed and employed the theory of harmonic maps
and the convex geometry of Grassmann manifolds, and obtained
Bernstein type results in general dimension and codimension.
Their main result says that a Bernstein result holds if the image
of the Gauss map is contained in a strictly convex distance ball.
Since the Riemannian sectional curvature of $\grs{n}{m}$ is
nonnegative, the maximal radius of such a convex ball is bounded.
In codimension 1, this in particular reproduces Moser's theorem,
and in this sense, their result is optimal. For higher
codimension, their result can be improved, for the following
reason. Since the sectional curvature of $\grs{n}{m}$ for $n,m
\ge 2$ is not constant, there exist larger convex sets than
geodesic distance balls, and it turns out that harmonic (e.g.
Gauss) maps with values in such convex sets can still be well
enough controlled. In this sense, the results of \cite{h-j-w}
could be improved by Jost-Xin \cite{j-x}, Wang \cite{wang} and
Xin-Yang \cite{x-y1}. In \cite{j-x}, the largest such
geodesically convex set in a Grassmann manifold
was found.
Formulating it somewhat differently, the harmonic map approach is based on the fact that the
composition of a harmonic map with a convex function is a subharmonic
function, and by using quantitative estimates for such subharmonic
functions, regularity and Liouville type results for harmonic maps
can be obtained. The most natural such convex function is the squared
distance from some point, when its domain is restricted to a suitably
small ball. As mentioned, the largest such ball on which a squared
distance function is convex was utilized in \cite{h-j-w}. As also
mentioned, however, this result is not yet optimal, and other convex functions
were systematically utilized in \cite{x-y1}. In that paper, also the
fundamental connection between estimates for the second fundamental
form of minimal submanifolds and estimates for their Gauss maps was
systematically explored. On this basis, the fundamental curvature
estimate technique, as developed by Schoen-Simon-Yau \cite{s-s-y}
and Ecker-Huisken \cite{e-h}, could be used in \cite{x-y1}.
Still, there remains a large quantitative gap between those positive
results and the counterexample of Lawson-Osserman. In this situation,
it could either be that Bernstein theorems can be found under more
general conditions, or that there exist other counterexamples in the
so far unexplored range.
In the present paper, we make a step towards closing this gap in
the positive direction. We identify a geometrically natural
function $v$ on a Grassmann manifold and a natural quantitative
condition under which the precomposition of this function with a
harmonic (Gauss) map is (strongly) subharmonic (Theorem
\ref{thm1}). When the precomposition of $v$ with the Gauss map of
a complete minimal submanifold is bounded, then that submanifold
is an entire graph of bounded slope. On one hand, this is the
first systematic example in harmonic map regularity theory where
this auxiliary function is not necessarily convex. On the other
hand, the Lawson-Osserman's counterexample can also be readily
characterized in terms of this function. Still, the range of
values for $v$ where we can apply our scheme is strictly
separated from the value of $v$ in that example. Therefore, still
some gap remains which should be explored in future work.
Our work also finds its natural position in the general regularity
theory for harmonic maps. Also, once we have a strongly
subharmonic function, we could derive Bernstein type results
within the frame work of geometric measure theory, by the
standard blow-down procedure and appeal to Allard's regularity
theorem \cite{a}. By building upon the work of many people on
harmonic map regularity, we can obtain more insight, however. In
particular, we shall use the iteration method of \cite{h-j-w},
we can explore the relation with curvature estimates, and we
shall utilize a version of the telescoping trick (Theorem
\ref{thm2}) to finally obtain a quantitatively controlled Gauss
image shrinking process (Theorem \ref{thm3} and Theorem
\ref{thm4}). In this way, we can understand why the submanifold
is flat as the Bernstein result asserts. More precisely, we
obtain the following Bernstein type result, which substantially
improves our previous results.
\begin{thm}\label{thm5}
Let $z^\a=f^\a(x^1,\cdots,x^n),\ \a=1,\cdots,m$, be smooth
functions defined everywhere in $\R^n$ ($n\geq 3,m\geq 2$).
Suppose their graph $M=(x,f(x))$ is a submanifold with parallel
mean curvature in $\R^{n+m}$. Suppose that there exists a number
$\be_0<3$ such that
\begin{equation}
\De_f=\Big[\det\Big(\de_{ij}+\sum_\a \f{\p f^\a}{\p x^i}\f{\p
f^\a}{\p x^j}\Big)\Big]^{\f{1}{2}}\leq \be_0.\label{be2}
\end{equation}
Then $f^1,\cdots,f^m$ has to be affine linear, i.e., it represents an
affine $n$-plane.
\end{thm}
The essential point is to show that $v:=\De_f$ is subharmonic when
$<3$. In fact, when $v \le \be_0 <3$, then $\Delta v \ge K_0 |B|^2$
where $K_0$ is a positive constant and $B$ is the second fundamental
form of $M$ in $\R^{n+m}$. This principle is not new. Wang \cite{wang}
has given conditions under which $\log v$ is subharmonic and has
derived Bernstein results from this, as indicated above. He only needs that $v$ be
uniformly bounded by some constant, not necessarily $<3$, but in
addition that there exist some $\delta >0$ such that for any two
eigenvalues $\la_i, \la_j$ with $i\neq j$, the inequality $|\la_i
\la_j|\le 1 -\delta$ holds (the latter condition means in geometric
terms that $df$ is strictly area decreasing on any two-dimensional subspace). Since subharmonicity of $\log v$ is a
weaker property than subharmonicity of $v$ itself, his computation is
substantially easier than ours, and our results cannot be deduced from
his. In fact, $v^2=\prod (1+\la_i^2)$, and while the condition of
\cite{j-x} which can be reformulated as $v^2$ being bounded away from
4 implies the condition of \cite{wang} so that the latter result
generalizes the former, the condition needed in the present paper is
only
the weaker one that $v^2$ be bounded away from 9.
In fact, somewhat more refined results can be obtained, as will
be pointed out in the final remarks of this paper.
\Section{Geometry of Grassmann manifolds}{Geometry of Grassmann manifolds}\label{s1}
Let $\R^{n+m}$ be an $(n+m)$-dimensional Euclidean space. Its
oriented $n$-subspaces constitute the Grassmann manifold
$\grs{n}{m}$, which is the Riemannian symmetric space of compact
type $SO(n+m)/SO(n)\times SO(m).$
$\grs{n}{m}$ can be viewed as a submanifold of some Euclidean space
via the Pl\"ucker embedding. The restriction of the
Euclidean inner product on $M$ is denoted by $w:\grs{n}{m}\times \grs{n}{m}\ra \R$
$$w(P,Q)=\lan e_1\w\cdots\w e_n,f_1\w\cdots\w f_n\ran=\det W$$
where $P$ is spanned by a unit $n$-vector $e_1\w\cdots\w e_n$, $Q$ is spanned by another unit $n$-vector $f_1\w\cdots
\w f_n$, and $W=\big(\lan e_i,f_j\ran\big)$. It is well-known that
$$W^T W=O^T \La O$$
with $O$ an orthogonal matrix and
$$\La=\left(\begin{array}{ccc}
\mu_1^2 & & \\
& \ddots & \\
& & \mu_n^2
\end{array}\right).$$
Here each $0\leq \mu_i^2\leq 1$. Putting $p:=\min\{m,n\}$, then
at most $p$ elements in $\{\mu_1^2,\cdots, \mu_n^2\}$ are not
equal to $1$. Without loss of generality, we can assume
$\mu_i^2=1$ whenever $i>p$. We also note that the $\mu_i^2$ can
be expressed as
\begin{equation}\label{di1a}
\mu_i^2=\frac{1}{1+\la_i^2}. \end{equation}
The Jordan angles between $P$ and $Q$ are defined by
$$\th_i=\arccos(\mu_i)\qquad 1\leq i\leq p.$$
The distance between $P$ and $Q$ is defined by
\begin{equation}\label{di}
d(P, Q)=\sqrt{\sum\th_i^2}. \end{equation}
Thus, (\ref{di1a}) becomes
\begin{equation}\label{di2}
\la_i=\tan\th_i. \end{equation}
In the sequel, we shall
assume $n\geq m$ without loss of generality. We use the
summation convention and agree on the ranges of indices:
$$1\leq i,j,k,l\leq n,\; 1\leq \a,\be,\g\leq m,\; a, b,\cdots =1,\cdots, n+m.$$
Now we fix $P_0\in \grs{n}{m}.$ We represent it by $ n $ vectors
$\eps_i,$, which are complemented by $ m $ vectors $ \eps_{n+\a} $,
such that $ \{\eps_i, \eps_{n+\a} \} $ form an orthonormal base of $
\ir{m+n} $.
Denote
$$\Bbb{U}:=\{P\in \grs{n}{m},\; w(P,P_0)>0\}.$$
We can span an arbitrary $P\in \Bbb{U}$ by $n$-vectors $f_i$:
$$f_i=\eps_i+Z_{i\a}\eps_{n+\a}.$$
The canonical metric in $\Bbb{U}$ can be described as
\begin{equation}\label{m1}ds^2 = tr (( I_n + ZZ^T )^{-1} dZ (I_m + Z^TZ)^{-1} dZ^T
),\end{equation} where $ Z = (Z_{i \a}) $ is an $ (n \times m)
$-matrix and $ I_n $ (res. $ I_m $) denotes the $ (n\times n)
$-identity (res. $ m \times m $) matrix. It is shown that
(\ref{m1}) can be derived from (\ref{di}) in \cite{x}.
For any $P\in\Bbb{U}$, the Jordan angles between $P$ and $P_0$
are defined by $\{\th_i\}$. Let $E_{i\a}$ be the matrix with $1$
in the intersection of row $i$ and column $\a$ and $0$ otherwise.
Then, $\sec\th_i\sec\th_\a E_{i\a}$ form an orthonormal basis of
$T_P\grs{n}{m}$ with respect to (\ref{m1}). Denote its dual frame
by $\om_{i\a}.$
Our fundamental quantity will be
\begin{equation}
v(\cdot, P_0):=w^{-1}(\cdot, P_0) \text{ on }\Bbb{U}.
\end{equation}
For
arbitrary $P\in \U$ determined by an $n\times m$ matrix $Z$, it
is easily seen that
\begin{equation}\label{v}
v(P,P_0)=\big[\det(I_n+ZZ^T)\big]^{\f{1}{2}}=\prod_{\a=1}^m
\sec\th_\a =\prod_{\a=1}^m \frac{1}{\mu_\alpha}.
\end{equation}
where $\th_1,\cdots,\th_m$ denote the Jordan angles between $P$
and $P_0$.
In this terminology, Hess$(v(\cdot, P_0)$ has been estimated in
\cite{x-y1}. By (3.8) in \cite{x-y1}, we have
\begin{eqnarray}\label{He}\aligned
\Hess(v(\cdot,P_0))&=\sum_{i\neq \a}v\ \om_{i\a}^2+\sum_\a
(1+2\la_\a^2)v\ \om_{\a\a}^2
+\sum_{\a\neq\be} \la_\a\la_\be v(\om_{\a\a}\otimes \om_{\be\be}+\om_{\a\be}\otimes\om_{\be\a})\\
&=\sum_{m+1\leq i\leq n,\a}v\
\om_{i\a}^2+\sum_{\a}(1+2\la_\a^2)v\ \om_{\a\a}^2
+\sum_{\a\neq \be}\la_\a\la_\be v\ \om_{\a\a}\otimes\om_{\be\be}\\
&\qquad\qquad+\sum_{\a<\be}\Big[(1+\la_\a\la_\be)v\Big(\f{\sqrt{2}}{2}(\om_{\a\be}
+\om_{\be\a})\Big)^2\\
&\hskip2in+(1-\la_\a\la_\be)v\Big(\f{\sqrt{2}}{2}(\om_{\a\be}-\om_{\be\a})\Big)^2\Big].
\endaligned
\end{eqnarray}
It follows that
\begin{equation}\label{hess}
v(\cdot,P_0)^{-1}\Hess(v(\cdot,P_0))
=g+\sum_\a 2\la_\a^2 \om_{\a\a}^2+\sum_{\a\neq \be}\la_\a\la_\be(\om_{\a\a}\otimes \om_{\be\be}+
\om_{\a\be}\otimes \om_{\be\a}).
\end{equation}
The canonical Riemannian metric on $\grs{n}{m}$ derived from
(\ref{di}) can also be described by the moving frame method. This will
be useful for understanding some of the sequel. Let
$\{e_i,e_{n+\a}\}$ be a local orthonormal frame field in
$\ir{n+m}.$ Let $\{\om_i,\om_{n+\a}\}$ be its dual frame field
so that the Euclidean metric is
$$g=\sum_{i}\om_i^2+\sum_{\a}\om_{n+\a}^2.$$
The Levi-Civita connection forms $\om_{ab}$ of $\ir{n+m}$ are
uniquely determined by the equations
$$\aligned
&d\om_{a}=\om_{ab}\wedge\om_b,\cr &\om_{ab}+\om_{ba}=0.
\endaligned$$
It is shown in \cite{x} that the canonical Riemannian metric on
$\grs{n}{m}$ can be written as
\begin{equation}\label{m2}
ds^2=\sum_{i,\ \a}\om_{i\, n+\a}^2. \end{equation}
\Section{Subharmonic functions}{Subharmonic functions}
Let $M^m\ra \R^{n+m}$ be an isometric immersion with second
fundamental form $B.$ Around any point $p\in M$, we choose an
orthonormal frame field $e_i,\cdots, e_{n+m}$ in $\R^{n+m},$ such
that $\{e_i\}$ are tangent to $M$ and $\{e_{n+\a}\}$ normal to
$M.$ The metric on $M$ is $g=\sum_i \om_i^2.$ We have the structure
equations
\begin{equation}\label{str}
\om_{i\ n+\a}=h_{\a ij}\om_j,
\end{equation}
where $h_{\a ij}$
are the coefficients of second fundamental form $B$ of $M$ in
$\R^{n+m}.$
Let $0$ be the origin of $\R^{n+m}$, $SO(m+n)$ be the Lie group
consisting of all orthonormal frames $(0;e_i,e_{n+\a})$,
$TF=\big\{(p;e_1,\cdots,e_n):p\in M,e_i\in T_p M,\lan
e_i,e_j\ran=\de_{ij}\big\}$ be the principle bundle of orthonormal
tangent frames over $M$, and
$NF=\big\{(p;e_{n+1},\cdots,e_{n+m}):p\in M,e_{n+\a}\in N_p M\big\}$
be the principle bundle of orthonormal normal frames over $M$. Then
$\bar{\pi}: TF\oplus NF\ra M$ is the projection with fiber
$SO(n)\times SO(m)$.
The Gauss map $\g: M\ra \grs{n}{m}$ is defined by
$$\g(p)=T_p M\in \grs{n}{m}$$
via the parallel translation in $\R^{n+m}$ for every $p\in M$. Then the following diagram commutes
$$\CD
TF \oplus NF @>i>> SO (n+m) \\
@V{\bar\pi}VV @VV{\pi}V \\
M @>{\g}>> \grs{n}{m}
\endCD$$
where $i$ denotes the inclusion map and $\pi: SO(n+m)\ra \grs{n}{m}$ is defined by
$$(0;e_i,e_{n+\a})\mapsto e_1\w\cdots\w e_n.$$
It follows that
\begin{equation}\label{edg}
|d\g|^2=\sum_{\a,i,j}h_{\a ij}^2=|B|^2.
\end{equation}
(\ref{hess}) was computed for the metric (\ref{m1}) whose
corresponding coframe field is $\om_{i \a}.$ Since (\ref{m1}) and
(\ref{m2}) are equivalent to each other, at any fixed point
$P\in\grs{n}{m}$ there exists an isotropic group action, i.e., an
$SO(n)\times SO(m)$ action, such that $\om_{i\a}$ is
transformed to $\om_{i\ n+\a}$, namely, there are a local tangent
frame field and a local normal frame field such that at the
point under consideration,
\begin{equation}\label{str2}
\om_{i\ n+\a}=\g^*\om_{i \a}.
\end{equation}
In conjunction with (\ref{str}) and (\ref{str2}) we obtain
\begin{equation}\label{hij}
\g^*\om_{i\a}=h_{\a ij}\om_j.
\end{equation}
By the Ruh-Vilms theorem \cite{r-v}, the mean curvature of $M$
is parallel if and only if its Gauss map is a harmonic map. Now,
we assume that $M$ has parallel mean curvature.
We define
\begin{equation}
v:=v(\cdot,P_0)\circ \g,
\end{equation}
This function $ v$ on $M$ will be the source of the basic
inequality for this paper. Its geometric significance is seen
from the following observation. If the $v-$ function has an upper
bound (or the $w-$function has a positive lower bound), $M$ can
be described as an entire graph on $\ir{n}$ by $f:\ir{n}\to
\ir{m}$, provided $M$ is complete. In this situation, $\la_i$ is
the singular values of $df$ and
\begin{equation}\label{v1}
v=\Big[\det\Big(\de_{ij}+\sum_\a \f{\p f^\a}{\p x^i}\f{\p
f^\a}{\p x^j}\Big)\Big]^{\f{1}{2}}
\end{equation}
Using the composition formula, in conjunction with (\ref{hess}),
(\ref{edg}) and (\ref{hij}), and the fact that $\tau(\g)=0$ (the
tension field of the Gauss map vanishes \cite{r-v}), we deduce the
important
formula of Lemma 1.1
in \cite{fc} and Prop. 2.1 in \cite{wang}.
\begin{pro}Let $M$ be an $n-$submanifold in $\ir{n+m}$ with parallel mean
curvature. Then
\begin{equation}\label{Dv}
\De v=v|B|^2+v\sum_{\a,j}2\la_\a^2h_{\a,\a j}^2 +v\sum_{\a\neq
\be,j}\la_\a\la_\be(h_{\a,\a j}h_{\be,\be j}+h_{\a,\be
j}h_{\be,\a j}),
\end{equation}
where $h_{\a,ij}$ are the coefficients of the second fundamental form
of $M$ in $\ir{n+m}$ (see (\ref{str}).
\end{pro}
A crucial step in this paper is to find a condition which
guarantees the strong subharmonicity of the $v-$ function on $M$.
More precisely, under a condition on $v$, we shall bound its Laplacian from below by a
positive constant times squared norm of the second
fundamental form.
Looking at the expression (\ref{Dv}), we group its terms according to the
different types of the indices of the coefficients of the second
fundamental form as follows.
\begin{equation}
v^{-1}\De v=
\sum_\a\sum_{i,j>m}h_{\a,ij}^2+\sum_{j>m}I_j+\sum_{j>m,\a<\be}II_{j\a\be}
+\sum_{\a<\be<\g}III_{\a\be\g}+\sum_\a IV_\a
\end{equation} where
\begin{equation}
I_j=\sum_\a(2+2\la_\a^2)h_{\a,\a j}^2+\sum_{\a\neq
\be}\la_\a\la_\be h_{\a,\a j}h_{\be,\be j},
\end{equation}
\begin{equation}
II_{j\a\be}=2h_{\a,\be j}^2+2h_{\be,\a j}^2+2\la_\a\la_\be h_{\a,\be j}h_{\be,\a j},
\end{equation}
\begin{equation}\aligned
III_{\a\be\g}=&2h_{\a,\be\g}^2+2h_{\be,\g\a}^2+2h_{\g,\a\be}^2\\
&+2\la_\a\la_\be h_{\a,\be\g}h_{\be,\g\a}+2\la_\be\la_\g h_{\be,\g\a}h_{\g,\a\be}+2\la_\g\la_\a h_{\g,\a\be}h_{\a,\be\g}
\endaligned
\end{equation}
and
\begin{equation}\aligned
IV_\a=&(1+2\la_\a^2)h_{\a,\a\a}^2+\sum_{\be\neq \a}\big(h_{\a,\be\be}^2+(2+2\la_\be^2)h_{\be,\be\a}^2\big)\\
&+\sum_{\be\neq \g}\la_\be\la_\g h_{\be,\be \a}h_{\g,\g \a}+2\sum_{\be\neq \a}\la_\a\la_\be h_{\a,\be\be}h_{\be,\be\a}.
\endaligned
\end{equation}
It is easily seen that
\begin{equation}\label{es1}
I_j=(\sum_\a \la_\a h_{\a,\a j})^2+\sum_\a (2+\la_\a^2)h_{\a,\a j}^2\geq 2\sum_\a h_{\a,\a j}^2.
\end{equation}
Obviously
\begin{equation}
II_{j\a\be}=\la_\a\la_\be(h_{\a,\be j}+h_{\be,\a j})^2+(2-\la_\a\la_\be)(h_{\a,\be j}^2+h_{\be,\a j}^2).
\end{equation}
$v=\Big(\prod_\a (1+\la_\a^2)\Big)^{\f{1}{2}}$ implies $(1+\la_\a^2)(1+\la_\be^2)\leq v^2$. Assume
$(1+\la_\a^2)(1+\la_\be^2)\equiv C\leq v^2$, then differentiating both sides implies
$$\f{\la_\a d\la_\a}{1+\la_\a^2}+\f{\la_\be d\la_\be}{1+\la_\be^2}=0.$$
Therefore
\begin{equation}
\aligned
d(\la_\a \la_\be)&=\la_\be d\la_\a+\la_\a d\la_\be\\
&=\big[\la_\be^2(1+\la_\a^2)-\la_\a^2(1+\la_\be^2)\big]\f{d\la_\a}{\la_\be(1+\la_\a^2)}\\
&=(\la_\be^2-\la_\a^2)\f{d\la_\a}{\la_\be(1+\la_\a^2)}.
\endaligned
\end{equation}
It follows that $(\la_\a,\la_\be)\mapsto \la_\a\la_\be$ attains its maximum at the point satisfying $\la_\a=\la_\be$,
which is hence $((C^{\f{1}{2}}-1)^{\f{1}{2}},(C^{\f{1}{2}}-1)^{\f{1}{2}})$.
Thus $\la_\a\la_\be\leq C^{\f{1}{2}}-1\leq v-1$ and moreover
\begin{equation}\label{es2}
II_{j\a\be}\geq (3-v)(h_{\a,\be j}^2+h_{\be,\a j}^2).
\end{equation}
\bigskip
\begin{lem}\label{l1}
$III_{\a\be\g}\geq
(3-v)(h_{\a,\be\g}^2+h_{\be,\g\a}^2+h_{\g,\a\be}^2)$.
\end{lem}
\begin{proof}
It is easily seen that
$$\aligned
III_{\a\be\g}-&(3-v)(h_{\a,\be\g}^2+h_{\be,\g\a}^2+h_{\g,\a\be}^2)\\
=&(\la_\a h_{\a,\be\g}+\la_\be h_{\be,\g\a}+\la_\g
h_{\g,\a\be})^2+(v-1-\la_\a^2)h_{\a,\be\g}^2\\
&\qquad+(v-1-\la_\be^2)h_{\be,\g\a}^2
+(v-1-\la_\g^2)h_{\g,\a\be}^2.\endaligned$$
If $\la_\a^2,\la_\be^2,\la_\g^2\leq v-1$, then
$III_{\a\be\g}-(3-v)(h_{\a,\be\g}^2+h_{\be,\g\a}^2+h_{\g,\a\be}^2)$
is obviously nonnegative definite. Otherwise, we can assume
$\la_\g^2>v-1$ without loss of generality, then
$(1+\la_\a^2)(1+\la_\be^2)(1+\la_\g^2)\leq v^2$ implies $\la_\a^2<v-1, \la_\be^2<v-1$.
Denote $s=\la_\a h_{\a,\be\g}+\la_\be h_{\be,\g\a}$, then by the Cauchy-Schwarz inequality,
$$\aligned
s^2&=(\la_\a h_{\a,\be\g}+\la_\be h_{\be,\g\a})^2\\
&=\Big(\f{\la_\a}{\sqrt{v-1-\la_\a^2}}\sqrt{v-1-\la_\a^2}h_{\a,\be\g}+\f{\la_\be}{\sqrt{v-1-\la_\be^2}}
\sqrt{v-1-\la_\be^2}h_{\be,\g\a}\Big)^2\\
&\leq\Big(\f{\la_\a^2}{v-1-\la_\a^2}+\f{\la_\be^2}{v-1-\la_\be^2}\Big)\big((v-1-\la_\a^2)h_{\a,\be\g}^2+(v-1-\la_\be^2)h_{\be,\g\a}^2\big)
\endaligned$$
i.e.
\begin{equation}
(v-1-\la_\a^2)h_{\a,\be\g}^2+(v-1-\la_\be^2)h_{\be,\g\a}^2\geq \Big(\f{\la_\a^2}{v-1-\la_\a^2}+\f{\la_\be^2}{v-1-\la_\be^2}\Big)^{-1}s^2.
\end{equation}
Hence
\begin{equation}\label{ineq3}\aligned
&III_{\a\be\g}-(3-v)(h_{\a,\be\g}^2+h_{\be,\g\a}^2+h_{\g,\a\be}^2)\\
\geq& (s+\la_\g h_{\g,\a\be})^2+\Big(\f{\la_\a^2}{v-1-\la_\a^2}+\f{\la_\be^2}{v-1-\la_\be^2}\Big)^{-1}s^2+(v-1-\la_\g^2)h_{\g,\a\be}^2\\
=&\Big[1+\Big(\f{\la_\a^2}{v-1-\la_\a^2}+\f{\la_\be^2}{v-1-\la_\be^2}\Big)^{-1}\Big]s^2+(v-1)h_{\g,\a\be}^2+2\la_\g
sh_{\g,\a\be}.
\endaligned
\end{equation}
It is well known that $ax^2+2bxy+cy^2$ is nonnegative definite if
and only if $a,c\geq 0$ and $ac-b^2\geq 0$. Hence the right hand
side of (\ref{ineq3})
is nonnegative definite if and only if
\begin{equation}\label{cond}
(v-1)\Big[1+\Big(\f{\la_\a^2}{v-1-\la_\a^2}+\f{\la_\be^2}{v-1-\la_\be^2}\Big)^{-1}\Big]-\la_\g^2\geq 0
\end{equation}
i.e.
\begin{equation}\label{cond2}
\f{1}{v-1-\la_\a^2}+\f{1}{v-1-\la_\be^2}+\f{1}{v-1-\la_\g^2}\leq \f{2}{v-1}.
\end{equation}
Denote $x=1+\la_\a^2$, $y=1+\la_\be^2$, $z=1+\la_\g^2$. Let $C$ be a
constant $\leq v^2$, denote
$$\Om=\big\{(x,y,z)\in \R^3:1\leq x,y<v,\; z>v,\; xyz=C\big\}$$
and $f:\Om\ra \R$
$$(x,y,z)\mapsto \f{1}{v-x}+\f{1}{v-y}+\f{1}{v-z}.$$
We claim $f\leq \f{2}{v-1}$ on $\Om$. Then (\ref{cond2}) follows
and hence
$$III_{\a\be\g}-(3-v)(h_{\a,\be\g}^2+h_{\be,\g\a}^2
+h_{\g,\a\be}^2)$$ is nonnegative definite.
We now verify the claim. For arbitrary $\ep>0$, denote
$$f_\ep=\f{1}{v+\ep-x}+\f{1}{v+\ep-y}+\f{1}{v+\ep-z},$$
then $f_\ep$ is obviously a smooth function on
$$\Om_\ep=\big\{(x,y,z)\in \R^3:1\leq x,y\leq v,\; z\geq v+2\ep,\; xyz=C\big\}.$$
The compactness of $\Om_\ep$ implies the existence of
$(x_0,y_0,z_0)\in \Om_\ep$ satisfying
\begin{equation}\label{sup}
f_\ep(x_0,y_0,z_0)=\sup_{\Om_\ep} f_\ep.
\end{equation}
Fix $x_0$, then (\ref{sup}) implies that for arbitrary $(y,z)\in \R^2$
satisfying $1\leq y\leq v,\; z\geq v+2\ep$ and $yz=\f{C}{x_0}$,
we have
$$f_{\ep,x_0}(y, z)=\f{1}{v+\ep-y}+\f{1}{v+\ep-z}\leq
\f{1}{v+\ep-y_0}+\f{1}{v+\ep-z_0}.$$ Differentiating both sides
of $yz=\f{C}{x_0}$ yields $\f{dy}{y}+\f{dz}{z}=0.$ Hence
\begin{equation}\aligned
&d\Big(\f{1}{v+\ep-y}+\f{1}{v+\ep-z}\Big)=\f{dy}{(v+\ep-y)^2}+\f{dz}{(v+\ep-z)^2}\\
=&\Big[\f{y}{(v+\ep-y)^2}-\f{z}{(v+\ep-z)^2}\Big]\f{dy}{y}=\f{((v+\ep)^2-yz)(y-z)}{(v+\ep-y)^2(v+\ep-z)^2}\f{dy}{y}.
\endaligned
\end{equation}
It implies that $f_{\ep,x_0}\left(y, \f{C}{yx_0}\right)$ is
decreasing in $y$ and $y_0=1.$ Similarly, one can derive
$x_0=1$. Therefore
$$\sup_{\Om_\ep} f_\ep=f_\ep(1,1,C)=\f{2}{v+\ep-1}+\f{1}{v+\ep-C}<\f{2}{v+\ep-1}.$$
Note that $f_\ep\ra f$ and $\Om\subset \lim_{\ep\ra
0^+}\Om_\ep$. Hence by letting $\ep\ra 0$ one can obtain $f\leq
\f{2}{v-1}.$
\end{proof}
\bigskip
\begin{lem}\label{l2}
There exists a positive constant $\ep_0$, such that if $v\leq 3$, then
$$IV_\a\geq \ep_0\big(h_{\a,\a\a}^2+\sum_{\be\neq \a}(h_{\a,\be\be}^2+2h_{\be,\be\a}^2)\big).$$
\end{lem}
\begin{proof}
For arbitrary $\ep_0\in [0,1)$, denote $C=1-\ep_0$, then
\begin{equation}\label{ineq4}
\aligned
&IV_\a-\ep_0\big(h_{\a,\a\a}^2+\sum_{\be\neq \a}(h_{\a,\be\be}^2+2h_{\be,\be\a}^2)\big)\\
=&(\sum_\be \la_\be h_{\be,\be \a})^2+(C+\la_\a^2)h_{\a,\a\a}^2+\sum_{\be\neq \a}\big[Ch_{\a,\be\be}^2+(2C+\la_\be^2)h_{\be,\be\a}^2+2\la_\a\la_\be h_{\a,\be\be}h_{\be,\be\a}\big].
\endaligned
\end{equation}
Obviously
$$\aligned
C\, h_{\a,\be\be}^2&+C^{-1}\la_\a^2\la_\be^2h_{\be,\be\a}^2+2\la_\a\la_\be h_{\a,\be\be}h_{\be,\be\a}\\
&\geq (C^{\f{1}{2}}h_{\a,\be\be}+C^{-\f{1}{2}}\la_\a\la_\be
h_{\be,\be\a})^2\geq 0,\endaligned$$ hence, the third term of the
right hand side of (\ref{ineq4}) satisfies
\begin{equation}\label{ineq2}
Ch_{\a,\be\be}^2+(2C+\la_\be^2)h_{\be,\be\a}^2+2\la_\a\la_\be h_{\a,\be\be}h_{\be,\be\a}\geq (2C+\la_\be^2-C^{-1}\la_\a^2\la_\be^2)h_{\be,\be\a}^2
\end{equation}
If there exist 2 distinct indices $\be,\g\neq \a$ satisfying
$$2C+\la_\be^2-C^{-1}\la_\a^2\la_\be^2\leq 0$$ and
$$2C+\la_\g^2-C^{-1}\la_\a^2\la_\g^2\leq 0,$$ then $\la_\a^2>C$ and
$$\la_\be^2\geq \f{2C^2}{\la_\a^2-C},\qquad \la_\g^2\geq \f{2C^2}{\la_\a^2-C}.$$
It implies
$$(1+\la_\a^2)(1+\la_\be^2)(1+\la_\g^2)\geq \f{(\la_\a^2+1)(\la_\a^2+2C^2-C)^2}{(\la_\a^2-C)^2}.$$
Define $f:x\in (C,+\infty)\mapsto
\f{(x+1)(x+2C^2-C)^2}{(x-C)^2}$, then a direct calculation shows
$$(\log f)'=\f{1}{x+1}+\f{2}{x+2C^2-C}-\f{2}{x-C}=\f{(x-C(2C+3))(x+C)}{(x+1)(x+2C^2-C)(x-C)}.$$ It follows that
$f(x)\geq f(C(2C+3))=\f{(2C+1)^3}{C+1}$, i.e.
\begin{equation}\label{ineq7}
v^2\geq (1+\la_\a^2)(1+\la_\be^2)(1+\la_\g^2)\geq \f{(2C+1)^3}{C+1}.
\end{equation}
If $C=1$, then $\f{(2C+1)^3}{C+1}=\f{27}{2}>9$; hence there is $\ep_1>0$, once
$\ep_0\leq \ep_1$, then $C=1-\ep_0$ satisfies $\f{(2C+1)^3}{C+1}>9$,
which causes a contradiction to $v^2\leq 9$.
Hence, one can find an index $\g\neq \a$, such that
\begin{equation}
2C+\la_\be^2-C^{-1}\la_\a^2\la_\be^2> 0\qquad \text{for arbitrary }\be\neq \a,\g.
\end{equation}
Denote $s=\sum_{\be\neq \g}\la_\be h_{\be,\be\a}$, then by using the
Cauchy-Schwarz inequality,
\begin{equation}\label{ineq5}\aligned
(C+\la_\a^2)h_{\a,\a\a}^2&+\sum_{\be\neq
\a,\g}(2C+\la_\be^2-C^{-1}\la_\a^2\la_\be^2)h_{\be,\be\a}^2\\
& \geq \Big(\f{\la_\a^2}{C+\la_\a^2}+\sum_{\be\neq
\a,\g}\f{\la_\be^2}{2C+\la_\be^2-C^{-1}\la_\a^2\la_\be^2}\Big)^{-1}s^2.
\endaligned\end{equation} Substituting (\ref{ineq5}) and (\ref{ineq2})
into (\ref{ineq4}) yields
\begin{equation}\label{ineq6}
\aligned
IV_\a&-\ep_0\big(h_{\a,\a\a}^2+\sum_{\be\neq \a}(h_{\a,\be\be}^2+2h_{\be,\be\a}^2)\big)\\
&\geq (s+\la_\g h_{\g,\g\a})^2+
\Big(\f{\la_\a^2}{C+\la_\a^2}+\sum_{\be\neq \a,\g}\f{\la_\be^2}{2C+\la_\be^2-C^{-1}\la_\a^2\la_\be^2}\Big)^{-1}s^2\\
&\qquad +(2C+\la_\g^2-C^{-1}\la_\a^2\la_\g^2)h_{\g,\g\a}^2\\
&\geq \Big[1+\Big(\f{\la_\a^2}{C+\la_\a^2}+\sum_{\be\neq
\a,\g}\f{\la_\be^2}{2C+\la_\be^2-C^{-1}\la_\a^2\la_\be^2}\Big)^{-1}\Big]s^2\\
&\qquad
+(2C+2\la_\g^2-C^{-1}\la_\a^2\la_\g^2)h_{\g,\g\a}^2+2\la_\g s
h_{\g,\g\a}.
\endaligned
\end{equation}
Note that when $m=2$, $s=\la_\a h_{\a,\a\a}$ and $\sum_{\be\neq
\a,\g}\f{\la_\be^2}{2C+\la_\be^2-C^{-1}\la_\a^2\la_\be^2}=0$.
The right hand side of (\ref{ineq6}) is nonnegative definite if
and only if
\begin{equation}\label{con1}
2C+2\la_\g^2-C^{-1}\la_\a^2\la_\g^2\geq 0
\end{equation}
and
\begin{equation}\label{con2}
\Big[1+\Big(\f{\la_\a^2}{C+\la_\a^2}+\sum_{\be\neq \a,\g}\f{\la_\be^2}{2C+\la_\be^2-C^{-1}\la_\a^2\la_\be^2}\Big)^{-1}\Big](2C+2\la_\g^2-C^{-1}\la_\a^2\la_\g^2)-\la_\g^2\geq 0.
\end{equation}
Assume $2C+2\la_\g^2-C^{-1}\la_\a^2\la_\g^2< 0$, then
$\la_\a^2>2C$ and $\la_\g^2> \f{2C^2}{\la_\a^2-2C}$, which implies
$(1+\la_\a^2)(1+\la_\g^2)\geq
\f{(\la_\a^2+1)(\la_\a^2+2C(C-1))}{\la_\a^2-2C}$. Define $f:x\in
(2C,+\infty)\mapsto \f{(x+1)(x+2C(C-1))}{x-2C}$, then
$$(\log f)'=\f{1}{x+1}+\f{1}{x+2C(C-1)}-\f{1}{x-2C}=
\f{x^2-4Cx-2C^2(2C-1)}{(x+1)(x+2C(C-1))(x-2C)}.$$ and hence
$$\min f=f\big(C(2+\sqrt{4C+2})\big)=2C^2+2C+1+2C\sqrt{4C+2}.$$
In particular, when
$C=1,\; \min f= 5+2\sqrt{6}>9$. There exists $\ep_2>0$, such that once
$\ep_0\leq \ep_2$, one can derive $\min f>9$ and moreover $v^2\geq
(1+\la_\a^2)(1+\la_\g^2)>9$, which contradicts $v\leq 3$.
Therefore (\ref{con1}) holds.
If $2C+\la_\g^2-C^{-1}\la_\a^2\la_\g^2\geq 0$, (\ref{con2})
trivially holds.
At last, we consider the situation when there exists $\g, \,
\g\neq\a$, such that
$$2C+\la_\g^2-C^{-1}\la_\a^2\la_\g^2<0.$$ In this case,
(\ref{con2}) is equivalent to
\begin{equation}\label{con}
\f{\la_\a^2}{C+\la_\a^2}+\sum_{\be\neq \a}\f{\la_\be^2}{2C+\la_\be^2-C^{-1}\la_\a^2\la_\be^2}\leq -1.
\end{equation}
Noting that
$$\f{\la_\be^2}{2C+\la_\be^2-C^{-1}\la_\a^2\la_\be^2}=\f{C}{C-\la_\a^2}-\f{2C^3}{(C-\la_\a^2)^2}\f{1}{1+\la_\be^2
+\f{\la_\a^2+C(2C-1)}{C-\la_\a^2}}$$ and let $x_\be=1+\la_\be^2$,
then (\ref{con}) is equivalent to
\begin{equation}
\f{x_\a-1}{x_\a+C-1}+\sum_{\be\neq \a}\Big[\f{C}{C+1-x_\a}-\f{2C^3}{(C+1-x_\a)^2}\f{1}{x_\be
-\f{x_\a+2C^2-C-1}{x_\a-C-1}}\Big]\leq -1.
\end{equation}
Denote
$$\aligned
\psi(x_\a)=\f{x_\a-1}{x_\a+C-1},&\qquad \varphi(x_\a)=\f{x_\a+2C^2-C-1}{x_\a-C-1},\\
\zeta(x_\a)=\f{C}{C+1-x_\a},&\qquad
\xi(x_\a)=\f{2C^3}{(C+1-x_\a)^2}.
\endaligned
$$
Let
\begin{equation}\label{Om}
\aligned\Om=\big\{&(x_1,\cdots,x_m)\in \R^m: x_\a>C+1,1\leq x_\be<\varphi(x_\a)\text{ for all }\be\neq \a,\g, \\
&\qquad x_\g>\varphi(x_\a), \prod_{\be}x_\be=v^2\big\}
\endaligned
\end{equation}
and define $f:\Om\ra \R$
$$(x_1,\cdots,x_m)\mapsto \psi(x_\a)+\sum_{\be\neq \a}\Big[\zeta(x_\a)-\f{\xi(x_\a)}{x_\be-\varphi(x_\a)}\Big].$$
We point out that in (\ref{Om}), $\a$ and $\g$ are fixed indices.
Now we claim
\begin{equation}\label{claim}
\sup_\Om f=\sup_\G f
\end{equation}
where
\begin{equation}\aligned
\G=\big\{&(x_1,\cdots,x_m)\in \R^m: x_\a\geq C+1, x_\be=1\text{ for all }\be\neq \a,\g,\\
&\qquad x_\g\geq \varphi(x_\a),\prod_\be x_\be=v^2\big\}\subset \Om.
\endaligned
\end{equation}
When $m=2$, obviously $\G=\Om$ and (\ref{claim}) is trivial. We
put
$$\varphi_\ep(x_\a)=\varphi(x_\a+\ep),\ \zeta_\ep(x_\a)=\zeta(x_\a+\ep),\ \xi_\ep(x_\a)=\xi(x_\a+\ep)$$
for arbitrary $\ep>0$. If $m\geq 3$, as in the proof of Lemma
\ref{l1}, we define
$$f_\ep=\psi(x_\a)+\sum_{\be\neq \a}\Big[\zeta_\ep(x_\a)-\f{\xi_\ep(x_\a)}{x_\be-\varphi_\ep(x_\a)}\Big],$$
then $f_\ep$ is well-defined on
$$\aligned\Om_\ep=\big\{&(x_1,\cdots,x_m)\in \R^m:x_\a\geq C+1,1\leq x_\be\leq \varphi_{2\ep}(x_\a)\text{ for all }\be\neq \a,\g,\\
&\qquad x_\g\geq \varphi_{\f{\ep}{2}}(x_\a),
\prod_{\be}x_\be=v^2\big\}.\endaligned$$
The compactness of $\Om_\ep$ enables us to find $(y_1,\cdots,y_m)\in \Om_\ep$, such that
\begin{equation}\label{mini}
f_\ep(y_1,\cdots,y_m)=\sup_{\Om_\ep} f_\ep.
\end{equation}
Denote $b=\varphi_\ep(y_\a)$, then (\ref{mini}) implies for arbitrary
$\be\neq \a,\g$ that
$$\f{1}{x_\be-b}+\f{1}{x_\g-b}\geq \f{1}{y_\be-b}+\f{1}{y_\g-b}$$
holds whenever $x_\be x_\g=y_\be y_\g$, $1\leq x_\be\leq
\varphi_{2\ep}(y_\a)$ and $x_\g\geq \varphi_{\f{\ep}{2}}(y_\a)$.
Differentiating both sides yields
$\f{dx_\be}{x_\be}+\f{dx_\g}{x_\g}=0$, thus
\begin{equation}\label{diff}\aligned
d\Big(\f{1}{x_\be-b}+\f{1}{x_\g-b}\Big)&=-\f{dx_\be}{(x_\be-b)^2}-\f{dx_\g}{(x_\g-b)^2}\\
&=\f{(b^2-x_\be x_\g)(x_\g-x_\be)}{(x_\be-b)^2(x_\g-b)^2}\f{dx_\be}{x_\be}.
\endaligned
\end{equation}
Similarly to (\ref{ineq7}), one can prove $y_\a
b^2=\f{y_\a(y_\a+\ep+2C^2-C-1)^2}{(y_\a+\ep-C-1)^2}>9$ when
$\ep_0\leq \ep_1$ (note that $C=1-\ep_0$) and $\ep_1$ is
sufficiently small. In conjunction with $y_\a x_\be x_\g=y_\a
y_\be y_\g\leq v^2<9$, we have $b^2-x_\be x_\g>0$. Hence
(\ref{diff}) implies $y_\be=1$ for all $\be\neq \a, \g$. In other
words, if we put
$$\aligned
\G_\ep=\big\{(x_1,\cdots,x_m)&\in \R^m: x_\a\geq C+1,\; x_\be=1\text{ for all }\be\neq \a,\g,\\
&x_\g\geq \varphi_{\f{\ep}{2}}(x_\a),\prod_\be x_\be=v^2\big\},
\endaligned$$
then $\max_{\Om_\ep}f_\ep=\max_{\G_\ep}f_\ep$. Therefore, (\ref{claim}) follows from
$\Om\subset \bigcup_{\ep>0}\Om_\ep$, $\G\subset \bigcup_{\ep>0}\G_\ep$ and $\lim_{\ep\ra 0}f_\ep=f$.
To prove (\ref{con2}), i.e. $f\leq -1$ , it is sufficient to show
on $\G$,
\begin{equation}
\psi(x_\a)+\zeta(x_\a)-\f{\xi(x_\a)}{\f{v^2}{x_\a}-\varphi(x_\a)}\leq
-1
\end{equation}
whenever $x_\a > C+1$ and $\f{v^2}{x_\a}>\varphi(x_\a)$. After a
straightforward calculation, the above inequality is equivalent to
\begin{equation}\label{con6}
x_\a^3+(2C^2-C-2)x_\a^2+(C^3-3C^2+C+1)x_\a-v^2(x_\a^2-(C+2)x_\a-(C^2-C-1))\geq 0.
\end{equation}
It is easily seen that if
\begin{equation}\label{con5}
\inf_{t^2-(C+2)t-(C^2-C-1)>0}\f{t^3+(2C^2-C-2)t^2+(C^3-3C^2+C+1)t}{t^2-(C+2)t-(C^2-C-1)}> 9.
\end{equation}
then (\ref{con6}) naturally holds and furthermore one can deduce that $IV_\a-\ep_0\big(h_{\a,\a\a}^2+\sum_{\be\neq \a}(h_{\a,\be\be}^2+2h_{\be,\be\a}^2)\big)$ is nonnegative definite.
When $C=1$, (\ref{con5}) becomes
\begin{equation}\label{con7}\inf_{t>\f{3+\sqrt{5}}{2}}\f{t^2(t-1)}{t^2-3t+1}>9.
\end{equation}
If this is true, one can find a positive constant $\ep_3$ to ensure (\ref{con5}) holds true whenever
$\ep_0\leq \ep_3$. Finally,, by taking $\ep_0=\min\{\ep_1,\ep_2,\ep_3\}$ we obtain the final conclusion.
(\ref{con7}) is equivalent to the property that
$h(t)=t^2(t-1)-9(t^2-3t+1)=t^3-10t^2+27t-9$ has no zeros on $\big(\f{3+\sqrt{5}}{2},+\infty\big)$. $h'(t)=3t^2-20t+27$ implies
$h'(t)<0$ on $\big(\f{3+\sqrt{5}}{2},\f{10+\sqrt{19}}{3}\big)$ and $h'(t)>0$ on $\big(\f{10+\sqrt{19}}{3},+\infty\big)$, hence
$$\inf_{t> \f{3+\sqrt{5}}{2}}h=h\big(\f{10+\sqrt{19}}{3}\big)=\f{187-38\sqrt{19}}{27}>0$$
and (\ref{con7}) follows.
\end{proof}
\bigskip
In conjunction with (\ref{es1}), (\ref{es2}), Lemma \ref{l1} and \ref{l2}, we can arrive at
\begin{thm}\label{thm1}
Let $M^n$ be a submanifold in $\R^{n+m}$ with parallel mean curvature, then for arbitrary $p\in M$ and $P_0\in \grs{n}{m}$,
once $v(\g(p),P_0)\leq 3$, then $\De\big(v(\cdot,P_0)\circ \g\big)\geq 0$ at $p$. Moreover,
if $v(\g(p),P_0)\le
q \be_0<3$, then there exists a positive constant $K_0$, depending only on $\be_0$, such that
\begin{equation}\label{Dv1}
\De\big(v(\cdot,P_0)\circ \g\big)\geq K_0|B|^2
\end{equation}
at $p$.
\end{thm}
We also express this result by saying that the function $v$ satisfying (\ref{Dv1}) is strongly
subharmonic under the condition $v(\g(p),P_0)\leq \be_0<3$.
\begin{rem}
If\, $\log v$ is a strongly subharmonic function, then $v$ is
certainly strongly subharmonic, but the converse is not
necessarily true. Therefore, the above result does not seem to follow
from Theorem 1.2 in \cite{wang}.
\end{rem}
\Section{Curvature estimates}{Curvature estimates}
Let $z^\a=f^\a(x^1,\cdots,x^n),\a=1,\cdots,m$ be smooth functions
defined on $D_{R_0}\subset \R^n$. Their graph $M=(x,f(x))$ is a
submanifold with parallel mean curvature in $\R^{n+m}$. Suppose
there is $\be_0\in [1,3)$, such that
\begin{equation}\label{slope}
\De_f=\Big[\det\Big(\de_{ij}+\sum_\a \f{\p f^\a}{\p x^i}\f{\p f^\a}{\p x^j}\Big)\Big]^{\f{1}{2}}\leq \be_0.
\end{equation}
Denote by $\eps_1,\cdots,\eps_{n+m}$ the canonical basis of
$\R^{n+m}$ and put $P_0=\eps_1\w\cdots\w\eps_n$. Then by
(\ref{slope}) $$v(\cdot,P_0)\circ \g\leq \be_0$$ holds
everywhere on $M$. Putting $v=v(\cdot,P_0)\circ \g$, Theorem
\ref{thm1} tells us
\begin{equation}\label{sub}
\De v\geq K_0(\be_0)|B|^2.
\end{equation}
Let $\eta$ be a
nonnegative smooth function on $M$ with compact support. Multiplying both sides of (\ref{sub}) by $\eta$ and integrating on $M$
gives
\begin{equation}\label{weak}
K_0\int_M |B|^2 \eta*1\leq -\int_M \n\eta\cdot\n v*1.
\end{equation}
$F:D_{R_0}\mapsto M$ defined by
$$x=(x^1,\cdots,x^n)\mapsto (x,f(x))$$
is obviously a diffeomorphism. $F_*\f{\p}{\p x^i}=\eps_i+\f{\p f^\a}{\p x^i}\eps_{n+\a}$ implies
$$\big\lan F_*\f{\p}{\p x^i},F_*\f{\p}{\p x^j}\big\ran=\de_{ij}+\sum_\a \f{\p f^\a}{\p x^i}\f{\p f^\a}{\p x^j}.$$
Hence
\begin{equation}
F^* g=\Big(\de_{ij}+\sum_\a \f{\p f^\a}{\p x^i}\f{\p f^\a}{\p x^j}\Big)dx^idx^j
\end{equation}
where $g$ is the metric tensor on $M$. In other words, $M$ is
isometric to the Euclidean ball $D_{R_0}$ equipped with the metric
$g_{ij}dx^i dx^j$ ($g_{ij}=\de_{ij}+\sum_\a \f{\p f^\a}{\p
x^i}\f{\p f^\a}{\p x^j}$). It is easily seen that for arbitrary
$\xi\in \R^n$,
\begin{equation}\label{eig1}
\xi^i g_{ij}\xi^j=|\xi|^2+\sum_\a \Big(\sum_i \f{\p f^\a}{\p x^i}\xi^i\Big)^2\geq |\xi|^2.
\end{equation}
On the other hand, $\De_f\leq \be_0$ implies $\prod_{i=1}^n
\mu_i\leq \be_0^2$, with $\mu_1,\cdots,\mu_n$ the eigenvalues of
$(g_{ij})$, thus
\begin{equation}\label{eig2}
\xi^i g_{ij}\xi^j\leq \be_0^2|\xi|^2\leq 9|\xi|^2.
\end{equation}
In local coordinates, the Laplace-Beltrami operator is
$$\De=\f{1}{\sqrt{G}}\f{\p}{\p x^i}\Big(\sqrt{G}g^{ij}\f{\p }{\p x^j}\Big).$$
Here $(g^{ij})$ is the inverse matrix of $(g_{ij})$, and $G=\det(g_{ij})=\De_f^2$. From
(\ref{slope}), (\ref{eig1}) and (\ref{eig2}) it is easily seen that
\begin{equation}\label{uniform}
\f{1}{3}|\xi|^2\leq \be_0^{-1}|\xi|^2\leq \xi^i(\sqrt{G}g^{ij})\xi^j\leq \be_0|\xi|^2\leq 3|\xi|^2.
\end{equation}
Following \cite{j} and \cite{j-x-y} we shall make use of the
following abbreviations: For arbitrary $R\in (0,R_0)$, let
\begin{equation}
B_R=\big\{(x,f(x)):x\in D_R\big\}\subset M.
\end{equation}
And for arbitrary $h\in L^\infty(B_R)$ denote
\begin{equation}
\aligned
&h_{+,R}\mathop{=}\limits^{\text{def.}}\sup_{B_R}h,\qquad
h_{-,R}\mathop{=}\limits^{\text{def.}}=\inf_{B_R}h,\qquad
\bar{h}_R\mathop{=}\limits^{\text{def.}}\aint{B_R}h=\f{\int_{B_R}h*1}{|\text{Vol}(B_R)|}\\
&|\bar{h}|_{p,R}\mathop{=}\limits^{\text{def.}}\Big(\aint{B_R}|h|^p\Big)^{\f{1}{p}}\;
(p\in (-\infty,+\infty).
\endaligned
\end{equation}
(\ref{uniform}) shows that $\De$ is a uniform elliptic operator. Moser's Harnack inequality \cite{m} for supersolutions
of elliptic PDEs in divergence form gives
\begin{lem}\label{Har}
For a positive superharmonic function $h$ on $B_R$ with $R\in
(0,R_0]$, $p\in (0,\f{n}{n-2})$ and $\th\in [\f{1}{2},1)$, we
have the following estimate
$$|\bar{h}|_{p,\th R}\leq \g_1 h_{-,\th R}.$$
Here $\g_1$ is a positive constant only depending on $n$, $p$ and
$\th$, but not on $h$ and $R$.
\end{lem}
(\ref{sub}) shows the subharmonicity of $v$, and therefore $v_{+,R}-v+\ep$ is a positive superharmonic function on $B_R$
for arbitrary $\ep>0$. With the aid of Lemma \ref{Har}, one can follow \cite{j} to get
\bigskip
\begin{cor}\label{c1}
There is a constant $\de_0\in (0,1)$, depending only on $n$, such that
$$v_{+,\f{R}{2}}\leq (1-\de_0)v_{+,R}+\de_0\bar{v}_{\f{R}{2}}.$$
\end{cor}
\bigskip
Denote by $G^\rho$ the mollified Green function for the Laplace-Beltrami operator on
$B_R$. Then for arbitrary $p=(y,f(y))\in B_R$, once
$$B_\rho(p)=\big\{(x,f(x))\in M: x\in D_R(y)\big\}\subset B_R$$
we have
$$\int_{B_R}\n G^\rho(\cdot,p)\cdot \n\phi*1=\aint{B_\rho(p)}\phi$$
for every $\phi\in H_0^{1,2}(B_R)$. The apriori estimates for mollified Green functions
of \cite{g-w} tell us
\bigskip
\begin{lem}\label{l4}
With $o:=(0,f(0))$, we have
\begin{equation}\label{Gr1}\aligned
0\leq G^{\f{R}{2}}(\cdot,o)\leq c_2(n)R^{2-n}&\qquad \text{on }B_R\\
G^{\f{R}{2}}(\cdot,o)\geq c_1(n)R^{2-n}&\qquad \text{on }B_{\f{R}{2}}.
\endaligned
\end{equation}
For arbitrary $p\in B_{\f{R}{4}}$,
\begin{equation}\label{Gr3}
G^\rho(\cdot,p)\leq C(n)R^{2-n}\qquad \text{on }B_R\backslash \bar{B}_{\f{R}{2}}.
\end{equation}
Moreover if $\rho\leq \f{R}{8}$,
\begin{equation}\label{Gr2}
\int_{B_R\backslash \bar{B}_{\f{R}{2}}}\big|\n G^\rho(\cdot,p)\big|^2*1\leq C(n)R^{2-n}.
\end{equation}
\end{lem}
\bigskip
Based on (\ref{weak}), Corollary \ref{c1} and Lemma \ref{l4}, we can
use the method of \cite{j} to derive a telescoping lemma a la
Giaquinta-Giusti \cite{g-g} and Giaquita-Hildebrandt \cite{g-h}.
\bigskip
\begin{thm}\label{thm2}
There exists a positive constant $C_1$, only depending on $n$ and $\be_0$, such that for arbitrary $R\leq R_0$,
\begin{equation}\label{tele}
R^{2-n}\int_{B_{\f{R}{2}}}|B|^2*1\leq C_1(v_{+,R}-v_{+,\f{R}{2}})
\end{equation}
Moreover, there exists a positive constnat $C_2$, only depending
on $n$ and $\be_0$, such that for arbitrary $\ep>0$, we can find $R\in
[\exp(-C_2\ep^{-1})R_0,R_0]$, such that
\begin{equation}
R^{2-n}\int_{B_{\f{R}{2}}}|B|^2*1\leq \ep.
\end{equation}
\end{thm}
\begin{proof} With
$$\om^R=R^{-2}\text{Vol}(B_{\f{R}{2}})G^{\f{R}{2}}(\cdot,o)\qquad \text{where }o=(0,f(0)),$$
then
$$\int_{B_R}\n \om^R\cdot \n \phi*1=R^{-2}\int_{B_{\f{R}{2}}}\phi*1.$$
Choosing $(\om^R)^2\in H_0^{1,2}(B_R)$ as a test function in (\ref{weak}), we obtain
$$\aligned
K_0 \int_{B_R}|B|^2(\om^R)^2*1&\leq -\int_{B_R}\n (\om^R)^2\cdot \n v*1=-2\int_{B_R}\n \om^R\cdot \om^R\n(v-v_{+,R})*1\\
&=-2\int_{B_R}\n \om^R\cdot\big(\n(\om^R(v-v_{+,R}))-(v-v_{+,R})\n \om^R\big)*1\\
&\leq -2\int_{B_R}\n \om^R\cdot \n\big(\om^R(v-v_{+,R})\big)*1\\
&=-2R^{-2}\int_{B_{\f{R}{2}}}\om^R(v-v_{+,R})*1.
\endaligned$$
By (\ref{Gr1}), there exist positive constants $c_3,c_4$, depending only on $n$, such that
$$\aligned
0\leq \om^R\leq c_4 \qquad &\text{on } B_R,\\
\om^R\geq c_3\qquad &\text{on }B_{\f{R}{2}}.
\endaligned$$
Hence
\begin{equation}\label{tele2}\aligned
\int_{B_{\f{R}{2}}}|B|^2*1&\leq -2K_0^{-1}c_4^{-1}c_3^2R^{-2}\int_{B_{\f{R}{2}}}(v-v_{+,R})*1\\
&\leq c_5(n,\be_0)R^{n-2}(v_{+,R}-\bar{v}_{\f{R}{2}}).
\endaligned
\end{equation}
By Corollary \ref{c1}, $v_{+,R}-\bar{v}_{\f{R}{2}}\leq \de_0^{-1}(v_{+,R}-v_{+,\f{R}{2}})$; substituting it into
(\ref{tele2}) yields (\ref{tele}).
For arbitrary $k\in \Bbb{Z}^+$, (\ref{tele}) gives
\begin{equation}\aligned
\sum_{i=0}^k (2^{-i}R_0)^{2-n}\int_{B_{2^{-i-1}R_0}}|B|^2*1&\leq C_1\sum_{i=0}^k(v_{+,2^{-i}R_0}-v_{+,2^{-i-1}R_0})\\
&=C_1(v_{+,R_0}-v_{+,2^{-k-1}R_0})\\
&\leq C_1(\be_0-1)\leq 2C_1
\endaligned
\end{equation}
For arbitrary $\ep>0$, we take
$$k=[2C_1\ep^{-1}],$$
then we can find $1\leq j\leq k$, such that
$$(2^{-j}R_0)^{2-n}\int_{B_{2^{-j-1}R_0}}|B|^2*1\leq \f{2}{k+1}C_1\leq \ep.$$
Since $2^{-j}\geq 2^{-k}\geq 2^{-2C_1\ep^{-1}}=\exp\big[-2(\log 2)C_1\ep^{-1}\big]$,
it is sufficient to choose $C_2=-2(\log 2)C_1$.
\end{proof}
\Section{Gauss image shrinking property}{A Gauss image shrinking
property}
\begin{lem}\label{l3}
For arbitrary $a>1$ and $\be_0\in [1,a)$, there exists a positive
constant $\ep_1=\ep_1(a,\be_0)$ with the following property. If
$P_1,Q\in \grs{n}{m}$ satisfies $v(Q,P_1)\leq b\leq \be_0$, then we can find $P_2\in \grs{n}{m}$, such that
$v(P,P_2)\leq a$ for every $P\in \grs{n}{m}$ satisfying $v(P,P_1)\leq b$, and
\begin{equation}
1\leq v(Q,P_2)\leq \left\{\begin{array}{cc}
1 & \text{if }b<\sqrt{2}(1+a^{-1})^{-\f{1}{2}}\\
b-\ep_1 & \text{otherwise.}
\end{array}\right.
\end{equation}
\end{lem}
\begin{proof}
Obviously $w(P,P)=1$ for every $P\in \grs{n}{m}$, which shows $\grs{n}{m}$ is a submanifold in a Euclidean
sphere via the Pl\"ucker embedding. Denote by $r(\cdot,\cdot)$ the restriction of the spherical distance on $\grs{n}{m}$,
then by spherical geometry, $w=\cos r$ and hence $v=\sec r$.
Denote $\a=\arccos(a^{-1})$ and $\be=\arccos(b^{-1})$. Now we put $\g=\a-\be$ and
\begin{equation}\label{dis}
c=\sec \g=(a^{-1}b^{-1}+(1-a^{-2})^{\f{1}{2}}(1-b^{-2})^{\f{1}{2}})^{-1}.
\end{equation}
Once $v(P_2,P_1)\leq c$, the triangle inequality implies
$$r(P,P_2)\leq r(P,P_1)+r(P_2,P_1)\leq \arccos(b^{-1})+\arccos(c^{-1})=\a$$
for every $P$ satisfying $v(P,P_1)\leq b$, and thus $v(P,P_2)\leq a$
follows.
If $b<\sqrt{2}(1+a^{-1})^{-\f{1}{2}}$, then a direct
calculation shows $\be<\f{\a}{2}$, hence $\g>\be$ and moreover
$v(Q,P_1)\leq b<c$. Thereby $P_2=Q$ is the required point.
Otherwise $b\geq \sqrt{2}(1+a^{-1})^{-\f{1}{2}}$ and hence $c\leq b$. Obviously one of the following two cases
has to occur:
\textit{Case I.} $v(Q,P_1)<c$. One can take $P_2=Q$ to ensure $v(\cdot,P_2)\leq a$ whenever $v(\cdot,P_1)\leq b$. In this case
\begin{equation}\label{below1}
b-v(Q,P_2)=b-1\geq \sqrt{2}(1+a^{-1})^{-\f{1}{2}}-1.
\end{equation}
\textit{Case II.} $v(Q,P_1)\geq c$. Denote by $\th_1,\cdots,\th_m$ the Jordan angles between $Q$ and $P_1$, and put
$L^2=\sum_{1\leq \a\leq m}\th_\a^2$, then as shown in \cite{w}, if we denote the shortest normal geodesic from
$Q$ to $P_1$ by $\g$, then the Jordan angles between $Q$ and $\g(t)$ are $\f{\th_1}{L}t,\cdots,\f{\th_m}{L}t$,
while the Jordan angles between $\g(t)$ and $P_1$ are $\f{\th_1}{L}(L-t),\cdots,\f{\th_m}{L}(L-t)$. Hence
$$\aligned
v(Q,\g(t))&=\prod_\a \sec\big(\f{\th_\a}{L}t\big),\\
v(\g(t),P_1)&=\prod_\a \sec\big(\f{\th_\a}{L}(L-t)\big).
\endaligned$$
Since $t\mapsto \prod_\a \sec\big(\f{\th_\a}{L}(L-t)\big)$ is a strictly decreasing function, there exists
a unique $t_0\in [0,L)$, such that $\prod_\a \sec\big(\f{\th_\a}{L}(L-t_0)\big)=c$. Now we choose
$P_2=\g(t_0)$, then $v(P_2,P_1)=c$ and
\begin{equation}\label{below2}
b-v(Q,P_2)=b-\prod_\a \sec\big(\f{\th_\a}{L}t_0\big).
\end{equation}
It remains to show $b-\prod_\a
\sec\big(\f{\th_\a}{L}t_0\big)$ is bounded from below by a
universal positive constant $\ep_2$. Once this holds true, in conjunction with (\ref{below1}) and (\ref{below2}),
\begin{equation}
\ep_1=\min\{\sqrt{2}(1+a^{-1})^{-\f{1}{2}}-1,\ep_2\}
\end{equation}
is the required constant.
$t_0$ can be regarded as a smooth function on
$$\Om=\big\{(b,\th_1,\cdots,\th_m)\in \R^{m+1},\sqrt{2}(1+a^{-1})^{-\f{1}{2}}\leq b\leq \be_0,
0\leq \th_\a\leq \f{\pi}{2},c\leq \prod_\a \sec(\th_\a)\leq b\big\}$$
which is the unique one satisfying
$$\prod_\a \sec\big(\f{\th_\a}{L}(L-t_0)\big)=c.$$
(By (\ref{dis}), $c$ can be viewed as a function of $b$.) The
smoothness of $t_0$ follows from the implicit function theorem.
Therefore $F:\Om\ra \R$
$$(\th_1,\cdots,\th_m)\mapsto b-\prod_\a \sec\big(\f{\th_\a}{L}t_0\big)$$
is a smooth function on $\Om$. $t_0<L$ implies $F>0$; then the compactness of $\Om$ gives $\inf_\Om F>0$,
and $\ep_2=\inf_\Om F$ is the required constant.
\end{proof}
\begin{rem}
$\ep_1$ is only depending on $a$ and $\be_0$, non-decreasingly
during the iteration process in Theorem \ref{thm4}.
\end{rem}
\begin{thm}\label{thm3}
Let $M=\big\{(x,f(x)):x\in D_{R_0}\subset \R^n\big\}$ be a graph with parallel mean curvature, and $\De_f\leq \be_0$
with $\be_0\in [1,3)$. Assume there exists
$P_0\in \grs{n}{m}$, such that $v(\cdot,P_0)\circ \g\leq b$ on $M$ with $1\leq b\leq \be_0$.
If $b<\f{\sqrt{6}}{2}$, then for arbitrary $\ep>0$, one can find a constant $\de\in (0,1)$ depending only on
$n$, $\be_0$ and $\ep$
such that
\begin{equation}\label{es3}
1\leq v(\cdot,P_1)\circ \g\leq 1+\ep\qquad \text{on }B_{\de R_0}
\end{equation}
for a point $P_1\in \grs{n}{m}$.
If $b\geq \f{\sqrt{6}}{2}$, then there are two constants $\de_0\in (0,1)$ and $\ep_1>0$, only depending on $n$ and $\be_0$, such that
\begin{equation}\label{es4}
1\leq v(\cdot,P_1)\circ \g\leq b-\f{\ep_1}{2}\qquad \text{on }B_{\de_0R_0}
\end{equation}
for a point $P_1\in \grs{n}{m}$.
\end{thm}
\begin{proof}
Let $H$ be a smooth function on $\grs{n}{m}$, then $h=H\circ \g$ gives a smooth function on $M$. Let $\eta$ be a
nonnegative smooth function on $M$ with compact support and $\varphi$ be a $H^{1,2}$-function on $M$, then
by Stokes' Theorem,
$$\aligned
0&=\int_M \text{div}(\varphi\eta\n h)*1\\
&=\int_M \varphi\n\eta\cdot \n h*1+\int_M \eta\n\varphi\cdot \n h*1+\int_M \varphi \eta\De h*1\\
&=\int_M \varphi\n \eta\cdot \n h*1+\int_M\n\varphi\cdot \n(\eta h)*1-\int_M h\n \varphi\cdot \n\eta*1+\int_M \varphi\eta \De h*1.
\endaligned$$
Hence
\begin{equation}\label{sh0}
\int_M \n\varphi\cdot \n(\eta h)*1=-\int_M \varphi\n\eta\cdot \n h*1+\int_M h\n\varphi\cdot \n\eta*1-\int_M \varphi\eta\De h*1.
\end{equation}
For arbitrary $R\leq R_0$, we take a cut-off function $\eta$ supported in the interior of
$B_R$, $0\leq \eta\leq 1$, $\eta\equiv 1$ on $B_{\f{R}{2}}$ and $|\n \eta|\leq c_0 R^{-1}$. For every
$\rho\leq \f{R}{8}$, denote by $G^\rho$ the mollified Green function on $B_R$. For arbitrary $p\in B_{\f{R}{4}}$,
inserting $\varphi=G^\rho(\cdot,p)$ into (\ref{sh0}) gives
\begin{equation}\label{sh}\aligned
&\int_{B_R}\n G^\rho(\cdot,p)\cdot \n(\eta h)*1\\
=&-\int_{B_R}G^\rho(\cdot,p)\n\eta\cdot \n h*1+\int_{B_R}h\n G^\rho(\cdot,p)\cdot \n\eta*1
-\int_{B_R}G^\rho(\cdot,p)\eta\De h*1.\endaligned
\end{equation}
We write (\ref{sh}) as
$$I_\rho=II_\rho+III_\rho+IV_\rho.$$
By the definition of mollified Green functions,
\begin{equation}\label{sh11}
I_\rho=\aint{B_\rho(p)}\eta h=\aint{B_\rho(p)}h.
\end{equation}
Hence
\begin{equation}\label{sh1}
\lim_{\rho\ra 0^+}I_\rho=h(p).
\end{equation}
Noting that $|d\g|^2=|B|^2$, we have $|\n h|\leq |\n^G H||d\g|=|\n^G H||B|$.
Here and in the sequel, $\n^G$ denotes the Levi-Civita connection on $\grs{n}{m}$. In
conjunction with (\ref{Gr3}), we have
\begin{equation}\label{sh2}\aligned
|II_\rho|&\leq \int_{T_R}G^\rho(\cdot,p)|\n \eta||\n h|*1\\
&\leq \sup_{T_R}G^\rho(\cdot,p)\sup_{T_R}|\n \eta|\sup_{\Bbb{V}}|\n^G H|\int_{B_R}|B|*1\\
&\leq C(n)R^{1-n}\sup_{\Bbb{V}}|\n^G H|\Big(\int_{B_R}|B|^2*1\Big)^{\f{1}{2}}\text{Vol}(B_R)^{\f{1}{2}}\\
&\leq c_1(n)\sup_{\Bbb{V}}|\n^G H|\Big(R^{2-n}\int_{B_R}|B|^2*1\Big)^{\f{1}{2}}.
\endaligned
\end{equation}
Here $T_R\mathop{=}\limits^{\text{def.}}B_R\backslash \bar{B}_{\f{R}{2}}$ and
$$\Bbb{V}=\{P\in \grs{n}{m}: v(P,P_0)\leq 3\},$$
which is a compact subset of $\Bbb{U}$.
As shown in Section \ref{s1}, there is a one-to-one
correspondence between the points in $\Bbb{U}$ and the
$n\times m$-matrices. And each $n\times m$-matrix can be viewed
as a corresponding vector in $\R^{nm}$. Define $T:\Bbb{U}\ra
\R^{nm}$
$$Z\mapsto \big(\det(I+ZZ^T)^\f{1}{2}-1\big)\f{Z}{\big(\tr(ZZ^T)\big)^{\f{1}{2}}}$$
Note that
$\big(\tr(ZZ^T)\big)^{\f{1}{2}}=(\sum_{i,\a}Z_{i\a}^2)^{\f{1}{2}}$
equals $|Z|$ when $Z$ is treated as a vector in $\R^{nm}$. Since
$t\in [0,+\infty)\mapsto
\left[\det\big(I+(tZ)(tZ)^T\big)\right]^{\f{1}{2}}$ is a strictly
increasing function and maps $[0,+\infty)$ onto $[1,+\infty)$,
$T$ is a diffeomorphism. By (\ref{v}), $|T(Z)|=v(P,P_0)-1$. Via
$T$, we can define the mean value of $\g$ on $B_R$ by
\begin{equation}
\bar\g_R=T^{-1}\Big[\f{\int_{B_R}(T\circ
\g)*1}{\text{Vol}(B_R)}\Big].
\end{equation}
Note that $T$ maps sublevel sets of $v(\cdot,P_0)$ onto Euclidean balls centered at the origin.
Hence the convexity of Euclidean balls gives
\begin{equation}
v(\bar{\g}_R,P_0)\leq \sup_{B_R}v(\cdot,P_0)\circ \g\leq b.
\end{equation}
The compactness of $\Bbb{V}$ ensures the existence of positive constants $K_1$ and $K_2$, such that for arbitrary
$X\in T\Bbb{V}$,
$$K_1|X|\leq |T_* X|\leq K_2|X|.$$
The classical Neumann-Poincar\'{e} inequality says
$$\int_{D_R}|\phi-\bar{\phi}|^2\leq C(n)R^2\int_{D_R}|D \phi|^2.$$
As shown above, $B_R$ can be regarded as $D_R$ equipped with the metric $g=g_{ij}dx^idx^j$, and the eigenvalues
of $(g_{ij})$ are bounded. Hence it is easy to get
$$\int_{B_R}|\phi-\bar{\phi}|^2*1\leq C(n)R^2\int_{B_R}|\n \phi|^2*1.$$
Here $\phi$ can be a vector-valued function.
Denote by $d_G$ the distance function on $\grs{n}{m}.$ Then, by
using the above Neumann-Poincar\`{e} inequality we have
\begin{equation}\label{poin}\aligned
\int_{B_R}d_G^2(\g,\bar\g_R)*1&\leq K_1^{-2}\int_{B_R}\big|T\circ \g-T(\bar{\g}_R)\big|^2*1\\
&\leq C(n)K_1^{-2}R^2\int_{B_R}\big|d(T\circ \g)\big|^2*1\\
&\leq C(n)K_1^{-2}K_2^2R^2\int_{B_R}|d\g|^2*1\\
&= C(n)K_1^{-2}K_2^2R^2\int_{B_R}|B|^2*1.
\endaligned
\end{equation}
Now we write
$$h=H\circ \g=H(\bar{\g}_R)+\big(H\circ \g-H(\bar{\g}_R)\big),$$
then
\begin{equation}\label{sh3}
III_\rho=H(\bar{\g}_R)\int_{B_R}\n G^\rho(\cdot,p)\cdot
\n\eta*1+\int_{T_R}\big(H\circ \g-H(\bar{\g}_R)\big)\n
G^\rho(\cdot,p)\cdot \n \eta*1.
\end{equation}
Similar to (\ref{sh11}),
\begin{equation}\label{sh31}
\lim_{\rho\to 0^+}H(\bar{\g}_R)\int_{B_R}\n G^\rho(\cdot,p)\cdot
\n \eta*1=\lim_{\rho\to
0^+}H(\bar{\g}_R)\aint{B_\rho(p)}\eta=H(\bar\g_R).
\end{equation}
The second term can be controlled by
\begin{equation}\label{sh321}\aligned
&\int_{T_R}\big(H\circ \g-H(\bar{\g}_R)\big)\n G^\rho(\cdot,p)\cdot \n \eta*1\\
\leq&\sup_{\Bbb{V}}|\n^G H|\sup_{T_R}|\n \eta|\int_{T_R}d_G(\g,\bar{\g}_R)|\n G^\rho(\cdot,p)|*1\\
\leq&c_0R^{-1}\sup_{\Bbb{V}}|\n^G H|\Big(\int_{B_R}d_G^2(\g,\bar{\g}_R)\Big)^{\f{1}{2}}
\Big(\int_{T_R}|\n G^\rho(\cdot,p)|^2*1\Big)^{\f{1}{2}}
\endaligned
\end{equation}
Substituting (\ref{Gr2}) and
(\ref{poin}) into (\ref{sh321}) yields
\begin{equation}\label{sh32}
\int_{T_R}\big(H\circ \g-H(\bar{\g}_R)\big)\n
G^\rho(\cdot,p)\cdot \n \eta*1\leq c_2(n)\sup_{\Bbb{V}}|\n^G
H|\Big(R^{2-n}\int_{B_R}|B|^2*1\Big)^{\f{1}{2}}.
\end{equation}
From (\ref{sh}), (\ref{sh1}), (\ref{sh2}), (\ref{sh3}), (\ref{sh31}) and (\ref{sh32}), letting
$\rho\ra 0$ we arrive at
\begin{equation}\label{sh4}
\aligned
h(p)\leq &H(\bar{\g}_R)+c_3(n)\sup_{\Bbb{V}}|\n^G H|\Big(R^{2-n}\int_{B_R}|B|^2*1\Big)^{\f{1}{2}}\\
&-\limsup_{\rho\ra 0^+}\int_{B_R} G^\rho(\cdot,p)\eta\De h*1.
\endaligned
\end{equation}
for every $p\in B_{\f{R}{4}}$.
The compactness of $\grs{n}{m}$ implies the existence of a positive constant $K_3$, such that
\begin{equation}\label{sh51}
\big|\n^Gv(\cdot,P)\big|\leq K_3\qquad \text{whenever }1\leq v(\cdot,P)\leq 3
\end{equation}
for arbitrary $P\in \grs{n}{m}$. Hence by
inserting $H=v(\cdot,P)$ into (\ref{sh4}) one can obtain
\begin{equation}\label{sh6}
\aligned
v(\g(p),P)\leq &v(\bar{\g}_R,P)+c_3K_3\Big(R^{2-n}\int_{B_R}|B|^2*1\Big)^{\f{1}{2}}\\
&-\limsup_{\rho\ra 0^+}\int_{B_R} G^\rho(\cdot,p)\eta\De \big(v(\cdot,P)\circ \g\big)*1.
\endaligned
\end{equation}
By Lemma \ref{l3}, if we put $P_1=\bar{\g}_R$, then $1\leq v(\cdot,P_1)\leq 3$
whenever $1\leq v(\cdot,P_0))\leq b$ provided that $b<\f{\sqrt{6}}{2}$, which implies $v(\cdot,\bar{\g}_R)\circ \g$ is a subharmonic function
on $B_R$. Letting $P=\bar{\g}_R$ in (\ref{sh6}) yields
\begin{equation}\label{sh7}
v(\g(p),\bar{\g}_R)\leq 1+c_3K_3\Big(R^{2-n}\int_{B_R}|B|^2*1\Big)^{\f{1}{2}}
\end{equation}
for all $p\in B_{\f{R}{4}}$. By Theorem \ref{thm2}, for every $\ep>0$, there is $\de\in (0,1)$, depending only
on $n,\be_0$ and $\ep$, such that
\begin{equation}\label{sh8}
R^{2-n}\int_{B_R}|B|^2*1\leq c_3^{-2}K_3^{-2}\ep^2
\end{equation}
for some $R\in [4\de R_0,R_0]$. Substituting (\ref{sh8}) into (\ref{sh7}) gives (\ref{es3}).
If $b\geq \f{\sqrt{6}}{2}$, we put $\ep_1=\ep_1(3,\be_0)$ as given in Lemma \ref{l3}. Then Theorem \ref{thm2} enables us
to find $R\in [4\de_0R_0,R_0]$ such that
\begin{equation}\label{sh53}
R^{2-n}\int_{B_R}|B|^2*1\leq \f{1}{4}c_3^{-2}K_3^{-2}\ep_1^2,
\end{equation}
where $\de_0$ only depends on $n$ and $\be_0$.
Applying Lemma \ref{l3}, one can find $P_1\in \grs{n}{m}$, such that
\begin{equation}\label{sh52}
v(\bar{\g}_R,P_1)\leq b-\ep_1
\end{equation}
and $1\leq v(\cdot,P_1)\leq 3$ whenever $1\leq v(\cdot,P_0)\leq
b$. Theorem \ref{thm1} ensures that $v(\cdot,P_1)\circ \g$ is a
subharmonic function on $B_R$. Taking $P=P_1$
in (\ref{sh6}) yields
\begin{equation}
v\big(\g(p),P_1\big)\leq v(\bar{\g}_R,P_1)+c_3K_3(\f{1}{4}c_3^{-2}K_3^{-1}\ep_1^2)^{\f{1}{2}}\leq
b-\f{\ep_1}{2}
\end{equation}
for all $p\in B_{\f{R}{4}}$. Here we have used (\ref{sh53}) and (\ref{sh52}). From the above inequality (\ref{es4}) immediately follows.
\end{proof}
\Section{Bernstein type results}{Bernstein type results}
Now we can start an iteration as in \cite{h-j-w} and \cite{g-j} to get the following estimates:
\bigskip
\begin{thm}\label{thm4}
Let $M=\big\{(x,f(x)):x\in D_{R_0}\subset \R^n\big\}$ be a graph with parallel mean curvature, and $\De_f\leq \be_0$
with $\be_0\in [1,3)$, then for arbitrary $\ep>0$, there exists $\de\in (0,1)$, only depending on $n$, $\be_0$ and $\ep$,
not depending on $f$ and $R_0$, such that
$$1\leq v(\cdot,\g(o))\circ \g\leq 1+\ep\qquad \text{on }B_{\de R_0},$$
where $o=(0,f(0))$. In particular, if $|Df|(0)=0$, then
$$\De_f\leq 1+\ep\qquad \text{on }D_{\de R_0}.$$
\end{thm}
\begin{proof}
Let $\{\eps_1,\cdots,\eps_{n+m}\}$ be canonical orthonormal basis of $\R^{n+m}$ and put $P_0=\eps_1\w\cdots\w\eps_n$.
Then $\De_f\leq \be_0$ implies $v(\cdot,P_0)\leq \be_0$ on $B_{R_0}$. If $\be_0<\f{\sqrt{6}}{2}$,
we put $Q_0=P_0$. Otherwise by Theorem \ref{thm3}, one can find $P_1\in \grs{n}{m}$, such that
\begin{equation}
v(\cdot,P_1)\circ \g \leq \be_0-\ep_1\qquad \text{on }B_{\de_0 R_0}
\end{equation}
with constants $\de_0$ and $\ep_1$ depending only on $n$ and $\be_0$. Similarly for each $j\geq 1$, if
$\be_0-j\ep_1<\f{\sqrt{6}}{2}$, then we put $Q_0=P_j$; otherwise Theorem \ref{thm3} enables us to
find $P_{j+1}\in \grs{n}{m}$ satisfying
\begin{equation}
v(\cdot,P_{j+1})\circ \g\leq \be_0-(j+1)\ep_1\qquad \text{on }B_{\de_0^{j+1}R_0}.
\end{equation}
Denoting
$$k=\big[(3-\f{\sqrt{6}}{2})\ep_1^{-1}\big]+1,$$
then obviously $\be_0-k\ep_1<\f{\sqrt{6}}{2}$. Hence there exists $Q_0\in \grs{n}{m}$, such that
\begin{equation}
v(\cdot,Q_0)\circ \g\leq b<\f{\sqrt{6}}{2}\qquad \text{on }B_{\de_0^kR_0}.
\end{equation}
Again using Theorem \ref{thm3}, for arbitrary $\ep>0$, there exists $\de_1\in (0,1)$, depending only on $n,\be_0$ and $\ep$, such that
\begin{equation}
v(\cdot,Q_1)\circ \g\leq \sqrt{2}(1+(1+\ep)^{-1})^{-\f{1}{2}}\qquad \text{on }B_{\de_1\de_0^kR_0}
\end{equation}
for a point $Q_1\in \grs{n}{m}$. With $r(\cdot,\cdot)$ as in the proof of Lemma \ref{l3}, then
$$r(\cdot,Q_1)\circ \g=\arccos v(\cdot,Q_1)^{-1}\circ \g\leq \f{1}{2}\arccos (1+\ep)^{-1}.$$
Using the triangle inequality we get
$$r(\cdot,\g(0))\circ \g\leq r(\cdot,Q_1)\circ \g+r(\g(0),Q_1)\circ \g\leq \arccos(1+\ep)^{-1}.$$
Thus $v(\cdot,\g(0))\circ \g\leq 1+\ep$ on $B_{\de_1\de_0^k R_0}$. It is sufficient to put $\de=\de_1\de_0^k$.
\end{proof}
Letting $R_0\ra +\infty$ we can arrive at a Bernstein-type theorem:
\bigskip
\begin{thm}\label{thm5}
Let $z^\a=f^\a(x^1,\cdots,x^n),\ \a=1,\cdots,m$, be smooth functions
defined everywhere in $\R^n$ ($n\geq 3,m\geq 2$). Suppose their graph $M=(x,f(x))$ is a
submanifold with parallel mean curvature in $\R^{n+m}$. Suppose that
there exists a number $\be_0<3$
with
\begin{equation}
\De_f=\Big[\det\Big(\de_{ij}+\sum_\a \f{\p f^\a}{\p x^i}\f{\p f^\a}{\p x^j}\Big)\Big]^{\f{1}{2}}\leq \be_0.\label{be2}
\end{equation}
Then $f^1,\cdots,f^m$ has to be affine linear (representing an affine $n$-plane).
\end{thm}
\noindent{\bf Final remarks}
\medskip
For any $P_0\in\grs{n}{m}$, denote by $r$ the distance function
from $P_0$ in $\grs{n}{m}$. The eigenvalues of $\Hess(r)$ were
computed in \cite{j-x}. Then define
\begin{equation*}
B_{JX}(P_0)=\big\{P\in \grs{n}{m}:\mbox{ sum of any two Jordan
angles between }P\mbox{ and }P_0<\f{\pi}{2}\big\}
\end{equation*} in the geodesic polar coordinate neighborhood around $P_0$ on the Grassmann
manifold. From (3.2), (3.7) and (3.9) in \cite{j-x} it turns out
that $\Hess(r)>0$ on $B_{JX}(P_0).$ Moreover, let $\Si\subset
B_{JX}(P_0)$ be a closed subset, then $\th_\a+\th_\be\le \be_0
<\f{\pi}2$ and
$$\Hess(r)\ge \cot\be_0\ g,$$
where $g$ is the metric tensor on $\grs{n}{m}.$ Hence, the
composition of the distance function with the Gauss map is a
strongly subharmonic function on $M$, provided the Gauss image of
the submanifold $M$ with parallel mean curvature in $\ir{n+m}$ is
contained in $\Si$. The largest sub-level set of $v(\cdot, P_0)$
in $B_{JX}(P_0)$ were studied in \cite{j-x}. The Theorem 3.2 in
\cite{j-x} shows that
$$\max\{w(P, P_0);\; P\in\p B_{JX}(P_0)\}=\f{1}{2}.$$ Therefore,
$$\{P\in \grs{n}{m},\; v(\cdot, P_0)<2\}\subset B_{JX}(P_0),$$ and
$$\{P\in \grs{n}{m};\; v(\cdot, P_0)=
2\}\bigcap\p B_{JX}(P_0)\neq\emptyset.$$
On the other hand, we can compute directly. From (\ref{He}) we
also have
$$\aligned
\Hess(v(\cdot,P_0))&=\sum_{m+1\leq i\leq n,\a}v\
\om_{i\a}^2+\sum_{\a}(1+2\la_\a^2)v\ \om_{\a\a}^2
+\sum_{\a\neq \be}\la_\a\la_\be v\ \om_{\a\a}\otimes\om_{\be\be}\\
&\qquad\qquad+\sum_{\a<\be}\Big[(1+\la_\a\la_\be)v\Big(\f{\sqrt{2}}{2}(\om_{\a\be}
+\om_{\be\a})\Big)^2\\
&\hskip2in+(1-\la_\a\la_\be)v\Big(\f{\sqrt{2}}{2}(\om_{\a\be}-\om_{\be\a})\Big)^2\Big].
\endaligned
$$
It follows that $v(\cdot, P_0)$ is strictly convex on
$B_{JX}(P_0).$ Moreover, if $\th_\a+\th_\be\le
\be_0<\f{\pi}{2},$ then
$$\Hess (v(\cdot, P_0))\ge (1-\tan\th_\a\tan\th_\be)v\ g=\f{\cos(\th_\a+\th_\be)}{\cos\th_\a\cos\th_\be}v\,g
\ge\cos\be_0 v\,g$$ where $g$ is the metric tensor of
$\grs{n}{m}$ and
$$\De v(\g(\cdot), P_0)\ge \cos\be_0 v|B|^2\ge \cos\be_0 |B|^2.$$
Now, we define
$$\Sigma(P_0)=B_{JX}(P_0)\bigcup\{P\in \grs{n}{m};\;
v(\cdot,P_0)<3\}\subset\grs{n}{m}.$$ The function $v(\cdot, P_0)$
is not convex on all of $\Si(P_0)$. But, its precomposition with the
Gauss map could be a strongly subharmonic function on $M$ under
suitable conditions.
Therefore, we could obtain a more general result: Let $M$ be a
complete submanifold in $\ir{n+m}$ with parallel mean curvature.
If its image under the Gauss map is contained in a closed subset
of $\Sigma(P_0)$ for some $P_0\in \grs{n}{m}$, then $M$ has to be
an affine linear subspace.
\bigskip\bigskip
\bibliographystyle{amsplain}
|
{
"timestamp": "2010-09-21T02:04:06",
"yymm": "1009",
"arxiv_id": "1009.3901",
"language": "en",
"url": "https://arxiv.org/abs/1009.3901"
}
|
\section{Introduction}
A \emph{hyperbolic surface group} is the fundamental group of a closed surface with negative Euler characteristic.
We will denote by $F_n$ the free group of rank $n$ with a fixed basis $\mathcal{A}_n=\{a_1,\ldots,a_n\}$.
A \emph{double of a free group}
is the fundamental group of a graph of groups where there are two free vertex groups and
at least one infinite cyclic edge group; here, each edge group is amalgamated along the copies of some word in the free group
(Figure~\ref{fig:double}).
If $U$ is a list of words in $F_n$,
we denote by $D(U)$ the double of $F_n$ where a cyclic edge group is glued along the copies of each word in $U$.
We study the existence of hyperbolic surface subgroups
in doubles of free groups. This is motivated by the following remarkable question due to Gromov.
\begin{QUE}[{Gromov~\cite[p. 277]{Gromov1993}}]\label{que:gromov}
Does every one-ended word-hyperbolic group have a hyperbolic surface subgroup?
\end{QUE}
Question~\ref{que:gromov} has been answered affirmatively for the following cases.
\begin{enumerate}
\item
Coxeter groups~\cite{GLR2004}.
\item\label{result:calegari} Graphs of free groups with infinite cyclic edge groups with nontrivial second rational homology~\cite{Calegari2009}.
\item
The fundamental groups of closed hyperbolic $3$-manifolds~\cite{KM2009}.
\end{enumerate}
A basic, but still captivating case is when the group is given as a double of a free group.
Using (2), Gordon and Wilton~\cite{GW2010} were able to construct explicit families of examples of
doubles that contain hyperbolic surface groups; they showed that those families virtually have nontrivial second rational homology.
The existence of a hyperbolic surface subgroup is not known for
doubles with trivial virtual second rational homology. This leads us
to the next question.
\begin{QUE}\label{que:double}
Does every one-ended double of a nonabelian free group have a hyperbolic surface subgroup?
\end{QUE}
Our first main result resolves Question~\ref{que:double} for rank-two case:
\begin{THM}\label{thm:main 1}
A double of a rank-two free group is one-ended if and only if it has a hyperbolic surface subgroup.
\end{THM}
Our second main result on Question~\ref{que:double}
is on the free groups in which every generator appears the same number
of times in the amalgamating words.
More precisely,
let $U$ be a list of words in $F_n$.
When approaching Question~\ref{que:double} for $D(U)$,
one can always assume that $U$ is \emph{minimal} in the sense that no automorphism of $F_n$ reduces the sum of the lengths of the words in $U$. This is because the isomorphism type of $D(U)$ is invariant under the automorphisms of $F_n$.
We say $U$ is \emph{$k$-regular} if each generator in $\mathcal{A}_n$ appears exactly $k$ times in $U$. Our second main result answers Question~\ref{que:double} affirmatively for a minimal, $k$-regular list of words.
\begin{THM}\label{thm:main 2}
Suppose $U$ is a minimal, $k$-regular list of words in $F_n$ when $n>1$.
If $D(U)$ is one-ended, then $D(U)$ contains a hyperbolic surface group.
\end{THM}
Here is an overview of our proof.
We first explain why Tiling Conjecture~\cite{KW2009,Kim2009} implies
an affirmative answer for Question~\ref{que:double} in Section~\ref{sec:polygonal}.
And then we reformulate Tiling Conjecture into a purely graph theoretic conjecture in Section~\ref{sec:whitehead}.
We resolve this graph theoretic conjecture in two special cases.
In Section~\ref{sec:regular}, we prove it for regular graphs and
deduce Theorem~\ref{thm:main 2}. Here we use the characterization of
perfect matching polytopes of graphs by Edmonds \cite{Edmonds1965a}.
In Section~\ref{sec:4vertex}, we prove it for 4-vertex graphs and
deduce Theorem~\ref{thm:main 1}.
\subsection*{Acknowledgement.}
Authors would like to thank Daniel Kr\'al' for pointing out Example~\ref{ex:connectivity}.
\section{Polygonality and Doubles of Free Groups}\label{sec:polygonal}
Kim and Wilton~\cite{KW2009} proved that the double along a \emph{polygonal} word contains a hyperbolic surface group (Theorem~\ref{thm:polygonal}).
Their proof relied on the subgroup separability of free groups and the normal form theorem for graphs of groups.
In this section, we give a self-contained geometric proof. Then we describe Tiling Conjecture and its implication.
\subsection{Basic definitions and notations}\label{subsec:defn}
Each word in $F_n$ can be written as $w=x_1 x_2\cdots x_l$ where $x_i\in \mathcal{A}_n\cup \mathcal{A}_n^{-1}$; each $x_i$ is called as a \emph{letter} of $w$, and the subscript of $x_i$ is taken modulo $l$.
We say that $w$ is \emph{cyclically reduced} if $x_{i+1}\ne x_i^{-1}$ for each $i=1,2,\ldots,l$.
With respect to the given basis $\mathcal{A}_n$, we denote the Cayley graph of $F_n$ by $\cay(F_n)$.
There is a natural free action of $F_n$ on $\cay(F_n)$, so that $\cay(F_n)/F_n$ is a bouquet of circles.
Let $\alpha_1,\ldots,\alpha_n$ denote the oriented circles in $\cay(F_n)/F_n$ corresponding to $a_1,\ldots,a_n$.
The loop obtained by a concatenation
$\alpha_i^p\alpha_j^q\cdots \alpha_k^r$ where $p,q,\ldots,r\in \Z$ is said to
\emph{read} the word $a_i^p a_j^q\cdots a_k^r$.
Given a list $U$ of nontrivial words $u_1,u_2,\ldots,u_r$ in $F_n$,
take two copies $\Gamma$ and $\Gamma'$ of $\cay(F_n)/F_n$.
To $\Gamma$ and $\Gamma'$, we glue a cylinder along the copies of the closed curve reading $u_i$, for each $i$.
Let $X(U)$ be the resulting space and let $D(U)=\pi_1(X(U))$ be the fundamental group of $X(U)$; see Figure~\ref{fig:double}.
In the literature, $D(U)$ is called a \emph{double of $F_n$ along $U$}, or simply a \emph{double}~\cite{BFMT2009}.
If we let $\mathcal{B}_n$ and $V=\{v_1,\ldots,v_r\}$ denote the copies of $\mathcal{A}_n$ and $U$ respectively, then a presentation of $D(U)$ is given as:
\[D(U) \cong \form{\mathcal{A}_n, \mathcal{B}_n,t_2,t_3,\ldots,t_r
\mid u_1=v_1, u_i^{t_i}=v_i\mbox{ for }i=2,\ldots,r}.\]
Since the isomorphism type of $D(U)$ does not change if some words in $U$ are replaced by their conjugates, we may always assume that every word in $U$ is cyclically reduced.
\begin{figure}[tb]%
\includegraphics[width=.6\textheight]{figDU}
\caption{A construction of $X(U)$, where $\pi_1(X(U))=D(U)$.
}
\label{fig:double}\end{figure}
\subsection{Non-positively curved cubical complexes}\label{subsec:npc}
We briefly summarize elementary facts on $\catz$-spaces; a standard reference for this subject is~\cite{Bridson1999}.
We denote by $\E^2$ the Euclidean plane.
Let $X$ be a geodesic metric space.
For a geodesic triangle $\Delta\subseteq X$,
there is a geodesic triangle $\Delta'\subseteq \E^2$ of the same side-lengths
and a length-preserving map $f\co \Delta\to\Delta'$.
We say that $X$ is a \emph{$\catz$-space} if $d_X(x,x')\le d_{\E^2}(f(x),f(x'))$ for every choice of $\Delta$, $f$ and $x,x'\in\Delta$.
A metric space $X$ is \emph{non-positively curved} if each point in
$X$ has a neighborhood which is a $\catz$-space.
We will need the following.
\begin{PROP}[see~{\cite[p.~201]{Bridson1999}}]\label{prop:pi1inj}
Let $X$ and $Y$ be complete geodesic spaces.
If $X$ is non-positively curved and $f\co Y\to X$ is locally an isometric embedding,
then $Y$ is non-positively curved and $f_*\co \pi_1(Y)\to\pi_1(X)$ is injective.
\end{PROP}
Let $I$ denote the unit interval.
A \emph{cube complex} is a piecewise-Euclidean cell complex $X$ inductively defined as follows: for all $k$, the $k$-skeleton $X^{(k)}$ is obtained from $X^{(k-1)}$ by attaching $k$-dimensional unit cubes $I^k$ such that the restriction of each attaching map to a $(k-1)$-face of $I^k$ is a $(k-1)$-dimensional attaching map.
If $X=X^{(2)}$, we say that $X$ is a \emph{square} complex.
A finite-dimensional cube complex is known to be a complete geodesic metric space~\cite{Bridson1991}.
For a cube complex $X$ and $v\in X^{(0)}$, $\link_X(x)$ is defined to be the set of unit vectors from $v$ toward $X$;
in particular, a link is naturally equipped with a piecewise-spherical metric.
We will only consider cube complexes that are finite-dimensional and locally compact.
Moreover, we always assume that given cube complexes are \emph{simple}, in the sense that no vertex has a link containing a bigon; hence, each link will be a simplicial complex~\cite{HW2008}.
A simplicial complex $L$ is a \emph{flag complex} if every complete subgraph of $L^{(1)}$ is the $1$-skeleton of some simplex in $L$.
Gromov gave a combinatorial formulation of non-positive curvature for a cube complex.
\begin{PROP}[Gromov~\cite{Gromov1987}]\label{prop:linkcondition}
A cube complex $X$ is non-positively curved if and only if the link of each vertex is a flag complex.
\end{PROP}
Recall that for a simplicial complex $L$ and a set of vertices $S$ in $L$,
a \emph{full subcomplex} $L'$ on $S$ is the maximal subcomplex of $L$ whose vertex set is $S$.
A map $f\co Y\to X$ between cube complexes is \emph{cubical} if $f$ maps each cube to a cube of the same dimension.
Locally an isometric cubical map has a combinatorial characterization as follows.
\begin{PROP}[\cite{Charney2000,CW2004}]\label{prop:localisom}
Let $X$ and $Y$ be cube complexes and $f\co Y\to X$ be a cubical map.
Then $f$ is locally an isometric embedding if the following are true for each vertex $y\in Y^{(0)}$.
\begin{enumerate}[(i)]
\item The induced map on the links $\link(f;y)\co\link_Y(y)\to\link_X(f(y))$ is injective.
\item The image of $\link(f;y)$ is a full subcomplex of $\link_X(f(y))$.
\end{enumerate}
\end{PROP}
\subsection{Polygonality}\label{subsec:polygonal}
We let $U$ be a list of cyclically reduced words $u_1,\ldots,u_r$ in $F_n$.
For a word $w = x_1x_2\ldots x_l\in F_n$,
$x_1x_2,x_2x_3,\ldots,x_{l-1}x_l,x_lx_1$
are called \emph{length-$2$ cyclic subwords} of $w$.
The \emph{Whitehead graph $W(U)$ of $U$} is constructed as follows~\cite{Whitehead1936}:
\begin{enumerate}[(i)]
\item
the vertex set of $W(U)$ is $\mathcal{A}_n\cup\mathcal{A}_n^{-1}$;
\item
For each length-$2$ cyclic subword $xy$ of a word in $U$,
we add an edge joining $x$ and $y^{-1}$ to $W(U)$.
\end{enumerate}
A \emph{polygonal disk} means a topological $2$-disk $P$ equipped with a graph structure on the boundary $\partial P\approx S^1$.
We let $Z(U)$ denote the presentation $2$-complex of $F_n/\fform{U}$. This means, $Z(U)$ is obtained from its 1-skeleton $\cay(F_n)/F_n$ by attaching a polygonal disk $D_i$ along the loop reading $u_i$ for each $i=1,2,\ldots,r$. Here, $\partial D_i$ is regarded as a $\vform{u_i}$-gon.
Let $\alpha_j$ denote the oriented loop in $Z(U)^{(1)}=\cay(F_n)/F_n$ reading $a_j$.
The link of the unique vertex in $Z(U)$ is seen to be the Whitehead graph of $U$, by identifying the incoming (outgoing, respectively) portion of $\alpha_j$ with the vertex $a_j$ ($a_j^{-1}$, respectively) in $W(U)$.
Let us fix a point $d_i$ in the interior of $D_i$ and triangulate $D_i$ so that each triangle contains $d_i$ and one edge of $\partial D_i$.
Remove a small open neighborhood of $d_i$ for each $i$, to get a square complex $Z'$; see Figure~\ref{fig:zprime} (a).
We obtain a square complex structure on $X(U)$ by taking two copies of $Z'$ and gluing the circles corresponding to the boundary of the neighborhood of each $d_i$.
The unique vertex of $Z(U)$ gives two special vertices of $X(U)$.
Note that the link of each special vertex is the barycentric subdivision $W(U)'$ of $W(U)$.
Since $W(U)$ has no loops, $W(U)'$ is a bipartite graph without parallel edges.
It follows from Proposition~\ref{prop:linkcondition} that $X(U)$ is non-positively curved.
\begin{figure}[t]%
\tikzstyle {a}=[red,postaction=decorate,decoration={%
markings,%
mark=at position .5 with {\arrow[red]{stealth};}}]
\tikzstyle {b}=[blue,postaction=decorate,decoration={%
markings,%
mark=at position .43 with {\arrow[blue]{stealth};},%
mark=at position .57 with {\arrow[blue]{stealth};}}]
\tikzstyle {v}=[draw,circle,fill=black,inner sep=0pt]
\subfloat[(a) $Z'$]{\begin{tikzpicture}[thick]
\foreach \i in {1,2,4}
\draw [a] (360/5*\i+90:1.5)--(360/5*\i+360/5+90:1.5);
\draw [b] (90:1.5)--(90+360/5:1.5);
\draw [b] (360/5*4+90:1.5)--(90+360/5*3:1.5);
\foreach \i in {0,...,4} {
\node [circle,draw,inner sep=10pt] at (0,0) (c) {};
\draw (c)-- (360/5*\i+90:1.5) node [v] {} ;
}
\node [inner sep=0.9pt] at (-90:1.5) {}; %
\end{tikzpicture}}\quad\quad\quad\quad\quad\quad
\subfloat[(b) $S'$]{\begin{tikzpicture}[thick]
\foreach \i in {0,2,4,5,7,9}
\draw [a] (360/10*\i+90:1.5)--(360/10*\i-360/10+90:1.5);
\foreach \i in {1,6}
\draw [b] (360/10*\i+90:1.5)--(360/10*\i-360/10+90:1.5);
\foreach \i in {2,7}
\draw [b] (360/10*\i+90:1.5)--(360/10*\i+360/10+90:1.5);
\node [circle,draw,inner sep=10pt] at (0,0) (c) {};
\foreach \i in {0,...,9} {
\draw (c)-- (360/10*\i+90:1.5) node [v] {} ;
}
\end{tikzpicture}}%
\caption{Square complex structures on $Z'$ and on $S'$.
A single and a double arrow denote the generators $a$ and $b$, respectively.
Figure (a) shows a punctured $D_i$ in $Z'$, divided into squares.
Figure (b) is a punctured $P_i$ in $S'$, where $\partial P_i\to\cay(F)/F$ reads $(b^{-1}aba^2)^2$.
}
\label{fig:zprime}
\end{figure}
A \emph{side-pairing} on polygonal disks $P_1,\ldots,P_m$ is an
equivalence relation on the sides of $P_1,\ldots,P_m$ such that each equivalence class consists of two sides, along with a choice of a homeomorphism between the two sides of each equivalence class.
For a given side-pairing $\sim$ on polygonal disks $P_1,\ldots,P_m$, one gets a closed surface $S=\coprod_i P_i/\sim$ by identifying the sides of $P_i$ by $\sim$. The surface $S$ is naturally equipped with a two-dimensional CW-structure.
A graph map $\phi\co G\to\cay(F_n)/F_n$ induces an orientation and a label by $\mathcal{A}_n$ on each edge $e$ of $G$, so that the oriented loop $\phi(e)$ reads the label of $e$.
An edge labeled by $a_i$ is called an \emph{$a_i$-edge}.
An \emph{immersion} is a locally injective graph map.
\begin{DEFN}[\cite{KW2009,Kim2009}]\label{defn:polygonal}
Let $U$ be a list of cyclically reduced words in $F_n$.
We say $U$ is \emph{polygonal} if there exist a side-pairing $\sim$ on some polygonal disks $P_1,P_2,\ldots,P_m$
and an immersion $S^{(1)}\to \cay(F_n)/F_n$ where $S=\coprod_i P_i/\sim$ such that the following hold:
\begin{enumerate}[(i)]
\item the composition $\partial P_i\to S^{(1)}\to\cay(F_n)/F_n$ reads a nontrivial power of a word in $U$ for each $i$;
\item the Euler characteristic $\chi(S)$ of $S$ is less than $m$.
\end{enumerate}
In this case, we call $S$ a \emph{$U$-polygonal surface}.
\end{DEFN}
\begin{REM}\label{rem:independence}
\begin{enumerate}
\item Polygonality has been defined
for a set of words~\cite{KW2009,Kim2009},
but we generalize
to a (possibly redundant) list of words.
The main implication of polygonality still holds,
as described in Theorem~\ref{thm:polygonal}.
\item
Polygonality of a list of words depends on the choice of a free-basis. An example given in~\cite{KW2009} is the word $w=abab^2ab^3$ in $F_2=\form{a,b}$. It was shown that while $w$ is not polygonal, the automorphism $(a\mapsto ab^{-2},b\mapsto b)$ maps $w$ to a polygonal word $ab^{-1}a^2b$.
\end{enumerate}
\end{REM}
\begin{THM}[\cite{KW2009,Kim2009}]\label{thm:polygonal}
If $U$ is a polygonal list of words in $F_n$, then $D(U)$ contains a hyperbolic surface group.
\end{THM}
\begin{proof}
Let $S$ be a closed surface obtained from a side-pairing $\sim$ on polygonal disks $P_1,P_2\ldots,P_m$,
equipped with immersions $\partial P_i\to S^{(1)}\to\cay(F_n)/F_n$ satisfying the conditions
in Definition~\ref{defn:polygonal}.
Choose a point $p_i$ in the interior of $P_i$ and triangulate $P_i$ so that $p_i$ is the common vertex, similarly to the triangulation of $D_i$ in $Z(U)$.
There is a natural extension $\phi\co S\to Z(U)$ of the immersion $S^{(1)}\to \cay(F)/F$.
In particular, $\phi$ respects the triangulation and is locally injective away from $p_1,\ldots,p_m$.
We obtain a square complex $S'$ from $S$ by taking out small open disks around $p_1,\ldots,p_m$; see Figure~\ref{fig:zprime} (b).
Similarly to what we have done for $Z'$, we glue two copies of $S'$ along the corresponding boundary components.
The resulting square complex $S''$ is a closed surface such that $\chi(S'')=2\chi(S')=2(\chi(S)-m)<0$.
With the square complex structure on $X(U)$ described previously, we have a locally injective cubical map $\phi ''\co S''\to X(U)$.
For a vertex $v\in S''^{(0)}$, $\link(f;v)$ embeds $\link_{S''}(v)\approx S^1$ onto a cycle in a link $W(U)'$ of $X(U)$.
Since each cycle in $W(U)'$ is a full subcomplex, Propositions~\ref{prop:pi1inj} and~\ref{prop:localisom} imply that $\phi''$ is locally an isometric embedding
and so, $\phi''_*$ is injective.
\end{proof}
\subsection{Tiling Conjecture and its implication}\label{subsec:split}
A list $U$ of words in $F_n$ is said to be \emph{diskbusting} if one cannot write $F_n=A\ast B$ in such a way that $A,B\ne\{1\}$ and each word in $U$ is conjugate into $A$ or $B$~\cite{Canary1993,Stong1997,Stallings1999}.
\begin{CON}[Tiling Conjecture; see \cite{KW2009,Kim2009}]\label{con:tiling}
A minimal and diskbusting list of cyclically reduced words in $F_n$ is polygonal when $n>1$.
\end{CON}
We note that $D(U)$ is one-ended if and only if $U$ is diskbusting~\cite{GW2010}.
By \cite{KW2009,Kim2009} and Theorem~\ref{thm:polygonal},
the double along a polygonal list contains a hyperbolic surface group.
Hence, if Tiling Conjecture is true, then every one-ended double of a nonabelian free group has a hyperbolic surface subgroup, answering Question~\ref{que:double}. Moreover, one would be able to precisely describe when doubles contain hyperbolic surface groups as follows.
\begin{PROP}\label{prop:tilingequiv}
Let $n>1$.
Suppose that every minimal and diskbusting list of cyclically reduced words in $F_m$ is polygonal for all $m=2,3,\ldots,n$.
Then for a list $U$ of cyclically reduced words in $F_n$, $D(U)$ contains a hyperbolic surface group if and only if $F_n$ cannot be written as
$F_n=G_1\ast G_2\ast\cdots G_n$ in such a way that
each $G_i$ is infinite cyclic and each word in $U$ is conjugate into one of
$G_1,\ldots,G_n$.
\end{PROP}
For the proof, we need the following:
\begin{LEM}\label{lem:cyclic}
A double of $\mathbb{Z}$ is virtually $\Z\times F_s$ for some $s\ge 0$.
\end{LEM}
\begin{proof}
We let $m_1,\ldots,m_k$ be given positive integers and $M$ be their least common multiple.
We consider a graph of spaces $X$ where there are two vertex spaces and $k$ edge spaces joining the two vertex spaces as follows. The vertex spaces are circles denoted by $\alpha$ and $\beta$, and each edge space $E_i$ is a cylinder whose boundary components are attached to $\alpha^{m_i}$ and $\beta^{m_i}$ for $i=1,\ldots,k$.
Then the double of $\mathbb{Z}$ along the words $m_1,\ldots,m_k$ is $\pi_1(X)$.
There exists a degree-$M$ cover $Y$ of $X$ with precisely two vertex spaces and $\sum_{i=1}^k m_i$ edge spaces;
the vertex spaces are circles projecting onto $\alpha^M$ and $\beta^M$, and $E_i$ lifts to $m_i$ cylinders whose attaching maps are homeomorphisms, for $i=1,\ldots,k$. Note that
$\pi_1(Y)\cong \Z\times F_s$ where $s + 1 = \sum_{i=1}^k m_i$.
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:tilingequiv}]
There exists a maximum $k$ such that $F_n=G_1\ast\cdots\ast G_k$ for some nontrivial groups $G_1,\ldots,G_k$ and each word in $U$ is conjugate into one of the $G_1,\ldots,G_k$. Note that $1\le k\le n$.
For the forward implication, suppose $k<n$.
Then we may assume that $G_1$ has rank $m > 1$.
Let $U_1$ be the list of all the words in $U$ conjugate into $G_1$.
Then suitably chosen conjugates of the words in $U_1$ form a diskbusting list $U_1'$ in the rank-$m$ free group $G_1$.
We note that $DU_1')\subseteq D(U_1') \subseteq D(U_1'\cup (U\setminus U_1)) \cong D(U)$; here, the second inclusion can be seen by Propositions~\ref{prop:pi1inj} and~\ref{prop:localisom}.
From the hypothesis, a free basis $\mathcal{B}$ of $G_1$ can be chosen so that $U_1'$ is polygonal as a list of words written in $\mathcal{B}$.
By Theorem~\ref{thm:polygonal}, $DU_1')$ contains a hyperbolic surface group; hence, so does $D(U)$.
For the backward implication, assume $k=n$ and we claim that $D(U)$ does not contain a hyperbolic surface group.
Since we are only interested in the isomorphism type of $D(U)$, we may assume that each word in $U$ is contained in one of $G_1,\ldots,G_n$, by taking conjugation if necessary.
By choosing the basis $\mathcal{A}_n$ of $F_n$ from the bases of $G_1,\ldots,G_n$,
one may write $\mathcal{A}_n=\{a_1,\ldots,a_n\}$
and $G_i = \form{a_i}$ for $i=1,\ldots,n$.
One sees that up to homotopy equivalence, $X(U)$ is obtained from graphs of spaces of the form considered in Lemma~\ref{lem:cyclic} by adding closed intervals and circles. So, $D(U)\cong\pi_1(X(U))$ can be written as a free product such that each free factor has a finite-index subgroup isomorphic to $\Z\times F_s$ for some $s\ge0$. In particular, $D(U)$ does not contain a hyperbolic surface group.
\end{proof}
\begin{REM}\label{rem:kw}
Tiling Conjecture would actually imply that the fundamental group of every one-ended graph of virtually free groups with virtually cyclic edge group either is virtually $\mathbb{Z}\times F_m$ for some $m>0$ or contains a hyperbolic surface group~\cite{KW2010pc}.
Moreover, since minimal diskbusting words are \emph{generic}~\cite{MS2003,BV2002}, Tiling Conjecture (Conjecture~\ref{con:tiling}) would imply that polygonal words are generic.
\end{REM}
\section{Combinatorial Formulation of Tiling Conjecture\label{sec:whitehead}}
Throughout this section, we let $U$ be a list of cyclically reduced words in $F_n$ for some $n>1$.
\subsection{Terminology on graphs}
We allow graphs to have parallel edges or loops;
a \emph{loop} is an edge with only one endpoint.
For a graph $G$, we write $V(G)$ and $E(G)$ to denote the vertex set and the edge set of $G$,
respectively. The \emph{degree} $\deg_G(v)$ of a vertex $v$ is the number of
edges incident with $v$, assuming that loops are counted twice.
A graph is \emph{$k$-regular} if every vertex has degree $k$,
and it is \emph{regular} if it is $k$-regular for some $k$.
A \emph{cycle} is a (finite) $2$-regular connected graph.
For a set $X$ of vertices, we write $\delta_G(X)$ to denote the set of edges having endpoints in both $X$ and $V(G)\setminus X$.
In particular,
$\delta_G(v)$ is the set of non-loop edges incident with $v$.
For two distinct vertices $x$ and $y$ of a graph $G$,
the \emph{local edge-connectivity} $\lambda_G(x,y)$ is the maximum number of
pairwise edge-disjoint paths from $x$ to $y$ in $G$. We omit the
subscript $G$ in $\deg_G$, $\delta_G$, and $\lambda_G$ if the underlying graph $G$ is clear from the context. Menger's theorem~\cite{Menger1927} states that
$\lambda(x,y)=\min\{ |\delta(X)| \co x\in X, y\not\in X\}$.
\subsection{Whitehead graph and the associated connecting map\label{subsec:connecting}}
The following characterization of a minimal set of words is given in~\cite[Section 8]{Berge1990}: a set $A$ of cyclically reduced words in $F_n$ is not minimal if and only if for some $i$, there exists a set $C$ of edges in the Whitehead graph $W(A)$ such that $\lvert C\rvert<\deg(a_i)$ and $W(A)\setminus C$ has no path from $a_i$ to $a_i^{-1}$. %
By Menger's theorem~\cite{Menger1927},
it follows that $A\subseteq F_n$ is minimal if and only if
\[\lambda(a_i,a_i^{-1})=\deg(a_i) \text{ for each }i.\]
Also, a minimal set $A\subseteq F_n$ is diskbusting if and only if $W(A)$ is connected~\cite{Whitehead1936,Stong1997,Stallings1999}.
These results on sets of words immediately generalize to lists of words as follows.
\begin{PROP}[\cite{Whitehead1936,Berge1990,Stong1997,Stallings1999}]\label{prop:whitehead}
A list $U$ of cyclically reduced words in $F_n$ is minimal and diskbusting if and only if $W(U)$ is connected and $\lambda(v,v^{-1})=\deg(v)$ for each
vertex $v$ of $W(U)$.
\end{PROP}
There is a canonical fixed point free involution $\mu$ on $\mathcal A_n\cup \mathcal A_n^{-1}$
such that $\mu(a)=a^{-1}$ for all $a\in A_n\cup \mathcal A_n^{-1}$.
For each vertex $v$ of $W(U)$, the \emph{connecting map $\sigma_v$ associated with $W(U)$ at $v$} is a bijection
from $\delta(v)$ to $\delta(\mu(v))$ defined as follows. For an edge $e$ given by $x_{i}x_{i+1}$ in a word $w=x_1x_2\ldots x_l$ in $U$,
$\sigma_{x_{i+1}^{-1}}$ maps the edge $e$ joining $x_{i}$ and $x_{i+1}^{-1}$
to the edge $f$ joining $x_{i+1}$ and $x_{i+2}^{-1}$ created by the consequently following length-2 cyclic subword $x_{i+1}x_{i+2}$ of $w$.
We assume that $x_{l+1}=x_1$ and $x_{l+2}=x_2$.
We note that if $\sigma_{y^{-1}}\circ\sigma_{x^{-1}}(e)$ is well-defined for an edge $e$ and vertices $x\ne y^{-1}$, then there exists a word $w$ in $U$ such that $xy$ is a length-2 cyclic subword of $w$ or $w^{-1}$. The proof of the following observation is now elementary.
\begin{LEM}\label{lem:reading}
Let $U$ be a list of cyclically reduced words in $F_n$.
In $W(U)$, consider an edge $f_0$ and vertices $x_1,x_2,\ldots,x_l$ where $l>0$, such that $x_{i+1}\ne x_i^{-1}$ for $i=1,\ldots,l$.
Suppose that
\[\sigma_{x_l^{-1}}\circ\sigma_{x_{l-1}^{-1}}\circ\cdots\circ\sigma_{x_1^{-1}}(f_0)\]
is well-defined and equal to $f_0$.
Then $x_1 x_2 \cdots x_l $ is a nontrivial power of a cyclic conjugation of a word in $U$.\qed
\end{LEM}
Connecting maps can be described in $Z(U)$.
The link of a vertex $p$ in a polygonal disk $P$ is called the \emph{corner} of $P$ at $p$.
Suppose an edge $e$ is incident with $a_i^{-1}$ in $W(U)$, where $e$ corresponds to the corner of a vertex $x$ in some $D_j$ attached to $Z(U)$.
Since we are assuming that every word in $U$ is cyclically reduced, there exists a unique $a_i$-edge $\alpha$ outgoing from $x$.
Choose the other endpoint $y$ of $\alpha$, and let $e'\in E(W(U))$ correspond to the corner of $D_j$ at $y$; see Figure~\ref{fig:connecting}.
Then we observe that $\sigma_{a_i^{-1}}(e)=e'$ and $\sigma_{a_i}(e')=e$.
\begin{figure}%
\tikzstyle {a}=[red,postaction=decorate,decoration={%
markings,%
mark=at position .5 with {\arrow[red]{stealth};}}]
\tikzstyle {b}=[blue,postaction=decorate,decoration={%
markings,%
mark=at position .43 with {\arrow[blue]{stealth};},%
mark=at position .57 with {\arrow[blue]{stealth};}}]
\tikzstyle {v}=[draw,shape=circle,fill=black,inner sep=0pt]
\tikzstyle {bv}=[black,draw,shape=circle,fill=black,inner sep=1pt]
\tikzstyle{every edge}=[-,draw]
\subfloat[(a) $D_j$]{
\begin{tikzpicture}[thick]
\foreach \i in {0,...,4}
\draw (360/5*\i+90:1.5) node [v] (v\i) {} ;
\draw[a] (v1)--(v2)
node[pos=0.7,bv] (ap1) {};
\draw[a] (v2)--(v3)
node[pos=.5] (alph) {}
node[pos=0.3,bv] (am) {}
node[pos=.7,bv] (ap2) {};
\draw[a] (v4)--(v0);
\draw [b] (v0)--(v1);
\draw [b] (v4)--(v3)
node[pos=0.7,bv] (bp) {};
\draw (v2) node [left] {$x$};
\draw (v3) node [right] {$y$};
\draw (alph) node [above] {${}_\alpha$};
\draw [bend left] (ap1) edge node [above] {$e$} (am)
(ap2) edge node [above] {$e'$} (bp);
\draw (am) node [below] {$a^{-1}$};
\draw (ap1) node [left] {$a$};
\draw (ap2) node [below] {$a$};
\draw (bp) node [right] {$b$};
\end{tikzpicture}
}
\quad\quad\quad\quad\quad\quad
\subfloat[(b) $W(U)$]{\begin{tikzpicture}[thick]
\draw (-1,-1) node [bv] (am) {} node[left] {$a^{-1}$}
-- node[midway,left] {$e$} (-1,1) node [bv] (ap) {} node[left] {$a$}
-- node[midway,above]{$e'$} (1,1) node [bv] (bp) {} node[right]{$b$}
(1,-1) node [bv] {} node[right]{$b^{-1}$};
\end{tikzpicture}}%
\caption{Each corner of a cell $D_j$ in $Z(U)$ corresponds an edge in $W(U)$.
Here, $F_2=\form{a,b}$ and $U=\{b^{-1}aba^2\}$.
In these two figures, we note that $\sigma_{a^{-1}}(e)=e'$ and $\sigma_{a}(e')=e$.
}
\label{fig:connecting}
\end{figure}
\subsection{Graph-theoretic formulation of Tiling Conjecture}
The polygonality was described in terms of Whitehead graphs~\cite[Propositions~17 and~21]{Kim2009}.
But this description required infinitely many graphs to be examined.
In the following lemma, we obtain a simpler formulation of polygonality requiring only one finite graph to be examined.
\begin{LEM}\label{lem:equiv}
Let $n>1$. A list $U$ of cyclically reduced words in $F_n$ is polygonal if and only if $W(U)$ has a nonempty list of cycles such that one of the cycles has length at least three and for each pair of edges $e$ and $f$ incident with a vertex $v$, the number of cycles in the list containing both $e$ and $f$ is equal to the number of cycles in the list containing both $\sigma_{v}(e)$ and $\sigma_{v}(f)$. Here, $\sigma_v$ denotes the connecting map associated with $W(U)$ at $v$.
\end{LEM}
We prove the necessity part by similar arguments to~\cite[Propositions~17 and 21]{Kim2009}.
The sufficiency part is what we mainly need for this paper.
\begin{proof}
We denote by $\mu$ the involution on the vertices of $W(U)$ defined by $\mu(a_i^{\pm1})=a_i^{\mp1}$.
To prove the necessity, assume $U$ is polygonal;
we can find a $U$-polygonal surface $S=\coprod_{1\le i\le m} P_i/\!\!\sim$ as in Definition~\ref{defn:polygonal}.
In particular, each edge in $S^{(1)}$ is oriented and labeled by $\mathcal{A}_n$. Put $S^{(0)}=\{v_1,\ldots,v_t\}$.
Fix $p_i$ in the interior of each $P_i$.
In Section 2.2, we have seen that there exists a map $\phi \co S\to Z(U)$ such that $\phi $ is locally injective away from $p_1,\ldots,p_m$.
Since $S$ is a closed surface and
$\phi$ is locally injective at $v_i$,
the image of each $\link_S(v_i)$
by $\phi $ is a cycle, say $C_i$, in $W(U)$.
Choose a vertex $v\in W(U)$ and two edges $e,f$ incident with $v$.
Without loss of generality, we may assume that $v=a^{-1}$ for some generator $a\in\mathcal{A}_n$ and $C_1,\ldots, C_{t'}$ is the list of the cycles among $C_1,\ldots,C_t$ which contain both $e$ and $f$.
Then for each $i=1,\ldots,t'$, there exists a unique $a$-edge $e_i$ outgoing from $v_i$. Let $v_{i'}$ be the endpoint of $e_i$ other than $v_i$.
There exist exactly two polygonal disks $Q_i$ and $R_i$ sharing $e_i$ in $S$, so that $\link(\phi;v_i)$ sends the corner of $Q_i$ at $v_i$ to $e$, and that of $R_i$ at $v_i$ to $f$. By the definition of a connecting map, $\link(\phi;v_{i'})$ maps the corners of $Q_i$ and $R_i$ at $v_{i'}$ to $\sigma_{a^{-1}}(e)$ and $\sigma_{a^{-1}}(f)$, respectively;
see Figure~\ref{fig:equiv}, which is similar to~\cite[Figure~7]{Kim2009}.
The correspondence $e\cup f \to \sigma_{a^{-1}}(e)\cup \sigma_{a^{-1}}(f)$ defines an involution on the list of length-$2$ subpaths of $ C_1,\ldots, C_t$. The conclusion follows.
\begin{figure}%
\tikzstyle {a}=[postaction=decorate,decoration={%
markings,%
mark=at position .5 with {\arrow[]{stealth};}}]
\tikzstyle {b}=[postaction=decorate,decoration={%
markings,%
mark=at position .43 with {\arrow[]{stealth};},%
mark=at position .57 with {\arrow[]{stealth};}}]
\tikzstyle {c}=[postaction=decorate,decoration={%
markings,%
mark=at position .4 with {\arrow[]{stealth};},%
mark=at position .5 with {\arrow[]{stealth};},
mark=at position .6 with {\arrow[]{stealth};}
}]
\tikzstyle {v}=[draw,shape=circle,fill=black,inner sep=0pt]
\tikzstyle {bv}=[black,draw,shape=circle,fill=black,inner sep=1pt]
\tikzstyle{every edge}=[-,draw]
\subfloat[(a) $S$]{
\begin{tikzpicture}[thick]
\draw (-3,0) node[] (v1) {}
[a] -- (-1.5,0) node [bv] (v2) {};
\node[left] at (-1.9,.3) {$v_i$};
\draw (v2)
[a]--
node[pos=.2,v] (e2) {}
node[pos=.5,above,black] {$e_i$}
node[pos=.8,v] (pe2) {}
(1.5,0) node [bv] (v3) {};
\node[right] at (1.9,.3) {$v_{i'}$};
\draw (v3)
[a]--
(3,0) node [] (v4) {};
\draw (-2.5,1.5) node[] (v5) {}
[b]--
node[pos=.2,v] (e1) {}
(v2);
\draw [orange,in=120,out=-30] (e1) edge node [above,black] {$e$} (e2);
\draw (v2)
[b]--
node[pos=.8,v] (e3) {}
(-2.5,-1.5) node[] (v6) {};
\draw [orange,out=-120,in=30] (e2) edge node [below,black] {$f$} (e3);
\draw (v3)
[c]--
node[pos=.8,v] (pe1) {}
(2.5,1.5) node[] (v7) {};
\draw [purple,in=60,out=210] (pe1)
edge
node [left,pos=.1,black,inner sep=10pt] {$\sigma_{a^{-1}}(e)$}
(pe2);
\draw (2.5,-1.5) node[] (v8) {}
[c]--
node[pos=.2,v] (pe3) {}
(v3);
\draw [purple,out=-60,in=150] (pe2)
edge
node [left,pos=.9,black,inner sep=10pt] {$\sigma_{a^{-1}}(f)$}
(pe3);
\draw[left] (0,1.2) node[] {$Q_i$};
\draw[left] (0,-1.2) node[] {$R_i$};
\node [inner sep=0.9pt] at (-90:1.5) {}; %
\end{tikzpicture}
}
\quad\quad
\subfloat[(b) $W(U)$]{\begin{tikzpicture}[thick]
\draw (-1.5,-1) node [bv] (bm) {} node[left] {$b^{-1}$}
[orange]-- node[pos=.3,above,black] {$f$} (0,-1) node [bv] (am) {} node [right,black]{$a^{-1}$};
\draw (am)
[orange]-- node[midway,right,black] {$e$}
(-1.5,1) node [bv] (bp) {} node[left,black]{$b$};
\draw
(1.5,1) node [bv] (cp) {} node [right,black] {$c$}
[purple]-- node [midway,above,black] {$\sigma_{a^{-1}}(f)$}
(0,1) node [bv] (ap) {} node [left,black] {$a$};
\draw
(ap)
[purple]-- node [midway, right,black] {$\sigma_{a^{-1}}(e)$}
(1.5,-1) node [bv] (cm) {} node [right,black] {$c^{-1}$};
\end{tikzpicture}}%
\caption{Consecutive corners in $S$ and their images by a connecting map.
$F_3 = \form{a,b,c}$, and single, double and triple arrows denote the labels $a,b$ and $c$, respectively.
}
\label{fig:equiv}
\end{figure}
For the sufficiency, consider a list of cycles $ C_1,\ldots, C_t$ in $W(U)$ satisfying the given condition.
For each $C_i$, let $V_i$ be a polygonal disk such that
$\partial V_i$ is a cycle of the same length as $C_i$.
We will regard $\partial V_i$ as the dual cycle of $ C_i$, in the sense that each edge of $\partial V_i$ corresponds to a vertex of $C_i$ and incident edges correspond to adjacent vertices.
Choose a linear order $\prec$ on $\{(v,e): e\in\delta(v)\}$ for each $v\in V(W(U))$ such that $(v,e)\prec (v,e')$ if and only if $(\mu(v),\sigma_v(e))\prec (\mu(v),\sigma_v(e'))$.
An edge $g$ of $\partial V_i$ will be labeled by $(a,\{e,f\})$ if the vertex $v$ of $W(U)$ corresponding to $g$ is labeled by $a$ or $a^{-1}$ for some $a\in \mathcal{A}_n$,
and $e$ and $f$ are the two edges of $C_i$ incident with $v$; see Figure~\ref{fig:equiv2} (a) and (b).
Considered as a side of $V_i$, $g$ will be given with a transverse orientation,
which is incoming into $V_i$ if $v\in\mathcal{A}_n$ and outgoing if $v\in\mathcal{A}_n^{-1}$.
If $w_e$ and $w_f$ denote the vertices of $g$ corresponding to $e$ and $f$ respectively,
and $(v,e)\prec (v,f)$, then we shall orient $g$ from $w_f$ to $w_e$.
Define a side-paring $\sim_0$ on $V_1,\ldots,V_t$ such that
$\sim_0$ respects the orientations, and moreover,
an incoming side labeled by $(a,\{e,f\})$
is paired with an outgoing side labeled by $(a,\{\sigma_{a}(e),\sigma_{a}(f)\})$ for each $a\in\mathcal{A}_n$ and $e,f\in\delta(a)$ where $e$ and $f$ are consecutive edges of some cycle $C_i$; the existence of such a side-pairing is guaranteed by the given condition.
Consider the closed surface $S_0 = \coprod_i V_i/\!\!\sim_0$.
Denote by $\eta$ and $\zeta$ the numbers of the edges and the faces in $S_0$, respectively.
Each edge in $S_0$ is shared by two faces, and each face has at least two edges;
moreover, at least one face has more than two edges by the given condition.
So, $2\zeta < \sum_i (\textrm{the number of sides in }V_i) = 2\eta$.
\begin{figure}%
\tikzstyle {a}=[red,postaction=decorate,decoration={%
markings,%
mark=at position 1 with {\arrow[red]{stealth};}}]
\tikzstyle {ab}=[black,postaction=decorate,decoration={%
markings,%
mark=at position .85 with {\arrow[black]{stealth};}}]
\tikzstyle {ab2}=[black,postaction=decorate,decoration={%
markings,%
mark=at position .75 with {\arrow[black]{stealth};}}]
\tikzstyle {b}=[blue,postaction=decorate,decoration={%
markings,%
mark=at position .85 with {\arrow[blue]{stealth};},%
mark=at position 1 with {\arrow[blue]{stealth};}}]
\tikzstyle {c}=[orange,postaction=decorate,decoration={%
markings,%
mark=at position .7 with {\arrow[orange]{stealth};},%
mark=at position .85 with {\arrow[orange]{stealth};},
mark=at position 1 with {\arrow[orange]{stealth};}
}]
\tikzstyle {v}=[draw,shape=circle,fill=black,inner sep=0pt]
\tikzstyle {gv}=[draw,shape=circle,fill=gray,inner sep=0pt]
\tikzstyle {bv}=[black,draw,shape=circle,fill=black,inner sep=1pt]
\tikzstyle{every edge}=[-,draw]
\subfloat[(a) $C_i$]{
\begin{tikzpicture}[thick]
\foreach \i in {0,...,4}
\draw (360/5*\i+90:1.5) node [v] (v\i) {} ;
\draw (0:0) node [] {${}_{(a,e)\prec (a,f)}$};
\draw (v1) node [left] {$a$} --(v2);
\draw (v2) node [left] {$a^{-1}$} --(v3) node [right] {$b^{-1}$};
\draw (v4) node [right] {$c^{-1}$} --(v0);
\draw (v0) node [above] {$b$} --(v1);
\draw[] (v0) -- node[pos=.4,below,gray] {${}_f$} (v1);
\draw[] (v1) -- node[pos=.5,right,gray] {${}_e$} (v2);
\draw (v4) --(v3);
\foreach \i in {1,2}
\draw (v\i) node [shape=circle,fill=red,inner sep=2pt] {};
\foreach \i in {3,0}
\draw (v\i) node [shape=circle,fill=blue,inner sep=2pt] {};
\foreach \i in {4}
\draw (v\i) node [shape=circle,fill=orange,inner sep=2pt] {};
\end{tikzpicture}
}
\quad
\subfloat[(b) $V_i$]{ \begin{tikzpicture}[thick]
\foreach \i in {0,...,4} {
\draw (360/5*\i+90:1.9) node [] (w\i) {} ;
\draw (360/5*\i+90:1.1) node [] (wp\i) {} ;
\draw (360/5*\i-90:1.5) node [v] (v\i) {} ;
}
\draw (v1) --(v2);
\draw (v2) --(v3);
\draw (v4) --(v0);
\draw (v0) --(v1);
\draw[ab] (v3) --(v4);
\draw[left] (v3) node[] {$w_f$};
\draw[left] (v4) node[] {$w_e$};
\draw[] (v3) -- node[right,pos=.6] {${}_{(a,\{e,f\})}$} (v4);
\draw[] (v3) -- node[left,pos=.7] {${}_{g}$} (v4);
\draw[] (v2) -- (v3);
\foreach \i in {1,2}
\draw (v\i) node [shape=circle,fill=black,inner sep=2pt] {};
\foreach \i in {3,0}
\draw (v\i) node [shape=circle,fill=black,inner sep=2pt] {};
\foreach \i in {4}
\draw (v\i) node [shape=circle,fill=black,inner sep=2pt] {};
\node [inner sep=0.9pt] at (-90:1.5) {}; %
\foreach \i in {1} {\draw[a] (w\i) -- (wp\i);};
\foreach \i in {2} {\draw[a] (wp\i) -- (w\i);};
\foreach \i in {3} {\draw[b] (wp\i) -- (w\i);};
\foreach \i in {4} {\draw[c] (wp\i) -- (w\i);};
\foreach \i in {0} {\draw[b] (w\i) -- (wp\i);};
\draw[] (v2) -- node [pos=.8,v] (pf1) {} (v3);
\draw[] (v4) -- node [pos=.6,v] (pf2) {} (v3);
\draw[] (v4) -- node [pos=.2,v] (pf3) {} (v0);
\draw[] (v4) -- node [pos=.4,v] (pf4) {} (v3);
\draw [gray,out=-90-18,in=-18+36] (pf1) node [v] {} edge node[gray,right,pos=.7] {${}_f$} (pf2);
\draw [gray,out=72,in=-18-36] (pf3) node [v] {} edge node[gray,right,pos=.3] {${}_e$} (pf4);
\end{tikzpicture}
}%
\\
\subfloat[(c) $S_0$]{
\begin{tikzpicture}[thick]
\draw (-1,2) node[] (v1) {}
-- node [pos=.7,gv] (fp) {}
(0,1) node [bv] (v5) {}
-- node [pos=.3,gv] (f) {}
(1,2) node [] (v2) {};
\draw (v5)
-- (0,-1) node [bv] (v6) {}
-- node [pos=.3,gv] (ep) {}
(-1,-2) node [] (v3) {};
\draw (v6)
-- node [pos=.3,gv] (e) {}
(1,-2) node [] (v4) {};
\draw (v6) node[below] {$q$};
\node[below] at (-1.2,1.7) {$V_j$};
\node[below] at (1.2,1.7) {$V_i$};
\draw[a] (-.3,0) node [] (v7) {}
-- (.3,0) node [] (v8) {};
\draw (v7) node [left] {${}_{(a,\{\sigma_a(e),\sigma_a(f)\})}$};
\draw (v8) node [right] {${}_{(a,\{e,f\})}$};
\draw[ab2] (v5) {}
-- node[pos=.2,gv] (u1) {}
node[pos=.8,gv] (u2) {}
(v6) {};
\draw [gray,out=-90-45,in=180] (fp) {} edge
node[gray,left,pos=.6] {${}_{\sigma_a(f)}$}
(u1) {};
\draw [gray,out=-45,in=0] (f) {} edge
node[gray,right,pos=.6] {${}_{f}$}
(u1) {};
\draw [gray,out=135,in=180] (ep) {} edge
node[gray,left,pos=.6] {${}_{f_1=\sigma_a(e)}$}
(u2) {};
\draw [gray,out=45,in=0] (e) {} edge
node[gray,right,pos=.6] {${}_{f_0=e}$}
(u2) {};
\node [inner sep=0.9pt] at (-90:1.5) {}; %
\end{tikzpicture}
}
\qquad
\subfloat[(d) $\link(q)$]{\begin{tikzpicture}[thick]
\foreach \i in {0,...,2} {
\draw (360/3*\i+90:2.5) node [] (v\i) {} ;
\draw (360/3*\i+90:2) node [v] (w\i) {} ;
\draw (0,0) -- (v\i);
}
\draw (0,-.2) node [below] {$q$};
\draw[c] (90 + 10:1.2)--(90-20:1.27);
\draw[a] (90-10+120 :1.2)--(90 +120 + 20:1.27);
\draw[b] (90 + 10-120 :1.2)--(90-20-120:1.27);
\draw [gray,out=-30,in=-150] (w1) edge node[pos=.5,below] {${}_{f_0}$} (w2);
\draw [gray,out=-30+120,in=-150+120] (w2) edge node[pos=.5,right] {${}_{f_2}$} (w0);
\draw [gray,out=-30-120,in=-150-120] (w0) edge node[pos=.5,left] {${}_{f_1}$} (w1);
\node [inner sep=0.9pt] at (-90:1.5) {}; %
\end{tikzpicture}}%
\caption{Constructing $V_i$ and $S_0$ from $C_i$ in the proof of Lemma~\ref{lem:equiv}.
In this example, we note from (d) that $f_1=\sigma_{a}(f_0), f_2=\sigma_{c^{-1}}(f_1)$ and $f_0=\sigma_{b^{-1}}(f_2)$.
}
\label{fig:equiv2}
\end{figure}
By the duality between $C_i$ and $V_i$, each corner of $V_i$ corresponds to an edge in $C_i$.
Then the link of a vertex $q$ of $S_0$ corresponds to the union of edges in $W(U)$ written as the following sequence
\[
f_0, f_1 = \sigma_{x_1^{-1}}(f_0), f_2 = \sigma_{x_2^{-1}}(f_1), \ldots, f_l = \sigma_{x_l^{-1}}(f_{l-1})
\]
so that $f_0 = f_l = \sigma_{x_l^{-1}}\circ\sigma_{x_{l-1}^{-1}}\circ\cdots\circ\sigma_{x_1^{-1}}(f_0)$
for some vertices $x_1,\ldots,x_l$ of $W(U)$; see Figure~\ref{fig:equiv2} (c).
By Lemma~\ref{lem:reading}, $x_1 \cdots x_l$ can be taken as a nontrivial power of a word in $U$.
We will follow the boundary curve $\alpha$ of a small neighborhood of $q$ with some orientation,
and whenever $\alpha$ crosses an edge of $S_0$ with the first component of the label being $a\in\mathcal{A}_n$, we record $a$ if the crossing coincides with the transverse orientation of the edge, and $a^{-1}$ otherwise.
Let $w_q\in F$ be the word obtained by this process.
Then $w_q = x_1 \cdots x_l$, up to taking an inverse and cyclic conjugations.
Let $S$ be a surface homeomorphic to $S_0$.
We give $S$ a $2$-dimensional cell complex structure, by letting the homeomorphic image of the dual graph of $S_0^{(1)}$ to be $S^{(1)}$.
In particular, the $2$-cells $P_1,\ldots,P_m$ in $S$ are the connected regions bounded by $S^{(1)}$.
The transverse orientations and the first components of the labels of the sides in $V_1,\ldots,V_t$ induce orientations and labels of the sides of $P_1,\ldots,P_m$.
By duality, the boundary reading of each $P_i$ in $S$ is of the form $w_q$ for some vertex $q$ of $S_0$; hence, $\partial P_i$ reads a nontrivial power of a word in $U$.
Finally, if we let $\nu$ be the number of the vertices in $S_0$, then
\[\chi(S) - m = \chi(S_0) - \nu = -\eta + \zeta < 0.\qedhere\]
\end{proof}
\begin{REM}\label{rem:polynomial time}
There is a polynomial-time algorithm to decide whether a list of words in a free group is polygonal~\cite{Kim2009}. We note that diskbusting property can also be determined in polynomial time~\cite{Whitehead1936,Stong1997,Stallings1999,RVW2007}.
\end{REM}
A graph is \emph{non-acyclic} if it contains at least one cycle.
We now restate Tiling Conjecture combinatorially as follows.
\begin{CON}\label{con}
Let $G=(V,E)$ be a non-acyclic graph with a fixed point free involution $\mu:V\to V$ and a bijection $\sigma_v:\delta(v)\to \delta(\mu(v))$ for every vertex $v$ such that $\lambda(v,\mu(v))=\deg(v)$ and $\sigma_{\mu(v)}=\sigma_v^{-1}$.
Then there exists a nonempty list of cycles of $G$ such that for each pair of edges $e$ and $f$ incident with a vertex $v$, the number of cycles in the list containing both $e$ and $f$ is equal to the number of cycles in the list containing both
$\sigma_{v}(e)$ and $\sigma_{v}(f)$. Moreover, the list can be
required to contain at least one cycle of length greater than two if
$G$ has a connected component which has at least four vertices.
\end{CON}
\begin{PROP}\label{prop:equiv}
Let $n'>1$.
Tiling Conjecture holds for all $n=2,\ldots, n'$ if and only if
Conjecture~\ref{con} holds for graphs on $2n'$ vertices.
\end{PROP}
\begin{proof}
(Conjecture~\ref{con} $\Rightarrow$ Tiling Conjecture)
Let $2\le n\le n'$ and let $U$ be a minimal and diskbusting list of
cyclically reduced words in $F_n$.
If Conjecture \ref{con} holds for $2n'$, then it holds for $2n$ because
we can add isolated vertices.
By Proposition~\ref{prop:whitehead}, the
connected graph $W(U)$ is equipped with the fixed point free
involution $\mu(v)=v^{-1}$ on $V(W(U))$ and the associated
connecting map $\sigma_v$ at each vertex $v$ such that
$\lambda(v,\mu(v))=\deg(v)$ and $\sigma_{\mu(v)}=\sigma_v^{-1}$. Note
that $W(U)$ is non-acyclic;
because otherwise $\deg(v)=\lambda(v,\mu(v))\le 1$ for each vertex $v$
and therefore $W(U)$ would be disconnected, as
$W(U)$ has at least four vertices.
The conclusion of Conjecture~\ref{con} along with Lemma~\ref{lem:equiv} implies that $U$ is polygonal.
(Tiling Conjecture $\Rightarrow$ Conjecture~\ref{con})
We let $G, \mu, \sigma_v$ be as in the hypothesis of Conjecture~\ref{con}
such that $|V(G)|=2n'$. Let $n=n'$.
Since for each vertex $v$, $v$ and $\mu(v)$ belong to the same
connected component of $G$, we may
assume that $G$ is connected by taking a non-acyclic component of $G$.
If $|V(G)|=2$, then the list of all bigons is a desired collection of cycles.
So we assume $G$ is connected and $|V(G)|\ge 4$.
Label the vertices of $G$ as $a_1,a_1^{-1},\ldots, a_n,a_n^{-1}$ so that $a_i^{-1}=\mu(a_i)$. Then $G$ can be regarded as the Whitehead graph of a list $U$ of cyclically reduced words in $F_n$.
Proposition~\ref{prop:whitehead} implies that $U$ is minimal and diskbusting, as well. As we are assuming Tiling Conjecture for $F_n$, $U$ is polygonal. Lemma~\ref{lem:equiv} completes the proof.\end{proof}
In Sections~\ref{sec:regular} and~\ref{sec:4vertex}, we will prove Conjecture~\ref{con} for
regular graphs and four-vertex graphs, respectively.
This amounts to proving Tiling Conjecture for $k$-regular lists of words and for rank-two free groups.
\section{Regular Graph and Proof of Theorem~\ref{thm:main 2}}\label{sec:regular}
We will prove that Conjecture~\ref{con} holds for regular graphs.
It turns out that we can prove a slightly stronger theorem.
\begin{THM}\label{thm:regular}
Let $k>1$.
Let $G=(V,E)$ be a $k$-regular graph with a fixed point free
involution $\mu:V\to V$
such that
$\lambda(v,\mu(v))=k$
for every vertex $v\in V$.
Then there exists
a nonempty list of cycles of $G$
with positive integers $m_1$, $m_2$ such that
every edge is in exactly $m_1$ cycles in the list
and
each adjacent pair of edges
is contained in exactly $m_2$ cycles in the list.
\end{THM}
Theorem~\ref{thm:main 2} is now an immediate corollary of the following.
\begin{COR}\label{cor:regularw}
A minimal, diskbusting, $k$-regular list of words in $F_n$ is polygonal when $n>1$.
\end{COR}
\begin{proof}[Proof of Corollary~\ref{cor:regularw}]
Let $U$ be such a list.
By Proposition~\ref{prop:whitehead}, $W(U)$ satisfies the hypotheses of Theorem~\ref{thm:regular},
and moreover, $W(U)$ is connected and $k$-regular.
Since $W(U)$ has $2n$ vertices and $n>1$, it has two adjacent edges $e$ and $f$, not parallel to each other.
By Theorem~\ref{thm:regular}, there must be a cycle in the list
containing both $e$ and $f$ and that cycle must have length at least three.
Lemma~\ref{lem:equiv} completes the proof.
\end{proof}
A graph $H$ is a \emph{subdivision} of $G$ if
$H$ is obtained from $G$ by replacing each edge by a path of length at
least one.
We remark that Conjecture~\ref{con} is also true for all subdivisions of $k$-regular graphs if $k>1$,
because every edge appears the same number of times in Theorem~\ref{thm:regular}.
Let us start proving Theorem~\ref{thm:regular}.
A graph $G=(V,E)$ is called a \emph{$k$-graph} if it is $k$-regular
and $|\delta(X)|\ge k$ for every subset $X$ of $V$ with $|X| $ odd.
In particular if $k>0$, then every $k$-graph must have an even number of
vertices, because otherwise $\lvert\delta(V(G))\rvert\ge k$.
It turns out that every $k$-regular
graph
with the properties required by Conjecture~\ref{con} is a $k$-graph.
\begin{LEM}\label{lem:kgraph}
Let $G=(V,E)$ be a $k$-regular graph with a fixed point free involution $\mu$
such that
$\lambda(v,\mu(v))=k$
for every vertex $v\in V$.
Then
$G$ is a $k$-graph.
\end{LEM}
\begin{proof}
Supposes $X\subseteq V$ and $|X|$ is odd.
Then there must be $x\in X$ with $\mu(x)\notin X$
because $\mu$ is an involution such that $\mu(v)\neq v$ for all
$v\in V$.
Then there exist $k$ edge-disjoint paths from $x$ to $\mu(x)$
and therefore $|\delta(X)|\ge k$.
\end{proof}
By the previous lemma, it is sufficient to consider $k$-graphs in order to prove Theorem~\ref{thm:regular}.
By using the characterization of the perfect matching polytope by Edmonds~\cite{Edmonds1965a},
Seymour~\cite{Seymour1979a} showed the following theorem. This is
also explained in Corollary~7.4.7 of the book by Lov\'asz and
Plummer~\cite{LP1986}.
A \emph{matching} is a set of edges in which no two are adjacent.
A \emph{perfect matching} is a matching meeting every vertex.
\begin{THM}[Seymour~\cite{Seymour1979a}]
Every $k$-graph is \emph{fractionally $k$-edge-colorable}.
In other words, every $k$-graph has a nonempty list of perfect matchings $M_1$, $M_2$,
$\ldots$, $M_\ell$ such that
every edge is in exactly $\ell/k$ of them.
\end{THM}
For sets $A$ and $B$, we write $A\Delta B=(A\setminus B)\cup
(B\setminus A)$.
\begin{LEM}\label{lem:regularlist}
Let $k>1$.
Every $k$-graph has a nonempty list of cycles
such that
every edge appears in the same number of cycles
and
for each pair of adjacent edges $e$, $f$,
the number of cycles in the list containing both $e$ and $f$ is
identical.
\end{LEM}
\begin{proof}
Let $M_1$, $M_2$, $\ldots$, $M_\ell$ be a nonempty list of perfect
matchings of a $k$-graph $G=(V,E)$ such that
each edge appears in $\ell/k$ of them.
Then for distinct $i,j$,
the set $M_i\Delta M_j$ induces a subgraph of $G$
such that every vertex has degree $2$ or $0$.
Thus each component of the subgraph $(V,M_i\Delta M_j)$ is a cycle.
Let $C_1$, $C_2$, $\ldots$, $C_m$ be the list of cycles appearing
as a component of the subgraph of $G$ induced by $M_i\Delta
M_j$ for each pair of distinct $i$ and $j$. We allow repeated cycles.
This list is nonempty because $k>1$ and so there exist $i, j$
such that $M_i\neq M_j$.
Since each edge is contained in exactly $\ell/k$ of $M_1$, $M_2$,
$\ldots$, $M_\ell$,
every edge is in exactly $\frac{\ell}{k}(\ell-\frac{\ell}{k})$
cycles in the list.
For two adjacent edges $e$ and $f$,
since no perfect matching contains both $e$ and $f$,
there are $(\ell/k)^2$ cycles in $C_1$, $C_2$, $\ldots$, $C_m$ using
both $e$ and $f$.
\end{proof}
Lemmas~\ref{lem:kgraph} and \ref{lem:regularlist} clearly imply Theorem~\ref{thm:regular}.
We also note that even the minimality assumption can be lifted for rank-two free groups:
\begin{COR}\label{cor:kregular4}
Let $U$ be a $k$-regular list of cyclically reduced words in $F_2$.
Then $U$ is diskbusting if and only if $U$ is polygonal; in this case,
$D(U)$ contains a hyperbolic surface group.
\end{COR}
\begin{proof}
We note that a $k$-regular $4$-vertex graph is always a $k$-graph.
For the sufficiency, we recall that if $U$ is diskbusting in $F_2$,
then $W(U)$ is connected~\cite{Stong1997,Stallings1999}.
Since a connected $4$-vertex graph contains at least one pair of
incident edges which are not parallel,
Lemma~\ref{lem:regularlist} implies that
$W(U)$ contains a list of cycles, not all bigons, such that each pair of incident edges appears the same number of times in the list.
Lemma~\ref{lem:equiv} proves the claim.
For the necessity, we note that the proof of the sufficiency part of
Proposition~\ref{prop:tilingequiv} shows if $U$ is not diskbusting in $F_2$, then $D(U)$ does not contain a hyperbolic surface group.
\end{proof}
\section{Graphs on four vertices}\label{sec:4vertex}
Let $G$ be a graph with a fixed point free
involution $\mu:V(G)\to V(G)$ and
a bijection $\sigma_v:\delta (v) \to \delta (\mu(v))$ for each vertex
$v$ so that
$\lambda(v,\mu(v))=\deg(v)$ and
$\sigma_{\mu(v)}=\sigma_v^{-1}$.
For a vertex $w$ of $G$,
a permutation $\pi$ on $\delta(w)$ is called \emph{$w$-good}
if $\{e,\sigma_w(\pi(e))\}$ is a matching of $G$ for every edge $e$
incident with $w$. Note that $\{e,f\}$ is a matching of $G$ if and
only if either $e=f$ or $e$, $f$ share no vertex.
In particular, if $x$ is an edge joining $w$ and $\mu(w)$, then
$\sigma_w(\pi(x))=x$.
A permutation $\pi$ on a set $X$ induces a permutation $\pi^{(2)}$ on
$2$-element subsets of $X$ such that
$\pi^{(2)}(\{x,y\})=\{\pi(x),\pi(y)\}$ for all distinct $x,y\in X$.
A $w$-good permutation $\pi$ on $\delta (w)$ is \emph{uniform}
if
$\pi^{(2)}$ has a list of orbits $X_1$, $X_2$, $\ldots$, $X_t$
satisfying the following.
\begin{enumerate}[(i)]
\item If $\{x,y\}\in X_i$, then $x$ and $y$ do not share a vertex
other than $w$ or $\mu(w)$ in $G$.
\item
There is a constant $c>0$ such that for every edge $e\in \delta(w)$,
\[
\lvert \{ (X_i,F): 1\le i\le t,~F\in X_i \text{ and } e\in F\} \rvert = c.\]
\end{enumerate}
The following lemma shows that
in order to prove Conjecture~\ref{con} for $4$-vertex graphs,
it is enough to find a $w$-good uniform
permutation on the edges incident with a vertex $w$ of minimum degree.
\begin{LEM}\label{lem:inductive}
Let $G$ be a connected $4$-vertex graph with a fixed point free
involution $\mu:V(G)\to V(G)$
such that
$\lambda(v,\mu(v))=\deg(v)$ for each vertex $v$.
Let $w$ be a vertex of $G$ with the minimum degree.
Let $\sigma_w:\delta(w)\to
\delta(\mu(w))$ be a bijection.
If there is a $w$-good uniform permutation $\pi$ on
$\delta(w)$,
then
$G$ admits a nonempty list of cycles satisfying the following properties.
\begin{enumerate}[(a)]
\item For distinct edges $e_1,e_2\in \delta(w)$,
the number of cycles in the list containing both $e_1$ and $e_2$
is equal to the number of cycles in the list containing both
$\sigma_{w}(e_1)$ and $\sigma_{w}(e_2)$.
\item There is a constant $c_1>0$ such that each edge appears in
exactly $c_1$ cycles in the list.
\item There is a constant $c_2>0$ such that
for a vertex $v\in V(G)\setminus\{w,\mu(w)\}$ and each pair of distinct edges $e_1,e_2\in \delta(v)$,
exactly $c_2$ cycles in the list contain both $e_1$ and $e_2$.
\item The list contains a cycle of length at least three.
\end{enumerate}
\end{LEM}
\begin{proof}
We say that a list of cycles is \emph{good} if it satisfies (a),
(b), (c), and (d).
We proceed by induction on $|E(G)|$.
Let $u$ be a vertex of $G$ other than $w$ and $\mu(w)$.
If $\deg(u)=\deg(w)$, then the conclusion follows by
Theorem~\ref{thm:regular}.
Therefore we may assume that $\deg(u)>\deg (w)$.
There should exist an edge $e$ joining $u$ and $\mu(u)$.
Moreover $G\setminus e$ is connected because
otherwise $G$ would not have $\deg(w)$ edge-disjoint paths from $w$
to $\mu(w)$.
By the induction hypothesis, $G\setminus e$ has a good list of
cycles $C_1'$, $C_2'$, $\ldots$, $C_s'$.
Note that we use the fact that $\deg (u)>\deg(w)$ so that
$G\setminus e$ has $\deg_{G\setminus e}(v)$ edge-disjoint paths from $v$ to
$\mu(v)$ for each vertex $v$ of $G\setminus e$.
Let $c_1'$, $c_2'$ be the constants given by (b) and (c), respectively,
for the list $C_1'$, $C_2'$, $\ldots$, $C_s'$ of cycles of $G\setminus e$.
Since $\pi$ is $w$-good uniform, $\pi^{(2)}$ has a list of
orbits $X_1$, $X_2$, $\ldots$, $X_t$ satisfying (i) and (ii),
where each edge in $\delta(w)$ appears $c$ times in this list.
Suppose that $\{x,y\}\in X_i$. Then $\{\pi(x),\pi(y)\}\in
X_i$.
If $x,y\in \delta(\mu(w))$, then
we let $C_{xy}$ be a cycle formed by two edges $x=\sigma_w(\pi(x))$
and $y=\sigma_w(\pi(y))$.
If $x,y\notin \delta(\mu(w))$,
let $C_{xy}$ be a list of two cycles, one formed by three edges
$e$, $x$, $y$, and
the other formed by three edges $e$, $\sigma_w(\pi(x))$, $\sigma_w(\pi(y))$.
If exactly one of $x$ and $y$, say $y$, is incident with $\mu(w)$, then
let $C_{xy}$ be the cycle formed by four edges $e$, $x$,
$y=\sigma_w(\pi(y))$, $\sigma_w(\pi(x))$.
Since $x$ and $y$ never share $u$ or $\mu(u)$ by (i),
$C_{xy}$ always consists of one or two cycles of $G$.
Let $C_1,C_2,\ldots,C_p$ be the list of all cycles in $C_{xy}$ for each
member $\{x,y\}$ of $X_i$ for all $i=1,2,\ldots,t$. Notice that we allow
repetitions of cycles.
We claim that the list $C_1,C_2,\ldots,C_p$ satisfies (a).
For each occurrence of $x,y\in \delta(w)$ in a cycle in the
list,
there is a corresponding $i$ such that $\{x,y\}\in X_i$. Since $X_i$ is an
orbit, there is $\{x',y'\}\in X_i$ where $\pi(x')=x$ and $\pi(y')=y$.
Then the list contains cycles in $C_{x'y'}$ for $X_i$.
This proves the claim because
$\sigma_w(x)=\sigma_w(\pi(x'))$ and $\sigma_w(y)=\sigma_w(\pi(y'))$.
By (ii) of the definition of a uniform permutation,
for each edge $f$ incident with $w$,
there are $c$ cycles in the list $C_1,C_2,\ldots,C_p$
containing the edge $f$ of $G$.
Notice that whenever an edge $f$ in
$C_{xy}$ is in $\delta(\{w,\mu(w)\})$, $C_{xy}$
contains $e$ and $\sigma_w(\pi(f))$ by the construction.
Therefore every edge incident with $w$ or $\mu(w)$
appears $c$ times
in the list $C_1,C_2,\ldots,C_p$.
We now construct a good list of cycles for $G$ as follows:
We take $c_2'$ copies of $C_1,C_2,\ldots, C_p$,
$c$ copies of $C_1',C_2',\ldots, C_s'$,
and $cc_2'$ copies of cycles formed by $e$ and another edge
$f\neq e$
joining $u$ and $\mu(u)$.
We claim that this is a good list of cycles of $G$.
It is trivial to check (a).
For distinct edges $e_1,e_2$ incident with $u$,
the list contains $cc_2'$ cycles containing both of them, verifying (c).
Let $a$ be the number of edges in $\delta(u)$ incident with $w$ or $\mu(w)$
and let $b$ be the number of edges joining $u$ and $\mu(u)$.
By (c) on $G\setminus e$, we have $c_1'=c_2'(a+b-2)$.
Finally to prove (b), every edge incident with $w$
or $\mu(w)$ appears $cc_2'+cc_1'=cc_2'(a+b-1)$ times in the list
and
the edge $e$ appears $acc_2'+(b-1)cc_2'=cc_2'(a+b-1)$ times in the list.
An edge $f\neq e$ joining $u$ and $\mu(u)$ appears
$cc_1'+cc_2'=cc_2'(a+b-1)$ times.
\end{proof}
\subsection{Lemma on Odd Paths and Even Cycles}
To find a $w$-good uniform permutation of $\delta(w)$,
we need a combinatorial lemma on a set of
disjoint odd paths and even cycles.
The \emph{length} of a path or a cycle is the number of its edges.
\begin{LEM}\label{lem:oddpath}
Let $D$ be a directed graph with at least
four vertices such that
each component is a
directed path of odd length
or a directed cycle of even length.
Suppose that every vertex of in-degree $0$ or out-degree $0$ in $D$
is colored with red or blue,
while the number of red vertices of in-degree $0$ is equal to the number of
red vertices of out-degree $0$.
We say that a graph is \emph{good} if at most half of all the
vertices
are blue and at most half of all the vertices are red.
We say that a directed path or a cycle is \emph{long} if its length is at least three.
A directed path or a cycle is said to be \emph{short} if it is not long.
A \emph{R-R} path denotes a directed path starting with a red vertex and ending
with a red vertex. Similarly we say \emph{R-B} paths, \emph{B-B} paths,
\emph{B-R} paths.
A set of paths is called \emph{monochromatic} if it has no blue
vertex or no red vertex.
If $D$ is good,
then
$D$ can be partitioned into good subgraphs,
each of which is one of eight types listed below. (See Figure~\ref{fig:types}.)
\begin{enumerate}
\item A short R-R path,
a short B-B path, and possibly a short cycle.
\item A monochromatic path and one or two short cycles.
\item A short cycle, a B-R path, and an R-B path.
\item At least two short cycles.
\item A long monochromatic path
and monochromatic short paths, possibly none.
\item
A B-R path, a R-B
path, and monochromatic short paths, possibly none.
\item A long cycle
and monochromatic short paths, possibly none.
\item A long cycle and a short cycle.
\end{enumerate}
\end{LEM}
We remark that in a subgraph of type (5), we require that
the long path is monochromatic and the set of short
paths monochromatic, but we allow the long path to have a
color unused in short paths.
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edge[e] (r1);
\node [v] (r2) [w,right of=r1] {};
\node [v] [above of=r2] {}
edge[<-] node[midway] (end) {} (r2)
edge[e] (r2);
\draw [ddd] (start)--(end);
}}\\%
\subfloat[(5)]{\tikz[showbg]{
\node[g] (bl){};
\node[v] (b1) [right of=bl] {}
edge[<-] (bl);
\node[v] (b2) [above of=b1] {}
edge[<-,dotted] (b1);
\node[ge] [left of=b2] {}
edge[<-] (b2);
\node [g] (br) [w,right of=b1] {};
\node [ge] [above of=br] {}
edge[<-] node[midway](start) {} (br);
\node [g] (br2) [w,right of=br] {};
\node [ge] [above of=br2] {}
edge[<-] node[midway](end) {} (br2);
\draw [ddd] (start)--(end);
}}\,%
\subfloat[(6)]{\tikz[showbg]{
%
\node[r] (bl) {};
\node[b] (b1) [above of=bl] {}
edge[<-,dotted] (bl);
\node[b] (bll) [w,left of=bl]{};
\node[r] (b1) [above of=bll] {}
edge[<-,dotted] (bll);
\node [g] (br) [w,right of=bl] {};
\node [ge] [above of=br] {}
edge[<-] node[midway](start) {} (br);
\node [g] (br2) [w,right of=br] {};
\node [ge] [above of=br2] {}
edge[<-] node[midway](end) {} (br2);
\draw [ddd] (start)--(end);
}}\,%
\subfloat[(7)]{\tikz[showbg]{
\node[v] (bl){};
\node[v] (b1) [right of=bl] {}
edge[<-] (bl);
\node[v] (b2) [above of=b1] {}
edge[<-,dotted] (b1);
\node[v] [left of=b2] {}
edge[<-] (b2)
edge[e] (bl);
%
%
%
%
\node [g] (br) [w,right of=b1] {};
\node [ge] [above of=br] {}
edge[<-] node[midway](start) {} (br);
\node [g] (br2) [w,right of=br] {};
\node [ge] [above of=br2] {}
edge[<-] node[midway](end) {} (br2);
\draw [ddd] (start)--(end);
}}
\subfloat[(8)]{\tikz[showbg]{
\node[v] (bl){};
\node[v] (b1) [right of=bl] {}
edge[<-] (bl);
\node[v] (b2) [above of=b1] {}
edge[<-,dotted] (b1);
\node[v] [left of=b2] {}
edge[<-] (b2)
edge[e] (bl);
%
%
%
%
\node [v] (br) [w,right of=b1] {};
\node [v] [above of=br] {}
edge[<-] (br)
edge[e] (br);
}}
\caption{Description of eight types of good subgraphs}
\label{fig:types}
\end{figure}
\begin{proof}
We proceed by induction on $|V(D)|$.
If $D$ has a subgraph $H$ that is a disjoint union of a short R-R
path and a short B-B path,
then $D\setminus V(H)$, the subgraph obtained by removing vertices
of $H$ from $D$,
is still good.
If $D=H$, then we have nothing to prove.
If $|V(D)\setminus V(H)|= 2$, then
$D$ is the disjoint union of a short R-R path, a short B-B path, and
a short cycle, and therefore $D$ is a directed graph of type (1).
If $|V(D)\setminus V(H)|\ge 4$, then
$H$ is a good subgraph of type (1).
Then we apply the induction
hypothesis to get a partition for $D\setminus V(H)$.
Therefore we may assume that $D$ has
no pair of a short B-B path and a short R-R path.
By symmetry, we may assume that $D$ has no short R-R path.
Then in each component, the number of red vertices is at most half
of the number of vertices. Thus, in order to check whether some
disjoint union of components is good, it is enough to count blue vertices.
Suppose that $D$ has a short cycle and a short B-B path.
We are done if $D$ is a graph of type (2).
Thus we may assume that $D$ has at least
eight vertices.
Let $X$ be the set of vertices in the pair of a short cycle and a short B-B
path. Then the subgraph of $D$ induced on $X$ is a subgraph of type (2).
Because $X$ has two blue vertices and two uncolored vertices,
$D\setminus X$ is good and has at least
four vertices.
By the induction hypothesis, we obtain a good partition of
$D\setminus X$. This together with the subgraph induced by $X$ is
a good partition of $D$.
We may now assume that either $D$ has no short cycles, or $D$ has no
short B-B path.
\noindent (Case 1) Suppose that $D$ has no short cycles.
The subgraph of $D$ consisting of all components other than short B-B paths can be partitioned
into good subgraphs $P_1$, $P_2$, $\ldots$, $P_k$ of type (5),
(6), or (7),
because the number of R-B paths is equal to the number of B-R paths.
We claim that short B-B paths can be
assigned to those subgraphs while maintaining each $P_i$ to
be good.
Suppose that $P_i$ has $2b_i$ blue vertices and $2n_i=|V(P_i)|$.
Notice that $b_i$ and $n_i$ are integers.
Let $x$ be the number of short B-B paths in $D$.
Since $D$ is good,
$2(2x+\sum_{i=1}^k 2b_i) \le \sum_{i=1}^k 2n_i+2x$
and therefore $x\le \sum_{i=1}^k (n_i-2b_i)$.
Each $P_i$ can afford to have $n_i-2b_i$ short B-B paths
to be good.
Overall all $P_1,\ldots,P_k$ can afford $\sum_{i=1}^k (n_i-2b_i)$ short B-B
paths; thus consuming all short B-B paths.
This proves the claim.
\noindent (Case 2)
Suppose $D$ has short cycles but has no short B-B paths.
If $D$ has at least two short cycles, then we can take all short
cycles as a subgraph of type (4)
and the subgraph of $D$ consisting of all components other than short cycles can be decomposed into
subgraphs, each of which is type (5), (6), or (7).
Thus we may assume $D$ has exactly one short cycle.
Since $D$ has at least
four vertices, $D$ must have
a subgraph $P$ consisting of components of $D$ that is one of the following type:
a monochromatic path,
a long cycle,
or a pair of a B-R path and an R-B path.
Then $P$ with the short cycle forms a good subgraph of type (2), (8),
or (3), respectively.
The subgraph of $D$ induced by all the remaining components
can be decomposed into subgraphs of type (5), (6), and (7).
\end{proof}
\subsection{Finding a Good Uniform Permutation}
Let $G$ be a connected $4$-vertex graph with a fixed point free
involution $\mu:V(G)\to V(G)$ such that $\lambda(v,\mu(v))=\deg(v)$
for each vertex $v$.
Let $w$ be a vertex of $G$ with the minimum degree
and
let $u$ be a vertex
of $G$ other than $w$ and $\mu(w)$.
Let $\sigma_w:\delta(w)\to\delta(\mu(w))$ be a bijection.
Let $e_1,e_2,\ldots,e_m$ be the edges incident with $w$
and let $f_1,f_2,\ldots,f_m$ be the edges incident with $\mu(w)$
so that $f_i=\sigma_w(e_i)$.
We construct an auxiliary directed graph $D$ on
the disjoint union of $\{e_1,e_2,\ldots,e_m\}$
and $\{f_1,f_2,\ldots,f_m\}$ as follows:
\begin{enumerate}[(i)]
\item For all $i\in \{1,2,\ldots,m\}$,
$D$ has an edge from $f_i$ to $e_i$.
\item If $e_i$ and $f_j$ denote the same edge in $G$,
then $D$ has an edge from $e_i$ to $f_j$.
\end{enumerate}
\begin{figure}
\centering
\tikzstyle{v}=[circle,solid, solid, fill=black, inner sep=0pt, minimum width=4pt]
\tikzstyle{every edge}=[-,draw]
\tikzstyle{l}=[circle,inner sep=0pt,fill=white]
\begin{tikzpicture}
%
\node [v] at (-3,0) (v0) {};
\node [v] at (-3,2) (v1) {}
edge (v0)
edge [bend right] (v0)
edge [bend left] (v0);
\node [v] at (-1,0) (v2) {}
edge node [l,pos=0.3] {$f_7$} (v0)
edge [bend right] node [l,pos=0.55] {$f_2$} (v0)
edge [bend left] node [l,pos=0.3,below=0.5pt] {$f_6$} (v0)
edge [in=-45-10,out=180-45+10] node [l,pos=0.3] {$f_5$} (v1)
edge [in=-45+10,out=180-45-10] node [l,pos=0.25,above] {$f_1$} (v1);
\node [v] at (-1,2) (v3) {}
edge node [l,pos=0.30] {$e_3$} node [l,pos=0.7] {$f_4$} (v2)
edge [bend left] node [l,right=.5pt,pos=0.25] {$e_2$} node [right=0.5pt,l,pos=0.7] {$f_3$} (v2)
edge [out=180+45+10,in=45-10] node [l,pos=.18] {$e_4$} (v0)
edge [out=180+45-10,in=45+10] node [l,pos=.34] {$e_5$} (v0)
edge [bend left] node [l,pos=0.5] {$e_1$} (v1)
edge node [l,pos=0.3] {$e_6$} (v1)
edge [bend right] node [l,above,pos=0.3] {$e_7$} (v1);
\draw (v0) node [left] {$\mu(u)$};
\draw (v1) node [left] {$u$};
\draw (v2) node [right] {$\mu(w)$};
\draw (v3) node [right] {$w$};
%
%
\tikzstyle{box}=[draw, shape=diamond,solid, color=black, inner sep=2pt, minimum width=2pt]
\tikzstyle{r}=[ rounded corners=3,draw, solid, color=red, inner sep=4pt, minimum width=4pt]
\tikzstyle{re}=[ rounded corners=3,draw, solid, color=red, fill=red!50, inner sep=4pt, minimum width=4pt]
\tikzstyle{b}=[draw, solid, color=blue, inner sep=4pt, minimum width=4pt]
\tikzstyle{be}=[draw, solid, color=blue, fill=blue!50, inner sep=4pt, minimum width=4pt]
\tikzstyle{every edge}=[->,>=stealth,draw,thick]
\tikzstyle{eb}=[->,thick,color=blue,out=120,in=-120,draw,>=stealth]
\tikzstyle{er}=[->,thick,color=red,out=120,in=-120,draw,>=stealth]
\foreach \x in {1,6,7} { \node [r] at (\x,2) (e\x) {} ;}
\foreach \x in {4,5} { \node [b] at (\x,2) (e\x) {}; };
\foreach \x in {2,3} { \node [box] at (\x,2) (e\x) {}; };
\foreach \x in {2,6,7} { \node [r] at (\x,0) (f\x) {} edge (e\x); };
\foreach \x in {1,5} { \node [b] at (\x,0) (f\x) {} edge (e\x); };
\foreach \x in {3,4} { \node [box] at (\x,0) (f\x) {} edge
(e\x); };
\foreach \x in {1,2,3,4,5,6,7} {
\node [above=2pt] at (e\x) {$e_{\x}$};
\node [below=2pt] at (f\x) {$f_{\x}$};
}
\node at (e2) {} edge (f3);
\node at (e3) {} edge (f4);
\end{tikzpicture}
\caption{A graph and its auxiliary directed graph at $w$}
\label{fig:ex}
\end{figure}
We have an example in Figure~\ref{fig:ex}.
It is easy to observe the following.
\begin{itemize}
\item Every vertex in $\{e_1,e_2,\ldots,e_m\}$ of $D$
has in-degree $1$.
\item Every vertex in $\{f_1,f_2,\ldots,f_m\}$ of $D$
has out-degree $1$.
\item A vertex $e_i$ of $D$ has out-degree $1$ if the edge $e_i$ of
$G$ is incident with $\mu(u)$,
and out-degree $0$ if otherwise.
\item A vertex $f_i$ of $D$ has in-degree $1$ if the edge $f_i$ of
$G$ is incident with $u$,
and in-degree $0$ if otherwise.
\end{itemize}
By the degree condition, $D$ is the disjoint union of odd
directed paths
and even directed cycles.
Let $r$ be the number of edges of $G$ joining $u$ and $w$
and let $b$ be the number of edges of $G$ joining $\mu(u)$ and $w$.
For each $i$, we color $e_i$
red
if it is incident with $u$
and
blue
if it is incident with $\mu(u)$.
Similarly for each $i$, we color $f_i$
blue
if it is incident with $u$
and
red
if it is incident with $\mu(u)$.
Clearly there are $r$ red vertices and $b$ blue vertices in $\{e_1,e_2,\ldots,e_m\}$.
Let $r'$ be the number of edges of $G$ joining $\mu(u)$ and $\mu(w)$
and let $b'$ be the number of edges of $G$ joining $u$ and $\mu(w)$.
We claim that $r'=r$ and $b'=b$.
Of course, there are $r$ red vertices and $b$ blue vertices in $\{f_1,f_2,\ldots,f_m\}$.
Since $\deg w=\deg \mu(w)$ and $\deg u=\deg \mu(u)$, we have
$r+b'=b+r'$ and $r+b=r'+b'$.
We deduce that $r=r'$ and $b=b'$.
We also assume that $G$ has $\deg(u)$ edge-disjoint paths
from $u$ to $\mu(u)$.
Therefore $|\delta (\{u,w\})|\ge |\delta(u)|$
and $|\delta(\{u,\mu(w)\})|\ge |\delta(u)|$.
This implies that
$b+b+(m-r-b)\ge b+r$ and
$r+r+(m-r-b)\ge b+r$.
Thus
\[
2r\le m \text{ and } 2b\le m.
\]
From now on, our goal is to describe a $w$-good permutation $\pi$ on $\delta(w)$
from a directed graph $D$ with a few extra edges.
\begin{LEM}\label{lem:good}
Let $D'$ be a directed graph obtained by adding one edge
from each vertex of out-degree $0$
to a vertex of in-degree $0$ with the same color
so that every vertex has in-degree $1$ and out-degree $1$ in $D'$.
Let $\pi$ be a permutation on $\delta (w)=\{e_1,e_2,\ldots,e_m\}$
so that
$\pi(e_i)=e_j$ if and only if
$D'$ has a directed walk from $e_i$ to $e_j$ of length two.
Then $\pi$ is $w$-good.
\end{LEM}
Let us call such a directed graph $D'$ a \emph{completion} of $D$.
A completion of $D'$ always exists,
because the number of red vertices of in-degree $0$ is equal to the
number of red vertices of out-degree $0$.
Clearly there are $r! \, b! $ completions of $D$.
\begin{proof}
It is enough to show that
if $D'$ has an edge $e$ from $e_i$ to $f_j$,
then $\{e_i,f_j\}$ is a matching of $G$.
If
$e\in E(D)$, then
$e_i=f_j$ and therefore $\{e_i,f_j\}=\{e_i\}$ is a matching of $G$.
If
$e\notin E(D)$, then
$e_i$ and $f_j$ should have the same color
and therefore $e_i$ and $f_j$ do not share any vertex.
\end{proof}
Out of $r!\,b!$ completions of $D'$, we wish to find a completion $D'$ of $D$ so that the
$w$-good permutation induced by $D'$ is uniform.
\begin{LEM}\label{lem:uniformtype}
If $D$ is a directed graph of type (1), (2), $\ldots$, (8) described
in Lemma~\ref{lem:oddpath}, then $D$ has a completion $D'$ so that
the induced $w$-good permutation is uniform.
\end{LEM}
\begin{proof}
We claim that for each type of a directed graph $D$, there is
a completion $D'$ of $D$ such that
its induced $w$-good permutation $\pi$ on $\delta(w)$ is uniform. Recall that a
$w$-good permutation $\pi$ is uniform if
$\pi^{(2)}$ has a list of orbits $X_1$, $X_2$, $\ldots$, $X_t$
satisfying the following conditions:
\begin{enumerate}[(i)]
\item If $\{x,y\}\in X_i$, then $x$ and $y$ do not share a vertex
other than $w$ or $\mu(w)$ in $G$.
\item
There is a constant $c>0$ such that for every edge $e\in \delta(w)$,
\[
\lvert \{ (X_i,F): 1\le i\le t,~F\in X_i \text{ and } e\in F\} \rvert = c.\]
\end{enumerate}
\medskip\noindent Case 1: Suppose that $D$ is of type (1) or (4) with $k$ components.
Then There is a unique completion $D'$ of $D$.
It is easy to verify that the list of all orbits of $\pi^{(2)}$
satisfies the conditions (i) and (ii) where $c=k-1$.
\medskip\noindent Case 2:
Suppose that $D$ is of type (2).
Then $D$ consists of a monochromatic path $P$ and one or two
short cycles.
A completion $D'$ of $D$ is unique, as it is obtained by adding an
edge from the terminal vertex of $P$ to the initial vertex of $P$.
Let $\pi$ be the permutation of $\delta(w)$ induced by $D'$.
Let $x_1,x_2,\ldots,x_m$ be the edges in $\delta(w)$ that are in
$P$ such that $\pi(x_i)=x_{i+1}$ for all $i=1,2,\ldots,m$ where $x_{m+1}=x_1$.
Let $y_1\in \delta(w)$ be the vertex in
the first short cycle such that $\pi(y_1)=y_1$. If $D$ has two
cycles, then let $y_2\in\delta(w)$
be the vertex in the second short cycle such that $\pi(y_2)=y_2$.
Then
$O_j=\{\{x_i,y_j\}: 1\le i\le m\}$ is an orbit of $\pi^{(2)}$
satisfying (i).
If $m>1$, then $O_P=\{\{x_i,x_{i+1}\}: 1\le i\le m\}$ is an orbit of $\pi^{(2)}$
satisfying (i) in which each $x_i$ appears twice if $m>2$ and each
$x_i$ appears once if
$m=2$.
If $D$ has only one cycle, then each $x_i$ appears once and $y_1$
appears $m$ times in $O_1$.
So if $m=1$, then $O_1$ satisfies (i) and (ii).
If $m=2$, then $O_1$ and $O_P$ form a list of orbits of $\pi^{(2)}$
satisfying (i) and (ii). If $m>2$, then a list of two copies of $O_1$ and
$(m-1)$ copies of $O_P$ satisfies (i) and (ii).
If $D$ has two short cycles, then in $O_1$ and $O_2$,
each $x_i$ appears twice and each $y_j$ appears $m$ times.
Notice that $\{\{y_1,y_2\}\}$ is an orbit of $\pi^{(2)}$.
If $m=1$, then a list of $O_1$, $O_2$, and $\{\{y_1,y_2\}\}$ satisfies
(i) and (ii).
If $m=2$, then a list of $O_1$ and $O_2$ satisfies (i) and (ii).
If $m>3$, then
a list of
two copies of $O_1$,
two copies of $O_2$,
and $(m-2)$ copies of $O_P$ satisfies (i) and (ii).
\medskip\noindent Case 3: If $D$ is of type (3), then $D$ has a unique
completion $D'$.
Let $\pi$ be the permutation of $\delta(w)$ induced by $D'$.
Let $y\in \delta(w)$ be a vertex of $D$ in the
short cycle such that $\pi(y)=y$.
Let $x_1,x_2,\ldots,x_m\in \delta(w)$ be the vertices on the long cycle in
$D'$ such that $\pi(x_i)=x_{i+1}$ for all $i=1,2,\ldots,m$ where
$x_{m+1}=x_1$. Since $D$ has two paths, $m>1$.
Then $O_P=\{\{x_i,x_{i+1}\}:i=1,2,\ldots,m\}$
and
$O_C=\{\{y,x_i\}: i=1,2,\ldots,m\}$ are orbits of $\pi^{(2)}$.
In $O_P$, each $x_i$ appears twice if $m>2$ and once if $m=2$.
In $O_C$ each $x_i$ appears once and $y$ appears $m$ times.
Now it is routine to create a list of orbits satisfying (i) and (ii)
by taking copies of $O_C$ and copies of $O_P$.
\medskip\noindent Case 4: Suppose that $D$ is of type (5) having both red and
blue vertices or $D$ is of type (7) or (8).
Let $D'$ be a completion of $D$ obtained by making each path
of $D$
to be a cycle of $D'$.
Let $x_1,x_2,\ldots,x_m\in \delta(w)$ be vertices in the long cycle
of $D'$ so that $\pi(x_i)=x_{i+1}$ for all $i=1,2,\ldots,m$ where
$x_{m+1}=x_1$.
Let $y_1,y_2,\ldots,y_k\in \delta(w)$ be vertices in short cycles of $D'$
such that $\pi(y_i)=y_i$.
Since $D$ is good, $k\le m$.
Let
$O_j=\{\{x_i,y_j\}: i=1,2,\ldots,m\}$ for $j=1,2,\ldots,k$
and
$O_P=\{\{x_i,x_{i+1}\}: i=1,2,\ldots,m\}$ where
$x_{m+1}=x_{1}$.
In the list of $O_1$, $O_2$, $\ldots$, $O_k$,
each $x_i$ appears $k$ times
and each $y_j$ appears $m$ times.
In $O_P$, each $x_i$ appears twice if $m>2$ and once if $m=2$.
To satisfy (i) and (ii), we can take a list of two copies of each $O_j$
for $j=1,2,\ldots,k$
and copies of $O_P$.
\medskip\noindent Case 5:
Suppose that $D$ is a directed graph of type (5) not having both red
and blue,
or $D$ is a directed graph of type (6).
Then $D$ has a completion $D'$ consisting of a single cycle.
Let $\pi$ be the permutation of $\delta(w)$ induced by $D'$.
Let $x_1,x_2,\ldots,x_m\in\delta(w)$ be vertices in $D$ such that
$\pi(x_i)=x_{i+1}$ for all $i=1,2,\ldots,m$.
We
$O_P=\{\{x_i,x_{i+\lfloor m/2\rfloor}\}: i=1,2,\ldots,m\}$ where
$x_{j+m}=x_{j}$ for all $j=1,\cdots,\lfloor m/2\rfloor$.
Then in $O_P$, each $x_i$ appears twice if $m$ is odd and once if
$m$ is even.
Moreover, since all the vertices of the same color appear
consecutively in $D'$ and the number of vertices of the same color
is at most half of $m$, $O_P$ never contains a pair $\{x_i,x_j\}$ of
vertices of the same color, red or blue.
Therefore $O_P$ satisfies (i) and (ii).
This completes the proof.
\end{proof}
\begin{LEM}\label{lem:uniform}
There exists a completion $D'$ of $D$ so that the $w$-good
permutation induced by $D'$ is uniform.
\end{LEM}
\begin{proof}
By Lemma~\ref{lem:oddpath}, $D$ can be partitioned into good
subgraphs $D_1$, $D_2$, $\ldots$, $D_t$ of type (1), (2), $\ldots$, (8).
Lemma~\ref{lem:uniformtype} shows that each $D_i$
admits a completion that induces a $w$-good uniform permutation
$\pi_i$
with a list $L_i$ of orbits of $\pi_{i}^{(2)}$ satisfying (i) and
(ii).
Let us assume that each vertex of $D_i$ appears $c_i>0$ times in
$L_i$.
Let $c=\operatorname{lcm}(c_1,c_2,\ldots,c_t)$.
Then let $L$ be the list of orbits obtained by
taking $c/c_i$ copies of $L_i$
for each $i=1,2,\ldots,t$.
Then $L$ satisfies (i) and (ii).
This proves the lemma.
\end{proof}
Now we are ready to prove Conjecture~\ref{con} for $4$-vertex graphs:
\begin{THM}\label{thm:4vertex}
Let $G$ be a connected $4$-vertex graph with a fixed point free
involution $\mu:V(G)\to V(G)$ and a
bijection $\sigma_v:\delta (v) \to \delta (\mu(v))$ for each vertex
$v$ such that
$\lambda(v,\mu(v))=\deg(v)$ and
$\sigma_{\mu(v)}=\sigma_v^{-1}$.
Then $G$ has a nonempty list of cycles
satisfying the following.
\begin{enumerate}[(a)]
\item For each pair of edges $e$ and $f$ incident with a vertex $v$,
the number of cycles in the list containing both $e$ and $f$
is equal to the number of cycles in the list containing both
$\sigma_{v}(e)$ and $\sigma_{v}(f)$.
\item Each edge of $G$ appears in the same number of cycles in the
list.
\item The list contains a cycle of length at least three.
\end{enumerate}
\end{THM}
\begin{proof}
Let $w$ be a vertex of minimum degree.
By Lemma~\ref{lem:uniform}, $G$ has a $w$-good uniform permutation
$\pi$ on $\delta (w)$.
By Lemma~\ref{lem:inductive},
$G$ has a nonempty list of cycles satisfying (a), (b), and (c).
\end{proof}
We remark that Conjecture~\ref{con} is true for subdivisions of connected $4$-vertex graphs because of (b).
By Proposition~\ref{prop:equiv} and Theorem~\ref{thm:4vertex}, we verify Tiling Conjecture for rank-two free groups:
\begin{COR}\label{cor:f2}
A minimal and diskbusting list of cyclically reduced words in $F_2$ is polygonal.
\end{COR}
Now Theorem~\ref{thm:main 1} is an immediate consequence of the following.
\begin{COR}\label{cor:f2 2}
For a list $U$ of words in $F_2$, the following are equivalent.
\begin{enumerate}
\item
The list $U$ is diskbusting.
\item
$D(U)$ contains a hyperbolic surface group.
\item
$D(U)$ is one-ended.
\end{enumerate}
\end{COR}
\begin{proof}
(1)$\Leftrightarrow$(3) is well-known and stated in~\cite{GW2010}, for example.
(1)$\Rightarrow$(2) follows from Corollary~\ref{cor:f2} and Theorem~\ref{thm:polygonal}.
By putting $n=2$ in the Proposition~\ref{prop:tilingequiv}, we have (2)$\Rightarrow$(1).
\end{proof}
\section{Final Remarks}\label{sec:final}
\subsection*{Minimality assumption in Tiling Conjecture}
A graph $G$ is \emph{$2$-connected} if $|V(G)|>2$,
$G$ is connected, and $G\setminus x$ is connected for every vertex $x$.
It is well-known that a list $U$ of cyclically reduced words in $F_n$ is diskbusting if and only if $W(\phi(U))$ is $2$-connected
for some $\phi\in\aut(F_n)$~\cite{Stong1997,Stallings1999}.
However, the minimality assumption in
Tiling Conjecture cannot be weakened to the
$2$-connectedness of the Whitehead graph;
this is equivalent to saying that $\lambda(v,\mu(v))=\deg(v)$
in Conjecture~\ref{con} cannot be relaxed to $2$-connectedness.
Daniel Kr\'al' \cite{Kral2010} kindly provided us Example~\ref{ex:connectivity} showing why this relaxation is not possible.
\begin{EX}\label{ex:connectivity}
Let $G$ be a $4$-vertex graph shown in Figure~\ref{fig:connectivity}.
For a vertex $v$ and edges $e\in \delta(v)$ and $f\in \delta(\mu(v))$, we let $\sigma_v(e)=f$ if and only if the number written on $e$ near $v$ coincides with the number written on $f$ near $\mu(v)$.
Actually, $G$ is the Whitehead graph of $a (ab^{-1})^3 b^{-2}$ with the associated connecting maps $\sigma_v$.
While $G$ is $2$-connected, one can verify that $G$ does not have a list of cycles satisfying the conclusion of Conjecture~\ref{con}. Note that $\lambda(a,\mu(a)) = 3 < 4 = \deg(a)$.
\end{EX}
\begin{figure}
\centering
\tikzstyle{v}=[circle,solid, solid, fill=black, inner sep=0pt, minimum width=4pt]
\tikzstyle{every edge}=[-,draw]
\tikzstyle{l}=[circle,inner sep=0pt,fill=white]
\begin{tikzpicture}
%
\node [v] at (-3,0) (v0) {};
\node [v] at (-3,2) (v1) {}
edge node [l,pos=0.2] {${}_1$} node [l,pos=0.8] {${}_2$} (v0)
;
\node [v] at (-1,0) (v2) {}
edge node [l,pos=0.2] {${}_1$} node [l,pos=0.8] {${}_3$} (v0)
edge [bend right] node [l,pos=0.2] {${}_5$} node [l,pos=0.8] {${}_1$} (v0)
edge [bend left] node [l,pos=0.2] {${}_2$} node [l,pos=0.8] {${}_4$} (v0)
;
\node [v] at (-1,2) (v3) {}
edge node [l,pos=0.2] {${}_4$} node [l,pos=0.8] {${}_3$} (v2)
edge [bend left] node [l,pos=0.2] {${}_5$} node [l,pos=0.8] {${}_4$} (v2)
edge [bend left] node [l,pos=0.2] {${}_3$} node [l,pos=0.8] {${}_4$} (v1)
edge node [l,pos=0.2] {${}_2$} node [l,pos=0.8] {${}_3$} (v1)
edge [bend right] node [l,pos=0.2] {${}_1$} node [l,pos=0.8] {${}_2$} (v1);
\draw (v0) node [left] {$\mu(a)$};
\draw (v1) node [left] {$a$};
\draw (v2) node [right] {$\mu(b)$};
\draw (v3) node [right] {$b$};
\end{tikzpicture}
\caption{Example~\ref{ex:connectivity}.}
\label{fig:connectivity}
\end{figure}
\subsection*{Control over positive degrees}
The following lemma states that Conjecture~\ref{con} can be strengthened to require each edge to appear the same number of times.
\begin{LEM}\label{lem:uniformnumber}
Suppose Conjecture~\ref{con} is true.
If $G$ is connected and has at least four vertices, then
the list of cycles in the conclusion of Conjecture~\ref{con} can be chosen so that each edge appears the same number of times.
\end{LEM}
\begin{proof}
Let $G$ be a given graph.
We claim that $G$ is $2$-connected.
Suppose not and let $x$ be a vertex such that $G\setminus x$ is disconnected.
Let $C$ be a component of $G\setminus x$ containing $\mu(x)$
and $D$ be a component of $G\setminus x$ other than $C$.
Since $G$ is connected, $x$ has an edge incident with a vertex in
$D$
and therefore $G$ can not have $\deg(x)$ edge-disjoint paths from
$x$ to $\mu(x)$, a contradiction.
This proves the claim.
Let $e_1,e_2,\ldots,e_m$ be the list of edges of $G$.
Let $G'$ be a graph obtained from $G$ by replacing each edge with a
path of length $m$.
Let $v_{ij}$ be the $j$-th internal vertex of the path of $G'$ representing
$e_i$
where $j=1,2,\ldots,m-1$.
We extend $\mu$ of $G$ to obtain $\mu'$ of $G'$ so that
$\mu'(v_{i,j})=v_{j,i-1}$ and
$\mu'(v_{j,i-1})=v_{i,j}$ for all $1\le j<i\le m$.
Since $G$ is $2$-connected, for each pair of edges $e$ and $f$ of $G$,
there is at least one cycle containing both $e$ and $f$.
Thus in $G'$, there are two edge-disjoint paths from $v_{i,j}$ to
$v_{j,i-1}$ for all $1\le j<i\le m$.
So we can apply Conjecture~\ref{con} to $G'$ and deduce that
each edge of $G$ is used the same number of times
because the number of cycles passing $v_{i,j}$
is equal to the number of cycles passing $v_{j,i-1}$ for all $1\le
j<i\le m$.
\end{proof}
Suppose $U$ is a polygonal list of
cyclically reduced words $u_1,\ldots,u_r$ in $F_n$.
There exists a closed $U$-polygonal surface $S$ obtained by a side-pairing on polygonal disks $P_1,\ldots,P_m$ equipped with an immersion $S^{(1)}\to \cay(F_n)/F_n$ as in Definition~\ref{defn:polygonal}.
We shall orient each $\partial P_i$ so that each $\partial P_i\to S^{(1)}\to \cay(F_n)/F_n$ reads a positive power of a word in $U$.
Partition $\mathcal{P}=\{P_1,\ldots,P_m\}$ into $\mathcal{P}_1,\ldots,\mathcal{P}_r$ so that
each $P_i\in \mathcal{P}_j$ reads a power of $u_j$.
If the polygonal disks in $\mathcal{P}_j$ read $u_j^{c_1},u_j^{c_2},\ldots,u_j^{c_k}$,
we say $c_1+c_2+\cdots+c_k$ is
the \emph{positive degree of $u_j$} with respect to the partition $\mathcal{P}=\mathcal{P}_1\cup\cdots\cup\mathcal{P}_r$.
\begin{PROP}\label{prop:uniformnumber}
Let $U$ be a minimal and diskbusting list of cyclically reduced words $u_1,\ldots,u_r$ in $F_n$ for some $n>1$.
We assume that either Tiling Conjecture is true, or $n=2$.
Then there exists a $U$-polygonal surface $S$ such that
the positive degree of every word in $U$ is the same
with respect to
a suitable partition of the polygonal disks in $S$.
\end{PROP}
\begin{proof}
Suppose that $W(U)$ has a list of cycles satisfying the conclusion of Conjecture~\ref{con} and each edge appears the same number of times, say $s$, in the list.
We consider a $U$-polygonal surface $S$ as in the proof of Lemma~\ref{lem:equiv}.
We define $\mathcal{P}_j$ to be the set of polygonal disks on $S$ so that in the construction of $S$, the corners of the polygonal disks in $\mathcal{P}_j$ correspond to the edges in $W(U)$ that are coming from $u_j$.
Then every word in $U$ has the positive degree $s$ with respect to this natural choice of a partition of the polygonal disks.
Hence, the proof follows from Part (b) of Theorem~\ref{thm:4vertex} and Lemma~\ref{lem:uniformnumber}.
\end{proof}
\subsection*{Non-virtually geometric words}
Let $H_n$ denote a $3$-dimensional handlebody of genus $n$.
A word $w$ in $F_n$ can be realized as an embedded curve $\gamma\subseteq H_n$.
A word $w$ is said to be \emph{virtually geometric}
if there exists a finite cover $p\co H'\to H_n$ such that
$p^{-1}(\gamma)$ is homotopic to a $1$-submanifold on the boundary of $H'$~\cite{GW2010}.
Using Dehn's lemma, Gordon and Wilton~\cite{GW2010} proved that if $w\in F_n$ is diskbusting and virtually geometric, then $D(\{w\})$ contains a surface group;
this also follows from the fact that a minimal diskbusting geometric word is polygonal~\cite{Kim2009}.
On the other hand, Manning provided examples of minimal diskbusting, non-virtually
geometric words as follows.
\begin{THM}[Manning~\cite{Manning2009}]\label{thm:manning}
If the Whitehead graph of a word $w$ in $F_n$ is non-planar, $k$-regular and $k$-edge-connected for some
$k\ge3$, then $w$ is not virtually geometric.
\end{THM}
Here, a graph $G$ is said to be \emph{$k$-edge-connected} if $\lvert\delta(X)\rvert \ge k$ for all $\emptyset\neq X\subsetneq V(G)$. So, if $W(U)$ is $k$-regular and $k$-edge-connected for a list $U$ of words in $F_n$, then $U$ is minimal and diskbusting.
Hence even for the words provided by Manning, Theorem~\ref{thm:main 2} finds hyperbolic surface groups in the corresponding doubles:
\begin{COR}\label{cor:regular}
If the Whitehead graph of a list $U$ of words in $F_n$ is $k$-regular and $k$-edge-connected for some
$k\ge3$, then $U$ is polygonal. In particular, $D(U)$ contains a hyperbolic surface group.
\end{COR}
\subsection*{Existence of separable surface subgroups}
A subgroup $H$ of a group $G$ is said to be \emph{separable} if $H$ coincides with the intersection of all the finite-index subgroups of $G$ containing $H$.
If every finitely generated subgroup of $G$ is separable, we say $G$ is \emph{subgroup separable}.
The \emph{Virtual Haken Conjecture} for a closed hyperbolic $3$-manifold $M$ asserts that there exists a $\pi_1$-injective, homeomorphically embedded, closed hyperbolic surface in some finite cover of $M$~\cite{Scott1978}; this is a main motivation for Question~\ref{que:gromov}. If $\pi_1(M)$ contains a \emph{separable} hyperbolic surface subgroup, then it is known that a closed hyperbolic surface $\pi_1$-injectively embeds into a finite cover of $M$~\cite{Scott1978}.
So, it is natural to augment Question~\ref{que:gromov} as follows.
\begin{QUE}\label{que:gromov2}
Does every one-ended word-hyperbolic group contain a separable
hyperbolic surface group?
\end{QUE}
Since $X(U)$ has a non-positively curved square complex structure and also decomposes a graph of free groups with cyclic edge groups, $D(U)$ is subgroup separable by~\cite{Wise2000}.
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\section{Introduction}
Let $D$ be a division algebra finite-dimensional over its center $K$. Then,
$$
\SK(D) \ = \ \{ d\in D^* \mid\Nrd_D(d) = 1\} \, \big / \, [D^*,D^*],
$$
where $\Nrd_D$ denotes the reduced norm and $[D^*,D^*]$ is the
commutator group of the group of units $D^*$ of~$D$.
If $D$ has a unitary involution $\tau$ (i.e., an involution $\tau$ on
$D$ with $\tau|_K \ne \operatorname{id}$), then the unitary $\SK$ for $\tau$~on~$D$~
is
\begin{equation}\label{unitarydef}
\SK(D, \tau) \ = \ \Sigma'_\tau(D) \, \big/ \, \Sigma_\tau(D),
\end{equation}
where
$$
\Sigma _\tau'(D) \ = \ \{d\in D^*\mid \Nrd_D(d) = \tau(\Nrd_D(d))\}
\qquad \text{and} \qquad \Sigma_\tau(D) \ = \ \big\langle \{ d\in D^*\mid
d = \tau(d)\}\big\rangle.
$$
The groups $\SK(D)$ and $\SK(D,\tau)$ are of considerable interest
as subtle invariants of $D$, and as reduced Whitehead groups for certain
algebraic groups (cf.~\cite{tits}, \cite{platsurvey}, \cite{gille}).
In this paper we will prove formulas for $\SK(\mathsf{E})$ and $\SK(\mathsf{E}, \tau)$
for $\mathsf{E}$ a semiramified graded division algebra $\mathsf{E}$ of finite rank over
its center. In view of the isomorphisms in \cite[Th.~4.8]{hazwadsworth} and
\cite[Th.~3.5]{I}, the formulas for $\mathsf{E}$ imply analogous formulas for
$\SK$ and unitary $\SK$ for a tame semiramified division algebra $D$ over a
Henselian valued field $K$. The formulas thus obtained in the Henselian case
generalize ones given by Platonov for $\SK(D)$ and Yanchevski\u\i\
for $\SK(D,\tau)$ for bicyclic decomposably
semiramified division algebras over iterated Laurent fields.
Most of our work will be in the unitary setting,
which is not as well developed as the nonunitary setting.
Ever since Platonov gave examples of division algebras
with $\SK(D)$ nontrivial there has been ongoing interest in
$\SK$. Platonov showed in \cite{platrat} that
nontriviality of $\SK(D)$ implies that the algebraic group
group $\text{SL}_1(D)$ (with $K$-points $\{d\in D\mid \Nrd_D(d) = 1\}$)
is not a rational variety. Also, Voskresenski\u\i\ showed in
\cite{V1} and \cite[Th., p.~186]{V2} that $\SK(D)\cong \text{SL}_1(D)/R$,
the group of \mbox{$R$-equivalence} classes of the variety $\text{SL}_1(D)$.
The corresponding unitary result, $\SK(D, \tau) \cong
\text{SU}_1(D, \tau)/R$ was given in \cite[Remark, p.~537]{yy} and
\cite[Th.~5.4]{cm}. More recently, Suslin in \cite{suslin1} and \cite{suslin2}
has related $\SK(D)$ to certain $4$-th cohomology groups associated to
$D$, and has conjectured that whenever the Schur index~$\operatorname{ind}(D)$
is not square-free then $\SK(D\otimes_KL)$ is nontrivial for some
field $L\supseteq K$. (This has been proved by Merkurjev
in \cite{merkconj1} and \cite{merkconj2} if $4|\operatorname{ind}(D)$, but remains
open otherwise.) Nonetheless, explicit computable formulas for $\SK(D)$
and $\SK(D,\tau)$ have remained elusive, and are principally available,
when $\operatorname{ind}(D) >4$, only for algebras over Henselian fields
(cf.~\cite{ershov} and \cite[Th.~3.4]{hazwadsworth}) and quotients of
iterated twisted polynomial algebras (cf.~\cite[Th.~5.7]{hazwadsworth}).
Platonov's original examples with nontrivial $\SK$
in \cite{platTannaka}
and \cite{platonov} were division
algebras $D$ over a twice iterated
Laurent power series field $K = k(((x))((y))$, where $k$ is a
local or global field
or an infinite algebraic extension of such a field. His $K$ has a
naturally associated rank $2$ Henselian valuation which extends uniquely
to a valuation on $D$. With respect to this valuation, his
$D$ is tame and \lq\lq decomposably semiramified" and, in addition, its residue division algebra
$\overline D$ is a field with $\overline D = L_1\otimes _kL_2$, where each $L_i$ is
cyclic Galois over $k$. His basic formula for such $D$ is:
\begin{equation}\label{platformula}
\SK(D) \, \cong\, \operatorname{Br}(\overline D/k) \big/ \big[\operatorname{Br}(L_1/k)\cdot \operatorname{Br}(L_2/k)\big],
\end{equation}
where $k$ is any field,
$\operatorname{Br}(k)$ is the Brauer group of $k$, and for a field $M\supseteq k$,
$\operatorname{Br}(M/k)$ denotes the relative Brauer
group
$\ker(\operatorname{Br}(k) \to \operatorname{Br}(M))$, a subgroup of $\operatorname{Br}(k)$.
That $D$ is tame and semiramified means that\break
${[\overline D \! :\! \overline K] = | \Gamma_D \! :\! \Gamma_K| = \sqrt{[D\! :\! K]}}$ and
$\overline D$ is a field separable (hence abelian Galois) over $\overline K$,
where $\Gamma_D$ is the value group the valuation on $D$. We say that $D$ is
{\it decomposably semiramified} (abbreviated~DSR) if $D$~is a tensor product
of cyclic tame and semiramified division algebras.
Using \eqref{platformula} with $k$ a global field or an algebraic
extension of a
global field, Platonov showed in \cite{plat76} that every finite
abelian group and some infinite abelian groups
of bounded torsion appear as $\SK(D)$ for suitable~$D$.
Shortly after Platonov's work, Yanchevski\u\i\ obtained in \cite{yannounc},
\cite{y}, \cite{yinverse}
similar results for
the unitary $\SK$ for similar types of division algebras, namely
$D$ decomposably semiramified over $K = k((x))((y))$, with $k$ any
field, given that $D$ has a unitary involution $\tau$ with fixed field $K^\tau
= \ell((x))((y))$ for some field $\ell \subseteq k$ with
$[k\! :\! \ell] = 2$. Yanchevski\u\i's key formula (when $\overline D =
L_1\otimes_k L_2$ as above) is:
\begin{equation}\label{yanchformula}
\SK(D,\tau) \cong \operatorname{Br}(\overline D/k;\ell) \big/ \big[\operatorname{Br}(L_1/k;\ell)\cdot
\operatorname{Br}(L_2/k;\ell)\big],
\end{equation}
where for a field $M\supseteq k$, $\operatorname{Br}(M/k;\ell) =
\ker\big(\text{cor}_{k\to \ell}\colon
\operatorname{Br}(M/k) \to \operatorname{Br}(\ell)\big)$; this is the subgroup of $\operatorname{Br}(k)$ consisting
of the classes of central simple $k$-algebras split by $M$ and having a unitary involution
$\tau$ with fixed field ${k^\tau = \ell}$.
He used this in \cite{yinverse} with $k$~and~$\ell$ global fields
to show that any finite abelian group is realizable as $\SK(D,\tau)$.
He obtained remarkably similar analogues for the unitary $\SK$ to
other results of Platonov for the nonunitary $\SK$, but generally with
substantially more difficult and intricate proofs.
Ershov showed in \cite{e} and \cite{ershov} that the natural setting
for viewing Platonov's examples of nontrivial $\SK(D)$ is that of
tame division algebras $D$ over a Henselian valued field $K$.
(Platonov considered his $K$
in a somewhat cumbersome way as a field with complete discrete
valuation with residue field which also has a complete discrete valuation.)
The Henselian valuation on $K$ has a unique extension to a valuation
on~$D$, and Ershov gave exact sequences that describe $\SK(D)$ in terms
of various data related to the residue division ring $\overline D$. In particular
he showed (combining \cite[p.~69, (6) and Cor.~(b)]{ershov}) that if
$D$ is DSR (with $K$ Henselian), then
\begin{equation}\label{DSRSK1}
\SK(D) \, \cong\, \widehat H^{-1}(\operatorname{Gal}(\overline D/\overline K), \overline D^*).
\end{equation}
More recently, there has been work on associated graded rings of
valued division algebras, see especially \cite{hwcor}, \cite{mounirh},
\cite{tignolwadsworth}. The tenor of this work has been that for
a tame division algebra $D$ over a Henselian valued field, most
of the structure of $D$ is inherited by its associated graded ring
$\operatorname{{\sf gr}}(D)$, while $\operatorname{{\sf gr}}(D)$ is often much easier to work with
than $D$ itself. This theme was applied quite recently by
R.~Hazrat and the author in \cite{hazwadsworth} and \cite{I} to
calculations of $\SK$ and unitary $\SK$. It was shown in
\cite[Th.~4.8]{hazwadsworth} that if $D$ is tame over $K$ with respect to
a Henselian valuation, then $\SK(D) \cong\SK(\operatorname{{\sf gr}}(D))$; the corresponding
result for unitary $\SK$ was proved in \cite[Th.~3.5]{I}. Calculations of
$\SK$ in the graded setting are significantly easier and more transparent
than in the original ungraded setting, allowing almost effortless recovery
of Ershov's exact sequences, with some worthwhile improvements. Notably,
it was shown in \cite[Cor.~3.6(iii)]{hazwadsworth} that if
$K$ is Henselian and
$D$ is tame and semiramified (but not necessarily DSR), then there is an
exact sequence
\begin{equation}\label{hwsemiram}
H\wedge H \, \longrightarrow\, \widehat H^{-1}(H, \overline D^*) \, \longrightarrow\, \SK(D)
\, \longrightarrow\, 1, \qquad
\text{where}\qquad H \,=\, \operatorname{Gal}(\overline D/\overline K) \, \cong\,
\Gamma_D/\Gamma_K.
\end{equation}
When $D$ is DSR,
the image of $H\wedge H$ in $\widehat H^{-1}(H, \overline D^*)$ is trivial, yielding
\eqref{DSRSK1}. Then, Platonov's formula
\eqref{platformula} is obtained from \eqref{DSRSK1} via the following isomorphism: For a field
$M = L_1\otimes_k L_2$ where each $L_i$ is cyclic Galois over $k$,
\begin{equation}\label{bicycliccohom}
\widehat H^{-1}(\operatorname{Gal}(M/k), M^*) \ \cong \ \operatorname{Br}(M/k)/\big[\operatorname{Br}(L_1/k)
\cdot \operatorname{Br}(L_2/k) \big].
\end{equation}
See \eqref{bicyclicbrel}--\eqref{njnj} below for a short proof of
\eqref{bicycliccohom} using facts about abelian crossed products.
When $D$ is semiramified but not DSR, the contribution of the first
term in \eqref{hwsemiram} can be better understood in terms of the
$I\otimes N$ decomposition of $D$: Our semiramified $D$ is equivalent
in $\operatorname{Br}(K)$ to $I\otimes_K N$, where $I$~is inertial (= unramified) over
$K$ and $N$ is DSR, so $\overline N \cong \overline D$ and $\Gamma_N = \Gamma_D$. Thus,
the $\widehat H^{-1}$ term in~\eqref{hwsemiram} coincides with
$\SK(N)$. We will show in Cor.~\ref{henselcor}(i) below that the image of
$H\wedge H$ in $\widehat H^{-1}(H, \overline D^*)$
is expressible in terms of parameters describing the residue algebra
$\overline I$ of $I$, which is central simple over $\overline K$ and
split by the field~$\overline D$. This $\overline I$ does not show up within
$D$ or $\overline D$, but nonetheless has significant influence on the
structure of $D$. (For example, it determines whether $D$ can be a crossed product
or nontrivially decomposable---see \cite[pp.~162--166, Remarks~5.16]{jw}.
In \cite{jw} DSR algebras were called \lq\lq nicely semiramified," and abbreviated
NSR. We prefer the more descriptive term decomposably semiramified.)
Also,
$\overline I$~is not uniquely determined by~$D$, but determined only modulo the
group $\operatorname{Dec}(\overline D/\overline K)$ of simple $\overline K$-algebras which
\lq\lq decompose according to~$\overline D$\,"---see \S3 below for the definition
of $\operatorname{Dec}(\overline D/\overline K)$. In the bicyclic case where $D$ is semiramified and
$K$ Henselian and $\overline D \cong L_1\otimes _{\overline K} L_2$ with each
$L_i$ cyclic Galois over~$\overline K$, we will show in Cor.~\ref{henselcor}(ii)
that
\begin{equation}\label{ubicyclicsemiram}
\SK(D) \ \cong \ \operatorname{Br}(\overline D/\overline K) \big/ \big[\operatorname{Br}(L_1/\overline K)
\cdot \operatorname{Br}(L_2/\overline K)\cdot\langle[\overline I]\rangle\big],
\end{equation}
which is a natural generalization of Platonov's formula \eqref{platformula}.
The principal aim of this paper is to prove unitary versions of the
results described above for nonunitary~$\SK$, especially \eqref{DSRSK1},
\eqref{bicycliccohom}, and \eqref{ubicyclicsemiram}. The
unitary versions of these are, respectively, Th.~\ref{unitaryDSR}(i),
Prop.~\ref{unitarybicyclic}, and Th.~\ref{main}(ii). Along the way, it
will be necessary to develop a unitary version of the $I\otimes N$
decomposition for semiramified division algebras. This is given in Prop.~
\ref{uINdecomp}. In the final section we will apply some of these formulas
to give an example where the natural map $\SK(D,\tau) \to \SK(D)$ is
not injective.
This paper is a sequel to \cite{I}, which describes the equivalence of the
graded setting and the Henselian valued setting for computing
unitary $\SK$, and has calculations of $\SK(D,\tau)$ for several cases other
than the semiramified one considered here. However, the present paper can be read
independently of \cite{I}.
We will work here primarily with
graded division algebras, where the calculations are more transparent
than for valued algebras.
Some basic background on the graded objects is given in \S\ref{graded}.
But we reiterate that by \cite[Th.~3.5]{I} every result in the graded
setting yields a corresponding result for tame division algebras over
Henselian valued fields. While what is proved here is for a rather
specialized type of algebra, we note that detailed knowledge of $\SK$ in special cases
sometimes has wider consequences. See, e.g., the paper~\cite{rty}
where Suslin's conjecture is reduced to the case of cyclic algebras.
See also \cite[Th.~4.11]{w}, where
the proof of nontriviality of a cohomological invariant of Kahn
uses a careful analysis of $\SK(D)$ for the $D$ in Platonov's
original example.
From the perspective of algebraic groups, it is perhaps unsurprising that
there should be results for the unitary $\SK$ similar to those in the
nonunitary case. For, $\text{SL}_1(D)$ is a group of inner
type $A_{n-1}$ where $n = \deg(D)$, and $\text{SU}_1(D, \tau)$ is a group
of outer type $A_{n-1}$ (cf.~\cite[Th.~(26.9)]{kmrt}).
Nonetheless, the similarities in
formulas for $\SK(D,\tau)$ given in Yanchevski\u\i's work
and in \cite{I} and here to those for $\SK(D)$ seem quite striking.
Likewise, the results by
Rost on $\SK(D)$ for biquaternion algebras (see \cite[\S17A]{kmrt}) and
by Merkurjev in \cite{merkurjev}
for arbitrary algebras of degree $4$, have a unitary analogue
proved by Merkurjev in \cite{merk2}.
This suggests that a further
analysis of the unitary $\SK$ would be worthwhile, notably to
investigate whether there are unitary versions of the deep results
by Suslin \cite{suslin2} and
Kahn \cite{kahn} relating $\SK(D)$ to higher \'etale cohomology groups.
\section{Graded division algebras and simple algebras}\label{graded}
We will be working throughout with graded algebras
graded by a torsion-free abelian group. We now set
up the terminology for such algebras and recall some
of the basic facts we will use frequently.
Let $\Gamma$ be a torsion-free abelian group, and let
$\mathsf{R}$ be a ring graded by $\Gamma$, i.e., $\mathsf{R} = \bigoplus
_{\gamma\in \Gamma}\mathsf{R}_\gamma$, where each $\mathsf{R}_\gamma$ is an
additive subgroup of $\mathsf{R}$ and $\mathsf{R}_\gamma \cdot \mathsf{R}_\delta
\subseteq \mathsf{R}_{\gamma +\delta}$ for all $\gamma, \delta\in \Gamma$.
The homogeneous elements of $\mathsf{R}$ are those lying in
$\bigcup_{\gamma\in \Gamma}\mathsf{R}_\gamma$. If $r\in \mathsf{R}_\gamma$, $r\ne 0$,
then we write $\deg(r) = \gamma$. The grade set of
$\mathsf{R}$ is $\Gamma_\mathsf{R} = \{ \gamma \in \Gamma\mid
\mathsf{R}_\gamma \ne \{0\}\}$. (We work only with gradings by
torsion-free abelian groups because we are interested
in the associated graded rings determined by
valuations on division algebras; for such rings the
grading is indexed by the value group of the valuation, which is
torsion-free abelian.) If $\mathsf{R}' = \bigoplus_{\gamma\in \Gamma}
\mathsf{R}'_\gamma$ is another graded ring, a {\it graded ring
homomorphism} $\varphi\colon \mathsf{R}\to \mathsf{R}'$ is a
ring homomorphism such that $\varphi(\mathsf{R}_\gamma) \subseteq
\mathsf{R}'_\gamma$ for all $\gamma\in \Gamma$. If $\varphi$~is an
isomorphism, we say that $\mathsf{R}$ and $\mathsf{R}'$ are graded
ring isomorphic, and write $\mathsf{R} \cong_g \mathsf{R}'$.
For example, if $a\in \mathsf{R}$ is homogeneous and $a\in \mathsf{R}^*$,
the group of units of $\mathsf{R}$, then the map $\intt(a)\colon
\mathsf{R}\to\mathsf{R}$ given by $r\mapsto ara^{-1}$ is a graded ring
automorphism of $\mathsf{R}$.
A graded ring $\mathsf{E} = \bigoplus_{\gamma \in \Gamma} \mathsf{E}_\gamma$
is said to be a {\it graded division ring} if every
nonzero homogeneous element of $\mathsf{E}$ lies in the
multiplicative group $\mathsf{E}^*$ of units of $\mathsf{E}$. See
\cite{hwcor} for background on graded division ring and
proofs of the properties mentioned here.
Notably (as $\Gamma$ is torsion-free abelian), $\mathsf{E}$ has no
zero divisors, $\mathsf{E}^*$~consists entirely of homogeneous
elements, $\Gamma _\mathsf{E}$ is a subgroup of $\Gamma$, \,
$\mathsf{E}_0$ is a division ring, and each nonzero
homogeneous component
$\mathsf{E}_\gamma$ of $\mathsf{E}$ is a $1$-dimensional left and right
$\mathsf{E}_0$-vector space. Furthermore, if $\mathsf{M}$ is any left graded
$\mathsf{E}$- module (i.e., an $\mathsf{E}$-module such that
$\mathsf{M} = \bigoplus_{\gamma \in \Gamma}\mathsf{M}_\gamma$ with
$\mathsf{E}_\gamma \!\cdot\! \mathsf{M}_\delta \subseteq \mathsf{M}_{\gamma +\delta}$
for all~$\gamma, \delta \in \Gamma$), then $\mathsf{M}$ is a free
$\mathsf{E}$-module with a homogeneous base, and any two such bases
have the same cardinality; this cardinality is called the
dimension of $\mathsf{M}$ and denoted $\dim_\mathsf{E}(\mathsf{M})$.
Any such $\mathsf{M}$ is therefore called a left graded $\mathsf{E}$-vector
space.
A commutative graded division ring $\mathsf{T} = \bigoplus
_{\gamma \in \Gamma}\mathsf{T}_\gamma$ is called a {\it graded field}.
Such a $\mathsf{T}$ is an integral domain;
let $q(\mathsf{T})$ denote the quotient field of $\mathsf{T}$.
A graded ring $\mathsf{A}$ which is a
$\mathsf{T}$-algebra is called a {\it graded $\mathsf{T}$-algebra} if the module
action of $\mathsf{T}$ on $\mathsf{A}$ makes $\mathsf{A}$ into a graded
$\mathsf{T}$-module. When this occurs, $\mathsf{T}$ is graded isomorphic
to a graded subring of the center of $\mathsf{A}$, which is denoted
$Z(\mathsf{A})$. { \it All graded $\mathsf{T}$-algebras considered in this
paper are assumed to be finite-dimensional graded
$\mathsf{T}$-vector spaces.} Note that if $\mathsf{A}$ is a graded
\mbox{$\mathsf{T}$-algebra},
then $\mathsf{A} \otimes _\mathsf{T} q(\mathsf{T})$ is a $q(\mathsf{T})$-algebra of the same dimension.
That is, $[\mathsf{A}\! :\!\mathsf{T}] = [\mathsf{A} \otimes _\mathsf{T} q(\mathsf{T})\! :\! q(\mathsf{T})]$,
where $[\mathsf{A}\! :\!\mathsf{T}]$~denotes $\dim_\mathsf{T}(\mathsf{A})$ and
$[\mathsf{A} \otimes _\mathsf{T} q(\mathsf{T})\! :\! q(\mathsf{T})] =
\dim_{q(\mathsf{T})}(\mathsf{A} \otimes _\mathsf{T} q(\mathsf{T}))$.
Note that if $\mathsf{A}$ and $\mathsf{B}$ are graded algebras over a graded
field $\mathsf{T}$ then $\mathsf{A} \otimes _\mathsf{T} \mathsf{B}$ is also a graded
$\mathsf{T}$-algebra with $(\mathsf{A} \otimes _\mathsf{T} \mathsf{B})_\gamma
= \sum _{\delta\in \Gamma} \mathsf{A}_\delta \otimes_{\mathsf{T}_0} \mathsf{B}_{\gamma - \delta}$
for all $\gamma \in \Gamma$. Clearly, $\Gamma_{\mathsf{A} \otimes _\mathsf{T} \mathsf{B}}
= \Gamma_\mathsf{A} + \Gamma_\mathsf{B}$. Also, if $C$ is a finite-dimensional
${\mathsf{T}_0}$-algebra, then $C\otimes _{\mathsf{T}_0} \mathsf{A}$ is a graded
$\mathsf{T}$-algebra with $(C\otimes _{\mathsf{T}_0} \mathsf{A})_\gamma = C\otimes _{\mathsf{T}_0}
\mathsf{A}_\gamma$ for all $\gamma \in \Gamma$, and $\Gamma_{C\otimes _{\mathsf{T}_0} \mathsf{A}}
= \Gamma_\mathsf{A}$.
A graded $\mathsf{T}$-algebra $\mathsf{A}$ is said to
be {\it simple} if it has no homogeneous two-sided ideals
except $\mathsf{A}$ and~$\{0\}$. $\mathsf{A}$ is called a
{\it central simple} $\mathsf{T}$-algebra if in addition its center
$Z(\mathsf{A})$ is $\mathsf{T}$.
The theory of simple graded algebras
is analogous to the usual theory of finite-dimensional simple
algebras. This is described in \cite[\S 1]{hwcor}, where proofs of
the following facts can be found. There is a graded
Wedderburn Theorem for simple graded algebras: Any such
$\mathsf{A}$ is graded isomorphic to $\mathsf{End}_\mathsf{E}(\mathsf{M})$ for some
finite-dimensional graded vector space $\mathsf{M}$ over a graded
division algebra $\mathsf{E}$, and $\mathsf{E}$ is unique up to graded
isomorphism. Also, while $\mathsf{A}_0$ need not be simple, it is
always semisimple, and ${\mathsf{A}_0\cong \prod_{j= 1}^s M_{\ell_j}
(\mathsf{E}_0)}$ for some $\ell_j\times \ell_j$ matrix rings
over $\mathsf{E}_0$
(see the proof of Lemma~\ref{zerosimple} below).
We write $[\mathsf{A}]$ for the equivalence class of~$\mathsf{A}$ under
the equivalence relation $\sim_g$ given by:
$\mathsf{A} \sim_g \mathsf{A}'$ iff
$\mathsf{A}\cong_g \mathsf{End}_\mathsf{E}(\mathsf{M})$ and ${\mathsf{A}' \cong_g
\mathsf{End}_\mathsf{E}(\mathsf{M}')}$ for the same graded division algebra~
$\mathsf{E}$. The Brauer group (of graded algebras)
for $\mathsf{T}$ is
$$
{\mathsf{Br}(\mathsf{T}) = \{[\mathsf{A}]\mid \mathsf{A} \text{ is
a graded central simple $\mathsf{T}$-algebra}\}},
$$
\vfill\eject
\noindent with the
well-defined group operation $[\mathsf{A}]\cdot [\mathsf{A}'] =
[\mathsf{A} \otimes_\mathsf{T} \mathsf{A}']$. When $\mathsf{A} \cong_g \mathsf{End}_\mathsf{E}(\mathsf{M})$ as
above, then $[\mathsf{A}] = [\mathsf{E}]$, and up to graded isomorphism
$\mathsf{E}$ is the only graded division algebra
with $\mathsf{A} \sim_g \mathsf{E}$.
There is a graded version
of the Double Centralizer Theorem, see \cite[Prop.~1.5]{hwcor}
and also the Skolem-Noether Theorem, see \cite[Prop.~1.6]{hwcor}.
We recall the latter, since it has an added condition not appearing
in the ungraded version.
\begin{proposition}[{\cite[Prop.~1.6(b),(c)]{hwcor}}]\label{grSN}
Let $\mathsf{A}$ be a central simple graded algebra over the graded field~$\mathsf{T}$,
and let $\mathsf{B}$ and $\mathsf{B}'$ be simple graded $\mathsf{T}$-subalgebras of $\mathsf{A}$. Let
$\mathsf{C} = C_\mathsf{A}(\mathsf{B})$, the centralizer of $\mathsf{B}$ in $\mathsf{A}$,
and let $\mathsf{Z} = Z(\mathsf{C}) = Z(\mathsf{B})$ and
$\mathsf{C}' = C_\mathsf{A}(\mathsf{B}')$. Let $\alpha\colon
\mathsf{B} \to \mathsf{B}'$ be a graded $\mathsf{T}$-algebra isomorphism. Then there is
a homogeneous $a\in A^*$ such that $\alpha(b) = aba^{-1}$ for all $b\in \mathsf{B}$
if and only if there is a graded $\mathsf{T}$-algebra isomorphism
$\gamma\colon \mathsf{C} \to \mathsf{C}'$ such that $\gamma|_Z = \alpha|_\mathsf{Z}$.
Such a $\gamma$ exists whenever $\mathsf{C}_0$ is a division ring.
\end{proposition}
If $\mathsf{E}$ is a graded division algebra over a graded field
$\mathsf{T}$, we write $[\mathsf{E}\! :\!\mathsf{T}]$ for $\dim_\mathsf{T}(\mathsf{E})$.
A basic fact is the Fundamental Equality
\begin{equation}\label{fun}
[\mathsf{E}\! :\! \mathsf{T}] \ = \ [\mathsf{E}_0\! :\!{\mathsf{T}_0}] \, |\Gamma_\mathsf{E}\! :\!\Gamma_\mathsf{T}|,
\end{equation}
where $|\Gamma_\mathsf{E}\! :\!\Gamma_\mathsf{T}|$ denotes the index in
$\Gamma_\mathsf{E}$ of its subgroup $\Gamma_\mathsf{T}$. Also,
it is known that $Z(\mathsf{E}_0)$ is abelian Galois over
${\mathsf{T}_0}$, and there is a well-defined group
epimorphism
\begin{equation}\label{Theta}
\Theta_\mathsf{E} \colon\Gamma_\mathsf{E} \to \operatorname{Gal}(Z(\mathsf{E}_0)/{\mathsf{T}_0})
\quad\text{ given by } \Theta_\mathsf{E}(\gamma)(z) = aza^{-1}
\text { for any $z\in Z(\mathsf{E}_0)$ and $a\in \mathsf{E}_\gamma\setminus
\{0\}$}.
\end{equation}
Clearly, $\Gamma_\mathsf{T}\subseteq \ker(\Theta_\mathsf{E})$, so
$\Theta_\mathsf{E}$ induces an epimorphism of finite groups
$\overline \Theta_\mathsf{E}\colon\Gamma_\mathsf{E}/\Gamma_\mathsf{T} \to \operatorname{Gal}(Z(\mathsf{E}_0)/\mathsf{E}_0)$.
The terminology for different cases in \eqref{fun} is carried
over from valuation theory: We say that a graded
field $\mathsf{S}\supseteq \mathsf{T}$ is {\it inertial over $\mathsf{T}$}
if $[\mathsf{S}_0\! :\!{\mathsf{T}_0}] = [\mathsf{S}\! :\!\mathsf{T}]<\infty$ and the
field $\mathsf{S}_0$ is separable over ${\mathsf{T}_0}$. When this occurs,
$\Gamma_\mathsf{S} = \Gamma_\mathsf{T}$, and
the graded monomorphism $\mathsf{S}_0\otimes _{\mathsf{T}_0} \mathsf{T} \to
\mathsf{S}$ given by multiplication in $\mathsf{S}$ is surjective
by dimension count; so $\mathsf{S}\cong_g \mathsf{S}_0\otimes _{\mathsf{T}_0}\mathsf{T}$.
At the other extreme, we say that a graded field
$\mathsf{J}\supseteq \mathsf{T}$
is {\it totally ramified over $\mathsf{T}$} if
$|\Gamma_\mathsf{J}\! :\!\Gamma_\mathsf{T}|= [\mathsf{J}\! :\!\mathsf{T}] <\infty$.
When this occurs, $\mathsf{J}_0 = {\mathsf{T}_0}$ and, more generally,
for any $\gamma \in \Gamma_\mathsf{T}$, we have $\mathsf{J}_\gamma = \mathsf{T}_\gamma$
since $\dim_{\mathsf{T}_0}(\mathsf{J}_\gamma) = \dim_{\mathsf{J}_0}(\mathsf{J}_\gamma) = 1
= \dim_{\mathsf{T}_0}(\mathsf{T}_\gamma)$.
There is an extensive theory of finite-degree graded field
extensions; \cite{hwalg} is a good reference for what we
need here. Notably, there is a version of Galois theory:
For graded fields $\mathsf{T} \subseteq \mathsf{F}$,
with $[\mathsf{F}\! :\!\mathsf{T}] <\infty$, the (graded) Galois group
of $\mathsf{F}$ over $\mathsf{T}$ is defined to be:
\begin{equation*}
\mathsf{Gal}(\mathsf{F}/\mathsf{T}) \ = \ \{\psi\colon \mathsf{F} \to \mathsf{F}\mid
\psi \text{ is a graded field automorphism of $\mathsf{F}$ and \
$\psi|_\mathsf{T} = \operatorname{id}$}\}.
\end{equation*}
Galois theory for graded fields follows easily from the
classical ungraded theory since for the quotient fields
of $\mathsf{F}$ and $\mathsf{T}$
we have $q(\mathsf{F}) \cong \mathsf{F} \otimes _\mathsf{T} q(\mathsf{T})$,
so $[q(\mathsf{F})\! :\! q(\mathsf{T})] = [\mathsf{F}\! :\! \mathsf{T}]$,
and there is a canonical isomorphism
${\mathsf{Gal}(\mathsf{F}/\mathsf{T}) \to \operatorname{Gal}(q(\mathsf{F})/q(\mathsf{T}))}$ (the usual Galois
group) given by $\psi \mapsto \psi \otimes \operatorname{id}_{q(\mathsf{T})}$
(see \cite[Cor.~2.5(d), Th.~3.11 ]{hwalg}). Thus, $\mathsf{F}$~is
Galois over $\mathsf{T}$ iff $q(\mathsf{F})$ is Galois over $q(\mathsf{T})$,
iff $|\mathsf{Gal}(\mathsf{F}/\mathsf{T})| = [\mathsf{F}\! :\!\mathsf{T}]$, iff $\mathsf{T}$
is the fixed ring of $\mathsf{Gal}(\mathsf{F}/\mathsf{T})$. This will arise
here primarily in the inertial case: Suppose $\mathsf{S}$ is a graded
field which contains and is inertial over~$\mathsf{T}$, with
$[\mathsf{S}\! :\!\mathsf{T}]<\infty$. For any $\psi\in \mathsf{Gal}(\mathsf{S}/\mathsf{T})$
clearly the restriction $\psi|_{\mathsf{S}_0}$ lies in
$\operatorname{Gal}(\mathsf{S}_0/{\mathsf{T}_0})$. Moreover, as
$\mathsf{S} \cong_g \mathsf{S}_0 \otimes_{\mathsf{T}_0} \mathsf{T}$, for any
$\rho \in \operatorname{Gal}(\mathsf{S}_0/{\mathsf{T}_0})$ we have $\rho \otimes \operatorname{id}_\mathsf{T}
\in \mathsf{Gal}(\mathsf{S}/\mathsf{T})$. Thus, the restriction map
$\psi \mapsto \psi|_{\mathsf{S}_0}$ yields a canonical
isomorphism $\mathsf{Gal}(\mathsf{S}/\mathsf{T}) \to \operatorname{Gal}(\mathsf{S}_0/{\mathsf{T}_0})$. Hence,
as $[\mathsf{S}\! :\!\mathsf{T}] = [\mathsf{S}_0\! :\!{\mathsf{T}_0}]$, $\mathsf{S}$ is Galois over
$\mathsf{T}$ iff $\mathsf{S}_0$ is Galois over ${\mathsf{T}_0}$.
Just as in the ungraded case, we can use Galois graded
field extensions to build central simple graded algebras.
If $\mathsf{F}$ is a Galois graded field extension of
$\mathsf{T}$, set $G = \mathsf{Gal}(\mathsf{F}/\mathsf{T})$ and take any $2$-cocycle
$f\in Z^2(G, \mathsf{F}^*)$. Then we can build a crossed product
graded algebra $\mathsf{B} = (\mathsf{F}/\mathsf{T}, G, f) = \bigoplus_{\sigma \in G}
\mathsf{F} x_\sigma$ with multiplication given by $(ax_\sigma)(bx_\rho)
= a \sigma(b) f(\sigma,\rho) x_{\sigma\rho}$ for all $a,b \in \mathsf{F}$,
$\sigma, \rho\in G$. The grading is given by viewing
$\mathsf{B}$ as a left graded $\mathsf{F}$-vector space with
$(x_\sigma)_{\sigma\in G}$ as a homogeneous base with
$\deg(x_\sigma) =
\frac 1{|G|}\sum _{\rho\in G}\deg(f(\sigma,\rho))$.
A short calculation shows that $\deg(f(\sigma,\tau)\,x_{\sigma\tau})
= \deg(x_\sigma) + \deg(x_\tau)$ for all $\sigma,\tau\in G$; it follows
easily that $\mathsf{B}$ is a graded $\mathsf{T}$-algebra.
Indeed, $\mathsf{B}$ is a simple graded algebra with $Z(\mathsf{B})
\cong_g \mathsf{T}$. Conversely, if $\mathsf{A}$~is any central
simple graded
$\mathsf{T}$-algebra containing~$\mathsf{F}$ as a strictly maximal graded
subfield (i.e., ${[\mathsf{F}\! :\!\mathsf{T}] = \deg(\mathsf{A}) \,(=
\sqrt{\dim_\mathsf{T}(\mathsf{A})}\,\,)}$, then by the graded Double
Centralizer Theorem
$C_\mathsf{A}(\mathsf{F})=\mathsf{F}= Z(\mathsf{F})$;
so the graded Skolem-Noether
Theorem, Prop.~\ref{grSN} above, applies to the graded
isomorphisms in $G$, which yields that
$\mathsf{A}\cong_g (\mathsf{F}/\mathsf{T},G,f)$
for some $f\in Z^2(G, \mathsf{F}^*)$. From this one deduces, as
in the ungraded case,
that $\mathsf{Br}(\mathsf{F}/\mathsf{T}) \cong H^2(G,\mathsf{F}^*)$, where
$\mathsf{Br}(\mathsf{F}/\mathsf{T})$ denotes the kernel of the canonical
map $\mathsf{Br}(\mathsf{T}) \to \mathsf{Br}(\mathsf{F})$ given by $[\mathsf{A}] \mapsto
[\mathsf{A}\otimes_\mathsf{T} \mathsf{F}]$. In particular, if $\mathsf{Gal}(\mathsf{F}/\mathsf{T})$,
is cyclic, say with generator $\sigma$, then for any
$b\in \mathsf{T}^*$ we have the graded cyclic algebra
$\mathsf{C} = (\mathsf{F}/\mathsf{T}, \sigma, b) = \bigoplus_{i=0}^{r-1}\mathsf{F} y^i$,
in which $ya = \sigma(a) y$ for all $a\in \mathsf{F}$ and $y^r = b$,
where $r = [\mathsf{F}\! :\!\mathsf{T}]$. For the grading, we view $\mathsf{C}$ as a
left graded $\mathsf{F}$-vector space with homogeneous base
$(1, y, y^2, \ldots, y^{r-1})$ with
$\deg(y^i) = \frac ir \deg(b)$. Then $\mathsf{C}$ is a
central simple graded $\mathsf{T}$-algebra.
There are also norm maps in the graded setting: If $\mathsf{T} \subseteq \mathsf{F}$
are graded fields with $[\mathsf{F}\! :\!\mathsf{T}] <\infty$, then because
$\mathsf{F}$ is a free module the norm $N_{\mathsf{F}/\mathsf{T}}\colon \mathsf{F} \to \mathsf{T}$
can be defined by $c\mapsto \det(\lambda_c)$, where for
$c\in \mathsf{F}$, $\lambda _c\in \operatorname{Hom}_\mathsf{T}(\mathsf{F},\mathsf{F})$ is the map $a\mapsto ca$.
Clearly, $N_{\mathsf{F}/\mathsf{T}}(c) = N_{q(\mathsf{F})/q(\mathsf{T})}(c)$, where $N_{q(\mathsf{F})/q(\mathsf{T})}$
is the usual norm for the quotient fields. Also, if $c\in \mathsf{F}$ is
homogeneous, say $c\in \mathsf{F}_\gamma$, then $N_{\mathsf{F}/\mathsf{T}}(c) \in
\mathsf{T}_{[\mathsf{F}:\mathsf{T}]\gamma}$. Likewise, if $\mathsf{B}$ is a central simple
graded $\mathsf{T}$-algebra, then it is known that $\mathsf{B}$ is an Azumaya algebra
of constant rank $[\mathsf{B}\! :\!\mathsf{T}]$ over~$\mathsf{T}$; hence there is a reduced
norm map $\Nrd_{\mathsf{B}}\colon \mathsf{B} \to \mathsf{T}$. It is easy to see that
for the central ring of quotients $q(\mathsf{B})= \mathsf{B}\otimes _\mathsf{T} q(\mathsf{T})$ of $\mathsf{B}$, we have
$q(\mathsf{B})$ is a central simple algebra over the field
$q(\mathsf{T})$, and it is known
(see \cite[proof of Prop.~3.2(i)]{hazwadsworth}) that for any $b\in \mathsf{B}$,
$\Nrd_{\mathsf{B}}(b) = \Nrd_{q(\mathsf{B})}(b)$, where $\Nrd_{q(\mathsf{B})}\colon q(\mathsf{B})
\to q(\mathsf{T})$ is the reduced norm for $q(\mathsf{B})$. As usual, $b\in \mathsf{B}^*$ iff
$\Nrd_\mathsf{B}(b)\in \mathsf{T}^*$. Also, if $b\in \mathsf{B}_\gamma$, then
$\Nrd_\mathsf{B}(b) \in \mathsf{T}_{\deg(\mathsf{B}) \gamma}$.
Now assume further that $\mathsf{B}$ is a graded division algebra,
so that all its units are homogeneous. Then
for the commutator group $[\mathsf{B}^*, \mathsf{B}^*]$ of $\mathsf{B}$,
we have
$[\mathsf{B}^*, \mathsf{B}^*] \subseteq \{b \in \mathsf{B}\mid \Nrd_\mathsf{B}(b) = 1\}
\subseteq \mathsf{B}_0^*$. We define
\begin{equation}\label{grsk}
\SK(\mathsf{B}) \ = \ \{b \in \mathsf{B}\mid \Nrd_\mathsf{B}(b) = 1\}\,\big/\, [\mathsf{B}^*,\mathsf{B}^*].
\end{equation}
The fact that both terms in the right quotient lie in $\mathsf{B}_0^*$ often
makes that calculation of $\SK(\mathsf{B})$ much more tractable in this graded
setting than for ungraded division algebras.
We need terminology for some types of simple graded algebras
and graded division algebras over a graded field~$\mathsf{T}$.
A central simple graded $\mathsf{T}$-algebra $\mathsf{I}$ is said to be
{\it inertial} (or unramified) if $[\mathsf{I}_0\! :\!\mathsf{T}_0] =
[\mathsf{I}\! :\!\mathsf{T}]$. When this occurs, the injective
graded $\mathsf{T}$-algebra homomorphism
$\mathsf{I}_0 \otimes _{\mathsf{T}_0} \mathsf{T} \to \mathsf{I}$ is surjective by
dimension count. So, $\Gamma_\mathsf{I} = \Gamma_\mathsf{T}$ and
$\mathsf{I}\cong_g \mathsf{I}_0\otimes _{\mathsf{T}_0} \mathsf{T}$.
Hence, $\mathsf{I}_0$ must be a central simple
${\mathsf{T}_0}$-algebra. Moreover, if we let $D$ be the
${\mathsf{T}_0}$-central division algebra with $\mathsf{I}_0\cong
M_\ell(D)$, then $D\otimes_{\mathsf{T}_0} \mathsf{T}$ is clearly a graded
division algebra over $\mathsf{T}$ which is also inertial over $\mathsf{T}$,
and $D\otimes_{\mathsf{T}_0} \!\mathsf{T}\sim_g \mathsf{I}$ (see Lemma~\ref{zerosimple}
below).
The principal focus of this paper is on calculating
$\SK$ and unitary $\SK$ for semiramified
graded division algebras.
Let $\mathsf{E}$ be a central graded division algebra over
a graded field $\mathsf{T}$. This $\mathsf{E}$ is said to be
{\it semiramified} if ${[\mathsf{E}_0\! :\!{\mathsf{T}_0}] =
|\Gamma_\mathsf{E}\! :\!\Gamma_\mathsf{T}| = \deg(\mathsf{E})}$
and $\mathsf{E}_0$ is a field. Since $\mathsf{E}_0 = Z(\mathsf{E}_0)$, $\mathsf{E}_0$ is
abelian Galois over~${\mathsf{T}_0}$ and the epimorphism
$\overline\Theta_\mathsf{E}\colon \Gamma_\mathsf{E}/\Gamma_\mathsf{T}\to
\operatorname{Gal}(\mathsf{E}_0/{\mathsf{T}_0})$ (see \eqref{Theta}) must be an isomorphism as
${|\Gamma_\mathsf{E}/\Gamma_\mathsf{T}| = [\mathsf{E}_0\! :\! {\mathsf{T}_0}] = |\operatorname{Gal}(\mathsf{E}_0/{\mathsf{T}_0})|}$.
Furthermore, $\mathsf{E}$ has the graded subfield
$\mathsf{E}_0\mathsf{T} \cong_g\mathsf{E}_0\otimes _{\mathsf{T}_0} \!\mathsf{T}$, which is inertial and
Galois over $\mathsf{T}$ with $\mathsf{Gal}(\mathsf{E}_0\mathsf{T}/\mathsf{T})
\cong\operatorname{Gal}(\mathsf{E}_0/{\mathsf{T}_0})$. Because $[\mathsf{E}_0\mathsf{T} \! :\! \mathsf{T}] =
\deg(\mathsf{E})$, the graded Double Centralizer Theorem
\cite[Prop.~1.5]{hwcor} shows that $C_\mathsf{E}(\mathsf{E}_0\mathsf{T}) = \mathsf{E}_0\mathsf{T}$,
and hence $\mathsf{E}_0\mathsf{T}$ is a maximal graded subfield of $\mathsf{E}$;
thus, $\mathsf{E}$ is a graded abelian crossed product, as will be
discussed in \S 3.
There is a significant special class of semiramified graded
division algebras which are building blocks for all semiramified algebras.
We say that a $\mathsf{T}$-central graded division algebra $\mathsf{N}$
is {\it decomposably semiramified} (abbreviated DSR)
if $\mathsf{N}$ has a maximal graded subfield $\mathsf{S}$ which is
inertial over $\mathsf{T}$ and another maximal graded subfield ~
$\mathsf{J}$ which is totally ramified over $\mathsf{T}$. The
graded Double Centralizer Theorem yields that
${[\mathsf{S}\! :\! \mathsf{T}] = [\mathsf{J}\! :\!\mathsf{T}] = \deg(\mathsf{N})}$. We thus have
\begin{equation}\label{DSRineqs}
\deg(\mathsf{N})\, = \, [\mathsf{J}\! :\!\mathsf{T}] \, = \, |\Gamma_\mathsf{J}\! :\! \Gamma_\mathsf{T}|
\, \le \, |\Gamma_\mathsf{N}\! :\! \Gamma_\mathsf{T}| \quad \text{ and }
\quad \deg(\mathsf{N}) \,=\, [\mathsf{S}\! :\!\mathsf{T}]\, = \,[\mathsf{S}_0\! :\!{\mathsf{T}_0}]
\,\le [\mathsf{N}_0\! :\!{\mathsf{T}_0}].
\end{equation}
Since $|\Gamma_\mathsf{N}\! :\!\Gamma_\mathsf{T}| \,[\mathsf{N}_0\! :\!{\mathsf{T}_0}] = [\mathsf{N}\! :\!\mathsf{T}]
= \deg(\mathsf{N})^2$, the inequalities in \eqref{DSRineqs} must
be equalities, showing that $\mathsf{N}_0 = \mathsf{S}_0$ and $\Gamma_\mathsf{N}=
\Gamma_\mathsf{J}$, hence $\mathsf{N}$ is semiramified. We call such an $\mathsf{N}$
decomposably semiramified because it is always decomposable
into a tensor product of cyclic semiramified graded division
algebras (see Prop.~\ref{DSRdecomp} below for the unitary
analogue to this). The older term for such algebras
is nicely semiramified (NSR).
While our focus in this paper is on central graded division
algebras we will often take tensor products of such algebras,
obtaining simple graded algebras which may have zero divisors.
The next lemma allows us to recover information
about the graded division algebra Brauer equivalent to such a
tensor product.
\begin{lemma}\label{zerosimple}
Let $\mathsf{B}$ be a central simple graded algebra over
the graded field $\mathsf{T}$. Let $\mathsf{D}$ be the graded division
algebra Brauer equivalent to $\mathsf{B}$. Suppose $\mathsf{B}_0$ is a
simple ring. Then,
\begin{enumerate}[\upshape (i)]
\item
$\mathsf{B} \!\cong_g\! M_\ell(\mathsf{D})$ for some $\ell$, where the matrix
ring $M_\ell(\mathsf{D})$ is given the standard grading in
which\break ${\big(M_\ell(\mathsf{D})\big)_\gamma = M_\ell(\mathsf{D} _\gamma)}$ for
all $\gamma \in \Gamma_\mathsf{D}$. Hence, $\mathsf{B}_0\cong M_\ell(\mathsf{D}_0)$,
$\Gamma_\mathsf{B} =\Gamma'_\mathsf{B}= \Gamma_\mathsf{D}$, and $\Theta_\mathsf{B} = \Theta_\mathsf{D}$,
where ${\Gamma_\mathsf{B}' = \{\deg(b) \mid b\in B^* \text{ and } b
\text{ is homogeneous}\}}$, and
\begin{equation}\label{simpleTheta}
\Theta_\mathsf{B}\colon
\Gamma_\mathsf{B}' \to \operatorname{Gal}(Z(\mathsf{B}_0)/{\mathsf{T}_0}) \text{ is
given by } \deg(b) \mapsto \intt(b)|_{Z(\mathsf{B}_0)},
\text{ for any homogeneous $b\in \mathsf{B}^*$}
\end{equation}
where $\intt(b)$ denotes conjugation by $b$.
\item
$\mathsf{B}$ is a graded division algebra if and only if $\mathsf{B}_0$
is a division ring.
\end{enumerate}
\end{lemma}
\begin{proof}
(i) By the graded Wedderburn Theorem \cite[Prop.~1.3]{hwcor},
$\mathsf{B}\cong_g \mathsf{End}_\mathsf{D}(\mathsf{V})$ for some right graded
vector space~$\mathsf{V}$ of $\mathsf{D}$. The grading on
$\mathsf{End}_\mathsf{D}(\mathsf{V})$ is given by
$$
\big(\mathsf{End}_\mathsf{D}(\mathsf{V})\big)_\varepsilon
\ = \ \{f\in \mathsf{End}_\mathsf{D}(\mathsf{V}) \mid f(\mathsf{V}_\delta) \subseteq
\mathsf{V}_{\varepsilon+\delta} \text{ for all }\delta \in \Gamma_\mathsf{V}\}.
$$
Take a homogeneous
$\mathsf{D}$-base $(v_1, \ldots, v_\ell)$ of $\mathsf{V}$, and let
$\gamma_i = \deg(v_i)$, for $1\le i\le \ell$; then,
$\Gamma_\mathsf{V} = \bigcup_{i = 1}^\ell \, \gamma_i + \Gamma_\mathsf{D}$.
Let $\delta_1 +\Gamma_\mathsf{D}, \ldots, \delta_s+\Gamma_\mathsf{D}$ be the
distinct cosets of $\Gamma_\mathsf{D}$ appearing in $\Gamma_\mathsf{V}$,
and let $t_j$ be the number of $i$ with $\gamma_i\in
\delta_j+ \Gamma_\mathsf{D}$. So, $t_1 + \ldots + t_s = \ell$.
By replacing each $v_i$ by a
$\mathsf{D}^*$-multiple of it, we may assume that
$\deg(v_i) = \delta_j$ whenever $\gamma_i\in \delta_j+\Gamma_\mathsf{D}$.
Then, we can reindex $(v_1, \ldots, v_\ell) =
(v_{1\mspace{1mu} 1}, \ldots, v_{1\mspace{1mu} t_1}, \ldots, v_{s\mspace{1mu} 1},
\ldots, v_{s \mspace{1mu} t_s})$ with $\deg(v_{jk}) =
\delta_j$ for all~$j,k$. Then,
$\mathsf{V}_{\delta_j} = \mathsf{D}_0\text{-span}(v_{j1}, \ldots, v_{jt_j})$
for $j = 1, 2, \ldots, s$, and
\begin{align*}
\textstyle
\big(\mathsf{End}_\mathsf{D}(\mathsf{V})\big)_0\, &= \,
\big\{f\in \mathsf{End}_\mathsf{D}(\mathsf{V}) \mid f(\mathsf{V}_\varepsilon)
\subseteq \mathsf{V}_\varepsilon\text{ for all }\varepsilon \in \Gamma_\mathsf{V}\big\} \\
&\cong \
\textstyle\prod\limits_{j=1}^s\operatorname{End}_{\mathsf{D}_0}(\mathsf{D}_0\text{-span}
(v_{j\mspace{1mu} 1}, \ldots, v_{j\mspace{1mu} t_j})) \
\cong \ \prod\limits_{j=1}^s M_{t_j}(\mathsf{D}_0).
\end{align*}
This is a direct product of $s$ simple algebras. Since
we have assumed that $\mathsf{B}_0$ is simple, we must have $s=1$,
i.e., all the $v_i$ have degree $\delta_1$. It is then clear that
when we use the base $(v_1, \ldots , v_\ell)$ for the
isomorphism $\mathsf{End}_\mathsf{D}(\mathsf{V}) \cong M_\ell(\mathsf{D})$, the grading
on $M_\ell(\mathsf{D})$ induced by the isomorphism is the standard
grading. Thus, $\mathsf{B} \cong_g M_\ell(\mathsf{D})$ and hence
$\mathsf{B}_0\cong M_\ell(\mathsf{D}_0)$ and $\Gamma_\mathsf{B} = \Gamma_\mathsf{D}$.
Then, $\Gamma'_\mathsf{B} = \Gamma'_{M_\ell(\mathsf{D})} = \Gamma_\mathsf{D}$ and,
when we identify $Z(\mathsf{B}_0)$ with $Z(M_\ell(\mathsf{D}_0))$
and with $Z(\mathsf{D}_0)$, clearly $\Theta_\mathsf{B} = \Theta_{M_\ell(\mathsf{D})}
= \Theta_\mathsf{D}$.
(ii) If $\mathsf{B}$ is a graded division algebra, then every nonzero
homogeneous element of $\mathsf{B}$ lies in $\mathsf{B}^*$. In particular,
$\mathsf{B}_0\setminus\{0\} \subseteq \mathsf{B}^*$, so $\mathsf{B}_0$ is a division
ring. Conversely, suppose $\mathsf{B}_0$ is a division ring.
Since $\mathsf{B}_0$ is then simple, part (i) applies, showing
that for some graded division algebra $\mathsf{D}$, we have
$\mathsf{B}\cong_g M_\ell(\mathsf{D})$ where $\mathsf{B}_0 \cong M_\ell(\mathsf{D}_0)$.
Necessarily $\ell =1$, as $\mathsf{B}_0$ is a division ring.
\end{proof}
\begin{corollary}\label{ItensorE}
Let $\mathsf{I}$ and $\mathsf{E}$ be central graded division algebras over
a graded field $\mathsf{T}$, with $\mathsf{I}$ inertial, and let~$\mathsf{D}$~be
the graded division algebra with $\mathsf{D} \sim_g \mathsf{I} \otimes_\mathsf{T}
\mathsf{E}$. Then, $\mathsf{D}_0 \sim \mathsf{I}_0\otimes _{\mathsf{T}_0} \mathsf{E}_0$, $Z(\mathsf{D}_0)
\cong Z(\mathsf{E}_0)$, $\Gamma_\mathsf{D} = \Gamma_\mathsf{E}$, and $\Theta_\mathsf{D} =
\Theta_\mathsf{E}$.
\end{corollary}
\begin{proof}
Let $\mathsf{B} = \mathsf{I}\otimes_\mathsf{T} \mathsf{E}$. Since $\mathsf{I} \cong_g \mathsf{I}_0
\otimes _{\mathsf{T}_0} \!\mathsf{T}$, we have $\mathsf{B} \cong_g \mathsf{I}_0 \otimes
_{\mathsf{T}_0} \mathsf{E}$. Hence,
${\mathsf{B}_0\cong \mathsf{I}_0 \otimes _{\mathsf{T}_0} \mathsf{E}_0}$,\break
${Z(\mathsf{B}_0) \cong Z(\mathsf{I}_0) \otimes _{\mathsf{T}_0} Z(\mathsf{E}_0) \cong Z(\mathsf{E}_0)}$,
and $\Gamma_\mathsf{B} = \Gamma_\mathsf{E}$. Moreover, $\mathsf{B}_0$ is simple
as $\mathsf{I}_0$ is central
simple over~${\mathsf{T}_0}$, so
Lemma~\ref{zerosimple} applies to $\mathsf{B}$. In particular,
$\Gamma'_\mathsf{B} = \Gamma_\mathsf{B}$ and $\Theta_\mathsf{B} = \Theta_\mathsf{E}$.
Since $\mathsf{D}$ is the graded division algebra with
$\mathsf{D} \sim_g \mathsf{B}$, the Lemma yields $\mathsf{D}_0 \sim \mathsf{B}_0 \cong
\mathsf{I}_0\otimes_{\mathsf{T}_0} \mathsf{E}_0$, so $Z(\mathsf{D}_0) \cong Z(\mathsf{B}_0) \cong
Z(\mathsf{E}_0)$, and $\Gamma_\mathsf{D} = \Gamma_\mathsf{B} = \Gamma_\mathsf{E}$, and
$\Theta_\mathsf{D} = \Theta_\mathsf{B} = \Theta_\mathsf{E}$.
\end{proof}
\section{Abelian crossed products and nonunitary $\SK$
for semiramified algebras}
\label{abcp}
Let $M$ be a finite degree abelian Galois extension of a field $K$, and let
$H = \operatorname{Gal}(M/K)$. Let ${X(M/K) = \text{Hom}(H, \mathbb{Q}/\mathbb{Z})}$, the character group
of $H$. Take any cyclic decomposition $H = \langle
\sigma_1 \rangle \times \ldots \times \langle \sigma_k \rangle$,
and let $r_i$ be the order of $\sigma_i$ in $H$. Let $(\chi_1 , \ldots, \chi_k)$
be the base of $X(M/K)$ dual to $(\sigma_1, \ldots, \sigma_k)$; so $\chi_i(\sigma_j) =
\delta_{ij}/r_i +\mathbb{Z}$, where $\delta_{ij} = 1$ if $j = i$ and $= 0$ if $j\ne i$.
Let $L_i$ be the fixed field of $\ker(\chi_i)$. So,
$M = L_1 \otimes_K \ldots \otimes_K L_k$, and for each $i$, $L_i$ is cyclic
Galois over $K$ with $[L_i\! :\! K] = r_i$ and $ \operatorname{Gal}(L_i/K) =
\langle \sigma_i|_K\rangle$. Let $A$ be any central simple $K$-algebra
containing $M$ as a strictly maximal subfield (i.e., $M$ is a
maximal subfield of $A$ with $ [M\! :\! K] = \deg(A)$). By the Double Centralizer
Theorem, the centralizer $C_A(M)$ is $M$.
Recall that
every algebra class in $\operatorname{Br}(M/K)$ is represented by a unique
such $A$. By
Skolem-Noether, for each $i$ there is $z_i \in A^*$ with
$\intt(z_i)|_M = \sigma_i$, where
$\intt(z_i)$ denotes conjugation by $z_i$. Set
$$
u_{ij} \, = \,z_iz_jz_i^{-1}z_j^{-1}\ \ \text{and} \ \ b_i \, = \, z_i^{r_i}
.
$$
Since $\intt(u_{ij})|_M = \sigma_i\sigma_j\sigma_i^{-1}\sigma_j^{-1} = \operatorname{id}_M$ and
$\intt(b_i)|_M = \sigma^{r_i} = \operatorname{id}_M$, all the $u_{ij}$ and $b_i$
lie in $C_A(M)^* = M^*$.
Take the index set $\mathcal I=\prod_{i=1}^k \{0,1,2,\dots,r_i-1\}
\subseteq \mathbb{Z}^k$. For $\bold i = (i_1, \ldots , i_k)\in \mathcal I$,
set $\sigma^{\bold i} = \sigma_1^{i_1} \ldots \sigma_k^{i_k}$ and
$z^{\bold i} = z_1^{i_1}\ldots z_k^{i_k}$. So,
$\intt(z^{\bold i})|_M = \sigma^{\bold i}$ and, as the map
$\bold i \mapsto \sigma^{\bold i}$ is a bijection
$\mathcal I \to H$, we
have the crossed product decomposition
$$
A \ = \ \textstyle \bigoplus\limits_{\bold i\, \in \, \mathcal I} \,
Mz^{\bold i}.
$$
For $\bold i , \bold j \in \mathcal I$, if we set
$\bold i
\bold *
\bold j$ to be the element of $\mathcal I$ congruent to
$\bold i + \bold j$ mod $r_1\mathbb{Z} \times \ldots \times r_k\mathbb{Z}$ in
$\mathbb{Z}^k$, and set
$$
f(\sigma ^{\bold i}, \sigma ^{\bold j}) \ = \ z^{\bold i} z^{\bold j}
(z^{ \bold i \bold * \bold j})^{-1} \ \in \ M^*,
$$
then $f\in Z^2(H, M^*)$ and the multiplication in $A$ is given by
$$
a z^{\bold i} \,\cdot \, c z^{\bold j} \ =
\ a\sigma^{\bold i}(c)
f(\sigma ^{\bold i}, \sigma ^{\bold j})\, z^{ \bold i \bold *
\bold j}, \ \ \text{ for all}\ \ a,c \in M \ \text { and }
\bold i, \bold j
\in \mathcal I.
$$
Since each $f(\sigma ^{\bold i}, \sigma ^{\bold j})$ is expressible as a computable
product of the $u_{ij}$ and the $b_i$ and their images under~$H$,
the multiplication for $A$ is completely determined by $M$, $H$, and the
$u_{ij}$ and $b_i$. Thus, we write
${A = A(M/K, \boldsymbol\sigma, \mathbf u,\mathbf b)}$, where $\boldsymbol\sigma = (\sigma_1, \ldots, \sigma_k)$,
$\mathbf u= (u_{ij})_{i=1, \,j=1}^{k\ \ \ \,k}$, and $\mathbf b =
(b_1, \ldots, b_k)$.
It is easy to check (cf.~ \cite[Lemma~1.2]{as} or \cite [p.~423]{tignol})
that the $u_{ij}$ and the $b_i$ satisfy the
following relations, for all $i,j,\ell$,
\begin{equation}\label{urels}
u_{ii} \, = \, 1, \ \ u_{ji} \, = \, u_{ij}^{-1}, \ \
\sigma_i(u_{j\ell}) \sigma_j(u_{\ell i}) \sigma_{\ell}(u_{ij}) \ = \
u_{j\ell} u_{\ell i} u_{ij}
\end{equation}
and
\begin{equation}\label{brel}
N_{M/M^{\langle \sigma_i\rangle}}(u_{ij}) \ = \ b_i/\sigma_j(b_i),
\end{equation}
where $M^{\langle \sigma_i\rangle}$ is the fixed field of
$M$ under $\langle \sigma_i\rangle$.
It is known (cf.~\cite[Th.~1.3]{as}) that
for any family of
$u_{ij}$ and $b_i$ in $M^*$ satisfying \eqref{urels}
and \eqref{brel} there is
a central simple $K$-algebra $A(M/K, \boldsymbol\sigma, \mathbf u, \mathbf b)$.
\begin{lemma}\label{onen}
Let $A=A(M/K,\boldsymbol\sigma,\mathbf u,\mathbf b)$ as above, and let
$B=A(M/K,\boldsymbol\sigma,\bf v, \bf c)$.
Then, there is a well-defined abelian crossed product
$A(M/K,\boldsymbol\sigma,\bf w, \bf d)$ where
$w_{ij}=u_{ij}v_{ij}$ and $d_i=b_ic_i$ for all $i,j$.
Moreover,
$A\otimes_K B \sim A(M/K,\boldsymbol\sigma,\bf w, \bf d)$
$($Brauer equivalent$)$.
\end{lemma}
\begin{proof}
Because the $u_{ij}$ and $b_i$ satisfy \eqref{urels} and \eqref{brel} as do the $v_{ij}$
and $c_i$, and the $\sigma_i$ and the norm maps are multiplicative,
the $w_{ij}$ and $d_i$ also satisfy \eqref{urels} and \eqref{brel}.
Therefore $A(M/K,\boldsymbol\sigma,\bf w, \bf d)$
is a well-defined abelian crossed product.
We have the $2$-cycle $ f\in Z^2(H,M^*)$ representing $A$
defined as above by,
$f(\sigma ^{\bold i}, \sigma ^{\bold j}) \ =
\ z^{\bold i} z^{\bold j} (z^{
\bold i \bold * \bold j})^{-1}$.
The relations $z_i^{r_i}=b_i$ and $[z_i,z_j]=u_{ij}$ are
encoded in $f$ by
\begin{equation}\label{cocycle}
f(\sigma_i^{\ell},\sigma_i)=
\begin{cases}
1, &\text{if $0\leq \ell \leq r_i-2$}\\
b_i, &\text{if $\ell=r_i-1$}
\end{cases}
\text{\,\,\,\,\,\,\,\, and \,\,\,\,\, }
f(\sigma_i,\sigma_j)=
\begin{cases}1 &\text{if }i< j\\
u_{ij} &\text{if $i>j$}.
\end{cases}
\end{equation}
We likewise build a cocycle $g\in Z^2(H,M^*)$ for
$B=A(M/K,\boldsymbol\sigma,\bf v,\bf c)$. Then, the cocycle $f\!\cdot \! g$ satisfies
conditions corresponding to those for $f$ in \eqref{cocycle}, so
$f\!\cdot \! g$ is
a cocycle
for ${C=A(M/K,\boldsymbol\sigma, \bf w, \bf d)}$ where ${w_{ij}=u_{ij}v_{ij}}$
and
$d_i=b_ic_i$. From the group isomorphism
${H^2(H,M^*)\cong\operatorname{Br}(M/K)}$
it follows that ${A\otimes_K B \sim C}$.
\end{proof}
In Tignol's terminology in \cite{tignol}, a central simple
$K$-algebra containing $M$ as a strictly maximal subfield
{\it decomposes according to $M$} if
$A\cong (L_1/K, \sigma_1, b_1)\otimes_K \ldots \otimes _K
(L_k/K, \sigma_k, b_k)$ for some $b_1, \ldots, b_k \in K^*$.
Clearly then, $A \cong A(M/K,\boldsymbol\sigma,\bold 1,\mathbf b)$, i.e., each
$u_{ij}= 1$. Conversely, for any algebra
$A(M/K,\boldsymbol\sigma,\bold 1,\mathbf b)$
(i.e., the $z_i$ commute with each
other), each $z_j$ centralizes
$b_i = z_i^{r_i}$, so $b_i\in M^H = K$ and the algebra
decomposes
according to $M$. The collection of such algebras yields an
important
distinguished subgroup $\operatorname{Dec}(M/K)$ of $\operatorname{Br}(M/K)$, i.e.
\begin{align}\label{decdef}
\begin{split}
\operatorname{Dec}(M/K) \ &= \ \{\,[A] \in \operatorname{Br}(M/K)\,|\ A\text{ decomposes according to
$M$}\}\\ &= \
\{\,[ A(M/K,\boldsymbol\sigma,\mathbf u,\mathbf b)] \ |\
\text{every }u_{ij} = 1 \text{ and every } b_i\in K^*\,\}.
\end{split}
\end{align}
Since $\operatorname{Br}(L_i/K) =
\{ \,[(L_i/K,\sigma_i, b)] \, |\ b\in K^*\,\}$,
we have also $\operatorname{Dec}(M/K) = \prod_{i = 1}^k\operatorname{Br}(L_i/K)
\subseteq\operatorname{Br}(M/K)
$. Tignol
also
also points out in \cite[p.~426]{tignol} a homological
characterization:
From the short exact sequence
of trivial $H$-modules $0 \to \mathbb{Z} \to \mathbb{Q} \to \mathbb{Q}/\mathbb{Z} \to 0$
the long exact cohomology sequence yields the connecting
homomorphism
$\delta\colon H^1(H, \mathbb{Q}/\mathbb{Z}) \to H^2(H, \mathbb{Z})$, which is an
isomorphism since
$H^i(H, \mathbb{Q}) = 1$ for $i\ge 1$ as $\mathbb{Q}$ is uniquely divisible. For any
$\chi\in X(M/K) = H^1(H, \mathbb{Q}/\mathbb{Z})$ and any $c\in K^* =
H^0(H, M^*)$
it is known (cf. ~\cite [p.~204, Prop.~2]{serre}) that under the cup
product pairing
${{\raise 0.9pt\hbox{$\,\scriptstyle\cup\,$}}\colon H^2(H,\mathbb{Z}) \times H^0(H, M^*) \to H^2(H,M^*) =
\operatorname{Br}(M/K)}$,\break
we have ${\delta(\chi){\raise 0.9pt\hbox{$\,\scriptstyle\cup\,$}} c = [(N/K,\rho|_N,c)]}$, where $N$
is the fixed field of
$\ker(\chi)$ and $\rho \in H$ is determined by\break ${\chi(\rho) =
(1/|\chi|) +\mathbb{Z}\in \mathbb{Q}/\mathbb{Z}}$. Thus, the algebra class
$[(L_1/K, \sigma_1, b_1)\otimes_K \ldots \otimes _K
(L_k/K, \sigma_k, b_k)]$ in $\operatorname{Br}(M/K)$ corresponds to
${(\delta(\chi_1) {\raise 0.9pt\hbox{$\,\scriptstyle\cup\,$}} b_1) +}$ $\ldots +(\delta(\chi_k)
{\raise 0.9pt\hbox{$\,\scriptstyle\cup\,$}} b_k)$ in
$H^2(H, M^*)$. Since the cup product is bimultiplicative and
$X(M/K) = \langle \chi_1, \ldots , \chi_k\rangle$, we have
\begin{equation}\label{dec}
\operatorname{Dec}(M/K) \ = \ \big\langle \operatorname{im}\big({\raise 0.9pt\hbox{$\,\scriptstyle\cup\,$}}\colon H^2(H, \mathbb{Z}) \times
H^0(H, M^*)
\to H^2(H, M^*)\,\big)\big\rangle \ = \ \textstyle \prod\limits_
{\substack{K \subseteq L \subseteq M \\ \operatorname{Gal}(L/K)
\text{ cyclic}}}
\operatorname{Br}(L/K),
\end{equation}
showing that $\operatorname{Dec}(M/K)$ is independent of the choice of the
$\sigma_i$
and the $L_i$. (Actually, Tignol uses \eqref{dec} as his definition
of $\operatorname{Dec}(M/F)$, and proves in \cite[Cor.~1.4]{tignol} that this is
equivalent to the definition given here in \eqref{decdef}.)
The case when $H$ is bicyclic is of particular interest, i.e.,
$H=\langle\sigma_1\rangle\times
\langle\sigma_2\rangle$ and $M=L_1\otimes_K L_2$. Then, for
any
algebra $A=A(M/K,\boldsymbol\sigma, \mathbf u,\mathbf b)$, if we set $u=u_{12}$,
then
$u$ determines all the $u_{ij}$ as $u_{21}=u_{12}^{-1}$ and
$u_{11}=u_{22}=1$. We write, for short, $A=A(u,b_1,b_2)$.
The conditions in \eqref{brel} can then be restated:
\begin{equation}\label{bicyclicbrel}
b_1\in M^{\langle\sigma_1\rangle}\, =\, L_2,
\quad b_2\in M^{\langle\sigma_2\rangle} \, =\, L_1,
\quad N_{M/L_2}(u) \, =\, b_1/\sigma_2(b_1),
\quad N_{M/L_1}(u) \, =\, \sigma_1(b_2)/b_2.
\end{equation}
Note that $N_{M/K}(u) = N_{L_2/K}(b_1/\sigma_2(b_1)) =1$.
An
easy
calculation~(cf.~\cite[Th.~1.4]{as}) shows that
\begin{equation} \label{ghgt}
\begin{split}
A(u,b_1,b_2)\, \cong \, A(u',b_1',b_2') \text{ if and only if there exist
$c_1,c_2 \in M^*$ such that}\qquad\qquad\qquad\qquad\\
u'\, =\, \big[c_1/\sigma_2(c_1)\big]\big[\sigma_1(c_2)/c_2\big]u,
\quad
b_1'\, =\, N_{M/L_2}(c_1)b_1, \quad \text{and \, }
b_2'\, =\, N_{M/L_1}(c_2)b_2. \ \ \quad\qquad\qquad
\end{split}
\end{equation}
These observations can be formulated homologically:
Recall that $\widehat{H}^{-1}(H,M^*)=\ker(N_{M/K})/I_H(M^*)$,
where
$\ker(N_{M/K})=\{m\in M^* \mid N_{M/K}(m)=1 \}$ and, as
$H = \langle \sigma_1 \rangle \times \langle \sigma_2\rangle$,
$$
I_H(M^*) \ = \ \big\{\,[a/\sigma_1(a)]\,[b/\sigma_2(b)]\mid
a,b \in M^* \big\}.
$$
We define a map
\begin{equation}
\eta\colon\operatorname{Br}(M/K) \longrightarrow \widehat{H}^{-1}(H,M^*)
\text{ \ \ given by \ \ } \big[A(u,b_1,b_2)\big]\mapsto uI_H(M^*).
\end{equation}
By~(\ref{ghgt}) above $\eta$ is well-defined, and
Lemma~\ref{onen}
shows that $\eta$ is a group homomorphism.
Given any $u\in M^*$ with $N_{M/K}(u) = 1$, Hilbert 90
gives $b_1 \in L_2^*$ and $b_2\in L_1^*$ so that the
conditions in \eqref{bicyclicbrel} are satisfied and the
algebra $A(u,b_1,b_2)$ exists. Therefore $\eta$
is surjective.
By~(\ref{ghgt}),
$$
\ker(\eta) \ = \ \big\{[A(u,b_1,b_2)] \mid u=1 \big \}
\ = \ \operatorname{Dec}(M/K),
$$
so $\eta$ yields an isomorphism
\begin{equation}\label{njnj}
\operatorname{Br}(M/K)\big / \operatorname{Dec}(M/K) \ \cong \ \widehat{H}^{-1}(\operatorname{Gal}(M/K),M^*)
\qquad\text{whenever $M$ is bicyclic over $K$}.
\end{equation}
This isomorphism is known (see, e.g., \cite[Remarque, pp.~
427--428]{tignol}); indeed, it follows by comparing
Draxl's formula \cite[Kor.~8, p.~133]{draxlSK} for $\SK$
of the division algebras considered by Platonov
in \cite{platonov} with Platonov's formula
in \cite[Th.~4.11, Th.~4.17]{platonov}. I learned
of this description of the isomorphism from Tignol.
Its relevance for $\SK$ calculations is shown
in the next proposition, which is the graded
version of \eqref{DSRSK1} and \eqref{platformula}
above.
\begin{proposition}\label{NSRSK}
Suppose $\mathsf{N}$ is a DSR central graded division algebra
over the graded field $\mathsf{T}$.
Then,
\begin{enumerate}[\upshape (i)]
\item
$\SK(\mathsf{N}) \ \cong \ \widehat H^{-1}(H,\mathsf{N}_0^*)\ \text{ where }
H = \operatorname{Gal}(\mathsf{N}_0/\mathsf{T}_0)$.
\item If
$\mathsf{N}_0 \cong L_1 \otimes_{\mathsf{T}_0} L_2$ with each
$L_i$ cyclic Galois over $\mathsf{T}_0$, then
$$
\SK(\mathsf{N}) \ \cong \ \operatorname{Br}(\mathsf{N}_0/\mathsf{T}_0) \big /
\operatorname{Dec}(\mathsf{N}_0/\mathsf{T}_0).$$
\end{enumerate}
\end{proposition}
\begin{proof}
(i) was given in \cite[Cor.~3.6(iv)]{hazwadsworth}, and (ii) follows from
(i) and \eqref{njnj} above.
\end{proof}
We will generalize Prop.~\ref{NSRSK} in Th.~\ref{goth} below
by giving formulas for $\SK(\mathsf{E})$ when
$\mathsf{E}$ is semiramified but not necessarily DSR. For this we need,
first, a graded version of the abelian crossed products
described at the beginning of this section.
Second, we need a graded version of the $I\otimes N$
decomposition for semiramified division algebras over a
Henselian valued field. Here $I$ is inertial
and $N$ is DSR. (See~\cite[Lemma~5.14, Th.~5.15]{jw} for the valued
$I\otimes N$ decomposition.)
Here is the graded version of abelian crossed products.
Let $\mathsf{B}$ be a
central simple graded algebra over a graded field $\mathsf{T}$.
Assume that $\mathsf{B}$ contains a maximal graded subfield
$\mathsf{S}$ with ${[\mathsf{S} \! :\!\mathsf{T}] = \deg(\mathsf{B}) \ (=\sqrt{[\mathsf{B}\! :\!\mathsf{T}]}\, )}$
such that $\mathsf{S}$ is Galois over $\mathsf{T}$ and $H = \mathsf{Gal}(\mathsf{S}/\mathsf{T})$
is abelian. We have $C_\mathsf{B}(\mathsf{S}) = \mathsf{S}$ by the graded Double Centralizer
Theorem. For any cyclic decomposition $H= \langle \sigma_1\rangle \times
\ldots \times \langle \sigma_k\rangle$, the graded Skolem-Noether Theorem,
Prop.~\ref{grSN}, is available
as $C_\mathsf{B}(\mathsf{S}) = \mathsf{S} = Z(\mathsf{S}) $; it shows that for each
$i$ there is $y_i\in \mathsf{B}^*$ with
$y_i$~homogeneous and $\intt(y_i)|_\mathsf{S} = \sigma_i$. Set
$c_i = y_i^{r_i}$ where $r_i$ is the order of $\sigma_i$ in $H$, and
set $v_{ij} = y_iy_jy_i^{-1} y_j^{-1}$. Then, each
$c_i \in C_\mathsf{B}(\mathsf{S})^* = \mathsf{S}^*$
with $\deg(y_i) = \frac 1{r_i}\deg(c_i)$,
and each $v_{ij} \in \mathsf{S}_0^*$. For each ${\bold i = (i_1, \ldots, i_k)\in
\mathcal I=\prod_{j=1}^k
\{0,1,2,\dots,r_j-1\}}$, set $y^{\bold i} = y_1^{i_1}\ldots y_k^{i_k}$.
Then, $\intt(y^{\bold i})|_\mathsf{S} = \sigma^{\bold i}$, and we have
\begin{equation}\label{grabcrprod}
\mathsf{B} \ = \ \textstyle\bigoplus\limits_{\bold i \in \mathcal I}
\, \mathsf{S}\, y^{\bold i}.
\end{equation}
For, the sum in the equation is direct since $\mathsf{B} \otimes _\mathsf{T}
q(\mathsf{T}) = \bigoplus_{ \bold i \in \mathcal I}
(\mathsf{S}\otimes _\mathsf{T} q(\mathsf{T}))\, y^\bold i$ by the ungraded case.
Then equality holds in \eqref{grabcrprod} by dimension count.
Note that $\mathsf{B}$ is a left graded $\mathsf{S}$-vector space with homogeneous
base $\big(y^{\bold i}\big)_{\bold i \in \mathcal I}$, and
\begin{equation}
\deg(y^{\bold i}) \ =\ \textstyle \sum\limits _{j= 1}^k\frac {i_j}{r_j} \deg(c_j).
\end{equation}
So,
\begin{equation}\label{gradeinfo}
\textstyle \Gamma_\mathsf{B} \, = \, \big \langle \frac1{r_1}\deg(c_1), \ldots,
\frac1{r_k}\deg(c_k)\big\rangle + \Gamma_\mathsf{S} \qquad \text{and}\qquad
\text{each}\ \ B_\delta \, = \, \bigoplus\limits_{\bold i \in \mathcal I}\,
S_ {(\delta - \deg(y^{\bold i}))}y^{\bold i}.
\end{equation}
Since $\mathsf{B}$ is determined as a graded $\mathsf{T}$-algebra by $\mathsf{S}$,
the~$\sigma_i$, the~$v_{ij}$, and the $c_i$, we write $\mathsf{B} =
\mathsf{A}(\mathsf{S}/\mathsf{T}, \boldsymbol\sigma, \bold v, \bold c)$, where $\boldsymbol\sigma = (\sigma_1, \ldots,
\sigma_k)$, $\bold v = (v_{ij})_{i = 1,j=1}^{k \ \ \ k}$, and
${\bold c = (c_1, \ldots, c_k)}$.
Note that the $v_{ij}$ and the $c_i$ satisfy the identities
corresponding to \eqref{urels} and \eqref{brel}. Conversely, given any
$v_{ij} \in \mathsf{S}_0^*$ and $c_i \in \mathsf{S}^*$ satisfying those identities
there is a central simple graded $\mathsf{T}$ algebra
$\mathsf{A}(\mathsf{S}/\mathsf{T}, \boldsymbol\sigma, \bold v,\bold c)$. This is obtainable as
$\mathsf{B} = \bigoplus _{\bold i \in \mathcal I}\mathsf{S}\,y^{\bold i}$ within
the ungraded abelian crossed product $A = A(q(\mathsf{S})/q(\mathsf{T}), \boldsymbol\sigma,
\bold v, \bold c)$, with the grading on $\mathsf{B}$ determined by
that on $\mathsf{S}$ and $\deg(y_i) = \frac 1{r_i}\deg(c_i)$, as described above.
To see that $\mathsf{B}$ is a graded ring, one uses that each $\sigma\in H$ is a
(degree-preserving) graded automorphism of
$\mathsf{S}$ and that $\deg(y^{\bold i}\cdot y^{\bold j}) = \deg(y^{\bold i})
+\deg(y^{\bold j})$ for all $\bold i, \bold j\in \mathcal I$, since all
the $v_{ij}$ have degree $0$.
This $\mathsf{B}$ is graded simple, since any nontrivial proper homogeneous ideal
would localize to a nontrivial proper ideal of the simple $q(\mathsf{T})$-algebra $A$.
\begin{remark}\label{abeliancpprod}
The graded analogue to Lemma~\ref{onen} holds, with the same
proof, since for $\mathsf{S}$~Galois over $\mathsf{T}$, we have
$\mathsf{Br}(\mathsf{S}/\mathsf{T}) \cong H^2(\mathsf{Gal}(\mathsf{S}/\mathsf{T}), \mathsf{S}^*)$.
\end{remark}
The graded abelian crossed products we work with here will
have $\mathsf{S}$ inertial over $\mathsf{T}$ and will be semiramified,
as described in the next lemma.
\begin{lemma} \label{incp}
Let $\mathsf{S}$ be an inertial graded field extension of $\mathsf{T}$ with
$\mathsf{S}$ abelian Galois over $\mathsf{T}$. Let\break ${H = \mathsf{Gal}(\mathsf{S}/\mathsf{T})=
\langle \sigma_1\rangle \times \ldots \times \langle \sigma_k\rangle}$ as above
with $r_i$ the order of $\sigma_i$, and let
$\mathsf{B} = \mathsf{A}(\mathsf{S}/\mathsf{T}, \boldsymbol\sigma, \bold v, \bold c)$ be a graded abelian
crossed product. Let $\delta_i = \frac 1{r_i}\deg(c_i) \in \Gamma_\mathsf{B}$
and $\overline {\delta_i} = \delta_i + \Gamma_\mathsf{T} \in \Gamma_\mathsf{B}/ \Gamma_\mathsf{T}$.
Then, $\mathsf{B}$ is a semiramified graded division algebra
if and only if each $\overline{\delta_i}$ has order $r_i$ and $\overline{\delta_1},
\ldots, \overline{\delta_k}$ are independent in $\Gamma_\mathsf{B} /\Gamma_\mathsf{T}$.
When this occurs, $\mathsf{B}_0= \mathsf{S}_0$ and $\Gamma_\mathsf{B}/\Gamma_\mathsf{T}
= \langle\overline{\delta_1}\rangle\times \ldots \times \langle\overline{\delta_k}\rangle
\cong H$.
\end{lemma}
\begin{proof}
Since $\mathsf{S}$ is inertial and Galois over $\mathsf{T}$, $\mathsf{S}_0$ is Galois over
$\mathsf{T}_0$ with $\operatorname{Gal}(\mathsf{S}_0/\mathsf{T}_0) \cong \mathsf{Gal}(\mathsf{S}/\mathsf{T}) = H$. We identify
$H$ with $\operatorname{Gal}(\mathsf{S}_0/\mathsf{T}_0)$.
We have $\mathsf{S}_0\subseteq \mathsf{B}_0$ and $[\mathsf{S}_0\! :\! \mathsf{T}_0] =
[\mathsf{S}\! :\!\mathsf{T}] = \deg(\mathsf{B})$.
Suppose $\mathsf{B}$ is a semiramified graded division algebra.
Then, ${[\mathsf{B}_0\! :\! \mathsf{T}_0] = \deg(\mathsf{B}) = [\mathsf{S}_0\! :\!\mathsf{T}_0]}$,
so ${\mathsf{B}_0 = \mathsf{S}_0}$.
Since $\mathsf{B}$ is semiramified,
the epimorphism $\overline\Theta_\mathsf{B}\colon \Gamma_\mathsf{B}/\mathsf{T} \to H$ is an
isomorphism, as noted in \S2.
When we represent ${\mathsf{B} = \bigoplus_{\bold i \in \mathcal I}\mathsf{S} y^{\bold i}}$
as above, since $\intt(y_i) = \sigma_i$ and
$\deg(y_i) = \frac 1{r_i}\deg(c_i) = \delta_i$,
we have $\overline \Theta_\mathsf{B}(\overline{\delta _i}) = \sigma_i$.
Hence, $\overline{\delta_i}$ has the same order $r_i$ as $\sigma_i$, and
$$
\Gamma_\mathsf{B}/\Gamma_\mathsf{T} \ = \ \overline \Theta_\mathsf{B}^{\, -1} (H)
\ = \ \overline \Theta_\mathsf{B}^{\, -1}(\langle\sigma_1\rangle) \times\ldots \times
\overline \Theta_\mathsf{B}^{\, -1}(\langle\sigma_k\rangle)\ = \ \langle\overline{\delta_i}\rangle
\times \ldots \times \langle \overline{\delta_k}\rangle,
$$
so the $\overline{\delta_i}$ are independent in $\Gamma_\mathsf{B}/\Gamma_\mathsf{T}$.
Conversely, suppose each $\overline{\delta_i}$ has order $r_i$ and
the $\overline{\delta_i}$ are independent in $\Gamma_\mathsf{B}/\Gamma_\mathsf{T}$. Then,
$$
|\Gamma_\mathsf{B}\! :\!\Gamma_\mathsf{T}|\!\mspace{1mu} \ \ge \ \textstyle\prod\limits_{i = 1} ^ k
|\langle\overline{\delta_i}\rangle| \ = \ r_1\ldots r_k \ = \ |H| \ = \ \deg(\mathsf{B}).
$$
Hence,
\begin{equation}\label{B0}
[\mathsf{B}_0\! :\! \mathsf{T}_0] \ = \ [\mathsf{B}\! :\!\mathsf{T}] \big/ |\Gamma_\mathsf{B}\! :\!\Gamma_\mathsf{T}| \
\le \,\mspace{1mu} \deg(\mathsf{B})^2/\deg(\mathsf{B}) \, = \ [\mathsf{S}_0\! :\!\mathsf{T}_0].
\end{equation}
Since $\mathsf{S}_0 \subseteq \mathsf{B}_0$, \eqref{B0} shows that $\mathsf{B}_0 = \mathsf{S}_0$,
so equality holds in \eqref{B0}. Since $\mathsf{B}_0$ is a field, $\mathsf{B}$
is a graded division algebra by Lemma~\ref{zerosimple}(ii), and it is
semiramified by the equality in \eqref{B0}.
\end{proof}
Observe that if $\mathsf{E}$ is any semiramified graded $\mathsf{T}$-central
division algebra, then $\mathsf{E}$ is a graded abelian crossed product
as described in Lemma~\ref{incp}.
For, $\mathsf{E}_0\mathsf{T}$ is a maximal graded subfield
of $\mathsf{E}$ which is inertial and Galois over $\mathsf{T}$ with $\mathsf{Gal}(\mathsf{E}_0\mathsf{T}/\mathsf{T})
\cong \operatorname{Gal}(\mathsf{E}_0/{\mathsf{T}_0})$, which is abelian.
\bigskip
\begin{proposition}\label{INdecomp}
Let $\mathsf{E}$ be a semiramified central graded division
algebra over the graded field $\mathsf{T}$. Then,
\begin{enumerate}[\upshape (i)]
\item
There exist graded $\mathsf{T}$-central division algebras
$\mathsf{I}$ and $\mathsf{N}$ such that
$\mathsf{I}$ is inertial, $\mathsf{N}$ is DSR, and
$\mathsf{E} \sim_g \mathsf{I}\otimes_\mathsf{T} \mathsf{N}$ in $\mathsf{Br}(\mathsf{T})$. When this occurs,
$\mathsf{N}_0\cong \mathsf{E}_0$, $\Gamma_N = \Gamma_E$, $\Theta_\mathsf{N}
= \Theta_\mathsf{E}$, and $\mathsf{E}_0$ splits $\mathsf{I}_0$.
\item For any other decomposition
$\mathsf{E} \sim_g \mathsf{I}'\otimes_\mathsf{T} \mathsf{N}'$ with $\mathsf{I}'$ inertial
and $\mathsf{N}'$ DSR, we have $\mathsf{I}_0' \equiv \mathsf{I}_0\
(\operatorname{mod}\
\operatorname{Dec}(\mathsf{E}_0/\mathsf{T}_0)\,)$.
\end{enumerate}
\end{proposition}
We do not give a proof of Prop.~\ref{INdecomp} because it is a simpler
version of the proof of the analogous unitary result, which is
Prop.~\ref{uINdecomp} below. Also, Prop.~\ref{INdecomp} is the
graded analogue of a known result for semiramified division algebras
over Henselian valued fields, \cite[Lemma~5.14, Th.~5.15]{jw}, and the graded result
given here is deducible from the Henselian one.
\begin{lemma}\label{morit}
For the semiramified graded division algebra
$\mathsf{E}=\mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T},\boldsymbol\sigma,\bf v,\bf c)$ as above, write\break
${\mathsf{E}
\sim_{g} \mathsf{I} \otimes_T \mathsf{N}}$ with $\mathsf{I}$ inertial and $\mathsf{N}$
DSR; so
$[\mathsf{I}_0] \in \operatorname{Br}(\mathsf{E}_0/\mathsf{T}_0)$. If $\mathsf{I}_0 \sim
A(\mathsf{E}_0/\mathsf{T}_0,\boldsymbol\sigma,\mathbf u,\mathbf b)$, then
by changing the chioce of
the $y_i\in \mathsf{E}^*$ inducing $\sigma_i$ on $\mathsf{E}_0\mathsf{T}$
we have $\mathsf{E} = \mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T},\boldsymbol\sigma,\bf u,\bf e)$
with the same $\bf u$ as for~$\mathsf{I}_0$.
\end{lemma}
\begin{proof}
Let $\mathsf{J}$ be a maximal graded subfield of $\mathsf{N}$ which is
totally
ramified over $\mathsf{T}$, so $\Gamma_\mathsf{N}=\Gamma_\mathsf{J}$. Because
$\mathsf{N}$~is
semiramified, the map $\Theta_\mathsf{N}\colon\Gamma_\mathsf{N}/\Gamma_\mathsf{T}
\rightarrow \operatorname{Gal}(\mathsf{N}_0/\mathsf{T}_0)$ is an isomorphism. But also
$\mathsf{N}_0=\mathsf{E}_0$. Thus, for each $i$, we can choose $x_i \in
\mathsf{J}^*$
with $\Theta_\mathsf{N}(\deg(x_i))=\sigma_i|_{\mathsf{E}_0}$.
Let
$d_i=x_i^{r_i} \in (\mathsf{N}_0\mathsf{T})^*=(\mathsf{E}_0\mathsf{T})^*$. Then,
$\mathsf{N}\cong_g\mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T},\boldsymbol\sigma,\bold w,\bold d)$, where
each
$w_{ij} =x_ix_jx_i^{-1} x_j^{-1}=1$, as all the $x_i$ lie in the graded
field
$\mathsf{J}$. Let $\mathsf{I}_0'=A(\mathsf{E}_0/\mathsf{T}_0,\boldsymbol\sigma,\mathbf u,\bold b)$,
which is
Brauer equivalent to $\mathsf{I}_0$. Then set
$\mathsf{I}'=\mathsf{I}_0'\otimes_{\mathsf{T}_0}\mathsf{T}$, which is an inertial
$\mathsf{T}$-algebra
with $\mathsf{I}' \sim_g \mathsf{I}$. Since $\mathsf{I}' \otimes_\mathsf{T} \mathsf{N}
\sim_g
\mathsf{I}\otimes_\mathsf{T} \mathsf{N}\sim_g \mathsf{E}$, we may without any loss
replace
$\mathsf{I}$ by $\mathsf{I}'$. Then, as ${\mathsf{I}_0\cong
A(\mathsf{E}_0/\mathsf{T}_0,\boldsymbol\sigma,\mathbf u,\bold b)}$, clearly
$\mathsf{I}\cong_g\mathsf{I}_0\otimes_{\mathsf{T}_0}\mathsf{T}
\cong_g\mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T},\boldsymbol\sigma,\mathbf u,\bold b)$.
Let $\mathsf{E}' = \mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T},\boldsymbol\sigma,\bold u,\bold e)$,
where each $e_i = b_id_i$, and let $y_1', \ldots ,
y_k'$ be the associated generators of $\mathsf{E}'$ over
$\mathsf{E}_0\mathsf{T}$. Then,
$\mathsf{E}\sim_g\mathsf{I} \otimes_\mathsf{T} \mathsf{N} \sim_g\mathsf{E}'$,
by Remark~\ref{abeliancpprod}, as $u_{ij}w_{ij} = u_{ij}$.
Note that for each~$i$, ${\deg(e_i) = \deg(d_i)}$, as
$\deg(b_i) = 0$. Hence, $\Gamma_{\mathsf{E}'} = \Gamma_\mathsf{N}$ by
\eqref{gradeinfo}. Furthermore, $\mathsf{E}'$ is a semiramified
graded division algebra since $\mathsf{N}$ is, because Lemma~\ref{incp}
shows that this is determined by the $\deg(e_i)$, resp.~
$\deg(d_i)$. Because $\mathsf{E}'$ is a graded division algebra (not just a
graded simple
algebra), as is $\mathsf{E}$, from $\mathsf{E}\sim_g\mathsf{E}'$
the
uniqueness in the graded Wedderburn
Theorem~\cite[Prop.~1.3]{hwcor} yields
a graded $\mathsf{T}$-isomorphism $\eta\colon
\mathsf{E} \to \mathsf{E}'$. By the graded Skolem-Noether Theorem,
Prop.~\ref{grSN}, $\eta$ can be chosen so that $\eta|_{\mathsf{E}_0\mathsf{T}}
=\operatorname{id}$. Then replacing the $y_i$ by $\eta^{-1}(y_i')$ in the
presentation of $\mathsf{E}$ changes each~$v_{ij}$ to $u_{ij}$.
\end{proof}
\begin{theorem}\label{goth}
Suppose $\mathsf{E}$ is a semiramified $\mathsf{T}$-central graded division
algebra, and take any decomposition
${\mathsf{E}\sim_g\mathsf{I}\otimes_\mathsf{T}
\mathsf{N}}$ where $\mathsf{I}$ is an inertial graded $\mathsf{T}$-algebra and
$\mathsf{N}$ is DSR. Then,
\begin{enumerate}[\upshape (i)]
\item Since $\mathsf{I}_0 \in \operatorname{Br}(\mathsf{E}_0/\mathsf{T}_0)$ with
$\mathsf{E}_0$ abelian Galois over ${\mathsf{T}_0}$,
we can write
$\mathsf{I}_0 \sim
A(\mathsf{E}_0/\mathsf{T}_0,\boldsymbol\sigma,\mathbf u,\bold b)$ in $\operatorname{Br}({\mathsf{T}_0})$.
Then,
$$
\SK(\mathsf{E}) \ \cong \ \widehat{H}^{-1}(H,\mathsf{E}_0^*)\big /
\big \langle \operatorname{im} \{ u_{ij} \mid 1\leq i,j \leq k \}
\big \rangle, \ \text{ where } H = \operatorname{Gal}(\mathsf{E}_0/\mathsf{T}_0).
$$
\item If
$\mathsf{E}_0 \cong L_1 \otimes_{\mathsf{T}_0} L_2$ with each
$L_i$ cyclic Galois over $\mathsf{T}_0$, then
$$
\SK(\mathsf{E}) \ \cong \ \operatorname{Br}(\mathsf{E}_0/\mathsf{T}_0) \big /
\big [\operatorname{Dec}(\mathsf{E}_0/\mathsf{T}_0)\cdot\langle[\mathsf{I}_0]\rangle\big ],$$
where $\operatorname{Dec}(\mathsf{E}_0/\mathsf{T}_0) = \operatorname{Br}(L_1/{\mathsf{T}_0}) \cdot \operatorname{Br}(L_2/{\mathsf{T}_0})$.
\end{enumerate}
\end{theorem}
\begin{proof} The definition of $\SK$ for graded division algebras
is given in \eqref{grsk} above.
(i) We have\break ${H = \operatorname{Gal}(\mathsf{E}_0/{\mathsf{T}_0}) \cong \mathsf{Gal}(\mathsf{E}_0 \mathsf{T}/\mathsf{T})}$.
Since $\mathsf{E}$ is semiramified,
$H\cong \Gamma_\mathsf{E}/\Gamma_\mathsf{T}$ via $\overline \Theta_\mathsf{E}^{-1}$ (see~\S 2).
By \cite[Cor.~3.6(ii)]{hazwadsworth}
there is an exact sequence
\begin{equation}\label{ppr}
0\ \longrightarrow \ H \textstyle\wedge H \
\stackrel{\Phi}{\longrightarrow} \ \widehat{H}^{-1}(G,\mathsf{E}_0^*) \
\stackrel{\Psi}{\longrightarrow} \ \SK(\mathsf{E}) \ \longrightarrow \ 0.
\end{equation}
The maps in~(\ref{ppr}) are given as follows: Let
$\ker(\Nrd_\mathsf{E})=\{a\in \mathsf{E}^* \mid \Nrd_\mathsf{E}(a) =1 \}
\subseteq
\mathsf{E}_0^*$, and let $\ker(N_{\mathsf{E}_0/\mathsf{T}_0})=\{a \in \mathsf{E}_0^*
\mid
N_{\mathsf{E}_0/\mathsf{T}_0}(a) =1 \}$. Because $\mathsf{E}$ is semiramified,
by~\cite[Remark~2.1(iii), Lemma~2.2]{I},
${\ker(\Nrd_\mathsf{E})=\ker(N_{\mathsf{E}_0/\mathsf{T}_0})}$. For every $\rho \in
H$, choose any $y_\rho\in \mathsf{E}^*$ with $\intt(y_\rho)|_{\mathsf{E}_0}=\rho$.
The map
$\Phi$ is given by: for $\rho,\pi \in H$,
$$
\Phi(\rho\wedge \pi) \ = \ y_\rho y_\pi y_\rho^{-1} y_\pi ^{-1} I_H(\mathsf{E}_0^*) \ \in
\ker(N_{\mathsf{E}_0/\mathsf{T}_0})\big /I_H(\mathsf{E}_0^*) \ = \
\widehat{H}^{-1}(H,\mathsf{E}_0^*).
$$
The map $\Psi$ is given by: for $a\in \ker(N_{\mathsf{E}_0/\mathsf{T}_0})$,
$$
\Psi\big(a\,I_H(\mathsf{E}_0)^*\big) \ = \
a\,[\mathsf{E}^*,\mathsf{E}^*] \ \in \ker(\Nrd_\mathsf{E})\big / [\mathsf{E}^*,\mathsf{E}^*]
\ = \ \SK(\mathsf{E}).
$$
By Lemma~\ref{morit}, we can assume
$\mathsf{E}= \mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T},\boldsymbol\sigma, \mathbf u, \bf c)$ (with
the same $u_{ij}$ as for $\!\mathsf{I}_0$). Since\break ${H \cong
\mathsf{Gal}(\mathsf{E}_0\mathsf{T}/\mathsf{T})=
\langle\sigma_1\rangle \times \ldots \times
\langle\sigma_k \rangle}$, we have
$H\wedge H =\langle \sigma_i
\wedge \sigma_j \mid 1\leq i,j \leq k \rangle$. There are
$y_1,\dots,y_k \in \mathsf{E}^*$, with
$\intt(y_i)|_{\mathsf{E}_0}=\sigma_i$ and
$y_iy_j y_i^{-1} y_j^{-1}=u_{ij}$. So we can take $y_{\sigma_i}=y_i,
1\leq i \leq
k$, yielding for the $\Phi$ in~(\ref{ppr}),
$\Phi(\sigma_i\wedge
\sigma_j)=u_{ij}I_H(\mathsf{E}_0^*)\in \widehat{H}^{-1}(H,\mathsf{E}_0^*)$.
Thus,
$\operatorname{im}(\Phi)=\langle \operatorname{im}(u_{ij})\mid 1\leq i,j \leq k \rangle$,
and
part (i) follows from the exact
sequence~(\ref{ppr}).
(ii) When $\mathsf{E}_0=L_1\otimes_{\mathsf{T}_0}L_2$, $H=\operatorname{Gal}(\mathsf{E}_0/\mathsf{T}_0)$
has
rank $2$, say $H=\langle\sigma_1\rangle \times
\langle \sigma_2
\rangle$. So, $H\wedge H=\langle \sigma_1\wedge \sigma_2
\rangle$
and $\operatorname{im} (\Phi)=\langle u_{12} I_H(\mathsf{E}_0^*)\rangle$. As we
saw in
discussion of~(\ref{njnj}) above, the isomorphism
$$
\operatorname{Br}(\mathsf{E}_0/\mathsf{T}_0)\big
/\operatorname{Dec}(\mathsf{E}_0/\mathsf{T}_0) \,\longrightarrow \,
\widehat{H}^{-1}(H,\mathsf{E}_0^*),
$$
maps $[\,\mathsf{I}_0]+\operatorname{Dec}(\mathsf{E}_0/\mathsf{T}_0)$ to $u_{12}I_H(\mathsf{E}_0^*)$.
Thus using part~(i),
\begin{equation*}
\SK(\mathsf{E}) \ \cong \ \widehat{H}^{-1}(H,\mathsf{E}_0^*) \big / \big
\langle \operatorname{im}(u_{12}) \big \rangle \ \cong \ \operatorname{Br}(\mathsf{E}_0/\mathsf{T}_0)
\big / \big
[\operatorname{Dec}(\mathsf{E}_0/\mathsf{T}_0)\cdot\langle[\,\mathsf{I}_0]\rangle\big ].\qedhere
\end{equation*}
\end{proof}
For any division algebra $D$ over a Henselian valued field
$F$, the valuation on $F$ extends uniquely to a valuation
on $D$, and we write $\overline D$ for its residue division algebra
and $\Gamma_D$ for its value group.
Recall the isomorphism $\SK(D) \cong \SK(\operatorname{{\sf gr}}(D))$ for a tame
such $D$,
proved in \cite[Th.~4.8]{hazwadsworth}. By using this isomorphism,
Th.~\ref{goth} yields the following:
\begin{corollary}\label{henselcor}
Let $F$ be field with Henselian
valuation $v$, and let $D$ be an $F$-central division algebra
which $($with respect to the unique extension of $v$ to
$D$$)$ is tame and semiramified. Take any decomposition
$D\sim I\otimes_F N$, where $I$ and $N$ are $F$-central
division algebras with $I$ inertial and $N$ DSR.
\begin{enumerate}[\upshape (i)]
\item
Since $\overline I \in \operatorname{Br}(\overline D/\overline F)$ with
$\overline D$ abelian Galois over $\overline F$,
we can write
$\overline I \sim A(\overline D/\overline F, \boldsymbol\sigma, \mathbf u, \mathbf b)$ in
$\operatorname{Br}(\overline F)$. Then,
$$
\SK(D) \ \cong \ \widehat{H}^{-1}(H,\overline D^*)\big /
\big \langle \operatorname{im} \{ u_{ij} \mid 1\leq i,j \leq k \}
\big \rangle, \ \ \text{ where } \ H \,=\, \operatorname{Gal}(\overline D/\overline F).
$$
\item If
$\overline D \cong L_1 \otimes_{\overline F} L_2$ with each
$L_i$ cyclic Galois over $\overline F$, then
$$
\SK(D) \ \cong \ \operatorname{Br}(\overline D/\overline F) \big /
\big [\operatorname{Dec}(\overline D/\overline F)\cdot\langle[
\overline I]\rangle\big ],$$
where $\operatorname{Dec}(\overline D/\overline F) = \operatorname{Br}(L_1/\overline F) \cdot \operatorname{Br}(L_2/\overline F)$.
\end{enumerate}
\end{corollary}
\begin{proof}
(That $D$ is tame and semiramified means
$[\overline D\! :\! \overline F] = |\Gamma_D\! :\! \Gamma_F| = \sqrt{[D\! :\! F]}$ and
$\overline D$ is a field separable over $\overline F$.)
Let $\mathsf{T} = \operatorname{{\sf gr}}(F)$, the associated graded ring of $F$ with
respect to the filtration on it induced by the valuation
(cf.~\cite{hwcor} or \cite{hazwadsworth}). Since $F$ is a field,
$\mathsf{T}$ is a graded field with ${\mathsf{T}_0} = \overline F$ and ${\Gamma_\mathsf{T} = \Gamma_F}$.
Since $v$ is Henselian, it has unique extensions
to valuations on $D$, $I$, and $N$; with respect to these valuations,
let ${\mathsf{E} = \operatorname{{\sf gr}}(D)}$, $\mathsf{I} = \operatorname{{\sf gr}}(I)$, and $\mathsf{N} = \operatorname{{\sf gr}}(N)$. These are graded
division rings, with $\mathsf{E}_0 = \overline D$, ${\mathsf{I}_0 = \overline I \sim
A(\mathsf{E}_0/ {\mathsf{T}_0}, \boldsymbol\sigma, \mathbf u, \mathbf b)}$, and $\mathsf{N}_0 = \overline N \cong \overline D = \mathsf{E}_0$.
Moreover, as $D$, $I$, and $N$ are each tame over~$F$,
it follows by \cite[Prop.~4.3]{hwcor} that
$\mathsf{T}$ is the
center of $\mathsf{E}$, $\mathsf{I}$, and $\mathsf{N}$, and
$[\mathsf{E}\! :\! \mathsf{T}] = [D\! :\! F]$, $[\,\mathsf{I}\! :\! \mathsf{T}] = [I\! :\! F]$, and
$[\mathsf{N}\! :\! \mathsf{T}] = [N\! :\! F]$.
Since $I$ is inertial over $F$, we have $\mathsf{I}$ is inertial over $\mathsf{T}$.
That $N$ is DSR means (cf.~\cite[p.~149]{jw}, where the term NSR is used)
that $N$ has maximal subfields $S$ and $J$ with $S$ inertial over~$F$
and $J$ totally ramified of radical type over $F$. Then,
$\operatorname{{\sf gr}}(S)$ and $\operatorname{{\sf gr}}(J)$ are maximal graded subfields of
$\mathsf{N}$ with $\operatorname{{\sf gr}}(S)$ inertial over~$\mathsf{T}$ and $\operatorname{{\sf gr}}(J)$ totally
ramified over $\mathsf{T}$. So, $\mathsf{N}$ is DSR. Similarly, $\mathsf{E}$ is
semiramified since $D$~is tame and semiramified.
Let $\operatorname{Br}_t(F)$ be the tame part of the
Brauer group $\operatorname{Br}(F)$. From the isomorphism $\operatorname{Br}_t(F) \cong
\mathsf{Br}(\mathsf{T})$ given by \cite[Th.~5.3]{hwcor}, we obtain
$\mathsf{E} \sim_g \mathsf{I} \otimes_\mathsf{T} \mathsf{N}$ from $D\sim I\otimes_FN$.
Thus, Th.~\ref{goth} applies to $\mathsf{E}$ with the decomposition
$\mathsf{E} \sim_g \mathsf{I} \otimes _\mathsf{T} \mathsf{N}$, and the assertions of
Cor.~\ref{henselcor} follow immediately as $\SK(D)
\cong \SK(\mathsf{E})$ by \cite[Th.~4.8]{hazwadsworth}.
\end{proof}
\begin{example}\label{cyclicex}
Take any integer $n\geq 2$ and let $K$ be any
field containing a primitive $n^2$-root of unity
$\omega$. Let $\mathsf{T}=K[x,x^{-1},y,y^{-1}]$, the
Laurent polynomial
ring,
graded as usual by $\mathbb Z \times \mathbb Z$ with
$\mathsf{T}_{(k,\ell )}=Kx^ky^\ell$; in particular,
${\mathsf{T}_0} = K$. (So~$\mathsf{T}\cong_{g}
\operatorname{{\sf gr}}\big(K((x))((y))\big)$
where the iterated Laurent power series ring $K((x))(y))$ is
given
its usual rank~$2$ Henselian valuation.) Take any
$a,b\in K^*$ such
that $\big [K(\sqrt[n]{a},\sqrt[n]{b}\,)\! :\! K\big]=n^2$, and let
$\mathsf{E}$ be
the graded symbol algebra $\mathsf{E}=(ax^n,by^n,\mathsf{T})_\omega$, of
degree $n^2$.
That is, $\mathsf{E}$ is the graded central simple $\mathsf{T}$-algebra with
homogenous generators $i$ and $j$ such that $i^{n^2}=ax^{n}$,
$j^{n^2}=by^{n}$, and $ij=\omega ji$, and
$\deg(i)=(\frac 1n,0)$,
$\deg(j)=(0,\frac 1n)$. Then, $\Gamma_\mathsf{E}=(\frac 1n\mathbb Z)\times
(\frac 1n \mathbb Z)$,
and $\mathsf{E}_0=K(i^nx^{-1},j^ny^{-1})\cong K(\sqrt[n]{a},
\sqrt[n]{b}\,)$.
Since $\mathsf{E}_0$ is a field,
by Lemma~\ref{zerosimple}(ii) $\mathsf{E}$~is a graded division ring,
which is
clearly semiramified. We can write $\mathsf{E}_0=L_1
\otimes_K
L_2$ where $L_1=K(\sqrt[n]{a}\,)$ and $L_2=K(\sqrt[n]{b}\,)$, and
$H=\operatorname{Gal}(\mathsf{E}_0/K)=\langle \sigma_1 \rangle \times
\langle \sigma_2
\rangle$ where $\sigma_1(\sqrt[n]{a})=\omega^n\sqrt[n]{a}$,
$\sigma_1(\sqrt[n]{b})=\sqrt[n]{b}$ and
$\sigma_2(\sqrt[n]{b})=\omega^n\sqrt[n]{b}$,
$\sigma_2(\sqrt[n]{a})=\sqrt[n]{a}$.
Since $\intt(j^{-1})|_{\mathsf{E}_0} = \sigma_1$ and $\intt(i)|_{\mathsf{E}_0} = \sigma_2$,
we can express $\mathsf{E}$ as a graded abelian crossed product with
$y_1 = j^{-1}$ and $y_2 = i$,
obtaining $\mathsf{E} = \mathsf{A}(\mathsf{T}(\sqrt[n]{a},\sqrt[n]{b})/\mathsf{T}, \boldsymbol\sigma, \bold u,
\bold d)$, where $u_{11}=u_{22}=1$, $u_{12}=\omega$, $u_{21}=\omega^{-1}$,
and
$d_1=1/(y\sqrt[n]{b})$, $d_2 = x\sqrt[n]{a}$.
Graded symbol algebras
satisfy
the same multiplicative rules in the graded Brauer group as
do the
usual ungraded symbol algebras in the Brauer group.
(This follows, e.g., by the injectivity of the scalar extension map
$\mathsf{Br}(\mathsf{T}) \to \operatorname{Br}(q(\mathsf{T}))$, cf.~\cite[p.~90]{hwcor}.) Thus, in
$\operatorname{Br}(\mathsf{T})$, we have
\begin{align*}
\mathsf{E} \, & \sim_{g} (a,b,\mathsf{T})_\omega \otimes_\mathsf{T} (x^n,b,\mathsf{T})_\omega
\otimes_\mathsf{T} (a,y^n,\mathsf{T})_\omega \otimes_\mathsf{T}(x^n,y^n,\mathsf{T})_\omega\\
& \sim_{g}(a,b,\mathsf{T})_\omega \otimes_\mathsf{T} (x,b,\mathsf{T})_{\omega^n} \otimes
(a,y,\mathsf{T})_{\omega^n}.
\end{align*}
(The last two terms are symbol algebras of degree $n$.) Thus,
$\mathsf{E}\sim_{g}\mathsf{I} \otimes_\mathsf{T} \mathsf{N}$
where $\mathsf{I}=(a,b,\mathsf{T})_\omega$ and $\mathsf{N}=(x,b,\mathsf{T})_{\omega^n}\otimes_\mathsf{T}
(a,y,\mathsf{T})_{\omega^n}$. Then,
$\mathsf{I} \cong_{g}\mathsf{I}_0\otimes_{\mathsf{T}_0}\mathsf{T}$,
where
$\mathsf{I}_0=(a,b,\mathsf{T}_0)_\omega=A(K(\sqrt[n]{a},\sqrt[n]{b}\,)/K,
\boldsymbol\sigma,\mathbf u,\bold b)$,
with the same $\bold u$ as for $\mathsf{E}$ and
$b_1=1/\sqrt[n]{b}$,
$b_2=\sqrt[n]{a}$. So, $\mathsf{I}$ is an inertial central simple
graded
$\mathsf{T}$-algebra. We have $\mathsf{N}_0$ is the field
$K(\sqrt[n]{a},\sqrt[n]{b}\,)$, so $\mathsf{N}$ is a graded division
algebra by Lemma~\ref{zerosimple}(ii).
$\mathsf{N}$ is DSR since it has the inertial maximal graded
subfield
$\mathsf{T}(\sqrt[n]{a},\sqrt[n]{b}\,)=\mathsf{N}_0\mathsf{T}$ and the totally
ramified
maximal graded subfield $\mathsf{T}(\sqrt[n]{x},\sqrt[n]{y}\, )$.
As a graded abelian crossed product,
$\mathsf{N} \cong_g \mathsf{A}(\mathsf{T}(\sqrt[n]{a},\sqrt[n]{b})/\mathsf{T}, \boldsymbol\sigma,
\bold 1, \bold c)$, where $c_1 = 1/y$, $c_2 =x$.
Let
$M=K(\sqrt[n]{a},\sqrt[n]{b}\,)$. By Prop.~\ref{NSRSK},
$$
\SK(\mathsf{N}) \ \cong \ \widehat{H}^{-1}(H,M^*) \ \cong \
\operatorname{Br}(M/K) \big / \operatorname{Dec}(M/K),
$$
where $H=\operatorname{Gal}(M/K)$ and $\operatorname{Dec}(M/K) = \operatorname{Br}(K(\sqrt[n] a\,)/K)\cdot \operatorname{Br}
(K(\sqrt[n] b\,)/K)$; but, by Th.~\ref{goth},
\begin{equation}\label{cyclicSK1}
\SK(\mathsf{E}) \ \cong \ \widehat{H}^{-1}(H,M^*)\big /
\langle \operatorname{im}(\omega) \rangle \ \cong \ \operatorname{Br}(M/K)\big/
\big [\operatorname{Dec}(M/K)\cdot\langle[(a,b,K)_\omega]\rangle \big ].
\end{equation}
This example is the graded version of Platonov's example
in~\cite{platcyclic} and \cite{plat76}
of a cyclic algebra with nontrivial $\SK$, where
$K$ is a suitably chosen global field. (Platonov worked
with the Henselian valued
ground field ${K'=K((x))((y))}$ in place of the graded field $\mathsf{T}=\operatorname{{\sf gr}}(K')$
considered
here.) In~\cite[Th.~2]{plat76} the added term distingushing
$\SK(\mathsf{E})$ from
$\SK(\mathsf{N})$ is omitted. This error is corrected in~\cite[p.~536,
footnote~1]{yy} and in \cite[p.~70]{ershov},
giving the first isomorphism of \eqref{cyclicSK1} but not the second.
\end{example}
\section{Unitary graded $I\otimes N$ decomposition}\label{unitaryIN}
The goal for \S\S 4--7 is to give a unitary version of the formulas for
$\SK$ in Prop.~\ref{NSRSK} and Th.~\ref{goth} for semiramified graded
division algebras with graded unitary involution. In this section we
consider abelian crossed products with unitary involution and
prove a unitary analogue to the $I\otimes N$ decomposition of
Prop.~\ref{INdecomp}.
A {\it unitary involution} on a central simple algebra $A$ over a
field $K$ is a ring antiautomorphism $\tau$ of~$A$ such that
$\tau^2 = \operatorname{id}_A$ and $\tau|_K \ne \operatorname{id}$. (Such a $\tau$ is also called
an involution on $A$ of the second kind.) Let ${F = K^\tau = \{ c\in K\mid
\tau(c) = c\}}$, which is a subfield of $K$ with $[K\! :\! F] = 2$ and
$K$ Galois over~$F$ with $\operatorname{Gal}(K/F) = \{\tau|_K, \operatorname{id}_K\}$. Our $\tau$
is also called a {\it unitary $K/F$-involution}.
The unitary
$\SK(A,\tau)$ is defined just as for $\SK(D, \tau)$ in \eqref{unitarydef}.
Recall
(see \cite[Prop.~(17.24)(2)]{kmrt}) that if $\tau'$ is another unitary
$K/F$-involution
on $A$, then $\SK(A,\tau') = \SK(A,\tau)$. Thus, we will freely pass
from one unitary $K/F$-involution on $A$ to another when convenient.
In the unitary setting generalized dihedral Galois groups often arise
where abelian Galois groups appear in the nonunitary setting.
A group $G$ is said to be {\it generalized dihedral} with respect to a
subgroup $H$ if $|G\! :\! H| = 2$ and for some $\theta \in G\setminus H$,
$\theta^2 = 1$ and $\theta h \theta^{-1} = h^{-1}$ for every $h\in H$.
Equivalently, every element of $G\setminus H$ has order $2$.
See \cite[\S2.4]{I} for some remarks on such groups. Note that $H$ is
necessarily abelian. If $H$ is cyclic, we say that $G$ is {\it dihedral}.
(This includes the trivial cases where $|H| = 1$ or $2$.)
For fields $F \subseteq K\subseteq M$, we say that $M$ is
{\it $K/F$-generalized dihedral} if $[M\! :\! F] <\infty$,
$M$ is Galois over $F$,
and $G = \operatorname{Gal}(M/F)$ is generalized dihedral with respect to its
subgroup $H = \operatorname{Gal}(M/K)$.
\begin{lemma}\label{unitarycp} Let $F \subseteq K \subseteq M$ be
fields, and suppose $M$ is $K/F$-generalized dihedral.
Let $A$ be a central simple $K$-algebra containing
$M$ as a strictly maximal subfield. Let $G =
\operatorname{Gal}(M/F)$ and $H = \operatorname{Gal}(M/K)$, and fix any
$\theta \in G\setminus H$
$($so $\theta^2 = \operatorname{id}_M$$)$. Then,
the following conditions are equivalent:\begin{enumerate}
\item[{\rm(i)}]
$A$ has a unitary $K/F$-involution.
\item[{\rm(ii)}]
$A$ has a unitary $K/F$-involution $\tau$ such that
$\tau|_M = \theta$.
\item[{\rm(iii)}]
$A \cong A(M/K, \boldsymbol\sigma, \mathbf u, \mathbf b)$ where $($in addition to
conditions
\eqref{urels} and \eqref{brel}$)$
\begin{equation}\label{unitaryconds}
u_{ij} \cdot \sigma_i\sigma_j\theta(u_{ij}) \ = \ 1 \ \ \
\text{and} \ \ \ \ b_i = \theta(b_i)
\ \ \ \ \text{for all} \ i,j.
\end{equation}
\end{enumerate}
The $A$ in {\rm (iii)}, has a unitary $K/F$-involution $\tau$
with $\tau|_M = \theta$ and $\tau(z_i) = z_i$ for each of the
standard generators $z_i$ of $A$.
\end{lemma}
\begin{proof}
Note that as $\theta\notin H$ and $K$ is Galois over $F$, we
have $\theta(K) = K$ and
$\operatorname{Gal}(K/F) = \{ \operatorname{id}_K, \theta|_K\}$.
(i) $\Rightarrow$ (ii) This is a special case of a
substantial result
\cite[Th.~4.14]{kmrt} on simple subalgebras with compatible
involutions. For the
convenience of the reader we give a short direct proof.
Let $\rho$ be a unitary $K/F$-involution on~ $A$, so
$\rho |_K=\theta|_K$.
Since $\rho\theta$ is a $K$-linear homomorphism
$M\rightarrow A$, by the
Skolem-Noether
Theorem, there is $y \in A^*$ with $\intt(y)|_M = \rho\theta$.
For any $a\in M$, as $\rho^2 = \theta^2 = \operatorname{id}|_M$,
we have
$$
\rho(y) a\rho(y)^{-1} \ = \ \rho(y^{-1} \rho(a) y) \ =
\ \rho(\rho \theta)^{-1} \rho(a) \ = \ \rho\theta(a)
\ = \ yay^{-1}.
$$
Therefore, letting $c = y^{-1} \rho(y)$, we have $c\in
C_A(M)^* = M^*$ and
$\rho(y) = yc$. Hence,
$$
y \, = \, \rho^2(y) \, = \, \rho(yc) \, = \, \rho(c) yc
\, = \, \rho(c) \rho\theta(c) y \, = \, \rho(c\mspace{1mu}\theta(c))y;
$$
so, $c\mspace{1mu}\theta(c) = 1$. Since $\theta^2 = \operatorname{id}|_M$, by Hilbert 90
applied to the
quadratic extension $M/M^\theta$ there is $d\in M^*$ with
$c = d\mspace{1mu} \theta(d)^{-1}$.
Let $z = yd$. Then, as $\theta(c) = \theta(d)d^{-1}$,
$$
\rho(z) \, = \, \rho(d)yc \, = \,
\rho(d)\rho\theta(c)\, y
\, = \, \rho\theta(d)\, y \, = \, yd \, = \, z.
$$
Let $\tau = \rho\circ\intt(z)$, which is an involution on $A$, as
$\rho(z) = z$.
Then, $\tau|_M = \rho\intt(z)|_M = \rho\intt(y)|_M = \rho^2
\theta = \theta$, as desired.
(ii) $\Rightarrow$ (iii) Let $\tau$ be a unitary
$K/F$-involution on $A$ such that
$\tau|_M =\theta$. For any $\sigma \in H$, we claim that
there is
$z\in A^*$ with $\intt(z)|_M = \sigma$ and $\tau(z) = z$.
For this, first apply
Skolem-Noether to obtain $y\in A^*$ with $\intt(y)|_M = \sigma$.
For any $a\in M$ we have,
as $\tau \sigma ^{-1} \tau = \sigma$ on $M$ since $\tau\sigma^{-1}|_M \in G \setminus H$,
$$
\tau(y) a \tau(y)^{-1} \, = \ \tau(y^{-1} \tau(a) y) \, =
\ \tau\sigma^{-1}\tau(a) \, = \ \sigma(a) \, = \ yay^{-1}.
$$
Hence, $\tau(y) = cy$, where $c \in C_A(M)^* = M^*$. Now,
$$
y \, = \, \tau^2(y) \ =\ \tau(cy) \ = \ \tau(y) \tau(c)
\ = \
cy\theta(c) \ = \ c\mspace{1mu}\sigma \theta(c)\, y,
$$
so $c\,\sigma\theta(c) = 1$. Since $\sigma\theta$ has order $2$,
Hilbert 90
applied to the quadratic extension $M/M^{\sigma\theta}$ shows
that there is
$d\in M^*$ with $c = d\,\sigma\theta(d)^{-1}$. Let $z = dy$. Then,
$\intt(z)|_M = \intt(y)|_M = \sigma$ and
$$
\tau(z) \, = \, cy\theta(d) \, = \, [d\,\sigma\theta(d)^{-1}]\,
\sigma\theta(d) \,y \, =
\, z,
$$
proving the claim. Thus, with our cyclic decomposition
$H = \langle \sigma_1\rangle \times\ldots \times \langle \sigma_k
\rangle$,
we can choose $z_1, \ldots, z_k\in A^*$ with $\intt(z_i)|_M =
\sigma_i$
and $\tau(z_i) = z_i$. Then, for $b_i = z_i^{r_i} \in M^*$,
we have
$\theta(b_i) = \tau(b_i) = \tau(z_i^{r_i}) = b_i$. Also, for
$u_{ij} = z_iz_jz_i^{-1} z_j^{-1}$, we have
$$
\sigma_i\sigma_j\theta(u_{ij}) \, = \ z_iz_j\,
\tau(z_i z_j
z_i^{-1} z_j^{-1})z_j^{-1} z_i^{-1} \, =
\ z_iz_j(z_j^{-1} z_i^{-1}
z_jz_i)z_j^{-1} z_i^{-1}
\, = \ z_jz_iz_j^{-1} z_i^{-1} \, = \ u_{ij}^{-1},
$$
so $u_{ij}\,\sigma_i\sigma_j\theta(u_{ij}) = 1$. Thus, $A\cong
A(M/K, \boldsymbol\sigma, \mathbf u, \bold b)$
with the $u_{ij}$ and $b_i$ satisfying the equations in
\eqref{unitaryconds}.
(iii) $\Rightarrow$ (i) Assume $A = A(M/K, \boldsymbol\sigma, \mathbf u, \bold b)$
where the
$u_{ij}$ and $b_i$ satisfy the conditions in
\eqref{unitaryconds}. Take
$z_1, \ldots , z_k\in A^*$ with $\intt(z_i)|_M = \sigma$,
$z_i^{r_i} = b_i$ and
$z_iz_jz_i^{-1} z_j^{-1} = u_{ij}$. We show that there is a unitary
$K/F$-involution
$\tau$ on $A$ satisfying (and determined by)
$\tau|_M = \theta$ and
$\tau(z_i) = z_i$ for each $i$. Basically, this is a matter
of checking that the $\tau$ just described is compatible
with the defining
relations of $A$. Here is a more complete argument, based
on the
description of $A(M/K, \sigma_i, u_{ij}, b_i)$ given in the proof of
\cite[Th.~1.3]{as}.
First, take any ring $B$ with an automorphism $\sigma$,
and let $B[y;\sigma]$ be the twisted polynomial ring
$\{\sum c_iy^i\,|\ c_i\in B\}$ with the multiplication
determined by
$yc = \sigma(c) y$ for all $c\in B$. It is easy to check that
an involution
$\rho$ on $B$ extends to an involution $\rho'$ on $B[y;\sigma]$
with
$\rho'(y) = y$ iff $\sigma\rho\sigma = \rho$. Also, for
$d\in B^*$, an
automorphism $\eta$ of $B$ extends to an automorphism
$\eta'$ of
$B[y;\sigma]$ with $\eta'(y) = dy$ iff $\intt(d) \sigma\eta =
\eta \sigma$.
Here, let $B_0 = M$,
$B_1 = B_0[y_1;\sigma_1^*]$, \ldots, $B_\ell =
B_{\ell-1}[y_\ell;\sigma_\ell^*]$,
\ldots, $B_k = B_{k-1}[y_k;\sigma_k^*]$, where $\sigma_1^* =
\sigma_1$ and for~$\ell>1$,
the automorphism
$\sigma_\ell^*$ of $B_{\ell-1}$ is defined by $\sigma_\ell^*|_M =
\sigma_\ell$ and $\sigma_\ell^*(y_i)
= u_{\ell i} y_i$ for $1\le i<\ell$. (One checks inductively
using the identities in \eqref{urels} that
for $1\le i \le \ell-1$,
$\sigma_\ell^*$
satisfies $\intt(u_{\ell i})\sigma_i^*\sigma_\ell^* = \sigma_\ell^*
\sigma_i^*$ on~$B_{i-1}$, hence
$\sigma_\ell^*$ extends from $B_{i-1}$ to $B_i$;
thus, $\sigma_\ell^*$ is an automorphism of $B_{\ell-1}$.)
Define inductively involutions
$\tau_i$ on $B_i$ by $\tau_0 = \theta$ and for $\ell >0$,
$\tau_\ell |_{B_{\ell-1}}
= \tau_{\ell-1}$ and $\tau_\ell(y_\ell) = y_\ell$. Given
$\tau_{\ell-1}$, the
condition for the existence of~$\tau_\ell$ is that
$\sigma_\ell^* \tau_{\ell-1}\sigma_\ell^* = \tau_{\ell-1}$.
For this, note first that
$\sigma_\ell^* \tau_{\ell-1}\sigma_\ell^*|_M = \sigma_\ell
\theta \sigma_\ell = \theta =
\tau_{\ell-1}|_M$ as $G$ is generalized dihedral.
Furthermore, for $1\le i<\ell$,
$$
\sigma_\ell^* \tau_{\ell-1}\sigma_\ell^*(y_i) \, = \
\sigma_\ell^*\tau_{\ell-1}(u_{\ell i} y_i)
\, = \ \sigma_\ell^*\big(y_i\, \theta(u_{\ell i})\big) \, = \
\sigma_\ell^*[\sigma_i\theta(u_{\ell i})\, y_i]
\, = \, [\sigma_\ell\sigma_i\theta(u_{\ell i})] u_{\ell i}\, y_i \, = \ y_i
\, = \ \tau_{\ell-1}(y_i).
$$
Thus, $\sigma_\ell^* \tau_{\ell-1}\sigma_\ell^*$ agrees with
$\tau_{\ell-1}$
throughout $B_{\ell-1}$, as needed. By induction, we have
the involution
$\tau_k$ on $B_k$. As pointed out in \cite[p.~79]{as},
$A\cong B_k/I$,
where $I$ is the two-sided ideal of $B_k$ generated by
$\{ y_i^{r_i}-b_i \, |\ 1\le i\le k\}$. Since
$\tau_k(b_i) = \theta(b_i) = b_i$,
$\tau_k$ maps each generator of $I$ to itself. Therefore,
$\tau_k$ induces an involution
$\tau$ on $A \cong B_k/I$ which clearly restricts to
$\theta$ on $M$; so
$\tau$ is a unitary $K/F$-involution on $A$.
\end{proof}
We write $\operatorname{Br}(M/K;F)$ for the subgroup of $\operatorname{Br}(M/K)$ of
algebra classes $[A]$
such that $A$ has a unitary $K/F$-involution. By Albert's
theorem
\cite[Th.~3.1(2)]{kmrt}, $\operatorname{Br}(M/K;F)$ is the kernel of the
corestriction map
$\text{cor}_{K\to F}\colon \operatorname{Br}(M/K) \to \operatorname{Br}(M/F)$.
For $M$ a $K/F$-generalized
dihedral extension of $F$, as above, there is
in addition a
corresponding subgroup of
$\operatorname{Dec}(M/K)$. For this, note first that for any field $L$
with
$K \subseteq L\subseteq M$ and $L$ cyclic
Galois over $K$, say $\operatorname{Gal}(L/K) = \langle \sigma \rangle$,
$L$ is $K/F$-dihedral,
so Lemma~\ref{unitarycp} (with $k = 1$) implies that
$\operatorname{Br}(L/K;F) = \{[(L/K, \sigma, b)]\mid b\in F^*\}$. For
$H = \operatorname{Gal}(M/K)
= \langle \sigma_1\rangle \times \ldots \times\langle
\sigma_k \rangle$ and $(\chi_1, \ldots, \chi_k)$ the base
of $X(M/K)$ dual to $(\sigma_1, \ldots, \sigma_k)$, and $L_i$ the
fixed field of $\ker(\chi_i)$,
as at the beginning of \S\ref{abcp}, define
\begin{equation}
\operatorname{Dec}(M/K;F) \ = \ \{ [(L_1/K, \sigma_1, b_1)\otimes_K\ldots
\otimes _K
(L_k/K,\sigma_k, b_k)]\mid \text {each } b_i\in F^*\} \
\subseteq \ \operatorname{Br}(M/K;F)\}.
\end{equation}
Note that $\operatorname{Dec}(M/K;F)$ is generated as a group by the
image under the cup
product of $H^2(H,\mathbb{Z})\times F^*$. Thus $\operatorname{Dec}(M/K;F)$ is
independent of
the choice of cyclic decomposition of $H$, and we have
analogously to~\eqref{dec},
\begin{equation}\label{Decformula}
\operatorname{Dec}(M/K;F) \ = \ \textstyle\prod\limits _{i = 1}^k
\operatorname{Br}(L_i/K;F)
\ = \ \prod \limits_
{\substack{K \subseteq L \subseteq M \\ \operatorname{Gal}(L/K)
\text{ cyclic}}}
\operatorname{Br}(L/K;F).
\end{equation}
For the rest of this section we fix a graded field
$\mathsf{T}$ and a graded subfield $\mathsf{R}\subseteq \mathsf{T}$
such that $[\mathsf{T}\! :\!\mathsf{R}] =2$ and $\mathsf{T}$ is inertial and
Galois over $\mathsf{R}$. Let $\psi$ be the nonidentity graded
$\mathsf{R}$-automorphism of $\mathsf{T}$, and let~$\psi_0$~be the
restriction $\psi|_{\mathsf{T}_0}$.
Thus, $\Gamma_\mathsf{T} = \Gamma_\mathsf{R}$,
$[\mathsf{T}_0\! :\!\mathsf{R}_0]= 2$, $\mathsf{T} \cong_g \mathsf{T}_0\otimes_{\mathsf{R}_0} \mathsf{R}$,
$\mathsf{T}_0$ is Galois over $\mathsf{R}_0$, and $\psi$ on $\mathsf{T}$ corresponds
to $\psi_0 \otimes \operatorname{id}_\mathsf{R}$ on $ \mathsf{T}_0
\otimes_{\mathsf{R}_0} \mathsf{R}$. We are interested in central simple
graded $\mathsf{T}$-algebras $\mathsf{A}$ with graded unitary
$\mathsf{T}/\mathsf{R}$-involutions $\tau$. This means that $\tau$ is a
degree-preserving ring antiautomorphism of $\mathsf{A}$ with
$\tau^2 = \operatorname{id}_\mathsf{A}$ and the ring of invariants $\mathsf{T}^\tau=
\mathsf{R}$; the last condition is equivalent to $\tau|_\mathsf{T} = \psi$. Suppose now
that $\mathsf{A}$ is a graded division algebra. Set $\tau_0
= \tau|_{\mathsf{A}_0}$, which is a unitary involution on
$\mathsf{A}_0$, as $\tau_0|_{\mathsf{T}_0} = \psi_0 \ne \operatorname{id}$ and
$\mathsf{T}_0 \subseteq Z(\mathsf{A}_0)$. Just as for any graded division
algebra, $Z(\mathsf{A}_0)$ is abelian Galois over $\mathsf{T}_0$. But the
presence of the involution~$\tau$ implies further that
$Z(\mathsf{A}_0)$ is actually $\mathsf{T}_0/\mathsf{R}_0$-generalized dihedral,
by \cite[Lemma~4.6(ii)]{I}.
A central graded division algebra $\mathsf{N}$ over $\mathsf{T}$ is said to
be {\it decomposably semiramified for $\mathsf{T}/\mathsf{R}$}
(abbreviated DSR for $\mathsf{T}/\mathsf{R}$) if $\mathsf{N}$ has a
unitary graded $\mathsf{T}/\mathsf{R}$-involution $\tau$ and a
maximal graded subfield $\mathsf{M}$ inertial over~$\mathsf{T}$ and another
maximal graded subfield $\mathsf{J}$ with $\mathsf{J}$ totally ramified
over~$\mathsf{T}$ and $\tau(\mathsf{J}) = \mathsf{J}$. When this occurs,
$\mathsf{N}$~is semiramified with $\mathsf{N}_0 = \mathsf{M}_0$, a field,
which as just noted is $\mathsf{T}_0/\mathsf{R}_0$-generalized dihedral.
Also, $\Gamma_\mathsf{N} = \Gamma_\mathsf{J}$ and $\Theta_\mathsf{N}$ induces an
isomorphism $\Gamma_\mathsf{N}/\Gamma_\mathsf{T} \cong \operatorname{Gal}(\mathsf{N}_0/
\mathsf{T}_0)$. Furthermore, as $\mathsf{M} = \mathsf{M}_0\mathsf{T} = \mathsf{N}_0\mathsf{T}$, we have
$\tau(\mathsf{M}) = \mathsf{M}$.
\begin{example}\label{DSRex}
Let $L$ be any cyclic Galois field extension of ${\mathsf{T}_0}$ with
$L$ dihedral over ${\mathsf{R}_0}$. (That is, $L$ is Galois over ${\mathsf{R}_0}$
and there is $\theta \in \operatorname{Gal}(L/{\mathsf{R}_0})\setminus\operatorname{Gal}(L/{\mathsf{T}_0})$ with
$\theta^2 = \operatorname{id}_\mathsf{L}$ and
$\theta h\theta^{-1} = h^{-1}$ for every $h\in \operatorname{Gal}(L/{\mathsf{T}_0})$.
Thus, the group $\operatorname{Gal}(L/{\mathsf{R}_0})$ is either dihedral or isomorphic to
$\mathbb{Z}/2\mathbb{Z}$ or $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$.) Let $r =
[L\! :\!{\mathsf{T}_0}]$, and take any $b\in \mathsf{R}^*$ with the image
of $\deg(b)$ having order $r$ in $\Gamma_\mathsf{T}/r\Gamma_\mathsf{T}$. Take
any generator $\sigma$ of $\operatorname{Gal}(L/{\mathsf{T}_0})$, and let
$\sigma$ denote also its canonical extension $\sigma\otimes \operatorname{id}_\mathsf{T}$
in $\mathsf{Gal}((L\otimes _{\mathsf{T}_0} \mathsf{T})/\mathsf{T})$. Let
$$
\mathsf{N} \, = \,((L\otimes _{\mathsf{T}_0} \!\mathsf{T})/\mathsf{T}, \sigma, b),
\text{ a cyclic graded algebra over $\mathsf{T}$}.
$$
We show that $\mathsf{N}$ is a central graded division algebra over $\mathsf{T}$
of degree $r$, and $\mathsf{N}$ is DSR for $\sT/\sR$. For, letting
$L\mathsf{T}$ denote $L\otimes _{\mathsf{T}_0} \!\mathsf{T}$, note that $L\mathsf{T}$ is a
graded field which is inertial over $\mathsf{T}$ and is Galois over
$\mathsf{T}$ with ${\mathsf{Gal}(L\mathsf{T}/\mathsf{T}) = \langle \sigma\rangle}$. Our $\mathsf{N}$
is $\bigoplus_{i=0}^{r-1}L\mathsf{T} z^i$, where $zcz^{-1} =
\sigma(c)$ for all $c\in L\mathsf{T}$, and $z^r = b$, with the grading
on~$\mathsf{N}$ extending that on $L\mathsf{T}$ by setting $\deg(z)=
\frac 1 r \deg(b)$. A graded cyclic $\mathsf{T}$-algebra is
always graded simple with center~$\mathsf{T}$.
Note that for $j\in \mathbb{Z}$,
if $j\deg(b)/r\in \Gamma_{L\mathsf{T}} = \Gamma_\mathsf{T}$,
then $j\deg(b) \in r\Gamma_\mathsf{T}$,
so by hypothesis $r\,|\, j$.
Hence,
$$
\mathsf{N}_0 \,=\,\textstyle \sum\limits_{i=0}^{r-1}(L\mathsf{T})_{-i\deg(b)/r}\,z^i
\ =\,
(L\mathsf{T})_0 \,=\, L.
$$ Since $\mathsf{N}_0$~is a division ring, the
simple graded algebra~$N$~is a graded division ring,
by Lemma~\ref{zerosimple}(ii).
Also, as
${[L\mathsf{T}\! :\! \mathsf{T}] = [L\! :\! {\mathsf{T}_0}] = r = \deg(\mathsf{N})}$,
$L\mathsf{T}$~is a maximal graded subfield of
$\mathsf{N}$ which is inertial over $\mathsf{T}$. Take any
$\theta\in \operatorname{Gal}(L/{\mathsf{R}_0})$ with ${\theta|_{\mathsf{T}_0} = \psi_0}$,
and let $\theta$ denote
also its canonical extension $\theta \otimes \operatorname{id}|_\mathsf{R}$
to $\mathsf{Gal}(L\mathsf{T}/\mathsf{R})$. Define a map $\tau\colon \mathsf{N}\to \mathsf{N}$
by
$$
{\tau(\textstyle\sum\limits_{i=0}^{r-1} c_i z^i) \,
=\, \sum\limits_{i=0}^{r-1} z^i\theta (c_i) \,=\,
\sum\limits_{i=0}^{r-1} \sigma^i \theta(c_i) z^i}.
$$ Since $\theta|_\mathsf{T} = \psi$,
$\theta^2 = \operatorname{id}$,
and $\theta \sigma \theta ^{-1} = \sigma^{-1}$ (as $L$ is
${\mathsf{T}_0}/{\mathsf{R}_0}$-dihedral), it is easy to check
that $\tau$ is a graded $\sT/\sR$-involution of $\mathsf{N}$. Moreover,
if we let $\mathsf{J} = \bigoplus_{i=0}^{r-1} \mathsf{T} z^i= \mathsf{T}[z]$, then
$\mathsf{J}$ is a maximal graded subfield of $\mathsf{N}$, and the hypothesis
on $\deg(b)$ assures that $\mathsf{J}$ is totally ramified over $\mathsf{T}$;
also $\tau(\mathsf{J}) = \mathsf{J}$. This verifies that $\mathsf{N}$~
is DSR for $\sT/\sR$. Note that $\mathsf{N}_0 = L$ and
$\Gamma_\mathsf{N} = \langle \frac 1r \deg(b)\rangle + \Gamma_\mathsf{T}$.
\end{example}
\begin{lemma}\label{DSRprod}
Let $\mathsf{N}$ and $\mathsf{N}'$ be graded division algebras which
are each DSR for $\mathsf{T}/\mathsf{R}$. Suppose
$\mathsf{N}_0$ and $\mathsf{N}'_0$ are linearly disjoint over
$\mathsf{T}_0$ and $\Gamma_\mathsf{N} \cap\Gamma_{\mathsf{N}'} = \Gamma_\mathsf{T}$.
Then, $\mathsf{N}\otimes _\mathsf{T} \mathsf{N}'$ is a graded division algebra
which is DSR for~$\sT/\sR$. Also, $(\mathsf{N}\otimes_\mathsf{T}\mathsf{N}')_0
\cong \mathsf{N}_0 \otimes _{\mathsf{T}_0} \mathsf{N}_0'$ and
$\Gamma_{\mathsf{N}\otimes _\mathsf{T}\mathsf{N}'} = \Gamma_\mathsf{N} + \Gamma_{\mathsf{N}'}$.
\end{lemma}
\begin{proof}
Let $\mathsf{B} = \mathsf{N}\otimes_\mathsf{T} \mathsf{N}'$, which is a
central simple graded $\mathsf{T}$-algebra, since this is true
for $\mathsf{N}$ and $\mathsf{N}'$ by \cite[Prop.~1.1]{hwcor}.
For each $\gamma\in \Gamma_\mathsf{T}$ choose a
nonzero $t_\gamma\in \mathsf{T}_\gamma$.
Then,
$$
\mathsf{B}_0 \ = \textstyle\sum \limits_{\gamma\in \Gamma_\mathsf{N}\cap
\Gamma_{\mathsf{N}'}}\mathsf{N}_\gamma\otimes
_{\mathsf{T}_0}\! \mathsf{N}'_{-\gamma} \ = \ \sum\limits_{\gamma\in \Gamma_\mathsf{T}}
\mathsf{N}_0 \,t_\gamma\otimes _{\mathsf{T}_0} \!\mathsf{N}'_0\,t_\gamma^{-1} \ =
\ \mathsf{N}_0 \otimes_{\mathsf{T}_0}\!\mathsf{N}'_0.
$$
The linear disjointness hypothesis assures that
$\mathsf{B}_0$ is a field, and hence $\mathsf{B}$ is a
graded division ring, by Lemma~\ref{zerosimple}(ii).
Moreover, by dimension count $\mathsf{B}_0 \mathsf{T}$
is a graded maximal subfield of $\mathsf{B}$ which is inertial over~
$\mathsf{T}$. Let $\tau$ be a graded $\sT/\sR$-involution of
$\mathsf{N}$, and let $\mathsf{J}$ be a graded maximal subfield of
$\mathsf{N}$ with $\tau(\mathsf{J}) = \mathsf{J}$. Take $\tau'$ and $\mathsf{J}'$
correspondingly for $\mathsf{N}'$. Then,
$\mathsf{J}\sJ' = \mathsf{J}\otimes_\mathsf{T}\mathsf{J}'$ and $\tau \otimes \tau'$
is a graded $\sT/\sR$-involution on $\mathsf{B}$ with
$(\tau\otimes \tau')(\mathsf{J}\sJ') = \mathsf{J}\sJ'$. Moreover,
$\mathsf{J}\sJ'$ is a maximal graded subfield of $\mathsf{B}$ by dimension count,
and, as $\Gamma_\mathsf{J}\cap \Gamma_{\mathsf{J}'} = \Gamma_\mathsf{N}\cap \Gamma_{\mathsf{N}'} =
\Gamma_\mathsf{T}$, we have
$$
|\Gamma_{\mathsf{J}\sJ'}\! :\! \Gamma_\mathsf{T}|
\, \ge\, |\Gamma_{\mathsf{J}} +\Gamma_{\mathsf{J}'}\! :\!\Gamma_\mathsf{T}|\, = \, |\Gamma_\mathsf{J}\! :\!\Gamma_\mathsf{T}|
\cdot |\Gamma_{\mathsf{J}'}\! :\! \Gamma_\mathsf{T}|\, =\, [\mathsf{J}\! :\!\mathsf{T}]\cdot [\mathsf{J}'\! :\!\mathsf{T}]
\, =\, [\mathsf{J}\sJ'\! :\!\mathsf{T}].
$$
Hence, $\mathsf{J}\sJ'$ is totally
ramified over~$\mathsf{T}$. Thus, $\mathsf{B}$ is DSR for $\sT/\sR$.
\end{proof}
The next proposition shows that all graded division algebras
$\mathsf{N}$
which are DSR for $\sT/\sR$ are obtainable from those in
Ex.~\ref{DSRex} by iterated application of Prop.~\ref{DSRprod}.
This justifies the term \lq\lq decomposably semiramified" for such $\mathsf{N}$.
\begin{proposition}\label{DSRdecomp}
Let $\mathsf{N}$ be a graded division algebra which is DSR for
$\sT/\sR$. Take any decomposition ${\mathsf{N}_0 = L_1 \otimes _{\mathsf{T}_0}
\ldots \otimes _{\mathsf{T}_0}\!L_k}$ with each
$L_i$ cyclic Galois over ${\mathsf{T}_0}$, and choose correspondingly\break
${\sigma_1 \ldots, \sigma_k\in \mathsf{Gal}(\mathsf{N}_0\mathsf{T}/\mathsf{T}) \cong
\operatorname{Gal}(\mathsf{N}_0/{\mathsf{T}_0})}$ such that $\sigma_i|_{L_j}
= \operatorname{id}$ whenever $j\ne i$ and $\operatorname{Gal}(L_i/{\mathsf{T}_0}) = \langle
\sigma_i|_{L_i}\rangle$ for each $i$. $($So ${\mathsf{Gal}(\mathsf{N}_0\mathsf{T}/\mathsf{T})=
\langle \sigma_1\rangle \times \ldots \times \langle \sigma_k\rangle}$.$)$
Let $r_i$ be the order of $\sigma_i$.
For each $i$ choose $\gamma_i\in \Gamma_\mathsf{N}$ with $\Theta_\mathsf{N}(\gamma_i)
= \sigma_i$. Then, there exist $b_1,\ldots, b_k \in \mathsf{R}^*$
such that $\deg(b_i) = r_i \gamma_i$ and
$$
\mathsf{N} \ \cong_g (L_1\mathsf{T}/\mathsf{T}, \sigma_1, b_1) \otimes _\mathsf{T}
\ldots \otimes _\mathsf{T} (L_k\mathsf{T}/\mathsf{T}, \sigma_k, b_k)
\ \cong_g \ \mathsf{A}(\mathsf{N}_0\mathsf{T}/\mathsf{T}, \boldsymbol\sigma,\boldsymbol 1, \mathbf b).
$$
\end{proposition}
\begin{proof}
Since $\mathsf{N}$ is DSR for $\sT/\sR$, there is a graded
$\sT/\sR$-involution $\tau$ of $\mathsf{N}$ and a maximal graded
subfield $\mathsf{J}$ of $\mathsf{N}$ with $\mathsf{J}$ totally ramified
over $\mathsf{T}$ and $\tau(\mathsf{J}) = \mathsf{J}$. As noted earlier,
we have $\Gamma_\mathsf{J} = \Gamma_\mathsf{N}$. Since $\tau$ is a graded
automorphism of $\mathsf{J}$ of order $2$, the fixed set
$\mathsf{S} = \mathsf{J}^\tau = \{a\in \mathsf{J}\mid \tau(a) = a\}$ is
a graded subfield of $\mathsf{J}$ with $2 = [\mathsf{J}\! :\!\mathsf{S}] =
[\mathsf{J}_0\! :\!\mathsf{S}_0]\,|\Gamma_\mathsf{J}\! :\!\Gamma_\mathsf{S}|$.
Since $\mathsf{S}_0 \cap{\mathsf{T}_0} = {\mathsf{R}_0} \subsetneqq {\mathsf{T}_0} =
\mathsf{J}_0\cap {\mathsf{T}_0}$ we have $\mathsf{S}_0\subsetneqq\mathsf{J}_0$, so
$[\mathsf{J}_0\! :\!\mathsf{S}_0] = 2$, and hence $\Gamma_\mathsf{S} = \Gamma_\mathsf{J}
\ (= \Gamma_\mathsf{N})$.
Thus, for each $i$ there is a nonzero $x_i \in \mathsf{S}_{\gamma_i}$,
and for any such $x_i$,
$\intt(x_i)|_{\mathsf{N}_0\mathsf{T}} = \sigma_i$ as
$\Theta_\mathsf{N}(\gamma_i) = \sigma_i$. Let $b_i = x_i^{r_i} \in \mathsf{S}^*$.
Then, $\Theta_\mathsf{N}(\deg(b_i)) = \sigma_i^{r_i} = \operatorname{id}$,
so $\deg(b_i) \in \ker(\Theta_\mathsf{N}) = \Gamma_\mathsf{T}$;
hence, ${b_i \in \mathsf{J}_{\deg(b_i)} = \mathsf{T}_{\deg(b_i)}}$ as
$\mathsf{J}$ is totally ramified over $\mathsf{T}$. Therefore,
$b\in \mathsf{S}^* \cap \mathsf{T}= \mathsf{R}^*$. Let $\mathsf{C}_i$ be the
graded $\mathsf{T}$-subalgebra of~$\mathsf{N}$ generated by
$L_i$ and $x_i$. Since $\intt(x_i)|_{L_i\mathsf{T}} = \sigma_i|_{L_i\mathsf{T}}$,
there is a graded $\mathsf{T}$-algebra epimorphism
$(L_i\mathsf{T}/\mathsf{T},\sigma_i,b_i) \to \mathsf{C}_i$, which is a graded
isomorphism as the domain is graded simple. Since the
$x_i$ all lie in the graded field $\mathsf{S}$
and $\sigma_i|_{L_j\mathsf{T}} = \operatorname{id}$ for $j\ne i$, the distinct
$\mathsf{C}_i$ centralize each other. Hence, there is a graded
$\mathsf{T}$-algebra homomorphism
${(L_1\mathsf{T}/\mathsf{T},\sigma_1,b_1) \otimes _\mathsf{T} \ldots \otimes _\mathsf{T}
(L_k\mathsf{T}/\mathsf{T},\sigma_k,b_k)\to \mathsf{N}}$ which is injective as
the domain is graded simple, and surjective by dimension
count. Clearly also, ${(L_1\mathsf{T}/\mathsf{T},\sigma_1,b_1) \otimes _\mathsf{T} \ldots \otimes _\mathsf{T}
(L_k\mathsf{T}/\mathsf{T},\sigma_k,b_k)\cong_g \mathsf{A}(\mathsf{N}_0\mathsf{T}/\mathsf{T}, \boldsymbol\sigma,
{\boldsymbol 1}, \mathbf b)}$.
\end{proof}
\begin{proposition}\label{uINdecomp}
Let $\mathsf{E}$ be a semiramified central graded division
algebra over $\mathsf{T}$, and suppose $\mathsf{E}$ has a graded
$\sT/\sR$-involution, where $\mathsf{T}$ is inertial over $\mathsf{R}$.
Then, $\mathsf{E}_0$ is ${\mathsf{T}_0}/{\mathsf{R}_0}$-generalized dihedral
and
\begin{enumerate}
\item[{\rm(i)}]
$\mathsf{E} \sim_g \mathsf{I} \otimes _\mathsf{T} \mathsf{N}$ in $\mathsf{Br}(\mathsf{T})$
for some $\mathsf{T}$-central graded
division algebras $\mathsf{I}$ and $\mathsf{N}$ with $\mathsf{I}$
inertial and $\mathsf{N}$~DSR~for~$\sT/\sR$.
\item[{\rm(ii)}]
Take any decomposition
$\mathsf{T} \sim_g \mathsf{I}' \otimes _\mathsf{T} \mathsf{N}'$ in $\mathsf{Br}(\mathsf{T})$
with graded $\mathsf{T}$-central division algebras $\mathsf{I}'$
and $\mathsf{N}'$ with $\mathsf{I}'$~inertial and
$\mathsf{N}'$ DSR for $\sT/\sR$. Then,
$\mathsf{N}_0'\cong \mathsf{E}_0$, $\Gamma_{\mathsf{N}'}= \Gamma_\mathsf{E}$, $\Theta_{\mathsf{N}'} =
\Theta_\mathsf{E}$, and ${[\,\mathsf{I}_0'] \in \operatorname{Br}(\mathsf{E}_0/{\mathsf{T}_0};{\mathsf{R}_0})}$.
Furthermore, $\mathsf{I}_0'$ is uniquely determined modulo
$\operatorname{Dec}(\mathsf{E}_0/{\mathsf{T}_0};{\mathsf{R}_0})$.
\end{enumerate}
\end{proposition}
\begin{proof} (i)
Since $\mathsf{E}$ is semiramified, $\mathsf{E}_0\mathsf{T}$ is an inertial
maximal graded subfield of $\mathsf{E}$. Moreover, as
$\mathsf{E}$~has a graded $\sT/\sR$-involution, $\mathsf{E}_0$
is ${\mathsf{T}_0}/{\mathsf{R}_0}$-generalized dihedral, by \cite[Lemma~4.6(ii)]{I}.
Because $\mathsf{E}$ has an inertial graded maximal subfield,
it is a graded abelian crossed product: Say
$\mathsf{E}_0 = L_1\otimes_{\mathsf{T}_0}\ldots\otimes _{\mathsf{T}_0} L_k$, where
each field $L_i$~is cyclic Galois over ${\mathsf{T}_0}$ (so
dihedral over ${\mathsf{R}_0}$). Then $G = \mathsf{Gal}(\mathsf{E}_0\mathsf{T}/\mathsf{T})
\cong \operatorname{Gal}(\mathsf{E}_0/{\mathsf{T}_0})$ has a corresponding
cyclic decomposition
$G = \langle \sigma_1\rangle\times \ldots \times \langle\sigma_k\rangle$,
where each $\sigma_i|_{L_j\mathsf{T}} = \operatorname{id}$ for $j\ne i$,
and $\sigma_i|_{L_i\mathsf{T}}$ generates $\mathsf{Gal}(L_i\mathsf{T}/\mathsf{T})$.
Let $r_i = |\langle\sigma_i\rangle|=[L_i\! :\!{\mathsf{T}_0}]$.
By Lemma~\ref{incp}, $\mathsf{E} = \mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T}, \boldsymbol\sigma, \mathbf u,\mathbf b)$
where each $u_{ij} \in \mathsf{E}_0^*$, $b_i \in \mathsf{E}_0\mathsf{T}^*$,
$\frac 1{r_i}\deg(b_i) + \Gamma_\mathsf{T}$ has order $r_i$ in
$\Gamma_\mathsf{E}/\Gamma_\mathsf{T}$, and
\begin{equation}\label{relGamma}
\textstyle\Gamma_\mathsf{E}/\Gamma_\mathsf{T} \, = \
\langle\frac 1{r_1}\deg(b_1) + \Gamma_\mathsf{T} \rangle\times \ldots \times
\langle\frac 1{r_k}\deg(b_k) + \Gamma_\mathsf{T} \rangle.
\end{equation}
So, $\deg(b_i) \in \Gamma_{\mathsf{E}_0\mathsf{T}} = \Gamma_\mathsf{T} = \Gamma_\mathsf{R}$ and
the image of $\deg(b_i)$ has order $r_i$ in $\Gamma_\mathsf{T}/r_i\Gamma_\mathsf{T}$.
For each $i$, choose $c_i\in \mathsf{R}^*$ with $\deg(c_i) =
\deg(b_i)$. Let
$$
\mathsf{N} \, = \ \mathsf{C}_1\otimes _\mathsf{T} \ldots \otimes_\mathsf{T} \mathsf{C}_k,
\ \ \text{where each}\ \ \mathsf{C}_i = (L_i\mathsf{T}/\mathsf{T}, \sigma_i, c_i).
$$
By Ex.~\ref{DSRex} each $\mathsf{C}_i$ is DSR for $\sT/\sR$ with
$(\mathsf{C}_i)_0 \cong L_i$ and $\Gamma_{\mathsf{C}_i} = \langle\frac1{r_i}
\deg( c_i)\rangle +\Gamma_\mathsf{T} =\langle\frac1{r_i}
\deg(b_i)\rangle +\Gamma_\mathsf{T}$. It follows by
induction on~$k$ using Lemma~\ref{DSRprod}
and \eqref{relGamma} that
$\mathsf{N}$ is a graded division algebra which is DSR for $\sT/\sR$.
Choose $z_i\in \mathsf{C}_i^*$ with $\intt(z_i)|_{L_i\mathsf{T}} = \sigma_i$
and $z_i^{r_i} = c_i$.
Then, when we view $z_i\in \mathsf{N}^*$, we have $\intt(z_i) = \sigma_i$
on all of $\mathsf{N}_0\mathsf{T}$. Since further $z_iz_j= z_jz_i$ for all
$i,j$, our
$\mathsf{N}$ is the graded abelian crossed product
$\mathsf{N} = \mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T}, \boldsymbol\sigma, {\boldsymbol 1}, \bf c)$. For its
opposite algebra $\mathsf{N}^\text{op}$ we then have
$\mathsf{N}^\text{op} \cong_g \mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T}, \boldsymbol\sigma, {\boldsymbol 1}, \bf d)$
where each $d_i = c_i^{-1}$.
Let $\widehat \mathsf{I} = \mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T}, \boldsymbol\sigma, \mathbf u, \bf e)$
where each $e_i =b_id_i=b_i c_i^{-1}\in\mathsf{E}_0^*$. The $u_{ij}$ and $b_i$
satisfy conditions \eqref{urels} and \eqref{brel}, as do the
$c_i$ with the corresponding $u_{ij}=1$; hence the $u_{ij}$ here and~
$e_i$ satisfy \eqref{urels} and \eqref{brel}; also, $\deg(u_{ij}) = 0$
for all $i,j$.
So, $\widehat \mathsf{I}$ is a well-defined graded abelian crossed product.
By Remark~\ref{abeliancpprod},
we have $\widehat \mathsf{I} \sim_g\mathsf{E} \otimes_\mathsf{T} \mathsf{N}^\text{op}$.
There are homogeneous $x_1, \ldots, x_k\in \widehat \mathsf{I}^{\, *}$
such that $\intt(x_i)|_{\mathsf{E}_0\mathsf{T}} = \sigma_i$, $x_i^{r_i}= e_i$, and
$x_ix_jx_i^{-1} x_j^{-1} = u_{ij}$ for all $i,j$. Then,
$\deg(x_i) = \frac 1 {r_i}\deg(e_i) = 0$; hence,
$\deg(x^{\bold i}) = 0$ for each $\bold i \in
\mathcal I=\prod_{i=1}^k
\{0,1,2,\dots,r_i-1\}$. Thus, in $\widehat\mathsf{I} = \bigoplus
_{\bold i \in \mathcal I}\mathsf{E}_0\mathsf{T} x^{\bold i}$ we have
$\widehat \mathsf{I}_{\,0} =\bigoplus_{\bold i \in \mathcal I}
\mathsf{E}_0 x^{\bold i} \cong A(\mathsf{E}_0/{\mathsf{T}_0}, \boldsymbol\sigma, \mathbf u, \bold e)$,
which is a central simple ${\mathsf{T}_0}$-algebra with
$\dim_{\mathsf{T}_0}(\,\,\widehat \mathsf{I}_{\,0}) = [\mathsf{E}_0\! :\!{\mathsf{T}_0}]^2 =
\dim_\mathsf{T}\big(\,\widehat \mathsf{I}\,\big)$. Hence, $\widehat \mathsf{I}$
is inertial over~$\mathsf{T}$. Since $\widehat\mathsf{I}$ is simple,
by Lemma~\ref{zerosimple} $\widehat\mathsf{I}\cong_g M_\ell(\mathsf{I})$
for a graded division algebra $\mathsf{I}$ with $\widehat\mathsf{I}_{\,0} \cong
M_\ell(\mathsf{I}_0)$. Then, $[\mathsf{I}_0\! :\! {\mathsf{T}_0}] =
\frac 1{\ell^2}\dim_{\mathsf{T}_0}(\,\,\widehat \mathsf{I}_{\,0})=
\frac 1{\ell^2}\dim_\mathsf{T}\big(\,\widehat \mathsf{I}\,\big) =
[\mathsf{I}\! :\!\mathsf{T}]$, showing that $\mathsf{I}$ is inertial over $\mathsf{T}$.
Since $\mathsf{I} \sim_g \widehat \mathsf{I}$, we have in~$\mathsf{Br}(\mathsf{T})$,
$$
[\mathsf{E}] \, = \, [\mathsf{E}]\,[\mathsf{N}]^{-1}[\mathsf{N}] \, = \,
[\mathsf{E} \otimes_\mathsf{T}\mathsf{N}^\text{op}] \, [\mathsf{N}] \, = \,[\,\widehat\mathsf{I}\,]\,[\mathsf{N}]
\, = \,[\,\mathsf{I}\,]\,[\mathsf{N}] \,=\, [\,\mathsf{I} \otimes_\mathsf{T} \mathsf{N}\,],
$$
i.e., $\mathsf{E} \sim_g \mathsf{I} \otimes _\mathsf{T} \mathsf{N}$, proving
(i). Also, $\mathsf{N}$ has a graded $\sT/\sR$-involution $\tau_\mathsf{N}$,
which is also a graded involution for~$\mathsf{N}^\text{op}$, and
$\mathsf{E}$ has a graded $\sT/\sR$-involution $\tau_\mathsf{E}$. So,
$\tau = \tau_\mathsf{E} \otimes \tau_\mathsf{N}$ is a graded $\sT/\sR$-involution
on $\widehat\mathsf{I}\,$, and $\tau_0 = \tau|_{\,\widehat \mathsf{I}_{\,0}}$ is a
${\mathsf{T}_0}/{\mathsf{R}_0}$-involution on $\widehat\mathsf{I}_{\,0}$.
So, in $\operatorname{Br}({\mathsf{T}_0})$ we have
$[\,\mathsf{I}_0] = [\,\widehat \mathsf{I}_{\,0}] \in \operatorname{Br}(\mathsf{E}_0/{\mathsf{T}_0};{\mathsf{R}_0})$.
(ii) Take any decomposition $\mathsf{E} \sim_g \mathsf{I}' \otimes
\mathsf{N}'$ as in (ii).
Since $\mathsf{I}'$ is inertial and
$\mathsf{E}$ is the graded division algebra with $\mathsf{E}\sim_g
\mathsf{I}'\otimes _\mathsf{T} \mathsf{N}'$, Cor.~\ref{ItensorE} yields
$\mathsf{E}_0\sim \mathsf{I}'_0 \otimes_{\mathsf{T}_0} \mathsf{N}'_0$ and $ \mathsf{E}_0 =Z(\mathsf{E}_0)
\cong Z(\mathsf{N}'_0) = \mathsf{N}'_0$, so $\mathsf{E}_0$~splits~
$\mathsf{I}'_0$; furthermore, $\Gamma_\mathsf{E} = \Gamma_{\mathsf{N}'}$
and $\Theta_\mathsf{E} = \Theta_{\mathsf{N}'}$.
We now use the $b_i$, $c_i$, $\mathsf{N}$, and $\mathsf{I}$ of part (i).
Because $\mathsf{N}'$ is DSR with $\mathsf{N}_0'\cong \mathsf{E}_0$ and
$\Theta_{\mathsf{N}'}(\frac 1 {r_i}\deg(c_i)) = \Theta_{\mathsf{E}}
(\frac 1 {r_i}\deg(b_i)) = \sigma_i$, by Prop.~\ref{DSRdecomp}
there exist $c_1', \ldots, c_k'\in \mathsf{R}^*$ with
${\deg(c_i') = \deg(c_i)}$ such that
$\mathsf{N}' \cong_g\mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T}, \boldsymbol\sigma, {\boldsymbol 1}, \bold c')$.
Let $\mathsf{B} = \mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T}, \boldsymbol\sigma, {\boldsymbol 1}, \bold f)$ where
each $f_i = c_i c_i^{\prime -1} \in \mathsf{R}_0^*$. So, in
$\mathsf{Br}(\mathsf{T})$, $\mathsf{B} \sim_g \mathsf{N}\otimes _\mathsf{T}
\mathsf{N}^{\prime \,\text{op}}
\sim_g\mathsf{I}'\otimes _\mathsf{T} \mathsf{I}\,^\text{op}$. Because $\deg(f_i) = 0$ for
each $i$, the argument for $\widehat\mathsf{I}$ in (i) shows that $\mathsf{B}$
is inertial over $\mathsf{T}$ with
$$
\mathsf{B}_0 \, \cong \ A(\mathsf{E}_0/{\mathsf{T}_0}, \boldsymbol\sigma, {\boldsymbol 1}, \bold f) \
\cong \ (L_1/{\mathsf{T}_0}, \sigma_1,f_1)\otimes _{\mathsf{T}_0} \ldots
\otimes _{\mathsf{T}_0}(L_k/{\mathsf{T}_0}, \sigma_k, f_k).
$$
Thus, $[\mathsf{B}_0] \in \operatorname{Dec}(\mathsf{E}_0/{\mathsf{T}_0};\mathsf{R}_0)$, as each
$f_i \in \mathsf{R}_0^*$ (see Ex.~\ref{DSRex}).
Let
$\mathsf{C}$ be the graded division algebra with
$\mathsf{C} \sim_g \mathsf{B} \sim_g \mathsf{I}' \otimes _\mathsf{T}\mathsf{I}^\text{op}$.
Since $\mathsf{B}_0$ is simple and $\mathsf{I}'$ is inertial,
Lemma~\ref{zerosimple} and Cor.~\ref{ItensorE}
yield $\mathsf{C}_0 \sim \mathsf{B}_0$ and
$\mathsf{C}_0 \sim (\mathsf{I}'\otimes_\mathsf{T}\mathsf{I}^\text{op})_0\cong\mathsf{I}'_0\otimes _{\mathsf{T}_0} \mathsf{I}_0^\text{op}$;
so, in $\operatorname{Br}({\mathsf{T}_0})$,
$$
[\,\mathsf{I}'_0]\, = \, [\mathsf{C}_0] \, [\,\mathsf{I}_0] \, =
\, [\mathsf{B}_0] \, [\,\mathsf{I}_0] \, = \,
[\mathsf{B}_0]\,[\,\widehat\mathsf{I}_{\,0}] \, \in \operatorname{Br}(\mathsf{E}_0/{\mathsf{T}_0};{\mathsf{R}_0}).
$$
Since $[\mathsf{B}_0] \in
\operatorname{Dec}(\mathsf{E}_0/{\mathsf{T}_0};{\mathsf{R}_0})$, we have ${\mathsf{I}'_0 \equiv \mathsf{I}_0\
(\operatorname{mod} \ \operatorname{Dec}(\mathsf{E}_0/{\mathsf{T}_0};{\mathsf{R}_0})\mspace{1mu})}$. This yields the uniqueness
of $\mathsf{I}'_0$ modulo $\operatorname{Dec}(\mathsf{E}_0/{\mathsf{T}_0};{\mathsf{R}_0})$ independent of
the choice of decomposition of $\mathsf{E}$ as $\mathsf{I}' \otimes _\mathsf{T}
\mathsf{N}'$.
\end{proof}
\begin{remark}
The $\mathsf{I} \otimes \mathsf{N}$ decomposition described in Prop.~\ref{uINdecomp}
for $\mathsf{E}$ semiramified actually holds more generally for $\mathsf{E}$ inertially
split (with graded $\sT/\sR$-involution), i.e., when $\mathsf{E}$ has a
maximal graded subfield inertial over~$\mathsf{T}$. One then has
$\mathsf{N}_0 \cong Z(\mathsf{E}_0)$ and $\mathsf{I}_0 \otimes _{\mathsf{T}_0} Z(\mathsf{E}_0) \sim \mathsf{E}_0$.
See \cite[Lemma~5.14, Th.~5.15]{jw} for the nonunitary nongraded Henselian valued
analogue of this.
\end{remark}
\section{Galois cohomology with twisted coefficients}\label{twist}
Where $\widehat H^{-1}(H,M^*)$ occurs in formulas for $\SK$ as in \S\ref{abcp},
analogous formulas for the unitary $\SK$ involve $\widehat H^{-1}(G,
\widetilde {M^*})$ for a twisted action of $G$ on the multiplicative group
$M^*$. In this section, we recall the relevant twisted action, and give some
calculations concerning $\widehat H^{-1}$ which will be used later.
The cohomology with twisted action also allows us to give a new
interpretation of Albert's corestriction condition for an algebra
to have a unitary involution, see Prop.~\ref{relbriso} below.
Let $G$ be a profinite group with a closed subgroup $H$ with
$|G\!\! :\! \!H| = 2$. From the mappping\break
${G/H \xrightarrow{\sim} \mathbb{Z}/2\mathbb{Z} \xrightarrow{\sim} \operatorname{Aut}(\mathbb{Z})}$
we obtain a nontrivial discrete $G$-module structure on
$\mathbb{Z}$ for which for $g\in G$, $j\in \mathbb{Z}$,
$$
g*j \ = \ \begin{cases} \ \ j, &\text {if } g\in H,
\\-j, &\text{if } g\notin H.
\end{cases}
$$
Let $\widetilde \mathbb{Z}$ denote $\mathbb{Z}$ with this new $G$-action.
Then, for any discrete $G$-module $A$ we have an
associated discrete $G$-module
$\widetilde A = A \otimes _\mathbb{Z} \widetilde\mathbb{Z}$. That is, $\widetilde A = A$
as an abelian group, but the $G$-action
on $\widetilde A$ (denoted by $*$, while $\cdot$~denotes the
$G$-action on $A$) is given by
\begin{equation}\label{tildeseq}
g*a \ = \ \begin{cases} \ \ g\cdot a, &\text {if } g\in H,
\\-g\cdot a, &\text{if } g\notin H,
\end{cases}
\quad \text{for all $g\in G$, $a\in A$}.
\end{equation}
So, the actions of $H$ on $\widetilde A$ and on $A$ coincide, and
$\widetilde{\widetilde A\,} = A$ as $G$-modules. The cohomology of
such modules is discussed in \cite[Appendix]{ae},
\cite[\S30.B]{kmrt}, \cite[\S5]{hkrt}. Notably,
there is a canonical short exact sequence of $G$-modules
$$
0 \ \longrightarrow \ \widetilde A \ \longrightarrow \
\Ind_{H\to G}(A) \ \longrightarrow \ A \
\longrightarrow \ 0
$$
Since Shapiro's Lemma says that
$\widehat H^i(G, \Ind_{H\to G}(A))
\cong \widehat H^i(H,A)$ for all $i\in \mathbb{Z}$, this yields
a long exact sequence of Tate cohomology groups:
\begin{equation}\label{longexact}
\ldots \ \longrightarrow \ \widehat H^{i-1}(G, A) \ \longrightarrow \widehat H^i(G, \widetilde A)
\ \longrightarrow \ \widehat H^i(H, A) \ \longrightarrow \ \widehat H^i(G,A) \
\longrightarrow \ \widehat H^{i+1}(G, \widetilde A) \
\longrightarrow \ \ldots
\end{equation}
(This is stated in \cite[(30.10)]{kmrt} and \cite{ae} for nonnegative
indices, but it is valid for $i<0$ as well.) For the trivial
$G$-module $\mathbb{Z}$ we have $|H^1(G, \widetilde \mathbb{Z})| = 2$, as
\eqref{longexact} shows, and each connecting homomorphism
$\delta\colon \widehat H^{i-1}(G,A) \to\widehat H^i(G, \widetilde A)$ is
given by the cup product with the nontrivial element of
$H^1(G, \widetilde \mathbb{Z})$.
We will invoke the twisted cohomology typically in the following
setting: Let $F \subseteq K \subseteq M$ be fields with
$[K\! :\! F] = 2$, and $M$ Galois over $F$. Let $G = \operatorname{Gal}(M/F)$
and $H = \operatorname{Gal}(M/K)$, which is a closed subgroup of
$G$ of index $2$. Then, $M^*$ is a discrete $G$-module, and
$\widetilde{M^*}$ denotes $M^*$ with the twisted $G$-action
relative to $H$
described above. Recall that $\operatorname{Br}(M/K;F)$ denotes the subgroup of
$\operatorname{Br}(M/K)$ consisting of classes of central simple $K$-algebras
split by $M$ and having a unitary $K/F$-involution.
\begin{prop}\label{relbriso}
$H^2(G, \widetilde{M^*}) \cong \operatorname{Br}(M/K; F)$.
\end{prop}
\begin{proof}
Part of the long exact sequence \eqref{longexact} is
\begin{equation}\label{H2seq}
H^1(G, M^*)\ \longrightarrow \ H^2(G,\widetilde{M^*})\ \ \longrightarrow \ \ H^2(H,M^*)
\ \ \overset \operatorname{cor}\longrightarrow \ \ H^2(G,M^*)
\end{equation}
By Albert's theorem \cite[Th.~3.1(2)]{kmrt}, for $[A] \in \operatorname{Br}(M/K)$,
the algebra $A$ has a $K/F$-involution iff $\operatorname{cor}_{K\to F}(A)$
is split. Thus, in the isomorphism $\operatorname{Br}(M/K)\cong
H^2(H,M^*)$, $\operatorname{Br}(M/K;F)$ maps isomorphically to
${\ker\big(H^2(H,M^*) \overset \operatorname{cor} \longrightarrow H^2(G,M^*)\big)}$.
Because $H^1(G, M^*) = 0$ by the homological Hilbert 90,
the exact sequence \eqref{H2seq} above yields the desired
isomorphism.
\end{proof}
\begin{remark}\label{Hiformulas}
Here are formulas for $\widehat H^i(G, \widetilde{M^*})$ for small
$i$, which are easily derived from
standard group cohomology formulas and
\eqref{longexact} above. We assume $[M\! :\! K]< \infty$, and
let $\theta$ be any element of $G\setminus H$. So,
$\operatorname{Gal}(K/F) = \{\operatorname{id}_K,\theta|_K\}$. We write $b^{1-\theta}$ for
$b/\theta(b)$.
\begin{align*}
\quad\text{(i)}& & H^1(G, \widetilde{M^*}) \ &\cong \
F^*\big/N_{K/F}(K^*)
\ \cong \ \widehat H^0(\operatorname{Gal}(K/F), K^*).\\
\text{(ii)} & &
H^0(G, \widetilde{M^*}) \ &\cong \ \{c\in K^*\mid N_{K/F}(c) = 1\}.\\
\text{(iii)}& &
\widehat H^0(G,\widetilde{M^*}) \ &\cong \
\{ c\in K^*\mid N_{K/F}(c) = 1\}
\, \big/ \, \{N_{M/K}(m)^{1-\theta}\mid m \in M^*\}
\qquad\qquad\qquad\qquad\qquad\qquad\\
& & & = \
\{ b^{1-\theta}\mid b\in K^*\} \, \big/ \,
\{N_{M/K}(m)^{1-\theta}\mid m \in M^*\}.
\end{align*}
We will be working particularly with $\widehat H^{-1}(G, \widetilde{M^*})$.
For this, let $\widetilde N\colon\widetilde{M^*} \to K^*$ be given by
$$
\widetilde N(m) \,= \ \textstyle \prod \limits_{g\in G}g*m
\ = \ \prod\limits_{h\in H}h(m) \cdot (\theta h)(m)^{-1} \
= \ N_{M/K}(m)\big/\theta(N_{M/K}(m)).
$$
So, $\widetilde N$ is the norm map for $\widetilde{M^*}$ as a $G$-module.
Note that
\begin{equation}\label{kerNtilde}
\ker(\widetilde N) \, = \ \{m\in M^*\mid N_{M/K}(m) \in F^*\}.
\end{equation}
Also, let
\begin{equation}\label{IsubG}
I_G(\widetilde {M^*})\, = \ \big\langle \,(g*m)m^{-1}\mid m\in M^*,
g\in G\big\rangle \ = \ \big\langle\,h(m)/m, \, h\theta(m) \,m\mid
m\in M^*, h\in H\big\rangle.
\end{equation}
Then, by definition,
\begin{equation}\label{Hhat-1}
\widehat H^{-1}(G, \widetilde{M^*}) \,\cong \,
\ker(\widetilde N) \big/ I_G(\widetilde {M^*}).
\end{equation}
\end{remark}
In the following useful lemma, part (ii) is an abstraction of an
argument of Yanchevski\u\i \ \cite[proof of~Cor.~4.13]{y}.
\begin{lemma}\label{Hdihedral}
Let $D$ be a finite dihedral group,
i.e., $D = \langle h,\theta\rangle$
where $\theta^2 = 1$, $\theta\ne 1$, and
$\,\theta h\theta^{-1} = h^{-1}$,
and $h$ has finite order.
Let~$H = \langle h \rangle$. Let $A$ be a $D$-module
such that $H^1(H,A) = 0$ and $H^1(\langle \theta \rangle,
A^H) = 0$. Let ${A^\theta = \{ a\in A \mid
\theta\cdot a = a\}}$
and $N_H(a) = \sum_{h\in H} h\!\cdot \! a$. Then,
\begin{enumerate}[\upshape (i)]
\item
$A^H + A^{\theta} \,=\, \{ a\in A\mid a - \theta \!\cdot \! a \in A^H\}$.
\item
$A^\theta + A^{h\theta} \,=\,
\{a\in A\mid N_H(a) \in A^\theta\}\, = \, A^\theta + A^{\theta h}$.
\item
The map $\operatorname{cor}_{\langle \theta \rangle \to D} \times
\operatorname{cor}_{\langle h\theta \rangle \to D}\colon
\widehat H^{-1}(\langle \theta \rangle, \widetilde A) \times
\widehat H^{-1}(\langle h\theta\rangle, \widetilde A)
\to \widehat H^{-1}(D, \widetilde A)$ is surjective.
\end{enumerate}
\end{lemma}
\begin{proof} (i)
We have the short exact sequence of $\langle \theta \rangle$-modules
$0 \to A^H \to A \to A/A^H \to 0$. Since
$H^1(\langle\theta\rangle,A^H) = 1$, the long exact cohomology sequence
shows that $A^\theta$ maps onto $(A/A^H)^\theta$, which yields (i).
(ii) Note that for $a\in A$, $N_H(\theta\!\!\cdot \!\! a) =
\sum_{k\in H}(k\theta)\!\cdot \! a
= \sum _{k\in H}(\theta k^{-1}) \!\cdot \! a = \theta \!\cdot \! N_H(a)$.
The left inclusion $\subseteq$ in~(ii) follows immediately. For the
inverse inclusion, take $a\in A$ with $N_H(a) \in A^\theta$.
Then, ${N_H(a-\theta\!\cdot \! a) =
N_H(a) - \theta\!\cdot \! N_H(a) = 0}$.
Since $H^1(H,A) = 0$, with $H = \langle h \rangle$, there is
$c\in A$ with\break
${a-\theta\!\cdot \! a = c- h\!\cdot \! c}$. So,
\begin{align*}
0 \ &= \ a - \theta\!\cdot \! a + \theta\!\cdot \!(a - \theta\!\cdot \! a) \ = \
c- h\!\cdot \! c + \theta\!\cdot \! c - (\theta h) \!\cdot \! c \\
& = \ c-h\!\cdot \! c + (h\theta h)\!\cdot \! c -(\theta h)\!\cdot \! c \
= \ [c- (\theta h)\!\cdot \! c] - h\!\cdot \! [c- (\theta h)\!\cdot \! c],
\end{align*}
i.e., $c- (\theta h)\!\cdot \! c\in A^H$. Since the group action
of $\langle \theta h \rangle$ on $A^H$ coincides with the
action of $\langle \theta \rangle$ on $A^H$, we have
$H^1(\langle \theta h \rangle, A^H) \cong H^1(\langle \theta \rangle,
A^H) = 0$. Therefore, part (i) applies, with $\theta h$
replacing $\theta$. Thus, we can write $c = d+e$
with $d\in A^H$ and $e\in A^{\theta h}$, hence
$\theta \!\cdot \! e = h \!\cdot \! e= (h\theta)\!\cdot \!(\theta\!\cdot \! e)$.
Now, as $d = h\!\cdot \! d$,
$$
a - \theta \!\cdot \! a \ = \ c- h\!\cdot \! c \ = \ e- h\!\cdot \! e
\ = \ e - \theta \!\cdot \! e,
$$
showing that $a+ \theta\!\cdot \! e \in A^\theta$. Thus,
$a = [a +\theta \!\cdot \! e] -\theta \!\cdot \! e \in
A^\theta + A^{h\theta}$,
completing the proof of the first equality in (ii). Since
$\theta h = h^{-1} \theta$, the second equality in
(ii) follows from the first by replacing $h$ by $h^{-1}$.
(iii) We have $\widehat H^{-1}(\langle \theta \rangle, \widetilde A)
\cong A^\theta\big/ \{a+ \theta \!\cdot \! a\mid a\in A\}$,
$\widehat H^{-1}(\langle h\theta \rangle, \widetilde A)
\cong A^{h\theta}\big/ \{a+ (h\theta) \!\cdot \! a\mid a\in A\}$,
and
$$
\widehat H^{-1}(D, \widetilde A) \ \cong \ \{a\in A \mid
N_H(a) \in A^\theta\} \,\big/\, \langle a-k\!\cdot \! a, \
a+(k\theta)\!\cdot \! a \mid a\in A, \ k\in H\rangle.
$$
The map $\operatorname{cor}_{\langle \theta \rangle \to D}\colon
\widehat H^{-1} (\langle \theta\rangle, \widetilde A)
\to \widehat H^{-1}(D, \widetilde A)$ arises from the inclusion
$A^\theta \hookrightarrow \{ a\in A\mid N_H(a) \in A^\theta\}$;
likewise for $\operatorname{cor}_{\langle h\theta \rangle \to D}\colon
\widehat H^{-1} (\langle h\theta\rangle, \widetilde A)
\to \widehat H^{-1}(D, \widetilde A)$. Thus, the surjectivity
asserted in part (iii) is immediate from part (ii).
\end{proof}
\begin{proposition}\label{gendihedralprop}
Let $F\subseteq K \subseteq M$ be fields with $[M\! :\! F] <\infty$
and $M$ a $K/F$-generalized dihedral exten-sion. Let
$G = \operatorname{Gal}(M/F)$ and $H = \operatorname{Gal}(M/K)$. Take any
$\theta \in G\setminus H$. Then there is an exact sequence:
\begin{equation}\label{gendihedralexact}
\textstyle\prod\limits_{h\in H}\widehat H^{-1}(\langle h\theta
\rangle , \widetilde {M^*}) \ \longrightarrow \ \widehat H^{-1} (G, \widetilde {M^*}) \
\longrightarrow
\ \ker(\widetilde N) \, \big/ \,\Pi \ \longrightarrow \ 1
\end{equation}
where $\ker(\widetilde N) = \{ m\in M^*\mid N_{M/K}(m) \in F^*\}$
and $\Pi = \prod_{h\in H}M^{*h\theta}$. In particular, if
$M/K$ is cyclic Galois, then $\ker(\widetilde N) /\Pi = 1$.
\end{proposition}
\begin{proof}
Here, $M^{*h\theta} = \{ m\in M^*\mid h\theta(m) = m\}$.
We have
$\widehat H^{-1}(G, \widetilde{M^*}) \cong \ker(\widetilde N)
\big/I_G(\widetilde {M^*})$ as in \eqref{kerNtilde}--\eqref {Hhat-1}.
For any $h\in H$ and $m \in M^*$,
$$
m/h(m)\, = \, [m\cdot \theta(m)]\big/[\theta(m) \cdot h(m)] \, = \,
[m\cdot \theta(m)]\big/[\theta(m) \cdot h\theta(\theta(m))] \, \in
M^{*\theta} M^{*h \theta}
$$
and $m\!\cdot \! h\theta(m)\in M^{*h\theta}$. Hence, by \eqref{IsubG},
\begin{equation} \label{IGeq}
I_G(\widetilde{M^*}) \,\subseteq \,
\textstyle\prod\limits_{h\in H}M^{*h\theta} \, = \,\Pi.
\end{equation}
Thus,
there is a well-defined epimorphism $\zeta\colon
\widehat H^{-1}(G,\widetilde{M^*})\to \ker(\widetilde N)
\, \big/ \, \Pi$, with
${\ker(\zeta) = \Pi\big/I_G(\widetilde{M^*})}$.
Now, for $h\in H$, we have $\widehat H^{-1}(\langle h\theta\rangle,
\widetilde{M^*}) \cong M^{*h\theta}\big/N_{M/M^{\langle h\theta
\rangle}}(M^*)$. So, $\prod_{h\in H}
\widehat H^{-1}(\langle h\theta \rangle, \widetilde{M^*})$ clearly maps
onto $\ker(\zeta)$, proving the exactness of
\eqref{gendihedralexact}. If $H$ is cyclic, then $G$ is dihedral, and
$\ker(\widetilde N) = \Pi$ by Lemma~\ref{Hdihedral}(ii).
\end{proof}
\begin{remark}
In the context of Prop.~\ref{gendihedralprop}, suppose
$H = \langle h_1, \ldots, h_m \rangle$. Then, the following
lemma shows that
\begin{equation}\label{productongens}
\textstyle \prod\limits_{h\in H}M^{*h \theta} \ = \
\prod\limits_{(\varepsilon_1, \ldots , \varepsilon_m)
\in \{0,1\}^m} M^{*\,h_1^{\varepsilon_1}\ldots h_m^{\varepsilon_m}
\theta},
\end{equation}
so the left term in \eqref{gendihedralexact} could be
replaced by $\prod\limits_{(\varepsilon_1, \ldots , \varepsilon_m)
\in \{0,1\}^m} \widehat H^{-1}(\langle h_1^{\varepsilon_1}\ldots
h_m^{\varepsilon_m}\theta \rangle, \widetilde {M^*})$. One can see by looking
at examples that the product on the right in \eqref{productongens}
is minimal in that if we delete any of the terms in that product, then
the equality no longer holds in general.
\end{remark}
\begin{lemma}\label{generators}
Let $G = \langle H, \theta \rangle$ be a generalized dihedral
group, where $H$ is an abelian subgroup of $G$ with
$|G\! :\! H| = 2$, $\theta$ has order $2$, and
$\theta h \theta = h^{-1}$ for all $h\in H$.
Let $A$ be any $G$-module. Suppose $H =
\langle h_1, \ldots, h_m\rangle$. Then,
$$
\textstyle \sum\limits_{h\in H} A^{h\theta} \ = \
\sum\limits_{(\varepsilon_1, \ldots \varepsilon_m) \in
\{0,1\}^m} A^{h_1^{\varepsilon_1}\ldots h_m^{\varepsilon_m}\theta}.
$$
\end{lemma}
\begin{proof}
This follows from \cite[Lemma~4.9]{I} (with $A$ for $U$,
$H$ for the abelian group $A$ and $W_h = A^{h\theta}$
for all $h\in H$), once we establish that
$A^{h\theta} \subseteq A^{k\theta} + A^{k^2h^{-1} \theta}$
for all $h,k\in H$. For this, take any $a\in A^{h\theta}$.
Then $\theta(a) = h^{-1}(a)$. Hence,
$k^2h^{-1}\theta(k\theta(a)) = k^2h^{-1} k^{-1}(a) = k\theta(a)$,
showing that $k\theta(a) \in A^{k^2h^{-1}\theta}$.
Thus $a = [a+ k\theta(a)] - k\theta(a) \in A^{k\theta}
+ A^{k^2h^{-1} \theta}$, proving the required inclusion.
\end{proof}
\section{Unitary relative Brauer Groups, bicylic case}\label{ubicyclic}
In this section we prove a unitary version of
the formula
$\operatorname{Br}(M/K)\big/\operatorname{Dec}(M/K) \cong \widehat{H}^{-1}(\operatorname{Gal}(M/K),M^*)$,
for $M$ a bicyclic Galois extension of $K$, see \eqref{njnj} above. The unitary version was
inspired by the result of Yanchevski\u\i~ \cite[Prop.~5.5]{y}, which was a
key part of his proof in~\cite[Th.~A]{yinverse} that any finite abelian
group can be realized as the unitary $\SK$ of some division algebra with
involution of the second kind.
Let $F\subseteq K \subseteq M$ be fields with
$[K\! :\! F] = 2$ and $K$ Galois over $F$, and
$M = L_1\otimes_KL_2$ with each $L_i$ cyclic
Galois over $F$.
Assume $M$ is $K/F$-generalized
dihedral,
as described at the beginning of \S \ref{unitaryIN}.
Let $G = \operatorname{Gal}(M/F)$ and $H = \operatorname{Gal}(M/K)$, and
choose and fix an element $\theta\in G\setminus H$. So,
$\operatorname{Gal}(K/F) = \{\theta|_K, \operatorname{id}_K\}$.
To simplify notation,
let $\sigma$ (not $\sigma_1$) be a fixed generator of $\operatorname{Gal}(M/L_2)$,
and $\rho$ (not $\sigma_2$) a fixed generator of $\operatorname{Gal}(M/L_1)$;
so, $H = \langle\sigma\rangle \times \langle\rho\rangle$.
Let $n = [L_1\! :\! K]$, which is the order of $\sigma$ in $H$,
and let $\ell = [L_2\! :\! K]$, which is the order of $\rho$.
As in Prop.~\ref{gendihedralprop}, let
$$
\ker(\widetilde{N}) \,= \,\{ a \in M^*\mid N_{M/K}(a)\in F^* \}
$$
and
\begin{equation}\label{nott}
\textstyle{\Pi}\ = \ \textstyle{\prod}_{h\in H}M^{*h\theta}
\, = \ M^{*\theta} M^{*\rho\theta} M^{*\sigma\theta} M^{*\rho\sigma\theta}.
\end{equation}
(See \eqref{productongens} for the second equality.)
\begin{proposition}\label{thmainm}
We have
$$
\operatorname{Br}(M/K;F)\big/
\operatorname{Dec}(M/K;F) \,\cong \,\ker(\widetilde{N})/\Pi.
$$
\end{proposition}
\begin{proof} This follows by combining the formulas for unitary $\SK$
given in \cite[Prop.~5.5]{y} with the Henselian version of
the formula in \cite[Cor.~4.11]{I}.
However, we give a direct proof avoiding the use of
Yanchevski\u\i's special unitary conorms, since we
will later need
an explicit description of the isomorphism.
Define a map
$$
\Psi\colon\operatorname{Br}(M/K;F) \longrightarrow \ker(\widetilde N)/\Pi
$$
as follows: By
Lemma~\ref{unitarycp}, a Brauer class in $\operatorname{Br}(M/K;F)$ is
represented by an algebra $A=A(u,b_1,b_2)$, where
$u,b_1,b_2$ satisfy the conditions in \eqref{bicyclicbrel} and
$b_1 \in L_2^{*\theta}, b_2 \in L_1^{*\theta}$, and
$u\,\rho\sigma\theta(u)=1$. By Hilbert 90 (for the group
$\langle \rho \sigma \theta \rangle$), there is $q \in M^*$
with $u=q /\rho \sigma \theta(q)$. Define
$$
\Psi\big(A(u,b_1,b_2)\big)\, =\ q\,\Pi\,\in \,
\ker(\widetilde N)/\Pi.
$$
We will show that $\Psi$ is a well-defined, surjective
homomorphism with kernel $\operatorname{Br}(L_1/K;F) \operatorname{Br}(L_2/K;F)$, which equals
$\operatorname{Dec}(M/K;F)$ (see \eqref{Decformula}).
For the well-definition of $\Psi$, first note that
$$
1\, =\, N_{M/K}(u)\, =\, N_{M/K}(q/\rho\sigma \theta(q))\, =\,
N_{M/K}(q)\big/N_{M/K}(\theta(q))\, =\,
N_{M/K}(q)\big/\theta(N_{M/K}(q)),
$$
so, $q\in \ker(\widetilde N)$. Also, given $u$, the choice
of $q$ with $q/\rho \sigma \theta(q)=u$ is unique up to a
multiple in~$M^{*\rho \sigma \theta}$. Since
$M^{*\rho \sigma \theta}\subseteq \Pi$,
$\ \Psi\big(A(u,b_1,b_2)\big)$ is independent of the choice
of $q$ from $u$. Now, suppose
${A(u,b_1,b_2)\cong A(u',b_1',b_2')}$, with $u,b_1,b_2$ and
$u',b_1',b_2'$ each satisfying the conditions of
Lemma~\ref{unitarycp}(iii). We have the presentation
$A(u,b_1,b_2)=\bigoplus_{i=0}^{n-1}\bigoplus_{j=0}^{\ell-1}Mx^iy^j$,
where
$\intt(x)|_M=\sigma$, $x^n = b_1$,
$\intt(y)|_M=\rho$, $y^\ell = b_2$, and $xyx^{-1}y^{-1}=u$, so,
(see \eqref{bicyclicbrel})
\begin{equation}\label{bicyclicbrels}
b_1\in M^{\langle\sigma\rangle}\, =\, L_2,
\quad b_2\in M^{\langle\rho\rangle} \, =\, L_1,
\quad N_{M/L_2}(u) \, =\, b_1/\rho(b_1),
\quad N_{M/L_1}(u) \, =\, \sigma(b_2)/b_2.
\end{equation}
The conditions of Lemma~\ref{unitarycp}(iii)
we are also assuming
are that
\begin{equation}\label{unitaryeqs}
b_1 \,\in \, L_2^{\theta},\qquad b_2\in L_1^{\theta},
\qquad\text{and} \qquad u\,\rho\sigma\theta(u) \,=\, 1.
\end{equation}
The corresponding conditions in \eqref{bicyclicbrels} and
\eqref{unitaryeqs} hold for $b_1'$, $b_2'$ and $u'$.
By
Lemma~\ref{unitarycp},
there is a $K/F$-involution $\tau$ of $A=A(u,b_1,b_2)$ with
$\tau|_M=\theta, \tau(x)=x, \tau(y)=y$. We have an
isomorphism
${A(u,b_1,b_2)\cong A(u',b_1',b_2')}$, and by Skolem-Noether
there is such an isomorphism which restricts to the identity
on $M$. Therefore,
there exist $x'$ and
$y'$ in $A^*$ such that
$\intt(x')|_M = \sigma$,
$x'^n=b_1'$, $\intt(y')|_M = \rho$, $y'^\ell=b_2'$,
and $x'y'x'^{-1}y'^{-1}=u'$. Since $\intt(x')|_M = \intt(x)|_M$
there is $c_1\in C_A(M)^* = M^*$ with $x' = c_1 x$, and likewise
$c_2\in M^*$ with $y'= c_2 y$. By simplifying the expressions
$b_1' = (c_1x)^n$, $b_2' = (c_2y)^\ell$, and $u'=(c_1x)(c_2y) (c_1x)^{-1}
(c_2y)^{-1}$, we find that
\begin{equation}\label{prime}
b_1'\, = \, N_{M/L_2}(c_1) \,b_1, \qquad b_2' \,=\, N_{M/L_1}(c_2) \,b_2,
\qquad u' \, =\, \big(c_1/\rho(c_1)\big) \big(\sigma(c_2)/c_2\big) \,u.
\end{equation}
By Lemma~\ref{unitarycp}, there is a
$K/F$-involution $\tau'$ on $A$ with $\tau'(x')=x', \tau'(y')=y'$, and
$\tau'|_M=\theta$. Since $\tau'\tau^{-1}$ is a $K$-automorphism of $A$,
there exists $e\in A^*$ with $\tau'=\intt(e)\tau$. Because
$\tau'|_M=\tau|_M$, $e \in C_A(M)=M$. The condition that
$\tau'^2=\operatorname{id}_A$ implies that $e/\theta(e)\in K^*$. Since
$e/\theta(e)\big(\theta (e/\theta(e))\big)=1$, Hilbert 90
for $K/F$ shows
that there is $d\in K^*$ with $d/\theta(d)=e/\theta(e)$. By replacing
$e$ by $e/d$, we may assume that $\theta(e)=e$. The conditions that
$c_1 x=\tau'(c_1 x)=\intt(e)\tau(c_1x)$ and
$c_2 y=\tau'(c_2 y)=\intt(e)\tau(c_2 y)$ yield
$$
c_1\,=\,\sigma\theta(c_1)\,e/\sigma(e) \qquad\text{and}\qquad
c_2\,=\,\rho\theta(c_2)\,e/\rho(e),
$$
hence,
\begin{equation}\label{rhoc1}
\rho(c_1)\,=\,\rho\sigma\theta(c_1)\,\rho (e)/\rho\sigma(e) \qquad\text{and}\qquad
\sigma(c_2)\,=\,\rho\sigma\theta(c_2)\,\sigma(e)/\rho\sigma(e).
\end{equation}
The equations \eqref{rhoc1} yield
\begin{equation}\label{t6}
c_1/\rho(c_1)\, =\,
\big(c_1/\rho\sigma\theta(c_1)\big)\big(\rho\sigma(e)/\rho(e)\big)
\qquad\text{and} \qquad
\sigma(c_2)/c_2\,=\,
\big(\rho\sigma\theta(c_2)/c_2\big)\big(\sigma(e)/\rho\sigma(e)\big).
\end{equation}
Let $\widetilde{q}=(c_1/c_2)\sigma(e)$. Then,
using (\ref{t6}), \eqref{prime} and $\theta(e) = e$,
\begin{align}\label{t7}
\begin{split}
\widetilde{q}/\rho\sigma\theta(\widetilde{q}) \ &= \
\big(c_1/\rho\sigma\theta(c_1)\big)\big(\rho\sigma\theta(c_2)/c_2\big)
\big(\sigma(e)/\rho\sigma\theta\sigma(e)\big)\\
&= \
\big(c_1/\rho(c_1)\big)\,\big(\rho(e)/\rho\sigma(e)\big)\,
\big(\sigma(c_2)/c_2\big)\, \big(\rho\sigma(e)/\sigma(e)\big)\,
\big(\sigma(e)/\rho\theta(e)\big)\\
&= \
\big(c_1/\rho(c_1)\big)\,\big(\sigma(c_2)/c_2\big) \ = \ u'/u.
\end{split}
\end{align}
When $q \in M^*$ is chosen so that $q/\rho\sigma\theta(q)=u$,
set $q'=\widetilde{q}q$; then (\ref{t7}) shows
that $q'/\rho\sigma\theta(q')=u'$. We check that $\widetilde q \in \Pi$:
We have (see~(\ref{prime}) and \eqref{unitaryeqs})
$N_{M/L_2}(c_1)=b_1'/b_1 \in L_2^{*\theta}$. Therefore, by
Lemma~\ref{Hdihedral}(ii) applied to the dihedral group
$\langle \sigma, \theta\rangle = \operatorname{Gal}(M/L_2^\theta)$,
$c_1 \in M^{*\theta} M^{*\sigma \theta}\subseteq \Pi$.
Likewise, $c_2 \in M^{*\theta} M^{*\rho \theta}\subseteq \Pi$
as
$N_{M/L_1} (c_2)=b_2'/b_2 \in L_1^{*\theta}$. Finally, since
$\theta(e)=e$, we have $\sigma(e)=\sigma\theta(e)=
\sigma\theta\sigma^{-1}(\sigma(e))=\sigma^2\theta(\sigma(e))$. So,
$\sigma(e)\in M^{*\sigma^2\theta} \subseteq \Pi$.
Thus, ${q' \equiv q \pmod{\Pi}}$,
which shows that $\Psi$ is well-defined independent of the
choice of presentation of $A$ as $A(u,b_1,b_2)$ with
$u,b_1,b_2$ as in Lemma~\ref{unitarycp}(iii).
For the surjectivity of $\Psi$, take any
$q\in \ker(\widetilde{N})$ and set $u=q/\rho\sigma\theta(q)$.
So, $u\,\rho\sigma\theta(u) = 1$. Furthermore,
as $N_{M/K}(q)\in F^*$,
$$
N_{M/K}(u)\,=\,N_{M/K}(q)/N_{M/K}(\rho\sigma\theta(q))\,=\,
N_{M/K}(q)/\theta(N_{M/K}(q))\,=\,1.
$$
Since $N_{L_2/K}(N_{M/L_2}(u))=N_{M/K}(u)=1$, by Hilbert 90
for $L_2/K$ there is $b_1 \in L_2^*$ with
${b_1/\rho(b_1)=N_{M/L_2}(u)}$. Then,
$$
b_1/\rho(b_1)\,=\,N_{M/L_2}(q)\big/N_{M/L_2}(\rho\sigma\theta(q))\,=\,
N_{M/L_2}(q)/\rho\theta(N_{M/L_2}(q)).
$$
Hence,
$$
1\,=\,(b_1/\rho(b_1))\,\rho\theta(b_1/\rho(b_1))\,=\,
(b_1/\theta(b_1))/\rho(b_1/\theta(b_1)),
$$
which shows that $b_1/\theta(b_1) \in L_2^\rho = K$.
By Lemma~\ref{Hdihedral}(i) applied to the dihedral group
${\operatorname{Gal}(L_2/F) = \langle\rho|_{L_2},\theta|_{L_2}\rangle}$, it follows that
$b_1=k\widehat{b_1}$ with $k\in K^*$ and
$\widehat{b_1} \in L_2^{*\theta}$. By replacing $b_1$ with
$\widehat{b_1}$, we may assume that $b_1 \in L_2^{*\theta}$.
Likewise, there is $b_2 \in L_1^{*\theta}$ with
$N_{M/L_1}(u^{-1})=b_2/\sigma(b_2)$. Then, as $u,b_1,b_2$
satisfy the conditions of \eqref{bicyclicbrel} (where $\sigma_1 = \sigma$
and $\sigma_2=\rho$) the algebra $A(u,b_1,b_2)$ exists, and
by Lemma~\ref{unitarycp}
$[A(u,b_1,b_2)] \in \operatorname{Br}(M/K;F)$. Clearly,
$\Psi[A(u,b_1,b_2)]=q\,\Pi$.
Finally, we determine $\ker(\Psi)$:
If $[B] \in \operatorname{Br}(L_1/K;F)$ then we can assume that $B$ has
$L_1$ as a maximal subfield. Then, by Lemma~\ref{unitarycp},
$B\cong (L_1/K,\sigma,b_1)$, where $b_1 \in K^{*\theta} = F^*$.
Likewise, for
any ${[C] \in \operatorname{Br}(L_2/K;F)}$, we have
$C \thicksim (L_2/K,\rho,b_2)$ for some $b_2\in F^*$. Then,
$$
\big[B\otimes_K C\big]=\big[(L_1/K,\sigma,b_1)\otimes_K
(L_2/K,\rho,b_2)\big]=\big[A(1,b_1,b_2)\big] \in \ker(\Psi),
$$
since when $u = 1$ we can take $q = 1$.
So $\operatorname{Br}(L_1/K;F)\operatorname{Br}(L_2/K;F) \subseteq \ker(\Psi)$. For the
reverse inclusion, take any
$A=A(u,b_1,b_2)$ with $[A] \in \ker(\Psi)$. Since
$[A]\in \operatorname{Br}(M/K;F)$, by Lemma~\ref{unitarycp} we may assume that
$b_1 \in L_2^{*\theta}, b_2 \in L_1^{*\theta}$ and
$u\,\rho\sigma\theta(u)=1$. Since, $[A]\in \ker(\Psi)$, we have
$u=q/\rho\sigma\theta(q)$ with $q\in \Pi$, so
$q=q_\theta q_{\rho\theta} q_{\sigma\theta}q_{\rho\sigma\theta}$, where
$q_\theta \in M^{*\theta}, q_{\rho\theta} \in M^{*\rho \theta}$,
$q_{\sigma\theta}\in M^{*\sigma\theta}$, and $q_{\rho\sigma\theta}\in
M^{*\rho\sigma\theta}$. Thus,
\begin{align*}
u\ &= \ q/\rho\sigma\theta(q)\ = \
\big(q_\theta/\rho\sigma(q_\theta)\big)
\big(q_{\rho\theta}/\sigma(q_{\rho\theta})\big)
\big(q_{\sigma\theta}/\rho(q_{\sigma\theta})\big)\\
&= \ \big(q_\theta q_{\rho\theta}/\sigma(q_\theta q_{\rho \theta})
\big)\big(q_{\sigma\theta}\sigma(q_\theta)/
\rho(q_{\sigma\theta}\sigma(q_\theta)\big)
\ = \ \big(c_2/\sigma(c_2)\big)\big(\rho(c_1)/c_1\big),
\end{align*}
where $c_2=q_\theta q_{\rho\theta}$ and
$c_1=(q_{\sigma\theta}\sigma(q_\theta))^{-1}$. Then
by~(\ref{ghgt}), $A =A(u,b_1,b_2) \cong A(u',b_1',b_2')$
where\break
${u'=(c_1/\rho(c_1))(\sigma(c_2)/c_2)u=1}$, and
$b_1'=N_{M/L_2}(c_1)b_1$ and $b_2'=N_{M/L_1}(c_2)b_2$.
Since $c_2\in M^{*\theta}M^{*\rho\theta}$,
an easy calculation or an application of
Lemma~\ref{Hdihedral}(ii)
for the dihedral group $\operatorname{Gal}(M/L_1^\theta) = \langle \rho, \theta\rangle$
shows\break that $N_{M/L_1}(c_2) \in L_1^{*\theta}$.
Therefore, $b_2'= N_{M/L_1}(c_2)b_2
\in L^*_1{^\theta}$, as
$b_2\in L_1^{*\theta}$.
But also, as in \eqref{bicyclicbrels},\break
${\sigma(b_2')/b_2'=N_{M/L_1}(u')=N_{M/L_1}(1)=1}$. Hence,
$b_2' \in L_1^{*\theta}\cap L_1^{*\sigma}=K^{*\theta}=F^*$.
Likewise, as $q_{\sigma\theta} \in M^{*\theta}$ and $\sigma(q_\theta)
\in M^{*\sigma^2\theta} \subseteq M^{*\theta}M^{*\sigma\theta}$ (see
\eqref{productongens}), we have $c_1 \in M^{*\theta}M^{*\sigma\theta}$.
Therefore, an
easy calculation or Lemma~\ref{Hdihedral}(ii)
for the dihedral group $\operatorname{Gal}(M/L_2^\theta) = \langle\sigma,\theta\rangle$
shows that $N_{M/L_2}(c_1)\in L_2^{*\theta}$.
So, arguing just as for $b_2'$, we find that $b_1' \in F^*$. Thus,
$$
A \ \cong \ A(1,b_1',b_2') \ \cong \
(L_1/K,\sigma,b_1')\otimes_K(L_2/K,\rho,b_2'),
$$
and since the $b_i' \in F^*$,
$\big[(L_1/K,\sigma,b_1')\big] \in \operatorname{Br}(L_1/K;F)$ and
$\big[(L_2/K,\rho,b_2')\big] \in \operatorname{Br}(L_2/K;F)$, by
Lemma~\ref{unitarycp}. Thus, $\ker(\Psi)=\operatorname{Br}(L_1/K;F)\operatorname{Br}(L_2/K;F)
=\operatorname{Dec}(M/K;F)$.
\end{proof}
This yields our unitary analogue to \eqref{njnj} above.
\begin{proposition} \label{unitarybicyclic}
For $M$ bicyclic Galois over $K$ with $M$
$K/F$-generalized dihedral, setting $G = \operatorname{Gal}(M/F)$, $H=\operatorname{Gal}(M/K)$,
and $\theta$ any element of $G\setminus H$ as above,
there is an exact sequence
\begin{equation}
\textstyle \prod\limits_{h\in H} \widehat H^{-1}(\langle h\theta\rangle,
\widetilde{M^*}) \
\longrightarrow \ \widehat H^{-1}(G, \widetilde {M^*}) \ \longrightarrow \
\operatorname{Br}(M/K;F)/\operatorname{Dec}(M/K;F) \ \longrightarrow\, 0
\end{equation}
\end{proposition}
\begin{proof}
This follows from Prop.~\ref{thmainm} and Prop.~\ref{gendihedralprop}.
\end{proof}
\section{Semiramfied algebras}\label{semiram}
We now apply the results of the preceding sections
to the calculation of unitary $\SK$ for semiramified
graded division algebras with graded $\sT/\sR$-involution
Throughout this section, fix a graded field $\mathsf{T}$
and a graded subfield $\mathsf{R}$ of $\mathsf{T}$ with
$[\mathsf{T}\! :\!\mathsf{R}] = 2$ and $\mathsf{T}$ Galois over $\mathsf{R}$, say with
$\mathsf{Gal}(\mathsf{T}/\mathsf{R}) = \{\operatorname{id},\psi\}$. Assume further that
$\mathsf{T}$ is inertial over $\mathsf{R}$. Thus, $\Gamma_\mathsf{T} = \Gamma_\mathsf{R}$,
$[{\mathsf{T}_0}\! :\!{\mathsf{R}_0}] = 2$, ${\mathsf{T}_0}$ is Galois over with $\operatorname{Gal}
({\mathsf{T}_0}/{\mathsf{R}_0}) = \{ \operatorname{id} , \psi_0\}$, where $\psi_0 = \psi|_{\mathsf{T}_0}$,
and $\psi = \psi_0\otimes \operatorname{id}_\mathsf{R}$ when we identify
$\mathsf{T}$ with ${\mathsf{T}_0} \otimes_{\mathsf{R}_0} \mathsf{R}$.
By definition, for a central simple graded division algebra
$\mathsf{B}$ over $\mathsf{T}$ with a graded unitary $\sT/\sR$-involution $\tau$,
the unitary
$\SK$ is given by
$$
\SK(\mathsf{B}, \tau) \ = \ \Sigma_\tau'(\mathsf{B}) \,\big / \,
\Sigma_\tau(\mathsf{B}),
$$
where
$$
\Sigma_\tau'(\mathsf{B}) \ = \ \{b\in \mathsf{B}^*\mid \Nrd_\mathsf{B}(b) \in \mathsf{R}\}\qquad\text{and}\qquad
\Sigma_\tau(\mathsf{B})\ = \ \big\langle \{ b\in \mathsf{B}^*|\ \tau(b) = b\}\big\rangle
$$
We are assuming that $\sT/\sR$ is
inertial because otherwise $\sT/\sR$ is totally ramified and $\SK(\mathsf{B},\tau) = 1$,
by \cite[Th.~4.5]{I}. It is known by \cite[Lemma~2.3(iii)]{I} that
$[\mathsf{B}^*,\mathsf{B}^*] \subseteq \Sigma _\tau(\mathsf{B})$, so $\SK(\mathsf{B},\tau)$
is an abelian group. Also, if $\tau'$ is another graded
$\sT/\sR$-involution on $\mathsf{B}$, then
$\Sigma_{\tau'}'(\mathsf{B}) = \Sigma_\tau'(\mathsf{B})$
and $\Sigma_{\tau'}(\mathsf{B}) = \Sigma_\tau(\mathsf{B})$,
so $\SK(\mathsf{B},\tau') =\SK(\mathsf{B},\tau)$. The easy proof is analogous
to the ungraded proof given in \cite[Lemma~1]{yin}.
Let $\mathsf{E}$ be a semiramified
$\mathsf{T}$-central graded division algebra. So, as we have seen,
$\mathsf{E}_0$ is a field abelian Galois over ${\mathsf{T}_0}$, and
$\overline\Theta_\mathsf{E} \colon \Gamma_\mathsf{E}/\Gamma_\mathsf{T} \to
\operatorname{Gal}(\mathsf{E}_0/{\mathsf{T}_0})$ is a canonical isomorphism.
Suppose $\mathsf{E}$ has a graded $\sT/\sR$-involution $\tau$;
so $\tau|_{\mathsf{T}_0} = \psi_0$. We have seen in Prop.~\ref{uINdecomp}
that $\mathsf{E}_0$ is then a ${\mathsf{T}_0}/{\mathsf{R}_0}$-generalized dihedral Galois
extension. Let $H = \operatorname{Gal}(\mathsf{E}_0/{\mathsf{T}_0})$ and $G = \operatorname{Gal}
(\mathsf{E}_0 /{\mathsf{R}_0})$, and let $\overline \tau = \tau|_{\mathsf{E}_0}
\in G\setminus H$. For each $\gamma\in \Gamma_\mathsf{E}$ choose and
fix $x_\gamma \in \mathsf{E}_\gamma$ with $x_\gamma\ne 0$ and $\tau(x_\gamma)
= x_\gamma$. (Such $x_\ga$ exist, by \cite[Lemma~4.6(i)]{I}.) Our starting
point is the formula proved in \cite[Th.~4.7]{I}
\begin{equation}\label{USK1PiX}
\SK(\mathsf{E}, \tau) \ \cong \ \big(\Sigma_\tau(\mathsf{E})' \cap \mathsf{E}_0^*\big)\, \big/
\big(\Sigma_\tau(\mathsf{E}) \cap \mathsf{E}_0^*\big) \ = \
\ker(\widetilde N) \big /
\big(\, \Pi \cdot X\,\big),
\end{equation}
where
\begin{align*}
\ker(\widetilde N) \ &= \ \{ a\in \mathsf{E}_0^*
\mid N_{\mathsf{E}_0/{\mathsf{T}_0}}(a) \in {\mathsf{R}_0}\}; \\
\Pi \ & = \textstyle \prod \limits _{h\in H}\mathsf{E}_0^{*h\overline \tau},
\quad\text{where} \ \ \mathsf{E}_0^{*h\overline \tau} \ = \
\{ a\in \mathsf{E}_0^*\mid h\overline \tau(a) = a\};\qquad\qquad\qquad\qquad\qquad\\
X \ & = \ \big\langle x_\gax_\de x_{\ga+\de}^{-1} \mid
\gamma, \delta \in \Gamma_\mathsf{E}\big\rangle \ \subseteq \ \mathsf{E}_0^*.
\end{align*}
Note that
$H$ maps $\ker(\widetilde N)$ (resp.~$\Pi$) to itself, so $H$
acts on $\ker(\widetilde N) /\Pi$. But this action is trivial
since $I_H(\ker(\widetilde N)) \subseteq I_G(\widetilde \mathsf{E}_0^*)
\subseteq \Pi$ (see \eqref{IGeq} above).
\begin{theorem}\label{unitaryDSR}
Suppose $\mathsf{E}$ is DSR for $\sT/\sR$, i.e.,
in addition to the hypotheses above, $\mathsf{E}$
has a maximal
graded subfield $\mathsf{J}$ with $\tau(\mathsf{J}) = \mathsf{J}$. Then,
\begin{enumerate}[\upshape (i)]
\item
$\SK (\mathsf{E}, \tau)\cong \ker(\widetilde N)/\Pi$, and there is an
exact sequence
$$
\textstyle \prod\limits _{h\in H}
\widehat H^{-1}(\langle h\overline \tau\rangle, \widetilde \mathsf{E}_0^*) \ \longrightarrow \
\widehat H^{-1}(G, \widetilde\mathsf{E}_0^*) \ \longrightarrow \ \SK(\mathsf{E}, \tau) \ \longrightarrow
\ 1.
$$
\item
If $\mathsf{E}_0 = L_1 \otimes _{\mathsf{T}_0} L_2$ with each $L_i$ cyclic Galois
over ${\mathsf{T}_0}$, then
$$
\SK(\mathsf{E}, \tau) \ \cong \ \operatorname{Br}(\mathsf{E}_0/{\mathsf{T}_0};{\mathsf{R}_0}) \big/
\operatorname{Dec}(\mathsf{E}_0/{\mathsf{T}_0};{\mathsf{R}_0}).
$$
\end{enumerate}
\end{theorem}
\begin{proof}
(i) The first formula for $\SK(\mathsf{E}, \tau)$ was given
in \cite[Cor.~4.11]{I}. The point is that the $x_\ga$ can all
be chosen in~$\mathsf{J}$; then $X \subseteq \mathsf{J}_0^{*\tau}
= {\mathsf{R}_0}^* \subseteq \Pi$, so the $X$ term in \eqref{USK1PiX}
drops out.
The exact sequence in (i)
then follows by Prop.~\ref{gendihedralprop}. Part (ii)
is immediate from~(i) and Prop.~\ref{thmainm}.
\end{proof}
Note that Th.~\ref{unitaryDSR} is the unitary analogue to
Prop.~\ref{NSRSK} for nonunitary $\SK$ in the DSR case.
To improve the formula \eqref{USK1PiX}
in the manner of Th.~\ref{unitaryDSR} for $\mathsf{E}$ semiramified but not DSR
we need more information on the contribution of
the $X$ term. This contribution is measured by $(\Pi\cdot X )/\Pi$.
For $\gamma \in \Gamma_\mathsf{E}$ we write
$\overline \gamma$ for $\gamma + \Gamma_\mathsf{T}\in \Gamma_\mathsf{E}/\Gamma_\mathsf{T}$.
\begin{proposition}\label{Gammafn}
There is a well-defined $2$-cocycle $g\in Z^2(
\Gamma_\mathsf{E}/\Gamma_\mathsf{T} , \ker(\widetilde N) / \Pi)$ given by
\begin{equation}
g(\overline \gamma, \overline \delta) \ = \ x_\gamma x_\delta x_{\gamma+\delta}^{-1} \, \Pi.
\end{equation}
This $g$ is independent of the choice of nonzero symmetric elements
$x_\gamma, x_\delta , x_{\gamma + \delta}$ in $\mathsf{E}_\gamma, \mathsf{E}_\delta, \mathsf{E}_{\gamma+\delta}$.
Furthermore,
for all $\overline \gamma,\overline \delta \in \Gamma_\mathsf{E}/\Gamma_\mathsf{T}$
and $i,j, k, \ell\in \mathbb{Z}$, we have
\begin{equation}\label{gformula}
g(i\overline \gamma+ j\overline \delta, k\overline \gamma + \ell\overline \delta) \ = \ g(\overline \gamma,
\overline \delta)^\Delta\quad \text{where}\ \ \Delta =
\det\left(\begin{smallmatrix}
i&j\\ k&\ell
\end{smallmatrix}\right).
\end{equation}
$($In particular, $g(\overline \gamma, \overline \gamma) = 1 \, \Pi$ and $g(\overline \delta , \overline \gamma)
= g(\overline \gamma, \overline\delta) ^{-1}$.$)$ Moreover, $\langle\operatorname{im}(g) \rangle =
\big(\Pi\cdot X\big) /\Pi$,
which is a finite group.
\end{proposition}
\begin{proof}
For $\gamma, \delta\in \Gamma_\mathsf{E}$, set
$$
c_{\gamma,\delta} \, = \, x_\gamma x_\delta x_{\gamma +\delta}^{-1} \, \in \mathsf{E}_0^*.
$$
Note that $c_{\gamma, \delta} \in\ker(\widetilde N)$, since it is a product
of $\tau$-symmetric elements of $\mathsf{E}^*$.
For notational convenience we work with the
function
$$
f\colon \Gamma_\mathsf{E} \times \Gamma_\mathsf{E} \,\longrightarrow \, \ker(\widetilde N) /\Pi\quad
\text{given by} \ \ f(\gamma, \delta) \, = \, c_{\gamma, \delta}\, \Pi.
$$
Thus, $g(\overline \gamma, \overline \delta) = f(\gamma, \delta)$
We first show that the definition of $f$ is independent of
the choices made of $x_\gamma, x_\delta, x_{\gamma + \delta}$.
Fix $\gamma$ and $\delta$ in $\Gamma_\mathsf{E}$ for the moment. Take any
$a\in \mathsf{E}_0^*$ with
$\tau(ax_\gamma ) = a x_\gamma$. Then,\break
${ax_\ga = \tau(ax_\ga) = x_\ga\overline \tau(a)
= \Theta_\mathsf{E}(\gamma)( \overline \tau(a)) x_\ga}$;
so ${a = \Theta_\mathsf{E}(\gamma)(\overline\tau(a))}$, i.e.
$a\in \mathsf{E}_0^{*\Theta_\mathsf{E}(\gamma)\overline \tau}\subseteq \Pi$.
Hence, if we let $x_\ga' =a x_\ga$, then
$x_\ga'x_\dex_{\ga+\de}^{-1} \equiv x_\gax_\de x_{\ga+\de}^{-1}\ (\operatorname{mod}\ \Pi)$.
Likewise, if we take any $b\in \mathsf{E}_0^*$ with
${\tau(bx_\de) = bx_\de}$, then $\Theta_\mathsf{E}(\gamma)(b) \in
\mathsf{E}_0^{*\Theta_E(2\gamma+\delta) \overline \tau}\subseteq \Pi$
so ${x_\gax_\de'x_{\ga+\de}^{-1} = \Theta_\mathsf{E}(\gamma)(b) x_\gax_\dex_{\ga+\de}^{-1}\equiv
x_\gax_\dex_{\ga+\de}^{-1} \ (\operatorname{mod} \ \Pi)}$. Again, for\break
$d\in \mathsf{E}_0^*$
with $\tau(dx_{\ga+\de}) = dx_{\ga+\de}$, we have $d\in \mathsf{E}_0^{*\Theta_\mathsf{E}(\gamma + \delta)
\overline \tau}\subseteq \Pi$, so for $x_{\ga+\de}' = dx_{\ga+\de}$,
we have\break
${x_\gax_\dex_{\ga+\de}^{\prime-1} \equiv
x_\gax_\de x_{\ga+\de}^{-1}\ (\operatorname{mod} \ \Pi)}$. Thus, each such change does not affect
the value of $f(\gamma, \delta)$, and we are free to make such
changes when convenient.
We prove further identities for the function $f$ which hold for all
$\gamma, \delta, \varepsilon \in \Gamma_\mathsf{E}$ and $i,j,k,\ell \in \mathbb{Z}$:
$$
\qquad (\text{i})\qquad f(\gamma+\beta, \delta) \, = \, f(\gamma, \delta) \,=\, f(\gamma, \delta + \beta)
\quad\text{for any}\ \ \beta \in \Gamma_\mathsf{T}. \qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\ \,
$$
For, as $\Gamma_\mathsf{R} = \Gamma_\mathsf{T}$, there is a nonzero $a \in \mathsf{R}_\beta$.
Since $a\in Z(\mathsf{E})$ and $\tau(a) = a$, we could have chosen
$x_{\gamma+\beta} = ax_\ga$, $x_{\delta+\beta} = ax_\de$, and
$x_{\gamma+\delta +\beta}= ax_{\ga+\de}$. Then,
$$
f(\gamma+\beta,\delta) \, = \,(ax_\ga) x_\de (ax_{\ga+\de})^{-1}\,\Pi \, = \,
x_\gax_\dex_{\ga+\de}^{-1} \, \Pi \, =\, f(\gamma, \delta),
$$
and likewise $f(\gamma, \delta+\beta) = f(\gamma, \delta)$. This proves (i),
which shows that the $g$ of the Prop. is well-defined.
$$
\qquad\,(\text{ii})\qquad f(i\gamma,j\gamma) \, = \, 1\,\Pi.\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad
$$
For, we can choose $x_{i\gamma} = x_\ga^i$, $x_{j\gamma} = x_\ga^j$, and
$x_{i\gamma +j\gamma} = x_\ga^{i+j}$.
Then, $c_{i\gamma,j\gamma} = 1$.
$$
\qquad(\text{iii})\qquad f(\delta,\gamma)\ = \ f(\gamma,\delta)^{-1}.
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\qquad
$$
For, by applying $\tau$ to the equation $x_\gax_\de = c_{\gamma,\delta} x_{\ga+\de}$,
we obtain
$$
x_\de x_\ga\, =\, x_{\ga+\de}\overline\tau(c_{\gamma, \delta})\, =\, \Theta_\sE(\gamma +\delta)
(\overline \tau(c_{\gamma, \delta}))x_{\delta+\gamma},
$$
yielding $c_{\delta,\gamma} = \Theta_\sE(\gamma+\delta)(\overline \tau(c_{\gamma,\delta}))$,
so $c_{\delta,\gamma} c_{\gamma,\delta}\in \mathsf{E}_0^{*\Theta_\sE(\gamma+\delta)\overline \tau}
\subseteq \Pi$. Formula (iii) then follows.
$$
\qquad(\text{iv})\qquad f(\gamma,\delta) f(\gamma+\delta, \varepsilon) \,= \,
f(\gamma, \delta+\varepsilon) f(\delta, \varepsilon), \qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\qquad\quad \ \
$$
i.e., $f\in Z^2(\Ga_\sE, \ker(\widetilde N) /\Pi)$, since
$\Ga_\sE$ (acting via $\Theta_\sE(\Ga_\sE) = H$) acts trivially on
$\ker(\widetilde N) /\Pi$. This identity follows
from $(x_\gax_\de)x_\varepsilon = x_\gamma(x_\de x_\varepsilon)$, which yields
$c_{\gamma, \delta}\, c_{\gamma+ \delta, \varepsilon} =
\Theta_\sE(\gamma)(c_{\delta,\varepsilon}) \,c_{\gamma, \delta +\varepsilon}$.
Then (iv) follows, given the trivial action of
$H$ on $\ker(\widetilde N) /\Pi$.
$$
\qquad(\text{v})\qquad f(\gamma + \delta, \delta) \,=\, f(\gamma, \delta)
\quad \text {and} \quad f(\gamma, \gamma + \delta) \,=\, f(\gamma, \delta).
\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad
$$
For, as $\tau(\xdx_\gax_\de) = \xdx_\gax_\de$, we can take
$x_{\gamma+2\delta} = \xdx_\gax_\de$. Then,
$$
\xdx_\gax_\de\, = \, c_{\delta,\gamma} \,c_{\delta + \gamma, \delta} \,x_{\gamma + 2\delta}
\, = \, c_{\delta,\gamma} \,c_{\delta + \gamma, \delta} \,\xdx_\gax_\de.
$$
Hence, $1\,\Pi = f(\delta,\gamma) f(\delta+\gamma, \delta)$, so
$f(\delta+\gamma, \delta) = f(\delta, \gamma) ^{-1} = f(\gamma,\delta)$, using (iii).
This proves the first formula in (v), and the second formula
follows analogously, or from the first by using (iii).
$$
\qquad(\text{vi})\qquad f(\gamma + j\delta,\delta) \, =\, f(\gamma,\delta)
\, = \, f(\gamma, j\gamma +\delta) \quad\text{for all} \,\, j\in \mathbb{Z}.
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
$$
This follows from (v) by induction on $j$.
$$
\quad\ \ \ \, (\text{vii})\qquad f(i\gamma, j\delta) \, = \, f(\gamma, \delta) ^{ij} .
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\qquad\qquad\ \
\quad
$$
For, by (iv) with $j\delta$ for $\delta$ and $\delta$ for $\varepsilon$,
$$
f(\gamma, j\delta) f(\gamma+ j\delta,\delta) \, = \,
f(\gamma,(j+1)\delta)f(j\delta,\delta),
$$
which by (vi) and (ii) reduces to
$ f(\gamma, j\delta) f(\gamma, \delta) = f(\gamma, (j+1) \delta)$. Then (vii) for
$i = 1$ follows by induction on $j$ with the initial
case $j = 0$ given by (ii). From the $i = 1$ case the result for
arbitrary $i$ follows by using~(iii).
$$
\quad\ \ \, (\text{viii})\qquad
f(i \gamma+ j \delta, k \gamma + \ell \delta) \ = \ f( \gamma,
\delta)^\Delta\quad \text{where}\ \ \Delta =
\det\left(\begin{smallmatrix}i&\,j\\ k&\ell
\end{smallmatrix}\right).
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\ \
$$
For this note first that this is true if $i = 0$, as
$$
f(j\delta,k\gamma+\ell\delta) \, = \, f(\delta,k\gamma+\ell\delta)^j
\, = \, f(\delta, k\gamma)^j \, =\, f(\gamma,\delta)^{-jk},
$$
by (vii), (vi), (vii), and (iii).
Analogously, (viii) is true if $k = 0$. To verify
(viii) in general, we argue by induction on
$|i| + |k|$. By invoking (iii) and interchanging
$i\gamma + j \delta$ with $k\gamma + \ell\delta$ if necessary,
we can assume $|i| \le |k|$. We can assume
$|i|\ge 1$, since the case $|i| = 0$ is already done.
Let $\eta = \pm1$, with the sign chosen so that $|k - \eta i|
= |k| - |i|$. Since $|i|+|k-\eta i|= |k|<|i|+|k|$, we have
by (vi) and induction,
\begin{align*}
f(i\gamma+ j\delta, k\gamma + \ell \delta) \, &= \,
f\big(i\gamma + j\delta, (k\gamma + \ell\delta\big)
- \eta(i\gamma + j\delta))\,
=\,f\big(i\gamma + j\delta, (k- \eta i)\gamma + (\ell - \eta j)\delta\big)
\\ &= \, f(\gamma,\delta)^{\Delta'}\quad\text{where} \ \ \Delta' =
\det\left(\begin{smallmatrix}i &j\\
k-\eta i & \ \ell-\eta j\end{smallmatrix}\right) =
\det\left(\begin{smallmatrix}i &j\\
k & \,\ell\end{smallmatrix}\right).
\end{align*}
Thus, (viii) is proved, and when (viii) is restated in terms of
$g$, it is formula~\eqref{gformula}. It is clear from the definition and
well-definition of $g$ that $\langle \operatorname{im}(g) \rangle= (\Pi\cdot X)/\Pi$. This
abelian group is finite since the domain of $g$
is finite, and each $g(\overline \gamma,
\overline \delta)$ has finite order by formula~\eqref{gformula}. Identity (iv) above
shows that $f$ is a $2$-cocycle, so $g$ is also a $2$-cocycle.
\end{proof}
Remark. If the finite abelian group $\Gamma_\mathsf{E}/\Gamma_\mathsf{T}$ has exponent $e$,
then formula \eqref{gformula} shows that $\langle\operatorname{im}(g)\rangle$ has exponent
dividing $e$. So, we have the crude upper bound
$|\langle\operatorname{im}(g)\rangle| \le e^{|\Gamma_\mathsf{E}/\Gamma_\mathsf{T}|^2}$.
We can now prove a formula for unitary $\SK$ of semiramified graded
algebras. This is a unitary analogue to Th.~\ref{goth} above.
\begin{theorem}\label{main}
Let $\mathsf{E}$ be a semiramified $\mathsf{T}$-central graded division algebra
with a unitary graded $\sT/\sR$-involution $\tau$, where
$\mathsf{T}$ is unramified over $\mathsf{R}$. Take any decomposition
$\mathsf{E} \sim_g \mathsf{I} \otimes _\mathsf{T} \mathsf{N}$ where $\mathsf{I}$ is inertial
with ${[\mathsf{I}_0]\in \operatorname{Br}(\mathsf{E}_0/{\mathsf{T}_0};{\mathsf{R}_0})}$ and $\mathsf{N}$ is DSR
for $\sT/\sR$, as in Prop.~\ref{uINdecomp} above. Then,
\begin{enumerate}
\item[{\rm(i)}] $\SK(\mathsf{E},\tau) \cong
\big(\ker(\widetilde N)/\Pi\big)\big/\langle\operatorname{im}(g)\rangle$, where
$g$ is the function of Prop.~\ref{Gammafn}. If
$\mathsf{I}_0 \sim A(\mathsf{E}_0/{\mathsf{T}_0}, \boldsymbol\sigma, \mathbf u, \bold b)$ as in
Lemma~\ref{unitarycp}$($iii$)$ with $\theta = \tau|_{\mathsf{E}_0}$, then
$\operatorname{im}(g)$ is computable from the~$u_{ij}$.
\item[{\rm(ii)}]
If $\mathsf{E}_0\cong L_1\otimes_{\mathsf{T}_0} L_2$ with each $L_i$ cyclic Galois over
${\mathsf{T}_0}$, then
\begin{equation*}
\SK(\mathsf{E},\mathsf{T}) \ \cong \
\operatorname{Br}(\mathsf{E}_0/{\mathsf{T}_0};{\mathsf{R}_0})\big/[\operatorname{Dec}(\mathsf{E}_0/{\mathsf{T}_0};{\mathsf{R}_0})\cdot
\langle[\mathsf{I}_0]\rangle].
\end{equation*}
\end{enumerate}
\end{theorem}
\begin{proof}
(i) From \eqref{USK1PiX} and Prop.~\ref{Gammafn},
we have
$$
SK(\mathsf{E}, \tau) \ \cong \ \big( \ker(\widetilde N) /\Pi\big) \big /
\big[(\Pi\cdot X)/\Pi\big]
\ \cong \ \big(\ker(\widetilde N)/\Pi\big)\big/\langle\operatorname{im}(g)\rangle.
$$
It remains to relate $\operatorname{im}(g)$ to the $u_{ij}$ describing $\mathsf{I}_0$.
We have $\mathsf{I}_0\sim A(\mathsf{E}_0/{\mathsf{T}_0}, \boldsymbol\sigma, \mathbf u,\bold b)$, as in
Lemma~\ref{unitarycp}(iii), with $\theta = \overline \tau = \tau|_{{\mathsf{T}_0}}$.
Since $\mathsf{N}$ is DSR for~$\sT/\sR$ with $\mathsf{N}_0\cong\mathsf{E}_0$
and $\Theta_\mathsf{N} = \Theta_\mathsf{E}$ by
Prop.~\ref{uINdecomp}, Prop.~\ref{DSRdecomp} yields
$\mathsf{N}\cong_g \mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T}, \boldsymbol\sigma,\boldsymbol 1, \bold c)$,
with each $c_i \in \mathsf{R}^*$\break with
$\deg(c_i)= r_i \gamma_i$ for some $\gamma_i \in \Gamma_\mathsf{N} = \Gamma_\mathsf{E}$
with $\Theta_\mathsf{E}(\gamma_i) = \sigma_i$. Therefore, by
Remark~\ref{abeliancpprod},\break
$\mathsf{E}\sim_g \mathsf{E}'$, where ${\mathsf{E}' = \mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T}, \boldsymbol\sigma, \mathbf u, \bold d)}$,
with the same $\mathbf u$ as for $\mathsf{I}_0$ and each $d_i = b_i c_i\in \mathsf{E}_0^*\mathsf{R}^*$.
So,\break ${\tau(d_i) = \tau(c_i) \tau(b_i) = c_i b_i = b_i c_i = d_i}$.
Since $\mathsf{N}$ is a semiramified graded division algebra and
${\deg(d_i) = \deg(c_i)}$ for each $i$, Lemma~\ref{incp} applied
to $\mathsf{N}$ and to $\mathsf{E}'$ shows that $\Gamma_{\mathsf{E}'} = \Gamma_\mathsf{N}$
and $\mathsf{E}'$ is a semiramified graded division algebra.
Therefore, as $\mathsf{E}$ and $\mathsf{E}'$ are each graded division algebras
with $\mathsf{E}\sim_g \mathsf{E}'$, we have
$\mathsf{E} \cong_g \mathsf{E}'$ by the graded Wedderburn Theorem.
So, we may assume $\mathsf{E} = \mathsf{E}' = \mathsf{A}(\mathsf{E}_0\mathsf{T}/\mathsf{T}, \boldsymbol\sigma, \mathbf u, \bold d)$.
Take $y_1, \ldots, y_k \in \mathsf{E}^*$ with $\intt(y_i)|_{\mathsf{E}_0\mathsf{T}} = \sigma_i$,
$y_i^{r_i} = d_i$, and $y_i y_j y_i^{-1} y_j^{-1} = u_{ij}$. Now,
the graded field $\mathsf{E}_0\mathsf{T}$ is $\sT/\sR$ generalized dihedral,
and $\theta = \tau|_{\mathsf{E}_0\mathsf{T}}$ lies in $\mathsf{Gal}(\mathsf{E}_0 \mathsf{T}/\mathsf{R}) \setminus
\mathsf{Gal}(\mathsf{E}_0\mathsf{T}/\mathsf{T})$. Therefore, the proof of
Lemma~\ref{unitarycp}~(iii)~$\Rightarrow$~(i) shows that there is a
graded $\sT/\sR$-involution $\tau'$ of $\mathsf{E}$ with each $y_i =\tau'(y_i)$
and $\tau'|_{\mathsf{E}_0\mathsf{T}} = \theta$. Since $\SK(\mathsf{E},\tau) = \SK(\mathsf{E},\tau')$
we may replace~$\tau$ by $\tau'$, so each $y_i = \tau(y_i)$, while
$\overline \tau$ is unchanged.
Fix any
$ \eta\in \Gamma_\mathsf{E}/\Gamma_\mathsf{T}$, and let
$\sigma_{\eta} = \overline\Theta_E(\eta) \in H$. Take the unique
$\bold i \in \mathcal I$ with $\sigma^{\bold i} = \sigma_{\eta}$
(notation as in~\S\ref{abcp}),
let $\gamma =\deg(y^{\bold i})\in \Gamma_\mathsf{E}$, and set
$y_\gamma = y^{\bold i}$. Since $\Theta_\mathsf{E}(\gamma) =
\intt(y_\gamma)|_{\mathsf{E}_0} = \overline \Theta_\mathsf{E}(\eta)$ and
$\overline \Theta_\mathsf{E}\colon \Gamma_\mathsf{E}/\Gamma_\mathsf{T} \to H$ is an
isomorphism for $\mathsf{E}$ semiramified (see \S\ref{graded}), $\eta = \overline \gamma$ in
$\Gamma_\mathsf{E}/\Gamma_\mathsf{T}$.
Since $\tau(y_i) = y_i$ for each $i$,
$\tau(y_\gamma)$ is the product of the $y_i$ appearing in $y_\gamma$ but with
the order reversed. Hence, the commutator identities show that
$\tau(y_\gamma) = a_\gamma y_\gamma$ where $a_\gamma$ in $\mathsf{E}_0$ is a computable product
of the $u_{ij}$ and their conjugates under the $y_i$. Since
each $y_\ell u_{ij}y_\ell^{-1} = \sigma_\ell(u_{ij})$, $a_\gamma$ is a
computable product of terms $\sigma_\ell(u_{ij})$.
(For example, ${\tau(y_1 y_2y_3) = y_3 y_2y_1 =
[u_{32}\sigma_2(u_{31})u_{21}] y_1y_2y_3}$.)
By applying $\tau$ to the equation
$\tau(y_\gamma) = a_\gamma y_\gamma$, we find
$$
a_\gamma \, \sigma_\eta\tau(a_\gamma) \, = \, 1.
$$
Therefore, from Hilbert 90 for the quadratic extension
$\mathsf{E}_0/\mathsf{E}_0^{\sigma_\eta\tau}$, there is $t_\gamma\in \mathsf{E}_0^*$ with
$$
t_\gamma \,[\sigma_\eta\tau(t_\gamma)]^{-1} \, = \, a_\gamma.
$$
Then, $\tau(t_\gamma y_\gamma) = t_\gamma y_\gamma$, so for the $x_\gamma$ in $X$
we can set $x_\gamma =
t_\gamma y_\gamma$. Now take any $\zeta \in
\Gamma_\mathsf{E}/\Gamma_\mathsf{T}$ and
carry out the same process for $\zeta $
as we have just done for $\eta$, obtaining
$\delta \in \Gamma$ with $\overline \delta = \zeta$, and $y_\delta$ with
$\deg(y_\delta) = \delta$ and $\intt(y_\delta)|_{\mathsf{E}_0} = \sigma_\zeta$,
then determining $a_\delta$, $t_\delta$, $x_\delta$.
Then set $y_{\gamma+\delta} = y_\gamma y_\delta$, so
${\intt(y_{\gamma+\delta})|_{\mathsf{E}_0} = \sigma_\eta\sigma_\zeta}$.\break
Let ${a_{\gamma+\delta} =
\tau(y_{{\gamma+\delta}}) y_{\gamma+\delta}^{-1} \in \mathsf{E}_0^*}$. Since
$a_{\gamma+\delta} \sigma_\eta\sigma_\zeta \tau(a_{\gamma+\delta}) = 1$, by Hilbert 90 there is
$t_{\gamma+\delta}\in \mathsf{E}_0^*$ with\break
${t_{\gamma+\delta}[\sigma_\eta\sigma_\zeta\tau(b_{\gamma+\delta})]
^{-1} = a_{\gamma+\delta}}$.
Then set $x_{\gamma+\delta} = t_{\gamma+\delta} y_{\gamma+\delta}$, so that $\tau(x_{\gamma+\delta}) = x_{\gamma+\delta}$.
By the definition of the function $g$ of Prop.~\ref{Gammafn}, we have
in $\ker(\widetilde N)/\Pi$,
$$
g(\eta, \zeta) \,=\, x_\gamma x_\delta x_{\gamma+\delta}^{-1} \,\Pi
=\, (t_\gamma y_\gamma) (t_\delta y_\delta)(t_{\gamma+\delta} y_\gamma y_\delta)^{-1} \,\Pi\,
= \, t_\gamma \sigma_\eta(t_\delta) t_{\gamma+\delta}^{-1}\,\Pi.
$$
Since the $t$'s are determined by the $a$'s, which are determined by
the $u_{ij}$, this shows that $\operatorname{im}(g)$ is determined by the $u_{ij}$.
(ii) Suppose now that $\mathsf{E}_0 = L_1\otimes_{\mathsf{T}_0} L_2$ with each
$L_i$ cyclic Galois over ${\mathsf{T}_0}$, and let $\sigma = \sigma_1$
and $\rho = \sigma_2$, as in \S\ref{ubicyclic}. The isomorphism
\begin{equation}\label{unitarybicyc}
\operatorname{Br}(M/K;F)\big/
\operatorname{Dec}(M/K;F) \,\cong \,\ker(\widetilde{N})/\Pi
\end{equation}
of Prop.~\ref{thmainm} maps $[\mathsf{I}_0] = [A(u,b_1,b_2)]$
to $q\,\Pi$, where $q\in \mathsf{E}_0^*$ with $u = q[\rho\sigma\overline \tau(q)]^{-1}$.
Take standard generators $y_1,y_2$ of $A(u,b_1,b_2)$. As noted for (i),
we can assume after modifying $\tau$ (without changing $\overline\tau$)
that $\tau(y_1) = y_1$ and $\tau(y_2) = y_2$. Let
$\gamma = \deg(y_1)$ and $\delta = \deg(y_2)$ in $\Gamma_\mathsf{E}$, so
$\Theta_\mathsf{E}(\gamma) = \intt(y_1)|_{\mathsf{E}_0} = \sigma$ and
${\Theta_\mathsf{E}(\delta) = \intt(y_2)|_{\mathsf{E}_0} = \rho}$. Since
$\Gamma_\mathsf{E}/\Gamma_\mathsf{T} \cong H = \langle\sigma,\rho \rangle$, we have
$\Gamma_\mathsf{E}/\Gamma_\mathsf{T} = \langle\overline \gamma, \overline \delta \rangle$. As
$\tau(y_1) = y_1$, we can take $x_\gamma = y_1$, and likewise
$x_\delta = y_2$. Because $\tau(y_2y_1) = uy_2y_1 =
q[\rho\sigma\overline \tau(q)]^{-1} y_2y_1$, we have
$\tau(q y_2y_1) = qy_2y_1$; thus, we can take
$x_{\delta+\gamma} = qy_2y_1$. Then,
$$
g(\overline \delta, \overline \gamma) \, = \, x_\delta x_\gamma x_{\delta +\gamma}^{-1} \,\Pi
\, = \, y_2y_1(qy_2y_1)^{-1}\,\Pi \, = \, q^{-1} \,\Pi.
$$
Since $\overline\delta$ and $\overline\gamma$ generate $\Gamma_\mathsf{E}/\Gamma_\mathsf{T}$ formula~\eqref{gformula}
shows that $\operatorname{im}(g) = \langle g(\overline\delta, \overline \gamma)\rangle = \langle q^{-1}\Pi\rangle
= \langle q\,\Pi \rangle$. Therefore, the isomorphism of \eqref{unitarybicyc}
maps $\langle[\mathsf{I}_0]\rangle$ to $\langle q\,\Pi\rangle = \langle\operatorname{im}(g) \rangle$. Thus, the
isomorphism asserted for (ii) follows from~(i).
\end{proof}
\begin{example}
Here is a unitary version of Ex.~\ref{cyclicex}. Take any integer
$n\ge 2$, and let $F\subseteq K$ be fields with $[K\! :\! F] = 2$,
$K$ Galois over $F$, and $K = F(\omega)$ where $\omega$ is a primitive
$n^2$-root of unity. Suppose further that for the nonidentity element
$\psi_0$ of $\operatorname{Gal}(K/F)$ we have $\psi_0(\omega) = \omega^{-1}$.
(For example,
we could take $K = \mathbb{Q}(\omega)$, the $n^2$-cyclotomic extension of
$\mathbb{Q}$, and $F = K\cap \mathbb{R}$.) Let $\mathsf{T} = K[x,x^{-1},y,y^{-1}]$, the Laurent
polynomial ring, with its usual grading by $\mathbb{Z}\times\mathbb{Z}$;
so, $\mathsf{T}$ is a graded field. Let
$\mathsf{R} = F[x,x^{-1},y,y^{-1}]$, which is a graded subfield of $\mathsf{T}$
with $[\mathsf{T}\! :\!\mathsf{R}] = 2$, $\mathsf{T}$ Galois over $\mathsf{R}$, and
$\mathsf{T}$ inertial over $\mathsf{R}$. Also, $\mathsf{Gal}(\sT/\sR) = \{\psi, \operatorname{id}_\mathsf{T}\}$,
where $\psi = \psi_0\otimes \operatorname{id} _\mathsf{R}$ on $\mathsf{T} = {\mathsf{T}_0} \otimes_{\mathsf{R}_0} \mathsf{R}$.
Take any $a,b \in F^*$ such that $[K(\sqrt[n]a,\sqrt[n]b\,)\! :\! K] = n^2$,
and let $M = K(\sqrt[n]a,\sqrt[n]b\,)$. Then, it is easy to check that
$M$ is $K/F$-generalized dihedral. (One can think of such field extensions
$M/F$ as the generalized dihedral analogue to Kummer extensions.)
Indeed, $\psi_0$ on $K$ extends to $\theta\in \operatorname{Gal}(M/F)$
given by $\theta (\sqrt[n]a) =\sqrt[n]a$, $\theta(\sqrt[n]b) =
\sqrt[n]b$, and $\theta|_K = \psi_0$; so, $\theta^2 = \operatorname{id}_M$,
and for $h\in \operatorname{Gal}(M/K)$, we have $\theta h\theta = h^{-1}$.
As in
Ex.~\ref{cyclicex}, take the graded symbol algebra
$\mathsf{E} = (ax^n,by^n,\mathsf{T})_\omega$ of degree $n^2$, with its generators $i,j$
satisfying $i^{n^2} = ax^n$, $j^{n^2} = by^n$, $ij = \omega ji$.
For $\sigma_1, \sigma_2$ as in Ex.~\ref{cyclicex}, it was noted there
that $\mathsf{E} = \mathsf{A}(M\mathsf{T}/\mathsf{T}, \boldsymbol\sigma, \bold u, \bold d)$ where
$u_{12} = \omega$, and $d_1 = 1/(y\sqrt[n]b)$ and $d_2 = x\sqrt[n]a$.
We extend $\theta$ to an element of $\mathsf{Gal}(M\mathsf{T}/\mathsf{R})$ by setting
$\theta|_\mathsf{R} = \operatorname{id}$. Since $\theta(d_1) =d_1$, $\theta(d_2) =d_2$,
and $u_{12}\, \sigma_1\sigma_2\theta(u_{12}) = \omega\omega^{-1} = 1$, the
graded version of Lemma~\ref{unitarycp} shows that there is a
graded $\mathsf{T}/\mathsf{R}$-involution $\tau$ on $\mathsf{E}$ given by $\tau(j^{-1})
= j^{-1}$, $\tau(i) = i$, and $\tau|_{M\mathsf{E}} = \theta$.
That is, $\tau$ is
the $\mathsf{R}$-linear map $\mathsf{E} \to \mathsf{E}$ such that
$\tau(c\,i^\ell j^m) = \psi(c) j^m i^\ell$ for all $c\in \mathsf{T}$,
$\ell,m\in \mathbb{Z}$. We have
the decomposition of $\mathsf{E}$ noted in Ex.~\ref{cyclicex},
$$
\mathsf{E}\,\sim_g\, \mathsf{I} \otimes_\mathsf{T} \mathsf{N}\qquad \text{where} \qquad
\mathsf{I} \, = \, (a,b, \mathsf{T})_\omega \quad \text{and} \quad
\mathsf{N} \, = \, (x,b,\mathsf{T})_{\omega^n}\otimes_\mathsf{T} (a,y,\mathsf{T})_{\omega^n}.
$$
These $\mathsf{I}$ and $\mathsf{N}$ are $\mathsf{T}$-central graded division algebras
with $\mathsf{I}$ inertial and $\mathsf{N}$ DSR. Furthermore, as\break
${a,b,x,y\in \mathsf{R}^*}$,
there are unitary graded $\sT/\sR$-involutions $\tau_\mathsf{I}$ on $\mathsf{I}$
and $\tau_\mathsf{N}$ on $\mathsf{N}$ defined analogously to $\tau$ on $\mathsf{E}$. So,
by Th.~\ref{unitaryDSR}(ii)
$$
\SK(\mathsf{N}, \tau_\mathsf{N}) \ \cong \
\operatorname{Br}\big(M/K;F\big)\big/
\operatorname{Dec}\big(M/K;F\big), \quad\text{where}\quad M \, =
\,K(\sqrt[n]a, \sqrt[n]b\,),
$$
with $\operatorname{Dec}\big(M/K;F\big) =
\operatorname{Br}\big(K(\sqrt[n]a\,)/K;F\big) \cdot
\operatorname{Br}\big (K(\sqrt[n]b\,)/K;F\big)$ by \eqref{Decformula}.
Since $\mathsf{I}_0 \cong(a,b,K)_\omega$,
Th.~\ref{main}(ii) yields
$$
\SK(\mathsf{E}, \tau) \ \cong \ \operatorname{Br}(M/K;F)\big/
\big[\operatorname{Dec}(M/K;F)\cdot \langle(a,b,K)_\omega \rangle
\big].
$$
Note that $\mathsf{E}$ is semiramified, but it may or may not be DSR. Indeed, by
Prop.~\ref{uINdecomp}(ii) $\mathsf{E}$ is DSR if and only if
${\mathsf{I}_0 \in \operatorname{Dec}\big(M/K;F\big)}$; the formulas
above show that this holds if and only if the obvious surjection\break
${\SK(\mathsf{N}, \tau_N) \to \SK(\mathsf{E}, \tau)}$ is an isomorphism.
Note also that $\operatorname{Dec}(M/K;F)$ may be strictly smaller than
$\operatorname{Dec}(M/K) \cap \operatorname{Br}(M/K;F)$, i.e., there may be an algebra in $\operatorname{Br}(M/K)$
which decomposes according to $M$ and has a $K/F$-involution, but in any
decomposition the factors do not have $K/F$-involutions. Examples of this
are given in Remark~\ref{indecs} below.
For an
ungraded version of this example, let $K$, $F$, $a$, and $b$ be as above;
then
let $K' = K((x))((y))$ and
$F' = F((x))((y))$, and ${D = (ax^n,by^n,K')_\omega}$. Then,
with respect to the usual rank $2$ Henselian valuations $v_{K'}$ on
$K'$ and $v_{F'}$ on $F'$, $K'$ is inertial of degree $2$ over $F'$.
Furthermore, with respect to the valuation $v_D$ on $D$ extending
$v_{K'}$ on $K'$,
$D$ is a semiramified $K'$-central division algebra
with a unitary $K'/F'$-involution $\tau_D$ defined just as for $\tau$
on $\mathsf{E}$. For the associated graded ring $\operatorname{{\sf gr}}(D)$ of $D$
determined by $v_D$, we have
$\operatorname{{\sf gr}}(D) \cong_g \mathsf{E}$, so by \cite[Th.~3.5]{I}
$\SK(D, \tau_D) \cong \SK(\mathsf{E}, \tau)$.
\end{example}
\section{Noninjectivity}
For any $\mathsf{T}$-central graded division algebra $\mathsf{B}$
with unitary $\sT/\sR$-involution $\tau$,
there are well-defined canonical homomorphisms
\begin{equation}\label{alpha}
\alpha\colon \SK(\mathsf{B}, \tau) \to \SK(\mathsf{B})
\quad\text{given by} \ \
a\,\Sigma_\tau(\mathsf{B}) \mapsto \tau(a) a^{-1}\, [\mathsf{B}^*,\mathsf{B}^*]
\ \ \text{for\ } a\in \Sigma_\tau'(\mathsf{B}), \qquad \ \ \,
\end{equation}
\vskip -0.15truein
and
\vskip-.25truein
\begin{equation*}
\beta\colon \SK(\mathsf{B}) \to \SK(\mathsf{B}, \tau) \quad \text{given by}
\ \ b \, [\mathsf{B}^*,\mathsf{B}^*] \mapsto b\, \Sigma_\tau(\mathsf{B})
\ \ \text{for\ } b\in \mathsf{B}^* \ \text {with} \ \Nrd_\mathsf{B}(b) = 1.
\
\end{equation*}
It is easy to check that $\beta \circ \alpha$ and
$\alpha \circ \beta$ are each the squaring map. As
pointed out in \cite[Lemma, p.~185]{y}, since the
exponent of the abelian group $\SK(\mathsf{B},\tau)$ divides
$\deg(\mathsf{B})$, if $\deg(\mathsf{B})$ is odd, then $\alpha$ must be injective.
It seems to have been an open question up to now whether
$\alpha$ is always injective, even when $\deg(\mathsf{B})$ is even.
We now settle this question by using some of the results above
to give examples of $\mathsf{B}$ of
degree $4$ with $\alpha$~not injective.
We thank J.-P.~Tignol for pointing out the relevance of
indecomposable division algebras of degree $8$ and exponent $2$,
and for calling his paper \cite{tigtri} to our attention.
Let $F$ be a field with $\operatorname{char}(F) \ne 2$. Let $M = F(\sqrt a, \sqrt b,
\sqrt c\,)$
with $a,b,c\in F^*$ and $[M\! :\! F] = 8$. Let ${K = F(\sqrt a\,)}$.
We write $\operatorname{Br}_2(F)$ for the $2$-torsion subgroup of $\operatorname{Br}(F)$,
and set ${\operatorname{Br}_2(M/F) = \operatorname{Br}(M/F)\cap \operatorname{Br}_2(F)}$, $\operatorname{Br}_2(M/K;F)
= \operatorname{Br}(M/K;F) \cap \operatorname{Br}_2(K)$, etc. Note that as $\operatorname{Gal}(M/F)$ is an
elementary
abelian $2$-group, $M$ is a $K/F$-generalized dihedral extension.
Also, $\operatorname{res}_{F\to K}$ maps $\operatorname{Br}_2(M/F)$ to $\operatorname{Br}(M/K;F)$, since for
$[A] \in \operatorname{Br}_2(M/F)$, $\operatorname{cor}_{K\to F} [A\otimes_F K]
= [A]^{[K: F]} = 1$ in $\operatorname{Br}(F)$, so by Albert's Theorem
$A\otimes_F K$ has a
unitary $K/F$-involution.
\begin{proposition}\label{Br2}
There is an exact sequence:
\begin{equation}\label{decexact}
0 \ \longrightarrow \ \operatorname{Br}_2(M/F)\big/\operatorname{Dec}(M/F) \ \longrightarrow \
\operatorname{Br}(M/K;F)\big/ \operatorname{Dec}(M/K;F)\ \longrightarrow \
\operatorname{Br}(M/K)\big/ \operatorname{Dec}(M/K)
\end{equation}
\end{proposition}
\begin{proof}
The kernel of the right map in \eqref{decexact} is
$\big[\operatorname{Br}(M/K;F) \cap \operatorname{Dec}(M/K)\big]\big/
\operatorname{Dec}(M/K;F)$. So, the
exactness of \eqref{decexact} is equivalent to two assertions:
\begin{equation*}
\qquad(\text{a}) \quad \operatorname{Br}(M/K;F) \, \cap \, \operatorname{Dec}(M/K) \ = \ \operatorname{Br}_2(M/K;F).
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\end{equation*}
\vskip-7pt
\noindent and
\vskip-22pt
\begin{equation*}
\qquad(\text{b}) \quad \operatorname{Br}_2(M/F)\big/ \operatorname{Dec}(M/F) \ \cong \
\operatorname{Br}_2(M/K;F)\big/ \operatorname{Dec}(M/K;F)\qquad\qquad\qquad\qquad\qquad\qquad
\qquad\qquad\quad
\end{equation*}
The equality (a) is immediate from the fact that $\operatorname{Dec}(M/K) =\operatorname{Br}_2(M/K)$,
as $M$ is a biquadratic extension of $K$. (This is well-known, and is
deducible, e.g., by refining the argument in \cite[Prop.~16.2]{kmrt}.
It also appears in
\cite[Cor.~2.8]{tigtri} as the assertion that property P$_2$(2) holds for $K$.)
The isomorphism (b) appears in \cite[Prop.2.2]{tigtri} as the
isomorphism ${N_2(M/F) \cong M_2(M/K/F)}$, see the conmments on p.~14 of
\cite{tigtri}. Since the isomorphism (b) is somewhat buried in the general
arguments of
\cite{tigtri}, we give a short and direct proof of it:
If ${[A] \in \operatorname{Dec}(M/F)}$, then $A \sim Q_1\otimes_F Q_2
\otimes _F Q_3$, where $Q_1$ is the quaternion algebra $\quat arF$,
$Q_2 = \quat bsF$, and $Q_3 = \quat ctF$, for some $r,s,t\in F^*$.
So, ${A\otimes_FK \sim (Q_2 \otimes_F K)\otimes_K(Q_3 \otimes_FK)}$.\break
Here, $Q_2\otimes _F K$ has the unitary $K/F$-involution
$\eta \otimes \psi$, where $\eta$ is any involution of the first
kind
\break
on $Q_2$ and $\psi$ is the nonidentity $F$-automorphism of $K$.
So $[Q_2\otimes_FK] \in \operatorname{Br}(K(\sqrt b)/K;F)\subseteq \operatorname{Dec}(M/K;F)$;
likewise
$[Q_3\otimes_F K] \in
\operatorname{Br}(K(\sqrt c)/K;F)\subseteq \operatorname{Dec}(M/K;F)$, and hence
$[A\otimes_F K] \in \operatorname{Dec}(M/K;F)$. Thus, $\operatorname{res}_{F\to K}$
induces a well-defined map
${f\colon \operatorname{Br}_2(M/F)/\operatorname{Dec}(M/F) \to
\operatorname{Br}_2(M/K;F)/\operatorname{Dec}(M/K;F)}$. From Arason's long exact sequence (see, e.g.,
\cite[Cor.~30.12(1)]{kmrt} or \eqref{longexact} above)
$$
{\ldots \to H^2(F, \mu_2) \to H^2(K,\mu_2)
\to H^2(F, \mu_2) \to \ldots \ },
$$
$f$ is surjective. For injectivity of~$f$, take any $[A] \in
\operatorname{Br}_2(M/F)$ with $\operatorname{res}_{F\to K}[A] \in \operatorname{Dec}(M/K;F)$. We need to show
$[A]\in \operatorname{Dec}(M/F)$. We have $A\otimes _FK \sim Q_2' \otimes_K Q_3'$
where the $Q_i'$ are quaternion algebras over~$K$ with
$Q_2' \in \operatorname{Br}(K(\sqrt b\, )/K;F)$ and $Q_3'\in \operatorname{Br}(K(\sqrt c\, )/K;F)$. By
a result of Albert \cite[Prop.~2.22]{kmrt}, the quaternion algebra
$Q_2'$ with $K/F$-involution has the form
$Q_2' \cong Q_2^{\prime\prime} \otimes _F K$, where
$Q_2^{\prime\prime}$ is a quaternion algebra over $F$.
Then, $[Q_2^{\prime\prime}] \in \operatorname{Br}_2(K(\sqrt b\, )/F) =
\operatorname{Dec}(K(\sqrt b \,)/F)$, as noted for (a) above.
Likewise, $Q_3'\cong Q_3^{\prime\prime}\otimes_F K$, where
$[Q_3^{\prime\prime}] \in \operatorname{Dec}(K(\sqrt c\, )/F)$.
Since $[A\otimes _F Q_2^{\prime\prime} \otimes_F Q_3^{\prime\prime}]
\in \operatorname{Br}(K/F) = \operatorname{Dec}(K/F)$, we have
$$
[A] \,= \ [A\otimes _F Q_2^{\prime\prime} \otimes_F Q_3^{\prime\prime}]
\,[Q_2^{\prime\prime}]\,[Q_3^{\prime\prime}]
\ \in \ \operatorname{Dec}(K/F) \cdot \operatorname{Dec}(K(\sqrt b\, )/F) \cdot
\operatorname{Dec}(K(\sqrt c\, )/F)
\ \subseteq\, \operatorname{Dec}(M/F).
$$
Thus, $f$ is an isomorphism, proving (b).
\end{proof}
\begin{remark}\label{indecs}
The term $\operatorname{Br}_2(M/F)/\operatorname{Dec}(M/F)$ for $M/F$ triquadratic has arisen in the
study of indecomposable algebras $A$ of degree $8$ and exponent $2$.
Note first that for any $A$ of degree $8$ and exponent $2$,
by Rowen's theorem
\cite[Th.~6.2]{rowen} there is a triquadratic field extension $M$ of
the center $F$ of $A$, such that $M$~is a maximal subfield of~$A$.
If $A$ is indecomposable, then $[A]$ yields a nontrivial element of $\operatorname{Br}_2(M/F)/\operatorname{Dec}(M/F)$.
Examples of indecomposables if degree $8$ and exponent $2$ were
first given in \cite[Th.~5.1]{art}. Subsequently, Karpenko showed
in \cite[Cor.~5.4]{karpenko} that if $B$ is a division algebra with center $F$
of degree $8$ and exponent~$8$, and $F'$ is a field generically reducing the
exponent of $B$ to $2$, then $B\otimes_F F'$ is an indecomposable division
algebra of degree $8$ and exponent $2$. Also, K. McKinnie in
her thesis (unpublished), using lattice methods, gave another example of
indecomposables of degree $8$ and exponent $2$. There is a kind of converse
to this as well: Given a division algebra $A$ with $[A] \in
\operatorname{Br}_2(M/F)\setminus \operatorname{Dec}(M/F)$, Amitsur, Rowen, and Tignol showed in
\cite[Th.~3.3]{art} that the associated generic abelian crossed product algebra
$A'$ of $A$ is indecomposable of degree $8$ and exponent~$2$.
(It is not stated this way in \cite{art}, but made explicit in
\cite[\S~2]{tignol}.)
This $A'$ is the ring of quotients of a semiramified graded division
algebra $\mathsf{E}$ of the type considered in previous sections: $\mathsf{E}$ is
graded Brauer equivalent to $\mathsf{I} \otimes_\mathsf{T} \mathsf{N}$, where
$\mathsf{T}$ is a graded field with ${\mathsf{T}_0} \cong F$, $\mathsf{I}$ is an inertial graded
division algebra over $\mathsf{T}$ with $\mathsf{I}_0\cong A$, and
$\mathsf{N}$ is DSR over $\mathsf{T}$ with $\mathsf{N}_0 \cong M$.
\end{remark}
Using Prop.~\ref{Br2} we now construct biquaterion graded algebras where the
map $\alpha$ of \eqref{alpha} above is not injective.
\begin{example}\label{noninjex}
Let $M$ be a triquadratic extension of a field $F$ ($\operatorname{char}(F)\ne 2$) with
${\operatorname{Br}_2(M/F) /\operatorname{Dec}(M/F) \ne 0}$. (Such $F$ and $M$ exist, as noted
in Remark~\ref{indecs}.) Say $M = F(\sqrt a, \sqrt b, \sqrt c\, )$ for
$a,b,c\in F^*$. Let $K = F(\sqrt a\,)$, and let $H = \operatorname{Gal}(M/K)$.
Let $\mathsf{R} = F[x,x^{-1},y,y^{-1}]$, the Laurent polynomnial ring in indeterminates
$x$ and~$y$, with its usual grading in which $\mathsf{R}_{(k,\ell)} = Fx^ky^\ell$ for
all $(k,\ell) \in \mathbb{Z}\times \mathbb{Z}$. So, $\mathsf{R}$ is a graded field with $R_0 = F$
and $\Gamma_\mathsf{R} = \mathbb{Z}\times \mathbb{Z}$. Let $\mathsf{T} = K[x,x^{-1},y,y^{-1}]$, a graded
field with $[\mathsf{T}\! :\!\mathsf{R}] = 2$, and let $\mathsf{E} = \mathsf{Q}\otimes_\mathsf{T} \mathsf{Q}'$, where
$\mathsf{Q}$ and $\mathsf{Q}'$ are the following semiramified graded quaternion
division algebras
over $\mathsf{T}$: $\mathsf{Q} = \quat b x \mathsf{T}$, which is generated over $\mathsf{T}$ by
homogeneous elements $i$ and$j$ with relations $i^2= b$, $j^2 = x$,
and $ij = -ji$, with $\deg(i) = 0$ and $\deg(j) = (\frac12,0)$.
So, $\mathsf{Q}_0 \cong K(\sqrt b\, )$ and $\Gamma_\mathsf{Q} = \frac12\mathbb{Z}\times \mathbb{Z}$.
Likewise, set $\mathsf{Q}' = \quat c y\mathsf{T}$ with standard generators $i'$~and~$j'$,
with $\deg(i') = 0$ and $\deg(j') = (0,\frac12)$, so
$\mathsf{Q}_0 \cong K(\sqrt c\, )$ and $\Gamma_{\mathsf{Q}'} = \mathbb{Z}\times \frac12\mathbb{Z}$.
Since $\mathsf{Q} \cong \quat bx\mathsf{R} \otimes_ \mathsf{R}\mathsf{T}$, $\mathsf{Q}$~has the
graded $\mathsf{T}/\mathsf{R}$-involution $\tau_\mathsf{Q} = \eta \otimes \psi$, where
$\eta$ is the
canonical symplectic graded involution on $\quat bx\mathsf{R}$, for which $\eta(i) = -i$
and $\eta(j) = -j$, and $\psi$ is the nonidentity graded $\mathsf{R}$-automorphism
of $\mathsf{T}$. Likewise $\mathsf{Q}'$ has a graded $\mathsf{T}/\mathsf{R}$-involution $\tau_{\mathsf{Q}'}$ with
$\tau_{\mathsf{Q}'}(i') = -i'$ and $\tau_{\mathsf{Q}'}(j') = -j'$.
By Lemma~\ref{DSRprod}, $\mathsf{E}$~is a graded division algebra which is
DSR for $\sT/\sR$ with $\mathsf{E}_0 \cong \mathsf{Q}_0\otimes_ {\mathsf{T}_0} \mathsf{Q}_0'
\cong K(\sqrt b\, ) \otimes _K K(\sqrt c\, ) \cong M$ and
$\Gamma_\mathsf{E} = \Gamma_\mathsf{Q} + \Gamma_{\mathsf{Q}'} = \frac12\mathbb{Z}\times \frac12 \mathbb{Z}$;
our
graded $\sT/\sR$-involution on $\mathsf{E}$ is $\tau = \tau_\mathsf{Q} \otimes \tau_{\mathsf{Q}'}$.
(Explicitly, $\mathsf{S} = \mathsf{T}[i,i'] \cong_g M[x,x^{-1},y,y^{-1}]$ is a maximal
graded subfield of $\mathsf{E}$ with $\mathsf{S}$ inertial over $\mathsf{T}$, and
$\mathsf{J} = \mathsf{T}[j,j']\cong_g \mathsf{T}[\sqrt x, \sqrt x^{\,-1}, \sqrt y, \sqrt y^{\,-1}]$
is a maximal graded subfield of $\mathsf{E}$ which is totally ramified over
$\mathsf{T}$ with $\tau(\mathsf{J}) = \mathsf{J}$.) We claim that the following diagram is
commutative with all horizontal maps isomorphisms and vertical
maps described below:
\begin{equation}\label{diagram}
\begin{CD}
\operatorname{Br}(M/K;F)\big/\operatorname{Dec}(M/K;F) @>>> \ker(\widetilde N)/ \Pi @>>>\SK(\mathsf{E},\tau)\\
@VVV @VVV @V{\alpha}VV\\
\operatorname{Br}(M/K)\big/\operatorname{Dec}(M/K) @>>> \widehat H^{-1}(H,M^*)@>>> \SK(\mathsf{E})
\end{CD}
\end{equation}
The left vertical map is the map in Prop.~\ref{Br2}, whose kernel
is there shown to be isomorphic to\break
$\operatorname{Br}_2(M/F)/\operatorname{Dec}(M/F)$.
Since we have assumed this kernel is nontrivial, once the claim
is established the right vertical map $\alpha$, which is the map
of \eqref{alpha} must also have nontrivial kernel, as desired.
We now verify the claim. In the top line of \eqref{diagram},
$\ker(\widetilde N) = \{ a\in M^*\mid N_{M/K}(a) \in F\}$ and
${\Pi = \prod_{h\in H} M^{*h\overline \tau}}$, where $H = \operatorname{Gal}(M/K)$
and $\overline \tau = \tau|_{\mathsf{E}_0}$. The middle vertical map sends
$a\,\Pi \mapsto a/\overline\tau(a)\,I_H(M^*)$. It is well defined since
if $a\in \ker(\widetilde N)$, we have $N_{K/F}(a/\tau(a)) = N_{K/F}(a)/
\tau(N_{K/F}(a)) =1$, and
if $b\in M^{*h\overline \tau}$, then
${b/\overline \tau(b) = h\overline \tau(b)/\overline \tau(b) \in I_H(M^*)}$.
In the right rectangle of \eqref{diagram}, the top map sends
$a\,\Pi\mapsto a\Sigma_\tau(\mathsf{E})$, and the bottom map sends
$b\,I_H(M^*) \mapsto b\,[\mathsf{E}^*,\mathsf{E}^*]$, so the
right rectangle is clearly commutative. The horizontal maps
in this rectangle are the isomorphisms given in Th.~
\ref{unitaryDSR}(i) and Prop.~\ref{NSRSK}(i). For the left
vertical map take an arbitrary element of $\operatorname{Br}(M/K;F)$,
which has the form $[A]$, where $A = A(u,b_1,b_2)$ in the notation
of \S\ref{ubicyclic}, with $u, \ b_1, \ b_2$ satisfying
the relations in \eqref{urels} and \eqref{brel} and the added relations
in Lemma ~\ref{unitarycp}(iii), notably $u\, \sigma\rho \overline \tau(u)
= 1$. The horizontal map in the left rectangle is the isomorphism
of Th.~\ref{thmainm} which sends $[A]$ mod $\operatorname{Dec}(M/K;F)$ to
$q\, \Pi$ for any $q\in M^*$ with $q/\sigma\rho\overline \tau(q) = u$.
This is mapped downward to $u \,I_H(M^*)$, since
$q/\overline \tau(q) = u\,\sigma\rho\overline \tau(q)/\overline\tau(q) \equiv
u\ (\operatorname{mod}\ I_H(M^*))$. On the other hand, $[A]$ mod $\operatorname{Dec}(M/K;F)$
is mapped downward to $[A]$ mod $\operatorname{Dec}(M/K)$, which is mapped to the
right to $u\, I_H(M^*)$ by the isomorphism of \eqref{njnj}. Thus,
the left rectangle of \eqref{diagram} is commutative, and its
horizontal maps are isomorphisms, completing the proof of the claim.
\end{example}
\begin{remark}
For the preceding example with the $\alpha$ of \eqref{alpha}
noninjective, we have worked with graded division algebras.
There are corresponding examples of division algebras over
a Henselian valued field with the corresponding $\alpha$ not
injective, obtainable as follows: With fields $F \subseteq K \subseteq M$
as in Ex.~\ref{noninjex}, let $F' = F((x))((y))$, $K' = K((x))((y))$,
and $M' = M((x))((y))$, which are twice iterated Laurent power
series fields each with it standard Henselian valuation with
value group $\mathbb{Z}\times \mathbb{Z}$ (with right-to-left lexicographic ordering)
and residue fields $\overline{F'} \cong F$, $\overline{K'}\cong K$, and $\overline{M'}
\cong M$. Let $D = \quat bx{K'} \otimes_{K'} \quat cy{K'}$, which is
a division algebra over $K'$, and the Henselian valuation $v_{K'}$ on
$K'$ extends uniquely to a valuation $v_D$ on $D$, for which
$\overline D \cong M$ and $\Gamma_D = \frac12\mathbb{Z}\times \frac12 \mathbb{Z}$.
For the associated graded ring of $D$ determined by $v_D$, we have
$\operatorname{{\sf gr}}(D) \cong_g \mathsf{E}$ and, as $D$ is tame over $K'$,
$Z(\operatorname{{\sf gr}}(D)) = \operatorname{{\sf gr}}(K')\cong_g \mathsf{T}$, for the $\mathsf{E}$ and $\mathsf{T}$
of Ex.~\ref{noninjex}. Also, $\operatorname{{\sf gr}}(F')\cong_g\mathsf{R}$ for the $\mathsf{R}$
of Ex.~\ref{noninjex}.
This $D$ has a unitary $K'/F'$-involution $\tau_D$, since each
constituent quaternion algebra has such an involution.
Because the Henselian valuation $v_{F'}$ on $F'$ has a unique
extension to $K'$, namely $v_{K'}$, and $v_D$~is the unique extension
of $v_{K'}$ to $D$, we must have $v_D\circ \tau_D = v_D$.
Therefore, $\tau_D$ induces a graded involution
$\widetilde \tau$ on~$\mathsf{E}$, which is a unitary $\sT/\sR$-involution.
By \cite[Th.~3.5]{I} and \cite[Th.~4.8]{hazwadsworth},
$\SK(D,\tau_D) \cong \SK(\mathsf{E}, \widetilde \tau)$ and $\SK(D) \cong
\SK(\mathsf{E})$. These isomorphisms are compatible with the map
$\alpha_{\widetilde\tau}\colon\SK(\mathsf{E}, \widetilde \tau) \to \SK(\mathsf{E})$
and the corresponding map $\alpha_D\colon \SK(D, \tau_D) \to
\SK(D)$. Also, because $\widetilde \tau$ and the $\tau$ of
Ex.~\ref{noninjex} are each graded $\mathsf{T}/\mathsf{R}$-involutions
on $\mathsf{E}$, we have $\SK(\mathsf{E}, \widetilde \tau) \cong \SK(\mathsf{E}, \tau)$,
and it is easy to check that under this isomorphism
$\alpha_{\widetilde\tau}$ corresponds to the $\alpha$ of Ex.~
\ref{noninjex}. Since this $\alpha$ is not injective,
$\alpha_D$ is also noninjective.
\end{remark}
|
{
"timestamp": "2010-09-23T02:02:46",
"yymm": "1009",
"arxiv_id": "1009.3904",
"language": "en",
"url": "https://arxiv.org/abs/1009.3904"
}
|
\section{Introduction}
We have designed an inquiry-based laboratory activity on transiting extrasolar planets for an introductory college-level astronomy class.
In our work, ``inquiry" means ``teaching science as science is done": students learn about scientific concepts by figuring them out (as scientists do) instead of being given answers from a textbook or lecture. Inquiry activities can model different parts of the scientific method: for example, students can design and carry out an entire investigation from designing the question to presenting results to others, or they can simply draw conclusions from a supplied dataset. These sorts of activities enable students to learn content intertwined with research skills, like interpreting evidence, reasoning, thinking critically, conveying ideas, asking questions and figuring out how to answer them. These skills are useful in later science classes, and also are important life skills.
Like real science, this technique can involve some winding pathways. A student embarking on a self-designed investigation will likely hit dead ends or be confused by side issues, but with support he should eventually arrive at the desired conclusions. Inquiry activities are not completely open-ended; teachers carefully monitor students' progress and help guide students toward desired conclusions through a process we call ``facilitation.'' Investigation with facilitator support allows the students to internalize the content they are studying; instead of words on a page, the results are activities they did and conclusions they figured out. This process tends to be more engaging than following detailed step-by-step instructions or listening to a lecture, and studies have shown that the content learned lasts longer and is learned more thoroughly, as students have built up their own understanding instead of having it given to them \citep[e.g.][and references therein]{HSL, America}.
This activity was developed through the Professional Development Program (PDP) run by the Institute for Science and Engineer Educators (ISEE) at the University of California, Santa Cruz. Each team of PDP participants designs an activity and facilitates its teaching for a specific venue. These activities are ``backward-designed": each team carefully takes into consideration the goals for the particular activity and the needs of the students first, and prioritizes elements of the activity based on these. (For a more detailed discussion of the PDP, see Hunter et al., this volume; \citealt{PDP_description}.) This activity was designed to fit in as one week of the semester-long introductory astronomy laboratory course (Astro 1L) at Hartnell Community College in Salinas, California. Nicholas McConnell, Linda Strubbe, and Anne Medling were the primary designers and facilitators. Pimol Moth is the instructor of Astro 1L. Ryan Montgomery, Lynne Raschke, Lisa Hunter, and Barbara Goza provided additional support for the activity.
\section{The Venue and Activity}
Hartnell College is an accredited California Community College and Hispanic Serving
Institution located in Salinas, Monterey County, California, 120 miles south of
San Francisco. Of the College's 10,000+ students, 72\% are ethnic minority. More than 40\% of the College's students are non-native English speakers, and 64\% are first-generation college students. The majority of students enrolled in Astro 1L are from Hispanic backgrounds ($\sim$75-80\%), historically underrepresented in the sciences. Astro 1L is predominantly taken by students majoring in non-science
subjects, who use the course to fulfill their physical sciences general education
requirement.
Astro 1L is a semester-long course that consists of 3-hour weekly lab sessions.
Fall 2009 was the first time that Hartnell College implemented inquiry in Astro 1L. Two inquiry-based activities were included
in the Fall 2009 course: one on properties of lenses (Putnam et al., this volume),
and our activity, in which students investigate transiting
extrasolar planets. Transiting planets are those that cause a periodic dimming in the light from their host star as they pass between the star and us on Earth.
The students learn to generate light curves (plots of the star's brightness over time)
and learn about the properties of the extrasolar planets by interpreting the trends in
the light curves. Some important goals in Astro 1L are for students to gain an understanding of scientific processes, to view themselves as scientists, and to learn to interpret trends in data. The hands-on knowledge gained in the transiting planets activity complements the information about extrasolar planets that is presented in the Introduction to Astronomy companion lecture class.
\section{Activity Goals}
\label{sec:goals}
Here we describe our goals for the activity and rationales for choosing them. We mostly focused on goals that would have broad applicability to students' everyday lives: helping them to be curious, analytical, lifelong learners and helping them to communicate effectively with others. In particular, we aimed to have students:
\begin{small}
\begin{enumerate}
\item devise their own questions about planetary systems and revise initial questions to form investigable ones;
\item construct a light curve (plot of brightness versus time) from their own measurements;
\item deduce relative properties of planetary systems from transit light curves (e.g., the planet's radius and orbital inclination);
\item present their work clearly and coherently; and
\item connect the content of the activity to current transit searches (e.g., the Kepler mission) and see that scientific discoveries are ongoing.
\end{enumerate}
\end{small}
We chose to emphasize the process of asking questions because of how broadly valuable this skill is in life. We wanted to help students feel comfortable asking questions out loud in front of their peers. In the classroom and beyond, asking questions helps students take charge of their own learning: it pushes them to identify specific aspects they do not understand, thereby giving them an avenue toward finding out the answer. Questioning can motivate students to pursue their curiosity, encouraging their ongoing learning about the world around them. Furthermore, questions are the foundation of scientific inquiry and a crucial component of an authentic scientific experience. We hoped that having students devise their own questions to investigate later would help give them ownership of the scientific content.
Our second and third goals above combined scientific content and processes. We wanted students to get the scientific experience of taking their own data based on their own experimental set-up. We then wanted the students to plot the results for several reasons: to help them start to understand the connection between their model and their measurements, to understand why scientists make plots, and to see that their plots are essentially the same type that astronomers make to study actual transiting planet data. The third goal
represents the heart of the scientific content: students have to reason the same way astronomers do to understand the physical mechanism producing the different light curve shapes. These two goals push students to make, interpret, and compare observations, and to connect their results to physical objects and processes in outer space.
Our fourth goal of developing students' presentation skills was chosen to help students deepen their understanding of the science, and for its broad value in life outside the classroom.
Students reinforce their comprehension by organizing their results mentally and visually on a poster, planning how to explain the results, vocalizing those explanations, and responding to questions about their explanations. Learning to communicate ideas effectively can help students engage in and get what they need from their communities, including the one they create in the classroom.
Our final goal was to show students that scientific understanding is not static: that our knowledge is constantly being tested and revised. By connecting the activity to the current Kepler mission \citep{borucki}, we hoped to encourage students to follow real scientific discoveries in the news, and to feel empowered to understand them. Sharing with family and friends would keep reinforcing their understanding about transiting planets.
\section{Activity Timeline}
{\bf Introduction} (25 minutes)
We began by introducing ourselves and verbally reminding students about what it means to participate in an inquiry-based lab, which can be difficult and frustrating but also rewarding. We followed with a short slideshow presentation in which we described why astronomers study other planetary systems, defined a planetary transit and a light curve, showed examples of light curves, and stated that a light curve provides information about a planetary system. We also mentioned the Kepler mission.
\medskip
\noindent {\bf Questioning} (30 minutes)
We began by introducing table-top model planetary systems (built by NM, LS, and AM; see Figure~\ref{fig:model} and Appendix~\ref{app}) and asked the students to spend ten minutes playing with the models and additional materials. This was to familiarize the students with the available materials, and to establish a safe atmosphere for free experimentation and brainstorming. When a few minutes remained, we interjected with suggestions if students seemed to be overlooking particular variables (e.g. orbital inclination). Additionally, we provided a handout containing words like ``transit,'' ``brighter,'' and ``twice as big'' for inspiration if the students felt stuck with devising their own questions.
The next steps followed:
students individually brainstormed on paper their first impressions and ideas about transiting planets;
facilitators described the idea of refining impressions into specific investigable questions, and went through one example;
students individually refined their first impressions;
small groups continued refining, and selected their favorite questions;
groups shared their favorite questions with the class; and
instructors classified the questions as ``Investigable Today'' and ``Not Equipped'' (respectively, questions which could and could not reasonably be addressed with the model planetary systems).
With these steps, we aimed to build up students' abilities to devise specific, investigable questions, and to help students feel ownership of the questions they would ultimately investigate. It was crucial to demonstrate to students that all proposed questions were valuable, regardless of whether they were ultimately eligible for investigation. As students shared their questions with the class, we had an opportunity to assess how well we were achieving Goal \#1.
\begin{figure}[htbp]
\centering
\plotfiddle{mcconnelletal_fig1.eps}{1.5in}{0}{58}{58}{-173}{-173}
\caption{\textit{Left:} Side view of the model orbital system built for the transiting planets activity. All components except for the socket are held together with glue. \textit{Middle:} Photograph of one of the models used at Hartnell. \textit{Right:} Cut-away view of the model orbital system. The light bulb socket is fed through a hole cut into a foam core mount. Before the light bulb is attached, rubber bands are added on either side of the hole to hold the socket in place.}
\label{fig:model}
\end{figure}
\medskip
\noindent {\bf Investigations} (60 minutes)
We asked each student group (2-4 students) to select a question from the ``Investigable Today'' category. Next, we demonstrated how to use a digital light meter to measure brightness during a model transit, and demonstrated plotting those data as a light curve. We then instructed students to use the model planetary system, light meter, and additional materials to investigate the answer to their question.
By eavesdropping on and occasionally questioning students' investigations, we were able to assess their understanding of the relationship between their experimental set-ups and the resultant light curves they plotted (Goals \#2 and \#3). In particular, we wanted students to use the light curves as a tool to gain insight about transit phenomena, instead of regarding the light curves as the final product of their investigation. We looked for students to recognize the connection between unresolved, quantitative outputs of the light meter and a resolved image of the star's surface partially blocked by a planet. One common facilitation technique was to ask a student to replace the light meter with her eye and describe what she saw.
\medskip
\noindent {\bf Presentations} (45 minutes)
Still in their groups, students prepared and gave oral poster presentations on the results of their investigations (Goal \#4). Each group had three minutes to present their poster and findings, and three minutes for questions from the audience. We required each audience group to ask at least one or two questions per presentation (Goal \#1), and offered a list of suggested questions if they needed inspiration. The presentations gave us another opportunity to assess students' understanding of how light curves can be used to deduce properties of planetary systems (Goals \#2 and \#3).
\medskip
\noindent {\bf Synthesis} (10 minutes)
We concluded with a slideshow presentation to recap the activity. We described the ``Thinking Skills'' students had worked on: asking questions (Goal \#1), explaining their ideas to each other (Goal \#4), and designing and carrying out an experiment. We also described how different physical properties of planetary systems lead to different observed light curves, in order to reinforce students' understanding of the data and the physics (Goals \#2 and \#3). We finished by reminding them about current transit searches like the Kepler mission, and pointed them to two recent popular-level science articles on transits (Goal \#5).
\section{Instructor Reflection and Student Feedback}
The activity at Hartnell was overall very successful. Many students met many of our goals, and 90\% of students wrote positive responses on their anonymous post-activity surveys.
The students built their own investigable questions (Goal \#1) quite successfully. We did not ask to see their impressions or first attempts at questions, so unfortunately we could not watch the questions develop, but ultimately each group was able to share at least one relevant, coherent question with the class. Some examples of students' investigable questions included the following and varieties thereof:
\begin{small}
\begin{itemize}
\item ``How can the size of the planet be determined?''
\vspace{-0.05in}
\item ``How does the brightness of the star affect how well we can detect the planet?''
\vspace{-0.05in}
\item ``In what ways does the inclination of a planet affect the transit?'' and
\vspace{-0.05in}
\item ``Can you tell if a planet has a ring?''
\end{itemize}
\end{small}
Students also asked questions for which we were not equipped to support an investigation, like:
\begin{small}
\begin{itemize}
\item ``Does the size of a planet affect its orbital period around the star?'' and
\vspace{-0.05in}
\item ``Does the size of our telescope affect our ability to view the planet?''
\end{itemize}
\end{small}
A majority of groups selected questions related to planet size or orbital inclination, but a sizeable fraction selected other questions related to planetary rings, atmospheres, or reflected light from planets.
The student groups generally worked well together on their investigations. Often, each student had a role in the measurement-taking process (e.g., one moved the model planet, one used the light meter, one recorded the brightness). Most students successfully plotted at least one light curve based on their group's measurements (Goal \#2). Brightness measurements were easy for most, but many had difficulty in using the equipment to measure the phase angle, or in keeping a steady orbital rate to measure even time increments. This could be improved for future labs.
Some students were able to deduce relative planetary properties from light curves (Goal \#3). The most successful tended to be those who studied planetary size. They could generally explain that a larger planet blocks more light and therefore produces a larger dip in the light curve; only rarely, though, did a student turn this around and articulate that astronomers can measure planet sizes from light curves. Groups who studied orbital inclination usually realized that some planetary systems will not show transits to a given observer, and often realized that this fraction of systems is large. The groups studying rings, atmospheres, and reflected light did not tend to demonstrate strongly meaningful results. (The model planetary systems' limited ability to accurately depict these phenomena likely contributed to the students' difficulties in these studies.)
Students' presentations to their classmates showed room for improvement. Most posters did show a few light curves and diagrams of the experimental set-up or face of the star during transit. Yet most students described the various steps and wrong turns that they took, with little aim at providing final results and explanations. Due to time constraints, we did not offer much support for presentation preparation: only a quick skeleton of guidelines and 10-15 minutes to prepare. We successfully elicited questions from each group to ask the presenters, although the audience was often reluctant and relied upon the list of suggested questions we provided. When students asked their own questions, they were usually related to the asker's own experiment (e.g., ``Did you try putting in different sized planets?''), which indicates the asker's ownership of their subtopic.
As mentioned above, 90\% of students wrote something positive on their feedback forms. A large majority used the words ``interesting'' and/or ``fun'' to describe the activity. Students said that they liked working with the model and the light meter, they liked asking their own questions and designing their own experiments, and they liked working in groups. The most common negative comment was that students felt that they did not have enough time during the activity. Students also said that the lab was difficult or frustrating, some wished for more help and guidance from us, and some felt uncomfortable by our ``hovering'' as they worked.
A few sample student comments are:
\begin{small}
\begin{itemize}
\item ``I really liked that we had to think about how were going to answer the question that we chose; even though it is frustrating it feels good to think like that.''
\item ``It was a little difficult to concentrate for me personally because we kept getting checked on and asked what our solution was when we ourselves didn't know at the time.''
\item ``It made us think like real astronomers, build our own question; hypothesis and make our own data.''
\item ``Because it was our own experiment, I felt like I was a scientist.''
\end{itemize}
\end{small}
\section{Comparison of Pre- and Post-Semester Surveys Before and After Inquiry}
In order to assess the effectiveness of introducing inquiry into the laboratory, we conducted a survey of students enrolled in Astro 1L during the Spring 2009 semester (without inquiry) and during the Fall 2009 semester (with inquiry). Adapted from earlier research \citep{chemers}, the survey assessed students' levels of \emph{self-efficacy for science}, \emph{identity as a scientist}, and \emph{commitment to a science career}. In both semesters, students anonymously completed the survey during the first week of classes (pre-semester) and again at the end of the semester (post-semester). We expected students involved in the inquiry-based activities to report greater gains in confidence in their science skills and their interest in pursuing a career in science.
In the \emph{self-efficacy for science} construct, students responded to 13 declarative statements such as ``I am confident I can generate a research question to answer" on a scale from 1 (``not confident at all") to 5 (``absolutely confident"). For \emph{identity as a scientist} (5 items) and \emph{commitment to a science career} (7 items), students responded on a scale from 1 (``strongly disagree") to 5 (``strongly agree"). Initial analyses determined that the constructs had good psychometric properties, with a single factor and high internal consistency. The descriptive statistics are found in Table 1. Independent $t$-tests conducted to compare means show that there was a statistically significant increase in all of the post-semester constructs ($t > 2.60$) except for Fall \emph{commitment to a science career}. This implies that students in general were more confident in their science abilities, identified more as scientists, and were more committed to a science career at the end of both semesters. There is, however, not a statistically significant difference in the means between the two semesters. Given that during the period of this assessment, only two inquiry activities were introduced out of 18 total labs, we cannot reliably deduce the level of effectiveness of adding inquiry to the laboratory curriculum. Furthermore, because these were anonymous surveys, we cannot track changes for individual students. In the future, we will be better able to assess this when more inquiry activities are introduced, the data are analyzed over several semesters, and we track individual students' responses across the semester. We expect to find a subset of students for whom the inquiry laboratories inspire greater interest in science and greater confidence about participating in science.
\begin{table}[!ht]
\label{survey}
\caption{Results of surveys to analyze the effect on students of adding inquiry activities to the course. The `**' indicates a statistically significant difference between pre- and post-semester means. `SD' refers to the standard deviation of student responses, where `n' is the number of student responses included in the statistics.}
\smallskip
\begin{center}
{\small
\begin{tabular}{cccc|cccc}
\tableline
& \multicolumn{3}{c}{\textbf{Pre-semester}} & \multicolumn{3}{c}{\textbf{Post-semester}} & \\
\textbf{Spring 2009 (pre-inquiry)} & Mean & SD & n & Mean & SD & n & Indep \textit{t} \\
\tableline
\noalign{\smallskip}
Self-Efficacy for Science & 3.01 & 0.73 & 62 & 3.51 & 0.72 & 47 & -3.55** \\
Identity as a Scientist & 2.07 & 0.82 & 69 & 2.98 & 0.92 & 63 & -5.98** \\
Commitment to a Science Career & 2.11 & 0.99 & 69 & 2.60 & 1.17 & 63 & -2.66** \\
\tableline
\textbf{Fall 2009 (post-inquiry)} & Mean & SD & n & Mean & SD & n & Indep \textit{t} \\
\tableline
\noalign{\smallskip}
Self-Efficacy for Science & 2.73 & 0.87 & 65 & 3.40 & 0.85 & 56 & -4.26** \\
Identity as a Scientist & 2.12 & 0.92 & 68 & 2.72 & 0.94 & 61 & -3.68**\\
Commitment to a Science Career & 2.14 & 1.05 & 68 & 2.36 & 1.10 & 59 & -1.12\\
\tableline
\end{tabular}
}
\end{center}
\end{table}
\section{Suggestions for Future Implementation}
We designed our activity to meet our goals within the time
constraints. Here, we suggest modifications to the activity for
different time constraints or to achieve different goals.
Teachers may choose to focus the activity more on understanding the
differences between models and the physical systems they represent.
Because it is impossible on a tabletop to accurately represent the
astronomical distances in a solar system, students may be misled by
the scale of the models and, in particular, not realize that transit
searches are necessary because we cannot resolve the planet separately
from the star. To address these concerns, students should explicitly
consider the fact that they are using a model, and could discuss other possible models. These may include scaling for distances: e.g., a light bulb in the classroom (Sun), a marble in the parking lot (Jupiter), and a second light bulb a few thousand miles away (the nearest star). To address relative brightnesses, students can discuss a model in which a bright lightbulb (Sun) is next to a peppercorn (exoplanet). To further emphasize that astronomers
cannot observationally resolve planets, one might add wax paper to one end of the model ``telescopes" (toilet paper
rolls). Students then cannot see the planet but should notice that
the light gets dim periodically, and can then begin a discussion of light
curves. A computer model could also be discussed.
Teachers may choose to focus further on the process of questioning.
The facilitators could spend more time discussing what makes a
question ``investigable", and the students could even categorize
proposed investigation questions rather than the facilitators. To
help students generalize the questioning process to other aspects of their
lives, students could read a short article from a Voter Information
Packet, then discuss questions that they had and how they might find
the answers.
Building presentation skills is another direction to focus. One of
the simplest ways is to give students more time to prepare their
presentations. Another idea is to discuss explicitly what constitutes
a good explanation, giving students a framework to rely on or
a template to follow. Facilitators could also lead a class discussion
about what goes into an effective presentation. It would be helpful
to give students an opportunity to practice their presentations before
getting up in front of the class. If presentation
skills were a focus for an entire semester-long course, students could
likely improve significantly by giving presentations every week. Whether
students give presentations frequently or only once, it is worth giving
each student or group detailed feedback when possible.
A final suggestion is to focus more on awareness of current
scientific research. Reading a
popular science article, discussing it with classmates or as a class,
and presenting a summary of it are all good ways to get students more confident and familiar with talking about current scientific research.
\section{Implementation in a Lecture Course}
Here we describe specific modifications to the transiting planet inquiry for a different teaching environment (performed in Spring 2010): U.C. Santa Cruz's introductory astronomy course for non-majors (Astronomy 2), a large (250 student), lecture-based survey course.
In order to give students practice with scientific process skills, instructor Ryan Montgomery adapted the activity to be completed during one of the 70-minute (required) discussion sections. Two Teaching Assistants (TAs) assisted the students in each discussion section of $\sim$40 students.
The aim was to have students work outside of discussion section to complete segments of the inquiry activity that required little or no facilitation, maximizing the utility of the TAs during section. A website gave a brief introduction to the scientific content (the transit detection method) and showed a series of demonstration videos to familiarize the students with the model planetary systems. The website then asked student groups to complete a pre-lab assignment of generating questions based on the video demonstrations.
The pre-lab group activity was to be completed and turned in by each group to their TA at least 24 hours prior to their discussion section, providing time to sort and electronically post the questions. Before arriving at section, groups were to decide on a question that they wanted to investigate.
The discussion was then used solely for investigation, with a brief ($\sim$10 minute) sharing/synthesis segment at the end of the period. Formal student presentations were cut from this implementation; the goal of having students present their work was met later in the course, when they gave formal presentations as part of a different activity.
Overall this version of the inquiry activity was well received, and met the course content and process goals. Students were asked to rate the amount of content they learned, opportunities to practice the processes of science, their enjoyment, and the activity overall. On a scale from 1 to 5, students consistently rated the activity at 3.6, between ``Fair" (3) and ``Good" (4). By later comparing students' ratings of the inquiry activity with their post-course ratings of their TAs, we conclude that TA support is likely responsible for some of the overall effectiveness of the transiting planet inquiry.
We believe that the modified activity successfully retained the self-direction, ownership, and engagement that make inquiry activities valuable. We strongly encourage other lab and lecture-based courses to consider using an implementation of this transiting planet inquiry activity.
\acknowledgements The authors acknowledge the National Science Foundation Science and
Technology Center funding of the Center for Adaptive Optics, managed by the
University of California, Santa Cruz, under cooperative agreement No.
AST-9876783. This work was funded in part by the National Science Foundation,
through the Course, Curriculum and Laboratory Improvement program
(DUE \#0816754), and supported by the U.C.~Santa Cruz Institute for Scientist \& Engineer Educators. We thank the U.C.~Santa Cruz Physics Department for allowing us to borrow light meters for the activity.
|
{
"timestamp": "2012-06-14T02:03:57",
"yymm": "1009",
"arxiv_id": "1009.3940",
"language": "en",
"url": "https://arxiv.org/abs/1009.3940"
}
|
\section{Introduction}
With the advent of rapid optical follow-up observations of Gamma Ray
Bursts (GRBs) (e.g. Mundell et al. 2010, Rykoff et al. 2009), the confirmed
lack of bright optical flashes from most GRBs challenges a key prediction
of the standard fireball model in which a reverse shock should produce
bright, short-lived optical emission at early time (M\'esz\'aros \&
Rees 1999; Sari \& Piran 1999; Kobayashi 2000).
Although the lack of optical flash could be partially due to
late observations which are not prompt enough to catch early flashes,
it is not trivial how to explain events like GRB 090313 which exhibits
the onset of afterglow without signatures of optical flash.
At early time, reverse shock emission should dominate optical band
and a bright optical peak is expected to be observed when a fireball
starts to be decelerated. However, a distinctive reverse shock component
is detected only in a small fraction of GRBs (Melandri et al. 2008). Several
afterglows show a fattening in the light curves, interpreted as the signature
of the rapid fading of reverse shock combined with the gradual dominance
of forward shock emission (Akerlof et al. 1999; Sari \& Piran 1999).
Afterglow modeling of such flattening cases implies that the magnetic
energy density in a fireball, expressed as a fraction of the equipartition
value of shock energy, is much larger than in the forward shock (but it still
suggests a baryonic jet rather than a Poynting-flux dominated jet: Fan et al. 2002;
Zhang et al. 2003; Kumar \& Panaitescu 2003; Gomboc et al. 2008). Polarization
measurements in a rapid decay phase of GRB 090102 afterglow shows the
existence of large-scale magnetic fields in the revere shock region (Steele
et al. 2009\footnote{Mundell et al. 2007b found no ordered magnetic fields
or a very high magnetic energy density in the ejecta of GRB 060418. More
observations are needed to give a strong conclusion on the nature of the
ejecta (baryonic versus Poynting flux dominated) and the distribution of
magnetization degree.}). The lack of optical flashes in most GRBs may be
due to extreme magnetic field properties, either high magnetic energy densities
that suppress the reverse shock (Gomboc et al. 2008; Mimica et al. 2009) or
very low magnetic energy densities that cause shock energy to be radiated
at higher frequencies than the optical band due to synchrotron self-Compton
processes (Beloborodov 2005; Kobayashi et al 2007; Zou et al 2009).
Alternatively the light curve flattening could be the result of refreshed shocks
and episodes of energy injection (Rees $\&$ M\'esz\'aros 1998, Melandri et al. 2009).
A more conventional model would imply that the reverse shock emits photons at
frequencies much lower than the optical band. Synchrotron emission is
known to be sensitive to the properties of emitter.
Within this framework, which we term the {\em
low-frequency model}, a single peak in the early time optical light
curve is produced when both of the typical synchrotron frequencies of
forward and reverse shock lie below the optical band (Mundell et al. 2007a);
the single peak actually consists of photons equally contributed from
forward and reverse shock, the peak time represents the
the deceleration of a fireball and hence it provides a direct estimate
of the initial Lorentz factor.
In this paper, we discuss the lack of optical flashes in the context of
the low-frequency model. GRB 090313 is a typical case of a burst that
displays a rising and falling light curve, little temporal
structure, no strong spectral evolution and well-monitored
multi-wavelength behavior from early times. Here, we analyse its
multi-wavelength properties, place it into the wider context of GRBs
with single optically peaked light curves and use the characteristics of
the full sample to test the low-frequency model and its predations for
radio light curve evolution.
Throughout the paper we use the following conventions: the power-law
flux density is given as $F(\nu,t)\propto t^{-\alpha} \nu^{-\beta}$,
where $\alpha$ is the temporal decay index and $\beta$ is the spectral
slope; a positive value of $\alpha$ corresponds then to a decrease in
flux, while a negative value indicates an increasing in time of the
observed flux. We assume a standard cosmology with $H_0 =
70$~km~s$^{-1}$~Mpc$^{-1}$, $\Omega_{m} = 0.3$, and $\Omega_{\Lambda}=
0.7$; and all uncertainties are quoted at the $1\sigma$ confidence level
(cl), unless stated otherwise.
\section{Observations}
On 2009 March 13 at 09:06:27 UT (=T0) the Burst Alert Telescope (BAT;
Barthelmy et al. 2005) onboard {\it Swift} triggered on GRB 090313 (Mao et al. 2009a). The
BAT light curves showed a series of multiple peaks with the emission starting before
T0-100 s and a T$_{90}$ in the 15-350 keV band starting at $\sim$ T0-3.9 s for a
total duration of $78 \pm 19$ s (Mao et al. 2009b).
Spectroscopic observations performed with the Gemini South telescope
provided a redshift of z=3.375 for GRB 090313 (Chornock et al. 2009b), later
confirmed by from VLT with FORS (Th\"{o}ene et al. 2009) and X-shooter (de
Ugarte Postigo et al. 2010; who derive a refined redshift value of 3.3736 +/-
0.0004) observations. The estimated redshift for this afterglow confirmed again
that the near object reported by Berger (2009) is indeed too bright to be the host
galaxy of GRB 090313. Most likely this extended object is one of the two absorbing
systems spectroscopically detected (at redshift z=1.96 or z=1.80) along the line
of sight of GRB 090313 (de Ugarte Postigo et al. 2010). Radio observations
performed with the AMI Large Array (Pooley 2009abc), the VLA (Frail $\&$
Chandra 2009) and the WSRT (van der Horst $\&$ Kamble 2009ab) confirmed
the detection and fading nature of the afterglow.
This event displayed an average $\gamma$-ray fluence of $\sim$ 1.4
$\times$ 10$^{-6}$ erg cm$^{-2}$ (Mao et al. 2009b). The redshift of the
burst (correspondent to a luminosity distance of $\sim 2.9 \times
10^{4}$ Mpc) resulted in an isotropic energy estimate of $\sim 3.4
\times 10^{52}$~ergs in the 15--150 keV observed bandpass.
\subsection{{\it Swift}/XRT and {\it Swift}/UVOT data}
Due to Moon distance observing constraints there were no prompt XRT (Burrows et al. 2005) and UVOT (Roming et al. 2005) observations. Follow-up observations of the BAT error circle were possible only after $\sim$ 27 ks, showing a power-law decay in the X-ray (Mao $\&$ Margutti 2009) and a possible marginal detection in the UVOT-v and UVOT-b filters (Schady et al. 2009, Mao et al. 2009b).
\subsection{Optical and Infrared data}
The optical afterglow was discovered by the KAIT telescope (Chornock et
al. 2009a) and later confirmed by the GROND telescope at equatorial
coordinates (J2000) R.A. = 13$^{h}$13$^{m}$36.21$^{s}$; Dec =
+08$^{\circ}$05$^{'}$49.2$^{''}$ (Updike et al. 2009).
The 2-m Faulkes Telescope North (FTN) observed the optical afterglow of
GRB 090313 starting from 168 s after the burst (corresponding to 38 s in
the rest frame). Observations continued up to several weeks after the
burst with FTN, the 2-m Liverpool Telescope (LT) and the 2-m Faulkes
Telescope South (FTS) (see Table \ref{obslog0}). Late time observations
were also performed in order to better correct the entire data set from the
contribution of the nearby object, close to the position of the afterglow.
This object was found to have a constant flux equal to $\sim 1 \%$ of the
peak flux of the optical afterglow, not affecting the shape of the light curve
at early time.
The optical afterglow was observed also with the 1.5m telescope at the
Observatorio de Sierra Nevada (OSN), the 0.8m IAC telescope, the 1.23m
telescope at the Calar Alto Astronomical Observatory (CAHA) and the 0.5m
Mitsume telescope in the optical bands (R and I), plus the 2.5m Nordic
Optical Telescope (NOT) and the 3.5m CAHA telescope in the near infrared
bands (J and K). It was then possible to build the light curve for all
the filters as shown in Fig.\ref{figLC}. A log of the observations is
given in Table \ref{obslog0}, where we report the mid time, integration
time, magnitude and fluxes for all our detections at different
wavelengths. Afterglow detections reported in GCNs are also shown in
Fig.\ref{figLC}.
The optical data were calibrated using a common set of selected
catalogued stars present in the field of view. SDSS catalogued stars
were used for $r'$ and $i'$ filters, while USNO-B1 $R2$ and $I$
magnitudes have been used for the $R$ and $I$ filters respectively. $J$
and $K$ observations were calibrated with respect to the 2MASS
catalog. Next, the calibrated magnitudes were corrected for the Galactic
absorption along the line of sight ($E_{B-V} = 0.028$ mag; Schlegel et
al. 1998); the estimated extinctions in the different filters are $A_R$
$\sim$ $A_{r'}$ = 0.074 mag, $A_I$ $\sim$ $A_{i'}$ = 0.054 mag, $A_J$ =
0.025 mag, $A_H$ = 0.016 mag and $A_K$ = 0.010 mag. Corrected magnitudes
were then converted into flux densities, $F_{\nu}$ (mJy), following
Fukugita et al. (1996). Results are summarized in Table \ref{obslog0}.
\subsection{Radio, mm and sub-mm data}
Continuum observations at 870 $\mu$m were carried out using LABOCA bolometer
array, installed on the Atacama Pathfinder EXperiment (APEX$\footnote{This work is
partially based on observations with the APEX telescope. APEX is a collaboration
between the Max-Plank-Institut f\"ur Radioastronomie, the European Southern Observatory
and the Onsala Space Observatory.}$) telescope. Data were
acquired on 2009 March 17 and 24 during the ESO program 082.F-9850A, under
good weather conditions (zenith opacity values ranged from 0.24 to 0.33 at 870$\mu$m).
Observations were performed using a spiral raster mapping,
providing a fully sampled and homogeneously covered map in an area of diameter
$\simeq$12$'$, centered at the coordinates of the optical afterglow of GRB 090313.
The total on source integration time of the two combined epochs was $\simeq$ 4.6 hours.
Calibration was performed using observations of Saturn as well as CW-Leo, B13134,
G10.62, and G5.89 as secondary calibrators. The absolute
flux calibration uncertainty is estimated to be $\simeq$ 11\%. The telescope pointing
was checked every hour, finding an rms pointing accuracy of 1.8$^{\prime\prime}$. Data
were reduced using the BoA and MiniCRUSH softwares. Finally, the individual maps
were co-added and smoothed to a final angular resolution of ~27.6$^{\prime\prime}$. We
obtained a 3$\sigma$ detection upper limit of 14 mJy for each of the two epochs.
The radio afterglow of GRB 090313 was successfully detected by the AMI
Large Array $\sim$ 2.8 days after the burst (Pooley 2009a) and then
monitored up to $\sim$ 47 days (Pooley 2009bc) as reported in Table
\ref{obslog1}. After an initial upper limit at $\sim$ 1.7 days (van der
Horst $\&$ Kamble 2009a) a detection was reported also by the
Westerbork Synthesis Radio Telescope at $\sim$ 7.6 days (WSRT, van der
Horst $\&$ Kamble 2009b) and by the Very Large Array at $\sim$ 5.9 days
(VLT, Frail $\&$ Chandra 2009). In the mm band the afterglow was
detected with CARMA about one day (Bock et al. 2009) and then monitored
with the Plateau de Bure Interferometer (PdBI) up to $\sim$ 20 days
after the burst event. The radio observations are reported in Table
\ref{obslog1} where the original frequency range of the observation has
been specified.
\section{Results}
\subsection{BAT spectral and temporal analysis}
We re-binned the BAT light curve of GRB 090313 with dt bins of 16.384 s in order
to better appreciate the long faint tail visible up to 500 s after the burst onset. As
reported also by Mao et al. (2009b), the mask-weighted light curve (shown
in Fig. \ref{batlc}) displays a series of multiple peaks extending long
after t=T$_{90}$ at a much fainter level. The time-averaged spectrum is
best fitted by a simple power-law model with a photon index of 1.91
$\pm$ 0.29 (Mao et al. 2009b).
\subsection{Optical/X-ray light curve}
Observations performed with the Faulkes North Telescope, beginning $\sim$
170 s after the burst, showed the optical afterglow rising to a maximum at
$\sim$ 1 ks (Guidorzi et al. 2009). The peak was followed by a decay with windings
and flares (possibly due to the interaction with the circum-burst
material or late time central engine activities). Around $3\times 10^5$ s,
the magnitude became constant in each filter, revealing the
presence of an underlying object at the position of the optical
afterglow. This faint ($r'$ = $21.6 \pm 0.2$ and $i'$ = $21.1 \pm 0.2$) and
apparently extended object is only 2.3" away from the optical afterglow
as reported by Berger (2009). It was not possible to separate the
contributions from the two objects in the late-time co-added
observations.
We model the optical light curve with a broken power-law (to fit the
peak up to $\sim 10^{4}$ s) plus an additional component to model the bumps
visible after $\sim 1.4 \times 10^{4}$ s and a constant flux to model
the behavior at late times. The fit to the component representing the
optical peak at early time gives: $\alpha_{\rm rise} = -1.72 \pm 0.41$,
$\alpha_{\rm decay} = 1.25 \pm 0.08$ and t$_{\rm peak} = 1060.9 \pm
153.6$ s. For completeness the parameters of the component modeling the
sharp bump around $\sim 10^{4}$ s are: $\alpha_{\rm r,bump} = -83.8 \pm
8.4$, $\alpha_{\rm d,bump} = 3.0 \pm 0.8$ and t$_{\rm peak,bump} =
(14.0 \pm 0.3) \times 10^{3}$ s, (t/dt)$_{\rm peak} \sim 1$
($\chi^{2}$/dof = 769.4/77 $\sim$ 9.9). The high $\chi^{2}_{\rm red}$
for the optical fit is clearly driven by the uncertainty of the bump fit
and the variability of the data around $\sim 10^{5}$ s. However this
does not affect the goodness of the fit for the smooth early time
behavior, where the peak (rise and fall) is well constrained with
negligible variability as shown in Fig. \ref{figLC2}.
Our independent analysis shows that the X-ray light curve of GRB 090313
is well fitted by a simple broken power-law with $\alpha_{\rm 1} = 0.83
\pm 0.49$, $\alpha_{\rm 2} = 2.56 \pm 0.46$ and $t_{\rm break} \sim 9
\times 10^{4}$ s ($\chi^{2}$/dof = 43.17/43 $\sim 1.0$). The estimated values
for $\alpha_{\rm 1}$ and $\alpha_{\rm 2}$ could be the result of flares activity,
and the subsequent cessation, in the early XRT data. The X-ray light
curve and its fit are shown in Fig. \ref{figLC2} together with the
composite optical/infrared light curve. As we will explain in Section 3.4, the latter
has been built by re-scaling all the filters with respect to the
SDSS $i'$ band. On the bottom panel of this figure we show the
no-evolution of the optical spectral index $\beta_{\rm O}$ as derived from the
fit of the spectral energy distribution.
\subsection{X-ray spectral analysis}
The X-ray spectrum (Fig. \ref{figXrayspec}; from the {\it Swift}-XRT repository, Evans et al. 2007) can be fitted by an absorbed simple power law with a photon index $\Gamma_X = 2.14^{+0.12}_{-0.14}$ and an absorbing column density $N_{\rm H} = (2.99^{+0.77}_{-0.71}) \times 10^{22}$ cm$^{-2}$, in excess of the Galactic value of $2.1 \times 10^{20}$ cm$^{-2}$.
\subsection{Spectral energy distribution}
From our data and others published in GCNs we
estimate the flux for the infrared ($JHK$) and optical ($i'r'$) filters
at four different epochs (corresponding to T0+100 s, T0+600 s,
T0+2$\times 10^{3}$ s and T0+1.6$\times 10^{4}$ s in the rest frame of
the burst). At the redshift of the burst (z=3.374) the wavelength of the Lyman-alpha break
(121.6 nm) is redshifted to 532 nm, that corresponds roughly to the
central peak wavelength of the $V$ filter. However also the tail of the
$R$ filter could be affected by the absorption and for that reason we
decided to perform the fit of the optical spectral energy distribution
only up to 2$\times 10^{15}$ Hz. The results of the fit are shown in
Fig. $\ref{figSED}$ and reported in Table \ref{tabsed}. The afterglow
of GRB 090313 did not display any spectral evolution before and after
the peak in the light curve. Only a slight and insignificant change of
the spectral parameter $\beta_{\rm O}$ is recorded around 3ks (observed
frame) after the break. For this reason we built a composite
optical/infrared light curve (fixing the value of $\beta_{\rm O} = 1.2$)
using rigid shifts for each filter to report all the fluxes relative to the SDSS-$i$ band.
\section{Discussion}
Here we examine the properties of 19 GRBs including GRB 090313 that
exhibit a single-peaked optical light curve. Those are all the GRBs with published data that
show a clear rise and fall of their optical light curves. The observed and derived
properties of the sample are given in Table \ref{tabprop}. In this table we
report the parameters of the optical peak ($\alpha_{\rm rise}$, $\alpha_{\rm decay}$,
t$_{\rm peak}$ and $F_p$), together with the X-ray decay index
($\alpha_{\rm X}$) in the post optical peak phase \footnote{~the value of $\alpha_{\rm X}$ is taken from the
literature or from the XRT light curve repository (Evans et al. 2007).}, the duration (T$_{90}$),
redshift ({\it z}), initial Lorentz factor $\Gamma$
and isotropic energy (E$_{\rm iso}$) for each burst. We have assumed that the
optical peak time represents the fireball deceleration time. Following equation 1 in Molinari et al. 2007, the initial Lorentz factor
of GRB 090313 is give by
\begin{equation}
\Gamma \approx 80~n^{-1/8}
\left(\frac{E_{\rm iso}}{3.2 \times 10^{52}~{\rm erg}}\right)^{1/8}
\left(\frac{1+z}{4.375}\right)^{3/8}
\left(\frac{t_{\rm peak}}{1060~{\rm s}}\right)^{-3/8}
\end{equation}
where $n$ is the ambient density in protons/cm$^3$. For all the bursts in Table \ref{tabprop}
the ISM environment is favored in literature (i.e. Klotz et al 2008, Rykoff et al. 2009, Oates et
al. 2009, Melandri et al. 2009, Greiner et al. 2009); only GRB 080330 is better explained by a
wind-like medium (Guidorzi et al. 2009). For the wind medium $\rho=A R^{-2}$, the equation 1
is replaced by $\Gamma \sim 25 ~ (A/5 \times 10^{11} ~{\rm g~cm^{-1}})^{-1/4} (E/3.2 \times 10^{52}~{\rm erg})^{1/4}
[(1+z)/4.375]^{1/4} (t_{\rm peak}/1060 ~{\rm s})^{-1/4}$.
It is well accepted that the X-ray temporal decay of the majority of GRB
afterglow can be described by a canonical light curve, where the initial
X-ray emission (steep decay) is consistent with the tail of the gamma-ray
emission, followed by a shallow phase that leads into a power-law decay
phase (Nousek et al. 2006, Zhang et al. 2006, Tagliaferri et al. 2005,
O'Brien et al. 2006). In our sample also, no peaks are detected in
the X-ray light curves, all the X-ray light curves monotonically decay
from the beginning of the X-ray observations except X-ray flares.
It is known that about 50$\%$ of GRBs show flaring activities on top of the
canonical light curve. The narrow structure $\Delta t/ t <1$ indicates that it
originates from a physically distinct emitting region (e.g. late internal shocks).
X-ray observations started before an optical peak for GRB 990123, GRB 050730,
GRB 050820A, GRB 060418, GRB 060605, GRB 060607A, GRB 060904B,
GRB 070419A, GRB 074020, GRB 071031, XRF 080330 and GRB 080810, while
it started after an optical peak for GRB 061007, GRB 080603A, GRB 080129,
GRB 080710 and GRB 090313. We have no X-ray observations for XRF 020903
and XRF 030418. If an optical peak is due to the deceleration of a fireball,
X-ray emission from external shocks also should peak simultaneously.
The tail of the prompt emission or a different emission component
might mask the X-ray peak.
Four events: GRB 060418, GRB 060605, GRB 060607A and GRB 060904B
show X-ray flares around an optical peak, we tested whether the observed
optical peaks could be explained by the flare emission alone by
extrapolating the peak flux of the X-ray flare to the optical band
assuming a spectral index between the two bands of $\beta \sim 1$. In
all cases, the contribution of the X-ray flare to the optical light curve was
significantly lower than that observed, ruling out a flare origin for the
optical peaks.
\subsection{The origin of the optical peak}
Recent results on the naked eye optical flash from GRB 080319B (Racusin
et al. 2008; Bloom et al. 2009), where the observed optical peak coincided
in time with the prompt gamma-ray emission, provided motivation to consider
that the prompt gamma-ray emission is Inverse Compton (IC) of the optical
flash. The dominance of IC cooling could lead to the lack of prompt optical
flashes.\footnote{The full discussion on IC cooling effects (e.g. Nakar, Ando $\&$
Sari 2009) is beyond the scope of this paper. We here give a rough estimate on
how much $\epsilon_B$ would be necessary to suppress an optical flash. We
assume that the typical frequency of the reverse shock is in the optical band,
and that the shock emissions are in the fast cooling regime. If the IC cooling
is not important, the luminosities would peak at the typical synchrotron
frequencies, and the luminosities would be comparable at the onset of afterglow.
The flux ratio is about $\Gamma$ in the optical band (Kobayashi $\&$ Zhang 2003).
If the IC cooling is the dominant cooling mechanism of the electrons in the shock
regions, the bulk of the shock energy is radiated in high energy radiation
(possibly the 1st scattering component for the forward shock and the 2nd
scattering component for the reverse shock). The optical flux ratio could be
reduced roughly by a factor of $(\epsilon_e/\epsilon_B)^{1/6}$ (Kobayashi
et al. 2007). A very small $\epsilon_B \sim \epsilon_e/\Gamma^6$ is required
to explain the lack of optical flashes. In the slow cooling regime, the Compton
parameter is smaller for a given ratio $\epsilon_e/\epsilon_B$, the required
$\epsilon_B$ could be even smaller.} However, the basic problem of such IC
model is that if the low-energy seed emission is in the optical, while the
observed soft gamma-ray spectrum is the first IC component, then second
IC scattering would create a TeV component. The second IC component in the
TeV range should carry much more energy than the soft gamma-ray components.
This could cause an energy crisis problem, possibly violating upper limits from
EGRET and Fermi (Piran et al. 2009).
Rykoff et al. (2004) suggested a model in which single-peaked light
curves are caused by GRB radiation emerging from a wind medium
surrounding a massive progenitor. This model suggests that the rise of the
afterglow observed in the optical band can be ascribed to extinction
and the emission can be modeled with an attenuated power-law. A
consequence of this model is that at very early times some afterglows
will rise very steeply and the extinction observed in the optical band
should be much greater that in the infrared band. As shown in
Fig.\ref{figp1} we see a very steep rise only for GRB 061007, however
for this burst as for the other bursts on that figure, we do not have data to
model the peak in the infrared band. If we fit the afterglow peak of GRB 090313
with an attenuated power-law function (equation 1 in Rykoff et al. 2004) we
find values of the decay index and the attenuation time scale ($\alpha$ = 1.15
$\pm$ 0.03 and $\beta_{\rm t}$ = 1097 $\pm$ 117 s) consistent with the decay
index $\alpha$ obtained in section 3.2. With this $\beta_{\rm t}$ we derive
a mass loss rate ($\sim 10^{-3}$~M$_{\odot}$~yr$^{-1}$) which is slightly
higher than what is usually suggested for GRB progenitors. The Lorentz factor
that we assumed for this estimate is obtained from the peak time based on the
wind model; $\Gamma$ based on the ISM model is higher and it would results
in a higher mass loss rate. This is a similar result to the one found by Rykoff et al.
for GRB 030418. As the majority of the GRBs in our sample rise slowly or with
comparable $\alpha_{\rm rise}$ with respect GRB 090313 this will imply a higher
mass loss rate for all those bursts. This model will be further tested with future
simultaneous optical/IR light curves obtained at early time.
If the observed peak is due to the passage of the typical frequency of
the forward shock through the optical band, we would expect much slower
rise ($\alpha_{\rm rise} \sim$ -0.5) and strong color evolution around
the peak. These are not consistent with GRB 090313 observations
($\alpha_{\rm rise} \sim$ -1.7 and no color evolution). If the optical peak
is due to the deceleration of a fireball,
the typical frequency of the forward shock $\nu_{\rm m,fs}$ should be
below the optical band at the onset, otherwise, the forward shock
emission slowly rises until the typical frequency crosses the optical
band. Actually when this condition:
$\nu_{\rm m, fs} (t_{\rm peak }) < \nu_{\rm optical}$ is satisfied,
the forward and reverse shock emission peak at the same time, and
produce a single peak (Mundell et al. 2007a). We here consider such a
low-frequency model in detail.
The onset of the afterglow is expected to occur immediately after the prompt
emission if the reverse shock is in the thick shell regime, while there
should be a gap between the prompt gamma-ray emission and the onset if
the reverse shock is in the thin shell regime (Sari 1997). At the onset
of afterglow, the forward and reverse shock emission rise as $F\propto
t^{3}$ and $t^{3p-3/2}$, respectively in the thin shell case,
while they are as shallower as $t^{(3-p)/2}$ and $t^{1/2}$ for the thick
shell case. If the two emission components are comparable at the onset,
the rising index could be determined by the shallower component. The rising
index is expected to be $t^3$ for the thin shell case, and $t^{1/2}$ or shallower for
the thick shell case.
As we will discuss, most optical afterglows are classified into
the thin shell case. The fireball deceleration time is given by
$t_{peak} \sim 90~(1+z) E_{52}^{1/3}n^{-1/3}\Gamma_2^{-8/3}$ s
where we have scaled parameters as $E_{52}=E_{\rm iso}/10^{52}$ergs and
$\Gamma_2=\Gamma/100$. At the peak time $t_{\rm peak}$, the cooling
frequency and the typical frequencies of the forward and reverse shock
emission are given (Sari et al. 1998, Kobayashi $\&$ Zhang 2003) by
\begin{eqnarray}
\nu_{c} &\sim& 2.6\times 10^{18}
~(1+z)^{-1}\epsilon_{B,-3}^{-3/2} E_{52}^{-2/3}n^{-5/6}\Gamma_2^{4/3} ~{\rm Hz},\\
\nu_{\rm m, fs} &\sim& 5.4 \times 10^{13}
~(1+z)^{-1}\epsilon_{e,-2}^{2} \epsilon_{B,-3}^{1/2}
n^{1/2}\Gamma_2^{4} ~{\rm Hz},\\
\nu_{\rm m, rs} &\sim& 5.4 \times 10^{9}
~(1+z)^{-1}\epsilon_{e,-2}^{2} \epsilon_{B,-3}^{1/2}
n^{1/2}\Gamma_2^{2} ~{\rm Hz},
\end{eqnarray}
where $\epsilon_{e,-2}=\epsilon_{e}/10^{-2}$ and
$\epsilon_{B,-3}=\epsilon_{B}/10^{-3}$. For plausible parameters,
the typical frequency of the forward shock is actually below optical band
and the both shock emission is in the slow cooling regime.
The low typical frequencies provide an upper limit to the microscopic
parameter $\epsilon_e$. Requiring that the typical frequency of the
forward shock is below the optical band at the onset of afterglow, we obtain
\begin{equation}
\epsilon_e \leq 0.30 ~
\left(\frac{\epsilon_B}{0.003}\right)^{-1/4} ~
(1+z)^{-1/4}~
\left(\frac{t_{\rm peak}}{30 ~{\rm min}}\right)^{3/4} ~
\left(\frac{E_{\rm iso}}{10^{52} ~{\rm ergs}}\right)^{-1/4}
\left(\frac{\nu_{\rm opt}}{10^{15} ~{\rm Hz}}\right)^{1/2}
\end{equation}
The estimated values for the upper limit of $\epsilon_e$ for the GRBs in our
sample are reported in table \ref{tabprop}. The spread of the values of $\epsilon_B$ is large (from $\sim 10^{-4}$ to $\sim 10^{-1}$) and this could still be a significant uncertainty in the upper limits estimates, even if $\epsilon_e$ do not strongly depend from that parameter. GRB 020903 does not constrain
$\epsilon_e$ well, the typical upper limit is $\sim 0.08$, consistent with values
from later afterglow modeling (e.g. Panaitescu $\&$ Kumar 2002).
A highly magnetized fireball is another possibility to explain the lack of optical
flashes.\footnote{The reverse shock emission might be suppressed for high
magnetization: $\sigma = B^2/4\pi \rho c^2 \sim 0.1$ or larger where $B$ and $\rho$
are the rest-frame magnetic field strength and density, respectively (Mimica et al. 2009).
Assuming a mildly relativistic reverse shock, the critical magnetization could
correspond to $\epsilon_B \sim 0.1$.} However, Granot et al. (2010) recently argued
that in the thin shell case the magnetization of the GRB outflow at the deceleration
time is not high enough to suppress the reverse shock.
Most events in our sample are classified into the thin shell case. Even if
the reverse shock is suppressed by high magnetization, the same condition
$\nu_{\rm m,fs}(t_{\rm peak}) < \nu_{\rm optical}$ could be required to
avoid slowly rising forward shock emission after the onset of afterglow.
\subsection{Reverse and Forward Shocks: Relative Contributions}
In Fig. \ref{figp1} we plot the light curve rise index ($\alpha_{\rm
rise}$) against the time of the peak in the GRB rest frame (left panel)
and the ratio $t_{\rm peak}$/T$_{90}$ (right panel). In a recent work
Panaitescu $\&$ Vestrand (2008) classified the optical light curves into
'fast-rising with an early peak' and 'slow-rising with a late peak'. In our sample
an apparent weak anti-correlation can be seen between $\alpha_{\rm rise}$
and $t_{\rm peak}$; however the significance is very low ($\sim 12\%$) not
allowing any firm conclusion about the existence of this anti-correlation.
A simple fireball model predicts that the dynamics of a fireball is classified
into two cases: (1) thin shell fireballs (t$_{\rm peak} >$ T$_{90}$) produce a sharp
peak with rising index $\alpha_{\rm rise} \sim -3$; (2) thick shell fireballs ($t_{\rm peak}
\sim $T$_{90}$) have a wider peak with $\alpha_{\rm rise} \sim -1/2$ (where this value is a limit and the rise could be much shallower).
As shown in Fig. \ref{figp1} (right panel), most GRBs in the sample are classified
into the thin shell case, and the rising indexes are consistent with the
simple model or shallower. The simple reverse shock model assumes a
homogeneous fireball. However, as internal shock process requires, the
initial fireball could be highly irregular. The complex structure of
shell or energy injection in the post-prompt phase could make
the rising index shallower.
In Fig. \ref{figp1}, GRB 061007 stands out as a notable exception with
Rykoff et al. (2009) quoting a peculiarly steep rising index
($\alpha_{\rm rise} \sim$ -9). Mundell et al. (2007a)
showed that the afterglow is detected from gamma to optical wavelength, beginning
during the prompt emission as early as 70~s post-trigger. The softening of the
gamma-ray spectral index after 70~s further confirms the afterglow onset at this time
(Mundell et al. 2007a; Rykoff et al. 2009). The gamma ray light curve is dominated by a
multi-peaked flare between T=20 and 70~s, coincident with the steepest rising part of
the optical light curve and possible double optical peak. If the optical emission during
these prompt gamma-ray flares comprises a rising afterglow component with a
contemporaneous prompt (flaring) component superimposed, the underlying
afterglow rising index would be much shallower than the observed value.
In our small sample, the optical afterglow of XRFs tend to rise slowly
with a late peak. If we ignore XRFs and the peculiar case of GRB 061007
in fig \ref{figp1}, the anti-correlation between the rising index and
peak time is very weak or it might not exist.
GRB 990123 has a clear reverse shock component in early optical afterglow. Our low-frequency
model is not suitable to discuss this event, because it is considered to explain the lack of optical
flash. On the other hand, the simple reverse shock model still predicts that the rising index is $\sim 1/2$
for the reverse shock dominant thick shell case. We plot GRB 990123 in Fig. \ref{figp1} also to test
the simple model. The discrepancy might be due to the irregularity of the fireball.
An interesting comparison can be done with the peaks detected in the
high energy band by the {\it Fermi}/LAT. Ghisellini et
al. (2010) studied the emission observed at energies $>0.1$ GeV of
11 GRBs detected by the Fermi. They argue that the observed high
energy flux can be interpreted as afterglow emission shortly following
the start of the prompt emission. Most events show the onset of
afterglow during the prompt gamma-ray phase. This is quite a contrast
to what we have seen in our sample. The reason for this difference
might be that Fermi events tend to have very high Lorentz factors,
which allow to emit high energy photons without pair attenuation,
and the events are classified into the thick shell. The peak time
should be early and comparable to the duration of the prompt emission.
On the other hand, our sample (early optical observations) might be biased
towards the thin shell case, because early peaks are technically
difficult to catch.
\subsection{Radio Afterglow Modeling}
In the low frequency model, the characteristics of an optical peak: the peak time
$t_{\rm peak}$ and the peak flux $F_p$ can be used to predict the behavior of
early radio afterglow. At the onset of afterglow (peak time), the typical frequencies
and spectral peaks of reverse and forward shock are related as $\nu_{\rm m,rs}\sim\Gamma^{-2}\nu_{\rm m,fs}$
and $F_{\rm max,rs} \sim \Gamma F_{\rm max,fs}$, respectively (Kobayashi
\& Zhang 2003). Note that $F_p$ is a peak in the time domain, while $F_{max}$ is
a peak in the spectral domain. To produce bright forward shock emission,
$\nu_{\rm m, fs}$ should be close to optical band and we get $\nu_{\rm m,rs}\sim \Gamma^{-2}\nu_{\rm
opt}$ and $F_{\rm max,rs} \sim \Gamma F_p$. After the original fireball deceleration,
the typical frequency and spectral peak behave as $\nu_{\rm m,rs} \sim t^{-3/2}$
and $F_{\rm max,rs} \sim t^{-1}$. The typical frequency comes to the radio band at
$t\sim \chi^{2/3}(\nu_{\rm opt}/\nu_{\rm radio})^{2/3}
\Gamma^{-4/3} t_{\rm peak} \sim 2 \times10^3 \Gamma^{-4/3} t_{\rm peak}$
and the flux at that time is $F\sim \chi^{-(3p+1)/6} (\nu_{\rm opt}/\nu_{\rm radio})^{-2/3}\Gamma^{7/3}F_p
\sim 5 \times10^{-4}\Gamma^{7/3}F_p$ where $\chi=\nu_{\rm m,fs}/\nu_{\rm opt} < 1$
is a correction factor when $\nu_{\rm m,fs}$ is well below
the optical band and $F_p = \chi^{(p-1)/2}F_{\rm max,fs}$, in principle, $\chi$ could be determined from radio
observations, $\Gamma$ is estimated from the peak time $t_{\rm
peak}$ as shown in table \ref{tabprop}. In eq (5), the upper limit corresponds
to the case of $\chi = 1$. If $\chi$ is obtained from radio observations, the
right-hand side of the inequality with a correction factor of $\chi^{1/2}$
gives the value of $\epsilon_e$.
At low frequencies and early times, self-absorption takes an important
role and significantly reduces the flux. A simple estimate of the maximal
flux is the emission from the black body with the reverse shock
temperature (Sari \& Piran 1999; Kobayashi \& Sari 2000). The black body
flux at the peak time is
\begin{equation}
F_{\nu,BB} \sim \pi (1+z)\nu^2 \epsilon_e m_p \Gamma \left(\frac{R_\bot}{D_L}\right)^2
\end{equation}
where $R_\bot \sim 2 \Gamma ct_{\rm peak}$ is the observed size of the
fireball. This limit initially increases as $\sim t^{1/2}$, and then steepen
as $\sim t^{5/4}$ after $\nu_{\rm m,rs}$ crosses the observation frequency $\nu$.
In Fig. \ref{figradio} the dashed lines indicate the black body flux limit. Once
the reverse shock emission becomes dimmer than the limit, the flux decays
as $\sim t^{-(3p+1)/4}$. The combination of the increasing limit and decaying
flux shapes ``radio flare'' (Kulkarni et al. 1999).
The forward shock emission (thin solid)
evolves as t$^{1/2}$ before the passage of $\nu_{\rm m,fs}$ through the
radio band, and then decays as $t^{-3(p-1)/4}$. The forward shock peak
$F\sim \chi^{-(p-1)/2} F_p$ should happen around $t\sim \chi^{2/3}
(\nu_{\rm opt}/\nu_{\rm radio})^{2/3} t_{\rm peak} \sim 2 \times10^3 t_{\rm peak}$ s.
In the case of GRB 090313, assuming $\chi \sim 1$, the forward shock
peaks in the optical band $\nu_{\rm m,fs}$ $\sim$ $4.6 \times 10^{14}$ Hz with a flux density
F$_{\rm max,fs} \sim$ 2 mJy and a peak time corresponding to
$\Gamma \sim$100; therefore the reverse-shock peak flux at this
time occurs at $\nu_{\rm m,rs}$ (t$_{\rm peak}$) $\sim$ 46 $\times
10^{9}$ Hz and is F$_{\rm max,rs} \sim$ 200 mJy. Correcting for
synchrotron self-absorption, results in an observable flux density of
$\sim$ 4 mJy after 2.4 hours. After the deceleration time the
reverse-shock emission in the radio band decays as
$\sim t^{-2}$ (dot-dashed line
Fig. \ref{figradio}) and the emission at 1 day is about $\sim$ 20
$\mu$Jy for GRB~090313.
For GRB 090313, the forward shock emission is expected to peak in the
radio band around 12 days after the burst (assuming $\nu_{radio} =
1.5 \times 10^{10}$ Hz), with peak flux of 2 mJy
(solid line Fig. \ref{figradio}). Taking all these factors into
account, the resultant expected light curve of the radio afterglow of
GRB 090313 is shown in Fig. \ref{figradio}. The expected 15 GHz
and 100 GHz light curves (thick lines) are reasonably consistent with the
observations. The deviation of the 15 GHz estimates from the
observations might be partially due to a simplified synchrotron spectrum
which is described by a broken power law. The deviation of the 100 GHz
point around 1 day is much more apparent and might be due to an additional
emission component (e.g. late time central engine activity).
Since a realistic synchrotron spectrum is rounded at the break
frequencies, a more accurate estimate should give a light curve
rounded at the peak time. However, if this is the case, our simple model
further underestimates the 100 GHz flux. It is interesting that
the 15 GHz flux decays very slowly up to few ten days while X-ray
afterglow displayed a steep decay around $\sim$ 1 day, as shown in
Section 3.2. This might indicate different origins (e.g. emission regions) for
the two; the $\delta\alpha = \alpha_{2} - \alpha_{radio} > 2$ is indeed too
large to be explained assuming that the cooling frequency lies at the
X-ray frequencies at that time.
In Fig \ref{figradio2}, we show radio light curves expected for
our sample, which are evaluated by using early optical observations.
GRB 990123, XRF 020903, GRB 030418, GRB 060607A, GRB 070420
and GRB 080810 are excluded in the radio afterglow estimates. Since
GRB 990123 clearly shows a reverse shock component in the early
afterglow, it is not consistent with our model assumption. For the other
five events, the optical peak time or peak flux was not well constrained.
In future, we should be able to estimate radio afterglow light curves in real
time as soon as a single peaked optical light curve is detected. Depending
on the Lorentz factor at the time of the peak and on the energetics of the
burst, the shape of the radio will slightly change, displaying an early
peak/flash at $\sim$ 0.1 days and later on the peak of the forward shock
in the radio band peaking at about 2-10 days after the burst. Diffractive
scintillation might make the detection of radio flares difficult if the amplitude
of flares are order of unit. In cases similar to GRB~061007, in which the
optical forward and radio reverse shocks peak at early time and the
forward shock flux is large, the radio peak due to the the passage of
the forward shock typical frequency is expected to be very bright. Liang et al. (2010)
suggest a correlation, such that $\Gamma \propto E_{\rm iso}^{2/7}$,
therefore Fig. \ref{figradio2} can also be viewed in terms of increasing
E$_{\rm iso}$. The scatter in this correlation, however results in an
over-prediction of the the initial Lorentz factor for GRB 090313 ($\Gamma \sim$ 130) compared
with the value calculated directly from the light curves.
\section{Conclusions}
We have analysed multiwavelength observations of GRB~090313 and
similar 18 GRBs which exhibit a single-peaked optical light curve. We have
compared prompt and afterglow properties to test the standard
fireball model with amended microphysics parameters. The goal of the
study was to understand the origin of single optical peaks in
afterglow light curves and to explain the surprising lack of bright
optical flashes from reverse shocks that were predicted from
the standard fireball model.
Within this amended standard model, which
we term the {\em low-frequency model}, a single peak in the early
time optical light curve is produced when the typical synchrotron
frequencies of shock emission lie below the optical band.
We have shown that this condition is satisfied with plausible
microphysics parameter $\epsilon_e$; the single peak
consists of forward and reverse shock emission components, the
peak time represents the initial deceleration of the fireball at
the onset of the afterglow and the reverse shock emits most
photons at frequencies below the optical band. We find that:
\begin{itemize}
\item In the case of GRB~090313, no spectral evolution was observed at
the time of the optical peak,
the peak is considered to represent the onset of the GRB
afterglow (or fireball deceleration) and the initial Lorentz
factor of the ejecta was derived $\Gamma \sim 80$. The
Lorentz factors that were similarly derived for the other GRBs and
XRFs in the sample cover a wide range $40 < \Gamma <
450$.
\item The rising indexes of most optical light-curves
are consistent or shallower than the value of $F\sim t^3$
expected in the standard model. Although a simple reverse
shock model assumes a homogeneous fireball, the internal shock
model requires a highly irregular fireball. At the end of
the prompt gamma-ray phase, the fireball might still have
an irregular structure. The irregularity in the density
distribution or energy injection in post-prompt gamma-ray phase
could make the rising index shallower than the expected value.
In the small sample, the optical
afterglow of XRFs tend to rise slowly with a late peak.
\item We constrained the value of $\epsilon_e$ for the single-peak
events, found an average value of $< 0.08$ for the whole
sample. The values derived from early time light curve properties
are consistent with published values derived from late-time
afterglow modeling. However the large spread of values for $\epsilon_B$ could
affect the estimates of the upper limit for $\epsilon_e$.
\item Using the observed optical properties for our sample of GRBs, we
predicted the radio afterglow light curves for the low-frequency
model. Synchrotron self-absorption is important at early times in
shaping the radio light curve and masking the reverse shock
emission. This could result in an early detectable peak around
$\sim$ 0.1 days, though prompt radio observations might be
challenging. The forward shock peaks later around 2-10 days
after the burst. It is important to note that high energies and
Lorentz factors (as in the case of GRB 061007) could produce
bright optical and radio afterglows. We demonstrate the effectiveness
of this method in the case of GRB 090313. This is important for future
observations of GRBs afterglow in the radio band with new
facilities such as the EVLA, ALMA and LOFAR. The latter will have a very large
field of view, and prompt radio observations could be possible. However,
LOFAR will operate at low frequencies (below 250 MHz) and since synchrotron
self-absorption limit F$_{\nu,BB} \sim \nu^{2}$ is much lower, it could be
still difficult to catch prompt optical flares. Current radio sensitivities of ~50
$\mu$Jy are already adequate for detecting reverse and forward shock peaks
but with predicted sensitivities as low as 2.3 $\mu$Jy in a 2-hour integration
(Chandra et al. 2010) all radio light curves in our sample would be easily
observed from early to late time with instruments such as the EVLA and ALMA.
\end{itemize}
\section*{Acknowledgments}
We thank the anonymous referee for valuable comments and suggestions that improved
the paper. AM acknowledges funding from the Particle Physics and Astronomy Research
Council (PPARC). CGM is grateful for financial support from the Royal Society
and Research Councils (UK). AG acknowledges founding from the Slovenian
Research Agency and from the Centre of Excellence for Space Sciences and
Technologies SPACE-SI, an operation partly financed by the European Union,
European Regional Development Fund and Republic of Slovenia, Ministry of
Higher Education, Science and Technology. IdG is partially supported by
Ministerio de Ciencia e Innovaci\'on (Spain), grant AYA2008-06189-C03
(including FEDER funds), and by Consejer\'{i}a de Innovaci\'on, Ciencia
y Empresa of Junta de Andaluc\'{i}a (Spain).The Liverpool Telescope is
operated by Liverpool John Moores University at the Observatorio del
Roque de los Muchachos of the Instituto de Astrofisica de Canarias. The
Faulkes Telescopes, now owned by Las Cumbres Observatory, are operated
with support from the Dill Faulkes Educational Trust. This work is
partially based on observations carried out with the IRAM Plateau de Bure
Interferometer and observations collected at the German-Spanish Astronomical
Center, Calar Alto, jointly operated by the Max-Planck-Institut f\"ur Astronomie
Heidelberg and the Instituto de Astrof\'{i}sica de Andaluc\'{i}a (CSIC). IRAM is
supported by INSU/CNRS (France), MPG (Germany) and IGN (Spain). We
thank Calar Alto Observatory for allocation of director's discretionary time to this
program. We also would like to thank M.R Zapatero-Osorio for the acquisition and
reduction of NOT data.The research of JG and AJCT is suppported by the
Spanish programmes AYA2007-63677, AYA2008-03467/ESP and
AYA2009-14000-C03-01. This work made use of data supplied by the UK {\it
Swift} Science Data Centre at the University of Leicester.
|
{
"timestamp": "2010-09-24T02:01:15",
"yymm": "1009",
"arxiv_id": "1009.4361",
"language": "en",
"url": "https://arxiv.org/abs/1009.4361"
}
|
\section{Introduction}
Weakly bound states represent an interesting
field of research in atomic and molecular physics. The behavior of systems
near the binding threshold
is important in the study of ionization of atoms and molecules,
molecule dissociation and scattering collisions.
Moreover, the stability of atomic and molecular systems in external
electric, magnetic and laser fields
is of
fundamental importance in atomic and molecular physics and
has attracted considerable
experimental and theoretical attention over the past
decades\cite{science,magnetic,
Laser0,qiwei}.
A superintense laser field can change the nature of atomic and molecular
systems and their anions; the stabilization in superstrong
fields is accompanied by splitting of the electron distribution
into distinct lobes, with locations governed by the quiver
amplitude and polarization of the laser field. This localization
markedly alters electron-nucleus interactions as well as
reduces electron-electron repulsions and hence suppresses
autoionization. In molecules, it can also enhance chemical
bonding. This localization markedly
reduces the ionization probability and can enhance chemical bonding when the laser strength
becomes sufficiently strong and can give rise to new stable multiply charged
negative ions such as H$^{--}$, He$^-$ and H$_2^-$\cite{Laser0,Laser4,Laser5,Qi-Laser}.\\
In general, the energy is non-analytical, an analytic function is a function that is locally given by
a convergent power series, as a function of the Hamiltonian
parameters or a bound-state does not exist
at the threshold energy.
It has been suggested for some time, based on
large-dimensional models, that there are
possible analogies between
critical phenomena and singularities of the energy
\cite{stillinger1,katriel,dudley}.\\
Phase transitions are associated with singularities of the free energy. These
singularities occur only in the thermodynamic limit\cite{yanglee1,yanglee2}
where the dimension of the system approaches infinity. However calculations
are done only on finite systems. A Finite Size Scaling (FSS) approach is
needed in order to extrapolate results from finite systems to the
thermodynamic limit\cite{fisher}. FSS is not only a formal way to understand
the asymptotic behavior of a system when the size tends to infinity, but a
theory that also gives us numerical
methods\cite{widom,barber,privman,cardy,nightingale1,Peter1,Peter2} capable of obtaining
accurate results for infinite systems by studying the corresponding small
systems\cite{neirotti0,serra2,kais,snk1,snk2,nsk,qicun,kais1,review,adv,dipole,quadrupole}.
Applications include expansion in Slater-type basis functions\cite{adv},
Gaussian-type basis functions\cite{gto} and recently
finite elements\cite{fem-fss}.
\section{Criticality for Large-Dimensional Models}
Large dimension models were originally developed for
specific theories in the
fields of nuclear physics, critical phenomena and particle physics\cite{D3,witten}.
Subsequently, with the pioneering work of Herschbach\cite{DudleyJCP,dudley},
they found wide use in the field of
atomic and molecular physics\cite{D4}.
In this method one takes
the dimension of space, $D$, as a variable, solves the problem
at some
dimension $D \not= 3$ where the physics becomes much simpler,
and then uses perturbation
theory or other techniques to obtain an approximate
result for $D=3$\cite{dudley}. \\
It is possible to describe stability and symmetry breaking of
electronic structure configurations of atoms and molecules
as phase transitions and critical phenomena.
This analogy was revealed by
using dimensional scaling method and the large dimensional limit model of
electronic structure configurations\cite{D-He,D-N,D-H2,D-AB}. \\
To study the behavior of a given system near the critical
point, one has to rely on model calculations which are simple, capture the
main physics of the problem and which belong to the same universality class\cite{privman,cardy}.
Here we will illustrate the phase transitions and
symmetry breaking using the large dimension model.
In the
application of dimensional scaling to electronic structure, the large-D limit
reduces to a semi-classical electrostatic problem in which the electrons
are assumed to have fixed positions relative both to the nuclei and to each other
in the D-scaled space\cite{dudley}.
This configuration corresponds to the minimum
of an effective potential which includes Coulomb interactions as well
as centrifugal terms arising from the generalized D-dependence kinetic
energy. Typically, in the large-D regime the electronic structure
configuration undergoes symmetry breaking for certain ranges of nuclear
charges or molecular geometries\cite{frantz}.\\
In order to illustrate the analogy between symmetry breaking and phase transitions
we present as an example: the
results for the two-electron atoms in the
Hartree-Fock (HF) approximation\cite{D-He}.
In the HF approximation at the $D \rightarrow \infty$ limit,
the dimensional-scaled effective Hamiltonian
for the two-electron atom in an external weak electric
field $\cal E$ can be written as\cite{loeser,cabrera},
\begin{equation}
{\cal H_{\infty}} \, = \, \frac{1}{2} \left( \frac{1}{r_1^2}
\,+\,\frac{1}{r_2^2}
\right) \,-\, Z \, \left( \frac{1}{r_1} \,+\,\frac{1}{r_2}
\right) \,+\,
\frac{1}{\left( r_1^2 + r_2^2 \right)^{1/2}} \,-\,
{\cal E} \left( r_1 - r_2 \right)
\label{11}
\end{equation}
\noindent where $r_1$ and $r_2$ are the electron-nucleus radii,
and $Z$ is
the nuclear charge.
The ground state energy at the large-D limit is then given by
${ E}_{\infty}(Z,{\cal E}) \,=\, \min_{\{r_1,r_2\}} \;
{\cal H_{\infty}}$.
In the absence of an external electric field, ${\cal E}=0$,
Herschbach
and coworkers\cite{goodson}
have found that these equations have
a symmetric solution with the two electrons equidistant from
the nucleus,
with $r_1=r_2=r$. This symmetric solution represents a minimum in the region
where all the eigenvalues of the Hessian matrix
are positive, $Z \, \ge \, Z_c \,=\,
\sqrt{2}$.
For values of $Z$ smaller than $Z_c$,
the solutions
become
unsymmetrical with one electron much closer to the nucleus than the other
($r_1 \neq r_2 $). In order to describe this symmetry breaking, it is
convenient to introduce
new variables $(r, \eta)$ of the form:
$r_1 \,=\, r ;\; r_2 \,=\, (1-\eta) r$,
where $\eta =(r_1-r_2)/r_1 \ne 0 $ measures the deviation from the symmetric solution. \\
By studying the eigenvalues of the Hessian matrix, one finds that the
solution is a minimum of the effective potential
for the range, $1 \le Z \le Z_c$.
We now turn to the question of how to describe the system near the critical
point. To answer this question,
a complete mapping between this problem and
critical phenomena in statistical mechanics
is readily feasible with the following analogies:
\begin{center}
\begin{itemize}
\item nuclear charge $(Z) \leftrightarrow $ temperature $(T)$
\item external electric field $({\cal E}) \leftrightarrow $ ordering field $(h)$
\item ground state energy $({E_{\infty}}(Z,{\cal E})) \leftrightarrow
$ free energy $(f(T,h))$
\item asymmetry parameter $(\eta ) \leftrightarrow $ order parameter $(m)$
\item stability limit point $(Z_c,{\cal E}=0) \leftrightarrow
$ critical point $(T_c,h=0)$
\end{itemize}
\end{center}
Using the above scheme, we can define the critical
exponents $(\beta, \alpha, \delta$ and $\gamma)$ for the
electronic structure of the two electron atom in the following way:
\begin{equation}
\begin{array}{llll}
\eta(Z,{\cal E}=0) & \sim
& (- \Delta Z)^\beta & \;; \;{ \Delta Z
\rightarrow 0^-} \\
{E_{\infty}}(Z,{\cal E}=0) & \sim
& \mid \Delta Z \mid^{\alpha}&\; ;\;{\Delta Z
\rightarrow 0}\\
{\cal E}(Z_c,\eta) & \sim & \eta^\delta sgn(\eta)& \;;\;{\eta \rightarrow 0}\\
\frac{\partial \eta}{\partial {\cal E}}\left|_{{\cal E}=0} \right. &
\sim &
\mid \Delta Z \mid^{-\gamma}&\;;\;\Delta Z \rightarrow 0
\end{array}
\label{61}
\end{equation}
where $\Delta Z \equiv Z - Z_c$.
These critical exponents describe the nature of the singularities
in the above quantities at the critical charge $Z_c$.
The values obtained for these critical exponents are known as classical
or mean-field critical exponents:
$\beta \,=\, \frac{1}{2} \;\;;\;\; \alpha \,=\, 2, \;\;;\;\; \delta \,=\, 3
\;\;;\;\; \gamma \,=\, 1$. \\
This analogy between symmetry breaking and phase transitions
was also generalized to include the
large dimensional model of the N-electron atoms\cite{D-N}, simple diatomic
molecules\cite{D-H2,D-AA},
both linear and planar one-electron systems\cite{D-AB} as well as three-body Coulomb
systems of the general form $ABA$\cite{D-ABA}. \\
The above simple large-D picture helps to establish a
connection to phase
transitions. However, the next question to be addressed is:
How to carry out such an analogy to $D=3$?.
This question will be examined in the subsequent sections using the
finite size scaling approach.
\section{Finite Size Scaling: A Brief History}
Ice tea, boiling water and other aspects of two-phase coexistence are
familiar features of daily life. Yet phase transitions do not exist at all in finite
systems! They appear in the thermodynamic limit: The volume $V \rightarrow \infty$ and
particle number $N \rightarrow \infty$ in such a way that
their ratio, which is the density $\rho=N/V$, approaches a finite
quantity. In statistical mechanics, the existence of
phase transitions is associated with singularities of the free
energy per particle in some region of the thermodynamic space.
These singularities occur only in the {\it thermodynamic limit}\cite{yanglee1,yanglee2}.
This fact could be understood by examining the partition
function $Z$.
\begin{equation}
Z= \sum_{microstate \; \Omega} e^{-E(\Omega)/k_B T},
\end{equation}
where $E(\Omega)$ is the energies of the states, $k_B$ is the Boltzmann constant and
$T$ is the temperature. For a finite
system, the partition function is a finite sum of analytical terms, and
therefore it is itself an analytical function. The Boltzmann factor is an
analytical function of $T$ except at $T=0$. For $T> 0$,
it is necessary
to take an infinite
number of terms in order to obtain a singularity
in the thermodynamic limit\cite{yanglee1,yanglee2}.\\
In practice, real systems have a large but finite volume and particle numbers
($N \sim 10^{23}$),
and phase transitions are observed.
More dramatic even is the case of
numerical simulations, where sometimes systems with only a few number
(hundreds, or even
tens) of particles are studied, and ``critical" phenomena are still
present. Finite size scaling theory, which was pioneered by Fisher\cite{fisher},
addresses
the question of why finite systems apparently
describe phase transitions and what is the
relation of this phenomena with the true phase transitions in
corresponding infinite systems. Moreover,
finite-size scaling is not only a formal way to understand the
asymptotic
behavior of a system when the size tends to infinity. In fact, the theory
gives us numerical methods capable
of obtaining accurate results for infinite systems by studying
the corresponding small systems (see \cite{barber,privman,cardy} and
references therein).\\
In order to understand the main idea of finite size scaling, let us consider
a system defined in a $D$-dimensional volume $V$ of a linear dimension $L$
($V=L^D$). In a finite size system, If quantum effects are not taken into consideration,
there are in principle three length scales: The finite geometry characteristic size L,
the correlation length $\xi$, which may be defined as the length scale covering the
exponential decay $e^{-r/{\xi}}$ with distance $r$ of the correlation
function, and the microscopic length $a$ which governs the range of the interaction.
Thermodynamic quantities thus may depend on the dimensionless ratios
$\xi/a$ and $L/a$. The finite size scaling hypothesis assumes that,
close to the critical point, the microscopic length drops out.\\
If in the thermodynamic limit, $L \rightarrow \infty$,
we consider that there is only one parameter (say temperature $T$)
in the problem and the infinite system has a second order phase transition at a critical
temperature $T_c$, a thermodynamic quantity $G$ develops a singularity
as a function of the temperature $T$ in the form:
\begin{equation}
\label{limit}
G(T)=\lim_{L\to\infty} G_L(T)\sim\left|T-T_c
\right|^{-\rho}\;,
\end{equation}
\noindent whereas it is regular in the finite system,
$G_L(T)$ has no singularity.
When the size $L$ increases, the singularity of $G(T)$ starts to develop.
For example, if the correlation length diverges at $T_c$ as:
\begin{equation}
\label{xi}
\xi(T)=\lim_{L\to\infty} \xi_L(T)\sim\left|T-T_c
\right|^{-\nu}\;,
\end{equation}
\noindent then $\xi_L(T)$ has a maximum which becomes sharper and sharper,
then FSS ansatz assumes the existence of scaling function
$F_K$ such that:
\begin{equation}
\label{kn}
G_L(T)\sim G(T) F_K \left(\frac{L}{\xi(T)}\right)\;,
\end{equation}
\noindent where $F_K(y) \,\sim\, y^{\rho/\nu}$ for $y \sim 0^+$.
Since the FSS ansatz, Eq. (\ref{kn}), should be valid for any quantity
which exhibits
an algebraic singularity in the bulk, we can apply it to the correlation length
$\xi$ itself. Thus the correlation length in a finite system should have the
form:
\begin{equation}
\label{xin}
\xi_L(T)\sim L \phi_{\xi}(L^{1/\nu}|T-T_c|)\;.
\end{equation}
\noindent The special significance of this result was first realized by
Nightingale \cite{nightingale}, who showed how it could be reinterpreted as a
renormalization group transformation of the infinite system. The
phenomenological renormalization (PR) equation for finite systems
of sizes $L$ and $L'$ is given by:
\begin{equation}
\label{pr1}
\frac{\xi_L(T)}{L}=\frac{\xi_{L'}(T')}{L'}\,,
\end{equation}
\noindent and has a fixed point at $T^{(L,L')}$.
It is expected that the
succession of points $\left\{T^{(L,L')}\right\}$ will
converge to the true $T_c$ in the infinite size limit.
The finite-size scaling theory combined with transfer matrix calculations
had been, since the development of
the phenomenological renormalization in 1976 by Nightingale\cite{nightingale},
one of the most powerful
tools to study critical phenomena in two-dimensional lattice models.
For these models the partition function and all the
physical quantities of the system (free energy, correlation length,
response functions, etc) can be written as a function of the eigenvalues of
the transfer matrix\cite{thompson}. In particular, the free energy takes the
form:
\begin{equation}
\label{fe}
f(T)=-T \ln\lambda_1
\end{equation}
\noindent and the correlation length is:
\begin{equation}
\label{cl}
\xi(T)=-\frac{1}{ \ln\left(\lambda_2/\lambda_1\right)}
\end{equation}
\noindent where $\lambda_1$ and $\lambda_2$ are the largest and the second
largest eigenvalues of the transfer matrix.
In this context, critical
points are related with the degeneracy of these eigenvalues. For finite
transfer matrix the largest
eigenvalue is isolated (non degenerated) and phase transitions can occur
only in the limit $L \rightarrow \infty$ where the size of the transfer
matrix goes to infinity and the largest eigenvalues can be degenerated.
In the next section, we will see that these ideas of finite size scaling
can be generalize to quantum mechanics, in particular addressing the criticality
of the Schr\"odinger equation.
\section{Finite Size Scaling for the Schr\"odinger Equation}
The finite size scaling method is a systematic way to extract
the critical behavior of an infinite system from analysis on finite
systems\cite{adv}. It is efficient and accurate for the calculation of
critical parameters of the Schr\"odinger equation. Let's assume we
have the following Hamiltonian:
\begin{equation}
\label{h1}
{\cal H} \,=\, {\cal H}_0 \,+\, V_\lambda \;
\end{equation}
\noindent where ${\cal H}_0$ is $\lambda$-independent and $ V_\lambda$
is the $\lambda$-dependent term. We are interested in the study of
how the different properties of the system change when the value of
$\lambda$ varies.
A critical point, $\lambda_c$, will be defined as a point for which a
bound state becomes absorbed or degenerate with a continuum. \\
Without loss of generality,
we will assume that the Hamiltonian, Eq. (\ref{h1}), has a bound
state, $E_\lambda$,
for
$\lambda > \lambda_c $ which becomes equal to zero at
$\lambda = \lambda_c$. As in statistical mechanics,
we can define some critical exponents related
to the asymptotic behavior of different quantities near the critical point.
In particular, for the energy we can define the critical exponent $\alpha$
as:
\begin{equation}
\label{alphaiv}
E_\lambda \, \hbox{}_{\stackrel{\hbox{\normalsize$\sim$}}
{\hbox{\scriptsize$\lambda\to\lambda_c^+$}}} (\lambda - \lambda_c)^\alpha.
\end{equation}
The existence or absence of a bound state at the
critical point is related to the type of the singularity in the energy.
Using statistical mechanics terminology, we can associate
``first order phase transitions" with the existence of a normalizable
eigenfunction at the critical point. The absence of such a function
could be related to ``continuous phase transitions"\cite{adv}. \\
In quantum calculations, the variational method is widely used to
approximate the solution of the Schr\"odinger equation. To obtain
exact results one should expand the exact wave function in a
complete basis set and take the number of basis functions to
infinity. In practice, one truncates this expansion at some
order $N$. In
the present approach,
the finite
size corresponds not to the spatial dimension, as in statistical
mechanics, but to the number of elements
in a complete basis set used to expand the exact eigenfunction
of a given Hamiltonian.
We will compare two methods to obtain the matrix elements needed to apply
the FSS ansatz. The size of our system for the basis set expansion will
correspond to the dimension of the Hilbert space. For a given
complete basis set {$\Phi_n$}, the ground-state eigenfunction has
the following expansion:
\begin{equation}
\Psi_\lambda= \sum_{n}^{}a_n(\lambda)\psi_n,
\end{equation}
where n is the set of quantum numbers. We have to truncate the
series at order N and the expectation value of any general operator $O$ at
order N is given by:
\begin{equation}
\left<O\right>^{N} = \sum_{n,m}^N a_n^{(N)}a_m^{(N)}O_{n,m},
\end{equation}
\noindent where ${\cal O}_{n,m}$ are the matrix elements of ${\cal O}$
in the basis set $\{\psi_n\}$. \\
For the finite element method (FEM), the wavefunction
$\psi_n(r)$ in the $n$-th element is expressed in terms of local shape
functions. For our calculations, we use Hermite interpolation polynomials
with two nodes and three degrees of freedom.
This choice ensures the continuity of the wavefunction and its
first two derivatives.
Then in $n$-th element the wavefunction is\cite{fem-fss}:
\begin{equation}
\psi_n(r)= \sum_{i=1}^2 \left [
\phi_{i}(r) \psi_n^{i} +
\bar{\phi}_{i}(r) {\psi_n^{'i}} +
\accentset{=}{\phi}_{i}(r) {\psi_n^{''i}} \right],
\label{localpsi}
\end{equation}
with $\alpha$ indicating the nodal index of the element; $i=1$ for the
left and $i=2$ for the right border of the element. The functions
$\phi_{i}(r)$, $\bar{\phi_{i}}(r)$, and
$\accentset{=}{\phi}_{i}(r)$ are the (fifth degree) Hermite interpolation
polynomials.
Then $\psi_n^{i}$, $\psi_n^{'i}$, and
$\psi_n^{''i}$ are the undetermined values values of the wavefunction
and its first and second derivative on the nodal points.
The size for the case of solving the equation with the FEM will be the number
of elements used. \\
Since $\left< O \right>_\lambda$ is not analytical at
$\lambda=\lambda_c$, then we define a critical exponent, $\mu_O$, if the general operator has
the following relation:
\begin{equation}
\left< O \right>_\lambda \approx (\lambda - \lambda_c)^{\mu_O} \, \,
\, \, {for} \,\,\,\, \lambda \rightarrow \lambda_c^+,
\end{equation}
where $\lambda \rightarrow \lambda_c^+$ represents taking the limit
of $\lambda$ approaching the critical point from larger values of
$\lambda$. As in the FSS ansatz in statistical mechanics \cite{privman,derrida1}, we will assume
that there exists a scaling function for the truncated magnitudes
such that:
\begin{equation}
\left< O \right>_\lambda^{(N)} \sim \left< O \right>_\lambda F_O
(N|\lambda-\lambda_c|^\nu),
\end{equation}
with the scaling function $F_O$ being particular for different
operators but all having the same unique scaling exponent $\nu$.
To obtain the critical parameters, we define the following function:
\begin{equation}
\triangle_O(\lambda;N,N')=\frac{\ln(\left<O\right>_
\lambda^{N}/\left<O\right>_\lambda^{N'})} {\ln(N'/N)}.
\label{fourteen}
\end{equation}
At the critical point, the expectation value is related to $N$ as a
power law, $\left<O\right> \sim N^{\mu_O/\nu}$, and Eq.
(\ref{fourteen}) becomes independent of $N$. For the energy operator
$O=H$ and using the critical exponent $\alpha$ for the
corresponding exponent $\mu_O$ we have:
\begin{equation}
\triangle_H(\lambda_c;N,N')=\frac{\alpha}{\nu}.
\label{alphanu}
\end{equation}
In order to obtain the critical exponent $\alpha$ from numerical
calculations, it is convenient to define a new function\cite{adv}:
\begin{equation}
\Gamma_\alpha(\lambda,N,N')=\frac{\triangle_H(\lambda;N,N')}{\triangle_H(\lambda;N,N')-\triangle_{\frac{\partial
V_\lambda}{\partial \lambda}}(\lambda;N,N')},
\label{gammafunc}
\end{equation}
which at the critical point is independent of $N$ and $N'$ and takes
the value of $\alpha$. Namely, for $\lambda=\lambda_c$ and any
values of $N$ and $N'$ we have
\begin{equation}
\Gamma_\alpha(\lambda_c,N,N')=\alpha,
\end{equation}
and the critical exponent $\nu$ is readily given by Eq. (\ref{alphanu}).
\section{The Hulthen Potential}
To illustrate the application of the FSS method in quantum mechanics,
let us give an example of the criticality of the Hulthen potential.
The Hulthen potential behaves like a Coulomb potential for
small distances whereas for large distances it decreases exponentially so that the ``capacity''
for bound states is smaller than that of Coulomb potential. Thus, they have the same singularity
but shifted energy levels. They always lie lower in the Coulomb case than in the Hulthen case,
where there remains only space for a finite number of bound states\cite{flugge}.
Here, we present the FSS calculations using two methods: finite elements and basis set
expansion; each used
to obtaining quantum critical parameters for the Hulthen Hamiltonian.
First, we give the analytical solution, then FSS with
basis set expansion and finite element solution.
\subsection{Analytical Solution}
The Hulthen potential has the following form\cite{hulthen,flugge}:
\begin{equation}
V(r) = - \frac{\lambda}{a^2} \frac{{e}^{-r/a}}{1-{e}^{-r/a}}
\end{equation}
where $\lambda$ is the coupling constant, and $a$ is the scaling parameter.
For small values of $r/a$ the potential $V(r) \rightarrow - \frac{1}{a}\lambda/r$,
whereas for large values of $r/a$ the potential approaches zero
exponentially fast, therefore the \emph{scale a} in the potential regulates
the infinite number of levels that would otherwise appear with a
large-distance \emph{Coulomb} behavior.
Shr\"odinger radial differential equation in the dimensionless variable $r=r/a$ becomes:
\begin{equation}
\frac{1}{2}\frac{d^2 \chi}{d r^2} + ( - \alpha^2 + \lambda
\frac{{e}^{-r}}{1-{e}^{-r}} ) \chi = 0.
\end{equation}
We only consider the case for $l=0$ for the Hulthen potential.
Here we used the abbreviations $\alpha^2 = - E a^2 \geq 0 $ (in atomic units $m=\hbar=1$).
The complete solutions for the wavefunctions are written in term of
hypergeometric functions as follows\cite{flugge}:
\begin{equation}
\chi = N_0 e^{-\alpha r}(1-e^{-r}) _{2}F_{1} (2\alpha+1+n,1-n,2\alpha+1;e^{-r}),
\label{hulthenWaveFnc}
\end{equation}
where the normalization factor is given by
$N_0=[\alpha(\alpha+n)(2 \alpha+n)]^{\frac{1}{2}}[\Gamma(2\alpha+n)/\Gamma(2\alpha+1)\Gamma(n)]$.
It follows that the energy levels are given by:
\begin{equation}
E_{n} = - \frac{1}{a^2} \frac{(2\lambda-n^2)^2}{8 n^2}; \,\,\, n=1,2,3...,n_{max}.
\end{equation}
We can make the following comments concerning the energy levels
obtained for the Hulthen potential. There exists a \emph{critical}
value for the coupling $\lambda_{c}$ to have the
given energy levels, $\lambda_{c} = n^2 / 2$.
It follows directly from the first observation that the number of
levels $n_{max}$ allowed is \emph{finite} and it
depends on the size of the coupling constant $n^2_{max} \leq 2 \lambda$.
As $\lambda \rightarrow \infty$ the potential is well behaved, which
can be seen as follows: In this limit we get the obvious inequality
$ \alpha^2 \ll 2\lambda \Rightarrow \sqrt{2\lambda} \approx n$.
It follow that we can set $ \alpha \approx 0$ in Eq. (\ref{hulthenWaveFnc}) to obtain:
\begin{equation}
\chi_{\alpha \rightarrow 0} = (1-e^{-r}) _{2}F_{1} (1+n,1-n,1;e^{-r}),
\end{equation}
which is the wave function at threshold. This wave function is not normalizable as
expected when the energy exponent $\alpha=2$, $E \sim (\lambda-\lambda_c)^{\alpha}$.
For the ground state, the asymptotic limit of the probability density
for $r>>1$ and $\lambda \rightarrow \lambda_c$
becomes:
\begin{equation}
P(r) \sim e^{-r/\xi},\;\; \xi \sim |\lambda-\lambda_c|^{-\nu},
\end{equation}
with a characteristic length $\xi$ and exponent $\nu=1$.
The Hulthen potential has a finite capacity determined by the critical
coupling, $\lambda_{c}$. The potential admits bound states between the range
of values for the coupling: $\lambda = [1/2, \infty)$.
\begin{figure}[htp]
\begin{center}
\includegraphics*[width=300pt]{Gamma-basis-fem}
\caption{Plot of $\Gamma_\alpha$, obtained by FSS method, as a function of $\lambda$.
Using the number of basis N from 8 to 48 in steps of 2. For FEM the number
elements used were from 100 to 380 in steps of 20.}
\label{gamma}
\end{center}
\end{figure}
\subsection{Basis Set Expansion}
For the Hulthen potential, the wavefunction can be expanded
in the following Slater basis ( see Chapter 7 for details \cite{chapter7}):
\begin{equation}
\Phi_n(r)=\sqrt{\frac{1/4\pi}{(n+1)(n+2)}}e^{-r/2}L_n^{(2)}(r).
\end{equation}
$L_n^{(2)}(r)$ is the Laguerre polynomial of degree $n$ and order $2$.
The kinetic term can be obtained analytically. However, the
potential term need to be calculated numerically\cite{Edwin}.
Figure {1}, show the results for the
plot $\Gamma_\alpha(\lambda,N,N')$ as a function of
$\lambda$ with different $N$ and $N'$, all the curves will cross
exactly at the critical point.
\subsection{Finite Element Method}
The FEM is a numerical technique which gives approximate solutions to
differential equations. In the case of quantum mechanics, the
differential equation is formulated as a boundary value problem\cite{pepper,reddy}. For
our purposes, we are interested in solving the time-independent
Shr\"odinger equation with finite elements.
We will require our boundary conditions to be restricted to the
Dirichlet type. For this problem, we will use two interpolation methods:
linear interpolation and Hermite Interpolation polynomials to solve for this
potential.
We start by integration by parts and impose the boundary conditions
for the kinetic energy and reduce it to the weak form\cite{fem-fss}:
\begin{eqnarray}
&&\frac{1}{2}\int_0^{\infty} r^2 {\psi^*}'(r)\psi'(r) dr.
\nonumber \\
&& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{weak}
\end{eqnarray}
For the potential energy:
\begin{eqnarray}
&&\int_0^{\infty} r^2 {\psi^*}(r)\psi(r) \lambda \frac{-e^{-r}}{1-e^{-r}} dr.
\nonumber \\
&& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\label{hulthen}
\end{eqnarray}
We calculated the local matrix elements of the potential energy by using a four point Gaussian
Quadrature to evaluate the integral. We set the cutoff for the integration to
$r_c$. To include the integration to infinity, we added an infinite
element approximation. To do so, we
approximate the solution of the wave function in the region of $[r_c,\infty)$ to be an exponentially decaying
function with the form $\psi(r)=\psi(r_c)\,e^{-r}$.
The local matrices are then
assembled to form the complete solution and by invoking the variational
principle on the nodal values $\psi_i$ we obtain a
generalized eigenvalue problem representing the initial Schr\"odinger
equation:
\begin{equation}
H_{ij}|\psi_j \rangle=\epsilon U_{ij}|\psi_j \rangle.
\label{gev}
\end{equation}
Solution of Eq. (\ref{gev}) is achieved using standard numerical methods
(see Chapter 10 for details \cite{chapter10}).
\begin{figure}[htp]
\begin{center}
\includegraphics*[width=300pt]{extrp-basis-fem}
\caption{Extrapolated values for the critical exponents and the critical
parameter $\lambda$. The solid read dots at $1/N=0$ are the extrapolated critical
values. The left side is the basis set method while the right is the FEM
with Hermite interpolation polynomials.}
\label{bst}
\end{center}
\end{figure}
\subsection{Finite Size Scaling Results}
\begin{figure}[htp]
\begin{center}
\includegraphics*[width=300pt]{dataclaps-basis-fem}
\caption{Data collapse study of the basis set method and FEM. The left is
the basis set method and the right being the FEM.}
\label{collapse}
\end{center}
\end{figure}
The finite size scaling equations are valid only as
asymptotic expressions, but unique values of $\lambda_c$, $\alpha$, and $\nu$
can be obtained as a succession of values as a function of $N$. The lengths of
the elements are set $h=0.5$.
The plots of $\Gamma_\alpha$, figure
\ref{gamma}, the basis set expansion is giving values very close to
the analytical solution of the Hulthen potential. For the plot of
$\Gamma_\alpha$ for the FEM
estimation of $\lambda_c$ is producing results very close to the exact values using
Hermite interpolation. The
intersection of these curves indicate the $\lambda_c$ on the abscissa. The
ordinate gives the critical exponents $\alpha$ (in $\Gamma_\alpha$ plots).
In Figure \ref{bst}, we observed
the behavior of the pseudocritical parameters, $\lambda_c^{(N)}, \alpha_c^{(N)}, \nu_c^{(N)}$,
as a function of $1/N$.
The three curves monotonically
converge to limiting values for the Hermite interpolation and the basis set expansion. \\
To check the validity of our finite size scaling assumptions, we
performed a data collapse\cite{datta-collapse} calculation of the Hulthen potential.
In the data collapse analysis, we
examine the main assumption we have made in Eq. (17) for the existence
of a scaling function for each truncated magnitude
$\left<{\cal O } \right>^{(N)}_\lambda $ with a unique
scaling exponent $\nu$.
Since the
$\left<{\cal O } \right>^{(N)}_\lambda $ is analytical in $\lambda$,
then from Eq. (17) the asymptotic
behavior of the scaling function must have the form:
\begin{equation}
\label{ab}
F_{\cal O} (x) \, \sim \, x^{-\mu_{\cal O}/\nu} \;.
\end{equation}
For large values of $N$, at the $\lambda_c$, we have
\begin{equation}
\left< {\cal O } \right>^{(N)}(\lambda_c) \sim N^{-\mu_{\cal O}/\nu}.
\end{equation}
Because the same argument of regularity holds for the derivatives of the
truncated expectation values, we have:
\begin{equation}
\label{do}
\left. \frac{\partial^m \left<{\cal O } \right>^{(N)}}
{\partial \lambda^m} \right|_{\lambda=\lambda_c}
\sim N^{-(\mu_{\cal O}-m)/\nu},
\end{equation}
$\left<{\cal O } \right>^{(N)}$ is analytical in $\lambda$, then using
Eq. (\ref{do}), the Taylor expansion could be written as:
\begin{equation}
\label{te}
\left<{\cal O } \right>^{(N)}(\lambda)\sim N^{-\mu_{\cal O}/\nu}
G_{\cal O} (N^{1/\nu} (\lambda - \lambda_c)),
\end{equation}
\noindent where $G_{\cal O}$ is an analytical function of its argument.
This equivalent expression for the scaling of a given expectation value
has a correct form to study the data collapse in order to test FSS hypothesis.
If the scaling Eq. (17) or Eq. (\ref{te}) holds,
then near the critical point the
physical quantities will collapse to a single universal curve when plotted
in the appropriate form $\left<{\cal O } \right>^{(N)} N^{\mu_{\cal O}/\nu}$
against $N^{1/\nu} (\lambda - \lambda_c)$. If the operator ${\cal O}$ is the Hamiltonian
then we will have a data collapse when plotting
$E_0 N^{-\alpha/\nu}$
against $N^{1/\nu} (\lambda - \lambda_c)$.
In Figure \ref{collapse} we plot the
results corresponding to the basis set method (right panel) and Hermite interpolation
(left panel),
which have been calculated
with $\lambda_c=0.49999$, $\alpha=1.9960$ and $\nu=0.99910$ for the basis set method
and for the Hermite interpolation we have $\lambda_c=0.50000$, $\alpha=2.00011$ and
$\nu=1.000322$. The data collapse study do in fact support our FSS assumptions.
We have conveniently summarized our results for the critical parameters for
the analytical, linear interpolation, Hermite interpolation and the
basis set expansion in table \ref{table:results}.\\
\begin{table}[htp]
\begin{center}
\caption{Critical Parameters for the Hulthen Potential}
\centering
\begin{tabular}{c c c c c}
\hline\hline
$ $&Analytical& Linear & Hermite & Basis Set\\
\hline
$\lambda$ &0.5 (exact) & 0.50184 & 0.50000 & 0.49999\\
$\alpha$ &2 (exact) & 1.99993& 2.00011 &1.9960 \\
$\nu$ &1 (exact)& 1.00079& 1.00032& 0.99910\\
\hline\hline
\end{tabular}
\label{table:results}
\end{center}
\end{table}
We have successfully obtained the critical exponents and the critical
parameter for the Hulthen potential using FSS with the basis set method and the
FEM. The results are in excellent agreement with the analytical solution even for the
very simplistic linear interpolation used for the FEM calculations. However,
the ability of the FEM to describe the wavefunction locally in terms of
elements affords a very natural way to extend its use for FSS purposes.
\section{Finite Size Scaling and Criticality of M-Electron Atoms}
Let us examine the criticality of the N-electrons atomic Hamiltonian as a function
of the nuclear charge $Z$. The scaled Hamiltonian takes the form:
\begin{equation}
\label{ham}
{\cal H}(\lambda)= \sum_{i=1}^M \left[ -\frac{1}{2} \nabla_i^2 -\frac{1}{r_i}
\right ] + \lambda \sum_{i<j=1}^M \frac{1}{r_{ij}},
\end{equation}
\noindent where $r_{ij}$ are the interelectron distances, and $\lambda=1/Z $ is
the inverse of the nuclear charge. For this Hamiltonian, a critical point
means the value of the
parameter, $\lambda_c$, for
which a bound state energy becomes absorbed or degenerate
with the continuum.
To carry out the FSS procedure, one has to choose a convenient basis set
to obtain the two lowest eigenvalues and eigenvectors of the finite
Hamiltonian matrix. For $M=2$, one can choose the following
basis set functions:
\begin{equation}
\label{hylleraas}
\Phi_{ijk,\ell}(\vec{x}_1,\vec{x}_2) = \frac1{\sqrt2}\left(r_1^i \,r_2^j\,
e^{-(\gamma r_1+\delta r_2)} + \right.\nonumber
\left. r_1^j \,r_2^i\,e^{-(\delta r_1 + \gamma r_2)}\right)
\,\, r_{12}^k \,\;F_{\ell}(\theta_{12},{\bf\Omega})
\end{equation}
\noindent where $\gamma$ and $\delta$ are fixed parameters,
we have found numerically that
$\gamma = 2$ and $\delta = 0.15$ is a good choice for the ground state\cite{neirotti0},
$r_{12}$ is the interelectronic distance and $F_{\ell}(\theta_{12},{\bf\Omega})$
is a suitable function of the angle between the positions of the two electrons
$\theta_{12}$ and the Euler angles ${\bf\Omega}=(\Theta,\Phi,\Psi)$. This
function $F_{\ell}$ is different for each orbital-block of the Hamiltonian. For
the ground state $F_0 (\theta_{12},{\bf\Omega})= 1$ and
$F_1(\theta_{12},{\bf\Omega})=
\sin(\theta_{12})\cos(\Theta)$ for the $2p^2\; {}^3P$ state.
These basis sets are complete
for each $\ell$-subspace.
The complete
wave function is then a linear combination of these terms multiplied
by variational coefficients determined by matrix diagonalization \cite{neirotti0}.
In the truncated basis set at order $N$, all terms are included such that
$N\geq i+j+k$. Using FSS calculations with $N=6,7,8,\dots,13$ gives the
extrapolated values of $\lambda_c=1.0976 \pm 0.0004$
which is in excellent agreement with the best estimate of
$\lambda_c=1.09766079$ using large-order perturbation calculations\cite{ivanov}.
Since the critical charge $Z_c=1/\lambda_c \sim 0.91 $ indicates that the
hydrogen anion H$^{-}$ is stable, $Z=1> Z_c$.
For three-electron atoms, $M=3$, one can repeat the FSS procedure with the following
Hyllerass-type basis set\cite{serra2}:
\begin{equation}
\label{wfyd}
\Psi_{ijklmn}(\vec{x}_1,\vec{x}_2,\vec{x}_3) = {\cal C \; A}\left(r_1^i \,r_2^j
\, r_3^k r_{12}^l \, r_{23}^m \, r_{31}^n \;
e^{-\alpha (r_1+ r_2)} e^{-\beta r_3} \;\;\chi_1
\right),
\end{equation}
\noindent where the variational parameters,$\alpha=0.9$ and $\beta=0.1$,
were chosen to
obtain accurate results near the critical charge $Z \simeq 2$,
$\chi_1$ is the spin function
with spin angular moment 1/2:
\begin{equation}
\label{spin}
\chi_1 \,=\, \alpha(1) \beta(2) \alpha(3) \,-\, \beta(1) \alpha(2) \alpha(3),
\end{equation}
\noindent $ {\cal C}$ is a normalization constant and
${\cal A}$ is the usual three-particle antisymmetrizer operator\cite{serra2}.
The FSS calculations gives $\lambda_c=0.48 \pm 0.03$. Since $Z_c \sim 2.08$
the anions He$^{-}$ and H$^{--}$ are unstable. \\
One can extend this analysis and calculate the critical charges
for M-electron atoms in order to perform a systematic check of the stability of atomic
dianions. In order to have a stable doubly negatively charged atomic
ion one should require the surcharge, $S_e(N) \equiv N-Z_c(N) \geq 2$.
We have found that the
surcharge never exceeds two.
The maximal surcharge, $S_e(86)=1.48$, is
found for the closed-shell configuration of element Rn and can be
related to the peak of electron affinity of the element $N=85$.
The FSS numerical results for M-electron atoms show that
at most, only one electron
can be added to a free atom in the gas phase.
The second extra electron is not
bound by singly charged negative ion because the combined action of
the repulsive potential surrounding the isolated negative ion and
the Pauli exclusion principle. However, doubly charged
atomic negative ions might exist in
a strong magnetic field of the order few atomic units, where
$1 a.u.=2.3505\; 10^9 G$ and superintese laser fields.
\section{Conclusions}
In this chapter, we show how the finite size scaling
ansatz can be combined with the variational
method to extract information about critical behavior of
quantum Hamiltonians. This approach is based on
taking the number of elements
in a complete basis set or the finite element method
as the size of the system.
As in statistical mechanics, finite size scaling
can then be used directly to the Schr\"odinger equation.
This approach is general and gives very accurate results for
the critical parameters, for which the bound state energy
becomes absorbed or degenerate with a continuum.
To illustrate the applications in quantum calculations,
we present detailed calculations for the simple case of Hulthen potential
and few electron atoms. For atomic systems we have shown that
finite size scaling can be used to explain and predict the stability
of atomic anions: At most, only one electron can be added to a free
atom in the gas phase.\\
Recently, there has
been an ongoing experimental and theoretical search for doubly
charged negative molecular dianions\cite{science}.
In contrast to atoms, large molecular
systems can hold many extra electrons because the extra electrons
can stay well separated.
However, such systems are challenging from both
theoretical and experimental points of view.
The present finite size scaling approach might be useful
in predicting the general stability of molecular dianions. \\
The approach can be generalize to complex systems by calculating the matrix
elements needed for FSS analysis by ab initio, density functional methods,
orbital free density functional
(OF-DFT) \cite{princetorn,gavini}
approach, density matrices\cite{David1,David2} and other
electronic structure methods\cite{ortiz1}.
The implementation should be straightforward. We need to obtain the matrix elements
to calculate $\Gamma_a$ as a function of the number of
elements used in solving for the
system. In the finite element using mean field equations
(like Hartree-Fock or Kohn Sham methods) the solution region will be discretized into
elements composed of tetrahedrons. \\
The field of quantum critical phenomena in atomic and molecular physics is
still in its infancy and there are many open questions about
the interpretations of the results including whether or not these quantum
phase transitions really do exist.
The possibility of exploring these
phenomena experimentally in the field of
quantum dots\cite{Wang-Kais} and systems in superintense laser
fields\cite{Qi-Herschbach} offers an exciting
challenge for future research.
This finite size scaling approach is general and
might provide a powerful way
in determining critical parameters for the stability of atomic and molecular
systems in external fields, and for design and control electronic
properties of materials using
artificial atoms. \\
The critical exponents calculated with finite size scaling indicate the nature
of the transitions from bound to continuum states. Study of the
analytical behavior of the energy near the critical point show
that the open shell system, such as the lithium like atoms, is
completely different from that of a closed shell system, such as
the helium like atoms. The transition in the closed shell
systems from a bound state to a continuum resemble a ``first-order
phase transition", $E \sim (\lambda - \lambda_c)^1$,
while for the open shell system, the transition
of the valence electron to the continuum is a ``continuous phase
transition", $E \sim (\lambda - \lambda_c)^2$. For closed shell systems,
one can show that
${\cal H}(\lambda_c)$ has a square-integrable eigenfunction corresponding
to a threshold energy, the existence of a
bound state at the critical coupling constant $\lambda_c$ implies that for
$\lambda < \lambda_c$, $E(\lambda)$ approaches $E(\lambda_c)$
linearly in $(\lambda-\lambda_c)$ as $\lambda \rightarrow \lambda_c^-$.
However, for open shell systems, the wave function is not
square-integrable at at $\lambda_c$. This difference in critical exponents
might be helpful in developing a new atomic classification schemes based
on the type of phase transitions and criticality of the system.
\section{Acknowledgments}
I would like to thank Pablo Serra, Juan Pablo Neirotti,
Marcelo Carignano, Winton
Moy and Qi Wei for their valuable contributions
to this ongoing research of developing and applying finite size scaling to quantum
problems and Ross Hoehn for critical reading of the Chapter. I would like also to thank the Army Research Office (ARO)
for financial support of this project.
\newpage
|
{
"timestamp": "2010-09-23T02:02:23",
"yymm": "1009",
"arxiv_id": "1009.4393",
"language": "en",
"url": "https://arxiv.org/abs/1009.4393"
}
|
\section{Introduction}
Multiple planetary systems analogous to our Solar System
play a key role in understanding planet formation and evolution.
If planets in multiple systems display
transits as well { (e.g. Kepler-9, Holman et al. 2010)},
a very detailed analysis becomes possible, resulting in
a set of dynamical parameters; and even the internal density distribution
of the planets (Batyigin et al. 2009).
As of this writing, three multiple systems with a transiting component
have been discovered. The CoRoT-7 system has two orbiting super-Earths, one
showing transits (L\'eger et al. 2009, Queloz et al. 2009); HAT-P-7 hosts a hot
Jupiter in a polar or retrograde orbit and a long-period companion that can
either be a planet or a star (P\'al et al. 2008, Winn et al. 2009). But the
most
prominent example of such systems is HAT-P-13 (Bakos et al. 2009, Winn et al.
2010). The central star of this system
is a G4 dwarf with 1.22 M$_\odot$ mass and 1.56 R$_\odot$ radius. HAT-P-13b
is a 0.85 M$_J$ hot Jupiter on a 2.9 day orbit that has almost been
circularized. HAT-P-13c has a minimum mass of M$\sin i$=15.2 M$_J$ in a 428
day orbit with 0.69 eccentricity. Winn et al. (2010) predicted a possible transit
for the second planet, which, if confirmed, would make HAT-P-13 an extremely
special system.
In multiple planetary systems, the most important question is whether the
orbital planes are aligned.
If this is the case for HAT-P-13 b and c, the exact mass of companion
c can be derived. The $\Delta i$ mutual inclination may be derived from the
Transit Timing
Variations of HAT-P-13b (Bakos et al. 2009). A more stringent
constraint on coplanarity would be delivered
if HAT-P-13c also transits. In this case the coplanarity is highly probable, and
the radius and the orbit of planet c can be measured.
If the apsides are also aligned,
tidal dynamics can reveal planet b's internal structure, which is a
fascinating opportunity to extract unique information
on an exoplanet (Batygin et al. 2009, Fabricky 2009).
It has been unknown whether HAT-P-13c transits. Dynamical models of Mardling
(2010) suggest that the HAT-P-13 system is likely to be close to prograde
coplanar or have a mutual inclination between 130$^\circ$ and 135$^\circ$.
She interpreted the system geometry
as a result of early chaotic interactions. A hypothetical
d companion has been invoked at the early stages
of evolution { that should have escaped later and could explain the
vivid scattering history}.
Her argument
for coplanarity is that lower masses are favoured because of dynamical reasons,
although c's high inclination itself favours a large mutual inclination.
Winn et al. (2010) points to the observed small stellar obliquity $\psi_{*,b}$
as an indirect evidence of orbital alignment: in Mardling's model, after
having planet d escaped,
$\psi_{*,b}$ oscillates about a mean value of $\Delta i$. Thus,
observing small value for $\psi_{*,b}$ at any time, e.g. now, is unlikely
unless $\Delta i$ is small.
\begin{table*}
\caption{}
\centering\begin{tabular}{lllll}
\hline
Code & Telescope & CCD & FoV & resolution \\
\hline
K60 & Konkoly 0.6 Schmidt, Piszk\'estet\H{o}, Hungary & 1526$\times$1024 KAF & 25$^\prime\times$17$^\prime$ & 1.0$^{\prime \prime}$/pixel \\
K100 & Konkoly 1.0 RCC, Piszk\'estet\H{o}, Hungary & 1340$\times$1300
PI VersArray 1300b NTE& 7$^\prime \times$7$^\prime$ &
0.32$^{\prime\prime}$/pixel\\
SLN & INAF-OACt 0.91, Fracastoro, Italy & 1100$\times$1100 KAF1001E & 13$^\prime \times$13$^\prime$ & 0.77 $^{\prime \prime}$/pixel\\
TEN & 0.8 RCC Tenagra II, Arizona, USA & 1024$\times$1024 & 14.8$^\prime \times$14.8$^\prime$ & 0.81 $^{\prime \prime}$/pixel \\
LNO & Langkawi 0.5 RCC, Malaysia & 1024$\times$1024 SBIG 1001E CCD & 20$^\prime \times$20$^\prime$ & 1.2 $^{\prime\prime}$/pixel\\
SLT & Lulin 0.4 RCC, Taiwan & 3056$\times$3056 Apogee U9000 & 50.7$^\prime\times$50.7$^\prime$ & 0.99$^{\prime\prime}$/pixel\\
\hline
\end{tabular}
\end{table*}
\begin{table*}
\caption{Observations during the HAT-P-13c campaign. Telescope codes:
K60: Konkoly 60 cm Schmidt, TEN: Tenagra, SLT: Lulin, LNO: Langkawi,
SLN: INAF-OACt.
Observation windows and the number of photometry points are indicated.}
\centering\begin{tabular}{rllllll}
\hline
Date ~~~& K60 & TEN & SLT & LNO & SLN\\
\hline
2010--04--22& 20:23--22:23 (80)& &\\
04--25& 18:40--23:33 (190)~~~& 03:01--05:17 (101)~~~ & 12:17--15:03 (108)~~~ & \\
04--26& 18:43--21:27 (134)& 04:44--07:01 (100) & & 14:28--14:56 (34)\\
04--27 && 03:10--05:26 (85) &&& 20:46--23:56 (162)\\
04--28& 18:41--22:53 (139)& 05:30--06:52 (61)& &13:34--14:55 (23)& 20:37--00:08 (116)\\
04--29& 18:45--23:15 (349)& 04:55--06:49 (84)& & 13:42--15:32 (58)& 20:53--23:58 (128)\\
04--30& 18:43--23:17 (342)& 04:55--06:55 (51)&\\
05--01& 19:21--20:32 (66)& 05:39--06:17 (45) & 11:48--14:16 (55)&12:42--14:45 (60)\\
05--03& 19:25--22:50 (252)& &&\\
\hline
\end{tabular}
\end{table*}
\begin{figure*}
\begin{centering}
\includegraphics[width=17.5cm]{hat13all.eps}
\caption{Observations of HAT-P-13 between April 26--30.
Observations are shifted with $+$0.01 (V points) and $-$0.01 (R points) as
indicated. Different symbols are applied for the different
observatories: square: Langkawi, stars: Konkoly, triangles: INAF-OACt, circle:
Tenagra. The typical standard deviation is 0.0013 in R and 0.0014 in V.
A $\pm0.0015$ error bar is indicated in the upper right corner of the top panel.
}
\end{centering}
\end{figure*}
The refined orbital elements suggested
that the transit - if it happened - should have occurred around 2010 April 28, 17 UT,
(JD 2455315.2) with 1.9 days
FWHM of transit probability and a maximal duration of 14.9 hours
(Winn, 2010). We started monitoring of HAT-P-13 for further transits
in November 2009 and organised an international campaign in the 2 weeks
surrounding the expected transit { of HAT-P-13c}.
\section{Observations and Data Reduction}
The seasonal visibility of HAT-P-13 is quite unfavourable in April.
Hence the longest possible run at mid-northern latitudes
may last 3--4 hours after twilight with
observations ending at high (X$>$2)
airmass. Our data were collected at 5 observing sites with 6 telescopes,
and, due to the weather conditions,
30\%{} time coverage was reached. The telescope parameters and the log of the
observations is shown in
Table 1 and 2, respectively.
The observing strategy was the same in most observatories: a sequence of
RRRVVV was repeated continuously, while Tenagra Observatory measured the first
half of the light curve in R, and the second half in V.
Integration time was adjusted all
along the night to compensate for the air mass variation in
an effort to take advantage of the full dynamic range of the
camera. The average exposure time was about 65 s and 35 s
in the V and R bands, respectively. Each night several bias, dark and sky
flat images were taken for calibration.
Before the multisite campaign, we observed HAT-P-13 on 8 additional nights.
Two nights (2009-11-05/06 and 2010-02-21/22) included a transit of
HAT-P-13b, the rest acquired as out-of-transit observations.
{
In these observations, the K100 telescope was
also involved.
No transit signal exceeding a depth of 0.005 (3-sigma level)
was observed during the following out-of-transit observation runs:
2009--11--05/06, 23:03--03:45 UT (1 RCC),
2010--01--11/12, 01:41--04:29 (1 RCC),
2010--01--14/15, 21:41--23:19 (0.6 Schmidt),
2010--01--16/17, 22:20--03:39 (0.6 Schmidt),
2010--02--21/22, 18:32--02:19 (0.6 Schmidt),
2010--03--18/19, 19:08--00:03 (0.6 Schmidt),
2010--03--18/19, 20:08-23:37 (1 RCC)
2010--03--19/20, 21:38--00:11 (0.6 Schmidt),
2010--03--28/29, 18:30--00:16 (0.6 Schmidt).
Transits of HAT-P-13b were} analyzed with an automated image processing and aperture
photometry pipeline developed in
{\sc gnu-r}\footnote{r-project.org}
environment. { The flat image was constructed as the median of the normailzed flat frames (i.e. each acquired images were divided by the mean of their pixel values), and that similar procedures were performed for darks and bias.}
After the standard calibrations, star identification
was performed. Comparison stars were selected iteratively
for attaining the best S/N in the light curve. Finally,
3 comparison stars were used in all images { (2MASS J08392449+4723225, 2 MASS J08392164+4720500, 2MASS J08391779+4722238), to ensure the consistency of the
entire dataset. $J-K$ colors of the comparison stars are 0.419, 0.384 and 0.337,
quite close to $J-K=0.353$ of HAT-P-13.}
The data were corrected for systematics with {
the well-known parameter decorrelation technique (e.g. Robinson et al. 1995), in our
case applying the specific implementation of the
External Parameter Decorrelation (EPD) in constant mode (Bakos et al. 2010). The observed external parameters
were the PSF of stellar profiles and the local photometry of the
flat field image at the same $X,Y$ position where the stars were
observed. The variation of stellar profile is a known error source
which has been involved in most standard reduction pipelines of
exoplanet photometry. Considering the flat field image intensities
as an error source means assuming that dividing with the flat
field under/overestimates the neccessary correction by a factor of a
few 0.1\%{}. We experienced that most of the artificial patterns of the light curves is due to systematic residuals of flat field correction and could be well eliminated this way.} In the end, 6585 raw photometric points
were extracted. We omitted points out of the 5--95\%{}
quantile interval { of the measured fluxes}
and averaged the surviving points by 3.
This resulted in 1952 data points submitted to further analysis.
\section{Results}
\subsection{Significance analysis of the null detection}
In Fig. 1 we plot sample light curves from the multisite campaign.
The panels show the combined light curves from April 26, 27, 28, 29
and 30. Neither signs of ingress or egress nor
significant deviations from the average brightness have been observed.
These
features strongly suggest that all observations are out of
transit, and HAT-P-13c is
likely to be a non-transiting exoplanet.
What is the significance of this conclusion? The time coverage
of our data is 30\%. Thus the first answer could be that a
transit could happen anytime in 70\% of the time, i.e. when observations were
not done, and this null result is essentially insignificant.
But this conclusion is not correct and in fact,
our observations rule out the majority of transiting orbits for HAT-P-13c.
We did a numerical experiment to determine the
quantitative measure of the significance.
A set of
$10^5$ exoplanets were simulated on a similar orbit to HAT-P-13 (428 days
period around an 1.22~R$_\odot$, 1.56~R$_\odot$ star).
{
The radius of the planet was assumed to be 1.2 $R_J$, { which is the typical size
for} the most massive known exoplanets. With this choice, the density of
HAT-P-13c is 8.7 times of the Jupiter.
The orbital eccentricity of the model was $e=0.691$, the argument of periastron was
$\omega=176.7^\circ$, coefficients for quadratic limb darkening were
$\gamma_1=0.3060$, $\gamma_2=0.3229$ (planet and orbit parameters
from Bakos et al. 2009).
To include grazing transits, the value of the impact
parameter $b$ was allowed to be $>1$ and was drawn from an uniform distibution between 0 and 1.08. The transit time followed a uniform distribution in the April 26.5 UT and April 30.5 UT interval.}
{ In some possible planet configurations it is probable that data of a given run could have included only the bottom of the transit. This should be seen as a slight offset from the rest of the runs, but that this cannot be detected because of non-photometric conditions.}
What we are sure about is that ingress and egress phases were not detected
within our time coverage. Solely this information constrains the possible
orbits seriously in the transit time--impact parameter space.
{ Model transit light curves were sampled at the times of observation points (all data in Table 2), sorted to observation runs and the average level was individually subtracted. We added bootstrap noise to the individual points (the measured light curve errors were randomly added to the simulated values with subtitution). Then a $\chi^2$ test was applied to check whether the simulations are inconsistent with zero at the 99\% significance level. This way we identified those configurations of HATP-P-13c which should have been observed in our measurements (we call these observable configurations in the following). Because our observations are consistent with zero variation, observable configurations are explicitelly excluded by our data.
We identified that 72\%{} of the $10^5$ model transit configurations would have been
observable. Therefore the hypothesis of HAT-P-13c to be a transiting exoplanet can be rejected with 72\%{} confidence. By allowing the mean transit times to be distributed
normally around April 28 17 UT with 1.9 days standard deviation, the level
of significance turns out to be 70\%{}. The level of significance does not vary significantly in the range of orbits allowed by the parameter uncertainties in Bakos et al (2009), because the errors are rather small (3\%{} in $e$ and 0.3\%{} in $\omega$). We reduced the model light curves in amplitude to define the size limit where the detection efficiency starts decreasing significantly. The resulting significance was $65\%{}$ when the amplitude was reduced by 0.45. The planet size corresponding to this signal amplitude is 1.04 $R_J$, which is our detection limit. The conclusion is that roughly
three quaters of all possible transiting configurations are excluded by our observations.
}
This result does not mean that HAT-P-13c could not orbit on an aligned orbit
with HAT-P-13b. HAT-P-13c is quite far from the central star,
hence the star's apparent diameter
is 0.6 degrees as seen from the planet.
Thus, transiting configurations require the orbit to be in a thin region,
very close to our line of sight. There is a huge set of configurations
with HAT-P-13c on { an orbit close to that of planet b,} without displaying any transits.
In this case, Transit Timing Variation (TTV) { of HAT-P-13b} can reveal the orientation
of HAT-P-13c's orbital plane (Bakos et al. 2009).
\subsection{Transit Timing Variations of HAT-P-13b}
\begin{figure}
\includegraphics[bb=138 250 432 488, width=8cm]{1106fit.eps}
\includegraphics[bb=138 250 432 488, width=8cm]{0221fit.eps}
\caption{Model fit to the transit on November 05/06, 2009 (upper panel)
and February 21/22, 2010 (lower panel). V and R band data
are plotted with open and soild dots, respectively.}
\end{figure}
Before the suspected transit of HAT-P-13c, two transits of HAT-P-13b were
observed to refine the period and to search for Transit
Timing Variations (TTV). Data
from 2009-11-05/06 { (measured with the K100 telescope, Table 1)} and
2010-02-21/22 { (K60 telescope)} are plotted
in Fig 2. In November (upper panel in Fig. 2),
the sky was photometric
during the transit, but it was foggy in the evening and from
40 minutes after the egress phase. In February, 2010,
cirri were present
that significantly affected the V band data, but the R light curve
was well reconstructed with constant EPD (see lower panel in Fig. 2).
Times of minima were determined by fitting a model light curve, similarly
to Szab\'o et al. (2010).
For the November 2009 transit, both V and R data were included in the fitting,
while we used only the R curve for the February 2010 transit. (However,
even including the more noisy V curve does not change the mid-transit time
by more than 0.0004 days.)
To reduce the degrees of freedom in the fit, the shape of the model
was not adjusted; we used previously published parameters (Winn et al. 2010).
The model light curve was calculated with our transit simulator
(Simon et al. 2009, 2010). The model
was shifted in time, minimising the rms scatter of the measurements.
We determined new transit times as: BJD 2455141.5522$\pm$0.001 and
2455249.4508$\pm$0.002. Seven transit times were published by
Bakos et al. (2009) which were included in the TTV analysis.
Combining all data, we refined the period of HAT-P-13b to be
2.916293$\pm$0.000010 days, while the determined TTV diagram
is plotted in Fig. 3.
All points are consistent with
zero within the error bars.
It has to be noted that HAT-P-13b must exhibit some TTV,
because of the perturbations by HAT-P-13c.
HAT-P-13c causes 8.5 s
light-time effect (LITE) and perturbations in the orbit of HAT-P-13b.
On short ($\approx$1 yr) time scales, the LITE is dominant.
But the expected LITE is smaller than the ambiguity of our transit times
by a factor of 5, and therefore there is no chance for a positive detection
at this level of accuracy.
\begin{figure}
\includegraphics[width=8cm]{TTV.eps}
\caption{Transit Timing Variation of HAT-P-13b.}
\end{figure}
\section{Summary}
\begin{itemize}
\item{} A multisite campaign has been organised to observe HAT-P-13 around the expected
transit of HAT-P-13c. Two transits of HAT-P-13b were also observed.
\item{}
HAT-P-13c was not observed to transit. We concluded that HAT-P-13c is not a
transiting planet with 75\%{} significance.
\item{} The refined period of HAT-P-13b is 2.916293$\pm$0.000010 days.
The determined TTV is consistent with zero variation.
\end{itemize}
\begin{acknowledgements}
This project has been supported by
the Hungarian OTKA Grants K76816 and MB08C 81013, and the ``Lend\"ulet'' Young
Researchers' Program of the Hungarian Academy of Sciences. GyMSz was supported
by the `Bolyai' Research Fellowship of the Hungarian Academy of Sciences.
The 91cm telescope of the Serra La Nave station is supported by
INAF Osservatorio Astrofisico di Catania, Italy.
We acknowledge assistance of the queue observers, Karzaman Ahmad from LNO and
Hsiang-Yao Hsiao from Lulin Observatory. ZsK acknowledges the support of the
Hungarian OTKA grants K68626 and K81421.
\end{acknowledgements}
|
{
"timestamp": "2010-09-21T02:01:12",
"yymm": "1009",
"arxiv_id": "1009.3598",
"language": "en",
"url": "https://arxiv.org/abs/1009.3598"
}
|
\section{Introduction and the classification of\\Fano 3-folds}\label{s!intr}
A {\em Fano $3$-fold} $X$ is a normal projective 3-fold whose
anti\-canonical divisor $-K_X=A$ is \hbox{$\mathbb Q$-Cartier} and ample. We
eventually impose additional conditions on the singularities and class
group of~$X$, such as terminal, $\mathbb Q$-factorial, quasi\-smooth, prime
(that is, class group $\Cl X$ of rank~1) or $\Cl X=\mathbb Z\cdot A$, but more
general cases occur in the course of our arguments.
We study $X$ via its anticanonical graded ring
\[
R(X,A) = \bigoplus_{m \in \mathbb N} H^0(X,mA).
\]
Choosing generators of $R(X,A)$ embeds $X$ as a projectively normal
sub\-variety $X\subset\mathbb P(a_1,\dots,a_n)$ in weighted projective space.
The anti\-canonical ring $R(X,A)$ is known to be Gorenstein, and we say
that $X\subset\mathbb P(a_1,\dots,a_n)$ is projectively Gorenstein. The {\em
codimension} of $X$ refers to this anticanonical embedding. The
discrete invariants of a Fano 3-fold $X$ are its {\em genus} $g$
(defined by $g+2=h^0(X,-K_X)$) together with a basket of terminal cyclic
quotient singularities; for details see \ref{s!num} and \cite{ABR}.
In small codimension we can write down hypersurfaces, codimension~2
complete intersections and codimension~3 Pfaffian varieties fluently.
This underlies the classification of Fano $3$-folds in
codimension~$\le3$ (see \cite{C3f}, \cite{Fl} and \cite{APhD}): the
famous 95 weighted hypersurfaces, 85 codimension~2 families, and 70
families in codimension~3, of which 69 are $5\times5$ Pfaffians.
Gorenstein in codimension~4 remains one of the frontiers of science:
there is no automatic structure theory, and deformations are almost
always obstructed. Type~I projection and Kustin--Miller unprojection
(see \cite{KM}, \cite{PR}, \cite{Ki}) is a substitute that is sometimes
adequate. This paper addresses codimension~4 Fano $3$-folds in this vein.
The analysis of \cite{APhD}, \cite{A}, \cite{ABR}, \cite{GRDB} provides
145 numerical candidates for codimension~4 Fano $3$-folds. This paper
isolates 115 of these that can be studied using Type~I projections,
hence as Kustin--Miller unprojections. Our main result is
Theorem~\ref{th!main}: each of these 115 numerical candidates occurs in
at least two ways (the Tom and Jerry of the title), that give rise to
topologically distinct varieties $X$.
The reducibility of the Hilbert scheme of Fano 3-folds is a systematic
feature of our results, that goes back to Takagi's study of prime Fano
3-folds with basket of $\frac12(1,1,1)$ points (\cite{T}, Theorem~0.3).
He describes families of varieties having the same invariants, but
arising from different ``Takeuchi programs'', that is, different
Sarkisov links. Four of his numerical cases have codimension~4. The
first, No.~1.4 in the tables of \cite{T}, is our initial case
$X\subset\mathbb P^7(1^7,2)$; it projects to the $(2,2,2)$ complete
intersection, so has $7\times12$ resolution and is unrelated to Tom and
Jerry. Takagi's three other pairs of codimension~4 cases correspond to
our Tom and Jerry families as follows:
\begin{equation*}
\begin{array}{ccc}
X\subset\mathbb P^7(1^4,2^4): & \hbox{Tom$_1 =$ No.~2.2 (8 nodes)}, & \hbox{Jer$_{45} =$ No.~3.3 (9 nodes)},\\
X\subset\mathbb P^7(1^5,2^3): & \hbox{Tom$_1 =$ No.~5.4 (7 nodes)}, & \hbox{Jer$_{23} =$ No.~4.1 (8 nodes)},\\
X\subset\mathbb P^7(1^6,2^2): & \hbox{Tom$_1 =$ No.~4.4 (6 nodes)}, & \hbox{Jer$_{15} =$ No.~1.1 (7 nodes)}.
\end{array}
\end{equation*}
Each of these is prime. In our treatment, each of these numerical cases
admits one further Jerry family consisting of Fano 3-folds of Picard rank
$\ge2$.
Section~\ref{sec!anc} traces the origin of Tom and Jerry back to the
geometry of linear subspaces of $\Grass(2,5)$ and associated
unprojections to twisted forms of $\mathbb P^2\times\mathbb P^2$ and
$\mathbb P^1\times\mathbb P^1\times\mathbb P^1$; for more on this, see
Section~\ref{s!fmt}. Section~\ref{s!main} is a detailed discussion of
our Main Theorem~\ref{th!main}, whose proof occupies the rest of the
paper. Flowchart~\ref{ss!flchart} maps out the proof, which involves
many thousand computer algebra calculations. Section~\ref{s!fmt}
discusses the wider issue of codimension~4 formats, and serves as a
mathematical counterpart to the computer algebra of
Sections~\ref{s!fail}--\ref{s!grdb}. We do not elaborate on this point,
but Tom and Jerry star in many other parallel or serial unprojection
stories beyond Fano 3-folds or codimension~4, notably the diptych
varieties of \cite{aflip}.
We are indebted to a referee for several pertinent remarks that led to
improvements, and to a second referee who verified our computer algebra
calculations independently. This research is supported by the Korean
government WCU Grant R33-2008-000-10101-0.
\section{Ancestral examples}\label{sec!anc}
\subsection{Linear subspaces of $\Grass(2,5)$}\label{s!dP6}
A del Pezzo variety of degree~5 is an $n$-fold $Y^n_5\subset\mathbb P^{n+3}$
of codimension~3, defined by 5 quadrics that are Pfaffians of a
$5\times5$ skew matrix of linear forms. Thus $Y$ is a linear section of
Pl\"ucker $\Grass(2,5)\subset\mathbb P(\bigwedge^2V)$ (here $V=\mathbb C^5$). We want
to unproject a projective linear subspace $\mathbb P^{n-1}$ contained as a
divisor in $Y$ to construct a degree~6 del Pezzo variety
$X^n_6\subset\mathbb P^{n+4}$. The crucial point is the following.
\begin{lem} The Pl\"ucker embedding $\Grass(2,5)$ contains two families
of maximal linear subspaces. These arise from
\begin{enumerate}
\renewcommand{\labelenumi}{(\Roman{enumi})}
\item The $4$-dimensional vector subspace $v\wedge V\subset\bigwedge^2V$
for a fixed $v\in V$.
\item The $3$-dimensional subspace $\bigwedge^2U\subset\bigwedge^2V$ for
a fixed $3$-dimensional vector subspace $U\subset V$.
\end{enumerate}
\end{lem}
Thus there are two different formats to set up $\mathbb P^{n-1}\subset Y$.
Case~I gives $\mathbb P^3_v\subset\Grass(2,5)$. A section of $\Grass(2,5)$ by
a general $\mathbb P^7$ containing $\mathbb P^3_v$ is a 4-fold $Y^4$ whose
unprojection is $\mathbb P^2\times\mathbb P^2\subset\mathbb P^8$. Case~II gives
$\Grass(2,U)=\mathbb P^2_U\subset\Grass(2,5)$. A section of $\Grass(2,5)$ by a
general $\mathbb P^6$ containing $\mathbb P^2_U$ is a 3-fold $Y^3$ whose unprojection
is $\mathbb P^1\times\mathbb P^1\times\mathbb P^1\subset\mathbb P^7$.
The proof is a lovely exercise. Hint: use local and Pl\"ucker coordinates
\begin{equation}
\begin{pmatrix} 1&0&a_1&a_2&a_3 \\ 0&1&b_1&b_2&b_3 \end{pmatrix}
\quad\hbox{and}\quad
\begin{pmatrix} 1&a_1&a_2&a_3 \\ &b_1&b_2&b_3 \\ &&m_{12} & m_{13} \\
&&& m_{23} \end{pmatrix}
\end{equation}
with Pl\"ucker equations $m_{12}=a_1b_2-a_2b_1$, etc.; permute the
indices and choose signs pragmatically to make this true. Prove that in
Pl\"ucker $\mathbb P^9$, the tangent plane $m_{12}=m_{13}=m_{23}=0$ intersects
$\Grass(2,5)$ in the cone over the Segre embedding of $\mathbb P^1\times\mathbb P^2$.
\subsection{Tom$_1$ and Jer$_{12}$ in equations}\label{s!same}
Tom$_1$ is
\begin{equation}
\begin{pmatrix}
y_1&y_2&y_3&y_4 \\
& m_{23}& m_{24}& m_{25}\\
&& m_{34}& m_{35}\\
&&& m_{45}\\
\end{pmatrix} \label{eq!1.2}
\end{equation}
with $y_{1\dots4}$ arbitrary elements, and the six entries $m_{ij}$
linear combinations of a regular sequence $x_{1\dots4}$ of length four.
Expressed vaguely, there are ``two constraints on these six entries'';
these two coincidences take the simplest form when $m_{23}=m_{45}=0$. In
this case, the Pfaffian equations all reduce to binomials, and can be
seen as the $2\times2$ minors of an array: as a slogan,
\begin{equation}
\hbox{$4\times4$ Pfaffians of }
\left(\begin{smallmatrix}
y_1&y_2&y_3&y_4 \\
& 0 & m_{24}& m_{25}\\
&& m_{34}& m_{35}\\
&&& 0\\
\end{smallmatrix}\right)
=\hbox{$2\times2$ minors of }
\left(\begin{smallmatrix}
* & y_3 &y_4 \\
y_1 & m_{24}& m_{25}\\
y_2& m_{34}& m_{35}
\end{smallmatrix}\right).
\label{eq!1.4}
\end{equation}
That is, the $4\times 4$ Pfaffians on the left equal the five $2\times2$
minors of the array on the right. To see the Segre embedding of
$\mathbb P^2\times\mathbb P^2$ and its linear projection from a point, replace the
star entry by the unprojection variable $s$.
In a similar style, Jer$_{12}$ is
\begin{equation}
\begin{pmatrix}
m_{12} & m_{13}& m_{14}& m_{15}\\
& m_{23}& m_{24}& m_{25}\\
&& y_{34}& y_{35}\\
&&& y_{45}\\
\end{pmatrix}
\end{equation}
with $y_{34},y_{35},y_{45}$ arbitrary, and the seven entries $m_{ij}$
linear combinations of $x_{1\dots4}$. Vaguely, ``three constraints on
these seven entries''; most simply, these take the form
$m_{15}=m_{23}=0$, $m_{24}=m_{14}$. We leave you to see this as the
linear projection of $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$, starting from the
hint:
\begin{equation} \label{eq!cube1}
\hbox{$4\times4$ Pfaffians of }
\begin{pmatrix}
t&z_1&z_2&0 \\
& 0 & z_2& z_3\\
&& y_3& y_2\\
&&& y_1\\
\end{pmatrix}
=\hbox{$2\times2$ minors of }
\begin{picture}(30,30)(0,0)
\renewcommand{\arraycolsep}{.2em}
\put(8,-7){$\begin{matrix}
* &\frac{\quad}{\quad}& y_2 \\
\vert&&\vert \\
y_1 &\frac{\quad}{\quad}& z_3
\end{matrix}$}
\put(28,7){$\begin{matrix}
y_3 &\frac{\quad}{\quad}&z_1 \\
\vert&&\vert \\
z_2 &\frac{\quad}{\quad}& t
\end{matrix}$}
\qbezier(20,15)(23,17)(26,19)
\qbezier(20,-12)(23,-10)(26,-8)
\qbezier(52,15)(55,17)(58,19)
\qbezier(52,-12)(55,-10)(58,-8)
\end{picture}
\kern16mm
\end{equation}
that is, on the right, take $2\times 2$ minors of the three square faces
out of $t$, together with the ``diagonal'' minors $y_1z_1=y_2z_2=y_3z_3$,
then replace the star by an unprojection variable. Compare
\eqref{eq!cube}.
\subsection{General conclusions}
\begin{dfn} \rm \label{d!TJ}
Tom$_i$ and Jer$_{ij}$ are matrix formats that specify unprojection
data, namely a codimension~3 scheme $Y$ defined by a $5\times5$ Pfaffian
ideal, containing a codimension~4 complete intersection $D$. Given a
regular sequence $x_{1\dots4}$ in a regular ambient ring $R$ generating
the ideal $I_D$, the ideal of $Y$ is generated by the Pfaffians of a
$5\times5$ skew matrix $M$ with entries in $R$, subject to the
conditions \begin{description}
\item{Tom$_i$:} the 6 entries $m_{jk}\in I_D$ for all $j,k\ne i$; in
other words, the 4 entries $m_{ij}$ of the $i$th row and column are free
choices, but the other entries of $M$ are required to be in $I_D$. See
\eqref{eq!2.8} for an example.
\item{Jer$_{ij}$:} the 7 entries $m_{kl}\in I_D$ if either $k$ or $l$
equals $i$ or $j$. See \eqref{eq!2.9} for an example. The bound entries
are the {\em pivot} $m_{ij}$ and the two rows and columns through it.
The 3 free entries are the Pfaffian partners $m_{kl}$, $m_{km}$,
$m_{lm}$ of the pivot, where $\{i,j,k,l,m\}=\{1,2,3,4,5\}$. In $Y$, the
pivot vanishes twice on $D$.
\end{description}
\end{dfn}
Case~I in \ref{s!dP6} is the ancestor of our Tom constructions and~II
that of Jerry. Our main aim in what follows is to work out several
hundred applications of the same formalism to biregular models of Fano
3-folds, when our ``constraints''
\begin{equation}
m_{ij}=\hbox{linear combination of $x_{1\dots4}$}
\end{equation}
are not linear, do not necessarily reduce to a simple normal form, and
display a rich variety of colourful and occasionally complicated
behaviour.
Nevertheless, the same general tendencies recur again and again. Tom
tends to be fatter than Jerry. Jerry tends to have a singular locus of
bigger degree than Tom, and the unprojected varieties $X$ have different
topologies, in fact different Euler numbers. For example, $Y^4$ in
Case~I has two lines of transversal nodes; the $Y^3$ in Case~II has
three nodes. If we only look at 3-folds in \ref{s!dP6} (cutting $Y^4$ by
a hyperplane), the unprojected varieties $X$ are then the familiar del
Pezzo 3-folds of index 2, namely the flag manifold of $\mathbb P^2$ versus
$\mathbb P^1\times\mathbb P^1\times\mathbb P^1$; see Section~\ref{s!Jac} (especially
Remark~\ref{rk!2v3}) for the number of nodes (2 and 3 in the two cases)
via enumerative geometry. Tom equations often relate to extensions of
$\mathbb P^2\times\mathbb P^2$ such as the ``extrasymmetric $6\times6$ format'';
Jerry equations often relate to extensions of
$\mathbb P^1\times\mathbb P^1\times\mathbb P^1$ such as the ``rolling factors format'' (an
anticanonical divisor in a scroll) or the ``double Jerry format'';
Section~\ref{s!fmt} gives a brief discussion.
\section{The main result}\label{s!main}
\subsection{Numerical data of Fano 3-folds}\label{s!num}
Let $X$ be a Fano $3$-fold. As explained in \cite{ABR}, the numerical
data of $X$ consists of an integer genus $g\ge-2$ plus a basket
$\mathcal B=\{\frac1r(1,a,r-a)\}$ of terminal cyclic orbi\-fold points; this
data determines the Hilbert series $P_X(t)= \sum_{a\ge0}h^0(X,nA) t^n$ of
$R(X,A)$, and is equivalent to it. At present we only treat cases when
the ring is generated as simply as possible, and not (say) cases that
fall in a monogonal or hyperelliptic special case. The database
\cite{GRDB} lists cases of small codimension, including 145 candidate
cases in codimension~4 from Alt{\i}nok's thesis \cite{APhD}. We
sometimes say Fano \hbox{3-fold} to mean numerical candidate; the abuse of
terminology is fairly harmless, because practically all the candidates
in codimension $\le5$ (possibly all of them) give rise to quasismooth
Fano \hbox{3-folds}; in fact usually more than one family, as we now
relate.
\subsection{Type~I centre and Type I projection}\label{s!TyIc}
An orbifold point $P\in X$ of type $\frac1r(1,a,r-a)$ with $r\ge2$ is a
{\em Type~I centre} if its orbinates are restrictions of global forms
$x\in H^0(A)$, $y\in H^0(aA)$, $z\in H^0((r-a)A)$ of the same weight.
The condition means that after projecting, the exceptional locus of the
projection is a weighted projective plane $\mathbb P(1,a,r-a)$ that is
embedded projectively normally.
One may view a projection $P\in X\dasharrow Y\supset D$ in simple terms: in
geometry, as the map
$(x_1,\dots,x_n)\mapsto(x_1,\dots,\widehat{x_i},\dots,x_n)$ analogous to
linear projection $\mathbb P^n\dasharrow\mathbb P^{n-1}$ from centre
$P_i=(0,\dots,1,\dots,0)$; or in algebra, as eliminating a variable,
corresponding to passing to a graded subring
$k[x_1,\dots,\widehat{x_i},\dots,x_n]$; to be clear, the distinguishing
characteristic is not the eliminated variable $x_i$, rather the point
$P_i$ and the complementary system of variables $x_j$ that vanish there.
We take the more sophisticated view of \cite{CPR}, 2.6.3 of a projection
as an intrinsic biregular construction of Mori theory; namely a diagram
\begin{equation}
\renewcommand{\arraycolsep}{1pt}
\begin{array}{ll}
\kern2.5em P \in X & \subset \mathbb P(a_0,\dots,a_n)\\
\kern3.25em\nearrow \\
E \subset X_1 \\
\kern3.25em\searrow \\
\kern2.5em D \subset Y & \subset \mathbb P(a_0,\dots,\widehat{a_k},\dots,a_n)
\end{array}
\label{eq!proj}
\end{equation}
consisting of an extremal extraction $\sigma\colon X_1\to X$ centred at $P$
followed by the anticanonical morphism $\varphi\colon X_1\to Y$. In more
detail, we have the following result.
\begin{lem}\label{l!s-amp}
The Type I assumption implies that $-K_{X_1}$ is semiample.
The anticanonical morphism
$\varphi\colon X_1\to Y$ contracts only curves $C$ with $-K_{X_1}C=0$
meeting the exceptional divisor $E=\mathbb P(1,a,r-a)\subset X_1$
transversely in one point.
\end{lem}
\paragraph{Proof} A theorem of Kawamata \cite{Ka} (discussed also in
\cite{CPR}, 3.4.2) says that the $(1,a,r-a)$ weighted blowup $\sigma\colon
X_1\to X$ is the unique Mori extremal extraction whose centre meets the
$\frac1r(1,a,r-a)$ orbifold point $P\in X$. It has exceptional divisor
the weighted plane $E=\mathbb P(1,a,r-a)$ with discrepancy $\frac1r$. Thus $-K_{X_1}=-K_X-\frac1rE$, and the anticanonical ring of $X_1$ consists
of forms of weight $d$ in $R(X,-K_X)$ vanishing to order $\ge\frac dr$
on $E$. The homogenising variable $x_k$ of degree $r$
with $x_k(P)=1$ does not vanish at all, so is eliminated. By
assumption, the orbinates $x,y,z$ at $P$ are global forms of weights
$1,a,r-a$ vanishing to order exactly $\frac1r,\frac ar,\frac{r-a}r$, so
these extend to regular elements of $R(X_1,-K_{X_1})$. Locally at $P$,
appropriate monomials in $x,y,z$ base the sheaves $\mathcal O_X(d)$ modulo any
power of the maximal ideal $m_P$, so we can adjust the remaining
generators $x_l$ of $R(X,-K_X)$ to vanish to order $\ge \frac{\wt x_l}r$,
and so they lift to $R(X_1,-K_{X_1})$. It follows that $-K_{X_1}$ is
semi\-ample and the anti\-canonical morphism $\varphi\colon X_1\to Y$
takes $E$ isomorphically to $D\subset Y$.
\ifhmode\unskip\nobreak\fi\quad\ensuremath{\mathrm{QED}} \par\medskip
In our cases, $\varphi$ contracts a nonempty finite set of flopping
curves to singular points of $Y$ on $D$, and $Y$ is a codimension~3
Fano 3-fold. The anti\-canonical model $Y$ is not $\mathbb Q$-factorial because
the divisor $D\subset Y$ is not $\mathbb Q$-Cartier. It is the {\em midpoint of
a Sarkisov link} (compare \cite{CPR}, 4.1 (3)); we develop this idea in
Part~II. The ideal case is when each $\Gamma_i\subset X_1$ is a copy of $\mathbb P^1$
with normal bundle $\mathcal O(-1,-1)$, or equivalently, $Y$ has only
ordinary nodes on $D$. We prove that this happens generically in all
our families.
In other situations, Type~I allows $\varphi$ to be an isomorphism, typically
for $X$ of large index. At the other extreme, the Type~I condition on its
own does not imply that $-K_{X_1}$ is big, and $\varphi$ could be an elliptic
Weierstrass fibration over $D=\mathbb P(1,a,r-a)$, although this never happens
for codimension~4 Fano 3-folds. Also $\varphi$ might
contract a surface to a curve of canonical singularities of $Y$; then
$X\dasharrow Y$ is a ``bad link'' in the sense of \cite{CPR}, 5.5. We know
examples of this if $X$ is not required to be $\mathbb Q$-factorial and prime,
but none with these conditions.
\begin{exun} \rm
Consider the general codimension~2 complete intersection
\begin{equation}
X_{12,14}\subset\mathbb P(1,1,4,6,7,8)_{\Span{x,a,b,c,d,e}}.
\end{equation}
The coordinate point $P_e=(0,\dots,0,1)$ is necessarily contained in
$X$: near it, the two equations $f_{12}:be=F_{12}$ and
$g_{14}:ce=G_{14}$ express $b$ and $c$ as implicit functions of the
other variables, so that $X$ is locally the orbifold point
$\frac18(1,1,7)$ with orbinates $x,a,d$.
Eliminating $e$ from $f_{12},g_{14}$ projects $X_{12,14}$ birationally
to the hypersurface
$Y_{18}:(bG-cF=0)\subset\mathbb P(1,1,4,6,7)_{\Span{x,a,b,c,d}}$. Note that
$Y$ contains the plane $D=\mathbb P(1,1,7)_{\Span{x,a,d}}=V(b,c)$, and has in
general $24=\frac17\times12\times14$ ordinary nodes at the points
$F=G=0$ of $D$.
In this case, the Kustin--Miller unprojection of the ``opposite''
divisor $(b=F=0)\subset Y$ completes the 2-ray game on $X_1$ to a
Sarkisov link, in the style of Corti and Mella \cite{CM}: the flop
$X_1\to Y\leftarrow Y^+$ blows this up to a $\mathbb Q$-Cartier divisor, and
the unprojection variable $z_2=c/b=G/F$ then contracts it to a
nonorbifold terminal point $P_z\in
Z_{14}\subset\mathbb P(1,1,4,7,2)_{\Span{x,a,b,d,z}}$.
\end{exun}
\subsection{Main theorem}
Write $P\in X$ for the numerical type of a codimension~4 Fano 3-fold of
index~1 marked with a Type~I centre. There are 115 or 116 candidates for
$X$ (depending on how you count the initial case); some have two or
three centres, and treating them separately makes 162 cases for $P\in X$.
\begin{thm}\label{th!main}
Let $P\in X$ be as above; then the projected variety is realised as a
codimension~$3$ Fano $Y\subsetw\mathbb P^6$, and $Y$ can be made to contain a
co\-or\-dinate stratum $D=\mathbb P(1,a,r-a)$ of $w\mathbb P^6$ in several ways.
For every numerical case $P\in X$, there are several formats, at least
one Tom and one Jerry (see Definition~\ref{d!TJ}) for which the general
$D\subset Y$ only has nodes
on $D$, and unprojects to a quasi\-smooth Fano $3$-fold $X\subsetw\mathbb P^7$.
In different formats, the resulting $Y$ have different numbers of nodes
on $D$, so that the unprojected quasi\-smooth varieties $X$ have
different Betti numbers. Therefore in each of the $115$ numerical cases
for $X$, the Hilbert scheme has at least two components containing
quasismooth Fano $3$-folds.
\end{thm}
\subsection{Discussion of the result}
The theorem constructs around 320 different families of quasi\-smooth
Fano \hbox{$3$-folds}. We do not burden the journal pages with the
detailed lists, the {\em Big Table} in the Graded Ring Database
\cite{GRDB}; the case worked out in Section~\ref{exa!main} may be
adequate for most readers. Our data and the software tools for
manipulating them are available from \cite{GRDB}.
Our 162 cases for $P\in X$ project to $D\subset Y\subset w\mathbb P^6$; of the
69 codimension~3 families of Fanos $Y$ that are $5\times5$ Pfaffians, 67
are the images of projections, each having up to four candidate planes
$D\subset Y$. For each of the 162 candidate pairs $D\subset Y$, we study
5 Tom and 10 Jerry formats, of which at least one Tom and one Jerry
succeeds (often one more, occasionally two), so that
Theorem~\ref{th!main} describes around 450 constructions of pairs $P\in
X$ of quasismooth Fano $3$-folds with marked centre of projection,
giving around 320 different families of $X$.
Theorem~\ref{th!main} covers codimension~4 Fano 3-folds of index~1 for
which there exists a Type~I centre. If one believes the possible
conjecture raised in \cite{ABR}, 4.8.3 that every Fano 3-fold in the
Mori category (that is, with terminal singularities) admits a
$\mathbb Q$-smoothing, this also establishes the components of the Hilbert
scheme of codimension~4 Fano 3-folds in these numerical cases. The main
novelty of this paper (and this was a big surprise to us) is that in
every case, the moduli space has 2, 3 or 4 different components.
An important remaining question is which $X$ are prime. In some cases,
our Tom or Jerry matrices have a zero entry, possibly after massaging.
Then 3 of the Pfaffian equations are binomial, which implies that $X$
has class group of rank $\rho\ge2$. This happens in the ancestral
examples of Section~\ref{sec!anc} and the easier cases
\ref{s!t2}--\ref{s!j25} of Section~\ref{exa!main}. Our Big Table
confirms that if we set aside all these cases with a zero, each of our
numerical possibilities for Type~I centres $P\in X$ admits exactly one
Tom and one Jerry construction that is potentially prime. Compare
Takagi's cases discussed in Section~\ref{s!intr}. We return to this
question in Part~II.
\subsection{Flowchart} \label{ss!flchart}
Our proof in Sections~\ref{s!fail}--\ref{s!grdb} applies computer algebra
calculations and verifications to a couple of thousand cases; any of
these could in principle be done by hand. We go to the database for
candidates for $P\in X$, figure out the weights of the coordinates of
$D\subset Y\subset w\mathbb P^6$ and the matrix of weights, and list all
inequivalent Tom and Jerry formats. Section~\ref{s!fail} gives criteria
for a format to fail. In the cases that pass these tests,
Section~\ref{s!ns} contains an algorithm to produce $D\subset Y$ in the
given format, and to prove that it has only allowed singularities (that
is, only nodes on~$D$). Section~\ref{s!Jac} contains the Chern class
calculation for the number of nodes, so proving that the different
constructions build topologically distinct varieties.
Section~\ref{s!grdb} gives ``quick start-up'' instructions; do not
under any circumstances read the \verb!README! file.
\subsection{Further outlook}
The reducibility phenomenon appearing in this paper is characteristic
of Gorenstein in codimension~4; we have several current preprints and
work in progress addressing different aspects of this. See for example
\cite{Ki}.
This paper concentrates on 115 numerical cases of codimension~4 Fano
\hbox{3-folds} of index~1. Most of the remaining numerical cases from
Alt{\i}nok's list of 145 \cite{APhD} can be studied in terms of more
complicated Type~II or Type~IV unprojections, when the unprojection
divisor is not projectively normal; see \cite{Ki} for an introduction.
We believe that codimension~5 is basically similar: most cases have
two or more Type~I centres that one can project to smaller codimension,
leading to parallel unprojection constructions.
The methods of this paper apply also to other categories of varieties,
most obviously K3 surfaces and Calabi--Yau 3-folds. K3 surfaces are
included as general elephants $S\in|{-}K_X|$ in our Fano 3-folds,
although the K3 is unobstructed, so that passing to the elephant hides
the distinction between Tom and Jerry. We can also treat some of the
Fano 3-folds of index $>1$ of Suzuki's thesis \cite{S}, \cite{BS}; we
have partial results on the existence of some of these families, and
hope eventually to cover the cases not excluded by Prokhorov's
birational methods \cite{Pr}.
This paper uses Type~I projections $X\dasharrow Y$ to study the biregular
question of the existence and moduli of $X$; however, in each case, the
Kawamata blowup $X_1\to X$ initiates a 2-ray game on $X_1$, with the
anticanonical model $X_1\to Y$ and its flop $Y\leftarrow Y^+$ as first step. In
many cases, we know how to complete this to a Sarkisov link using Cox
rings, in the spirit of \cite{CPR}, \cite{CM}, \cite{BCZ} and \cite{BZ};
we return to this in Part~II.
\section{Extended example} \label{exa!main}
The case $g=0$ plus basket
$\bigl\{\frac12(1,1,1),\frac13(1,1,2),\frac14(1,1,3),\frac15(1,1,4)\bigr
\}$ gives the codimension~4 candidate $X\subset\mathbb P^7(1,1,2,3,3,4,4,5)$
with Hilbert numerator
\begin{equation}
1-2t^6-3t^7-3t^8-t^9+t^9+4t^{10}+6t^{11}+\cdots+t^{22}.
\end{equation}
It has three different possible Type~I centres, namely the $\frac13$,
$\frac14$ or $\frac15$ points. We project away from each of these,
obtaining consistent results; each case leads to four unprojection
constructions for $X$, two Toms and two Jerries:
\begin{description}
\item{from $\frac13$:} gives $\mathbb P(1,1,2)\subset Y\subset\mathbb P(1,1,2,3,4,4,5)$
with matrix of weights
\begin{equation}
\begin{pmatrix}
2&2&3&4 \\
&3&4&5 \\
&&4&5 \\
&&&6
\end{pmatrix} \qquad\hbox{and}\qquad
\begin{array}l
\hbox{Tom$_2$ has 13 nodes} \\
\hbox{Tom$_1$ has 14 nodes} \\
\hbox{Jer$_{45}$ has 16 nodes} \\
\hbox{Jer$_{25}$ has 17 nodes}
\end{array} \label{eq!3.2}
\end{equation}
\item{from $\frac14$:} gives $\mathbb P(1,1,3)\subset
Y\subset\mathbb P(1,1,2,3,3,4,5)$ with matrix of weights
\begin{equation}
\begin{pmatrix}
2&3&3&4 \\
&3&3&4 \\
&&4&5 \\
&&&5
\end{pmatrix} \qquad\hbox{and}\qquad
\begin{array}l
\hbox{Tom$_3$ has 9 nodes} \\
\hbox{Tom$_1$ has 10 nodes} \\
\hbox{Jer$_{35}$ has 12 nodes} \\
\hbox{Jer$_{15}$ has 13 nodes}
\end{array} \label{eq!3.3}
\end{equation}
\item{from $\frac15$:} gives $\mathbb P(1,1,4)\subset
Y\subset\mathbb P(1,1,2,3,3,4,4)$ with matrix of weights
\begin{equation}
\begin{pmatrix}
2&2&3&3 \\
&3&4&4 \\
&&4&4 \\
&&&5
\end{pmatrix} \qquad\hbox{and}\qquad
\begin{array}l
\hbox{Tom$_4$ has 8 nodes} \\
\hbox{Tom$_2$ has 9 nodes} \\
\hbox{Jer$_{24}$ has 11 nodes} \\
\hbox{Jer$_{14}$ has 12 nodes}
\end{array} \label{eq!3.4}
\end{equation}
\end{description}
Specifically, we assert that {\em in each of these 12 cases, if we pour
general elements of the ideal $I_D$ and general elements of the ambient
ring into the Tom or Jerry matrix $M$ as specified in
Definition~\ref{d!TJ}, the Pfaffians of $M$ define a Fano $3$-fold $Y$
having only the stated number of nodes on $D$, and the resulting $X$ is
quasismooth.} Section~\ref{s!ns} verifies this claim by cheap computer
algebra, although we work out particular cases here without such
assistance. Section~\ref{s!Jac} computes the number of nodes in each
case from the numerical data. Imposing the unprojection plane $D$ on the
general quasismooth $Y_t$ introduces singularities on $Y=Y_0$, nodes in
general, which are then resolved on the quasismooth $X_1$. Each node
thus gives a conifold transition, replacing a vanishing cycle $S^3$ by a
flopping line $\mathbb P^1$, and therefore adds 2 to the Euler number of $X$;
so the four different $X$ have different topology.
The unprojection formats and nonsingularity algorithms establish the
existence of four different families of quasismooth Fano 3-folds $X$.
The rest of this section analyses these in reasonably natural formats;
an ideal would be to free ourselves from unprojection and computer
algebra, although we do not succeed completely.
For illustration, work from $\frac13$; take
$X\subset\mathbb P^7(1,1,2,3,3,4,4,5)_{\Span{x,a,b,c,d,e,f,g}}$, and assume
that $P_d=(0,0,0,0,1,0,0,0)$ is a Type~1 centre on $X$ of type
$\frac13(1,1,2)$. The assumption means that $P\in X$ is quasismooth with
orbinates $x,a,b$. The cone over $X$ is thus a manifold along the
$d$-axis, and therefore, by the implicit function theorem, four of the
generators of $I_X$ form a regular sequence locally at $P_d$, with
independent derivatives, say $cd=\cdots$, $de=\cdots$, $df=\cdots$,
$dg=\cdots$ of degrees $6,7,7,8$. Eliminating $d$ gives the Type~I
projection $X\dasharrow Y$ where $Y\subset\mathbb P^6(1,1,2,3,4,4,5)$ has Hilbert
numerator
\begin{equation}
1-t^6-t^7-2t^8-t^9+t^{10}+2t^{11}+t^{12}+t^{13}-t^{19}.
\end{equation}
Let $Y$ be a $5\times5$ Pfaffian matrix with weights as in
\eqref{eq!3.2}. Since rows 2 and 3 have the same weights, we can
interchange the indices 2 and 3 throughout; thus Tom$_2$ is equivalent
to Tom$_3$, Jer$_{25}$ to Jer$_{35}$, and so on.
\subsection{Failure} \label{s!fa}
Some Tom and Jerry cases fail, either for coarse or for more subtle
reasons; for example, it sometimes happens that for reasons of weight,
one of the variables $x_i$ cannot appear in the matrix, so the variety
is a cone, which we reject. Section~\ref{s!fail} discusses failure
systematically.
In the present case $D=\mathbb P(1,1,2)_{\Span{x,a,b}}$, the generators of
$I_D=(c,e,f,g)$ all have weight $\ge3$, but $\wt m_{12},m_{13}=2$. Thus
requiring $m_{12},m_{13}\in I_D$ forces them to be zero, making the
Pfaffians $\Pf_{12.34}$ and $\Pf_{12.35}$ reducible. This kills Tom$_4$,
Tom$_5$, Jer$_{1i}$ for any $i$ and Jer$_{23}$. The same argument
says that Tom$_2$ has $m_{13}=0$ and Jer$_{25}$ has $m_{12}=0$, a key
simplification in treating them: a zero in $M$ makes three of the
Pfaffians binomial.
We see below that Jer$_{24}$ fails for an interesting new reason. The
other cases all work, as we could see from the nonsingularity algorithm
of Section~\ref{s!ns}. Tom$_2$ and Jer$_{25}$ are simpler, and we start
with them, whereas Tom$_1$ and Jer$_{45}$ involve heavier calculations;
they are more representative of constructions that possibly lead to
prime $X$.
\subsection{Tom$_2$}\label{s!t2} The analysis of the matrix proceeds as:
\begin{equation}
\renewcommand{\arraycolsep}{.35em}
\begin{pmatrix}
K_2 & 0 & c & e \\
& L_3 & M_4 & N_5 \\
&& f & g \\
&&& \Span{c,e,f,g}_6
\end{pmatrix} \mapsto
\begin{pmatrix}
b & 0 & c & e \\
& L_3 & M_4 & N_5 \\
&& f & g \\
&&& 0
\end{pmatrix}
\mapsto
\begin{pmatrix}
b & c & e \\
d & M & N \\
L & f & g
\end{pmatrix}
\label{eq!2.8}
\end{equation}
here $m_{13}=0$ is forced by low degree, $K_2$, $L_3$, $M_4$, $N_5$ are
general forms of the given degrees, that we can treat as tokens
(independent indeterminates), and the four entries
$m_{14},m_{15},m_{34},m_{35}$ are general elements of $I_D$ that we
write $c,e,f,g$ by choice of coordinates. Next, $m_{45}$ can be whittled
away to 0 by successive row-column operations that do not harm the
remaining format; seeing this is a ``crossword puzzle'' exercise that
uses the fact that $m_{13}=0$ and all the entries in Row~2 are general
forms. For example, subtracting a suitable multiple of Row~1 from Row~5
(and then the same for the columns) kills the $c$ in $m_{45}$, while
leaving $m_{15}$ and $m_{35}$ unchanged (because $m_{11}=m_{13}=0$) and
modifying $N_5$ by a multiple of $K_2$, which is harmless because $N_5$
is just a general ring element of weight~5.
The two zeros imply that all the Pfaffians are binomial, and, as in
\ref{s!same}, putting in the unprojection variable $d$ of weight 4
gives the $2\times2$ minors of the matrix on the right. The equations
describe $X$ inside the projective cone over
$w(\mathbb P^2\times\mathbb P^2)\subset\mathbb P(2,3^3,4^3,5^2)$ with vertex
$\mathbb P^1_{\Span{x,a}}$ as the complete intersection of three general forms
of degree $3,4,5$ expressing $L,M,N$ in terms of the other variables.
(It is still considerably easier to do the nonsingularity computation
after projecting to smaller codimension.)
\subsection{Jer$_{25}$}\label{s!j25} We start from
\begin{equation}
\begin{pmatrix}
0 & b & L_3 & f \\
& c & e & g \\
&& M_4 & \lambda_1 e \\
&&& \mu_3 c + \nu_2 e
\end{pmatrix}
\label{eq!2.9}
\end{equation}
where $m_{12}=0$ is forced by low degree, and we put tokens $b,L,M$ in
place of the free entries $m_{13},m_{14},m_{34}$. We have cleaned out
$m_{35}$ and $m_{45}$ as much as we can; the quantities
$b,L,M,\lambda,\mu,\nu$ are general ring elements of the given weights.
We have to adjoin $d$ together with unprojection equations for
$dc,de,df,dg$. There are various ways of doing this, including the
systematic method of writing out the Kustin--Miller homomorphism between
resolution complexes, that we use only as a last resort. An ad hoc
parallel unprojection method is to note that $g$ appears only as the
entry $m_{25}$, so we can project it out to a codimension~2 c.i.\
containing the plane $c=e=f=0$:
\begin{equation}
\begin{pmatrix} \mu b & \nu b-\lambda L & M \\ L & -b & 0 \end{pmatrix}
\begin{pmatrix} c \\ e \\ f \end{pmatrix}=0.
\end{equation}
The equations for $dc,de,df$ come from Cramer's rule, and we can
write the unprojection in rolling factors format:
\begin{equation}
\bigwedge^2\begin{pmatrix}
b & L & f & d\\
c & e & g & M
\end{pmatrix}
\quad\hbox{and}\quad
\begin{array}l
\mu b^2 + \nu bL-\lambda L^2+ df,\\
\mu bc + \nu cL-\lambda eL + Mf, \\
\mu c^2 + \nu ce-\lambda e^2 + Mg.
\end{array}
\label{eq!rf}
\end{equation}
The first set of equations of \eqref{eq!rf}, with the entries viewed as
indeterminates, defines
$w(\mathbb P^1\times\mathbb P^3)\subset\mathbb P(2,3,3,3,4,4,4,5)_{\Span{b,c,d,L,e,f,M,g}}$;
the second set is a single quadratic form evaluated on the rows, so
defines a divisor in the cone over this with vertex
$\mathbb P^1_{\Span{x,a}}$. Finally, setting $L,M$ general forms gives $X$ as
a complete intersection in this.
\subsection{Jer$_{24}$ fails}
The matrix has the form
\begin{equation}
\begin{pmatrix}
0 & b & c & L_4 \\
& c & f & g \\
&& e & M_5 \\
&&& \Span{c,e,f,g}_6
\end{pmatrix}
\mapsto
\begin{pmatrix}
0 & b & c & L_4 \\
& c & f & g \\
&& e & M_5 \\
&&& 0
\end{pmatrix}
\end{equation}
The entries in the rows and columns through the pivot $m_{24}=f$ are
general elements of the ideal $I_D=(c,e,f,g)$. As before, $m_{12}=0$ is
forced by degrees. Although \ref{s!fish}, (5) fails this for a
mechanical reason, we discuss it in more detail as an instructive case,
giving a perfectly nice construction of the unprojected variety $X$,
that happens to be slightly too singular. First, please check that the
entry $m_{45}$ can be completely taken out by row and column operations.
For example, to get rid of the $e$ term in $m_{45}$, add $\alpha_3$ times
Row~3 to Row~5; in $m_{25}$ this changes $g$ to $g+\alpha c$, that we
rename $g$.
One sees that the equations of the unprojected variety $X$ take the form
\begin{equation}
\bigwedge^2\begin{pmatrix} b&c&e&f \\ d&L&M&g \end{pmatrix}=0
\quad\hbox{and}\quad \left\{
\begin{matrix}
bf=c^2, \\ bg=cL, \\ dg=L^2.
\end{matrix}
\right.
\end{equation}
(exercise, hint: project out $f$ or $g$). In straight projective
space, these equations define $\mathbb P^1\times Q\subset\mathbb P^1\times\mathbb P^3$
where $Q\subset\mathbb P^3$ is the quadric cone. This is singular in
codimension~2, so the 3-fold $X$ cannot have isolated singularities.
\subsection{Tom$_1$}
The matrix and its clean form are
\begin{equation}
\begin{pmatrix}
b& K_2 & L_3 & M_4 \\
& c & e & g \\
&& f & \Span{c,e,f,g}_5 \\
&&& \Span{c,e,f,g}_6
\end{pmatrix}
\mapsto
\begin{pmatrix}
b& K & L & M \\
& c & e & g \\
&& f & \lambda_1e \\
&&& \mu_3c+\nu_2e
\end{pmatrix}
\end{equation}
where $K,L,M$ and $\lambda,\mu,\nu$ are general forms, that we treat as
tokens. We add a multiple of Column~2 to Column~5 to clear $c$ from
$m_{35}$, so we cannot use the same operation to clear $e$ from
$m_{45}$. The nonsingularity algorithm of Section~\ref{s!ns} ensures
that for general choices this has only nodes on $D$.
We show how to exhibit $X$ as a triple parallel unprojection from a
hypersurface in the product of three codimension~2 c.i.\ ideals (compare
\ref{s!xtra}). Since $g$ only appears as $m_{25}$, it is
eliminated by writing the two Pfaffians $\Pf_{12.34}$ and $\Pf_{13.45}$
as:
\begin{equation}
\begin{pmatrix} L & -K & b \\ \mu K & \nu K-\lambda L & M \end{pmatrix}
\begin{pmatrix} c \\ e \\ f \end{pmatrix}= 0;
\end{equation}
in the same way, $\Pf_{12.45}$ and $\Pf_{12.35}$ eliminate $f$:
\begin{equation}
\begin{pmatrix} M & \lambda b & -K \\ \mu b & M+\nu b &-L \end{pmatrix}
\begin{pmatrix} c \\ e \\ g \end{pmatrix} = 0.
\end{equation}
Cramer's rule applied to these gives the unprojection equations for $d$:
\begin{equation}
\begin{array}{l}
dc = KM + \nu bK -\lambda bL, \\
de = LM - \mu bK,
\end{array} \quad
\begin{array}{l}
df = -\mu K^2 + \nu KL - \lambda L^2, \\
dg = M^2 + \nu bM - \lambda\mu b^2.
\end{array}
\end{equation}
The combination eliminating $d$, $f$ and $g$ is
\begin{equation}
eKM - cLM - \lambda beL + \mu bcK + \nu beK = 0.
\label{eq!Z}
\end{equation}
This is a hypersurface $Z_{10} \subset
\mathbb P^4(1,1,2,3,4)_{\Span{x,a,b,c,e}}$ contained in the product ideal of
$I_d=(c,e)$, $I_f=(b,M_4)$, $I_g=(K_2,L_3)$. The unprojection planes
$\Pi_d$, $\Pi_f$, $\Pi_g$ are projectively equivalent to $\mathbb P(1,1,2)$,
$\mathbb P(1,1,3)$, $\mathbb P(1,1,4)$, but we cannot normalise all three of them to
coordinate planes at the same time. Their pairwise intersection is:
\begin{align*}
\Pi_d\cap \Pi_f &= \hbox{the 4 zeros of $M_4$ on the line $b=c=e=0$,} \\
\Pi_d\cap \Pi_g &= \hbox{the 3 zeros of $L_3$ on the line $c=e=K=0$,} \\
\Pi_f\cap \Pi_g &= \hbox{the 2 zeros of $K_2$ on the line $b=L=M=0$.}
\end{align*}
\paragraph{Nonsingularity based on \eqref{eq!Z}}
All the assertions we need for $Y$ and $X$ are most simply derived from
\eqref{eq!Z}. The linear system $|I_d\cdot I_f\cdot
I_g\cdot\mathcal O_{\mathbb P}(10)|$ of hypersurfaces through the three unprojection
planes has base locus the planes themselves, together with the curve
$(b=c=K_2=0)$, which is in the base locus because the term $eLM\in
I_d\cdot I_f\cdot I_g$ has degree~11 and so does not appear in the
equation of $Z$. This curve is a pair of generating lines
$(K=0)\subset\mathbb P(1,1,4)_{\Span{x,a,e}}$. One sees that for general
choices, one of the terms $cLM$ or $\lambda beL$ in $Z$ provides a nonzero
derivative $LM$ or $\lambda eL$ at every point along this curve away from
the three planes.
The singular locus of $Z$ on $\Pi_d=\mathbb P(1,1,2)$ is given by
\begin{equation}
\frac{\partial Z}{\partial c}=-LM+\mu bK =0, \quad
\frac{\partial Z}{\partial e}=KM-\lambda bL+\nu bK=0.
\end{equation}
For general choices, these are $21=\frac{7\times6}2$ reduced points of
$\mathbb P(1,1,2)$, including the 4 points of $\Pi_d\cap \Pi_f$ and the 3
points of $\Pi_d\cap \Pi_g$; after unprojecting $\Pi_f$ and $\Pi_g$,
this leaves 14 nodes of Tom$_1$, as we asserted in \eqref{eq!3.2}. The
calculations for the other planes are similar.
We believe that $Z_{10} \subset \mathbb P^4(1,1,2,3,4)$ has class group
$\mathbb Z^4$ generated by the hyperplane section $A=-K_Z$ and the three planes
$\Pi_d$, $\Pi_f$, $\Pi_g$, so that $X$ is prime.
\subsection{Jer$_{45}$} \label{s!J45}
The tidied up matrix is
\begin{equation}
\begin{pmatrix}
b & -L_2 & c & e \\
& M_3 & e & g \\
&& f & \lambda_2 c \\
&&& m_{45}
\end{pmatrix},
\label{eq!3.12}
\end{equation}
with pivot $m_{45}=\delta_3c+\gamma_2e+\beta_2f+\alpha_1g$; we use row and column
operations and changes of coordinates in $I_D=(c,e,f,g)$ to clean $c$
and $f$ out of $m_{24}$, but we cannot modify the pivot $m_{45}$ without
introducing multiples of $b,L,M$ into Row~4 or Row~5, spoiling the
Jer$_{45}$ format.
We get parallel unprojection constructions for $X$ by eliminating $f$ or
$g$ or both. First, subtract $\alpha$ times Row~2 from Row~4, and ditto
with the columns, to take $g$ out of $m_{45}$. This spoils the format by
$c\mapsto c-\alpha b\notin I_D$ in $m_{14}$, but does not change the
Pfaffian ideal. The new matrix only contains $g$ in $m_{25}$;
the two Pfaffians not involving it are $\Pf_{12.34}$ and the modified
$\Pf_{13.45}$, giving
\begin{equation}
\begin{pmatrix} M & L & b \\
\delta L+\lambda c-\alpha\lambda b & \gamma L-\alpha M & \beta L-e \end{pmatrix}
\begin{pmatrix} c \\ e \\ f \end{pmatrix}=0.
\label{eq!gless}
\end{equation}
Eliminating $f=m_{34}$ is similar, with $\Pf_{12.35}$ and modified
$\Pf_{12.45}$ giving
\begin{equation}
\begin{pmatrix} \lambda b & M & L \\
\delta b-\beta M & \gamma b+e-\beta L & \alpha b-c \end{pmatrix}
\begin{pmatrix} c \\ e \\ g \end{pmatrix}=0.
\label{eq!fless}
\end{equation}
We derive the unprojection equations for $d$ using Cramer's rule:
\begin{equation}
\begin{array}l
dc=-L(e-\beta L)-\gamma Lb+\alpha Mb, \\
de=M(e-\beta L)+\lambda b(c-\alpha b)+\delta Lb, \\
df=-\lambda L(c-\alpha b)-\delta L^2+\gamma LM-\alpha M^2, \\
dg=\lambda b(e-\beta L)+M(g-\delta b)+\gamma\lambda b^2+\beta M^2.
\end{array}
\label{eq!deqns}
\end{equation}
This is also a triple parallel unprojection, but with a difference: the
hypersurface $Z_{10}\subset\mathbb P(1,1,2,3,4)$ obtained by eliminating $f$
from \eqref{eq!gless} or $g$ from \eqref{eq!fless} or $d$ from the first
two rows of \eqref{eq!deqns} is now
\begin{equation}
e(e-\beta L)L+\delta cbL+\gamma ebL+\lambda bc(c-\alpha b)+M(ce-\beta cL-\alpha be)=0.
\label{eq!H}
\end{equation}
It is in the inter\-section of the three codimension~2 c.i.\
unprojection ideals $I_d=(c,e)$, $I_f=(b,e-\beta L)$, $I_g=(c-\alpha b,L)$,
but not in their product: the first 4 terms are clearly in the product
ideal. The interesting part is the bracket in the last term, which
cannot be in the product since it has terms of degree~2, but is in
$I_d\cap I_f\cap I_g$, because
\begin{equation}
c(e-\beta L)-\alpha be = e(c-\alpha b)-\beta Lc.
\end{equation}
The slogan is {\em like lines on a quadric}; the three ideals have
linear combinations of $b,c$ as first generator, and of $e,L$ as second
generator, like three disjoint lines $x=z=0$, $y=t=0$ and $x=t,y=z$ on
$Q:(xy=zt)$. One analyses the singularities of $Z_{10}$ from this much
as before; we believe that $\Cl Z=\Span{A,D_1,D_2,D_3}$, so that the
triple unprojection $X$ is prime.
\section{Failure}\label{s!fail}
We give reasons for failure following the introductory discussion in
Section~\ref{exa!main}; we don't need to treat all the possible tests in
rigorous detail, or the logical relations between them. For the
structure of our proof, the point of this section is merely to give
cheap preliminary tests to exclude all the candidates $D\subset Y$ that
will not pass the nonsingularity algorithm in Section~\ref{s!ns}.
\subsection{Easy fail at a coordinate point} \label{s!easy}
Consider a coordinate point $P_i=P_{x_i}\in Y$. In either of the
following cases, $P_i$ cannot be a hyperquotient point, let alone
terminal, and we can safely fail the candidate $D\subset Y$:
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item $x_i$ does not appear in the matrix $M$.
\item $x_i$ does not appear as a pure power in any entry of $M$, which
thus has rank zero at $P_i$.
\end{enumerate}
\subsection{Fishy zero in $M$ and excess singularity} \label{s!fish}
Suppose we can arrange that $m_{12}=0$, if necessary after row and
column operations; then the subscheme $Z=V(\{m_{1i},m_{2i}\mid
i=3,4,5\})$ is in the singular locus of $Y$. Indeed, the three Pfaffians
$\Pf_{12,ij}$ are in $I_Z^2$, so do not contribute to the Jacobian at
points of $Z$. The case that $\dim Z=0$ and $Z\subset D$ is perfectly
acceptable and happens in a fraction of our successful constructions
(see Tom$_2$ and Jer$_{25}$ in Section~\ref{exa!main}). Notice that
$\dim Z=0$ if and only if the 6 forms $m_{1i},m_{2i}$ make up a regular
sequence for $\mathbb P^6$; in the contrary case, the zero is {\em fishy}.
Thus any little coincidence between the six $m_{1i},m_{2i}$ fails
$D\subset Y$. The tests we implement are:
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\setcounter{enumi}{2}
\item Two collinear zeros in $M$; see \ref{s!fa} for an example.
\item Two of the $m_{1i},m_{2i}$ coincide; see Section~\ref{exa!main},
Jer$_{24}$.
\item An entry $m_{1i}$ or $m_{2i}$ is in the ideal generated by the
other five.
\end{enumerate}
In fact, the tricky point here is how to read our opening ``Suppose we
can arrange that $m_{12}=0$''. The row and column operations clearly need
a modicum of care to preserve the format (i.e., the entries we require
to be in $I_D$). The harder point is that we may need a particular
change of basis in $I_D$ for the zero to appear. For example, in the
Tom$_5$ format for $\mathbb P^2\subset Y\subset\mathbb P(1^6,2)$, with matrix of
weights
$\begin{smallmatrix}1&1&1&2\\&1&1&2\\&&1&2\\&&&2\end{smallmatrix}$, the
lowest degree Pfaffian is quadratic in three variables of weight~1, so
we can write it $xy-z^2$. Mounting this as a Pfaffian in these
coordinates, we can force a fishy zero, with two equal entries $z$
arising from the term $z^2$. (The same applies to several candidates,
but this is the only one that fails solely for this reason.)
\subsection{More sophisticated and ad hoc reasons for failure}
For the unprojected $X$ to have terminal singularities, $Y$ itself must
also: it is the anticanonical model of the weak Fano 3-fold $X_1$. We
can test for this at a coordinate point $P$ of index $r>1$: by Mori's
classification, $Y$ is either quasismooth at $P$, or a hyperquotient
singularity with local weights $\frac1r(1,a,r-a,0)$ or
$\frac14(1,1,3,2)$. Thus we can fail the candidate $D\subset Y$ if:
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\setcounter{enumi}{5}
\item A coordinate point off $D$ is a nonterminal hyperquotient singularity.
\item A coordinate point on $D$ is a nonterminal hyperquotient singularity.
\end{enumerate}
These tests dispatch most of the remaining failing candidates.
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\setcounter{enumi}{7}
\item Ad hoc fail. Just two cases have nonisolated singularities not
revealed by the elementary tests so far:
\begin{enumerate}
\item Tom$_4$ for $\mathbb P(1,2,3)\subset Y\subset\mathbb P(1^2,2,3^2,4^2)$ with weights
$\begin{smallmatrix}2&2&3&3\\&3&4&4\\&&4&4\\&&&5\end{smallmatrix}$\,;
\item Jer$_{12}$ for $\mathbb P(1,2,3)\subset Y\subset\mathbb P(1^2,2^2,3^2,4)$ with weights
$\begin{smallmatrix}2&2&2&3\\&3&3&4\\&&3&4\\&&&4\end{smallmatrix}$\,.
\end{enumerate}
Each of these has a $\frac{1}2(1,1,1,0;0)$ hyperquotient singularity at
the $\frac12$ point of $D$. Such a point may be terminal if it is an
isolated double point, but the format of the matrix prevents this. The
second case also fails at the index~4 point $P_7$ lying off $D$: it is a
hyperquotient singularity of the exceptional type $\frac{1}4(1,1,3,2;2)$
with the right quadratic part to be terminal. However, it lies on a
curve of double points along the line $\mathbb P(2,4)$ joining $P_7$ to the
$\frac12$ point on $D$: in local coordinates $x,a,e,b$ at $P_7$, the
equation is $xa=e^2+b\times\hbox{terms in $(x,a,e)^2$}$.
\end{enumerate}
\section{Nonsingularity and proof of Theorem~\ref{th!main}} \label{s!ns}
To prove Theorem~\ref{th!main}, we need to run through a long list of
candidate \hbox{3-folds} $D\subset Y \subset w\mathbb P^6$ with choice of
format Tom$_i$ or Jer$_{ij}$. We exclude many of these by the automatic
methods of Section~\ref{s!fail}. In every remaining case, we run a
nonsingularity algorithm to confirm that the candidate can be
unprojected to a codimension~4 Fano 3-fold $X$ with terminal
singularities (in fact, we conclude also quasismooth). For the proof of
Theorem~\ref{th!main}, we check that at least one Tom and one Jerry
works for each case $D\subset Y$.
We outline the proof as a pseudocode algorithm; our implementation is
discussed in Section~\ref{s!grdb}. The justification of the algorithm is
that it works in practice. A priori, it could fail, e.g., the singular
locus of $Y$ on $D$ could be more complicated than a finite set of
nodes, or all three coordinate lines of $D$ could contain a node, but by
good luck such accidents never happen.
\subsection{Nonsingularity analysis} \label{s!sings}
We work with any $D\subset Y$ not failed in Section~\ref{s!fail}. The
homogeneous ideal $I_Y$ is generated by the $4\times4$ Pfaffians of $M$.
Differentiating the 5 equations $\Pf$ with respect to the seven
variables gives the $5\times7$ Jacobian matrix $J(\Pf)$. Its ideal
$I_{\Sing Y}=\bigwedge^3J(\Pf)$ of $3\times3$ minors defines the
singular locus of $Y$; more precisely, it generates the ideal sheaf
$\mathcal I_{\Sing Y}\subset\mathcal O_{\mathbb P^6}$. Our claim is that the only
singularities of $Y$ lie on $D$, and are nodes. For this, we check that
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item $\Sing Y\subset D$, or equivalently $I_D\subset\Rad(I_{\Sing Y})$.
\item The restriction $\mathcal I_{\Sing Y}\cdot\mathcal O_D$ defines a reduced
subscheme of $D$.
\end{enumerate}
In fact (b) together with Lemma~\ref{l!nodes} imply that $Y$ has only
nodes. In practice, we may work on a standard affine piece of $D$
containing all the singular points: it turns out in every case that some
1-strata of $D$ is disjoint from the singular locus.
\subsection{Proof of Theorem~\ref{th!main}}
We start with the data for a candidate $P\in X\subset w\mathbb P^7$: a genus
$g\ge-2$ and a basket $\mathcal B$ of terminal quotient singularities, or
equivalently, the resulting Hilbert series (see \cite{ABR}). We give a
choice of 8 ambient weights $W_X$ of $w\mathbb P^7$ and a choice of Type~I
centre $P=\frac1r(1,a,r-a)$ from the basket. The Type~I definition
predicts that the ambient weights of $Y\subset w\mathbb P^6$ are $W_X
\setminus \{r\}$ and that $D=\mathbb P(1,a,r-a)$ can be chosen to be a
coordinate stratum of $w\mathbb P^6$. We analyse all possible Tom and Jerry
formats for $D\subset Y\subset w\mathbb P^6$.
\paragraph{Step 1} Set up coordinates $x_1,x_2,x_3,x_4$, $y_1,y_2,y_3$
on $w\mathbb P^6$; here $x_{1\dots4}$ is a regular sequence generating $I_D$,
and $y_1,y_2,y_3$ are coordinates on $D$.
\paragraph{Step 2} The numerics of \cite{CR} determine the weights
$d_{ij}$ of the $5\times5$ skew matrix $M$ from the Hilbert numerator of
$Y\subset w\mathbb P^6$.
\paragraph{Step 3} Set each entry $m_{ij}$ of $M$ equal to a general
form, respectively a general element of the ideal $I_D$ of the given
degree $d_{ij}$, according to the chosen Tom or Jerry format (see
Definition~\ref{d!TJ}).
Tidy up the matrix $M$ as much as possible while preserving its Tom or
Jerry format. Some entries of $M$ may already be zero. Use coordinate
changes on $w\mathbb P^6$ to set some entries of $M$ equal to single
variables. If possible, use row and column operations to simplify $M$
further. Check every zero of $M$ for failure for the mechanical reasons
discussed in \ref{s!fish}, followed by the other failing conditions of
\ref{s!easy}. Now any candidate that passes these tests actually works.
\paragraph{Step 4} Carry out the singularity analysis of \ref{s!sings}.
\paragraph{Step 5} Calculate the number of nodes as in
Section~\ref{s!Jac}; check that no two sets of unprojection data give
the same number of nodes.
\paragraph{Step 6 (optional)} Apply the Kustin--Miller algorithm
\cite{KM} to construct the equations of $X$. This is not essential to
prove that $X$ exists, but knowing the full set of equations is useful
if we want to put the equations in a codimension~4 format, for example
by projecting from another Type~I centre.
\section{Number of nodes} \label{s!Jac}
The unprojection divisor $D=V(x_{1\dots4})\subset \mathbb P^6$ is a
codimension~4 c.i., with conormal bundle $\mathcal I_D/\mathcal I_D^2$ the direct sum
of four orbifold line bundles $\mathcal O_D(-x_i)$ on $D$. The ideal sheaf
$\mathcal I_Y$ is generated by 5 Pfaffians that vanish on $D$, so each is
$\Pf_i=\sum a_{ij}x_j$. Thus the Jacobian matrix $\Jac$ restricted to
$D$ is the $5\times4$ matrix $(\overline a_{ij})$, where bar is restriction
mod $I_D=(x_{1\dots4})$; the induced homomorphism to the conormal bundle
\begin{equation}
\mathcal J\colon \bigoplus_5\mathcal O_{\mathbb P}(-\Pf_i)\twoheadrightarrow \mathcal I_Y/(\mathcal I_D\cdot\mathcal I_Y) \to
\mathcal I_D/\mathcal I_D^2
\label{eq!N}
\end{equation}
has generic rank~3. Its cokernel $\mathcal N$ is the conormal sheaf to $D$ in
$Y$. It is a rank~1 torsion free sheaf on $D$ whose second Chern class
$c_2(\mathcal N)$ counts the nodes of $Y$ on $D$. The more precise result is as
follows:
\begin{lem} \label{l!nodes}
\begin{enumerate}
\renewcommand{\labelenumi}{(\Roman{enumi})}
\item The cokernel $\mathcal N$ is an orbifold line bundle at points of $D$
where $\rank\mathcal J=3$, that is, at quasi\-smooth points of $Y$.
\item Assume that $P\in D$ is a nonsingular point (not orbifold), and
that $\rank\mathcal J=2$ at $P$ and $=3$ in a punctured neighbourhood of $P$ in
$D$; then $\mathcal N$ is isomorphic to a codimension~$2$ c.i.\ ideal $(f,g)$
locally at $P$. This coincides locally with the ideal
$\bigwedge^3\Jac\cdot\mathcal O_D$ generated by the $3\times3$ minors of the
Jacobian matrix.
\item Assume that $\bigwedge^3\Jac\cdot\mathcal O_D$ is reduced (locally the
maximal ideal $m_P$ at each point). Then $Y$ has an ordinary node at $P$.
\item If this holds everywhere then $c_2(\mathcal N)$ is the number of
nodes of $Y$ on $D$.
\end{enumerate}
\end{lem}
\paragraph{Proof} The statement is the hard part; the proof is just
commutative algebra over a regular local ring. The rank~1 sheaf $\mathcal N$ is
the quotient of a rank~4 locally free sheaf by the image of the
$5\times4$ matrix $\Jac=(\overline a_{ij})$, of generic rank~3. It is a line
bundle where the rank is 3, and where it drops to~2, we can use a
$2\times2$ nonsingular block to take out a rank~2 locally free summand.
The cokernel is therefore locally generated by 2 elements, so is locally
isomorphic to an ideal sheaf $(f,g)$, a c.i.\ because the rank drops
only at $P$.
The minimal free resolution of $\mathcal N$ is the Koszul complex of $f,g$; now
\eqref{eq!N} is also part of a free resolution of $\mathcal N$, so covers the
Koszul complex. This means that the matrix $\Jac=(\overline a_{ij})$ can be
written as its $2\times2$ nonsingular block and a complementary
$2\times3$ block of rank~1, whose two rows are $g\cdot v$ and $-f\cdot
v$ for $v$ a 3-vector with entries generating the unit ideal. Therefore
$\bigwedge^3\Jac$ generates the same ideal $(f,g)$.
If $(f,g)=(y_1,y_2)$ is the maximal ideal at $P\in D$ then the shape of
$\bigwedge^3\Jac$ says that two of the Pfaffians $\Pf_1,\Pf_2$ express
two of the variable $x_1,x_2$ as implicit functions; then a linear
combination $p$ of the remaining three has $\partial p/\partial x_3=y_1$
and $\partial p/\partial x_4=y_2$, so that $Y$ is a hypersurface with an
ordinary node at $P$. \ifhmode\unskip\nobreak\fi\quad\ensuremath{\mathrm{QED}}\par\medskip
We now show how to resolve $\mathcal N$ by an exact sequence involving direct
sums of orbifold line bundles on $D$, and deduce a formula for
$c_2(\mathcal N)$.
\paragraph{Tom$_1$}
The matrix is
\begin{equation}
M = \begin{pmatrix}
K & L & M & N \\
& m_{23} & m_{24} & m_{25} \\
&& m_{34} & m_{35} \\
&&& m_{45}
\end{pmatrix}
\end{equation}
where $m_{ij}$ are linear forms in $x_{1\dots4}\in\mathcal I_D$ with
coefficients in the ambient ring. When we write out $\Jac=(\overline a_{ij})$,
the only terms that contribute are the derivatives $\partial/ \partial
x_{1\dots4}$, with the $x_i$ set to zero; thus only the terms that are
exactly linear in the $x_i$ contribute. Since $\Pf_1$ is of order $\ge2$
in the $x_i$, the corresponding row of the matrix $J$ is zero and we
omit it in \eqref{eq!J4}. Moreover, the first row $K,L,M,N$ of $M$
provides a syzygy $\Sigma_1=K\Pf_2+L\Pf_3+M\Pf_4+N\Pf_5\equiv0$ between the
4 remaining Pfaffians. Hence we can replace $J$ by the resolution
\begin{equation}
\mathcal N \leftarrow \sum_{1\dots4} \mathcal O(-d_i) \leftarrow \sum_{j\ne 1} \mathcal O(-a_j) \leftarrow
\mathcal O(-\sigma_1) \leftarrow 0
\label{eq!J4}
\end{equation}
where $d_i=\wt x_i$, $a_j=\wt\Pf_j$ and $\sigma_1=\wt\Sigma_1$, and leave the
reader to think of names for the maps. Therefore $\mathcal N$ has total Chern
class
\begin{equation}
\prod_{i=1}^4 (1-d_ih) \times (1-\sigma_1 h) \Big/ \prod_{j\ne1} (1-a_j h)
\label{eq!c2}
\end{equation}
The number of nodes $c_2(\mathcal N)$ is then the $h^2$ term in the expansion
of \eqref{eq!c2}; recall that we view $h=c_1(\mathcal O_D(1))$ as an orbifold
class, so that $h^2=1/ab$ for $D=\mathbb P(1,a,b)$.
\paragraph{Jer$_{12}$} The pivot $m_{12}$ appears in three Pfaffians
$\Pf_i=\Pf_{12,jk}$ for $\{i,j,k\}=\{3,4,5\}$ as the term
$m_{12}m_{jk}$, together with two other terms $m_{1j}m_{2k}$ of order
$\ge2$ in $x_{1\dots4}$. The Jacobian matrix restricted to $D$ thus has
three corresponding rows that are $m_{jk}$ times the same vector
$\partial m_{12}/\partial x_{1\dots4}$. This proportionality gives three
syzygies $\Sigma_l$ between these three rows, yoked by a second syzygy $T$
in degree $t=\hbox{adjunction number}-\wt m_{12}$. In other words, the
conormal bundle has the resolution
\begin{equation}
\mathcal N\leftarrow\bigoplus_4\mathcal O(-d_i)\leftarrow\bigoplus_5\mathcal O(-a_j)\leftarrow
\bigoplus_3\mathcal O(-\sigma_l) \leftarrow\mathcal O(-t)\leftarrow 0,
\end{equation}
so that the total Chern class of $\mathcal N$ is the alternate product
\begin{equation}
\frac{\prod_4(1-d_ih) \prod_3(1-\sigma_lh)}{\prod_5(1-a_jh) (1-th)},
\end{equation}
with $c_2(\mathcal N)$ equal to the $h^2$ term in this expansion.
\begin{exa} \label{rk!2v3} \rm
We read the number of nodes mechanically from the Hilbert numerator, the
matrix of weights and the choice of format. As a baby example, the
``interior'' projections of the two del Pezzo 3-folds of degree 6
discussed in \ref{s!same} have 2 and 3 respective nodes. These numbers
are the coefficient of $h^2$ in the formal power series
\begin{equation}
\frac{(1-h)^4(1-3h)}{(1-2h)^4}=1+h+2h^2 \enspace\hbox{and}\enspace
\frac{(1-h^4)(1-3h)^3}{(1-2h)^5(1-4h)}=1+h+3h^2.
\end{equation}
As a somewhat more strenuous example, in \eqref{eq!3.2},
\begin{description}
\item{Tom$_1$} has $\wt x_{1\dots4}=3,4,4,5$,
$\wt\Pf_{2\dots5}=8,8,7,6$, $\Sigma_1=10$, so that
\[
c(\mathcal N) = \frac{\prod_{a\in [3,4,4,5,10]}(1-ah)}
{\prod_{b\in [6,7,8,8]}(1-bh)}=1+3h+28h^2,
\hbox{ giving $\frac{28}{1\cdot1\cdot2}=14$ nodes.}
\]
\item{Jer$_{25}$} has the same $x_i$, $\Pf_{1\dots5}=9,8,8,7,6$, $\Sigma_l=10,11,12$,
adjunction number = 19, $\wt m_{25}=5$, so
$c(\mathcal N) = \frac{\prod_{a \in [3,4,4,5,10,11,12]}(1-ah)}
{\prod_{b \in [6,7,8,8,9,14]}(1-bh)}=1+3h+34h^2$, giving
$\frac{34}{1\cdot1\cdot2}=17$ nodes.
\end{description}
Try the other cases in \eqref{eq!3.2}--\eqref{eq!3.4} as homework.
\end{exa}
\section{Computer code and the GRDB database}
\label{s!grdb}
A Big Table with the detailed results of the calculations proving
Theorem~\ref{th!main} is online at the Graded Ring Database webpage
\begin{quote}
\verb!http://grdb.lboro.ac.uk! \quad + \hbox{Downloads.}
\end{quote}
This website makes available computer code implementing our calculations
systematically, together with the Big Table they generate. The code is
for the Magma system \cite{Ma}, and installation instructions are
provided; at heart, it only uses primary elements of any computer algebra
system, such as poly\-nomial ideal calculations and matrix
manipulations. The code runs online in the Magma Calculator
\begin{quote}
\verb!http://magma.maths.usyd.edu.au/calc!
\end{quote}
All the data on the codimension~4 Fano 3-folds we construct is available
on webloc.\ cit.: follow the link to Fano 3-folds, select Fano index
$f=1$ (the default value), codimension~$=4$ and Yes for Projections of
Type~I, then submit. The result is data on the 116 Fano 3-folds with a
Type~I projection (the 116th is an initial case with $7\times12$
resolution, that projects to the complete intersection
$Y_{2,2,2}\subset\mathbb P^6$ containing a plane, so is not part of our story
here). The $+$ link reveals additional data on each Fano.
The computer code follows closely the algorithm outlined as the proof of
Theorem~\ref{th!main}. For each Tom and Jerry format, we build a matrix
with random entries; some of these can be chosen to be single variables,
since we assume $Y$ is general for its format. We use row and column
operations to simplify the matrix further without changing the format.
The first failure tests (fishy zeroes, cone points and points of
embedding dimension~6) are now easy, and inspection of the equations on
affine patches at coordinate points on $Y$ is enough to determine
whether their local quotient weights are those of terminal
singularities. An ideal inclusion test checks that the singularities lie
on $D$. By good fortune, in every case that passes the tests so far, the
singular locus lies on one standard affine patch of $D$. We pass to this
affine patch and check that $\mathcal I_{\Sing Y}\cdot\mathcal O_D$ defines a reduced
scheme there. We calculate the length of the quotient $\mathcal O_D/(\mathcal I_{\Sing
Y}\cdot\mathcal O_D)$ on this patch, providing an alternative to the
computation of Section~\ref{s!Jac} (and a comforting sanity check).
The random entries in the matrix are not an issue: our non\-singularity
requirements are open, so if one choice leads to a successful $D\subset
Y$, any general choice also works. The only concern is false negative
reports, for example, an alleged nonreduced singular locus on $D$. To
tackle such hiccups, if a candidate fails at this stage (in practice, a
rare occurrence), we simply rerun the code with a new random matrix; the
fact that the code happens to terminate justifies the proof.
The conclusion is that every possible Tom and Jerry format for every
numerical Type~I projection either fails one of the human-readable tests
of Section~\ref{s!fail} (and we have made any number of such hand
calculations), or is shown to work by constructing a specific example.
To complete the proof of Theorem~\ref{th!main}, we check that the final
output satisfies the following two properties:
\begin{enumerate}
\renewcommand{\labelenumi}{(\alph{enumi})}
\item Every numerical candidate admits at least one Tom and one Jerry
unprojection.
\item Whenever a candidate has more than one Type~I centre, the
successful Tom and Jerry unprojections of any two correspond one-to-one,
with compatible numbers of nodes: the difference in Euler number
computed by the nodes is the same whichever centre we calculate from;
compare \eqref{eq!3.2}--\eqref{eq!3.4}.
\end{enumerate}
The polynomial ideal calculations of Nonsingularity
analysis~\ref{s!sings} (that is, the inclusion $I_D\subset\Rad(I_{\Sing
Y})$ and the statement that $\mathcal I_{\Sing Y}\cdot\mathcal O_D$ is reduced) are
the only points where we use computer power seriously (other than to
handle hundreds of repetitive calculations accurately). In cases with 2
or 3 centres, even this could be eliminated by projecting to a complete
intersection and applying Bertini's theorem, as in
Section~\ref{exa!main}.
\section{Codimension~4 Gorenstein formats}\label{s!fmt}
The Segre embeddings $\mathbb P^2\times\mathbb P^2\subset\mathbb P^8$ and
$\mathbb P^1\times\mathbb P^1\times\mathbb P^1\subset\mathbb P^7$ are well known codimension~4
projectively Gorenstein varieties with $9\times16$ resolution.
Singularity theorists consider the affine cones over them to be rigid,
because they have no nontrivial infinitesimal deformations or small
analytic deformation. Nevertheless, both are sections of higher
dimensional graded varieties in many different nontrivial ways. Each of
these constructions appears at many points in the study of algebraic
surfaces by graded rings methods.
\subsection{Parallel unprojection and extra\-symmetric format}
\label{s!xtra}
The extra\-symmetric $6\times6$ format occurs frequently, possibly first
in Dicks' thesis \cite{DPhD}. It is a particular case of triple
unprojection from a hypersurface in the product of three codimension 2
c.i.\ ideals. Start from the ``undeformed'' $6\times6$ skew matrix
\begin{equation}
M_0=
\begin{pmatrix}
b_3&-b_2&x_1&a_3&a_2\\
&b_1&a_3&x_2&a_1\\
&&a_2&a_1&x_3\\
&&&-b_3&b_2\\
&&&&-b_1
\end{pmatrix}
\end{equation}
with the ``extrasymmetric'' property that the top right $3\times3$ block
is symmetric, and the bottom right $3\times3$ block equals minus the top
left block. So instead of 15 independent entries, it has only 9
independent entries and 6 repeats.
Direct computation reveals that the $4\times4$ Pfaffians of $M_0$ fall
under the same numerics: of its 15 Pfaffians, 9 are independent and 6
repeats. One sees they generate the same ideal as the $2\times2$ minors
of the $3\times3$ matrix
\begin{equation}
N_0=
\begin{pmatrix}
x_1&a_3+b_3&a_2-b_2\\
a_3-b_3&x_2&a_1+b_1\\
a_2+b_2&a_1-b_1&x_3
\end{pmatrix}
\end{equation}
Here $N_0$ is the generic $3\times3$ matrix (written as symmetric plus skew),
with minors defining Segre $\mathbb P^2\times\mathbb P^2$, and thus far we have not
gained anything, beyond representing $\mathbb P^2\times\mathbb P^2$ as a nongeneric
section of $\Grass(2,6)$.
However $M_0$ can be modified to preserve the codimension~4 Gorenstein
property while destroying the sporadic coincidence with
$\mathbb P^2\times\mathbb P^2$. The primitive one-parameter way of doing this is to
choose the triangle $(1,2,6)$ and multiply the entries
$m_{12},m_{16},m_{26}$ by a constant $r_3$. This gives
\begin{equation}
M_1=
\begin{pmatrix}
r_3b_3&-b_2&x_1&a_3&r_3a_2\\
&b_1&a_3&x_2&r_3a_1\\
&&a_2&a_1&x_3\\
&&&-b_3&b_2\\
&&&&-b_1
\end{pmatrix}
\end{equation}
One checks that the three Pfaffians $\Pf_{12.i6}$ for $i=3,4,5$ are
$r_3$ times others, whereas three other repetitions remain unchanged. So
the $4\times4$ Pfaffians of $M_1$ still defines a Gorenstein
codimension~4 subvariety with $9\times16$ resolution. We can view it as
the Tom$_3$ unprojection of the codimension~3 Pfaffian ideal obtained by
deleting the final column, with $x_3$ as unprojection variable.
If $r_3=\rho^2$ is a perfect square then floating the square root $\rho$ to
the complementary entries $m_{34},m_{35},m_{45}$ restores the original
extrasymmetry. In general this is a ``twisted form'' of $\mathbb P^2\times\mathbb P^2$:
changing the sign of $\rho$ swaps the two factors.
A more elaborate version of this depends on 8 parameters:
\begin{equation}
M_2=
\begin{pmatrix}
r_3s_0b_3&-r_2s_0b_2&x_1&r_2s_1a_3&r_3s_1a_2&\\
&r_1s_0b_1&r_1s_2a_3&x_2&r_3s_2a_1&\\
&&r_1s_3a_2&r_2s_3a_1&x_3&\\
&&&-r_0s_3b_3&r_0s_2b_2&\\
&&&&-r_0s_1b_1&
\end{pmatrix}
\label{eq!r1r2r3}
\end{equation}
Now the same three Pfaffians $\Pf_{12.i6}$ are divisible by
$r_3$, and the complementary three are divisible by $s_3$ with
the same quotient, so one has to do a little cancellation to
see the irreducible component. The necessity of cancelling
these terms (although cheap in computer algebra as the colon
ideal) has been a headache in the theory for decades, since it
introduces apparent uncertainty as to the generators of the ideal.
The right way to view this is as the triple parallel unprojection of
the hypersurface
\begin{equation}
V(a_1a_2b_3r_3s_3 + a_1a_3b_2r_2s_2 + a_2a_3b_1r_1s_1 + b_1b_2b_3r_0s_0)
\end{equation}
in the product ideal $\prod_{i=1}^3(a_i,b_i)$. Then
\[
x_1=\frac{a_2a_3r_1s_1+b_2b_3r_0s_0}{a_1}=
-\frac{a_2b_3r_2s_2+a_2b_3r_3s_3}{b_1},
\]
etc., and the ideal is generated by the Pfaffians of the three matrices
\begin{equation}
\left(
\begin{smallmatrix}
x_2&b_1r_0s_0&a_1r_3s_3&a_3 \\
&-a_1r_2s_2&-b_1r_1s_1&b_3 \\
&&x_3&a_2 \\
&&&b_2
\end{smallmatrix}\right), \
\left(
\begin{smallmatrix}
x_1&b_3r_0s_0&a_3r_2s_2&a_2 \\
&-a_3r_1s_1&-b_3r_3s_3&b_2 \\
&&x_2&a_1 \\
&&&b_1
\end{smallmatrix}\right), \
\left(
\begin{smallmatrix}
x_3&b_2r_0s_0&a_2r_1s_1&a_1 \\
&-a_2r_3s_3&-b_2r_2s_2&b_1 \\
&&x_1&a_3 \\
&&&b_3
\end{smallmatrix}\right).
\notag
\end{equation}
If the $r_i$ and $s_i$ are nonzero constants, one still needs the
square root of the discriminant $\prod_{i=0}^3 (r_is_i)$ to get back
to $\mathbb P^2\times\mathbb P^2$.
\subsection{Double Jerry}\label{s!nJ}
The equations of Segre $\mathbb P^1\times\mathbb P^1\times\mathbb P^1\subset\mathbb P^7$ are the
minors of a $2\times2\times2$ array; they admit several extensions, and
it seems most likely that there is no irreducible family containing them
all. One family consists of various ``rolling factors'' formats
discussed below; here we treat ``double Jerry''.
Start from the equations written as
\begin{equation}
\begin{array}{cl}
sy_i = x_jx_k & \hbox{for $\{i,j,k\}=\{1,2,3\}$}, \\
tx_i = y_jy_k & \hbox{for $\{i,j,k\}=\{1,2,3\}$}, \\
st = x_iy_i & \hbox{for $i=1,2,3$}.
\end{array}
\label{eq!cube}
\end{equation}
corresponding to a hexagonal view of the cube centred at vertex $s$
(with three square faces $\square sx_iy_kx_j$, and $t$ behind the
page, cf.~\eqref{eq!cube1}):
\begin{equation}
\renewcommand{\arraycolsep}{.125em}
\begin{matrix}
&&y_2 \\[-3pt]
x_3&&&&&x_1 \\[-3pt]
&& s \\[-5pt]
y_1&&&&&y_3 \\[-3pt]
&&x_2
\end{matrix}
\begin{picture}(0,0)(25,0)
\qbezier(-5,19)(-8,17)(-11,15)
\qbezier(5,19)(8,17)(11,15)
\qbezier(-5,-15)(-8,-13)(-11,-11)
\qbezier(5,-15)(8,-13)(11,-11)
\qbezier(-5,3)(-8,5)(-11,7)
\qbezier(5,3)(8,5)(11,7)
\qbezier(0,-4)(0,-8)(0,-12)
\qbezier(-17,6)(-17,2)(-17,-2)
\qbezier(0,-5)(0,-9)(0,-13)
\qbezier(17,6)(17,2)(17,-2)
\end{picture}
\end{equation}
Eliminating both $s$ and $t$ gives the codimension~2 c.i.
\begin{equation}
(x_1y_1 = x_2y_2 = x_3y_3) \subset\mathbb P^5,
\label{eq!xiyi}
\end{equation}
containing the two codimension~3 c.i.s $\mathbf x=0$ and $\mathbf y=0$ as divisors.
We can view $\mathbf x$ as a row vector and $\mathbf y$ a column vector, and the
two equations \eqref{eq!xiyi} as the matrix products
\begin{equation}
\mathbf x A\mathbf y = \mathbf x B\mathbf y = 0, \quad\hbox{where}\quad
A=\left(\begin{smallmatrix}
1&0&0 \\
0&-1&0 \\
0&0&0
\end{smallmatrix}\right)\!,\,
B=\left(\begin{smallmatrix}
0&0&0 \\
0&1&0 \\
0&0&-1
\end{smallmatrix}\right)\!.
\end{equation}
The unprojection equations for $s$ and $t$ separately take the form
\begin{equation}
t\mathbf x = (A\mathbf y) \times (B\mathbf y) \quad\hbox{and}\quad
s\mathbf y = (\mathbf x A) \times (\mathbf x B),
\end{equation}
where $\times$ is cross product of vectors in $\mathbb C^3$, with the
convention that the cross product of two row vectors is a column vector
and vice versa. For example, $\mathbf x A=(x_1,-x_2,0)$, $\mathbf x B=(0,x_2,-x_3)$
and the equations $s\mathbf y=(\mathbf x A) \times (\mathbf x B)$ giving the first line of
\eqref{eq!cube} are deduced via Cramer's rule from \eqref{eq!xiyi}.
We can generalise this at a stroke to $A,B$ general $3\times3$ matrices.
That is, for $\mathbf x$ a row vector and $\mathbf y$ a column vector, $\mathbf x A\mathbf y=\mathbf x
B\mathbf y=0$ is a codimension~2 c.i.; since these are general bilinear forms
in $\mathbf x$ and $\mathbf y$, it represents a universal solution to two elements
of the product ideal $(x_1,x_2,x_3)\cdot(y_1,y_2,y_3)$. It has two
single unprojections:
\begin{align}
\mathbf x A\mathbf y=\mathbf x B\mathbf y=0 \quad\hbox{and}\quad
s\mathbf y=(\mathbf x A) \times (\mathbf x B),
\label{eq!sby} \\
\mathbf x A\mathbf y=\mathbf x B\mathbf y=0 \quad\hbox{and}\quad
t\mathbf x=(A\mathbf y) \times (B\mathbf y),
\label{eq!tbx}
\end{align}
either of which is a conventional $5\times5$ Pfaffian, and a parallel
unprojection putting those equations together with a 9th {\em long
equation}
\begin{equation}
st = \hbox{something complicated}.
\end{equation}
The equation certainly exists by the Kustin--Miller theorem. It can be
obtained easily in computer algebra by coloning out any of
$x_1,x_2,x_3,y_1,y_2,y_3$ from the ideal generated by the eight
equations \eqref{eq!sby} and \eqref{eq!tbx}. Its somewhat amazing right
hand side has 144 terms, each bilinear in $x,y$ and biquadratic in
$A,B$. Taking a hint from $144=12\times12$, we suspect that it may have
a product structure of the form
\begin{equation}
\mathbf x(A \wedge B)\times(A\wedge B)\mathbf y,
\end{equation}
with ``$\times$'' and ``$\wedge$'' still requiring elucidation.
If the entries of $A$ and $B$ are constants, one gets back to
$\mathbb P^1\times\mathbb P^1\times\mathbb P^1$ after coordinate changes based on
the three roots $(\lambda_i:\mu_i)$ of the relative characteristic
equation $\det(\lambda A-\mu B)=0$ and the three eigenvectors
$v_i=\ker(\lambda_i A-\mu_i B)$. Swapping the roots permutes the
three factors.
The significance of the double Jerry parallel unprojection format is
that it covers any Jerry case where the pivot is one of the generators
of $I_D$. Indeed, if the regular sequence generating $I_D$ is
$s,x_1,x_2,x_3$, a Jerry matrix for $D$ is
\begin{equation}
\begin{pmatrix}
s&m_{13}&m_{14}&m_{15} \\
&m_{23}&m_{24}&m_{25} \\
&& y_3 & -y_2 \\
&&& y_1
\end{pmatrix} \quad\hbox{where}\quad
\begin{array}{l}
(m_{13},m_{14},m_{15})=\mathbf x A, \\[3pt]
(m_{23},m_{14},m_{15})=\mathbf x B.
\end{array}
\end{equation}
for some $3\times3$ matrices $A,B$. Unprojecting $D$ gives a double Jerry.
\subsection{Rolling factors format}\label{s!rf}
Rolling factors view a divisor $X\subset V$ on a normal projective
variety $V\subset\mathbb P^n$ as residual to a nice linear system. This
phenomenon occurs throughout the literature, with typical cases a
divisor on the Segre embedding of $\mathbb P^1\times\mathbb P^3$, or on a rational
normal scroll $\mathbb F$, or on a cone over a Veronese embedding. A divisor
$X\subset\mathbb P^1\times\mathbb P^3$ in the linear system
$|ah_1+(a+2)h_2|=|{-}K_V+bH|$ is of course defined by a single
bihomogeneous equation in the Cox ring of $\mathbb P^1\times\mathbb P^3$, but to get
equations in the homogeneous coordinate ring of Segre
$\mathbb P^1\times\mathbb P^3\subset\mathbb P^7$ we have to add $|2h_1|$. This is a type
of hyperquotient, given by one equation in a nontrivial eigenspace.
Dicks' thesis \cite{DPhD} discussed the generic pseudoformat
\begin{equation}
\begin{gathered}
\bigwedge^2 \begin{pmatrix} a_1&a_2&a_3&a_4 \\ b_1&b_2&b_3&b_4
\end{pmatrix}=0, \quad\hbox{and} \\[4pt]
\renewcommand{\arraycolsep}{.25em}
\begin{array}{rclcc}
m_1a_1+m_2a_2+m_3a_3+m_4a_4 &=& 0 \\
m_1b_1+m_2b_2+m_3b_3+m_4b_4 &\equiv&
n_1a_1+n_2a_2+n_3a_3+n_4a_4 &=& 0 \\
&& n_1b_1+n_2b_2+n_3b_3+n_4b_4 &=& 0.
\end{array}
\label{eq!roll}
\end{gathered}
\end{equation}
One sees that under fairly general assumptions the ``scroll'' $V$
defined by the first set of equations of \eqref{eq!roll} is
codimension~3 and Cohen--Macaulay, with resolution
\[
\mathcal O_V\leftarrow R\leftarrow 6R\leftarrow 8R\leftarrow 3R\ot0.
\]
On the right, the identity is a preliminary condition on quantities in
the ambient ring. If we assume (say) that $R$ is a regular local ring
and $a_i,b_i,m_i,n_i\in R$ satisfy it (and are ``fairly general''), the
second set defines an elephant $X\in|{-}K_V|$ (anticanonical divisor)
which is a codimension~4 Gorenstein variety with $9\times16$ resolution.
The identity in \eqref{eq!roll} is a quadric of rank~16. It is a little
close-up view of the ``variety of complexes'' discussed in \cite{Ki},
Section~10. To use this method to build genuine examples, we have to
decide how to map a regular ambient scheme into this quadric; there are
several different solutions. If we take the $a_i,b_i$ to be independent
indeterminates, the first set of equations gives the cone on Segre
$\mathbb P^1\times\mathbb P^3\subset\mathbb P^7$, and the second set consists of a single
quadratic form $q$ in 4 variables evaluated on the two rows, so that
$X\subset V$ is given by $q(\mathbf a)=\varphi(\mathbf a,\mathbf b)=q(\mathbf b)=0$, with $\varphi$
the associated symmetric bilinear form (cf.\ \eqref{eq!rf}). This format
seems to be the only commonly occurring codimension~4 Gorenstein format
that tends not to have any Type~I projection.
On the other hand, if there are coincidences between the $a_i,b_i$,
there may be other ways of choosing the $m_i,n_i$ to satisfy the
identity in \eqref{eq!roll} without the need to take $m_i,n_i$ quadratic
in the $a_i,b_i$: for example, if $a_2=b_1$, we can roll $a_1\to a_2$
and $b_1\to b_2$.
|
{
"timestamp": "2011-07-04T02:02:51",
"yymm": "1009",
"arxiv_id": "1009.4313",
"language": "en",
"url": "https://arxiv.org/abs/1009.4313"
}
|
\section{Introduction}
A remarkable outcome of the heavy ion experiments at
RHIC~\cite{Adamsa3,Adcoxa1,Arsena2,Backa2} is the large elliptic flow
observed in the collisions. Phenomenological hydrodynamical models
that fit the RHIC data appear to require that the quark gluon matter
has a very small value for the dimensionless ratio of the viscosity to
the entropy density~\cite{Romat1}. This ratio $\eta/s$, a measure of
the ``perfect fluidity'' of the system, is estimated to be $\lesssim
5/4\pi$~\cite{Teane1}, where $\eta/s = 1/4\pi$ is a conjectured
universal lower bound~\cite{PolicSS1,PolicSS2}. Its was shown
recently~\cite{HiranHKLN1,LappiV1,DrescDHN1} that the degree of
perfectness of the quark-gluon fluid produced at RHIC is sensitive to
details of the initial spatial distribution of the produced matter at
the onset of hydrodynamic flow.
An important feature of the hydrodynamic models is that they require
very early thermalization after the collision. Estimates for the
thermalization time, which range from $\tau_{\rm relax}\sim 0.6 -1$
fm~\cite{SongH1,LuzumR1,DusliMT1}, are difficult to reconcile with a
simple picture of thermalization arising from the rapid scattering of
quasi-particles at rates greater than the expansion rate of the fluid.
The uncertainty principle tells us that for $\tau_{\rm relax} \leq 1$
fm, modes with momenta $\sim 200$ MeV are not even on-shell, let alone
amenable to being described as quasi-particles undergoing scattering.
While a quasi-particle description is not essential to thermalization,
it is the simplest one, and other realizations are more complicated.
With regard to the issue of flow however, it is sufficient to note
that one requires primarily that matter be isotropic and (nearly)
conformal to obtain a closed form expression for the hydrodynamic
equations~\cite{ArnolLMY1}.
How isotropization and (subsequently) thermalization is achieved in
heavy ion collisions is an outstanding problem which requires that the
problem be considered {\it ab initio}. What this means it that one
needs to understand and compute the properties of the relevant degrees
of freedom in the nuclear wavefunctions and how these degrees of
freedom decohere in a collision to produce quark-gluon matter. An {\it
ab initio} approach to the problem can be formulated within the
framework of the Color Glass Condensate (CGC) effective field theory,
which describes the relevant degrees of freedom in the nuclei as
dynamical classical fields coupled to static color
sources~\cite{IancuV1,IancuLM3,GelisIJV1}. The computational power of
this approach is a consequence of the dynamical generation of a
semi-hard scale, the saturation scale~\cite{GriboLR1,MuellQ1}, which
allows a weak coupling treatment of the relevant degrees of
freedom~\cite{McLerV1,McLerV2,McLerV3} in the high energy
nuclear wavefunctions.
There has been significant progress recently in applying the CGC
effective field theory to studying the early time behavior of the
matter produced in heavy ion collisions. Inclusive quantities such as
the pressure and the energy density in this matter (called the
Glasma~\cite{LappiM1}) can be written as expressions that factorize
the universal properties of the nuclear wavefunctions (measurable for
instance in proton-nucleus or electron-nucleus collisions) from the
detailed dynamics of the matter in
collision~\cite{GelisLV3,GelisLV4,GelisLV5}. Key to this approach are
the quantum fluctuations around the classical fields in the
wavefunctions and in the collision. Quantum fluctuations that are
invariant under boosts can be isolated in universal functionals that
evolve with energy. There are however also quantum fluctuations that
are not boost invariant which are generated during the collision.
These quantum fluctuations can grow rapidly and therefore play a
significant role in the subsequent temporal evolution of the Glasma.
The problem of how to treat these so-called ``secular divergences'' of
perturbative series is very general and occurs in a wide variety of
dynamical systems~\cite{Golde1}. In particular, the role of time
dependent quantum fluctuations in heavy ion collisions bears a strong
analogy to their role in the evolution of the early
universe~\cite{AllahBCM1}. In the latter case, quantum fluctuations
around a rapidly decaying classical field, the inflaton, are enhanced
due to parametric resonance, and it is conjectured that this dynamics
termed ``preheating''~\cite{GreenKLS1} may lead to turbulent
thermalization~\cite{MichaT1} in the early universe.
It is therefore very important to understand the precise role of these
quantum fluctuations in heavy ion collisions to determine whether they
play an analogous role to that in the early universe in the
isotropization/thermalization of the system. Their computation in a
gauge theory is quite involved so for simplicity, we shall in this
paper first attack this problem in a scalar $\phi^4$ field theory.
Like QCD, the coupling is dimensionless in this theory and the fields
are self interacting. In addition, we choose initial conditions for
our study that are similar to those in the CGC treatment of heavy ion
collisions. It must be said at the outset that there are important
differences between the two theories and there is no {\it a priori}
guarantee that the lessons learnt in one case will translate
automatically to the other.
The CGC initial conditions, for weak couplings $g\ll 1$, specifically
lead to a power counting scheme where the leading contribution to
inclusive quantities is the classical contribution of order ${\cal
O}(1/g^2)$. Quantum corrections begin at ${\cal O}(1)$ and their
contribution can be expressed as real-time partial differential
equations for small fluctuations in the classical background, with
purely retarded initial conditions. We will show that there are modes
of the small fluctuation field that grow very rapidly and can become
as large as the classical field on time scales of interest in the
problem. We observe that there are two sorts of rapidly growing modes
of the fluctuation field. One are modes that enjoy parametric
resonance and grow exponentially. These modes are however localized in
a rather narrow resonance band. The zero mode and low lying modes grow
linearly and can also influence the temporal evolution of the system.
Both sorts of ``secular'' terms can be isolated and resummed to all
orders in perturbation theory. The resulting expressions are stable
and can be expressed as an ensemble average over a spectrum of quantum
fluctuations convolved with the {\it leading order} inclusive quantity
which, for a particular fluctuation field, is a functional of the
classical field shifted by that quantum fluctuation. We note that a
similar observation was made previously in the context of inflationary
cosmology~\cite{Son1,PolarS1,KhlebT1}.
The fact that one can express resummed expressions for the pressure
and energy density as ensemble averages over quantum fluctuations has
profound consequences. Without resummation, the relation between the
energy density and the pressure is not single valued. For the resummed
expressions, while the relation between the pressure and energy
density is not single valued at early times, it becomes so after a
finite evolution time. This development of an ``equation of state''
therefore allows one to write the conservation equation for the
resummed energy momentum tensor $T^{\mu\nu}$ as a closed form set of
equations, which are the equations of ideal hydrodynamics. This of
course suggests that the system behaves as a perfect fluid. If the
considerations in our paper can be applied to a gauge theory, the
result would have significant ramifications for the interpretation of
the heavy ion experiments and the extraction of $\eta/s$ in
hydrodynamical models.
The evolution of the system towards the equation of state
characteristic of hydrodynamic flow can be interpreted as arising from
a phase decoherence of the different classical trajectories of the
energy momentum tensor for different initial conditions given by the
ensemble of quantum fluctuations. For a scalar $\phi^4$ theory, the
frequency of the periodic classical trajectories is proportional to
the amplitude. Therefore, for different initial values of the
amplitude, the different trajectories are phase shifted. The ensuing
cancellations between trajectories results in the single valued
relation between the pressure and the energy density. While it appears
that decoherence can arise from the zero mode and near lying modes
alone, the inclusion of the resonant band significantly alters the
decoherence of the system. Similar behavior has been seen in models of
reheating after inflation \cite{ProkoR1,Frolo1,FeldeT1}. In
particular, one sees that quantum de-coherence of the inflaton field
leads to a transition from a dust--like equation of state to a
radiation dominated era.
It is interesting to ask whether the decoherence and concomitant
fluidity observed in our numerical simulations implies thermalization
of the system. We first investigate the behavior of the ensemble of
initial conditions in the {\it Poincar{\'e}} phase plane for the toy
case of uniform background field and fluctuations. One sees that the
initially localized trajectories spread around a close loop filling
the phase-space as one would expect for the phase-space density of a
micro-canonical ensemble. For the toy example considered, the ensemble
average of the trace of the energy momentum tensor can be expressed at
large times as the time average along a single trajectory in the {\it
Poincar{\'e}} phase plane. For the scale invariant $\phi^4$ theory,
this average is zero with the consequence that the energy momentum
tensor becomes traceless resulting in a single valued relation between
the energy density and the pressure.
Going beyond the toy example, for the general case of spatially
non-uniform fluctuations, there is no easy way to visualize trajectories on the {\it
Poincar{\'e}} phase plane because the system is infinite dimensional. However, because the numerical problem
is formulated on a lattice, one can look at a small sub-system on this
lattice and study its event-by-event energy fluctuations. Starting
from a Gaussian initial distribution, we see that the distribution
converges to an exponential form. One can also study the moments of
the energy distribution; these again demonstrate a rapid change from
initial transient values to stationary values. While the behavior is
close to those expected from a canonical thermal ensemble, it is
premature from our present studies to make definitive conclusions.
This will require a careful study of the effects of varying the
coupling and volume effects and will be left to a future study.
We note however that the formalism developed in our paper is well
suited to the study of thermalization of quantum systems\footnote{We
thank Giorgio Torrieri for bringing to our attention Berry's
conjecture and the accompanying literature on eigenstate
thermalization.}. It has been argued
previously~\cite{Deuts1,Sredn1,Jarzy1,RigolDO1} that quantum systems
will thermalize if they satisfy Berry's conjecture~\cite{Berry1}. This
conjecture states that the high lying quantum eigenstates of a system
whose classical behavior is chaotic and ergodic have a wavefunction
that behaves as a linear superposition of plane waves whose
coefficients are Gaussian random variables. When an inclusive
measurement is performed on such an eigenstate, one obtains results
that agree with the predictions of the micro-canonical equilibrium
ensemble, a property that has been dubbed ``eigenstate
thermalization'' in~\cite{Sredn1}. If the state at $t=0$ is a coherent
superposition of such eigenstates, the micro-canonical predictions
become valid only after the states in the superposition have
sufficiently decohered--thus for quantum systems where Berry's
conjecture apply, thermalization appears to be a consequence of
decoherence. The ensemble of quantum fluctuations included via the
resummation we develop in the section 2.6 leads to fields that have
precisely this behavior (see
eqs.~(\ref{eq:berry-1}-\ref{eq:berry-2})). Our interest ultimately is
in QCD, where the classical behavior of the system is believed to be
chaotic~\cite{BiroGMT1,HeinzHLMM1,KunihMOST1}. Because much of our
formalism can be extended to gauge theories, we anticipate that a
first principles treatment of thermalization is feasible.
This paper is organized as follows. In section 2, we introduce the
model scalar theory and the CGC-like initial conditions for its
temporal evolution. We then discuss the computation of $T^{\mu\nu}$ at
leading and next-to-leading order. The problem of secular divergences
is noted, and a stable resummation procedure is developed. A
simplified toy model is considered in section 3, wherein only
spatially uniform fluctuations are considered. The behavior of the
resummed pressure and energy density and their relaxation to an
equation of state is studied. These results are interpreted and
understood as a consequence of the decoherence of the system which
allows one to equate ensemble averages to a temporal average over
individual classical trajectories. For the longitudinally expanding
case, temporal evolution in the toy model displays the behavior of a
fluid undergoing ideal hydrodynamic flow. The full quantum field
theory is considered in section 4, where we compute {\it ab initio}
the spectrum of fluctuations. The full theory displays the same
essential features as the toy model studied in section 3, albeit the
interplay of linearly growing low lying momentum modes and the
resonant modes leads to a more complex temporal evolution. In this
section, we also investigate the dependence of the relaxation time on
the strength of the coupling constant. Then, we study the energy
distribution in a small subsystem, and its time evolution. We
conclude with a brief outlook. Much of the details of the computation
are given in appendices. In appendix A, we discuss the numerical
solution of the scalar field model, including the lattice
discretization, the computation of the quantum fluctuation spectrum
and the sensitivity of the results to the ultraviolet cut-off. The
stability analysis of linearized perturbations to the classical field
is considered in Appendix B. The resonance band is identified and the
Lyapunov exponents are computed explicitly. We also discuss the
relationship between decoherence and linear instabilities.
\section{Temporal evolution of $T^{\mu\nu}$}
In this section, we will consider a scalar field toy model whose
behavior mimics key features of the Glasma~\cite{LappiM1} description
of the early behavior of the quark-gluon matter produced in high
energy heavy ion collisions. In the CGC framework, strong color fields
are present in the initial conditions for the evolution of the Glasma.
In this situation, the leading order contribution is given by
classical fields, with higher order corrections coming from the {\sl
apparently} sub-leading quantum fluctuations. We consider the
stress-energy tensor in this scalar field model and discuss its
temporal evolution at leading (LO) and next-to-leading orders (NLO). We
show explicitly that there are contributions at NLO that can grow
with time and become larger than the LO terms. We end this section by
describing how these ``secular'' terms can be resummed and the results
expressed in terms of an average over a Gaussian ensemble of classical
fields.
\subsection{Scalar model with CGC-like initial conditions}
Our CGC inspired scalar model has the Lagrangean
\begin{equation}
{\cal L}\equiv \frac{1}{2}(\partial_\mu\phi)(\partial^\mu\phi)-\frac{g^2}{4!}\phi^4+J\phi\; ,
\label{eq:L}
\end{equation}
where $J$ is an external source. In the CGC framework, the source $J$
coupled to the gauge fields represents the color charge current
carried by the two colliding heavy ions. The current is zero at
positive proper time, corresponding to times after the collision has
taken place. We emulate this feature of the CGC in a simpler
coordinate system by taking the source $J$ to be nonzero only for
Cartesian time $x^0<0$, and parameterize it as\footnote{In the
numerical implementation of the model, the time dependent prefactor
is constrained to vanish when $x^0\to-\infty$ to ensure a free
theory in the remote past.}
\begin{equation}
J(x)\sim \theta(-x^0)\frac{Q^3}{g}\; .
\label{eq:J}
\end{equation}
At $x^0>0$, where $J$ is zero, the fields evolve solely via their
self-interactions, in an analogous fashion to the non-Abelian color
fields produced in the collision of two hadrons or nuclei.
In eq.~(\ref{eq:J}), we incorporated two additional features of the
CGC. The first feature corresponds to a strong external current $J$,
which follows from the power of the inverse coupling when $g\ll 1$;
weak coupling is essential to motivate an expansion in powers of
$g^2$. The other feature of the CGC that is emulated is that the
dimensionful parameter $Q$ in eq.~(\ref{eq:J}) plays a role analogous
to that of the saturation scale~\cite{GriboLR1,MuellQ1}, in the sense
that non-linear interactions are sizeable for modes $|{\boldsymbol k}|\lesssim Q$.
Note that a scalar field theory with a $\phi^4$ coupling in four
space-time dimensions is scale invariant at the classical level--the
coupling constant $g$ is dimensionless in the theory. In our model,
this scale invariance is broken by the coupling of the scalar field to
the external source $J$ containing the dimensionful scale $Q$. We may
therefore anticipate that all physical quantities are simply expressed
by the appropriate power of $Q$ times a prefactor that depends on $g$.
\subsection{$T^{\mu\nu}$ at leading order}
Because the source $J$ contains a power of the inverse coupling, the
power counting for Feynman diagrams indicates that the order of
magnitude of a given graph depends only on its number of external
lines and number of loops, but not on the number of sources $J$
attached to the graph~\cite{GelisV2,GelisV3}. For the
energy-momentum tensor of the theory, the various contributions can be
organized in a series in powers of $g^2$ as
\begin{equation}
T^{\mu\nu}
=
\frac{Q^4}{g^2}
\Big[
c_0+c_1 g^2 +c_2 g^4+\cdots
\Big]\; .
\end{equation}
In this expansion, the coefficients $c_0, c_1, c_2,\cdots$ are
themselves infinite series in the combination $gJ$ corresponding to an
infinite set of Feynman diagrams. This combination is parametrically
independent of $g$ because $J\sim g^{-1}$. More precisely, $c_0$
contains only tree diagrams, $c_1$ 1-loop diagrams, $c_2$ 2-loop
diagrams, and so on.
In our model, the leading order (tree level) contribution to the
energy-momentum tensor can be expressed solely in terms of a classical
solution $\varphi$ of the field equation of motion~\cite{GelisV2}.
Namely, one has
\begin{equation}
T^{\mu\nu}_{_{\rm LO}}(x)
=
c_0\frac{Q^4}{g^2}
=
\partial^\mu\varphi\partial^\nu\varphi
-
g^{\mu\nu}\,\Big[\frac{1}{2}(\partial_\alpha\varphi)^2-\frac{g^2}{4!}\varphi^4\Big]\; ,
\end{equation}
where
\begin{eqnarray}
&&\square \varphi +\frac{g^2}{3!}\varphi^3=J\; ,
\nonumber\\
&&\lim_{x^0\to -\infty}\varphi(x^0,{\boldsymbol x})=0\; .
\label{eq:EOM-LO}
\end{eqnarray}
Clearly, due to the non-linear term in the equation of motion, the
solution $\varphi$ (and hence the coefficient $c_0$) depends on $gJ$
to all orders, as stated previously. This LO energy momentum tensor
is conserved\footnote{Strictly speaking, this is true only at $x^0>0$.
At negative times, some energy is injected into the system by the
external source $J$.},
\begin{equation}
\partial_\mu T^{\mu\nu}_{_{\rm LO}}=0\; .
\end{equation}
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{phi4_LO.ps}}}
\end{center}
\caption{\label{fig:LO} Components of $T^{\mu\nu}_{_{\rm LO}}$ for a
spatially uniform external source. To perform this calculation, we
took in eq.~(\ref{eq:EOM-LO}) a source
$J=g^{-1}Q^3\theta(-x^0)e^{bQx^0}$ (with $g=1, b=0.1$ and $Q=2.5$),
that vanishes adiabatically in the remote past.}
\end{figure}
If the source $J$ is taken to be spatially homogeneous, then the
energy-momentum tensor evaluated at leading order has the simple
form
\begin{equation}
T^{\mu\nu}_{_{\rm LO}}(x)
=
\begin{pmatrix}
\epsilon_{_{\rm LO}} & 0& 0& 0\\
0& p_{_{\rm LO}}& 0& 0\\
0& 0& p_{_{\rm LO}}& 0\\
0& 0& 0& p_{_{\rm LO}}\\
\end{pmatrix}\; ,
\end{equation}
with the leading order energy density and pressure given by
\begin{eqnarray}
\epsilon_{_{\rm LO}}&=& \frac{1}{2}\dot\varphi^2 + \frac{g^2}{4!}\varphi^4\nonumber\\
p_{_{\rm LO}}&=& \frac{1}{2}\dot\varphi^2 - \frac{g^2}{4!}\varphi^4\; .
\label{eq:edens-LO}
\end{eqnarray}
One can easily check that the energy density $\epsilon_{_{\rm LO}}$ is
constant in time at $x^0>0$ (after the external source $J$ has been
switched off), while the pressure $p_{_{\rm LO}}$ is a periodic
function of time at $x^0>0$, as illustrated in the figure
\ref{fig:LO}. From the numerical computation, it is clear that at
this order of the calculation of $\epsilon_{_{\rm LO}}$ and $p_{_{\rm
LO}}$, one does not have a well defined (single valued)
relationship $\epsilon_{_{\rm LO}} = f(p_{_{\rm LO}})$. In other
words, {\sl there is no equation of state at leading order in $g^2$}.
This might appear problematic at the outset because one might expect
that the scale invariance of the theory would require the energy
momentum tensor to be traceless. As discussed further in section
\ref{sec:qavg}, this is not so for the case of a scalar theory.
\subsection{$T^{\mu\nu}$ at next to leading order}
\label{sec:NLO}
At next-to-leading order, the energy momentum tensor can be written as
\begin{eqnarray}
T^{\mu\nu}_{_{\rm NLO}}
&=&c_1 Q^4=
\partial^\mu\varphi\partial^\nu\beta+\partial^\mu\beta\partial^\nu\varphi
-g^{\mu\nu}\Big[\partial_\alpha\beta\partial^\alpha\varphi
-\beta V^\prime(\varphi)
\Big]
+\nonumber\\
&&
+
\!\int\!\frac{d^3{\boldsymbol k}}{(2\pi)^3 2 k}
\Big[
\partial^\mu a_{-{\boldsymbol k}}\partial^\nu a_{+{\boldsymbol k}}
\!-\!
\frac{g^{\mu\nu}}{2}\Big(\partial_\alpha a_{-{\boldsymbol k}}\partial^\alpha a_{+{\boldsymbol k}}-V^{\prime\prime}(\varphi)a_{-{\boldsymbol k}}a_{+{\boldsymbol k}}\Big)
\Big]\; ,
\nonumber\\
&&
\label{eq:NLO}
\end{eqnarray}
where for brevity we use the notation $V(\varphi)\equiv
g^2\varphi^4/4!$ with each prime denoting a derivative with respect to
$\varphi$. In this formula, $\beta$ and $a_{\pm{\boldsymbol k}}$ are small field
perturbations, that are defined by the following equations:
\begin{eqnarray}
&&\Big[\square+V^{\prime\prime}(\varphi)\Big]a_{\pm{\boldsymbol k}}=0\nonumber\\
&&\lim_{x^0\to-\infty}a_{\pm{\boldsymbol k}}(x)=e^{\pm ik\cdot x}\; ,\nonumber\\
&&\Big[\square+V^{\prime\prime}(\varphi)\Big]\beta =
-\frac{1}{2}V^{\prime\prime\prime}(\varphi)
\int\frac{d^3{\boldsymbol k}}{(2\pi)^3 2k}\;a_{-{\boldsymbol k}}a_{+{\boldsymbol k}}\nonumber\\
&&\lim_{x^0\to-\infty}\beta(x)=0\; .
\label{eq:fluctuations}
\end{eqnarray}
Because the classical field $\varphi$ is
spatially homogeneous in the toy model considered here, the equation of motion for $a_{\pm{\boldsymbol k}}$
simplifies to
\begin{equation}
\ddot{a}_{\pm{\boldsymbol k}}+({\boldsymbol k}^2+V^{\prime\prime}(\varphi))a_{\pm{\boldsymbol k}}=0\; ,
\end{equation}
and the field fluctuation $\beta$ depends only on time.
After some algebra, it is easy to check that the energy-momentum
tensor is also conserved at NLO\footnote{This result should be
self-evident because the conservation equation $\partial_\mu
T^{\mu\nu}=0$ is linear in the components of $T^{\mu\nu}$.
Therefore, it does not mix the different $g^2$ orders, requiring the
conservation equation to be satisfied for each order in $g^2$.} for
$x^0>0$,
\begin{equation}
\partial_\mu T^{\mu\nu}_{_{\rm NLO}}=0\; .
\label{eq:cons-NLO}
\end{equation}
The $00$ component of $T^{\mu\nu}_{_{\rm NLO}}$ in eq.~(\ref{eq:NLO})
gives us the energy density at NLO,
\begin{equation}
\epsilon_{_{\rm NLO}}
=
\dot\beta\dot\varphi+\beta V^\prime(\varphi)
+\frac{1}{2}
\int\frac{d^3{\boldsymbol k}}{(2\pi)^3 2k}\;\Big[
\dot{a}_{-{\boldsymbol k}}\dot{a}_{+{\boldsymbol k}}+({\boldsymbol k}^2+V^{\prime\prime}(\varphi))a_{-{\boldsymbol k}}a_{+{\boldsymbol k}}
\Big]\; .
\label{eq:edens_NLO}
\end{equation}
Given eqs.~(\ref{eq:fluctuations}), it is straightforward to verify
that this correction is also constant in time, $\dot\epsilon_{_{\rm
NLO}}=0$, in agreement with eq.~(\ref{eq:cons-NLO}). The $11$
component of eq.~(\ref{eq:NLO}) --the NLO pressure in the $x$
direction-- reads
\begin{equation}
p_{_{\rm NLO}}
=
\dot\beta\dot\varphi -\beta V^\prime(\varphi)
+\frac{1}{2}
\int\frac{d^3{\boldsymbol k}}{(2\pi)^3 2k}\;\Big[
\dot{a}_{-{\boldsymbol k}}\dot{a}_{+{\boldsymbol k}}-({\boldsymbol k}^2-2k_x^2+V^{\prime\prime}(\varphi))a_{-{\boldsymbol k}}a_{+{\boldsymbol k}}
\Big]\; .
\label{eq:pressure_NLO}
\end{equation}
Note that although the integrand is not rotationally invariant, the
result of the ${\boldsymbol k}$ integration is symmetric and the NLO pressures are
the same in all directions.
We evaluated numerically $\epsilon_{_{\rm NLO}}$ and $p_{_{\rm NLO}}$
for a coupling constant $g=1$, by first solving
eqs.~(\ref{eq:fluctuations}) for $\beta$ and for the $a_{\boldsymbol k}$'s (for a
discretized set of ${\boldsymbol k}$'s). The results of this calculation are shown
in the figure \ref{fig:NLO}.
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{phi4_NLO.ps}}}
\end{center}
\caption{\label{fig:NLO} Components of $T^{\mu\nu}_{_{\rm NLO}}$ for a
spatially uniform external source. This calculation was performed
for $g=1$.}
\end{figure}
From this evaluation, we see that the energy density at NLO is
constant at $x^0>0$, as we expected\footnote{This time independence
can be seen as a test of the accuracy of the numerical calculation,
because it results from a cancellation between several terms that
grow with time.}. We also notice that for $g=1$, the NLO correction
to the energy density is very small, of the order of $1.4\%$ of the LO
result\footnote{Indeed, since there is a prefactor $1/4!$ in our
definition of the interaction potential, $g=1$ corresponds to fairly
weak interactions.}. Thus, we conclude from this that for such a
value of the coupling, we have a well behaved perturbative expansion
for $\epsilon$. The NLO pressure however behaves quite differently.
Not only it is varying in time (hence no equation of state at NLO),
but it also has oscillations whose amplitude grows exponentially at
large $x^0$. Therefore, the NLO correction to the pressure eventually
becomes larger than the LO contribution, and the perturbative
expansion for the pressure in powers of $g^2$ breaks down\footnote{A
similar behavior was observed in a different context in
\cite{BoyanVHLS1}.}. Also noteworthy is the fact that at $x^0=0$,
$p_{_{\rm NLO}}$ is still a small correction to $p_{_{\rm LO}}$; it
only becomes large at later times.
\subsection{Interpretation of the NLO result}
The secular divergence of the pressure at NLO can be understood as a
consequence of the unstable behavior of $a_{\pm{\boldsymbol k}}(x)$ for some values
of ${\boldsymbol k}$. The stability analysis of small quantum fluctuations in
$\phi^4$ field theory is performed in appendix \ref{app:stability}.
From this study, one obtains the following results:
\begin{itemize}
\item[{\bf i.}] There is a range in $|{\boldsymbol k}|$ where the $a_{\pm{\boldsymbol k}}$'s diverge
exponentially in time, due to the phenomenon of parametric resonance.
\item[{\bf ii.}] The zero mode ${\boldsymbol k}=0$ fluctuation, $a_0$, diverges
linearly in time, a phenomenon closely related to the fact that the
oscillation frequency in a non-harmonic potential depends on the
amplitude of the oscillations.
\end{itemize}
In addition, one observes numerically that fluctuation modes in the
vicinity of ${\boldsymbol k}=0$, albeit not mathematically unstable, can attain
quite large values. (They appear to grow linearly for some time before
decreasing in value.)
Because of the existence of modes that grow in time, integrals such as
\begin{equation}
I(x^0)\equiv\int \frac{d^3{\boldsymbol k}}{(2\pi)^3 2k}\; a_{-{\boldsymbol k}}(x)a_{+{\boldsymbol k}}(x)\; ,
\label{eq:I}
\end{equation}
that appear in the components of $T^{\mu\nu}_{_{\rm NLO}}$ (see
eqs.~(\ref{eq:edens_NLO}) and (\ref{eq:pressure_NLO})) or in the right
hand side of the equation (eq.~(\ref{eq:fluctuations})) for $\beta$,
are divergent when $x^0\to+\infty$ as illustrated in the figure
\ref{fig:I}.
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{I.ps}}}
\end{center}
\caption{\label{fig:I} Numerical evaluation of the integral defined in
eq.~(\ref{eq:I}). The line denotes an exponential fit to the envelope.}
\end{figure}
In this plot, one can check that the envelope of the oscillations
grows exponentially, with a growth rate $\lambda\approx 2*\mu_{\rm
max}$ where $\mu_{\rm max}$ is the maximal Lyapunov exponent in the
resonance band. If the integral in eq.~(\ref{eq:I}) is evaluated with
an upper cutoff that excludes the resonance band from the integration
domain, then $I(x^0)$ grows only linearly, because now its behavior is
dominated by the soft fluctuation modes whose growth is linear.
Even though secular divergences in integrals such as eq.~(\ref{eq:I})
are present in eq.~(\ref{eq:edens_NLO}), they cancel in the
calculation of $\epsilon_{_{\rm NLO}}$ because the energy density in
our toy model is protected by the conservation of the energy momentum
tensor. However, they do not cancel in $p_{_{\rm NLO}}$ which explains
the divergent behavior displayed in fig.~\ref{fig:NLO}.
\subsection{Alternate form of $T^{\mu\nu}$ at NLO}
The secular divergence of the pressure at NLO suggests that the weak
coupling series for the pressure may be better behaved if one develops
a resummation scheme that captures the physics of the secular terms by
identifying their contribution and summing them to all orders in
perturbation theory. Before we do this, we shall discuss a general
formulation of the energy-momentum tensor at NLO which will help
formulate the problem of resumming secular terms.
In previous works~\cite{GelisV2,GelisV3}, we showed that the problem
of computing NLO corrections for {\it inclusive} quantities-such as
components of the energy momentum tensor in field theories with strong
sources could be formulated as an initial value problem. Specifically,
for the energy-momentum tensor, we can write the NLO contribution at
an arbitrary space-time point as the action of a functional operator
acting on the LO contribution,
\begin{equation}
T^{\mu\nu}_{_{\rm NLO}}(x)
=
\Big[
\int d^3{\boldsymbol u}\; \beta\cdot{\mathbbm T}_{\boldsymbol u}
+\frac{1}{2}\int d^3{\boldsymbol u} d^3{\boldsymbol v}\;\int\frac{d^3{\boldsymbol k}}{(2\pi)^3 2k}
[a_{+{\boldsymbol k}}\cdot{\mathbbm T}_{\boldsymbol u}][a_{-{\boldsymbol k}}\cdot{\mathbbm T}_{\boldsymbol v}]
\Big]
T^{\mu\nu}_{_{\rm LO}}(x)
\; ,
\label{eq:NLO-1}
\end{equation}
The operator ${\mathbbm T}_{\boldsymbol u}$ that appears in eq.~(\ref{eq:NLO-1}) is
the generator of shifts of the initial conditions
$\varphi_0,\partial_0\varphi_0$ (at $x^0=0$) of the classical field,
\begin{equation}
a\cdot{\mathbbm T}_{\boldsymbol u}
\equiv
a(0,{\boldsymbol u})\frac{\delta}{\delta\varphi_0({\boldsymbol u})}
+
\dot{a}(0,{\boldsymbol u})\frac{\delta}{\delta\partial_0\varphi_0({\boldsymbol u})}\; .
\end{equation}
The factor $T^{\mu\nu}_{_{\rm LO}}$ in the functional formulation of
eq.~(\ref{eq:NLO-1}) should therefore be considered as a functional of
the value of $\varphi,\dot\varphi$ at $x^0=0$. The full content of the
temporal NLO evolution of $T^{\mu\nu}$ is contained in
eq.~(\ref{eq:NLO-1}). One can check that this expression is exactly
equivalent to eq.~(\ref{eq:NLO})~\cite{GelisLV3}.
The expression in eq.~(\ref{eq:NLO-1}) has been obtained by splitting
the time evolution at $x^0=0$ such that the $x^0<0$ part of the time
evolution is described by the operator in the square brackets, and the
evolution at $x^0>0$ is hidden in the functional dependence of
$T^{\mu\nu}_{_{\rm LO}}$ with respect to the value of the classical
field $\varphi$ at $x^0=0$. The choice of $x^0=0$ for this split in
the time evolution is arbitrary and equivalent formulas can be
obtained with other choices\footnote{The splitting of the time
evolution in two halves need not be done at a constant $x^0$ and any
locally space-like hypersurface will suffice.}. Here, our choice is
motivated by the fact that $x^0$ is the time at which the
external source $J$ turns off. In view of the resummation we will use
later, it is important to note that the quantum field fluctuations
$\beta$ and $a_{\pm{\boldsymbol k}}$ are still small relative to the classical
field at the splitting time used in the formula. That this is true in
our case is transparent from the figure~\ref{fig:NLO}.
\subsection{Resummation of the NLO corrections}
As seen previously, the fixed order NLO calculation is not meaningful
after a certain time, because it gives a pressure that is larger than
the LO contribution. The NLO contribution (and likely any higher fixed
loop order contribution) has secular divergences because it involves
the {\sl linearized} equation of motion for perturbations to the
classical field $\varphi$. In other words, if $\psi\equiv\varphi+a$,
the NLO calculation approximates the dynamics of $\psi$ by
\begin{eqnarray}
\square\varphi+V^\prime(\varphi)&=&J\nonumber\\
\Big[\square+V^{\prime\prime}(\varphi)\Big]\,a&=&0\; ,
\label{eq:classical+quant}
\end{eqnarray}
on the grounds that the nonlinear terms in $a$ are formally of higher
order in $g^2$. Obviously, if the dynamics of $\psi$ was treated
exactly, by solving instead\footnote{Though this expression looks
identical to the first equation of eq.~(\ref{eq:classical+quant}),
the initial conditions for this equation are different, leading to a
different solution.}
\begin{equation}
\square\psi+V^\prime(\psi)=J\; ,
\end{equation}
we would not have any divergence because the $\psi^4$ potential would
prevent runaway growth of $\psi$. However, in order to achieve this
substitution, we must include in our calculation some contributions
that are of higher order in $g^2$. Thus, we seek a resummation that
restores the lost nonlinearity in the field fluctuations, while
keeping in full the LO and NLO contributions that we have already
calculated.
As we will argue in this section, a simple resummation that leads to
an energy-momentum tensor which is finite at all times consists in
starting from eq.~(\ref{eq:NLO-1}) and in exponentiating the operator
inside the square brackets,
\begin{equation}
T^{\mu\nu}_{\rm resum}(x)
\!\equiv\!
\exp\!\Big[
\int \!\!d^3{\boldsymbol u}\, \beta\cdot{\mathbbm T}_{\boldsymbol u}
+\frac{1}{2}\!\int\! d^3{\boldsymbol u} d^3{\boldsymbol v}\!\int\!\frac{d^3{\boldsymbol k}}{(2\pi)^3 2k}
[a_{+{\boldsymbol k}}\cdot{\mathbbm T}_{\boldsymbol u}][a_{-{\boldsymbol k}}\cdot{\mathbbm T}_{\boldsymbol v}]
\Big]
T^{\mu\nu}_{_{\rm LO}}(x)
\, ,
\label{eq:sum}
\end{equation}
If we Taylor expand the exponential, we recover the full
expressions for the LO and NLO contributions, plus an infinite series
of other terms that are of higher order in $g^2$,
\begin{equation}
T^{\mu\nu}_{\rm resum}(x)
=
\frac{Q^4}{g^2}
\Big[
\underbrace{c_0+c_1 g^2}_{\mbox{fully}} +\underbrace{c_2 g^4+\cdots}_{\mbox{partly}}
\Big]\; .
\end{equation}
From the form of eq.~(\ref{eq:sum}), it is not evident that the
exponentiation leads to a better behaved result; on the surface it appears
that we are including an infinite series of terms that are
increasingly pathological at large times. To see that the result is
now stable when $x^0\to+\infty$, let us consider some generic function
of the classical field at the point $x$, ${\tilde F}[\varphi(x)]$.
The field $\varphi(x)$ is itself a functional of the
values\footnote{Although we do not write that explicitly in order to
simplify the notation, $\varphi_0$ and $\dot\varphi_0$ may depend on
the position ${\boldsymbol x}$.} $\varphi_0$ of the field and $\dot\varphi_0$ of
its first time derivative at $x^0=0$. Thus, the quantity ${\tilde
F}[\varphi(x)]$ is implicitly a function of
$\varphi_0,\dot\varphi_0$,
\begin{equation}
{\tilde F}[\varphi(x)] \equiv F[\varphi_0,\dot\varphi_0]\; .
\end{equation}
Note now that the exponential of $\beta\cdot{\mathbbm
T}_{\boldsymbol u}$ is a translation operator when it acts on a functional
$F[\varphi_0({\boldsymbol u}),\dot{\varphi}_0({\boldsymbol u})]$,
\begin{equation}
\exp\Big[\int d^3{\boldsymbol u}\;
\beta\cdot{\mathbbm T}_{\boldsymbol u}\Big]\,F[\varphi_0,\dot{\varphi}_0]
=
F[\varphi_0+\beta,\dot{\varphi}_0+\dot{\beta}]\; .
\label{eq:ident-1}
\end{equation}
The first term in the exponential in eq.~(\ref{eq:sum}) therefore merely
shifts the initial conditions $\varphi_0,\dot\varphi_0$ at $x^0=0$ of
the classical field $\varphi$ (by amounts $\beta,\dot\beta$).
Similarly, the second term, that involves the exponential of an
operator that has two ${\mathbbm T}$'s, can be rewritten as a sum over
fluctuations of the initial classical field\footnote{An elementary
form of the identity,
\begin{equation*}
e^{\frac{\gamma }{2}\partial_x^2}\,f(x)
=
\int_{-\infty}^{+\infty}dz\;
\frac{e^{-z^2/2\gamma }}{\sqrt{2\pi\gamma }}\,f(x+z)
\; ,
\end{equation*}
can be proven by doing a Taylor expansion of the exponential in the
left hand side and of $f(x+z)$ in the right hand side. In this simple
example, one sees that an operator which is Gaussian in derivatives is
a {\sl smearing operator} that amounts to convoluting the target
function with a Gaussian. Another way of proving the formula is to
apply a Fourier transform to both sides of the equation.}
\begin{eqnarray}
&&
\exp\Big[
\frac{1}{2}\!\int\! d^3{\boldsymbol u} d^3{\boldsymbol v}\int\!\frac{d^3{\boldsymbol k}}{(2\pi)^3 2k}
[a_{+{\boldsymbol k}}\cdot{\mathbbm T}_{\boldsymbol u}][a_{-{\boldsymbol k}}\cdot{\mathbbm T}_{\boldsymbol v}]
\Big]\,F[\varphi_0,\dot{\varphi}_0]
=
\nonumber\\
&&\qquad\qquad=
\int [D\alpha D\dot\alpha]\,Z[\alpha,\dot\alpha]\,
F[\varphi_0+\alpha,\dot{\varphi}_0+\dot{\alpha}]\; ,
\label{eq:ident-2}
\end{eqnarray}
where the distribution $Z[\alpha,\dot{\alpha}]$ is Gaussian in
$\alpha({\boldsymbol x})$ and $\dot\alpha({\boldsymbol x})$, with 2-point correlations given by
\begin{eqnarray}
\big<\alpha({\boldsymbol x})\alpha({\boldsymbol y})\big>&=&
\int\frac{d^3{\boldsymbol k}}{(2\pi)^3 2k}\;a_{+{\boldsymbol k}}(0,{\boldsymbol x})a_{-{\boldsymbol k}}(0,{\boldsymbol y})\; ,
\nonumber\\
\big<\dot\alpha({\boldsymbol x})\dot\alpha({\boldsymbol y})\big>&=&
\int\frac{d^3{\boldsymbol k}}{(2\pi)^3 2k}\;\dot{a}_{+{\boldsymbol k}}(0,{\boldsymbol x})\dot{a}_{-{\boldsymbol k}}(0,{\boldsymbol y})\; .
\label{eq:gaussian}
\end{eqnarray}
Therefore, the energy-momentum tensor resulting from the resummation of
eq.~(\ref{eq:sum}) can be written as
\begin{eqnarray}
T_{\rm resum}^{\mu\nu}
=
\int [D\alpha({\boldsymbol x}) D\dot\alpha({\boldsymbol x})]\,Z[\alpha,\dot\alpha]\;
T^{\mu\nu}_{_{\rm LO}}[\varphi_0+\beta+\alpha]\; ,
\label{eq:sum1}
\end{eqnarray}
where $T^{\mu\nu}_{_{\rm LO}}[\varphi_0+\beta+\alpha]$ denotes the LO
energy-momentum tensor evaluated with a {\sl classical field} whose
initial condition at $x^0=0$ is $\varphi_0+\beta+\alpha$ (and likewise
for the first time derivative).
From eq.~(\ref{eq:sum1}), one can now see why the proposed resummation
cures the pathologies of the NLO contribution. While the fixed-order
NLO result involved linearized perturbations to the classical fields
(that are generically divergent when $x^0\to \infty$), in the resummed
expression these perturbations appear only as a shift of the initial
condition for the full {\sl non-linear equation of motion}. After this
resummation, the evolution of the perturbations at $x^0>0$ is no
longer linear--since the $\phi^4$ potential is bounded from below the
evolution is stable.
In addition to manifestly demonstrating the stable evolution demanded
by the underlying theory, eq.~(\ref{eq:sum1}) is a most useful
expression for a practical implementation of our resummation. It is
important to note however that the integral over ${\boldsymbol k}$ in the 2-point
correlations (eq.~(\ref{eq:gaussian})) that define the Gaussian
distribution of $\alpha$ and $\dot{\alpha}$ should be cut-off at a
value $\Lambda\sim g\varphi_0\sim Q$ in order to avoid ultraviolet
singularities. With such a cutoff, one can show that the sensitivity
to the value of the cutoff is of higher order in $g^2$, while at the
same time being large enough to include in the resummation all the
relevant unstable modes (the modes with $Q\lesssim |{\boldsymbol k}|$ are all
stable).
\section{$T_{\rm resum}^{\mu\nu}$ from spatially uniform fluctuations}
\label{sec:toy}
Before we proceed to a full 3+1-dimensional numerical evaluation of
eq.~(\ref{eq:sum1}) with an {\it ab initio} computation of
eq.~(\ref{eq:gaussian}), we shall first consider, as a warm-up
exercise, a computation including only spatially homogeneous
fluctuations. Albeit not realistic, this much simpler calculation will
be very instructive in understanding the effects of these fluctuations
on the behavior of the energy-momentum tensor.
\subsection{Setup of the problem}
For spatially homogeneous fluctuations, the main
simplification is that functional integrations over the fields
$\alpha$ and $\dot\alpha$ in eq.~(\ref{eq:sum1}) become ordinary
integrals over a pair of real numbers, with the Gaussian weight
\begin{equation}
Z(\alpha,\dot\alpha)\equiv
\exp\left[
-\left(
\frac{\alpha^2}{2\sigma_1}+\frac{\dot\alpha^2}{2\sigma_2}
\right)
\right]\; .
\label{eq:Zuniform}
\end{equation}
The two parameters $\sigma_{1,2}$ can be used in this toy calculation
to control the magnitude of the fluctuations. In the limit
$\sigma_{1,2}\to 0$, we recover the leading order result which of
course receives no contribution from the fluctuations.
The second important simplification in this toy calculation is that
since both the underlying classical field and the fluctuations are
spatially homogeneous, the field equation of motion is an ordinary
differential equation\footnote{\label{foot:jacobi}In this case, the
field equations can even be solved analytically. This can be
seen very simply: from energy conservation,$\frac{1}{2}\,
{\dot\varphi}^2 + V(\varphi)=E_0= V(\phi_{\rm max})$ (with $\phi_{\rm max}$
the amplitude of the oscillations of $\varphi(t)$), on gets
\begin{equation*}
t = {\rm const}+
\frac{1}{\sqrt{2}}
\int_{0}^{\varphi(t)}
\frac{d\psi}{\sqrt{V(\phi_{\rm max}) - V(\psi)}} \; .
\end{equation*}
For a $\phi^4$ potential, the integral in the right hand side is an
elliptic integral, and one can express $\varphi(t)$ as
\begin{equation*}
\varphi(t) = \phi_{\rm max}
\;{\rm cn}_{1/2}\,( g\phi_{\rm max} t/\sqrt{24} +{\rm const})
\; ,
\end{equation*}
where ${\rm cn}_{1/2}$ is the Jacobi elliptic function of the first
kind with the elliptic modulus $k=1/2$. This expression is periodic
with a period $T = {2\sqrt{24}} K(1/2)/{g\phi_{\rm max}}$, where
$K(1/2)\approx 1.85$ is the complete elliptic integral of the first
kind.}. One should note that the characteristic oscillation
frequency is directly proportional to the amplitude of $\varphi(t)$.
This property of the solution will be key in interpreting the results
that follow.
We now turn to the computation of the resummed pressure and energy
density in this toy model. From eqs.~(\ref{eq:edens-LO}) and
(\ref{eq:sum1}), the expressions for the energy density and the
pressure read
\begin{eqnarray}
\epsilon_{\rm resum}&=&\left<\frac{1}{2}\dot\varphi^2+V(\varphi)\right>_{\alpha,\dot\alpha}\; ,\nonumber\\
p_{\rm resum}&=&\left<\frac{1}{2}\dot\varphi^2-V(\varphi)\right>_{\alpha,\dot\alpha}\; ,
\end{eqnarray}
where $\varphi$ is the solution of the classical equation of motion
whose value at $x^0=0$ is $\varphi_0+\alpha$ and whose time derivative
at $x^0=0$ is $\dot\varphi_0+\dot\alpha$. The brackets
$\big<\cdots\big>_{\alpha,\dot\alpha}$ denote an averaging over all
possible values of $\alpha,\dot\alpha$ with the distribution of
eq.~(\ref{eq:Zuniform}).
\subsection{Energy momentum tensor}
In fig.~\ref{fig:Tmunu-LO}, we display the result of the toy model
calculation in the limit where we do not have fluctuations
($\sigma_{1,2}\to 0$). As anticipated, the result is equivalent to the
one displayed in fig.~\ref{fig:LO} for the leading order
calculation.
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{phi4_one.ps}}}
\end{center}
\caption{\label{fig:Tmunu-LO} Components of $T_{\rm LO}^{\mu\nu}$,
where when no quantum fluctuations are included.}
\end{figure}
In this figure, for reasons that will become obvious shortly, we have
represented the energy density divided by three. In
fig.~\ref{fig:uniform}, we show the results of the same calculation
performed with non-zero widths $\sigma_{1,2}$ for the Gaussian
distribution of fluctuations.
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{phi4_avg.ps}}}
\end{center}
\caption{\label{fig:uniform}Components of $T_{\rm resum}^{\mu\nu}$ obtained with a
Gaussian ensemble of spatially uniform quantum fluctuations.}
\end{figure}
We observe a striking difference of the resummed result compared to
the previous (LO) figure--the oscillations of the pressure are damped
and the value of the pressure relaxes to $\epsilon/3$. Subsequently,
one has a single-valued relationship between the pressure and the
energy density, namely, an {\sl equation of state} -- specifically, the
equation of state $\epsilon=3p$ of a scale invariant system in $1+3$
dimensions.
\subsection{Phase-space density}
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{phase_space.ps}}}
\end{center}
\caption{\label{fig:ps}Phase-space distribution of the ensemble of
classical fields at various stages of the time evolution.}
\end{figure}
It is also instructive to look at the phase-space density
$\rho_t(\varphi,\dot\varphi)$ of the points $(\varphi,\dot\varphi)$ as the
system evolves in time. This is shown in fig.~\ref{fig:ps}. At
$t=0$, we start with a Gaussian distribution of the initial
conditions, with a small dispersion around the average values
($\varphi=10$ and $\dot\varphi=0$ in our example).
Each initial condition then evolves independently according to the
classical equation of motion, and the corresponding trajectory in the
$(\varphi,\dot\varphi)$ plane is a closed loop\footnote{These loops
are constant energy curves $\frac{1}{2}\dot\varphi^2+V(\varphi)=H$.}
due to the periodicity of classical solutions. One observes that the
initially Gaussian-shaped cloud of points starts spreading around a
closed loop, to eventually fill it entirely when $x^0\to+\infty$. When
this asymptotic regime is reached, the density
$\rho_t(\varphi,\dot\varphi)$ depends only on the energy--roughly
speaking, the radial coordinate in the plot of figure~\ref{fig:ps}
and no longer on the angular coordinate.
A more formal way of phrasing the same result is to first note that
the time evolution of the phase-space density $\rho_t$ obeys the
Liouville equation,
\begin{equation}
\frac{\partial\rho_t}{\partial t}+\{\rho_t,H\} =0\; ,
\end{equation}
where $\{\cdot,\cdot\}$ is the classical Poisson bracket. Therefore,
if a stationary distribution is reached at late times, it can only
depend on $\varphi$ and $\dot\varphi$ via $H(\varphi,\dot\varphi)$.
The asymptotic behavior of the phase-space density in our toy model is
reminiscent of a micro-canonical equilibrium state, in which the
phase-space density is uniform on a constant energy
manifold\footnote{It should be noted here that a spatially homogeneous
field is very special regarding this issue; indeed, any non-linear
system with a single degree of freedom is ergodic. This is not
necessarily the case if there are more than one degrees of freedom,
as is the case in a full fledged field theory.}. In other words, all
micro-states that have the same energy are equally likely.
\subsection{Interpretation of the results}
We shall now discuss the physical interpretation of our results, first
discussing the decoherence of the temporal evolution of the fields and
their time derivatives, and subsequently, the impact of decoherence on
the relaxation of the pressure towards that of a scale invariant system.
\subsubsection{Decoherence time}
Of the previous numerical observations, the easiest to understand is
the spreading of the phase-space density around a closed orbit.
Because the oscillations are non-harmonic, the various points in the
plot of figure \ref{fig:ps} rotate at different speeds\footnote{The
assumption of a scale invariant theory simplifies some expressions
here, but is not crucial to the argument. The only requirement for
this phenomenon is that the frequency of the oscillations depends on
their amplitude; thus any non-harmonic potential will lead to
similar results.}; in a $\phi^4$ potential, the outer points rotate
faster than the inner ones. Therefore, as time increases, the cloud of
points spreads more and more due to this effect.
One can estimate the time necessary for the cloud of points to spread
over a complete orbit. This happens when the angular spread of the
points reaches the value $2\pi$. For one field configuration, this
angular variable is, up to a phase that depends on the initial
condition, $\theta = \omega t$, and the angular velocity $\omega$
depends only on the energy of that particular field configuration. (In our case, this phase is small for a narrow Gaussian distribution.) If
we consider two field configurations, their angular variable
difference $\Delta\theta$ increases linearly in time, $\Delta\theta
= \Delta\omega\, t$, where $\Delta\omega$ is the difference between
their angular velocities. In the case of a $g^2\phi^4/4!$ potential,
one can prove that (see the footnote \ref{foot:jacobi})
\begin{equation}
\omega =
\frac{\pi}{2\sqrt{3}}
\frac{g\phi_{\rm max}}{\int_{-1}^{+1}\frac{dx}{\sqrt{1-x^4}}}
\approx 0.346\,g\phi_{\rm max}\; ,
\end{equation}
where $\phi_{\rm max}$ is amplitude of the oscillations of the $\varphi$ field.
Thus, the angular shift between the two field configurations is also
$\Delta\theta \approx 0.346\,g\Delta\phi_{\rm max}\, t$, and this
shift reaches $2\pi$ in a time
\begin{equation}
t\approx \frac{18.2}{g\Delta\phi_{\rm max}}\; .
\label{eq:t-decoherence}
\end{equation}
After this time, the two fields have become completely incoherent. We
see that this time is inversely proportional to the coupling constant
$g$, and to the difference of the field amplitudes. Thus a narrow
initial Gaussian distribution will need a longer time to spread around
the orbit than a broader initial distribution.
\subsubsection{Equation of state from quantum averaging}
\label{sec:qavg}
Once we know that the phase-space density spreads uniformly on
constant energy curves, it is easy to understand why the pressure
relaxes towards $\epsilon/3$ when we let the initial conditions for
the classical field fluctuate. The trace of the energy-momentum tensor
(assuming 4 dimensions of space-time) is
\begin{equation}
T^\mu{}_\mu=
\varphi\left(\square\varphi+4\frac{V(\varphi)}{\varphi}\right)
-
\partial_\alpha(\varphi\partial^\alpha\varphi)\; .
\end{equation}
A scale invariant theory in four dimensions is a theory in which the
interaction potential obeys $V^\prime(\varphi)=4V(\varphi)/\varphi$. This is
the case of a $\phi^4$ interaction. Therefore, the first term in the
right hand side of the previous equation vanishes thanks to the
equation of motion of the classical field $\varphi$. This result shows
that the energy-momentum tensor of a single configuration of classical
field is not zero in our model, but is a total
derivative\footnote{There is an alternative ``improved'' definition
of the energy-momentum tensor that is explicitly
traceless~\cite{CallaCJ1}. However, while the energy density has a
single valued relation to the pressure, this pressure is not the
canonical pressure. As we shall discuss later in section
\ref{sec:ccj}, both definitions give a deviation from ideal
hydrodynamic flow, which is cured by the quantum averaging described
here.}. In our simplified toy model where the fields are spatially
homogeneous, the previous relation simplifies to
\begin{equation}
T^\mu{}_\mu=
-
\frac{d(\varphi\dot\varphi)}{dt}\; .
\end{equation}
When averaged over one period, the trace of the energy-momentum of
one classical field configuration vanishes because the classical
field is a periodic function of time,
\begin{equation}
\overline{T^\mu{}_\mu}\equiv \frac{1}{T}\int_t^{t+T}d\tau\;
T^\mu{}_\mu(\varphi(\tau),\dot\varphi(\tau))=0\; ,
\label{eq:time-avg}
\end{equation}
where the result is independent of $t$. When we calculate the
energy-momentum tensor averaged over fluctuations of the initial
conditions, we are in fact performing an ensemble average weighted by
the phase-space density $\rho_t(\varphi,\dot\varphi)$,
\begin{equation}
\left<T^\mu{}_\mu\right>_{\alpha,\dot\alpha}
=
\int d\varphi\, d\dot\varphi\; \rho_t(\varphi,\dot\varphi)\;T^\mu{}_\mu(\varphi,\dot\varphi)\; ,
\end{equation}
and the time dependence of the left hand side comes from that of the
density $\rho_t$.
It is convenient to
trade the integration variables $\varphi,\dot\varphi$ for energy/angle
variables $E,\theta$,
\begin{equation}
\left<T^\mu{}_\mu\right>_{\alpha,\dot\alpha}
=
\int dE d\theta\; \tilde\rho_t(E,\theta)\;
T^\mu{}_\mu(E,\theta)\; ,
\end{equation}
where $\tilde\rho_t$ is the phase-space density in the new system of
coordinates\footnote{$\tilde\rho_t$ is equal to the original $\rho_t$ times the
Jacobian of the change of variables.}. Our first result shows that
$\tilde\rho_t(E,\theta)\longrightarrow \tilde\rho_t(E)$, namely,
becomes independent of $\theta$ at late times, which enables us to
write the previous equation as
\begin{equation}
\left<T^\mu{}_\mu\right>_{\alpha,\dot\alpha}
\empile{\approx}\over{t\to \infty}
\int dE \; \tilde\rho_t(E)\;\int d\theta\;
T^\mu{}_\mu(E,\theta)\; .
\end{equation}
The crucial point here is that the integral over $\theta$ is simply
the integral over one orbit for a single classical field configuration
(eq.~(\ref{eq:time-avg})),
\begin{equation}
\int d\theta\;
T^\mu{}_\mu(E,\theta)
=
\frac{2\pi}{T}\int_t^{t+T}d\tau\;
T^\mu{}_\mu(\varphi(\tau),\dot\varphi(\tau))=0\; .
\end{equation}
Thus, we have proven that
\begin{equation}
\epsilon-3p=\left<T^\mu{}_\mu\right>_{\alpha,\dot\alpha}
\empile{\approx}\over{t\to \infty}0\; ,
\end{equation}
in agreement with what we have observed numerically. Moreover, from
the derivation of this result, it is clear that the time necessary to
reach this limit is the same as the time (in
eq.~(\ref{eq:t-decoherence})) necessary for the phase-space density to
become independent of the angular variable $\theta$.
\subsection{Effect of the longitudinal expansion}
We have thus far considered a system of strong fields enclosed in a
box of fixed volume. There is therefore no concept of hydrodynamical
flow in such a system. To fully understand the implications of
decoherence and relaxation of the pressure we have discussed
previously for hydrodynamical flow, we will now simply generalize
the toy problem of spatially uniform fields and fluctuations to a
system undergoing a boost invariant one dimensional expansion.
\subsubsection{Relaxation of the pressure}
The geometry of the one dimensional expansion (chosen to be the $z$
direction) is appropriate to describe the collision of two projectiles
(nuclei) at ultrarelativistic energies. The natural coordinates are
the proper time $\tau$ and rapidity $\eta$ defined by
\begin{eqnarray}
\tau&\equiv& \sqrt{t^2-z^2}\; ,\nonumber\\
\eta&\equiv& \frac{1}{2}\ln\left(\frac{t+z}{t-z}\right)\; .
\end{eqnarray}
In the spatial plane orthogonal to the $z$ axis, the coordinates are
denoted by ${\boldsymbol x}_\perp$. In this system of coordinates, the classical
equation of motion for a field $\varphi$ that depends only on proper
time is
\begin{equation}
\ddot\varphi+\frac{1}{\tau}\dot\varphi+\frac{g^2}{6}\varphi^3=0\;,
\label{eq:eom-expanding}
\end{equation}
where the dot now denotes a derivative with respect to $\tau$. The
analog of the the toy problem we discussed previously in the first part of
this section is to let the initial conditions of the field have
Gaussian fluctuations $\alpha,\dot\alpha$ that are independent of
$\eta$ and ${\boldsymbol x}_\perp$.
The components of the energy-momentum tensor in this system of
coordinates, averaged over the fluctuations of the initial conditions are
\begin{eqnarray}
\epsilon\equiv T^{\tau\tau}
&=&
\Big<\frac{1}{2}\dot\varphi^2+V(\varphi)\Big>_{\alpha,\dot\alpha}\; ,
\nonumber\\
p\equiv T^{xx}=T^{yy} = \tau^2 T^{\eta\eta}
&=&
\Big<\frac{1}{2}\dot\varphi^2-V(\varphi)\Big>_{\alpha,\dot\alpha}\; .
\end{eqnarray}
As in the fixed volume case, we shall use the distribution of
eq.~(\ref{eq:Zuniform}) for $\alpha,\dot\alpha$. The result of this
computation is shown in the figure \ref{fig:exp-phi4}.
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{expansion-phi4-3.ps}}}
\end{center}
\caption{\label{fig:exp-phi4}Numerical evaluation of $T^{\mu\nu}$ for
fields undergoing a boost invariant 1-dimensional expansion in a
$\phi^4$ theory with a Gaussian ensemble of spatially uniform
initial fluctuations.}
\end{figure}
The dots represent the energy density divided by 3, and one observes
that its time dependence is well described by a $\tau^{-4/3}$ decay
characteristic of boost invariant flow in ideal relativistic
hydrodynamics. If we do not include fluctuations of the initial
conditions, we observe that the pressure oscillates between positive
and negative values, with a decreasing envelope. Conversely, if we
average over an ensemble of initial conditions, we see the
oscillations of the pressure dampen quickly, ensuring that the
pressure approaches one third of the energy density.
These results are in sharp contrast to what one obtains for a $\phi^2$
potential. The results are shown in fig.~\ref{fig:exp-phi2}. In this
case, the fluctuations do not make the pressure converge to
$T^{00}/3$, and the latter decreases as $\tau^{-1}$ instead of
$\tau^{-4/3}$.
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{expansion-phi2.ps}}}
\end{center}
\caption{\label{fig:exp-phi2}Numerical evaluation of $T^{\mu\nu}$ for
fields undergoing a boost invariant 1-dimensional expansion in a
$\phi^2$ theory, with a Gaussian ensemble of uniform initial
fluctuations.}
\end{figure}
\subsubsection{Interpretation of the results for expanding fields}
From the equation of motion in eq.~(\ref{eq:eom-expanding}), we obtain
\begin{eqnarray}
\frac{d\epsilon}{d\tau} &=&
\dot\varphi\Big[\ddot\varphi+V^\prime(\varphi)\Big]
= -\frac{1}{\tau}\dot\varphi^2\nonumber\\
&=& -\frac{\epsilon+p}{\tau}\; .
\label{eq:exp-hydro}
\end{eqnarray}
This equation, which is valid for individual classical field
configurations at every time, is identical to Euler's equation for
boost-invariant ideal hydrodynamics. The difference with
hydrodynamics lies in the fact that hydrodynamics assumes the
existence of an equation of state $p=f(\epsilon)$ to ensure a closed
form expression in eq~(\ref{eq:exp-hydro}). In classical field
dynamics, one is not free to impose a relationship between $\epsilon$
and $p$ since they are both completely determined from the field
$\varphi$ and its derivative $\dot\varphi$. For instance, as seen in
fig.~\ref{fig:exp-phi4}, for a single classical solution, we do not
have a one-to-one correspondence between $\epsilon$ and $p$;
$\epsilon$ has a monotonous behavior while $p$ oscillates. What is
remarkable is that the ensemble average over the initial conditions
leads in a short time to a one-to-one correspondence $\epsilon=3p$,
which is precisely the equation of state one would use in boost
invariant hydrodynamics of a perfect fluid.
The mechanism whereby this relationship is reached is the same as in
the non-expanding case. As previously, one can prove that for a single
phase-space trajectory, the time averages of $\epsilon$ and $p$ obey a
relation identical to the expected equation of state, because the
trace of the energy momentum tensor is a total derivative. Then, by
using the fact that different initial conditions lead to different
oscillation frequencies, one gets the phase decoherence that enables
us to transform the ensemble average over the initial conditions into
a time average along one classical field trajectory. This decoherence
is the missing ingredient in the harmonic case-as we noted previously,
it arises for the $\phi^4$ theory because the angular velocity of the
phase space trajectory of an individual configuration depends on the
amplitude of the configuration.
From this result, it is very easy to obtain the $\tau^{-4/3}$ behavior of
the energy density. The ensemble
average of eq.~(\ref{eq:exp-hydro}) at late times is
\begin{equation}
\frac{d\epsilon}{d\tau}\empile{=}\over{\tau\to+\infty}
-\frac{4}{3}\epsilon\; ,
\end{equation}
which leads immediately to the observed behavior. Since both
$\epsilon$ and $p$ decrease like $\tau^{-4/3}$ even for a single
configuration (if one considers the envelope of the oscillations of
$p$), this means that at late times we have
\begin{equation}
\varphi\sim \tau^{-1/3}\; ,\quad \dot\varphi \sim \tau^{-2/3}\; .
\end{equation}
This behavior is seen from the simple ansatz $\varphi(\tau)\sim
\cos(f(\tau)) \tau^{-1/3}$, which, while inaccurate in detail,
qualitatively captures the right physics. For a $\varphi^4$ potential,
the frequency $\dot{f}\propto \varphi\sim \tau^{-1/3}$ for a $\phi^4$
potential; using this relation in our ansatz gives the stated result.
The fact that $\dot\varphi$ decreases faster than $\varphi$ is due to
the slowing down of the oscillations with time, as their amplitude
decreases.
\subsubsection{Callan-Coleman-Jackiw energy-momentum tensor}
\label{sec:ccj}
An alternate definition of the energy-momentum tensor was proposed by
Callan, Coleman and Jackiw (CCJ)~\cite{CallaCJ1}. Their expression is
explicitly traceless. They argued further that their form of the
stress energy tensor improved properties relative to the canonical one
with regard to renormalization. Let us briefly summarize the
differences between the usual definition of $T^{\mu\nu}$ and CCJ's.
With the canonical definition that we have used thus far, one has,
\begin{eqnarray}
T^{\mu\nu} &\equiv& (\partial^\mu\varphi)(\partial^\nu\varphi)-g^{\mu\nu}{\cal L}\nonumber\\
\epsilon&=&\frac{1}{2}\dot\varphi^2+V(\varphi)\nonumber\\
p&=&\frac{1}{2}\dot\varphi^2-V(\varphi)\nonumber\\
\epsilon-3p &=& \frac{d(\varphi\dot\varphi)}{d\tau}\nonumber\\
\frac{d\epsilon}{d\tau}&=&-\frac{\epsilon+p}{\tau}\; .
\end{eqnarray}
With this definition of $T^{\mu\nu}$, one obtains the equation for
Bjorken hydrodynamics automatically for each configuration of the
classical field. However, one gets $\epsilon=3p$ only through
decoherence, by averaging over an ensemble of initial conditions.
In comparison, with CCJ's definition of the energy-momentum tensor, one has
\begin{eqnarray}
T^{\mu\nu} &\equiv& (\partial^\mu\varphi)(\partial^\nu\varphi)-g^{\mu\nu}{\cal L}-\frac{1}{6}(\partial^\mu\partial^\nu-g^{\mu\nu}\square)\varphi^2\nonumber\\
\epsilon&=&\frac{1}{2}\dot\varphi^2+V(\varphi)\nonumber\\
p&=&\frac{1}{2}\dot\varphi^2-V(\varphi)-\frac{1}{6}\square\varphi^2\nonumber\\
\epsilon-3p &=& 0\nonumber\\
\frac{d\epsilon}{d\tau}&=&-\frac{\epsilon+p}{\tau}-\frac{1}{3\tau}\frac{d(\varphi\dot\varphi)}{d\tau}\; .
\end{eqnarray}
With this form of the energy momentum tensor, the equation of state is
satisfied for each classical field configuration, but not Bjorken's
hydrodynamic equation. It is only after an average over an ensemble of
initial conditions that the last term ($\sim
d(\varphi\dot\varphi)/d\tau$) in the last equation vanishes by
decoherence.
Since one requires simultaneously
\begin{eqnarray}
\frac{d\epsilon}{d\tau}+\frac{\epsilon+p}{\tau}&=&0\nonumber\\
\epsilon-3p&=&0 \; ,
\end{eqnarray}
for ideal hydrodynamical flow, one sees that there is no discrepancy
between the two descriptions despite the apparent differences. For
our choice of the energy momentum tensor, the reason why we didn't
have an ideal hydrodynamic behavior at the beginning of the evolution
was because of the lack of an equation of state. In the case of CCJ's
energy momentum tensor, it is because of a violation of the canonical
Euler equation. The net effect of the quantum averaging in each case
is to get rid of one or the other violation thereby ensuring ideal
hydrodynamical behavior. In the following section, we will discuss
only the canonical energy momentum tensor, for which the focus is on
obtaining the equation of state as the necessary condition for
hydrodynamical flow.
\section{Results from the full fluctuation spectrum}
In the previous section, we showed that averaging over an ensemble of
initial conditions for classical fields can lead the pressure to relax
towards one third of the energy density. However, this study was
oversimplified since we used only fluctuations that are uniform in
space, and their Gaussian distribution was set by hand. However,
quantum field theory {\sl predicts} what the spectrum of these
fluctuations is: one should average the LO energy-momentum
tensor\footnote{Since, at $x^0=0$, $\beta$ (see eq.~(\ref{eq:sum1})) is
a small shift that does not fluctuate, we have absorbed it into a
redefinition of the classical field $\varphi_0$.},
\begin{eqnarray}
T_{\rm resum}^{\mu\nu}
=
\left<
T^{\mu\nu}_{_{\rm LO}}[\varphi_0+\alpha]\right>_{\alpha,\dot\alpha}\; ,
\label{eq:sum1-1}
\end{eqnarray}
over space-dependent random Gaussian fields $\alpha$ and
$\dot\alpha$ that have the following variance:
\begin{eqnarray}
\big<\alpha({\boldsymbol x})\alpha({\boldsymbol y})\big>&=&
\int\frac{d^3{\boldsymbol k}}{(2\pi)^3 2k}\;a_{+{\boldsymbol k}}(0,{\boldsymbol x})a_{-{\boldsymbol k}}(0,{\boldsymbol y})
\nonumber\\
\big<\dot\alpha({\boldsymbol x})\dot\alpha({\boldsymbol y})\big>&=&
\int\frac{d^3{\boldsymbol k}}{(2\pi)^3 2k}\;\dot{a}_{+{\boldsymbol k}}(0,{\boldsymbol x})\dot{a}_{-{\boldsymbol k}}(0,{\boldsymbol y})\; ,
\label{eq:gaussian-1}
\end{eqnarray}
which leaves no freedom to handpick what fluctuations we use. From
these formulas, we can numerically compute {\it ab initio} the
behavior of the pressure. The only tunable quantities in the
calculation are then the scale $Q$ (or more generally the source $J$)
that controls the amount of energy injected into the system at $t<0$,
and the coupling constant $g$. Note that the above spectrum of
fluctuations for the initial condition for the field $\varphi$ is
equivalent to parameterizing the initial field
as\footnote{\label{foot:berry}If we recall that the $a_{\pm{\boldsymbol k}}$'s are
plane waves modified by the presence of the background field
$\varphi_0$, we observe that the fluctuating part of the initial
field is very similar to the form of the wavefunction of high lying
eigenstates for quantum systems that obey Berry's conjecture.}
\begin{equation}
\varphi(0,{\boldsymbol x})\equiv \varphi_0(0,{\boldsymbol x})+\int\frac{d^3{\boldsymbol k}}{(2\pi)^3 2k}\;
\Big[c_{\boldsymbol k}\, a_{+{\boldsymbol k}}(0,{\boldsymbol x})+c_{\boldsymbol k}^*\, a_{-{\boldsymbol k}}(0,{\boldsymbol x})
\Big]\; ,
\label{eq:berry-1}
\end{equation}
where the $c_{\boldsymbol k}$ are random Gaussian numbers with the following
variance
\begin{equation}
\big<c_{\boldsymbol k} c_{\boldsymbol l}\big>=0\; ,\qquad
\big<c_{\boldsymbol k} c_{\boldsymbol l}^*\big>= (2\pi)^3 |{\boldsymbol k}|\delta({\boldsymbol k}-{\boldsymbol l})\; .
\label{eq:berry-2}
\end{equation}
Details of the numerical lattice computation are relegated to the
appendix \ref{app:lattice}; in this section we focus on the results
from numerical simulations with this {\it ab initio} spectrum of
fluctuations.
\subsection{Numerical results}
Unless stated otherwise, the numerical results in this section are
obtained on a $12^3$ lattice\footnote{In some instances, we have also
performed simulations on a $20^3$ lattice and found only very small
differences as long as the physical scales are below the lattice
cutoff.}. The functional integration in eq.~(\ref{eq:sum1}) is
approximated by a Monte-Carlo average over 1000 configurations of the
initial conditions, distributed according to eqs.~(\ref{eq:berry-1})
and (\ref{eq:berry-2}).
In fig.~\ref{fig:pressure1}, we show the result of the
computation of the pressure averaged over the Gaussian ensemble of
initial conditions, for a value of the coupling\footnote{Since the
prefactor in the interaction potential is $g^2/4!$, a value $g=0.5$
corresponds to a very weak coupling strength.} $g=0.5$. We also show the
energy density divided by three on the same plot.
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{relax_all_g0.5.ps}}}
\end{center}
\caption{\label{fig:pressure1}Time evolution of the pressure averaged
over the initial fluctuations. All the resonant modes are included
in the simulation. The coupling constant is $g=0.5$.}
\end{figure}
All the quantities in this plot are expressed in lattice units, which
means that the horizontal axis is $t/a$ (where $a$ is the lattice
spacing) and the vertical axis should be understood as $\epsilon
a^4/3$ or $pa^4$. The lattice cutoff in this simulation is chosen to
be just above the upper limit of the parametric resonance window
($k/m_0=3^{-1/4}$ where $m_0^2=g^2\varphi_0^2/2$); therefore, all the
resonant modes take part in the dynamics of the system.
We observe that the ensemble averaged pressure relaxes towards
$\epsilon/3$. This plot, obtained with the spectrum of
fluctuations predicted by quantum field theory, is one of the central
results of this paper. One can qualitatively identify two stages in
this relaxation: (1) in the range $0\le t\lesssim 50$, the amplitude
of the pressure oscillations decreases very quickly to a moderate
value and, (2) from time $50$ onwards, one has a slower approach of
the pressure to $\epsilon/3$ that gets slowly rid of the residual
oscillations. We will observe again this two-stage time evolution when
we look at the fluctuations of the energy density.
\subsection{Influence of the resonant modes}
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{relax_less_g0.5.ps}}}
\end{center}
\caption{\label{fig:pressure2}Time evolution of the pressure averaged
over the initial fluctuations. The lattice cutoff is located below
the resonance band in order to exclude them from the simulation. The
coupling constant is $g=0.5$.}
\end{figure}
In section \ref{sec:toy}, we observed that the pressure relaxes
to $\epsilon/3$ even if only the mode ${\boldsymbol k}=0$ is included in the
simulation. This was understood as an consequence of the phase decoherence
that exists in a non-harmonic potential between classical solutions
that have slightly different amplitudes. When we include all the
${\boldsymbol k}$-modes of the fluctuations, the situation becomes more
complicated. In particular, the stability analysis of these
fluctuations (see the appendix \ref{app:stability}) indicates that in
addition to a linear instability of the soft modes due to the above
mentioned decoherence phenomenon, there are also exponentially
unstable modes in a narrow band of values ${\boldsymbol k}$.
In order to assess the role played in the time evolution by the modes
of the resonance band, we performed a second simulation with the same
physical parameters, but now with the lattice cutoff placed just below
the lower end of the resonance band. This makes certain that none of
the modes that exist on this lattice has an exponential instability.
Since the resonance band is quite narrow, this is a small change of
the cutoff in physical units because the cutoff in the earlier
simulation was just above the upper end of the resonance band.
However, one can see in the figure \ref{fig:pressure2} that excluding
the resonant modes leads to significant changes.
The final outcome, the relaxation of the pressure towards $\epsilon/3$, is not
changed, but the details of the time evolution of the pressure are
modified. Firstly, one observes a rather long delay during which the
oscillations of the pressure remain almost constant in amplitude.
Then, at a time of order $75$ in lattice units, these oscillations are
damped very quickly to very small wiggles around $\epsilon/3$. Except
for a brief relapse, the oscillations remain very small after this
time. In particular, the two-stage evolution that we observed with the
full spectrum is now replaced by the following two stages: (1) nothing
happens and, (2) very rapid relaxation that leaves almost no residual
oscillations.
Therefore, it appears that the resonant modes, even if their presence
or absence in the resummation does not change the final outcome, do
alter significantly the detailed time evolution of the pressure. At
this point, the precise role of the resonant modes is somewhat
unclear. It appears that the dynamics of the complete system is much
richer than what one can learn by studying the linearized evolution of
a single mode as done in the stability analysis of the appendix
\ref{app:stability}. This analysis does not capture the non-linear
couplings between the various modes (once the instabilities have made
them large) which gives them a big role in the late stage evolution of
the system. This certainly deserves further study.
\subsection{Dependence on the coupling constant}
The simulation that led to the result of fig.~\ref{fig:pressure1}
was performed with a value $g=0.5$ for the coupling constant -- a very
small value for our scalar field theory since there is also a factor
$1/4!$ in the interaction potential.
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{gdep-all.ps}}}
\end{center}
\caption{\label{fig:gdep}Time evolution of the pressure averaged over
the initial fluctuations for various values of the coupling
constant: $g=0.5, 1, 2, 4, 8$. All the resonant modes are included
in the simulation. See footnote ~\ref{footnote:latticeunits}. }
\end{figure}
We have studied the time evolution of the pressure for larger values
of the coupling constant: $g=1,2,4,8$, and the results are shown in
the figure \ref{fig:gdep}. Note that this computation is done at fixed
energy density. Indeed, because $Q$ is the only dimensionful parameter
of our model and there is a factor $1/g$ in the source $J$, the energy
density behaves at leading order as $\epsilon\propto Q^4/g^2$. Thus,
if we increase $g$ at constant $Q$, the energy density decreases. As
our goal is to assess the time at which the pressure obeys an equation
of state (thereby justifying a hydrodynamical description of the
system), the comparison of the relaxation for various couplings should
be done for systems that have the same energy density. Therefore, in
the comparison shown in fig.~\ref{fig:gdep}, the value of $Q$ has
been adjusted in each simulation such that the energy density remains
unchanged.
Fig.~\ref{fig:gdep} demonstrates that the relaxation time
decreases with increasing coupling constant $g$.
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{gdep-1.ps}}}
\end{center}
\caption{\label{fig:gdep1}Points: relaxation time (see text for
the definition used here) as a function of the coupling $g$. Line:
fit by a power law.}
\end{figure}
In fig.~\ref{fig:gdep1} we have represented the relaxation time, defined
here as the time necessary to reduce the initial oscillations of the
pressure by a factor 4, as a function of the coupling constant $g$ for
our set of values of $g$. One can fit all the points except the last
one ($g=8$) by a power law that suggests the following
dependence\footnote{\label{footnote:latticeunits}The axis of the figure \ref{fig:gdep} are in
lattice units. Thus, the horizontal axis is $t/a$ and the
vertical axis $pa^4$ or $\epsilon a^4/3$, where $a$ is the lattice
spacing. Since our model is scale invariant, the relaxation time
scales like $\epsilon^{-1/4}$. By eliminating $a$ between the
horizontal and vertical axis, it is easy to get the value of
$\epsilon^{1/4}t$. For $g=4$ we have $\epsilon a^4=200$ and the
relaxation time is $t/a\approx 30$, leading to $\epsilon^{1/4} t
\approx 113$ (this combination is $11$ for $g=8$). Then, from one's
favorite value of $\epsilon$ in GeV/fm${}^3$, it is easy to obtain
the relaxation time in fm's.}
\begin{equation}
t_{\rm relax}=\frac{\rm const}{g^{2/3}\epsilon^{1/4}}\; .
\end{equation}
The right most point in this plot is an outlier that does not follow
this power law, possibly because this value of the coupling is too
extreme for our approximations/resummations to make sense (for $g=8$,
the interaction strength $g^2/4!$ is significantly above 1).
\subsection{Energy density fluctuations}
The results we have shown thus far indicate that the pressure in the
system relaxes towards the equation of state $p=\epsilon/3$, at
relaxation times that decrease as the coupling constant increases.
However, this study does not in and of itself tell us much about the
nature of the state reached by the system. In particular, it does not
tell us whether the system reaches a state of local thermal
equilibrium.
Because we have a system of strong fields whose modes have large
occupation numbers, it is unlikely that the system can be described in
terms of quasi-particles that have a Bose-Einstein distribution. In
section \ref{sec:toy}, we observed that in the simple example studied
that the phase-space density reaches a stationary form reminiscent of
a micro-canonical equilibrium ensemble. Unfortunately, now that we
are looking at a full fledged quantum field theory, the phase-space is
infinite dimensional and whether the same behavior occurs is difficult
to assess numerically.
There are however signs of thermalization in the fluctuations of the
energy distribution in the system. For the system as a whole, energy
is conserved and will not fluctuate, regardless of whether the system
is in thermal equilibrium or not. However, as is well known for
canonical ensembles, by looking at energy fluctuations in a small
subsystem, one can learn something about the energy exchanges between
this subsystem and the rest of the system which acts as a heat bath.
In particular, the nature of the fluctuations in the energy
distribution of the subsystem can tell us whether it is in equilibrium
with the rest of the system. If this is the case, the fluctuations are
those of a canonical ensemble with a density operator
$\rho\equiv\exp(-\beta H)$.
We show in figs.~\ref{fig:Edist} and \ref{fig:Emoments} results
from a study for the smallest subsystem one can conceive of on a
lattice--a single lattice site.
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{Edist.ps}}}
\end{center}
\caption{\label{fig:Edist}Distribution of energy density at one
lattice site, at various times in the evolution. The coupling
constant is $g=0.5$.}
\end{figure}
In fig.~\ref{fig:Edist}, we display histograms of the values of
the energy on one site\footnote{In lattice units, this is simply the
value of $T^{00}$ at one given site.}, at various times in the
evolution. These curves are normalized so that their integral is
unity--they can be interpreted as probability
distributions for the value of the energy on one lattice site. At
$t=0$, this distribution is very close to a Gaussian, centered on the
mean energy density in the system. The width of this Gaussian is
entirely determined by the Gaussian spectrum of fluctuations in
eq.~(\ref{eq:gaussian}). At early times, the distribution first
remains Gaussian-like, but tends to broaden with time. Around
$t\approx 30$ in lattice units, we observe a rapid change of shape of
this distribution--the peak of the distribution shifts to lower values
of the energy and the tail extends much further at large energy. Once
this dramatic change of shape has taken place, the evolution of the
distribution is rather slow and a stationary distribution is reached
at late times.
The evolution in the energy distribution can be explored further by
looking at its moments defined by
\begin{equation}
C_n\equiv \frac{\big<E^n\big>}{\big<E\big>^n}\; .
\end{equation}
Higher moments are very sensitive to changes in the shape of the
distribution, especially the appearance of an extended tail that
signals broader energy fluctuations.
\begin{figure}[htbp]
\begin{center}
\resizebox*{8cm}{!}{\rotatebox{-90}{\includegraphics{moments_g0.5_basic.ps}}}
\end{center}
\caption{\label{fig:Emoments}Normalized moments
$\big<E^n\big>/\big<E\big>^n$ of the energy density distribution at
one lattice site, as a function of time. The coupling constant is
$g=0.5$.}
\end{figure}
We represent these moments as a function of time in
fig.~\ref{fig:Emoments}, up to $n=6$. They all start very close to 1
at $t=0$, which is the sign of a very narrow distribution with little
fluctuations. The rapid change of shape of the distribution around
$t\approx 30$ corresponds to a rapid increase of the moments. By
$t\approx 70$, the moments have reached nearly asymptotic values
modulo moderate residual oscillations.
It is interesting to compare the evolution of the energy distribution
at a single lattice site with the time evolution of the pressure in
fig.~\ref{fig:pressure1}. The initial rapid decrease of the pressure
oscillations is concomitant with the change of shape of the energy
distribution. The subsequent (slower) relaxation of the residual
oscillations of the pressure occurs after the energy has reached a
stationary distribution.
\section{Summary and Outlook}
We discussed in this paper a formalism which resums secular terms in a
weak coupling expansion of a scalar field theory with initial
conditions generated by strong sources. We showed that resummed
expressions, to all orders in perturbation theory, for inclusive
quantities could be expressed as an ensemble average of the
corresponding leading order classical quantities where the initial
classical field for each member of the ensemble is shifted by a
quantum fluctuation drawn from a Gaussian distribution. We showed that
this averaging caused the resummed pressure to relax to a single
valued relation with the energy density and interpreted this as
arising from the phase decoherence of individual classical
trajectories. We showed in a toy model that for an expanding system
our result leads to ideal hydrodynamical flow. We briefly addressed the
issue of thermalization--while our numerical results display features
similar to those of a canonical thermal ensemble, they differ slightly
in the particulars. A more systematic numerical study will likely be
able to shed further light on this important point. As noted in the
introduction, our system appears to satisfy Berry's conjecture which
has been argued to be an important requirement for the thermalization
of quantum systems. We plan to pursue this topic further in the future.
Finally, we note that to be fully relevant to heavy ion collisions,
our methods should be extended to gauge theories. We have shown
previously that the formalism outlined in section 2 here is also
applicable to a gauge theory~\cite{GelisLV2}. The spectrum of quantum
fluctuations~\cite{FukusGM1} is the essential ingredient here and work
on computing this quantity is well underway~\cite{DusliGSV1}.
\section*{Acknowledgements}
We would like to acknowledge useful discussions with Jean-Paul
Blaizot, Kenji Fukushima, Miklos Gyulassy, Tuomas Lappi, Larry
McLerran, Rob Pisarski, Andreas Schafer and Giorgio Torrieri. K.D's
and R.V's research was supported by DOE Contract No.
DE-AC02-98CH10886. F.G's work is supported in part by Agence Nationale
de la Recherche via the programme ANR-06-BLAN-0285-01. We thank the
Institute for Nuclear Theory at the University of Washington for its
hospitality. One of us (FG) would like to thank Brookhaven National
Laboratory as well as the Yukawa International Program for
Quark-Hadron Sciences at Yukawa Institute for Theoretical Physics
(Kyoto University) for partial support during the completion of this
work.
|
{
"timestamp": "2010-09-23T02:02:07",
"yymm": "1009",
"arxiv_id": "1009.4363",
"language": "en",
"url": "https://arxiv.org/abs/1009.4363"
}
|
\section{Introduction}
Historically, the formula of ADM mass in the case with metric form
$$ds^2=A(U)\eta_{\mu\nu}dx^\mu dx^\nu+B(U) \delta_{mn}dy^a dy^b, \eqno{(1.1)}$$
in which $U^2\equiv\delta_{mn}dy^a dy^b$ had been first derived [1,2]. Next, Lu [3] had derived a general formula of ADM mass in the case with metric form
$$ds^2=-A(U)dt^2+B(U)dU^2+C(U)U^2d\Omega_d^2 + D(U)\delta_{ij}dx^i dx^j, \eqno{(1.2)}$$
which has been wildly used in many literatures since then [4-7].
In this paper we want to consider the spacetime associated to the black D-brane under the Melvin magnetic field [8], which is [9]
$$ds_{10}^2 =\sqrt{1+ B^2 r^2} \left[H^{-1\over2}\left(-f(U)~dt^2+dz^2+dw^2+dr^2+ {r^2d\phi^2\over 1+B^2r^2}\right)+H^{1\over2} \left(f(U)~dU^2+U^2d\Omega_4^2\right) \right],\eqno{(1.3)}$$
However, as our metric does not fall in above class we have to derive a slightly general formula to calculate the ADM mass.
In section II we derive a more general formula which enable us to calculate the ADM mass in our cases. In section II we use this formula to evaluate the thermodynamical quantities of the black D-branes with magnetic field, which is dual to the finite temperature gauge theory under the magnetic field. We have found the Hawking-Page transition for sufficiently large magnetic field. The last section is devoted to a short conclusion.
\section{ADM Mass in More General Geometry}
Consider a general black p-brane with metric $g_{MN}= g^{(0)}_{MN}+h_{MN}$ in which $g^{(0)}$ is the D dimensional flat limit of the corresponding space-time metric. $h_{MN}$ is asymptotically zero but not necessarily small everywhere [1-3]. To first order in $h_{MN}$ the Einstein equation looks like
$$ R^{(1)}_{MN} -{1\over2} g^{(0)}_{MN} R^{(1)} = \kappa^2 \Theta_{MN}. \eqno{(2.1)}$$
The ADM mass per unit volume is defined as
$$M= \int d^{D-d-1}y ~\Theta_{00}.\eqno{(2.2)}$$
The general $R^{(1)}_{MN}$ has been given in [1,2] as
$$R^{(1)}_{MN}= {1\over2}\left({\partial^2h^P_{~M}\over \partial x^P\partial x^N}+{\partial^2h^P_{~N}\over \partial x^P\partial x^M}-{\partial^2h^P_{~P}\over \partial x^M\partial x^N}-{\partial^2h_{MN}\over \partial x^P\partial x^P}\right),\eqno{(2.3)}$$
where the indices are raised and lowered using the flat Minkowski metric. Using (2.1) and (2.3) we find that
$$ \kappa^2 \Theta_{00}= - {1\over2}{\partial^2h^0_{~0}\over \partial x^Q\partial x_Q}+ {1\over2}{\partial^2h\over \partial x^Q\partial x_Q}+ {1\over2}{\partial^2h^M_{~N}\over \partial x^M\partial x_N},~~~~h\equiv \sum_P h^P_{~P} \eqno{(2.4)}$$
We will consider the more general metric which has a following block form
$$ds^2= -A(U,r,..)dt^2+\left[B(U,r,..)dU^2+C(U,r,..)U^2d\Omega_{d-1}^2 \right]\hspace{6cm}$$
$$ \left[E(U,r,..)dr^2+F(U,r,..)r^2d\Omega_{D-1}^2 \right] + G(U,r,..)\sum_i ^{d_x}dx_i^2 +\cdot\cdot\cdot, \eqno{(2.5)}$$
In below we present a systematic procedure to find the ADM mass.
$\bullet$ {\bf Step 1 :} The first property we can see is that the first term in (2.4) will be canceled by the $h^0_{~0}$ term in second term, as $h= h^0_{~0}+\cdot\cdot\cdot$. Thus we conclude that $ \Theta_{00}$ does not depend on $A$.
$\bullet$ {\bf Step 2 :} To see how the $B$ and $C$ will appear in $ \Theta_{00}$ we first rewrite a part of line element in the coordinate as follow
$$BdU^2+CU^2d\Omega_{d-1}^2=(B-C)dU^2+C(dU^2+U^2d\Omega_{d-1}^2)\hspace{3.5cm}$$
$$={B-C\over U^2}\sum_{i=1}^d U_i U_j dU_i dU_j+C\sum_{i=1}^d dU_i^2, \eqno{(2.6)}$$
in which $U^2\equiv\sum_{i=1}^d U_i^2$. This implies following two results :
$$\sum_{i=1}^d {\partial^2h^i_{~i}\over \partial U^i\partial U_i} = \sum_{i=1}^d {\partial^2\over \partial U^i\partial U_i}\left({B-C\over U^2} U_i^2\right) + \sum_{i=1}^d {\partial^2 C \over \partial U^i\partial U_i}\hspace{4cm}$$
$$ = \sum_{i=1}^d \left[{\partial^2 \left({B-C\over U^2}\right)\over \partial U^i\partial U_i} U_i^2 + 4 {\partial \left({B-C\over U^2}\right)\over \partial U^i} U_i \right] + 2d{B-C\over U^2} + {\vec\nabla}_U^2 C. \eqno{(2.7)}$$
$$\sum_{i\ne j}^d {\partial^2h^i_{~j}\over \partial U^i\partial U_j} = \sum_{i\ne j}^d {\partial^2\over \partial U^i\partial U_j}\left({B-C\over U^2} U_i U_j\right) \hspace{6cm}$$
$$ = \sum_{i\ne j}^d \left[{\partial^2 \left({B-C\over U^2}\right)\over \partial U^i\partial U_j} U_i U_j \right]+ 2(d-1) \sum_{i}^d {\partial \left({B-C\over U^2}\right)\over \partial U^i} U_i + d(d-1) {B-C\over U^2}.\eqno{(2.8)}$$
Now, using the property
$$\sum_{i}^d {\partial f\over \partial U^i} U_i = \vec U_i \cdot \vec \nabla = U{\partial f\over \partial U},\eqno{(2.9)}$$
if f=f(U), then (2.7) and (2.8) implies following simple result
$${\partial^2h^M_{~N}\over \partial x^M\partial x_N} = U^2{\partial^2 \left({B-C\over U^2}\right)\over \partial U^2}+ 2(d+1) U {\partial \left({B-C\over U^2}\right)\over \partial U} + d(d+1) {B-C\over U^2}+ {\vec\nabla}_U^2 C, \eqno{(2.10)}$$
which is a part of third term in (2.4).
$\bullet$ {\bf Step 3 :} From (2.6) we see that
$$ h = (B-C) + d~C + \cdot\cdot\cdot. \eqno{(2.11)}$$
Thus Eq.(2.4) tell us that $B$ and $C$ will contribute following quantity to $ \Theta_{00}$
$${\partial^2h\over \partial x^Q\partial x_Q} = {\vec\nabla}^2(B-C) +d~ {\vec\nabla}^2 C\cdot\cdot\cdot ={\vec\nabla}_U^2(B-C)+({\vec\nabla}')_U^2(B-C)+ d~ {\vec\nabla}_U^2 C+d~({\vec\nabla}')_U^2 C+\cdot\cdot\cdot, \eqno{(2.12)}$$
in which ${\vec\nabla}_U^2$ is the Laplacian on the coordinate $U_i$ while $({\vec\nabla}')_U^2$ is that on the coordinates except $U_i$.
$\bullet$ {\bf Step 4 :} Using the formula
$${\vec\nabla}_U^2 f(U) = {1\over U^{d-1}}\partial_U\left(U^{d-1}\partial_Uf(U)\right)=\partial_U^2 f(U) +{d-1\over U}\partial_U f(U), \eqno{(2.13)}$$
we can substitute (2.10) and (2.12) into (2.4) to find that $B$ and $C$ will contribute following into $\kappa^2 \Theta_{00}$
$$ \kappa^2 \Theta_{00}= {1\over2}{d-1\over U^{d-1}} \partial_U\left(U^{d-1}C\right) + {1\over2} ({\vec\nabla}')_U^2 C+{1\over2}{d-1\over U^{d-1}} \partial_U\left(U^{d-2}(B-C)\right) + {1\over2} ({\vec\nabla}')_U^2(B-C)+\cdot\cdot\cdot.\eqno{(2.14)}$$
That coming from $E$ and $F$ has a similar formula after replacing with $U\rightarrow r$ and $d\rightarrow D$.
$\bullet$ {\bf Step 5 :} A simple observation form (2.4) could see that $G$ will contribute following into $\kappa^2 \Theta_{00}$
$$ \kappa^2 \Theta_{00}= {1\over2} d_x~({\vec\nabla}')_x^2 G+\cdot\cdot\cdot.\eqno{(2.15)}$$
Finally, using (2.14) and (2.15) we can find the complete value of $\kappa^2 \Theta_{00}$. After substituting it into (2.2) we then obtain the ADM mass.
\section{ADM Mass of Magnetic Black D-brane}
The non-extremal black D-brane we considered is described by the geometry (1.3). We express the corresponding metric in the Einstein frame as following
$$ds_{10}^2 =(1+ B^2 r^2)^{1/8} \left[H^{-3\over8}\left(-f(U)~dt^2+dz^2+dw^2+dr^2+ {r^2d\phi^2\over 1+B^2r^2}\right)+H^{5\over8} \left(f(U)~dU^2+U^2d\Omega_4^2\right) \right],\eqno{(3.1)}$$
$$f(U) = 1-{U_0^3\over U^3},~~~~~~~H = 1+ {U_0^3 \sinh^2\gamma \over U^3},\eqno{(3.2)}$$
After the calculation the ADM mass is
$$M= {1\over 2\kappa^2}\Omega_4 L_zL_w \pi R^2 U_0^3 \left[(3\sinh^2\gamma+4)\left({8\over 9}{(1+B^2R^2)^{9/8}-1\over B^2R^2}\right)\right].\eqno{(3.3)}$$
Here we assume that the coordinate $z$ and $w$ is compactified on the circles of circumference $L_z$ and $L_w$ respectively. Brane also is wrapped on the radius with $0\le r \le R$. Notice that terms which do not depend on the $U_0$ are infinite and have been dropped out from $M$, as they are that of the background and shall not be regarded as parts of the black D-brane mass.
The temperature and entropy could be easily calculated and results are
$$ T = {3\over 4\pi U_0\cosh\gamma },\hspace{2cm}\eqno{(3.4)}$$
$$ S ={4\pi\over 2\kappa^2}\Omega_4 L_zL_w \pi R^2 U_0^4 \cosh\gamma,\eqno{(3.5)}$$
The energy denotes that above extremality is
$$E ={1\over 2\kappa^2}\Omega_4 L_z L_w \pi R^2 U_0^3 \left[(3\sinh^2\gamma+4)\left({8\over 9}{(1+B^2R^2)^{9/8}-1\over B^2R^2}\right)-3\sinh\gamma\cosh\gamma\right],\eqno{(3.6)}$$
\section{Hawking-Page Phase Transition in Magnetic Black D-brane}
To describe the dual gauge theory form above black D-brane property we have to consider the near-extremal configuration of the black D-brane. This could be found by the following limits [7].
First, we define $ h^3 \equiv U_0^3 \cosh\gamma\sinh\gamma$. Next, we consider the following rescaling
$$U\rightarrow{U_{old}\over \ell^2},~~~~U_0\rightarrow{(U_0)_{old}\over \ell^2},~~~~h^3\rightarrow{h^3_{old}\over \ell^2},\eqno{(4.1)}$$
and taking $\ell \rightarrow 0$ while keeping the old quantities fixed [7].
In this limit we find that
$$ T = {3\over 4\pi U_0}\left({U_0^3\over h}\right)^{3/2},\hspace{7cm}\eqno{(4.2)}$$
$$ S ={4\pi\over 2\kappa^2}\Omega_4 L_zL_w \pi R^2 U_0^4\left({U_0^3\over h}\right)^{-3/2},\hspace{4.7cm}\eqno{(4.3)}$$
$$E ={1\over 2\kappa^2}\Omega_4 L_zL_w \pi R^2 U_0^3 \left[{5\over2}+{3h^3\over U_0^3}\left({8\over 9}{(1+B^2R^2)^{9/8}-1\over B^2R^2}-1\right)\right],\eqno{(4.4)}$$
The free energy $F=E-TS$ becomes
$$F ={1\over 2\kappa^2}\Omega_4 L_zL_w \pi R^2 U_0^3 \left[-{1\over2}+{3h^3\over U_0^3}\left({8\over 9}{(1+B^2R^2)^{9/8}-1\over B^2R^2}-1\right)\right].\eqno{(4.5)}$$
which becomes that in [4] when $B\rightarrow 0$ and free energy is negative. However, for a large $B$ the free energy becomes positive.
This means that, as noted first by Hawking and Page [10], a first order phase transition occurs at some critical temperature, above which an AdS black hole forms. On the other hand, at a lower temperature, the thermal gas in AdS dominates. On dual gauge theory side [11], Witten related the Hawking-Page phase
transition of black holes in AdS space with the confinement-deconfinement phase transition of field theory [12]. Thus we have seen that the magnetic field could produce the Hawking-Page transition and the corresponding dual gauge theory will show the confinement-deconfinement phase transition under large magnetic flux.
\section{Conclusion}
In this paper we have derived a formula which enable us to evaluated the ADM mass in more general cases. We use this formula to evaluate the thermodynamical quantities of the black D-branes with Melvin magnetic field, which is dual to the finite temperature gauge theory under the magnetic field. We have found the Hawking-Page transition for sufficiently large magnetic field. This means that the corresponding dual gauge theory will show the confinement-deconfinement phase transition under large magnetic flux. It is hoped that the formula derived in this paper could help us to evaluate the ADM mass to study the thermodynamical quantities of general black brane systems.
\\
\\
{\bf Acknowledgments} :We are supported in part by the Taiwan National Science Council.
\newpage
\begin{center} {\bf \Large References}\end{center}
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\item Rong-Gen Cai. and Nobuyoshi Ohta, ``On the Thermodynamics of Large N Noncommutative Super Yang-Mills Theory", Phys.Rev. D61 (2000) 124012 [hep-th/9910092]; T. Harmark, N. A. Obers, ``Phase Structure of Non-Commutative Field Theories and Spinning Brane Bound States," JHEP 0003 (2000) 024 [hep-th/9911169 ].
\item Steven S. Gubser,``Thermodynamics of spinning D3-branes," Nucl.Phys. B551 (1999) 667 [hep-th/9810225] ; Rong-Gen Cai, Kwang-Sup Soh, ``Critical Behavior in the Rotating D-branes", Mod.Phys.Lett. A14 (1999) 1895 [hep-th/9812121].
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F.~Dowker, J.~P.~Gauntlett, D.~A.~Kastor and J.~Traschen, ``The decay of magnetic fields in Kaluza-Klein theory,'' Phys.\ Rev.\ D52 (1995) 6929 [hep-th/9507143]; M.~S.~Costa and M.~Gutperle, ``The Kaluza-Klein Melvin solution in M-theory,'' JHEP 0103 (2001) 027 [hep-th/0012072].
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[hep-th/9802109].
\end{enumerate}
\end{document}
|
{
"timestamp": "2010-09-23T02:01:46",
"yymm": "1009",
"arxiv_id": "1009.4332",
"language": "en",
"url": "https://arxiv.org/abs/1009.4332"
}
|
\section{Introducing FN\,CMa}
FN\,CMa (HD\,53\,974) is a bright ($V=5.4$\,mag) B0.5\,III star and
visually double. Within about a century, the relative position of
components A and B, which are separated by $\sim$0.6 arcsec, has
changed marginally at most. A is brighter than B by about 1.2 mag.
\section{Observations and data reduction}
The ESO Science Archive contains 60 VLT/{\it UVES} echelle spectra of FN\,CMa
obtained within 1.4 hours for a study of interstellar medium, and three more
spectra from {\it FEROS} at the 2.2-m ESO/MPG telescope, La Silla. In 2009
and 2010, an additional 59 echelle spectra were secured with the {\it BESO}
spectrograph, a clone of {\it FEROS} mounted on the Bochum 1.5-m Hexapod
Telescope on Cerro Armazones.
As a result of variable seeing and imperfect guiding, some UVES spectra
contain a significantly higher fraction of light from component B than
others. Since the light combination is geometric it has no spectral
dependency, and thus, under the assumption that certain spectral features are
due to either A (e.g., Si\,{\sc iii} 4553) or B (e.g., He\,{\sc ii} 4540)
alone, a simple linear set of equations can be used for the disentangling of
the spectra from the two stars over the entire wavelength range. The
inferred spectral light ratios, between 0.75 and 0.85, are in good agreement
with the known magnitude difference.
\section{Results}
{\bf FN\,CMa B:} This component has a spectrum typical of mid-O main-sequence
stars. Compared to the B0.5III primary, it would be considerably
underluminous if the pair were physical. However, assuming an O subdwarf
companion does not help because, then, component B would be about 2 mag {\it
over}\/luminous.
Components A and B display the same set of interstellar Ca\,{sc ii} K lines
except that the redmost one is significantly stronger in B. Considering also
incipient emission in N\,{\sc iii} and H$\alpha$, we conclude that component B
is best described as an O6V((f)) background star.
\begin{figure}
\centering
\parbox{\textwidth}{%
\parbox{.52\textwidth}{\includegraphics[angle=270,width=.52\textwidth,clip]{s5-25_rivinius_fig1a.pdf}}%
\parbox{.48\textwidth}{\includegraphics[viewport=15 18 694 500,angle=0,width=.48\textwidth,clip]{s5-25_rivinius_fig1b.pdf}}%
}
\caption[]{
\centering
Left panel: {\it HIPPARCOS} photometry of FN\,CMa phased with a period
of 2.13\,h. \newline Right panel: The radial velocity curve of FN\,CMa\,Aa
(P\,=\,117.55\,d).}
\label{FNCMa}
\end{figure}
{\bf FN\,CMa A:} In the literature, FN\,CMa has a record of low-amplitude
photometric variability and modulated spectral line profiles. But there is no
consensus about its nature. Our analysis of the {\it HIPPARCOS} photometry
yields a period of 0.08866\,d (2.13\,h; see left panel of Fig.\ \ref{FNCMa})
with $\sim$0.02\,mag amplitude. The combination of spectral type, period, and
amplitude makes FN\,CMa a $\beta$ Cephei star candidate, as already suggested
by other observers. This is further supported by the rapid spectral
line-profile variability of component A (however, at just 1.4\,h, the {\it
UVES} data string is too short and the {\it BESO} spectra are not
sufficiently densely sampled to attempt an independent period determination).
In any case, given the spectral variations seen in component A, we attribute
the photometric variability to this component as well.
Much larger-amplitude long-term radial-velocity variability is apparent from
the {\it BESO} data: FN\,CMa\,A is itself an SB1 binary with the following
properties:
\vspace*{5mm}
\noindent{\footnotesize
\begin{center}
\begin{tabular}{lr}
Period [d] & 117.55 $\pm$ 0.33 \\
Periastron epoch [JD] & 2\,453\,779.5 $\pm$ 4 \\
Periastron longitude [deg] & 247 $\pm$ 7 \\
$e$ & 0.60 $\pm$ 0.05 \\
$K_1$ [km/s] & 49.8 $\pm$ 3.5 \\
$\gamma$ [km/s] & 5.9 $\pm$ 1.5 \\
\end{tabular}
\end{center}}
\vspace*{5mm}
\noindent
The radial-velocity curve of FN\,CMa\,Aa is shown in Fig.\ \ref{FNCMa} (right
panel). Its relatively large amplitude suggests that the so-far (directly)
undetected component FN\,CM\,Ab is a fairly massive star. However, it appears
too faint to be the carrier of the rapid variability.
\section{Discussion}
The high eccentricity and moderate orbital period of the subsystem
FN\,CMa\,Aa$+$Ab may enable searches for a tidal modulation of the
pulsation of component Aa. Since FN\,CMa is bright and situated in a
region with numerous other pulsating OB stars, it might be worthwhile
including it in the target lists of wide-angle asteroseismology
satellites such as BRITE.
\end{document}
|
{
"timestamp": "2010-09-21T02:00:22",
"yymm": "1009",
"arxiv_id": "1009.3512",
"language": "en",
"url": "https://arxiv.org/abs/1009.3512"
}
|
\section{Introduction}
When in a multi-phase system initially in a mixed state the
temperature is decreased
to values corresponding to
a coexisting region of the phase diagram,
domains of ordered
phases start to form and grow with time. The process is called phase
separation and is relevant for a large variety of
systems \cite{GUNTON}. In most of the cases studied theoretically,
the temperature or other
control parameters are assumed not depending on time and
space, but are instantaneously set to their final values for coexistence.
This assumption, reasonable in many situations, typically gives
rise to a self-similar growth behavior with a characteristic domain
size following a time power-law \cite{BRAY}. However, there are cases where
the dynamics of
the control parameter needs to be considered \cite{SAARLOS}
since it can
greatly affect the morphology of domains.
In binary alloys, for example, slow cooling is used to
produce optimal sequences of alternate bands of different materials
\cite{METAL}.
In
polymeric mixtures the possibility of controlling the demixing
morphology
by appropriate thermal driving has been studied in
Refs.~\cite{KREKHOV,YAMAMURA};
modulated patterns have been observed when a mixture is periodically
brought
above and below the critical point
\cite{TANAKA}. Other worth examples of
complex pattern formation due to the dynamics of the
control parameters occur
in crystal growth \cite{LANGER}, immersion-precipitation membranes
\cite{CHENG}, or in electrolyte diffusion in gels \cite{LIESEGANG,ANTAL}.
In this paper we study binary fluids
quenched by contact with cold walls
at temperatures below the critical value.
The behavior of binary fluids
in sudden quenches at homogeneous temperature is quite known
\cite{BRAY,YEOMANS}. For symmetric composition, the typical
interconnected pattern of spinodal decomposition is observed. In the
system here considered, phase separation is expected to start close
to the walls and develop in the inner of the system following the
temperature evolution. The dynamics of this process and the role of
the velocity field have not been explored too much, in spite of
their relevance for many of the systems mentioned above.
Two-dimensional studies of diffusive binary systems with cold
sharp fronts propagating at constant speed have shown the formation
of structures aligned on a direction depending on the speed
\cite{FURUKAWA,ANTAL,HANTZ,WAGNERFOARD,KREK2}. These results
are also supported by theoretical analysis
\cite{WAGNERFOARD,KREK2}. Lamellar-like structures have
been also found in numerical studies of two-dimensional
off-symmetrical binary systems with the temperature following a
fixed diffusive law \cite{BALL}. In a model with the temperature
dynamically coupled to the concentration field, point-like cold
sources have been shown to give rise to ring structures of alternate
phases \cite{ESPANOL}. On the other hand, more usual morphologies
have been found in cases with fixed thermal gradient \cite{JASNOW},
while complex phenomena such as sequential phase-separation cascades
have been observed when the control parameter is slowly
homogeneously changed \cite{VOLLMER}.
The effects of full coupling
between all thermo-hydrodynamic variables have been not considered
sofar.
The paper is organized as follows. In the next section the theoretical model
and the numerical method
are illustrated. The dynamics of our system is described by
mass, momentum, and energy equations with thermodynamics based
on a free-energy functional including gradient terms.
In Section III the results of our simulations
are shown.
We will explore the control parameter space by varying the viscosity
and the
thermal diffusivity. This will allow to analyze the differences with respect
to the behavior of binary fluids in instantaneous quenching.
The presentation will be focused on few cases typical for each regime.
A final discussion will follow in Section IV.
\section{The model}
We consider a binary mixture
with dynamical variables $T,${\bf v}$, n, \varphi$ which
are, respectively, the temperature, the velocity, the total density,
and the order parameter field being the concentration difference.
Equilibrium
properties are encoded in the free-energy
\begin{equation}
F=\int
(\psi(n,\varphi,T)+\frac{1}{2}M|{\bf \nabla}
\varphi|^{2})d{\bf r}
\end{equation}
where
\begin{equation}
\psi(n,\varphi,T)= e -
k_{B}T[n\ln(n)-\frac{n+\varphi}{2}\ln(\frac{n+\varphi}{2})-
\frac{n-\varphi}{2}\ln(\frac{n-\varphi}{2})]
\end{equation}
with $ e = n k_B T +
\frac{\lambda n}{4}(1-\frac{\varphi^{2}}{n^{2}})$
being the bulk internal energy
and the term
in square brackets the mixing entropy.
The gradient term in Eq.~(1) is a combination
of an internal energy gradient contribution proportional
to $K$ and of an entropic term proportional to
$-C$ \cite{ONUKI}, hence $M=K+CT$.
The system has a critical transition at $k_B T_c =
\lambda/2$ and the order parameter in the separated phases takes
the values $\varphi_{\pm}(T) = \pm \sqrt{3n^2(T_c/T -1)}$.
The dynamical equations are given by \cite{DGM}
\begin{equation}
\partial_t n=-\partial_{\alpha} (n v_{\alpha}), \label{mass}
\end{equation}
\vskip -0.4cm
\begin{equation}
\partial_t \varphi=-\partial_{\alpha} (\varphi
v_{\alpha})-2\partial_{\alpha} J_{\alpha}^d, \label{massdiff}
\end{equation}
\vskip -0.4cm
\begin{equation}
\partial_t (n v_{\beta}) = - \partial_{\alpha} (n v_{\alpha} v_{\beta})
-\partial_{\alpha}(\Pi_{\alpha \beta}-\sigma_{\alpha \beta}),
\label{momentum}
\end{equation}
\vskip -0.4cm
\begin{equation}
\partial_t \widehat{e}=-\partial_{\alpha}(\widehat{e}
v_{\alpha})-(\Pi_{\alpha \beta}-\sigma_{\alpha \beta})\partial_{\alpha}
v_{\beta}-\partial_{\alpha}J^{q}_{\alpha} \label{ener},
\end{equation}
where ${\bf J}^{d}$ and ${\bf J}^{q}$ are the
diffusion and heat currents, $\Pi_{\alpha \beta}$ is the reversible stress
tensor, $\sigma_{\alpha \beta}=\eta(\partial_{\alpha} v_{\beta} +
\partial_{\beta} v_{\alpha}) +(\zeta-2 \eta/d)\delta_{\alpha \beta}
\partial_{\gamma} v_{\gamma}$ is the
dissipative stress tensor with $\zeta, \eta$ being the bulk and
shear viscosities, respectively, $d$ the space dimension, and
$\widehat{{e}}=e + \frac{K}{2}|{\bf \nabla}\varphi|^{2} $ the
total internal energy density also including gradient
contributions. We have recently established the expressions for the
pressure tensor $\Pi_{\alpha \beta}$ and chemical potential $\mu$
\cite{GONN} following the approach of Ref.~\cite{ONUKI}. One finds
\begin{equation}
\Pi_{\alpha \beta}=\left(p - M\varphi
\nabla^{2}\varphi - M{|{\bf \nabla}\varphi|^{2}}/2 -
T\varphi{\bf \nabla}\varphi\cdot{\bf \nabla}
({M}/{T})\right)\delta_{\alpha \beta} +
M\partial_{\alpha}\varphi\partial_{\beta}\varphi
\end{equation}
where $p = - \psi +
n\partial \psi/\partial n + \varphi
\partial \psi/\partial \varphi$ and
$\mu=\partial \psi/\partial \varphi|_T - T {\bf \nabla} \cdot
[(M/T){\bf \nabla}\varphi]$.
Finally, in order to
completely set up the dynamical system,
phenomenological expressions for the currents are needed.
As usually, one takes
${\bf J}^{d}=-\mathcal{L}_{11}
{\bf \nabla}({\mu}/{T})+\mathcal{L}_{12}{\bf \nabla}
({1}/{T})$,
${\bf J}^{q}=-\mathcal{L}_{21}{\bf \nabla}({\mu}/{T})
+\mathcal{L}_{22} {\bf \nabla}({1}/{T}) $ where
$\mathcal{L}_{\alpha \beta}$ is the positively defined matrix of kinetic
coefficients with $\mathcal{L}_{11} = T \Gamma$ and $\mathcal{L}_{22}
= T^2 k$, $\Gamma$ and $k$ being the mobility and thermal diffusivity,
respectively, assumed constant \cite{DGM}.
In order to solve
Eqs.~(\ref{mass}-\ref{ener}) in $d=2$ we have developed a hybrid
lattice Boltzmann method (LBM) \cite{lall,xu,maren,STELLA}
where LBM \cite{LBM} is
used to simulate the continuity and Navier-Stokes equations
(\ref{mass}) and (\ref{momentum}) while finite-difference methods are
implemented to solve the convection-diffusion
and the energy equations (\ref{massdiff}) and (\ref{ener}). LBM
has been widely
used to study multi-phase/component fluids \cite{DUN} and, in particular,
hydrodynamic effects in phase ordering \cite{CATES}. It is defined
in terms of a set of distribution functions, $f_i({\bf r},t)$ with
$i=0,1,...,8$, located in each site ${\bf r}$ at each time $t$ of a
D2Q9 (2 space dimensions and 9 lattice velocities)
lattice where sites are connected to first and second neighbors
by lattice velocity vectors of modulus $|{\bf e}_i|=c$
($i=1,...,4$) and $|{\bf e}_i|=\sqrt{2}c$ ($i=5,...,8$),
respectively. The zero velocity vector ${\bf e}_0=0$ is also
included. The lattice speed is $c=\Delta x/\Delta t$ where $\Delta
x$ and $\Delta t$ are the lattice and time steps, respectively. The
distribution functions evolve according to a single relaxation time
Boltzmann equation \cite{bgk} supplemented by a forcing term
\cite{guo}
\begin{equation}\label{evoleqn}
f_i({\bf r}+{\bf e}_i\Delta t,t+\Delta t)-f_i({\bf r},t)=-\frac{\Delta t}
{\tau}[f_i({\bf r},t)-f_i^{eq}({\bf r},t)]+\Delta t F_i({\bf r},t),
\end{equation}
where $\tau$ is the relaxation parameter, $f_i^{eq}$ are the
equilibrium distribution functions, and $F_i$ are the forcing terms
to be properly determined.
The total density and the fluid momentum are given by the following
relations
\begin{equation}\label{moment}
n=\sum_if_i , \hspace{1.3cm} n{\bf v}=\sum_if_i{\bf e}_i
+ \frac{1}{2}{\bf F}\Delta t,
\end{equation}
where ${\bf F}$ is the force density acting on the fluid. The
$f_i^{eq}$ are expressed as a standard second order expansion in
the fluid velocity ${\bf v}$ of the Maxwell-Boltzmann distribution
functions \cite{qian}.
The forcing terms $F_i$ in Eq.~(\ref{evoleqn})
are expressed as a second order expansion
in the lattice velocity vectors \cite{LADD}.
The continuity and the Navier-Stokes equations
(\ref{mass}) and (\ref{momentum}) can be recovered by using a
Chapman-Enskog expansion
when the $F_i$ are given by
\begin{equation}\label{latticeforceterm}
F_i=\left(1-\frac{\Delta t}{2\tau}\right)\omega_i\left[
\frac{{\bf e}_i-{\bf v}}{c^2_s}+\frac{{\bf e}_i\cdot{\bf v}}{c^4_s}
{\bf e}_i\right]\cdot {\bf F}
\end{equation}
with the force density ${\bf F}$ having components
\begin{equation}
F_{\alpha}=\partial_{\alpha}(nc_s^2)-\partial_{\beta}\Pi_{\alpha
\beta} ,
\end{equation}
$c_s=c/\sqrt{3}$ being the speed of sound in the LBM,
$\omega_0=4/9$, $\omega_i=1/9$ for $i=1,...,4$, and $\omega_i=1/36$
for $i=5,...,8$.
We observe that in this formulation the pressure tensor is inserted
as a body force in the lattice Boltzmann equations. From the
Chapman-Enskog expansion it comes out that $\xi=\eta$ with
\begin{equation}
\eta = n c_s^2 \Delta t\left(\frac{\tau}{\Delta t}
- \frac{1}{2}\right).
\end{equation}
On the other hand, a two-step finite difference scheme is used for
the equations (\ref{massdiff}) and (\ref{ener}) (details on the
implementation of Eq.~(\ref{massdiff}) in the case of an isothermal
LBM can be found in Ref.~\cite{STELLA}). At walls, no-slip boundary
conditions are adopted for the LBM \cite{PHYSA}, the temperature is
set to fixed values
$T_b$ at the bottom wall and $T_u$ at the up wall, respectively,
and neutral wetting for the concentration is adopted.
This latter condition corresponds to impose
${\bf a} \cdot \nabla \varphi|_{walls}= 0$
and ${\bf a} \cdot \nabla (\nabla^2 \varphi)|_{walls}=0$, where ${\bf a}$ is an
inward normal unit vector to the walls. These conditions together ensure
${\bf a} \cdot \nabla \mu|_{walls}= 0$ so that the concentration
gradient is parallel to the walls and there is
no flux across the walls.
We have found this algorithm stable in a wide range of
temperatures, viscosities and thermal diffusivities. With
respect to thermal LBM for non-ideal fluids \cite{SOFO}
where lattice Boltzmann equations are used
to simulate the full set of macroscopic dynamical equations, the
present model allows to reduce the number of lattice velocities thus
speeding up the code and reducing the required memory \cite{STELLA}.
\section{Results and discussion}
In the following we will explore the parameter space keeping fixed the values
of $K=0.003, C=0, k_B T_c = 0.005,
\Gamma=0.1$, and $\mathcal{L}_{12}=\mathcal{L}_{21}=0$. We will use lattices
of size ranging from $256 \times 256$ to $1024 \times 1024$.
We have considered different values of $\eta$ and $k$.
Before focusing on the cases representative of the various regimes,
we will list all the runs we did in terms of dimensionless numbers.
Common numbers used in hydrodynamics are the Reynolds
and Peclet numbers $Re$ and $Pe$.
They are defined as $Re=v L/\nu$, where $\nu=\eta/n$ is the kinematic
viscosity, $Pe_{md}=v L/D$ for mass diffusion, where $D$ is the mass
diffusion coefficient, and $Pe_{td}=v L/k$
for thermal diffusion.
$L$ and $v$ are a typical length and velocity of the system.
In phase separation $L$ can be identified with the average size
of domains so that $Re$ and $Pe$ would depend on time (for a discussion see
Ref.~\cite{kendon}). It is therefore more convenient for our purposes to
introduce the Schmidt and Prandtl numbers $Sc$ and $Pr$
defined as $Sc=\nu/D$ and $Pr=\nu/k$, where
$D= |a| \Gamma$
with $a=(k_B T_c/n) (T/T_c-1)$ being the coefficient of the linear term
in the chemical potential $\mu$ \cite{KREKHOV,GONN}. Here
$T$ can be chosen as the
value of the temperature at the walls.
Table I contains a list of the runs we did,
reported in terms of $Sc$ and $Pr$.
It is also useful to evaluate the Mach number
$Ma=|{\bf v}|_{max}/c_s$ where
$|{\bf v}|_{max}$ is the maximum value of the fluid velocity
during evolution.
In all our simulations $Ma$ is always much smaller than $0.1$
(see in the following), and
the fluid results practically incompressible, as checked,
with $n \simeq 1$. For this reason we do not present in the paper
any result about the time evolution of the total density $n$.
First, as a benchmark for our
method, we consider the relaxation of a single
interface profile with $k=10^{-2}$ and $\eta=0.167$ ($\tau=1$).
This corresponds to a low viscosity regime as discussed in the following.
We started the simulation with a sharp concentration step with values
$\varphi_{-}(T_b)$ and $\varphi_{+}(T_u)$ and bulk temperature
$T/T_c=0.8$ keeping fixed the temperatures $T_b/T_c=0.8,
T_u/T_c=0.9$ at the bottom and up walls (Fig.~1 (a)). The system
reaches a stationary state with constant temperature gradient and
concentration profile as in Fig.~1 (b). The numerical values of
concentrations in the two bulk phases are in very good agreement
with the analytical expression for $\varphi_{\pm}(T({\bf r}))$
corresponding to the equilibrium values of $T({\bf r})$
shown in the related inset. This means that the concentration
field $\varphi$ is in local equilibrium. The
temperature of the up wall is then set to the same value of the
temperature of the bottom wall (Fig.~1 (c)).
Then, as it can be seen in Fig.~1 (d), the system
equilibrates at constant temperature with the expected concentration
profile. Spurious velocities
are of order $10^{-9}$
and result completely negligible. The test shows
that stationary states are well reproduced by our algorithm.
\subsection{Diffusive regime}
We describe our results for phase separation. We first
consider a case at very high viscosity with $\eta=6.5$ ($\tau=20$)
and symmetric composition
(Runs 1-8). Here
the effects of the velocity field are negligible. We set
$T_b/T_c=T_u/T_c=0.8$ and initial bulk temperature above $T_c$. As
it can be seen in Fig.~2, for thermal diffusivities $k \ge 10^{-1}$,
usual isotropic phase separation is observed.
In the range
$ k= 5 \times 10^{-4} \div 5 \times 10^{-2}$, in spite of the
neutral wetting condition on the boundaries, domains in the bulk
have interfaces preferentially parallel to thermal fronts.
For smaller values of $k$ domains grow
perpendicularly to the walls. These results agree with those of
Refs.~\cite{FURUKAWA,WAGNERFOARD,KREK2}
in purely diffusive models
where the same morphological
sequence was found by decreasing the speed of cold fronts moving
into a region with the mixed phase. However, also in absence of
hydrodynamic effects, our case is different since the
thermodynamics of the mixture is fully consistently treated and
temperature fronts have no sharp imposed profile.
We will now concentrate on cases at intermediate thermal diffusivities
where domains are parallel to the walls
and propagation fronts can be traced.
Concentration and temperature configurations at successive times
for $k=10^{-2}$ (Run 4a) are shown in Fig.~3 and Fig.~4, respectively.
In this case it is $Ma \simeq 5 \times 10^{-5}$.
The temperature fronts have typical diffusive profiles which slowly
relax to the equilibrium value imposed on the boundaries. In
order to be quantitative, we defined $y_T(t)$ as the distance from
the wall where the temperature assumes a fixed value (we chose
$T/T_c=0.88$) and measured this quantity in simulations with large
rectangular lattices.
The solution of the diffusion
equation with initial temperature $T_0$ and fixed boundary value
$T_w$ is $(T(y,t)-T_w)/(T_0-T_w)=erf{[y/(2 \sqrt{k t})]}$
which implies $y_T/\sqrt{k} \sim \sqrt{t}$.
In the inset of Fig.~5 it is shown, in simulations with different $k$, that
$y_T$ follows the standard diffusion behavior.
The time behavior of $y_T$ has been checked not depending on the specific
value of the ratio $T/T_c$ in the range $[0.8,1.0]$; by considering
a value of $T$ such that $T/T_c<1$ allows to track the position of the
temperature front for a longer time interval.
One can also consider the behavior of the fronts limiting the
regions with separated phases, clearly
observable in the first three snapshots of Fig.~3.
Their position can be defined as the distance $y_{\varphi}$
from the walls beyond which the condition
$\nabla \varphi \simeq 0$ is verified everywhere.
More precisely, we took $y_{\varphi}$ as the point beyond which
$|\nabla \varphi| < C$ with $C=\sqrt{2} \times 0.01$; the value of $C$
is chosen to match the maximum value of the fluctuations
of $|\nabla \varphi|$ in the initial
disordered state, where $|\varphi| < 0.01$.
(In the
last snapshot of Fig.~3 the two fronts propagating from up and down
have come close each other and more usual phase separation occurs in
the central region of the system.) We measured $y_{\varphi}$ on
rectangular lattices for different $k$ and observed
deviations
from
diffusive behavior
(see Fig.~5). We found that $y_{\varphi}$ grows by power law
with an exponent depending on $k$. Our fits give $y_{\varphi} \sim
t^{0.66}$ for $k=10^{-2}$ and exponents closer to $1/2$ for smaller
$k$.
We analyzed for different $k$ possible variations
of the typical values of fluid velocity but we did not find any.
Therefore
the change of the exponent of $y_{\varphi}$ cannot be attributed
to the velocity field.
Even if $y_{\varphi}$ moves faster than $y_T$ and at long times it results
$y_{\varphi} > y_T$, we checked that the relation $y_{\varphi} < y_{T_c}$
is always verified so that phase separation always occurs for $T < T_c$.
Since the phase separation is induced by the
temperature change, one could
have expected a similar behavior for $y_{\varphi}$ and $y_T$.
The discrepancy could be related to the broad character
of the temperature fronts which spreads the phase separated region.
We also observed that the width of lamellar domains decreases at
larger $k$, in agreement with Ref.~\cite{WAGNERFOARD}.
\subsection{Hydrodynamic regime}
At lower viscosities the evolution of morphology is very different
in the range with intermediate values of thermal diffusivity.
We will in particular illustrate in Fig.~6 the case with
$\eta=0.167$ ($\tau=1$) and $k=10^{-2}$ (Runs 19),
for which we found $Ma \simeq 5 \times 10^{-4}$.
This is the same thermal diffusivity of Fig.~3.
At this viscosity hydrodynamics is relevant.
Indeed, in
instantaneous quenching at constant temperature and $\eta=0.167$ we
observed the domain growth exponent to
assume the inertial value $2/3$ (at odd with the diffusive
high-viscosity value $1/3$ ) \cite{YEOMANS}.
The growth exponent was calculated by measuring the characteristic length
defined by the inverse of the first momentum of the structure
factor \cite{corb}.
The main effect due to hydrodynamics observable in Fig.~6 is
that domains do not grow aligned with
temperature fronts as it occurs for the same thermal diffusivity
at high viscosity.
Circular patterns are
stabilized by the flow \cite{YEOMANS} and an example is given in
Fig.~7.
A similar picture
occurs for other values of $k$ here not reported (see Table I).
On the other hand, the other thermal diffusivity regimes are less
affected by hydrodynamics.
When decreasing $k$, it is still possible to observe
domains growing with interfaces normal to the walls
as in the case at high viscosity (see Fig.~8 - Run 21b),
while at larger $k$ (Run 18) phase separation occurs isotropically
like in an instantaneous quenching.
The cases shown in Figs.~3 and 6
are typical of the high and low viscosity regimes.
At intermediate values of $\eta$ one can observe features common to the
two above cases (see Fig.~9 for $\eta=2.167$ - Run 11a).
Concerning the behavior of $y_T(t)$, we could not find relevant differences
by varying $\eta$ with respect to the case at
high viscosity.
Another effect induced by hydrodynamics is the formation
of structures in the inner part of the system
at earlier times than in the case at high
viscosity (compare Fig.~3 and Fig.~6). In the inner region we can observe
the typical interconnected pattern of spinodal decomposition but
with a characteristic length-scale different from that of domains
close to the walls. However, while the structures close to the
walls are
in local equilibrium, that is $\varphi({\bf r})=\varphi_{\pm}(T({\bf
r}))$, in the middle of the system the concentration field is such
that $|\varphi| < \varphi_{+}(T({\bf r}))$. A temporal regime
characterized by the presence of domains with two scales was found in
systems of different size (from $256 \times 256$ to $1024 \times
1024$) and $k=10^{-3} \div 10^{-2}$.
In order to characterize the two scales we analyzed the behavior of the
structure factor. In Fig.~10 the spherically averaged structure factor
is shown at two consecutive times for a system having the same
parameters of Fig.~6 and size $L=512$. Two peaks are observable at each time
that can be interpreted as related to
the existence of two different length scales with one
about twice longer than the other. The higher peak at smaller wave vector
corresponds to the larger domains close to the walls while
the other peak is related to the thinner domains in the inner of the system.
At increasing times, the two peaks
tend to merge.
Due to this
morphological evolution, in simulations at low viscosity,
the position of the phase separation
front $y_{\varphi}$ could be measured only for a short time interval
making not possible to determine the power-law behavior.
Finally, we show results for systems with asymmetric composition. In
Fig.~11 the evolution of two systems only differing for the value of
viscosity is shown. Lamellar patterns prevail at high viscosity
while circular droplets dominate at low viscosity ($\eta=0.167$). In
the latter case, again, two typical scales can be observed with thin
tubes of materials connecting larger domains. The behavior of
$y_T$ is similar to that of the symmetric case.
\section{Conclusions}
We have developed a numerical method for thermal
binary fluids described by continuity, Navier-Stokes, convection-diffusion,
and energy equations.
We have studied quenching by contact with external
walls, and we have shown how the pattern formation depends on thermal
diffusivity, viscosity, and composition of the system.
The evolution is very different from that observed in instantaneous
homogeneous quenching.
At high
viscosity, different orientations of domains are possible. In an intermediate
range of thermal diffusivities domains are parallel to the walls.
The fronts limiting the regions with separated domains
move towards the inner of the system
with a power law behavior
not always corresponding to that of the temperature fronts.
At low viscosity, the velocity field favors more circular
patterns, and domains are characterized by different
length-scales close to the walls and
in the inner of the system. Off-symmetrical
mixtures give more ordered patterns.
We conclude with two remarks on possible future directions of work.
The first one concerns the Soret effect, which
corresponds to have a mass diffusion current induced by thermal
gradients. This effect can become relevant in quenching very close
to the critical point where the ratio $D_T/D$ becomes large \cite{KREKHOV}.
Here $D_T$ is the thermal (mass) diffusion coefficient
($D_T=\mathcal{L}_{12}/T^2$ in our notation) and $D$ is the mass
diffusion coefficient defined at the beginning of Section III.
In order to have a
first idea on how the Soret effect can affect the pattern
morphology, we considered a case with $D_T/D = 20 $ corresponding to
the highest values for this ratio reported in literature \cite{KREKHOV}.
This would give $D_T= 2 \times 10^{-3}$,
taking for $D$ the value used in the runs of Section III.
We run simulations for this case.
We observed, in
the intermediate range of thermal diffusivity and at high viscosity,
the tendency of the system to exhibit more ordered lamellar patterns
(parallel to the walls). At higher thermal diffusivity isotropic
phase separation is found as usually, while at very low thermal
diffusivity ($k=10^{-4}$), parallel patterns are found instead of
perpendicular patterns. At low viscosity (we tested the case
corresponding to that of Fig.~6) hydrodynamics continues to favor
domains with more circular shape. We run also simulations with
$D_T=10^{-4}$, corresponding to a ratio $D_T/D \simeq 1$, without
finding relevant differences with the respect to the case with
$D_T=0$.
We also observe that the behavior of $y_{\varphi}$
could depend on our choice for
$\mathcal{L}_{12}$ and $\mathcal{L}_{21}$.
A more comprehensive analysis of the Soret effect will be
presented elsewhere.
Finally, the morphology
could be still richer in three dimensions, also due to the existence of more
hydrodynamic regimes \cite{BRAY},
so that three-dimensional simulations would complete
the picture given sofar.
\begin{acknowledgments}
GG warmly acknowledges discussions with A. J. Wagner
during his visit at North Dakota State University.
\end{acknowledgments}
|
{
"timestamp": "2010-09-21T02:02:31",
"yymm": "1009",
"arxiv_id": "1009.3735",
"language": "en",
"url": "https://arxiv.org/abs/1009.3735"
}
|
\section{Introduction}\label{in}
Many procedures in science, engineering, and medicine produce data in the form of shapes.
If one expects such a cloud to follow roughly a submanifold of a certain type,
then it is of utmost importance to describe the space of all possible submanifolds
of this type (we call it a shape space hereafter)
and equip it with a significant metric which is able to distinguish special features
of the shapes. Most of the metrics used today in data analysis and computer vision are of an ad-hoc
and naive nature; one embeds shape space in some Hilbert space or Banach space and uses the
distance therein. Shortest paths are then line segments, but they leave shape space quickly.
Riemannian metrics on shape space itself are a better solution.
They lead to geodesics, to curvature and diffusion.
Eventually one also needs statistics on shape space like means of clustered subsets of data
(called Karcher means on Riemannian manifolds) and standard deviations.
Here curvature will play an essential role; statistics on Riemannian
manifolds seems hopelessly underdeveloped just now.
\subsection{The shape spaces used in this work}
Thus, initially, by a shape we mean a
smoothly embedded surface in $N$ which is diffeomorphic to $M$.
The space of these shapes will be denoted $B_e=B_e(M,N)$ and viewed as the
quotient (see \cite{Michor102} for more details)
$$ B_e(M,N) = \on{Emb}(M,N)/\on{Diff}(M)$$
of the open subset $\on{Emb}(M,N)\subset C^\infty(M,N)$
of smooth embeddings of $M$ in $N$,
modulo the group of smooth diffeomorphisms of $M$.
It is natural to consider all possible {\it immersions} as well
as embeddings, and thus introduce the larger
space $B_i=B_i(M,N)$ as the quotient of the space of smooth immersions by the
group of diffeomorphisms of $M$ (which is, however, no longer a manifold,
but an orbifold with finite isotropy groups, see \cite{Michor102}).
\begin{equation*}
\xymatrix{
\on{Emb}(M,N) \ar@{^{(}->}[d] \ar@{->>}[r] &
\on{Emb}(M,N)/\on{Diff}(M) \ar@{^{(}->}[d] \ar@{=}[r] &
B_e(M,N) \ar@{^{(}->}[d] \\
{\on{Imm}}(M,N) \ar@{->>}[r] &
{\on{Imm}}(M,N)/\on{Diff}(M) \ar@{=}[r] & B_i(M,N)
}
\end{equation*}
More generally, a shape will be an element of the Cauchy completion (i.e., the metric completion
for the geodesic distance) of $B_i(M,N)$ with
respect to a suitably chosen Riemannian metric. This will allow for corners.
In practice, discretization for numerical algorithms will hide the need to go to the Cauchy
completion.
\subsection{Where this work comes from}
In \cite{Michor107}, Michor and Mumford have investigated a variety of Riemannian metrics on the
shape space
$$
B_i(S^1,\mathbb R^2)={\on{Imm}}(S^1,\mathbb R^2)/\on{Diff}(S^1)
$$
of unparametrized immersion of the circle into the plane.
In \cite[section~3.10]{Michor98} they found that the simplest such metric has vanishing geodesic distance;
this is the metric induced by $L^2(\text{arc length})$ on ${\on{Imm}}(S^1,\mathbb R^2)$:
\begin{align*}
G^0_f(h,k) &= \int_{S^1} \langle h(\theta),k(\theta) \rangle |f'(\theta)|\,d\theta ,
\\
f &\in {\on{Imm}}(S^1,\mathbb R^2), \quad h,k \in C^\infty(S^1,\mathbb R^2) =
T_f{\on{Imm}}(S^1,\mathbb R^2).
\end{align*}
In \cite{Michor102} they found that the vanishing geodesic distance phenomenon for the
$L^2$-metric occurs also in
the more general shape space ${\on{Imm}}(M,N)/\on{Diff}(M)$ where $S^1$ is replaced by a compact
manifold $M$ and Euclidean $\mathbb R^2$ is replaced by
Riemannian manifold $N$; it also occurs on the full diffeomorphism group $\on{Diff}(N)$, but not on
the subgroup $\on{Diff}(N,{\on{vol}})$ of volume preserving diffeomorphisms, where the geodesic
equation for the $L^2$-metric is the Euler equation of an incompressible fluid.
In \cite[sections 3, 4 and 5]{Michor107} three classes of metrics were investigated: Almost local metrics on planar curves,
Sobolev metrics on planar curves, and metrics induced from Sobolev metrics on the diffeomorphism group of the plane.
The results about almost local metrics from \cite[section~3]{Michor107} were generalized by the authors to the
case of surfaces in \cite{Michor118}.
Now we take up the investigations from \cite[section~4]{Michor107}. The
{\it immersion-Sobolev metric}
considered there is
\begin{align*}
G^{{\on{Imm}},p}_f(h,k) &=
\int_{S^1} \big(\langle h,k \rangle + A.\langle D_s^p h,
D_s^p k \rangle \big). ds
\\&
= \int_{S^1}\langle L_p(h),k\rangle ds
\qquad\text{ where }
\\
L_p(h) \text{ or } L_{p,f}(h) &= \big(I + (-1)^p A.D_s^{2p}\big)(h) \text{ and }
D_s=\frac{\partial_\theta}{|f_\theta|}.
\end{align*}
The interesting special case $p=1$ and $A \rightarrow \infty$ has been
studied in \cite{TrouveYounes2000, Younes1998} and
in \cite{Michor111} where an isometry
to an infinite dimensional Grassmannian with the Fubini-Study metric was described.
In this case, the metric reduces to:
$$ G^{{\on{Imm}},1,\infty}_f(h,k) = \int_{S^1} \langle D_s(h), D_s(k)
\rangle.ds$$
The cases $p=1,2$ and $A \rightarrow \infty$ have also been treated in \cite{MennucciYezzi2008},
where estimates on the geodesic distance are proven and the metric completion of the space of curves
is characterized.
In this work we generalize the immersion-Sobolev metrics from \cite[section~4]{Michor107}
to higher dimensions and to non-flat ambient space, namely to the shape space
$B_i(M,N)= {\on{Imm}}(M,N)/\on{Diff}(M)$ of surfaces of type $M$ in
$N$; here $M$ is a compact orientable
connected manifold of smaller dimension than $N$, for example a sphere $S^m, m<\dim(N)$.
\subsection{Riemannian metrics}\label{in:ri}
The tangent space $T_f {\on{Imm}}(M,N)$ at an immersion $f$ consists
of all vector fields along $f$:
$$T_f {\on{Imm}}(M,N) = \Gamma(f^*TN) \cong \{h \in C^\infty(M,TN): \pi_{TN} \circ h = f\}. $$
A Riemannian metric on ${\on{Imm}}(M,N)$ is a family of
positive definite inner products $G_f(h,k)$ where
$f \in {\on{Imm}}(M,N)$ and $h,k \in T_f{\on{Imm}}(M,N)$.
Each metric is {\it weak} in the sense that $G_f$, viewed as linear map from
$T_f{\on{Imm}}(M,N)$ into its dual consisting of distributional sections of $f^*TN$ is injective.
(But it can never be surjective.)
We require that our
metrics will be invariant under the action of $\on{Diff}(M)$, hence the
quotient map dividing by this action will be a Riemannian submersion.
This means that the tangent map of the quotient map ${\on{Imm}}(M,N)\to B_i(M,N)$ is
a metric quotient mapping between all tangent spaces.
Thus we will get Riemannian metrics on $B_i$.
For any $f\in {\on{Imm}}(M,N)$ those vectors in
$T_f{\on{Imm}}(M,N)$ which are $G_f$-perpendicuar to the
$\on{Diff}(M)$-orbit through $f$ are called
\emph{horizontal} (with respect to $G$).
They form the $G_f$-orthogonal space to the orbit. A priori we do not
know that it is a complementary space. For the metrics considered in this work it will turn out to be
a complement.
The simplest inner product on the tangent bundle to
${\on{Imm}}(M,N)$ is
$$ G^0_f(h,k) = \int_{M} \overline{g}(h, k) \, {\on{vol}}(f^*\overline{g}),$$
where $\overline{g}=\langle\quad,\quad \rangle$ is the Euclidean inner product on $N$.
Since the volume form ${\on{vol}}(f^*\overline{g})$ reacts equivariantly to the action of the
group ${\on{Diff}}(M)$, this metric is invariant,
and the map to the quotient $B_i$ is a Riemannian
submersion for this metric.
The $G^0$-horizontal vectors in $T_f{\on{Imm}}(M,N)$ are just those vector fields along $f$
which are pointwise $\overline{g}$-normal to $f(M)$; we will call them {\it normal} fields.
All of the metrics we will look at will be of the form (see section \ref{so}):
$$ G^P_f(h,k) = \int_{M} \overline{g}( P^f h, k)\, {\on{vol}}(f^*\overline{g})$$
where $P^f:T_f{\on{Imm}} \to T_f{\on{Imm}}$ is a positive bijective operator depending smoothly on $f$,
which is selfadjoint unbounded in the Hilbert space $T_f{\on{Imm}}$ with inner product $G^0_f$.
We will assume that $P$ is in addition equivariant with respect to reparametrizations, i.e.
$$P^{f\circ\varphi}=\varphi^* \circ P^f \circ (\varphi^{-1})^* = \varphi^*(P^f)\qquad \text{for all }\varphi\in\on{Diff}(M).$$
The $G^P$-horizontal vectors will be those
$h\in T_f\on{Emb}(M,N)=C^\infty(M,N)$ such that $P^fh$ is normal.
The tangent map of the quotient map $\on{Emb}(M,N)\to B_i(M,N)$ is then an isometry
when restricted to the horizontal spaces, just as in the finite dimensional situation.
Riemannian submersions have a very nice effect on geodesics: the geodesics on
the quotient space $B_i$ are exactly the images of the horizontal geodesics on the top
space ${\on{Imm}}$; by a horizontal geodesic we mean a geodesic whose tangent lies
in the horizontal bundle.
The induced metric is invariant under the action of $\on{Diff}(M)$ and therefore induces
a unique metric on $B_i$.
See for example \cite[section~1]{Michor118}.
Later in section \ref{la} we
shall consider the special case
$P^f=1+A\Delta^p$.
\subsection{Inner versus outer metrics}
The metrics studied in this work are induced from ${\on{Imm}}(M,N)$ on shape space.
One might call them \emph{inner metrics} since the differential operator governing the metric
is defined intrinsic to $M$. Intuitively, these metrics can be seen as describing some elastic
or viscous behaviour of the shape itself.
In contrast to these metrics, there are also metrics induced from $\on{Diff}(N)$ on shape space.
(The widely used LDDMM algorithm uses such a metric.)
The differential operator governing these metrics is defined on all of $N$, even outside of the shape.
Intuitively, these metrics can be seen as describing some elastic or viscous behaviour of the
ambient space $N$ that gets deformed as the shape changes.
One might call these metrics \emph{outer metrics}.
\subsection{Contributions of this work.}
\begin{itemize}
\item This work is the first to treat Sobolev inner metrics on spaces of immersed
surfaces and on higher dimensional shape spaces.
\item It contains the first description of how the geodesic equation can be
formulated in terms of gradients of the metric with respect to itself
when the ambient space is not flat.
To achieve this, a covariant derivative on some bundles over immersions is defined.
This covariant derivative is induced from the Levi-Civita covariant derivative on ambient space.
\item The geodesic equation is formulated in terms of this covariant derivative.
Well-posedness of the geodesic equation is shown under some
regularity assumptions that are verified for Sobolev metrics.
Well-posedness also follows for the geodesic equation on
diffeomorphism groups, where this result has not yet been obtained in that full generality.
\item To derive the geodesic equation, a variational formula for the Laplacian operator is developed.
The variation is taken with respect to the metric on the manifold where the Laplacian is
defined. This metric in turn depends on the immersion inducing it.
\item It is shown that Sobolev inner metrics separate points in shape space when the
order of the differential operator governing the metric is high enough.
(The metric needs to be as least as strong as the $H^1$-metric.)
Thus Sobolev inner metrics overcome the degeneracy of the $L^2$-metric.
\item The path-length distance of Sobolev inner metrics is compared to the Fr\'echet distance.
It would be desirable to bound F\'echet distance by some Sobolev distance.
This however remains an open problem.
\item Finally it is demonstrated in some examples that the geodesic equation
for the $H^1$-metric on shape space of surfaces in $\mathbb R^3$ can be solved numerically.
\end{itemize}
Big parts of this work can also be found, partly in more details, in the doctoral theses of Martin Bauer
\cite{Bauer2010} and Philipp Harms \cite{Harms2010}.
\section{Content of this work}
This work progresses from a very general setting to a specific one in three steps.
In the beginning, a framework for general inner metrics is developed.
Then the general concepts carry over to more and more specific inner metrics.
\begin{itemize}
\item First, shape space is endowed with a \emph{general inner metric}, i.e with a metric
that is induced from a metric on the space of immersions, but that is unspecified otherwise.
It is shown how various versions of the geodesic equation can be expressed using gradients of the
metric with respect to itself and how conserved quantities arise from symmetries.
(This is section~\ref{sh}.)
\item Then it is assumed that the inner metric is defined via an elliptic pseudo-differential operator.
Such a metric will be called a \emph{Sobolev-type metric}.
The geodesic equation is formulated in terms of the operator, and
existence of horizontal paths of immersions within each equivalence class of paths is proven.
(This is section~\ref{so}.)
Then estimates on the path-length distance are derived. Most importantly it is shown that
when the operator involves high enough powers of the Laplacian, then the metric does not have
the degeneracy of the $L^2$-metric. (This is section~\ref{ge}.)
\item Motivated by the previous results it is assumed that the elliptic pseudo-differential operator
is given by the \emph{Laplacian} and powers of it.
Again, the geodesic equation is derived. The formulas that are obtained are ready to be implemented numerically.
(This is section~\ref{la}.)
\end{itemize}
The remaining sections cover the following material:
\begin{itemize}
\item Section~\ref{no} treats some \emph{differential geometry of surfaces} that is needed in
this work. It is also a good reference for the notation that is used.
The biggest emphasis is on a rigorous treatment of the covariant derivative.
Some material like the adjoint covariant derivative is not found in standard text books.
\item Section~\ref{va} contains formulas for the \emph{variation} of the metric, volume form,
covariant derivative and Laplacian with respect to the immersion inducing them.
These formulas are used extensively later.
\item Section~\ref{su} covers the special case of \emph{flat ambient space}.
The geodesic equation is simplified and conserved momenta for the Euclidean motion group
are calculated. Sobolev-type metrics are compared to the Fr\'echet metric which is
available in flat ambient space.
\item Section~\ref{di} treats \emph{diffeomorphism groups} of compact manifolds as a
special case of the theory that has been developed so far.
\item In section~\ref{nu} it is shown in some examples that the geodesic equation
on shape space can be solved \emph{numerically}.
\end{itemize}
\section[Notation]{Differential geometry of surfaces and notation}\label{no}
In this section the differential geometric tools
that are needed to deal with immersed surfaces
are presented and developed.
The most important point is a rigorous treatment of the covariant derivative
and related concepts.
The notation of \cite{MichorH} is used.
Some of the definitions can also be found in \cite{Kobayashi1996a}.
A similar exposition in the same notation is \cite{Michor118}.
\subsection{Basic assumptions and conventions}\label{no:as}
\begin{ass*}
It is always assumed that $M$ and $N$ are connected manifolds
of finite dimensions $m$ and $n$, respectively.
Furthermore it is assumed that $M$ is compact,
and that $N$ is endowed with a Riemannian metric $\overline{g}$.
\end{ass*}
In this work, \emph{immersions} of $M$ into $N$ will be treated, i.e.
smooth functions $M \to N$ with injective tangent mapping at every point.
The set of all such immersions will be denoted by ${\on{Imm}}(M,N)$.
It is clear that only the case $\dim(M) \leq \dim(N)$ is of interest
since otherwise ${\on{Imm}}(M,N)$ would be empty.
Immersions or paths of immersions are usually denoted by $f$.
Vector fields on ${\on{Imm}}(M,N)$ or tangent vectors with foot point $f$, i.e., vector fields along $f$, will be called $h,k,m$, for example.
Subscripts like $f_t = \partial_t f = \partial f/\partial t$ denote differentiation
with respect to the indicated variable, but subscripts are also used to indicate the
foot point of a tensor field.
\subsection{Tensor bundles and tensor fields}\label{no:te}
The \emph{tensor bundles}
\begin{equation*}\xymatrix{
T^r_s M \ar[d] &
T^r_s M \otimes f^*TN \ar[d] \\
M & M
}\end{equation*}
will be used. Here $T^r_sM$ denotes the bundle of
$\left(\begin{smallmatrix}r\\s\end{smallmatrix}\right)$-tensors on $M$, i.e.
$$T^r_sM=\bigotimes^r TM \otimes \bigotimes^s T^*M,$$
and $f^*TN$ is the pullback of the bundle $TN$ via $f$, see \cite[section~17.5]{MichorH}.
A \emph{tensor field} is a section of a tensor bundle. Generally, when $E$ is a bundle,
the space of its sections will be denoted by $\Gamma(E)$.
To clarify the notation that will be used later,
some examples of tensor bundles and tensor fields are given now.
$S^k T^*M = L^k_{\on{sym}}(TM; \mathbb R)$ and
$\Lambda^k T^*M = L^k_{\on{alt}}(TM; \mathbb R)$
are the bundles of symmetric and alternating
$\left(\begin{smallmatrix}0\\k\end{smallmatrix}\right)$-tensors, respectively.
$\Omega^k(M)=\Gamma(\Lambda^k T^*M)$ is the space of differential forms,
$\mathfrak X(M)=\Gamma(TM)$ is the space of vector fields, and
$$\Gamma(f^*TN) \cong \big\{ h \in C^\infty(M,TN): \pi_N \circ h = f \big\}$$
is the space of \emph{vector fields along $f$}.
\subsection{Metric on tensor spaces}\label{no:me}
Let $\overline{g} \in \Gamma(S^2_{>0} T^*N)$ denote a fixed Riemannian metric on $N$.
The \emph{metric induced on $M$ by $f \in {\on{Imm}}(M,N)$} is the pullback metric
\begin{align*}
g=f^*\overline{g} \in \Gamma(S^2_{>0} T^*M), \qquad g(X,Y)=(f^*\overline{g})(X,Y) = \overline{g}(Tf.X,Tf.Y),
\end{align*}
where $X,Y$ are vector fields on $M$.
The dependence of $g$ on the immersion $f$ should be kept in mind.
Let $$\flat = \check g: TM \to T^*M \quad \text{and} \quad \sharp=\check g^{-1}: T^*M \to TM.$$
$g$ can be extended to the cotangent bundle $T^*M=T^0_1M$ by setting
$$g^{-1}(\alpha,\beta)=g^0_1(\alpha,\beta)=\alpha(\beta^\sharp)$$
for $\alpha,\beta \in T^*M$.
The product metric
$$g^r_s = \bigotimes^r g \otimes \bigotimes^s g^{-1}$$
extends $g$ to all tensor spaces $T^r_s M$, and
$g^r_s \otimes \overline{g}$ yields a metric on $T^r_s M \otimes f^*TN$.
\subsection{Traces}\label{no:tr}
The \emph{trace} contracts pairs of vectors and co-vectors in a tensor product:
\begin{align*}
\on{Tr}:\; T^*M \otimes TM = L(TM,TM) \to M \times \mathbb R
\end{align*}
A special case of this is the operator
$i_X$ inserting a vector $X$ into a co-vector or into a covariant factor of a tensor product.
The inverse of the metric $g$ can be used to define a trace
$$\on{Tr}^g: T^*M \otimes T^*M \to M \times \mathbb R$$
contracting pairs of co-vecors.
Note that $\on{Tr}^g$ depends on the metric whereas $\on{Tr}$ does not.
The following lemma will be useful in many calculations:
\begin{lem*}
\begin{equation*}
g^0_2(B,C)= \on{Tr}(g^{-1} B g^{-1} C) \quad \text{for $B,C \in T^0_2M$ if $B$ or $C$ is symmetric.}
\end{equation*}
(In the expression under the trace, $B$ and $C$ are seen as maps $TM \to T^*M$.)
\end{lem*}
\begin{proof}
Express everything in a local coordinate system $u^1, \ldots, u^{m}$ of $M$.
\begin{align*}
g^0_2(B,C)&=g^0_2\Big(\sum_{ik} B_{ik}du^i \otimes du^k,\sum_{jl}C_{jl}du^j \otimes du^l\Big) \\ & =
\sum_{ijkl} g^{ij}B_{ik}g^{kl}C_{jl} = \sum_{ijkl} g^{ji}B_{ik}g^{kl}C_{lj}
= \on{Tr}(g^{-1} B g^{-1} C)
\end{align*}
Note that only the symmetry of $C$ has been used.
\end{proof}
\subsection{Volume density}\label{no:vo}
Let ${\on{Vol}}(M)$ be the \emph{density bundle} over $M$, see \cite[section~10.2]{MichorH}.
The \emph{volume density} on $M$ induced by $f \in {\on{Imm}}(M,N)$ is
$${\on{vol}}(g)={\on{vol}}(f^*\overline{g}) \in \Gamma\big({\on{Vol}}(M)\big).$$
The \emph{volume} of the immersion is given by
$${\on{Vol}}(f)=\int_M {\on{vol}}(f^*\overline{g})=\int_M {\on{vol}}(g).$$
The integral is well-defined since $M$ is compact. If $M$ is oriented the volume
density may be identified with a differential form.
\subsection{Metric on tensor fields}\label{no:me2}
A \emph{metric on a space of tensor fields} is defined by integrating the appropriate metric on the
tensor space with respect to the volume density:
$$\widetilde{g^r_s}(B,C)=\int_M g^r_s\big(B(x),C(x)\big){\on{vol}}(g)(x)$$
for $B,C \in \Gamma(T^r_sM)$, and
$$\widetilde{g^r_s \otimes \overline{g}}(B,C) = \int_M g^r_s\otimes \overline{g} \big(B(x),C(x)\big){\on{vol}}(g)(x)$$
for $B,C \in \Gamma(T^r_sM \otimes f^*TN)$, $f \in {\on{Imm}}(M,N)$.
The integrals are well-defined because $M$ is compact.
\subsection{Covariant derivative}\label{no:co}
Covariant derivatives on vector bundles as explained in \cite[sections 19.12, 22.9]{MichorH}
will be used.
Let $\nabla^g, \nabla^{\overline{g}}$ be the \emph{Levi-Civita covariant derivatives} on $(M,g)$
and $(N,\overline{g})$, respectively.
For any manifold $Q$ and vector field $X$ on $Q$, one has
\begin{align*}
\nabla^g_X:C^\infty(Q,TM) &\to C^\infty(Q,TM), & h &\mapsto \nabla^g_X h \\
\nabla^{\overline{g}}_X: C^\infty(Q,TN) &\to C^\infty(Q,TN), & h &\mapsto \nabla^{\overline{g}}_X h.
\end{align*}
Usually the symbol $\nabla$ will be used for all covariant derivatives.
It should be kept in mind that $\nabla^g$ depends on the metric $g=f^*\overline{g}$ and therefore also on
the immersion $f$.
The following properties hold \cite[section~22.9]{MichorH}:
\begin{enumerate}
\item \label{no:co:ba}
$\nabla_X$ respects base points, i.e.
$\pi \circ \nabla_X h = \pi \circ h$, where $\pi$ is the projection
of the tangent space onto the base manifold.
\item
$\nabla_X h$ is $C^\infty$-linear in $X$. So for a tangent vector $X_x \in T_xQ$,
$\nabla_{X_x}h$ makes sense and equals $(\nabla_X h)(x)$.
\item
$\nabla_X h$ is $\mathbb R$-linear in $h$.
\item
$\nabla_X (a.h) = da(X).h + a.\nabla_X h$ for $a \in C^\infty(Q)$, the derivation property of $\nabla_X$.
\item \label{no:co:prop5}
For any manifold $\widetilde Q$ and smooth mapping
$q:\widetilde Q \to Q$ and $Y_y \in T_y \widetilde Q$ one has
$\nabla_{Tq.Y_y}h=\nabla_{Y_y}(h \circ q)$. If $Y \in \mathfrak X(Q_1)$ and $X \in \mathfrak X(Q)$ are $q$-related, then
$\nabla_Y(h \circ q) = (\nabla_X h) \circ q$.
\end{enumerate}
The two covariant derivatives $\nabla^g_X$ and $\nabla^{\overline{g}}_X$
can be combined to yield a covariant derivative $\nabla_X$ acting on
$C^\infty(Q,T^r_sM \otimes TN)$ by additionally requiring the following properties
\cite[section 22.12]{MichorH}:
\begin{enumerate}
\setcounter{enumi}{5}
\item $\nabla_X$ respects the spaces $C^\infty(Q,T^r_sM \otimes TN)$.
\item $\nabla_X(h \otimes k) = (\nabla_X h) \otimes k + h \otimes (\nabla_X k)$,
a derivation with respect to the tensor product.
\item $\nabla_X$ commutes with any kind of contraction (see \cite[section 8.18]{MichorH}).
A special case of this is
$$\nabla_X\big(\alpha(Y)\big)=(\nabla_X \alpha)(Y)+\alpha(\nabla_X Y) \quad
\text{for } \alpha\otimes Y :N \to T^1_1M.$$
\end{enumerate}
Property \eqref{no:co:ba} is important because it implies that $\nabla_X$
respects spaces of sections of bundles.
For example, for $Q=M$ and $f \in C^\infty(M,N)$, one gets
$$\nabla_X : \Gamma(T^r_s M \otimes f^* TN) \to \Gamma(T^r_s M \otimes f^* TN). $$
\subsection{Swapping covariant derivatives}\label{no:sw}
Some formulas allowing to swap covariant derivatives will be used repeatedly.
Let $f$ be an immersion, $h$ a vector field along $f$ and $X,Y$ vector fields on $M$.
Since $\nabla$ is torsion-free, one has \cite[section~22.10]{MichorH}:
\begin{equation}\label{no:sw:to}
\nabla_X Tf.Y-\nabla_Y Tf.X -Tf.[X,Y] = \on{Tor}(Tf.X,Tf.Y) = 0.
\end{equation}
Furthermore one has \cite[section~24.5]{MichorH}:
\begin{equation}\label{no:sw:r}
\nabla_X \nabla_Y h - \nabla_Y \nabla_X h - \nabla_{[X,Y]} h
= R^{\overline{g}} \circ (Tf.X,Tf.Y) h,
\end{equation}
where $R^{\overline{g}} \in \Omega^2\big(N;L(TN,TN)\big)$ is the Riemann curvature tensor of $(N,\overline{g})$.
These formulas also hold when $f:\mathbb R \times M \to N$ is a path of immersions,
$h:\mathbb R \times M \to TN$ is a vector field along $f$ and
the vector fields are vector fields on $\mathbb R \times M$.
A case of special importance is when one of the vector fields is $(\partial_t,0_M)$ and the
other $(0_\mathbb R,Y)$, where $Y$ is a vector field on $M$.
Since the Lie bracket of these vector fields vanishes,
\eqref{no:sw:to} and \eqref{no:sw:r} yield
\begin{equation}\label{no:sw:to2}
\nabla_{(\partial_t,0_M)} Tf.(0_{\mathbb R},Y)-\nabla_{(0_{\mathbb R},Y)} Tf.{(\partial_t,0_M)} = 0
\end{equation}
and
\begin{equation}\label{no:sw:r2}
\nabla_{(\partial_t,0_M)} \nabla_{(0_\mathbb R,Y)} h - \nabla_{(0_\mathbb R,Y)} \nabla_{(\partial_t,0_M)} h
\\= R^{\overline{g}} \big(Tf.(\partial_t,0_M),Tf.(0_\mathbb R,Y)\big) h .
\end{equation}
\subsection{Second and higher covariant derivatives}\label{no:co2}
When the covariant derivative is seen as a mapping
$$\nabla: \Gamma(T^r_s M) \to \Gamma(T^r_{s+1}M)\quad \text{or} \quad
\nabla : \Gamma(T^r_sM \otimes f^*TN) \to \Gamma(T^r_{s+1}M \otimes f^*TN),$$
then the \emph{second covariant derivative} is simply $\nabla\nabla=\nabla^2$.
Since the covariant derivative commutes with contractions,
$\nabla^2$ can be expressed as
$$\nabla^2_{X,Y} :=\iota_Y \iota_X \nabla^2 =
\iota_Y \nabla_X \nabla =
\nabla_X\nabla_Y -\nabla_{\nabla_XY} \qquad \text{for $X,Y\in \mathfrak X(M)$.}$$
Higher covariant derivates are defined accordingly as $\nabla^k$, $k \geq 0$.
\subsection{Adjoint of the covariant derivative}\label{no:co*}
The covariant derivative
$$\nabla: \Gamma(T^r_sM) \to \Gamma( T^r_{s+1}M)$$
admits an \emph{adjoint}
$$\nabla^*:\Gamma( T^r_{s+1}M)\to \Gamma(T^r_sM)$$
with respect to the metric $\widetilde{g}$, i.e.:
$$\widetilde{g^r_{s+1}}(\nabla B, C)= \widetilde{g^r_s}(B, \nabla^* C).$$
In the same way, $\nabla^*$ can be defined when $\nabla$ is acting on
$\Gamma(T^r_s M \otimes f^*TN)$.
In either case it is given by
$$\nabla^*B=-\on{Tr}^g(\nabla B), $$
where the trace is contracting the first two tensor slots of $\nabla B$.
This formula will be proven now:
\begin{proof}
The result holds for decomposable tensor fields $\beta \otimes B \in \Gamma(T^r_{s+1} M)$ since
\begin{align*} &
\widetilde {g^r_s}\Big(\nabla^*(\beta \otimes B),C\Big) =
\widetilde {g^r_{s+1}}\Big(\beta \otimes B,\nabla C\Big) =
\widetilde {g^r_{s}}\Big(B,\nabla_{\beta^\sharp} C\Big) \\&\qquad=
\int_M \mathcal L_{\beta^\sharp} g^r_{s}(B, C) {\on{vol}}(g) - \int_M g^r_s(\nabla_{\beta^\sharp} B,C) {\on{vol}}(g)
\\&\qquad=
\int_M -g^r_{s}(B, C) \mathcal L_{\beta^\sharp} {\on{vol}}(g) - \int_M g^r_s\big(\on{Tr}^g(\beta \otimes \nabla B),C\big) {\on{vol}}(g)
\\&\qquad=
\widetilde {g^r_{s}}\Big(-\on{div}(\beta^\sharp) B - \on{Tr}^g(\beta \otimes \nabla B),C\Big) \\&\qquad=
\widetilde {g^r_{s}}\Big(-\on{div}(\beta^\sharp) B + \on{Tr}^g((\nabla\beta) \otimes B)
-\on{Tr}^g(\nabla (\beta \otimes B)) ,C\Big)
\\&\qquad=
\widetilde {g^r_{s}}\Big(-\on{div}(\beta^\sharp) B + \on{Tr}^g(\nabla\beta) B
-\on{Tr}^g(\nabla (\beta \otimes B)) ,C\Big)\\&\qquad=
\widetilde {g^r_{s}}\Big(0-\on{Tr}^g(\nabla (\beta \otimes B)), C\Big)
\end{align*}
Here it has been used that $\nabla_X g=0$,
that $\nabla_{X}$ commutes with any kind of contraction and acts as a derivation
on tensor products \cite[section~22.12]{MichorH}
and that $\on{div}(X) = \on{Tr}(\nabla X)$ for all vector fields $X$ \cite[section~25.12]{MichorH}.
To prove the result for $\beta \otimes B \in \Gamma(T^r_{s+1} M \otimes f^*TN)$ one simply has
to replace $g^r_{s}$ by $g^r_{s} \otimes \overline{g}$.
\end{proof}
\subsection{Laplacian}\label{no:la}
The definition of the Laplacian used in this work is the \emph{Bochner-Laplacian}.
It can act on all tensor fields $B$ and is defined as
$$\Delta B = \nabla^*\nabla B = - \on{Tr}^g(\nabla^2 B).$$
\subsection{Normal bundle}\label{no:no}
The \emph{normal bundle} ${\on{Nor}}(f)$ of an immersion $f$ is a sub-bundle of $f^*TN$
whose fibers consist of all vectors that are orthogonal to the image of $f$:
$${\on{Nor}}(f)_x = \big\{ Y \in T_{f(x)}N : \forall X \in T_xM : \overline{g}(Y,Tf.X)=0 \big\}.$$
If $\dim(M)=\dim(N)$ then the fibers of the normal bundle are but the zero vector.
Any vector field $h$ along $f \in {\on{Imm}}$ can be decomposed uniquely
into parts {\it tangential} and {\it normal} to $f$ as
$$h=Tf.h^\top + h^\bot,$$
where $h^\top$ is a vector field on $M$ and $h^\bot$ is a section of the normal bundle ${\on{Nor}}(f)$.
\subsection{Second fundamental form and Weingarten mapping}\label{no:we}
Let $X$ and $Y$ be vector fields on $M$.
Then the covariant derivative $\nabla_X Tf.Y$ splits into tangential and a normal parts as
$$\nabla_X Tf.Y=Tf.(\nabla_X Tf.Y)^\top + (\nabla_X Tf.Y)^\bot = Tf.\nabla_X Y + S(X,Y).$$
$S$ is the \emph{second fundamental form of $f$}.
It is a symmetric bilinear form with values in the normal bundle of $f$.
When $Tf$ is seen as a section of $T^*M \otimes f^*TN$ one has $S=\nabla Tf$ since
$$S(X,Y) = \nabla_X Tf.Y - Tf.\nabla_X Y = (\nabla Tf)(X,Y).$$
The trace of $S$ is the \emph{vector valued mean curvature} $\on{Tr}^g(S) \in \Gamma\big({\on{Nor}}(f)\big)$.
\section{Shape space}\label{sh}
Briefly said, in this work the word shape means an \emph{unparametrized surface}.
(The term surface is used regardless of whether it has dimension two or not.)
This section is about the infinite dimensional space of all shapes.
First some spaces of parametrized and unparametrized surfaces are described,
and it is shown how to define Riemannian metrics on them.
The geodesic equation and conserved quantities arising from symmetries are derived.
The agenda that is set out in this section
will be pursued in section~\ref{so} when the arbitrary metric is replaced
by a Sobolev-type metric involving a pseudo-differential operator and later in
section~\ref{la} when the pseudo-differential operator is replaced by
an operator involving powers of the Laplacian.
\subsection{Riemannian metrics on immersions}\label{sh:im}\label{sh:na}\label{sh:ri}
The space of smooth immersions of the manifold $M$ into the manifold $N$
will be denoted by ${\on{Imm}}(M,N)$ or briefly ${\on{Imm}}$. It is a smooth Fr\'echet manifold containing
the space ${\on{Emb}}(M,N)$ of embeddings of $M$ into $N$ as an open subset \cite[theorem~44.1]{MichorG}.
Consider the following \emph{natural bundles of $k$-multilinear mappings}:
\begin{equation*}\xymatrix{
L^k(T{\on{Imm}};\mathbb R) \ar[d] & L^k(T{\on{Imm}};T{\on{Imm}} ) \ar[d] \\
{\on{Imm}} & {\on{Imm}}
}\end{equation*}
These bundles are isomorphic to the bundles
\begin{equation*}\xymatrix{
L\left(\widehat\bigotimes^k T{\on{Imm}};\mathbb R\right)\ar[d] &
L\left(\widehat\bigotimes^k T{\on{Imm}};T{\on{Imm}}\right)\ar[d]\\
{\on{Imm}} & {\on{Imm}}
}\end{equation*}
where $\widehat\bigotimes$ denotes the $c^\infty$-completed bornological tensor product of
locally convex vector spaces \cite[section~5.7, section~4.29]{MichorG}.
Note that $L(T{\on{Imm}};T{\on{Imm}})$ is not isomorphic to
$T^*{\on{Imm}} \;\widehat\otimes\; T{\on{Imm}}$
since the latter bundle corresponds to multilinear mappings with finite rank.
It is worth to write down more explicitly what some of these bundles of multilinear mappings are.
The \emph{tangent space to ${\on{Imm}}$} is given by
\begin{align*}
T_f{\on{Imm}} &= C^\infty_f(M,TN) := \big\{ h \in C^\infty(M,TN): \pi_N \circ h =f\big\}, \\
T{\on{Imm}} &= C^\infty_{{\on{Imm}}}(M,TN) := \big\{ h \in C^\infty(M,TN): \pi_N \circ h \in {\on{Imm}} \big\}.
\end{align*}
Thus $T_f{\on{Imm}}$ is the space of vector fields along the immersion $f$.
Now the \emph{cotangent space to ${\on{Imm}}$} will be described.
The symbol $\widehat\otimes_{C^\infty(M)}$ means that the tensor product
is taken over the algebra $C^\infty(M)$.
\begin{align*}
T^*_f{\on{Imm}} &= L(T_f{\on{Imm}};\mathbb R) = C^\infty_f(M,TN)' =
C^\infty(M)'\; \widehat\otimes_{C^\infty(M)} C^\infty_f(M,T^*N) \\
T^*{\on{Imm}} &= L(T{\on{Imm}};\mathbb R) = C^\infty(M)'\; \widehat\otimes_{C^\infty(M)} C^\infty_{{\on{Imm}}}(M,T^*N)
\end{align*}
The bundle $L^2_{\on{sym}}(T{\on{Imm}};\mathbb R)$ is of interest for the definition of a Riemannian
metric on ${\on{Imm}}$.
(The subscripts $_{\on{sym}}$ and $_{\on{alt}}$ indicate symmetric and alternating multilinear maps, respectively.)
Letting $\otimes_S$ denotes the symmetric tensor product and
$\widehat\otimes_S$ the $c^\infty$-completed bornological symmetric tensor product, one has
\begin{align*}
L^2_{\on{sym}}(T_f{\on{Imm}};\mathbb R) &= (T_f{\on{Imm}}\; \widehat\otimes_S\; T_f{\on{Imm}})' =
\big(C^\infty_f(M,TN) \; \widehat\otimes_S\; C^\infty_f(M,TN)\big)' \\ &=
\big(C^\infty_f(M,TN \; \otimes_S\; TN) \big)' \\&=
C^\infty(M)' \;\widehat\otimes_{C^\infty(M)} C^\infty_f(M,T^*N \; \otimes_S\; T^*N)
\\
L^2_{\on{sym}}(T{\on{Imm}};\mathbb R) &=
C^\infty(M)' \;\widehat\otimes_{C^\infty(M)} C^\infty_{{\on{Imm}}}(M,T^*N \;\otimes_S\; T^*N)
\end{align*}
A \emph{Riemannian metric $G$ on ${\on{Imm}}$} is a section of the bundle
$L^2_{\on{sym}}(T{\on{Imm}};\mathbb R)$
such that at every $f \in {\on{Imm}}$, $G_f$ is a symmetric positive definite bilinear mapping
$$G_f: T_f{\on{Imm}} \times T_f{\on{Imm}} \to \mathbb R.$$
Each metric is {\it weak} in the sense that $G_f$, seen as a mapping
$$G_f: T_f{\on{Imm}} \to T^*_f{\on{Imm}}$$
is injective. (But it can never be surjective.)
\subsection{Covariant derivative $\nabla^{\overline{g}}$ on immersions}\label{sh:cov}
The covariant derivative $\nabla^{\overline{g}}$ defined in section~\ref{no:co} induces a
\emph{covariant derivative over immersions} as follows.
Let $Q$ be a smooth manifold. Then one identifies
\begin{align*}
&h \in C^\infty\big(Q,T{\on{Imm}}(M,N)\big) && \text{and} && X \in \mathfrak X(Q)
\intertext{with}
&h^{\wedge} \in C^\infty(Q \times M, TN) && \text{and} && (X,0_M) \in \mathfrak X(Q \times M).
\end{align*}
As described in section~\ref{no:co} one has the covariant derivative
$$\nabla^{\overline{g}}_{(X,0_M)} h^{\wedge} \in C^\infty\big(Q \times M, TN).$$
Thus one can define
$$\nabla_X h = \left(\nabla^{\overline{g}}_{(X,0_M)} h^{\wedge}\right)^{\vee} \in C^\infty\big(Q,T{\on{Imm}}(M,N)\big).$$
This covariant derivative is torsion-free by section~\ref{no:sw}, formula~\eqref{no:sw:to}.
It respects the metric $\overline{g}$ but in general does not respect $G$.
It is helpful to point out some special cases of how this construction can be used.
The case $Q=\mathbb R$ will be important to formulate the geodesic equation.
The expression that will be of interest in the formulation of the
geodesic equation is $\nabla_{\partial_t} f_t$, which is
well-defined when $f:\mathbb R \to {\on{Imm}}$ is a path of immersions and $f_t: \mathbb R \to T{\on{Imm}}$ is its velocity.
Another case of interest is $Q = {\on{Imm}}$. Let $h, k, m \in \mathfrak X({\on{Imm}})$. Then the covariant
derivative $\nabla_m h$ is well-defined and tensorial in $m$.
Requiring $\nabla_m$ to respect the grading of the spaces of multilinear maps, to act as a derivation
on products and to commute with compositions of multilinear maps, one obtains
as in section~\ref{no:co} a covariant
derivative $\nabla_m$ acting on all mappings into
the natural bundles of multilinear mappings over ${\on{Imm}}$.
In particular, $\nabla_m P$ and $\nabla_m G$ are well-defined for
\begin{align*}
P \in \Gamma\big(L(T{\on{Imm}};T{\on{Imm}})\big), \quad
G \in \Gamma\big(L^2_{\on{sym}}(T{\on{Imm}};\mathbb R)\big)
\end{align*}
by the usual formulas
\begin{align*}
(\nabla_m P)(h) &= \nabla_m\big(P(h)\big) - P(\nabla_mh), \\
(\nabla_m G)(h,k) &= \nabla_m\big(G(h,k)) - G(\nabla_m h,k) - G(h,\nabla_m k).
\end{align*}
\subsection{Metric gradients}\label{sh:me}
The \emph{metric gradients} $H,K \in \Gamma\big(L^2(T{\on{Imm}};T{\on{Imm}})\big)$ are uniquely defined by the equation
$$(\nabla_m G)(h,k)=G\big(K(h,m),k\big)=G\big(m,H(h,k)\big),$$
where $h,k,m$ are vector fields on ${\on{Imm}}$ and
the covariant derivative of the metric tensor $G$ is defined as in the previous section.
(This is a generalization of the definition used in \cite{Michor107}
that allows for a curved ambient space $N \neq \mathbb R^n$.)
Existence of $H, K$ has to proven case by case for each metric $G$,
usually by partial integration.
For Sobolev metrics, this will be proven in sections~\ref{la:ad} and \ref{la:ge}.
\begin{ass*}
Nevertheless it will be assumed for now that the metric gradients $H,K$ exist.
\end{ass*}
\subsection{Geodesic equation on immersions}\label{sh:ge}
\begin{thm*}
Given $H,K$ as defined in the previous section and $\nabla$ as defined in
section~\ref{sh:cov}, the geodesic equation reads as
$$\nabla_{\partial_t} f_t=\frac12 H_f(f_t,f_t)-K_f(f_t,f_t).$$
\end{thm*}
This is the same result as in \cite[section~2.4]{Michor107}, but in a more general setting.
\begin{proof}
Let $f: (-\varepsilon,\varepsilon) \times [0,1] \times M \to N$ be a one-parameter
family of curves of immersions with fixed endpoints.
The variational parameter will be denoted by $s \in (-\varepsilon,\varepsilon)$ and
the time-parameter by $t \in [0,1]$.
In the following calculation, let $G_f$ denote $G$ composed with $f$, i.e.
$$G_f: \mathbb R \to {\on{Imm}} \to L^2_{{\on{sym}}}(T{\on{Imm}};\mathbb R).$$
Remember that the covariant derivative on ${\on{Imm}}$ that has been introduced in
section~\ref{sh:cov} is torsion-free so that one has
$$\nabla_{\partial_t}f_s - \nabla_{\partial_s}f_t=Tf.[\partial_t,\partial_s]+\on{Tor}(f_t,f_s) = 0.$$
Thus the first variation of the energy of the curves is
\begin{align*}
\partial_s \frac12 \int_0^1 G_f(f_t, f_t) dt &=
\frac12 \int_0^1 (\nabla_{\partial_s} G_f)(f_t, f_t)
+ \int_0^1 G_f(\nabla_{\partial_s} f_t, f_t) dt
\\&=
\frac12 \int_0^1 (\nabla_{f_s} G)(f_t, f_t)
+ \int_0^1 G_f(\nabla_{\partial_t} f_s, f_t) dt
\\&=
\frac12 \int_0^1 (\nabla_{f_s} G)(f_t, f_t) dt
+ \int_0^1 \partial_t\ G_f(f_s,f_t) dt \\&\qquad
- \int_0^1 (\nabla_{f_t} G)(f_s, f_t) dt
- \int_0^1 G_f(f_s,\nabla_{\partial_t} f_t) dt
\\&=
\int_0^1 G\Big(f_s,\frac12 H(f_t,f_t)+0-K(f_t,f_t)-\nabla_{\partial_t} f_t\Big) dt.
\end{align*}
If $f(0,\cdot,\cdot)$ is energy-minimizing, then one has at $s=0$ that
\begin{equation*}
\frac12 H(f_t,f_t)-K(f_t,f_t)-\nabla_{\partial_t} f_t =0. \qedhere
\end{equation*}
\end{proof}
\subsection[Geodesic equation on immersions]%
{Geodesic equation on immersions in terms of the momentum}\label{sh:gemo}
In the previous section the geodesic equation for the velocity $f_t$ has been derived.
In many applications it is more convenient to formulate the geodesic equation as an equation
for the momentum $G(f_t,\cdot) \in T^*_f{\on{Imm}}$.
$G(f_t,\cdot)$ is an element of the \emph{smooth
cotangent bundle}, also called \emph{smooth dual}, which is given by
$$G(T{\on{Imm}}) := \coprod_{f \in {\on{Imm}}} G_f(T_f{\on{Imm}}) =
\coprod_{f \in {\on{Imm}}} \{ G_f(h,\cdot): h \in T_f{\on{Imm}}\} \subset T^*{\on{Imm}}. $$
It is strictly smaller than $T^*{\on{Imm}}$ since at every $f \in {\on{Imm}}$
the metric $G_f: T_f{\on{Imm}} \to T^*_f{\on{Imm}}$ is injective but not surjective. It is called
smooth since it does not contain distributional sections of $f^*TN$, whereas $T_f^*{\on{Imm}}$ does.
\begin{thm*}
The geodesic equation for the momentum $p \in T^*{\on{Imm}}$ is given by
\begin{equation*}
\left\{\begin{aligned}
p &= G(f_t, \cdot) \\
\nabla_{\partial_t} p &= \frac12 G_f\big( H(f_t,f_t),\cdot\big),
\end{aligned}\right.
\end{equation*}
where $H$ is the metric gradient defined in section~\ref{sh:me}
and $\nabla$ is the covariant derivative action on mappings into $T^*{\on{Imm}}$ as
defined in section~\ref{sh:cov}.
\end{thm*}
\begin{proof}
Let $G_f$ denote $G$ composed with the path $f:\mathbb R\to{\on{Imm}}$, i.e.
$$G_f: \mathbb R \to {\on{Imm}} \to L^2_{{\on{sym}}}(T{\on{Imm}};\mathbb R).$$
Then one has
\begin{align*}
\nabla_{\partial_t} p &= \nabla_{\partial_t} \big(G_f(f_t,\cdot)\big) =
(\nabla_{\partial_t}G_f)(f_t,\cdot) + G_f(\nabla_{\partial_t} f_t,\cdot) \\&=
(\nabla_{f_t}G)(f_t,\cdot) + G_f\Big(\frac12 H(f_t,f_t) - K(f_t,f_t),\cdot\Big) \\&=
G_f\big(K(f_t,f_t),\cdot\big)+ G_f\Big(\frac12 H(f_t,f_t) - K(f_t,f_t),\cdot\Big) \qedhere
\end{align*}
\end{proof}
This equation is equivalent to \emph{Hamilton's equation} restricted to the smooth cotangent bundle:
\begin{equation*}
\left\{\begin{aligned}
p &= G(f_t, \cdot) \\
p_t &= (\on{grad}^{\omega} E)(p).
\end{aligned}\right.
\end{equation*}
Here $\omega$ denotes the restriction of the canonical symplectic form on $T^*{\on{Imm}}$ to the smooth
cotangent bundle and $E$ is the Hamiltonian
$$E: G(T{\on{Imm}}) \to \mathbb R, \quad E(p) = G^{-1}(p,p)$$
which is only defined on the smooth cotangent bundle.
\subsection{Shape space}\label{sh:sh}
${\on{Diff}}(M)$ acts smoothly on ${\on{Imm}}(M,N)$ and ${\on{Emb}}(M,N)$
by composition from the right.
For ${\on{Imm}}$, the action is given by the mapping
$${\on{Imm}}(M,N) \times {\on{Diff}}(M) \to {\on{Imm}}(M,N), \qquad (f,\varphi) \mapsto r(f,\varphi) = r^\varphi(f)= f \circ \varphi.$$
The tangent prolongation of this group action is given by the mapping
$$T{\on{Imm}}(M,N) \times {\on{Diff}}(M) \to T{\on{Imm}}(M,N), \qquad (h,\varphi) \mapsto Tr^\varphi(h) = h \circ \varphi.$$
\emph{Shape space} is defined as the orbit space with respect to this action.
That means that in shape space, two mappings differing only in their parametrization
will be regarded the same.
\begin{thm*}
Let $M$ be compact and of dimension $\leq n$.
Then ${\on{Emb}}(M,N)$ is the total space of a smooth principal fiber bundle
with structure group ${\on{Diff}}(M)$, whose base manifold
is a Hausdorff smooth Fr\'echet manifold denoted by
$$B_{e}(M,N) = {\on{Emb}}(M,N)/{\on{Diff}}(M).$$
However, the space
$$B_i(M,N) = {\on{Imm}}(M,N)/{\on{Diff}}(M)$$
is not a smooth manifold, but has singularities of orbifold type:
Locally, it looks like a finite dimensional orbifold times an infinite dimensional
Fr\'echet space.
\end{thm*}
The proof for immersions can be found in \cite{Michor40} and
the one for embeddings in \cite[section~44.1]{MichorG}.
As with immersions and embeddings, the notation $B_i, B_e$ will be used
when it is clear that $M$ and $N$ are the domain and target of the mappings.
\subsection{Riemannian metrics on shape space}\label{sh:rish}
We start with a metric $G$ on ${\on{Imm}}$.
The mapping $\pi:{\on{Imm}} \rightarrow B_i$ is a submersion of smooth manifolds,
that is, $T\pi:T{\on{Imm}} \rightarrow TB_i$ is surjective.
$$V=V(\pi):=\on{ker}(T\pi) \subset T{\on{Imm}}$$
is called the {\it vertical subbundle}.
The {\it horizontal subbundle}
is the $G$-orthogonal subspace of $V$:
$${\on{Hor}}={\on{Hor}}(\pi,G):=V(\pi)^\bot \subset T{\on{Imm}}.$$
It need not be a complement to $V$ (recall that the metric is weak; the complement could be in a
suitable completion of the tangent space).
For all metrics in this paper it will turn out to be a complement, however.
Then any vector $h \in T{\on{Imm}}$ can be decomposed uniquely in vertical and horizontal components as
$$h=h^{\on{ver}}+h^{{\on{hor}}}.$$
This definition extends to the cotangent bundle as follows:
An element of $T^*{\on{Imm}}$ is called horizontal when it annihilates all vertical vectors,
and vertical when it annihilates all horizontal vectors.
In the setting described so far, the mapping
\begin{equation*}
T_f \pi|_{{\on{Hor}}_f}:{\on{Hor}}_f\rightarrow T_{\pi(f)}B_i
\end{equation*}
is an isomorphism of vector spaces for all $f\in {\on{Imm}}$.
This isomorphism will be used to describe the tangent space to $B_i$.
If both ${\on{Imm}}$ and $B_i$ are Riemannian manifolds and
if this isomorphism is also an isometry for all $f\in {\on{Imm}}$,
then $\pi$ is called a {\it Riemannian submersion}.
In that case, the metric $G$ on ${\on{Imm}}$ is ${\on{Diff}}(M)$-invariant.
This means that $G=(r^\varphi)^* G$ for all $\varphi \in {\on{Diff}}(M)$, where
$r^\varphi$ denotes the right action of $\varphi$ on ${\on{Imm}}$ that was described in section~\ref{sh:sh}.
This condition can be spelled out in more details using the definition of $r^\varphi$ as follows:
\begin{align*}
G_f(h,k)=\big((r^\varphi)^* G\big)(h,k)
=G_{r^\varphi(f)}\big(Tr^\varphi(h),Tr^\varphi(k)\big)
=G_{f \circ \varphi}(h \circ \varphi,k \circ \varphi).
\end{align*}
The following theorem establishes the converse statement:
\begin{thm*}
Given a ${\on{Diff}}(M)$-invariant Riemannian metric on ${\on{Imm}}$,
there is a unique Riemannian metric on the quotient space $B_i$ such that the
quotient map $\pi:{\on{Imm}} \to B_i$ is a Riemannian submersion.
\end{thm*}
\begin{proof}
If the horizontal bundle ${\on{Hor}}_f$ is a complement to $V_f$ then $T_f\pi:{\on{Hor}}_f \to T_{\pi(f)}B_i$
is an isomorphism (off the orbifold singularities of $B_i$) and we can induce the metric on
$T_{\pi(f)}B_i$ which is independent of the choice of $f$ in the fiber over $\pi(f)$ by the the
${\on{Diff}}(M)$-invariance of the metric. If it is not a complement one has to consider the metric
quotient norm. See for example \cite[section 3]{Michor98}.
\end{proof}
\begin{ass*}
It will always be assumed that a ${\on{Diff}}(M)$-invariant metric $G$ on ${\on{Imm}}(M,N)$ is given
and that shape space $B_i$ is endowed with the unique metric such that
the quotient map is a Riemannian submersion.
\end{ass*}
\subsection{Riemannian submersions and geodesics}\label{sh:sub}
It follows from the general theory of Riemannian submersions that horizontal geodesics
in the top space correspond nicely to geodesics in the quotient space:
\begin{thm*}
Let $c:[0,1]\rightarrow {\on{Imm}}$ be a geodesic.
\begin{enumerate}
\item If $c'(t)$ is horizontal at one $t$, then it is horizontal at all $t$.
\item If $c'(t)$ is horizontal then $\pi \circ c$ is a geodesic in $B_i$.
\item If every curve in $B_i$ can be lifted to a horizontal curve in ${\on{Imm}}$,
then there is a one-to-one correspondence between curves in $B_i$ and horizontal curves in ${\on{Imm}}$.
This implies that instead of solving the geodesic equation on $B_i$ one can equivalently solve
the equation for horizontal geodesics in ${\on{Imm}}$.
\end{enumerate}
\end{thm*}
See \cite[section~26]{MichorH} for the proof.
\subsection{Geodesic equation on shape space}\label{sh:gesh}
Theorem~\ref{sh:sub} applied to the Riemannian submersion $\pi: {\on{Imm}} \to B_i$ yields:
\begin{thm*}
Assuming that every curve in $B_i$ can be lifted to a horizontal curve in ${\on{Imm}}$,
the geodesic equation on shape space is equivalent to
\begin{equation}\label{sh:gesh:eq1}
\left\{\begin{aligned}
f_t&=f_t^{{\on{hor}}}\in {\on{Hor}} \\
(\nabla_{\partial_t}f_t)^{{\on{hor}}} &= \Big(\frac12 H(f_t,f_t)-K(f_t,f_t)\Big)^{{\on{hor}}},
\end{aligned}\right.
\end{equation}
where $f$ is a horizontal curve in ${\on{Imm}}$, where $H,K$ are the metric gradients
defined in section~\ref{sh:me} and where $\nabla$ is the covariant derivative defined
in section~\ref{sh:cov}.
\end{thm*}
This is a consequence of the ${\on{Diff}}(M)$-invariance of the metric $G$
and the conservation of the reaparametrization momentum.
A general proof can be found in \cite[section~3.14]{Harms2010}.
It will be shown in section~\ref{so:ho2} that curves in $B_i$
can be lifted to horizontal curves in ${\on{Imm}}$ for the very general class of
Sobolev type metrics. Thus all assumptions and conclusions of the theorem hold.
\subsection[Geodesic equation on shape space]%
{Geodesic equation on shape space in terms of the momentum}\label{sh:geshmo}
As in the previous section, theorem~\ref{sh:sub} will be applied to the Riemannian submersion
$\pi: {\on{Imm}} \to B_i$.
But this time, the formulation of the geodesic equation in terms of the momentum will be used,
see section~\ref{sh:gemo}.
As will be seen in section~\ref{so:geshmo}, this is the most convenient formulation
of the geodesic equation for Sobolev-type metrics.
\begin{thm*}
Assuming that every curve in $B_i$ can be lifted to a horizontal curve in ${\on{Imm}}$,
the geodesic equation on shape space is equivalent to the set of equations
\begin{equation*}
\left\{\begin{aligned}
p &= G_f(f_t,\cdot) \in {\on{Hor}} \subset T^*{\on{Imm}}, \\
(\nabla_{\partial_t}p)^{{\on{hor}}} &= \frac12 G_f\big(H(f_t,f_t),\cdot)^{{\on{hor}}}.
\end{aligned}\right.
\end{equation*}
Here $f$ is a curve in ${\on{Imm}}$,
$H$ is the metric gradient defined in section~\ref{sh:me},
and $\nabla$ is the covariant derivative defined in section~\ref{sh:cov}.
$f$ is horizontal because $p$ is horizontal.
\end{thm*}
\section{Variational formulas}\label{va}
Recall that many operators like
$$g=f^*\overline{g}, \quad S=S^f, \quad {\on{vol}}(g), \quad \nabla=\nabla^g, \quad \Delta=\Delta^g, \quad \ldots$$
implicitly depend on the immersion $f$. In this section their derivative
with respect to $f$ which is called their \emph{first variation} will be calculated .
These formulas will be used to calculate the metric gradients that are needed
for the geodesic equation.
This section is based on \cite{Michor118}, see also \cite{Harms2010}.
Some but not all of the formulas were known before \cite{Besse2008, Michor102}.
More variational formulas can be found in \cite{Besse2008,Verpoort2008,Bauer2010}.
\subsection{Paths of immersions}\label{va:pa}
All of the differential-geometric concepts introduced in section \ref{no}
can be recast for a path of immersions instead of a fixed immersion.
This allows to study variations of immersions.
So let $f:\mathbb R \to \on{Imm}(M,N)$ be a path of immersions. By convenient calculus
\cite{MichorG}, $f$ can equivalently be seen as $f:\mathbb R \times M \to N$
such that $f(t,\cdot)$ is an immersion for each $t$.
The bundles over $M$ can be replaced by bundles over $\mathbb R \times M$:
\begin{equation*}\xymatrix{
\on{pr}_2^* T^r_s M \ar[d] &
\on{pr}_2^* T^r_s M \otimes f^*TN \ar[d] &
{\on{Nor}}(f) \ar[d]\\
\mathbb R \times M & \mathbb R \times M & \mathbb R \times M
}\end{equation*}
Here $\on{pr}_2$ denotes the projection $\on{pr}_2:\mathbb R \times M \to M$.
The covariant derivative $\nabla_Z h$ is now defined for vector fields $Z$ on $\mathbb R \times M$
and sections $h$ of the above bundles.
The vector fields $(\partial_t, 0_M)$ and $(0_{\mathbb R}, X)$, where $X$ is a vector field on $M$, are of
special importance.
In later sections they will be identified with $\partial_t$ and $X$
whenever this does not pose any problems.
Let
$$\on{ins}_t : M \to \mathbb R \times M, \qquad x \mapsto (t,x) .$$
Then by property~\ref{no:co:prop5} from section~\ref{no:co} one has for vector fields $X,Y$ on $M$
\begin{align*}
\nabla_X Tf(t,\cdot).Y &= \nabla_X T(f \circ \on{ins}_t) \circ Y = \nabla_X Tf \circ T\on{ins}_t \circ Y
\\&= \nabla_X Tf \circ (0_\mathbb R,Y) \circ \on{ins}_t
= \nabla_{T\on{ins}_t \circ X} Tf \circ (0_\mathbb R,Y)\\&
= \big(\nabla_{(0_\mathbb R,X)} Tf \circ (0_\mathbb R,Y)\big) \circ \on{ins}_t .
\end{align*}
This shows that one can recover the static situation at $t$ by using vector fields on $\mathbb R \times M$
with vanishing $\mathbb R$-component and evaluating at $t$.
\subsection{Directional derivatives of functions}
The following ways to denote directional derivatives of functions will be used, in particular in
infinite dimensions.
Given a function $F(x,y)$ for instance,
$$ D_{(x,h)}F \text{ will be written as a shorthand for } \partial_t|_0 F(x+th,y).$$
Here $(x,h)$ in the subscript denotes the tangent vector with foot point $x$ and direction $h$.
If $F$ takes values in some linear space, this linear space and its tangent space will be identified.
\subsection{Setting for first variations}\label{va:se}
In all of this chapter, let $f$ be an immersion and $f_t \in T_f{\on{Imm}}$ a tangent vector to $f$.
The reason for calling the tangent vector $f_t$ is that in calculations
it will often be the derivative of a curve of immersions through $f$.
Using the same symbol $f$ for the fixed immersion
and for the path of immersions through it, one has in fact that
$$D_{(f,f_t)} F = \partial_t F(f(t)).$$
\subsection{Variation of equivariant tensor fields}\label{va:ta}
Let the mapping $$F:{\on{Imm}}(M,N) \to \Gamma(T^r_s M)$$
take values in some space of tensor fields over $M$,
or more generally in any natural bundle over $M$, see \cite{MichorF}.
\begin{lem*}
If $F$ is equivariant
with respect to pullbacks by
diffeomorphisms of $M$, i.e.
$$F(f)=(\varphi^* F)(f)=\varphi^* \Big(F\big((\varphi^{-1})^*f\big)\Big) $$
for all $\varphi \in \on{Diff}(M)$ and $f \in {\on{Imm}}(M,N)$,
then the tangential variation of $F$ is its Lie-derivative:
\begin{align*}
D_{(f,Tf.f_t^\top)} F&=
\partial_t|_0 F\Big(f \circ Fl^{f_t^\top}_t\Big)=
\partial_t|_0 F\Big((Fl^{f_t^\top}_t)^* f\Big)\\&=
\partial_t|_0 \Big(Fl_t^{f_t^\top}\Big)^* \big(F(f)\big) = \mathcal L_{f_t^\top}\big(F(f)\big).
\end{align*}
\end{lem*}
This allows us to calculate the tangential variation of the pullback metric and
the volume density, for example.
\subsection{Variation of the metric}\label{va:me}
\begin{lem*}
The differential of the pullback metric
\begin{equation*}\left\{ \begin{array}{ccl}
{\on{Imm}} &\to &\Gamma(S^2_{>0} T^*M),\\
f &\mapsto &g=f^*\overline{g}
\end{array}\right.\end{equation*}
is given by
\begin{align*}
D_{(f,f_t)} g&= 2\on{Sym}\overline{g}(\nabla f_t,Tf) = -2 \overline{g}(f_t^\bot,S)+2 \on{Sym} \nabla (f_t^\top)^\flat
\\& = -2 \overline{g}(f_t^\bot,S)+ \mathcal L_{f_t^\top} g.
\end{align*}
\end{lem*}
Here $\on{Sym}$ denotes the symmetric part of the tensor field $C$ of type $\left(\begin{smallmatrix}0\\2\end{smallmatrix}\right)$ given by
$$\big(\on{Sym}(C)\big)(X,Y):=\frac12\big(C(X,Y)+C(Y,X)\big).$$
\begin{proof}
Let $f:\mathbb R \times M \to N$ be a path of immersions. Swapping covariant derivatives as in
section~\ref{no:sw}, formula \eqref{no:sw:to2} one gets
\begin{align*}
\partial_t\big(g(X,Y)\big) &= \partial_t\big( \overline{g}( Tf.X,Tf.Y ) \big)
= \overline{g}( \nabla_{\partial_t}Tf.X,Tf.Y ) + \overline{g}( Tf.X, \nabla_{\partial_t}Tf.Y )\\
&=\overline{g}( \nabla_X f_t,Tf.Y ) + \overline{g}( Tf.X, \nabla_Y f_t ) = \big(2 \on{Sym}\overline{g}(\nabla f_t,Tf)\big)(X,Y).
\end{align*}
Splitting $f_t$ into its normal and tangential part yields
\begin{align*}
2 \on{Sym}\overline{g}(\nabla f_t,Tf) &=
2 \on{Sym}\overline{g}(\nabla f_t^\bot + \nabla Tf.f_t^\top,Tf) \\&=
-2 \on{Sym}\overline{g}(f_t^\bot,\nabla Tf)+2 \on{Sym} g(\nabla f_t^\top,\cdot) \\&=
-2 \overline{g}(f_t^\bot,S)+2 \on{Sym} \nabla (f_t^\top)^\flat .
\end{align*}
Finally the relation
$$D_{(f,Tf.f_t^\top)} g = 2 \on{Sym} \nabla (f_t^\top)^\flat = \mathcal L_{f_t^\top} g $$
follows either from the equivariance of $g$
with respect to pullbacks by diffeomorphisms (see section~\ref{va:ta}) or directly from
\begin{align*}
(\mathcal L_Xg)(Y,Z)&=
\mathcal L_X\big(g(Y,Z)\big)-g(\mathcal L_XY,Z)-g(Y,\mathcal L_XZ)\\&=
\nabla_X\big(g(Y,Z)\big)-g(\nabla_XY-\nabla_YX,Z)-g(Y,\nabla_XZ-\nabla_ZX)\\&=
g(\nabla_YX,Z)+g(Y,\nabla_ZX)=
(\nabla_YX)^\flat(Z)+(\nabla_ZX)^\flat(Y)\\&=
(\nabla_YX^\flat)(Z)+(\nabla_ZX^\flat)(Y)=2 \on{Sym} \big(\nabla(X^\flat)\big)(Y,Z).\qedhere
\end{align*}
\end{proof}
\subsection{Variation of the inverse of the metric}\label{va:in}
\begin{lem*}
The differential of the inverse of the pullback metric
\begin{equation*}\left\{ \begin{array}{ccl}
{\on{Imm}} &\to &\Gamma\big(L(T^*M,TM)\big),\\
f &\mapsto &g^{-1}=(f^*\overline{g})^{-1}
\end{array}\right.\end{equation*}
is given by
\begin{align*}
D_{(f,f_t)} g^{-1} = D_{(f,f_t)} (f^*\overline{g})^{-1} =2 \overline{g}(f_t^\bot, g^{-1} S g^{-1}) + \mathcal L_{f_t^\top}(g^{-1})
\end{align*}
\end{lem*}
\begin{proof}
\begin{align*}
\partial_t g^{-1} &= - g^{-1} (\partial_t g ) g^{-1}
= -g^{-1} \big(-2 \overline{g}(f_t^\bot,S)+ \mathcal L_{f_t^\top} g\big) g^{-1} \\
& = 2 g^{-1} \overline{g}(f_t^\bot,S) g^{-1} -g^{-1} (\mathcal L_{f_t^\top} g) g^{-1}
= 2 \overline{g}(f_t^\bot,g^{-1} S g^{-1})+ \mathcal L_{f_t^\top} (g^{-1}) \qedhere
\end{align*}
\end{proof}
\subsection{Variation of the volume density}\label{va:vo}
\begin{lem*}
The differential of the volume density
\begin{equation*}
\left\{ \begin{array}{ccl}
{\on{Imm}} &\to &{\on{Vol}}(M),\\
f &\mapsto &{\on{vol}}(g)={\on{vol}}(f^*\overline{g})
\end{array}\right.\end{equation*}
is given by
\begin{equation*}
D_{(f,f_t)} {\on{vol}}(g) =
\on{Tr}^g\big(\overline{g}(\nabla f_t,Tf)\big) {\on{vol}}(g)=
\Big(\on{div}^{g}(f_t^{\top})-\overline{g}\big(f_t^{\bot},\on{Tr}^g(S)\big)\Big) {\on{vol}}(g).
\end{equation*}
\end{lem*}
\begin{proof}
Let $g(t) \in \Gamma(S^2_{>0}T^*M)$ be any curve of Riemannian metrics. Then
$$\partial_t {\on{vol}}(g)=\frac{1}{2}\on{Tr}(g^{-1}.\partial_t g){\on{vol}}(g).$$
This follows from the formula for ${\on{vol}}(g)$ in a local oriented chart
$(u^1,\ldots u^m)$ on $M$:
\begin{align*}
\partial_t{\on{vol}}(g)&=\partial_t \sqrt{\det( (g_{ij})_{ij})}\ du^1\wedge\cdots\wedge du^{m}\\
&=\frac{1}{2\sqrt{\det ((g_{ij})_{ij})}}\on{Tr}(\on{adj}(g) \partial_t g)\
du^1\wedge\cdots\wedge du^{m}\\
&=\frac{1}{2\sqrt{\det ((g_{ij})_{ij})}}\on{Tr}(\det((g_{ij})_{ij})g^{-1}\partial_t g)\
du^1\wedge\cdots\wedge du^{m}\\
&=\frac{1}{2}\on{Tr}(g^{-1}.\partial_t g){\on{vol}}(g)
\end{align*}
Now one can set $g = f^*\overline{g}$ and plug in the formula
$$\partial_t g=\partial_t (f^*\overline{g})=2\on{Sym}\overline{g}(\nabla f_t,Tf)$$
from \ref{va:me}.
This immediately proves the first formula:
\begin{align*}
\partial_t {\on{vol}}(g)&=\frac12 \on{Tr}\big(g^{-1}.2\on{Sym}\overline{g}(\nabla f_t,Tf) \big)
=\on{Tr}^g\big(\overline{g}(\nabla f_t,Tf) \big).
\end{align*}
Expanding this further yields the second formula:
\begin{align*}
\partial_t {\on{vol}}(g)&=\on{Tr}^g\Big(\nabla\overline{g}( f_t,Tf)-\overline{g}( f_t,\nabla Tf) \Big)\\&
=\on{Tr}^g\Big(\nabla\overline{g}( f_t,Tf)-\overline{g}( f_t,S) \Big)=-\nabla^*\overline{g}( f_t,Tf)-\overline{g}\big(f_t,\on{Tr}^g(S)\big)\\&
=-\nabla^*\big((f_t^{\top})^{\flat}\big)-\overline{g}\big(f_t^{\bot},\on{Tr}^g(S)\big)
=\on{div}(f_t^{\top})-\overline{g}\big(f_t^{\bot},\on{Tr}^g(S)\big).
\end{align*}
Here it has been used that
$$\nabla Tf = S \quad \text{and} \quad
\on{div}(f_t^\top) = \on{Tr}(\nabla f_t^\top)= \on{Tr}^g\big((\nabla f_t^\top)^\flat\big)
= -\nabla^*\big((f_t^\top)^\flat\big).$$
Note that by \ref{va:ta}, the formula for the tangential variation
would have followed also from the equivariance of the volume form with respect to pullbacks by
diffeomorphisms.
\end{proof}
\subsection{Variation of the covariant derivative}\label{va:co}
In this section, let $\nabla=\nabla^g=\nabla^{f^*\overline{g}}$ be the
Levi-Civita covariant derivative acting on vector fields on $M$.
Since any two covariant derivatives on $M$ differ by a tensor field,
the first variation of $\nabla^{f^*\overline{g}}$ is tensorial. It is given by the
tensor field $D_{(f,f_t)} \nabla^{f^*\overline{g}} \in \Gamma(T^1_2 M)$.
\begin{lem*}
The tensor field $D_{(f,f_t)}\nabla^{f^*\overline{g}}$ is determined by the following relation
holding for vector fields $X,Y,Z$ on $M$:
\begin{multline*}
g\big((D_{(f,f_t)} \nabla)(X, Y),Z\big) =
\frac12 (\nabla D_{(f,f_t)} g)\big( X \otimes Y \otimes Z
+ Y \otimes X \otimes Z - Z \otimes X \otimes Y \big)
\end{multline*}
\end{lem*}
\begin{proof}
The defining formula for the covariant derivative is
\begin{align*}
g(\nabla_X Y,Z)&= \frac12 \Big[ Xg(Y,Z)+Yg(Z,X)-Zg(X,Y)\\&\qquad
-g(X,[Y,Z])+g(Y,[Z,X])+g(Z,[X,Y]) \Big].
\end{align*}
Taking the derivative $D_{(f,f_t)}$ yields
\begin{multline*}
(D_{(f,f_t)}g)(\nabla_X Y,Z)+g\big((D_{(f,f_t)}\nabla)(X, Y),Z\big)\\
\begin{aligned}
=\frac12 \Big[ & X\big((D_{(f,f_t)}g)(Y,Z)\big)+Y\big((D_{(f,f_t)}g)(Z,X)\big)-Z\big((D_{(f,f_t)}g)(X,Y)\big)\\&
-(D_{(f,f_t)}g)(X,[Y,Z])+(D_{(f,f_t)}g)(Y,[Z,X])+(D_{(f,f_t)}g)(Z,[X,Y]) \Big].
\end{aligned}
\end{multline*}
Then the result follows by replacing all Lie brackets in the above formula by covariant derivatives using
$[X,Y]=\nabla_X Y - \nabla_Y X$
and by expanding all terms of the form $X\big((D_{(f,f_t}g)(Y,Z)\big)$ using
\begin{align*}
&X\big((D_{(f,f_t)}g)(Y,Z)\big)=\\&\qquad\qquad
(\nabla_X D_{(f,f_t)}g)(Y,Z)
+(D_{(f,f_t)}g)(\nabla_X Y,Z)
+(D_{(f,f_t)}g)(Y,\nabla_X Z).
\qedhere\end{align*}
\end{proof}
\subsection{Variation of the Laplacian}\label{va:la}
The Laplacian as defined in section \ref{no:la}
can be seen as a smooth section of the bundle $L(T{\on{Imm}};T{\on{Imm}})$ over ${\on{Imm}}$ since
for every $f \in {\on{Imm}}$ it is a mapping
$$\Delta^{f^*\overline{g}}:T_f{\on{Imm}} \to T_f{\on{Imm}}.$$
The right way to define a first variation is to
use the covariant derivative defined in section~\ref{sh:cov}.
\begin{lem*}
For $\Delta \in \Gamma\big(L(T{\on{Imm}};T{\on{Imm}})\big)$, $f \in {\on{Imm}}$ and $f_t,h \in T_f{\on{Imm}}$ one has
\begin{align*}
(\nabla_{f_t} \Delta)(h) &=
\on{Tr}\big(g^{-1}.(D_{(f,f_t)}g).g^{-1} \nabla^2 h\big)
-\nabla_{\big(\nabla^*(D_{(f,f_t)} g)+\frac12 d\on{Tr}^g(D_{(f,f_t)}g)\big)^\sharp}h \\&\qquad
+\nabla^*\big(R^{\overline{g}}(f_t,Tf)h\big)
-\on{Tr}^g\Big( R^{\overline{g}}(f_t,Tf)\nabla h \Big).
\end{align*}
\end{lem*}
\begin{proof}
Let $f$ be a curve of immersions and $h$ a vector field along $f$. One has
$$\Delta : {\on{Imm}} \to L(T{\on{Imm}};T{\on{Imm}}), \quad \Delta \circ f = \Delta^{f^*\overline{g}} : \mathbb R \to {\on{Imm}} \to L(T{\on{Imm}};T{\on{Imm}}). $$
Using property~\ref{no:co}.5 one gets
\begin{align*}
(\nabla_{f_t} \Delta)(h) &=
\big(\nabla_{\partial_t} (\Delta \circ f)\big)(h)=
\nabla_{\partial_t} \Delta h - \Delta \nabla_{\partial_t} h\\&=
-\nabla_{\partial_t} \on{Tr}^g(\nabla^2 h) - \Delta \nabla_{\partial_t} h \\&=
\on{Tr}\big(g^{-1} (D_{(f,f_t)} g) g^{-1} \nabla^2 h\big)- \on{Tr}^g(\nabla_{\partial_t} \nabla^2 h)
- \Delta(\nabla_{\partial_t} h).
\end{align*}
The term $\on{Tr}^g(\nabla_{\partial_t} \nabla^2 h)$ will be treated further.
Let $X,Y$ be vector fields on $M$ that are constant in time.
When they are seen as vector fields on $\mathbb R \times M$ then $\nabla_{\partial_t}X=\nabla_{\partial_t}Y=0$.
Using the formulas from section~\ref{no:sw} to swap covariant derivatives one gets
\begin{align*}
&(\nabla_{\partial_t}\nabla^2 h)(X,Y)=
\nabla_{\partial_t}(\nabla_X\nabla_Y h-\nabla_{\nabla_X Y}h)
\\&\qquad=
\nabla_X\nabla_{\partial_t}\nabla_Y h+R^{\overline{g}}(f_t,Tf.X)\nabla_Y h-\nabla_{\partial_t}\nabla_{\nabla_X Y}h
\\&\qquad=
\nabla_X\nabla_Y\nabla_{\partial_t} h+\nabla_X\big(R^{\overline{g}}(f_t,Tf.Y)h\big)
+R^{\overline{g}}(f_t,Tf.X)\nabla_Y h\\&\qquad\qquad
-\nabla_{\nabla_X Y}\nabla_{\partial_t}h-\nabla_{[\partial_t,\nabla_X Y]}h
-R^{\overline{g}}(f_t,Tf.\nabla_X Y)h.
\end{align*}
The Lie bracket is
\begin{align*}
[\partial_t,\nabla^{f^*\overline{g}}_X Y] = (D_{(f,f_t)}\nabla)(X,Y)
\end{align*}
since (now without the slight abuse of notation)
\begin{align*}
[(\partial_t,0_M),(0_\mathbb R,\nabla^{f^*\overline{g}}_X Y)]
&=\partial_s|_0\ TFl_{-s}^{(\partial_t,0_M)} \circ \nabla_X Y \circ Fl_s^{(\partial_t,0_M)}
\\&= \big(0_\mathbb R,(D_{(f,f_t)}\nabla)(X,Y)\big).
\end{align*}
Therefore
\begin{align*}
&(\nabla_{\partial_t}\nabla^2 h)(X,Y)=\\&\qquad=
(\nabla^2\nabla_{\partial_t} h)(X,Y)+\nabla_X\big(R^{\overline{g}}(f_t,Tf.Y)h\big)
+R^{\overline{g}}(f_t,Tf.X)\nabla_Y h\\&\qquad\qquad
-\nabla_{(D_{(f,f_t)}\nabla)(X,Y)}h-R^{\overline{g}}(f_t,Tf.\nabla_X Y)h
\\&\qquad=
(\nabla^2\nabla_{\partial_t} h)(X,Y)
+(\nabla_{Tf.X} R^{\overline{g}})(f_t,Tf.Y)h
+R^{\overline{g}}(\nabla_X f_t,Tf.Y)h\\&\qquad\qquad
+R^{\overline{g}}(f_t,\nabla_X Tf.Y)h
+R^{\overline{g}}(f_t,Tf.Y)\nabla_X h
+R^{\overline{g}}(f_t,Tf.X)\nabla_Y h\\&\qquad\qquad
-\nabla_{(D_{(f,f_t)}\nabla)(X,Y)}h-R^{\overline{g}}(f_t,Tf.\nabla_X Y)h
\\&\qquad=
(\nabla^2\nabla_{\partial_t} h)(X,Y)
+(\nabla_{Tf.X} R^{\overline{g}})(f_t,Tf.Y)h
+R^{\overline{g}}(\nabla_X f_t,Tf.Y)h\\&\qquad\qquad
+R^{\overline{g}}\big(f_t,(\nabla Tf)(X,Y)\big)h
+R^{\overline{g}}(f_t,Tf.Y)\nabla_X h
+R^{\overline{g}}(f_t,Tf.X)\nabla_Y h\\&\qquad\qquad
-\nabla_{(D_{(f,f_t)}\nabla)(X,Y)}h
\\&\qquad=
(\nabla^2\nabla_{\partial_t} h)(X,Y)
+ \nabla_X\big(R^{\overline{g}}(f_t,Tf.Y)h\big)
+R^{\overline{g}}(f_t,Tf.X)\nabla_Y h \\&\qquad\qquad
-\nabla_{(D_{(f,f_t)}\nabla)(X,Y)}h
\end{align*}
Putting together all terms one obtains
\begin{align*}
(\nabla_{f_t} \Delta)(h) &=
\on{Tr}\big(g^{-1} (D_{(f,f_t)} g) g^{-1} \nabla^2 h\big)
-\on{Tr}^g\Big( \nabla\big(R^{\overline{g}}(f_t,Tf)h\big) \Big)\\&\qquad
-\on{Tr}^g\Big( R^{\overline{g}}(f_t,Tf)\nabla h \Big)
+\nabla_{\on{Tr}^g(D_{(f,f_t)}\nabla)}h\\&=
\on{Tr}\big(g^{-1} (D_{(f,f_t)} g) g^{-1} \nabla^2 h\big)
+\nabla^*\big(R^{\overline{g}}(f_t,Tf)h\big) \\&\qquad
-\on{Tr}^g\Big( R^{\overline{g}}(f_t,Tf)\nabla h \Big)
+\nabla_{\on{Tr}^g(D_{(f,f_t)}\nabla)}h.
\end{align*}
It remains to calculate $\on{Tr}^g(D_{(f,f_t)}\nabla)$.
Using the variational formula for $\nabla$ from section~\ref{va:co}
one gets for any vector field $Z$ and a $g$-orthonormal frame $s_i$
\begin{align*}
&g\big(\on{Tr}^g(D_{(f,f_t)}\nabla),Z\big) \\&\qquad=
\frac12 \sum_i (\nabla D_{(f,f_t)} g)\big( s_i \otimes s_i \otimes Z
+ s_i \otimes s_i \otimes Z - Z \otimes s_i \otimes s_i \big) \\&\qquad=
-\big(\nabla^*(D_{(f,f_t)}g)\big)(Z) - \frac12 \on{Tr}^g(\nabla_Z D_{(f,f_t)}g) \\&\qquad=
-\big(\nabla^*(D_{(f,f_t)}g)\big)(Z) - \frac12\nabla_Z \on{Tr}^g(D_{(f,f_t)}g) \\&\qquad=
-\Big(\nabla^*(D_{(f,f_t)}g) + \frac12 d \on{Tr}^g(D_{(f,f_t)}g)\Big)(Z) \\&\qquad=
-g\Big(\big(\nabla^*(D_{(f,f_t)}g) + \frac12 d \on{Tr}^g(D_{(f,f_t)}g)\big)^\sharp,Z\Big) .
\end{align*}
Therefore
\begin{equation*}\label{va:la:eq2}
\on{Tr}^g(D_{(f,f_t)}\nabla) = -\big(\nabla^*(D_{(f,f_t)}g) + \frac12 d \on{Tr}^g(D_{(f,f_t)}g)\big)^\sharp.
\qedhere
\end{equation*}
\end{proof}
\section{Sobolev-type metrics}\label{so}
\begin{ass*}
Let $P$ be a smooth section of the bundle $L(T{\on{Imm}};T{\on{Imm}})$ over ${\on{Imm}}$
such that at every $f \in {\on{Imm}}$ the operator
$$P_f:T_f{\on{Imm}} \to T_f{\on{Imm}}$$
is an elliptic pseudo differential operator that is symmetric and positive with respect to
the $H^0$-metric on ${\on{Imm}}$,
$$H^0_f(h,k) = \int_M \overline{g}(h,k){\on{vol}}(g).$$
\end{ass*}
Note that an elliptic symmetric operator is self-adjoint by \cite[26.2]{Shubin1987}.
Then $P$ induces a metric on the set of immersions, namely
$$G^P_f(h,k)=\int_M \overline{g}(P_fh,k) {\on{vol}}(g) \quad \text{for} \quad f \in {\on{Imm}}, \quad h,k \in T_f{\on{Imm}}.$$
The metric $G^P$ is positive definite since $P$ is assumed to be positive with respect to the
$H^0$-metric.
In this section, the geodesic equation on ${\on{Imm}}$ and $B_i$ for the $G^P$-metric will be calculated
in terms of the operator $P$ and it will be proven that it is well-posed under some assumptions.
\subsection{Invariance of $P$ under reparametrizations}\label{so:in}
\begin{ass*}
It will be assumed that $P$ is invariant under
the action of the reparametrization group ${\on{Diff}}(M)$ acting on ${\on{Imm}}(M,N)$, i.e.
$$P=(r^{\varphi})^* P \qquad \text{for all } \varphi \in {\on{Diff}}(M).$$
\end{ass*}
For any $f \in {\on{Imm}}$ and $\varphi \in {\on{Diff}}(M)$ this means
$$P_f = (T_fr^{\varphi})^{-1} \circ P_{f \circ \varphi} \circ T_fr^{\varphi}.$$
Applied to $h \in T_f{\on{Imm}}$ this means
$$P_f(h) \circ \varphi = P_{f \circ \varphi}(h \circ \varphi).$$
The invariance of $P$ implies that the induced metric $G^P$ is invariant
under the action of ${\on{Diff}}(M)$, too. Therefore it induces a
unique metric on $B_i$ as explained in section~\ref{sh:rish}
\subsection{The adjoint of $\nabla P$}\label{so:ad}
The following construction is needed to express the metric gradient $H$ which is part of
the geodesic equation. $H_f$ arises from the metric $G_f$ by differentiating it with
respect to its foot point $f \in {\on{Imm}}$.
Since $G$ is defined via the operator $P$, one also needs to differentiate $P_f$ with respect to its
foot point. As for the metric, this is accomplished by the covariant derivate.
For $P \in \Gamma\big(L(T{\on{Imm}};T{\on{Imm}})\big)$ and $m \in T{\on{Imm}}$ one has
$$\nabla_m P \in \Gamma\big(L(T{\on{Imm}};T{\on{Imm}})\big), \qquad \nabla P \in \Gamma\big(L(T^2{\on{Imm}};T{\on{Imm}})\big).$$
See section~\ref{sh:cov} for more details.
\begin{ass*}
It is assumed that there exists a smooth \emph{adjoint}
$$\adj{\nabla P} \in \Gamma\big(L^2(T{\on{Imm}};T{\on{Imm}})\big)$$
of $\nabla P$ in the following sense:
\begin{equation*}
\int_M \overline{g}\big((\nabla_m P)h,k\big) {\on{vol}}(g)=\int_M \overline{g}\big(m,\adj{\nabla P}(h,k)\big) {\on{vol}}(g).
\end{equation*}
\end{ass*}
The existence of the adjoint needs to be checked in each specific example,
usually by partial integration.
For the operator $P=1+A\Delta^p$, the existence of the adjoint will be proven
and explicit formulas will be calculated in sections~\ref{la:ad} and \ref{la:ge}.
\begin{lem*}
If the adjoint of $\nabla P$ exists, then its tangential part is determined
by the invariance of $P$ with respect to reparametrizations:
\begin{align*}
\adj{\nabla P}(h,k)^\top &=\big(\overline{g}(\nabla Ph,k)-\overline{g}(\nabla h,Pk)\big)^\sharp \\
&=\on{grad}^g \overline{g}(Ph,k)-\big(\overline{g}(Ph,\nabla k)+\overline{g}(\nabla h,Pk)\big)^\sharp
\end{align*}
for $f \in {\on{Imm}}, h,k \in T_f{\on{Imm}}$.
\end{lem*}
\begin{proof}
Let $X$ be a vector field on $M$. Then
\begin{align*}
(\nabla_{Tf.X} P)(h) &=
(\nabla_{\partial_t|_0} P_{f\circ Fl_t^X})(h \circ Fl_0^X) \\&=
\nabla_{\partial_t|_0}\big(P_{f\circ Fl_t^X}(h \circ Fl_t^X)\big) -
P_{f\circ Fl_0^X} \big( \nabla_{\partial_t|_0}(h \circ Fl_t^X)\big) \\&=
\nabla_{\partial_t|_0}\big(P_f(h) \circ Fl_t^X\big) -
P_f \big( \nabla_{\partial_t|_0}(h \circ Fl_t^X)\big) \\&=
\nabla_X\big(P_f(h)) - P_f \big( \nabla_X h\big)
\end{align*}
Therefore one has for $m,h,k \in T_f{\on{Imm}}$ that
\begin{align*} &
\int_M g\big(m^\top,\adj{\nabla P}(h,k)^\top \big) {\on{vol}}(g) =
\int_M \overline{g}\big(Tf.m^\top,\adj{\nabla P}(h,k)\big) {\on{vol}}(g) \\&\qquad=
\int_M \overline{g}\big((\nabla_{Tf.m^\top} P)h,k\big) {\on{vol}}(g) \\&\qquad=
\int_M \overline{g}\big(\nabla_{m^\top}(Ph)-P(\nabla_{m^\top}h),k\big) {\on{vol}}(g) \\&\qquad=
\int_M \big(\overline{g}(\nabla_{m^\top}Ph,k)-\overline{g}(\nabla_{m^\top}h,Pk)\big) {\on{vol}}(g) \\&\qquad=
\int_M g\Big(m^\top,\big(\overline{g}(\nabla Ph,k)-\overline{g}(\nabla h,Pk)\big)^\sharp\Big) {\on{vol}}(g).
\qedhere
\end{align*}
\end{proof}
\subsection{Metric gradients}\label{so:me}
As explained in section~\ref{sh:ge}, the geodesic equation can be expressed
in terms of the metric gradients $H$ and $K$.
These gradients will be computed now. We shall use that $P_f$ is invertible on the space of smooth
sections. This follows because $P_f$ is an elliptic, self-adjoint, and positive operator, see the
beginning of the proof of theorem \ref{so:we} for a detailed argument.
\begin{lem*}
If $\adj{\nabla P}$ exists, then also $H$ and $K$ exist and are given by
\begin{align*}
K_f(h,m)&=P_f^{-1}\Big((\nabla_m P)h+\on{Tr}^g\big(\overline{g}(\nabla m,Tf)\big).Ph\Big) \\
H_f(h,k)&=P_f^{-1}\Big(\adj{\nabla P}(h,k)^\bot-Tf.\big(\overline{g}(Ph,\nabla k)+\overline{g}(\nabla h,Pk)\big)^\sharp
\\&\qquad -\overline{g}(Ph,k).\on{Tr}^g(S)\Big).
\end{align*}
\end{lem*}
\begin{proof}
For vector fields $m,h,k$ on ${\on{Imm}}$ one has
\begin{equation}\label{so:me:na}
\begin{aligned}
&(\nabla_m G^P)(h,k)=
D_{(f,m)} \int_M \overline{g}(Ph,k) {\on{vol}}(g) \\&\qquad\qquad
- \int_M \overline{g}\big(P(\nabla_m h),k\big) {\on{vol}}(g)
- \int_M \overline{g}(Ph,\nabla_m k) {\on{vol}}(g)
\\&\qquad=
\int_M D_{(f,m)}\overline{g}(Ph,k) {\on{vol}}(g)
+ \int_M \overline{g}(Ph,k) D_{(f,m)}{\on{vol}}(g)\\&\qquad\qquad
- \int_M \overline{g}\big(P(\nabla_m h),k\big) {\on{vol}}(g)
- \int_M \overline{g}(Ph,\nabla_m k) {\on{vol}}(g)
\\&\qquad=
\int_M \overline{g}\big(\nabla_m(Ph),k\big) {\on{vol}}(g)
+ \int_M \overline{g}(Ph,\nabla_m k) {\on{vol}}(g)\\&\qquad\qquad
+ \int_M \overline{g}(Ph,k) D_{(f,m)}{\on{vol}}(g)\\&\qquad\qquad
- \int_M \overline{g}\big(P(\nabla_m h),k\big) {\on{vol}}(g)
- \int_M \overline{g}(Ph,\nabla_m k) {\on{vol}}(g)
\\&\qquad=
\int_M \overline{g}\big((\nabla_m P)h,k\big) {\on{vol}}(g)
+ \int_M \overline{g}(Ph,k) D_{(f,m)}{\on{vol}}(g)
\end{aligned}
\end{equation}
One immediately gets the $K$-gradient
by plugging in the variational formula \ref{va:vo} for the volume form:
$$K_f(h,m)=P_f^{-1}\Big((\nabla_m P)h+\on{Tr}^g\big(\overline{g}(\nabla m,Tf)\big).Ph\Big).$$
To calculate the $H$-gradient, one rewrites equation~\eqref{so:me:na}
using the definition of the adjoint:
\begin{equation*}
\begin{aligned}
(\nabla_m G^P)(h,k)=
\int_M \overline{g}\big(m,\adj{\nabla P}(h,k)\big) {\on{vol}}(g) + \int_M \overline{g}(Ph,k) D_{(f,m)}{\on{vol}}(g).
\end{aligned}
\end{equation*}
Now the second summand is treated further using again the variational formula of the volume density
from section~\ref{va:vo}:
\begin{align*}
&\int_M \overline{g}(Ph,k) D_{(f,m)}{\on{vol}}(g)=
\int_M \overline{g}(Ph,k) \on{Tr}^g\big(\overline{g}(\nabla m,Tf)\big) {\on{vol}}(g) \\
&\qquad=
\int_M \overline{g}(Ph,k) \on{Tr}^g\big(\nabla\overline{g}(m,Tf)-\overline{g}(m,\nabla Tf)\big) {\on{vol}}(g) \\
&\qquad=
\int_M \overline{g}(Ph,k) \Big(-\nabla^*\overline{g}(m,Tf)-\overline{g}\big(m,\on{Tr}^g(S)\big)\Big) {\on{vol}}(g) \\
&\qquad=
-\int_M g^0_1\big(\nabla\overline{g}(Ph,k),\overline{g}(m,Tf)\big){\on{vol}}(g)-\int_M \overline{g}(Ph,k) \overline{g}\big(m,\on{Tr}^g(S)\big) {\on{vol}}(g) \\
&\qquad=
\int_M \overline{g}\big(m,-Tf.\on{grad}^g\overline{g}(Ph,k)-\overline{g}(Ph,k) \on{Tr}^g(S)\big) {\on{vol}}(g)
\end{align*}
Collecting terms one gets that
\begin{align*}
&G_f^P(H_f(h,k),m)=(\nabla_m G^P)(h,k)\\&\qquad=
\int_M \overline{g}\big(m,\adj{\nabla P}(h,k)-Tf.\on{grad}^g\overline{g}(Ph,k)-\overline{g}(Ph,k) \on{Tr}^g(S)\big) {\on{vol}}(g)
\end{align*}
Thus the $H$-gradient is given by
\begin{align*}
H_f(h,k)=P^{-1}\Big(\adj{\nabla P}(h,k)-Tf.\on{grad}^g\overline{g}(Ph,k) -\overline{g}(Ph,k).\on{Tr}^g(S)\Big)
\end{align*}
The highest order term $\on{grad}^g\overline{g}(Ph,k)$ cancels out when taking into
account the formula for the tangential part of the adjoint from section~\ref{so:ad}:
\begin{align*}
H_f(h,k)=P^{-1}\Big(&\adj{\nabla P}(h,k)^\bot-Tf.\big(\overline{g}(Ph,\nabla k)+\overline{g}(\nabla h,Pk)\big)^\sharp
\\& -\overline{g}(Ph,k).\on{Tr}^g(S)\Big).
\qedhere
\end{align*}
\end{proof}
\subsection[Geodesic equation]{Geodesic equation on immersions}\label{so:ge}
The geodesic equation for a general metric on ${\on{Imm}}(M,N)$ has been calculated
in section~\ref{sh:ge} and reads as
$$\nabla_{\partial_t} f_t = \frac12 H_f(f_t,f_t) - K_f(f_t,f_t). $$
Plugging in the formulas for $H,K$ derived in the last section yields the following theorem.
\begin{thm*}
The geodesic equation for a Sobolev-type metric $G^P$ on immersions is given by
\begin{align*}
\nabla_{\partial_t} f_t= &\frac12P^{-1}\Big(\adj{\nabla P}(f_t,f_t)^\bot-2.Tf.\overline{g}(Pf_t,\nabla f_t)^\sharp
-\overline{g}(Pf_t,f_t).\on{Tr}^g(S)\Big)
\\&
-P^{-1}\Big((\nabla_{f_t}P)f_t+\on{Tr}^g\big(\overline{g}(\nabla f_t,Tf)\big) Pf_t\Big).
\end{align*}
\end{thm*}
\subsection[Geodesic equation]{Geodesic equation on immersions in terms of the momentum}\label{so:gemo}
The geodesic equation in terms of the momentum has been calculated in section~\ref{sh:gemo}
for a general metric on immersions. For a Sobolev-type metric $G^P$, the momentum
$G^P(f_t,\cdot)$ takes the form
$$p=Pf_t\otimes{\on{vol}}(g): \mathbb R \to T^*{\on{Imm}}$$
since all other parts of the metric (namely the integral and $\overline{g}$)
are constant and can be neglected.
\begin{thm*}
The geodesic equation written in terms of the momentum for a Sobolev-type metric $G^P$
on ${\on{Imm}}$ is given by:
\begin{equation*}
\left\{\begin{aligned}
p&=Pf_t\otimes{\on{vol}}(g)
\\
\nabla_{\partial_t}p &= \frac12\big(\adj{\nabla P}(f_t,f_t)^\bot-2Tf.\overline{g}(Pf_t,\nabla f_t)^\sharp
-\overline{g}(Pf_t,f_t)\on{Tr}^g(S)\big)\otimes{\on{vol}}(g)
\end{aligned}\right.
\end{equation*}
\end{thm*}
\subsection{Well-posedness of the geodesic equation}\label{so:we}
It will be proven that the geodesic equation for a Sobolev-type metric $G^P$ on ${\on{Imm}}$ is well-posed
under some assumptions on $P$.
These assumptions are satisfied for the operator $1+A\Delta^p$ considered in section~\ref{la}.
It will also be shown that $(\pi,\exp)$ is a diffeomorphism from a neighbourhood
of the zero section in $T{\on{Imm}}$ to a neighbourhood of the diagonal in ${\on{Imm}} \times {\on{Imm}}$.
Before we can state the theorem, we have to introduce Sobolev completions of
the relevant spaces of mappings.
More information can be found in \cite{Shubin1987}, \cite{EichhornFricke1998}, and in
\cite{Eichhorn2007}.
We consider Sobolev completions
of $\Gamma(E)$, where $E \to M$ is a vector bundle.
First we choose a fixed (background) Riemannian metric $\hat g$ on $M$ and its
covariant derivative $\nabla^M$.
We equip $E$ with a (background) fiber Riemannian metric $\hat g^E$
and a compatible covariant derivative $\hat \nabla^E$.
Then the {\it Sobolev space} $H^k(E)$ is the Hilbert space completion
of the space of smooth sections $\Gamma(E)$ in the Sobolev norm
$$\|h\|_k^2 = \sum_{j=0}^k \int_M (\hat g^E\otimes \hat g^0_j)\big((\hat \nabla^E)^j h,
(\hat \nabla^E)^j h\big){\on{vol}}(\hat g).$$
This Sobolev space does not depend on the choices of $\hat g$, $\nabla^M$, $\hat g^E$ and
$\hat \nabla^E$ since $M$ is compact:
The resulting norms are equivalent.
We shall need the following results (see \cite{Eichhorn2007}, e.g.):
\newtheorem*{SL}{Sobolev lemma}
\newtheorem*{MP}{Module property of Sobolev spaces}
\begin{SL}
If $k>\dim(M)/2$ then the identy on $\Gamma(E)$ extends to a
injective bounded linear mapping $H^{k+p}(E)\to C^p(E)$ where $C^p(E)$ carries the
supremum norm of all derivatives up to order $p$.
\end{SL}
\begin{MP}
If $k>\dim(M)/2$ then pointwise evaluation $H^k(L(E,E))\times H^k(E)\to H^k(E)$ is
bounded bilinear. Likewise all other pointwise contraction operations are multilinear bounded
operations.
\end{MP}
This allows us to define Sobolev completions of ${\on{Imm}}$ and $T{\on{Imm}}$.
In the canonical charts for ${\on{Imm}}(M,N)$ centered at an immersion $f_0$,
every immersion corresponds to a section of
the vector bundle $f_0^*TN$ over $M$ (see \cite[section~42]{MichorG}).
The smooth Sobolev manifold ${\on{Imm}}^k(M,N)$ (for $k>\dim(M)/2+1$)
is constructed by gluing together the Sobolev completions
$H^k(f_0^*TN)$ of each canonical chart.
One has
$${\on{Imm}}^{k+1}(M,N) \subset {\on{Imm}}^k(M,N),\qquad \bigcap_{k}{\on{Imm}}^k(M,N) = {\on{Imm}}(M,N).$$
Similarly, Sobolev completions of the space $T{\on{Imm}} \subset C^\infty(M,TN)$ are defined
as $H^k$-mappings from $M$ into $TN$, i.e. $T{\on{Imm}}^k=H^k(M,TN)$.
\begin{ass}
$P,\nabla P$ and $\adj{\nabla P}^\bot$ are smooth sections of the bundles
\begin{equation*}\xymatrix{
L(T{\on{Imm}};T{\on{Imm}}) \ar[d] & L^2(T{\on{Imm}};T{\on{Imm}}) \ar[d] & L^2(T{\on{Imm}};T{\on{Imm}}) \ar[d] \\
{\on{Imm}} & {\on{Imm}} & {\on{Imm}},
}\end{equation*}
respectively.
Viewed locally in trivializations of these bundles,
$$P_f h, \qquad (\nabla P)_f (h,k), \qquad \big(\adj{\nabla P}_f(h,k)\big)^\bot$$
are pseudo-differential operators of order $2p$ in $h,k$ separately.
As mappings in the footpoint $f$ they are non-linear, and it is assumed that they are a
composition of operators of the following type:
\newline
\textrm{(a)} Local operators of order $l\le 2p$, i.e., nonlinear differential operators
$$A(f)(x)=A(x,\hat \nabla^{l}f(x),\hat \nabla^{l-1}f(x),\dots,\hat \nabla f(x), f(x)),$$
\newline
\textrm{(b)} Linear pseudo-differential operators of degrees $l_i$,
\newline
such that the total (top) order of the composition is $\le 2p$.
\end{ass}
\begin{ass}
For each $f\in {\on{Imm}}(M,N)$, the operator $P_f$
is an elliptic pseudo-differential operator of order $2p$ for $p>0$
which is positive and symmetric with respect to the $H^0$-metric on ${\on{Imm}}$, i.e.
$$\int_M \overline{g}(P_f h,k){\on{vol}}(g) = \int_M \overline{g}(h,P_f k){\on{vol}}(g) \qquad \text{for } h,k \in T_f{\on{Imm}}.$$
\end{ass}
\begin{ass}
$P$ is invariant under the action of ${\on{Diff}}(M)$.
(See section~\ref{so:in} for the definition of invariance.)
\end{ass}
\begin{thm*}
Let $p\ge 1$ and $k>\dim(M)/2+1$, and let
$P$ satisfy assumptions 1--3.
Then the initial value problem for the geodesic equation
\thetag{\ref{so:ge}}
has unique local solutions in the Sobolev manifold ${\on{Imm}}^{k+2p}$ of
$H^{k+2p}$-immersions. The solutions depend smoothly on $t$ and on the initial
conditions $f(0,\;.\;)$ and $f_t(0,\;.\;)$.
The domain of existence (in $t$) is uniform in $k$ and thus this
also holds in ${\on{Imm}}(M,N)$.
Moreover, in each Sobolev completion ${\on{Imm}}^{k+2p}$, the Riemannian exponential mapping $\exp^{P}$ exists
and is smooth on a neighborhood of the zero section in the tangent bundle,
and $(\pi,\exp^{P})$ is a diffeomorphism from a (smaller) neigbourhood of the zero
section to a neighborhood of the diagonal in
${\on{Imm}}^{k+2p}\times {\on{Imm}}^{k+2p}$.
All these neighborhoods are uniform in $k>\dim(M)/2+1$ and can be chosen
$H^{k_0+2p}$-open, for $k_0 > \dim(M)/2+1$. Thus both properties of the exponential
mapping continue to hold in ${\on{Imm}}(M,N)$.
\end{thm*}
This proof is partly an adaptation of \cite[section 4.3]{Michor107}.
It works in three steps: First, the geodesic equation
is formulated as the flow equation of a smooth vector field on each Sobolev completion
$T{\on{Imm}}^{k+2p}$.
Thus one gets local existence and uniqueness of solutions. Second, it is shown that
the time-interval where a solution exists does not depend on the order of the Sobolev space
of immersions. Thus one gets solutions on the intersection of all Sobolev spaces, which
is the space of smooth immersions. Third, a general argument involving the inverse function theorem
on Banach spaces proves the claims
about the exponential map.
\begin{proof} By assumption~1 the mapping
$P_f h$ is of order $2p$ in $f$ and in $h$ where $f$ is the footpoint of $h$.
Therefore
$f \mapsto P_f$ extends to a smooth section of the smooth Sobolev bundle
\begin{equation*}
L\big(T{\on{Imm}}^{k+2p};T{\on{Imm}}^k\mid {\on{Imm}}^{k+2p} \big) \to {\on{Imm}}^{k+2p},
\end{equation*}
where $T{\on{Imm}}^k\mid {\on{Imm}}^{k+2p}$ denotes the space of all $H^k$ tangent vectors
with foot point a $H^{k+2p}$ immersion, i.e., the restriction of the bundle $T{\on{Imm}}^k\to {\on{Imm}}^{k}$
to ${\on{Imm}}^{k+2p}\subset{\on{Imm}}^k$.
This means that $P_f$ is a bounded linear operator
$$P_f \in L\big(H^{k+2p}(f^*TN),H^k(f^*TN)\big)\quad\text{for}\quad f\in {\on{Imm}}^{k+2p}.$$
It is injective since it is positive.
As an elliptic operator,
it is an unbounded operator on the Hilbert completion of $T_f{\on{Imm}}$ with respect to the
$H^0$-metric, and a Fredholm operator $H^{k+2p}\to H^k$ for each $k$.
It is selfadjoint elliptic, thus by \cite[theorem 26.2]{Shubin1987} it has vanishing index.
Since it is injective, it is thus also surjective.
By the implicit function theorem on Banach spaces,
$f\mapsto P_f^{-1}$ is then a smooth section of the smooth Sobolev bundle
\begin{equation*}
L\big(T{\on{Imm}}^k \mid {\on{Imm}}^{k+2p} ; T{\on{Imm}}^{k+2p}\big) \to {\on{Imm}}^{k+2p}
\end{equation*}
As an inverse of an elliptic pseudodifferential operator,
$P_f^{-1}$ is also an elliptic pseudo-differential operator of order $-2p$.
By assumption~1 again, $(\nabla P)_f(m,h)$ and $\big(\adj{\nabla P}_f(m,h)\big)^\bot$
are of order $2p$ in $f,m,h$ (locally).
Therefore $f\mapsto P_f$ and $f\mapsto \adj{\nabla P}^\bot$
extend to smooth sections of the smooth Sobolev bundle
\begin{equation*}
L^2\big(T{\on{Imm}}^{k+2p}; T{\on{Imm}}^k \mid {\on{Imm}}^{k+2p}\big) \to {\on{Imm}}^{k+2p}
\end{equation*}
Using the module property of Sobolev spaces
and counting the order of all remaining terms in the geodesic equation
\ref{so:ge}, one obtains that the Christoffel symbols
\begin{align*}
\frac12 H_f(h,h)-K_f(h,h) &=\frac12P^{-1}\Big(\adj{\nabla P}(h,h)^\bot-2.Tf.\overline{g}(Ph,\nabla h)^\sharp
\\&\qquad
-\overline{g}(Ph,h).\on{Tr}^g(S)-(\nabla_{h}P)h-\on{Tr}^g\big(\overline{g}(\nabla h,Tf)\big) Ph\Big)
\end{align*}
extend to a smooth $(C^\infty)$ section
of the smooth Sobolev bundle
\begin{equation*}
L^2_{\text{sym}}\big(T{\on{Imm}}^{k+2p}; T{\on{Imm}}^{k+2p}\big) \to {\on{Imm}}^{k+2p}
\end{equation*}
Thus $h\mapsto \tfrac12 H(h,h)-K(h,h)$ is a smooth quadratic mapping $T{\on{Imm}} \to T{\on{Imm}}$ which
extends to smooth quadratic mappings $T{\on{Imm}}^{k+2p} \to T{\on{Imm}}^{k+2p}$ for each
$k\ge \frac{\dim(2)}2+1$.
The geodesic equation
$$\nabla^{\overline{g}}_{\partial_t}f_t = \frac12 H_f(f_t,f_t)-K_f(f_t,f_t)$$
can be reformulated using the linear connection $C^g:TN \times_N TN \to TTN$ (horizontal lift mapping)
of $\nabla^{\overline{g}}$,
see \cite[section~24.2]{MichorH}:
\begin{align*}
\partial_t f_t &=C\Big( \frac12 H_f(f_t,f_t)-K_f(f_t,f_t),f_t\Big).
\end{align*}
The right-hand side is a smooth vector field on $T{\on{Imm}}^{k+2p}$,
the geodesic spray. Note that the restriction to $T{\on{Imm}}^{k+1+2p}$ of the geodesic spray on
$T{\on{Imm}}^{k+2p}$ equals the geodesic spray there.
By the theory of smooth ODE's on Banach spaces, the flow of
this vector field exists in $T{\on{Imm}}^{k+2p}$ and is smooth in time and in the initial condition.
Consider a $C^\infty$ initial condition $h_0 \in T{\on{Imm}}$ with foot point $f_0\in {\on{Imm}}$.
Suppose the
trajectory $\on{Fl}^k_t(h_0)$ of
geodesic spray through these initial conditions in $T{\on{Imm}}^{k+2p}$ maximally exists for $t\in (-a_k,b_k)$,
and the trajectory $\on{Fl}^{k+1}_t(h_0)$ in $T{\on{Imm}}^{k+1+2p}$
maximally exists for $t\in(-a_{k+1},b_{k+1})$ with
$b_{k+1}<b_k$, say. By
uniqueness of solutions one has $\on{Fl}^{k+1}_t(h_0)=\on{Fl}^{k}_t(h_0)$ for
$t\in (-a_{k+1,}b_{k+1})$.
We now write $\hat\nabla$ for the covariant derivative induced by $\overline{g}$ on $N$ and the background
metric $\hat g$ on $M$.
Let $X$ be a vector field on $M$.
Applying $\nabla^{\overline{g}}_X=:\hat\nabla_X$ to the geodesic equation
and swapping covariant derivatives yields:
\begin{align*}
\nabla^{\overline{g}}_{\partial_t} \nabla^{\overline{g}}_{\partial_t} Tf.X
&=
\nabla^{\overline{g}}_{\partial_t} \nabla^{\overline{g}}_X f_t
=
\nabla^{\overline{g}}_{X} \nabla^{\overline{g}}_{\partial_t} f_t +R^{\overline{g}}(f_t,Tf.X)f_t
\\&=
\nabla^{\overline{g}}_{X} \big(\tfrac12 H_f(f_t,f_t)-K_f(f_t,f_t)\big)+R^{\overline{g}}(f_t,Tf.X)f_t
\tag{A}
\end{align*}
Note that $i_X\hat\nabla_{\partial_t}\nabla^{\overline{g}} f_t = \nabla^{\overline{g}}_{\partial_t}\nabla^{\overline{g}}_X f_t -
\nabla^{\overline{g}}_{\nabla^{\overline{g}}_{\partial_t}X} f_t = \nabla^{\overline{g}}_{\partial_t}\nabla^{\overline{g}}_X f_t-0$. Thus we can omit $X$ and
rewrite \thetag{A} as an equation for $Tf$.
We aim to rewrite equation \thetag{A} as a linear first order equation for the highest derivative
whose coefficients are given by $\on{Fl}^k_t(h_0)$ and thus exist beyond $(-a_{k+1},b_{k+1})$. For this we
have to pass to one (ore more) canonical chart for ${\on{Imm}}$ and the induced trivializations of all bundles as
before and in assumption \thetag{1}.
Then $f$ itself has values in a vector space and we may regard \thetag{A} as a vector valued 1-form on $M$.
So we rewrite \thetag{A} as:
\begin{align*}
\hat\nabla_{\partial_t} Tf &= \nabla^{\overline{g}} f_t
\\
\hat\nabla_{\partial_t} \nabla^{\overline{g}} f_t
&=
\nabla^{\overline{g}} \big(\tfrac12 H_f(f_t,f_t)-K_f(f_t,f_t)\big) + R^{\overline{g}}(f_t,Tf)f_t
\\&=
Y^1(f,f_t)(\hat\nabla^{2p}Tf) + Y^2(f,f_t)(\hat\nabla^{2p+1}f_t) + Y^3(f,f_t).
\tag{B}\end{align*}
We claim that
\thetag{B} consists of:
\newline $\bullet$
The smooth expression $Y^1(f,f_t)(\hat\nabla^{2p+1}f)$ which is \emph{linear} and of order 0 in
$\hat\nabla^{2p+1}f$ and where $Y^1(f,f_t)$ is of order $\le 2p$ in $f,f_t$; order here means that the
expression prolongs continuously to the corresponding Sobolev spaces.
\newline $\bullet$
The smooth expression $Y^2_X(f,f_t)(\hat\nabla^{2p+1} f_t)$ which is \emph{linear} and of order 0 in
$\hat\nabla^{2p+1}f_t$ and where $Y^2(f,f_t)$ is of order $\le 2p$ in $f,f_t$.
\newline $\bullet$
The smooth expression $Y^3_X(f,f_t)$ of order $\le 2p$ in $f,f_t$.
\newline
To see this
we claim that the highest derivatives of order $2p+1$ of $f$ and $f_t$ appear only linearly
in \thetag{A}.
This claim follows from assumption~1:
\newline
(a) For a local operator we can apply the chain rule:
The highest derivative of $f$ appears only linearly.
\newline
(b) For a linear pseudo differential operator $A$ of order $k$
the commutator $[\hat \nabla,A]$ is a pseudo-differential operator of order $k$ again.
\newline
On the left hand side of \thetag{B} we write $Tf=:u$ and $\hat\nabla f_t =: v$.
On the right hand side of \thetag{B} we write
$\hat \nabla^{2p} Tf = \hat \nabla^{2p}u$ and
$\hat \nabla^{2p+1}f_t = \hat \nabla^{2p} v$ for the highest derivatives only.
Then the system \thetag{B} becomes:
\begin{align*}
\hat\nabla_{\partial_t} u &= v
\\
\hat\nabla_{\partial_t} v
&=
Y^1(f,f_t)(\hat\nabla^{2p}u) + Y^2(f,f_t)(\hat\nabla^{2p}v) + Y^3(f,f_t).
\tag{C}\end{align*}
The coefficients $f,f_t$ in \thetag{C} exist for $t\in (-a_k,b_k)$ as $\on{Fl}^k_t(h_0)$.
Then \thetag{C} is a bounded and smooth inhomogeneous linear ODE for
$(u,v)\in \Omega^1(M,T{\on{Imm}}^{k+2p})$, i.e., in a Banach space.
This equation
therefore has a solution $(u(t),v(t))$ for all $t$ for which the
coefficients exists, thus for all $t\in (a_k,b_k)$, which is unique for the initial values $u_0 = T f_0$ and
$v_0 = \hat\nabla h_0$. The limit
$\lim_{t\nearrow b_{k+1}} (u(t),v(t))$ exists in $\Omega^1(M,T{\on{Imm}}^{k+2p})$
and by continuity it equals $(\hat \nabla Tf,\hat \nabla f_t)$
for $t=b_{k+1}$. Thus the flow line $\on{Fl}^k_t(h_0)$ was not
maximal and can be continued. So assuming $b_{k+1}<b_k$ leads to a contradiction, and thus
$(-a_{k+1},b_{k+1})=(-a_k,b_k)$.
Iterating this procedure one concludes that the flow line $\on{Fl}^{\infty}_t(h_0)$ exists in
$\bigcap_{k\ge \frac{\dim(M)}2+1} T{\on{Imm}}^{k+2p}= T{\on{Imm}}$.
It remains to check the properties of the Riemannian exponential mapping $\exp^P$.
It is given by $\exp^P_{f}(h)= c(1)$ where $c(t)$ is the geodesic emanating from
value $f$ with initial velocity $h$.
Let $k_0>\dim(M)/2+1$ and $k\ge k_0$.
On each space $T{\on{Imm}}^{k+2p}$, the properties claimed follow
from local existence and uniqueness of solutions to
the flow equation of the geodesic spray, from
the form of the geodesic equation $f_{tt}=\tfrac12 H(f_t,f_t)-K(f_t,f_t)$
when it is written down in a chart,
namely linearity in $f_{tt}$ and bilinearity in $f_t$,
and from the inverse function theorem
which holds on each of the Sobolev spaces ${\on{Imm}}^{k+2p}$.
See for example \cite[22.6 and 22.7,]{MichorH} for a detailed proof
which works without any change in notation.
${\on{Imm}}^{k_0+2p}$ contains ${\on{Imm}}^{k+2p}$ for $k>k_0$.
Since the spray on ${\on{Imm}}^{k_0+2p}$ restricts to the spray on each ${\on{Imm}}^{k+2p}$,
the exponential mapping $\exp^P$ and the inverse $(\pi,\exp^P)^{-1}$ on ${\on{Imm}}^{k_0+2p}$ restrict
to the corresponding mappings on each ${\on{Imm}}^{k+2p}$. Thus the neighborhoods of existence are
uniform in $k$ and can be chosen $H^{k_0+2p}$-open.
\end{proof}
\subsection{Momentum mappings}\label{so:mo}
Recall that by assumption, the operator $P$ is invariant under the action of the
reparametrization group $\on{Diff}(M)$. Therefore
the induced metric $G^P$ is invariant under this group action, too.
According to \cite[section~2.5]{Michor107} one gets:
\begin{thm*}
The reparametrization momentum, which is the momentum mapping
corresponding to the action of ${\on{Diff}}(M)$ on ${\on{Imm}}(M,N)$,
is conserved along any geodesic $f$ in ${\on{Imm}}(M,N)$:
\begin{align*}
&\forall X\in\mathfrak X(M): \int_M \overline{g}( Tf.X,Pf_t ) {\on{vol}}(g) \\
\intertext{or equivalently}
&g\bigl((Pf_t )^\top\bigr) {\on{vol}}(g) \in\Gamma(T^*M\otimes_M{\on{vol}}(M))
\end{align*}
is constant along $f$.
\end{thm*}
\subsection{Horizontal bundle}\label{so:ho}
The splitting of $T{\on{Imm}}$ into horizontal and vertical subspaces will be calculated
for Sobolev-type metrics $G^P$. See section~\ref{sh:rish} for the definitions.
By definition, a tangent vector $h$ to $f \in {\on{Imm}}(M,N)$ is horizontal if
and only if it is $G^P$-perpendicular to
the $\on{Diff}(M)$-orbits. This is the case if and only if
$\overline{g}( P_f h(x), T_x f .X_x) = 0$ at every point $x \in M$.
Therefore the horizontal bundle at the point $f$ equals
\begin{align*}
&\big\{h\in T_f{\on{Imm}}: P_fh(x)\,\bot\, T_x f(T_xM)\text{ for all
}x\in M\}
=\big\{h : (P_fh)^\top = 0\big\}.
\end{align*}
Note that the horizontal bundle consists of vector fields that are normal to $f$
when $P=\on{Id}$, i.e. for the $H^0$-metric on ${\on{Imm}}$.
Let us work out the $G^P$-decomposition of $h$ into vertical and
horizontal parts. This decomposition is written as
\begin{equation*}
h= Tf.h^{\text{ver}} + h^{\text{hor}}.
\end{equation*}
Then
\begin{align*}
P_fh = P_f (Tf.h^{\text{ver}}) + P_f h^{\text{hor}} \quad \text{with} \quad
(P_fh)^\top = (P_f (Tf.h^{\text{ver}}))^\top + 0.
\end{align*}
Thus one considers the operators
\begin{align*}
&P_f^\top :\mathfrak X(M) \to \mathfrak X(M), &\qquad P_f^\top (X) &= \big(P_f(Tf.X)\big)^\top,
\\
&P_{f,\bot}:\mathfrak X(M)\to \Gamma\big({\on{Nor}}(f)\big) \subset C^\infty(M,TN), & P_{f,\bot}(X) &=
\big(P_f(Tf.X)\big)^\bot.
\end{align*}
The operator $P_f^\top $ is unbounded, positive and symmetric on the Hilbert completion
of $T_f{\on{Imm}}$ with respect to the $H^0$-metric since one has
\begin{align*}
\int_M g(P_f^\top X,Y){\on{vol}}(g) &=
\int_M \overline{g}(Tf.P_f^\top X,Tf.Y){\on{vol}}(g)
\\&
= \int_M \overline{g}(P_{f}(Tf.X),Tf.Y){\on{vol}}(g)
\\&
= \int_M g(P_f^\top Y,X){\on{vol}}(g),
\\
\int_M g(P_f^\top X,X){\on{vol}}(g) &=
\int_M \overline{g}(P_{f}(Tf.X),Tf.X){\on{vol}}(g) \; > 0 \quad\text{ if }X\ne 0.
\end{align*}
Let $\sigma^{P_f}$ and $\sigma^{P_f^\top}$ denote the principal symbols of $P_f$ and $P_f^\top$,
respectively. Take any $x \in M$ and $\xi\in T^*_xM\setminus\{0\}$.
Then $\sigma^{P_f}(\xi)$ is symmetric, positive definite on $(T_{f(x)}N,\overline{g})$.
This means that one has for any $h,k \in T_{f(x)}N$ that
\begin{equation*}
\overline{g}\big(\sigma^{P_f}(\xi)h,k\big) = \overline{g}\big(h,\sigma^{P_f}(\xi)k\big), \qquad
\overline{g}\big(\sigma^{P_f}(\xi)h,h\big) > 0 \text{ for } h \neq 0.
\end{equation*}
The principal symbols $\sigma^{P_f}$ and $\sigma^{P_f^\top}$ are related by
\begin{equation*}
g\big(\sigma^{P_f^\top}(\xi)X,Y\big) = \overline{g}\big(Tf.\sigma^{P_f^\top}(\xi)X,Tf.Y\big)
= \overline{g}\big(\sigma^{P_f}(\xi)Tf.X,Tf.Y\big),
\end{equation*}
where $X,Y \in T_xM$.
Thus $\sigma^{P_f^\top}(\xi)$ is symmetric, positive definite on $(T_xM,g)$.
Therefore $P_f^\top $ is again elliptic, thus it is selfadjoint,
so its index (as operator $H^{k+2p}\to H^{k}$) vanishes. It
is injective (since positive) with vanishing index (since self-adjoint elliptic, by \cite[theorem 26.2]{Shubin1987})
hence it is bijective and thus invertible by the open mapping theorem.
Thus it has been proven:
\begin{lem*}
The decomposition of $h \in T{\on{Imm}}$ into its vertical and horizontal components
is given by
\begin{align*}
h^{\text{ver}} &= (P_f^\top )^{-1}\big((P_fh)^\top\big),
\\
h^{\text{hor}} &= h - Tf.h^{\text{ver}} = h - Tf.(P_f^\top )^{-1}\big((P_fh)^\top\big).
\end{align*}
\end{lem*}
\subsection{Horizontal curves}\label{so:ho2}
To establish the one-to-one correspondence between horizontal
curves in ${\on{Imm}}$ and curves in shape space that has been described in theorem~\ref{sh:sub},
one needs the following property:
\begin{lem*}
For any smooth path $f$ in ${\on{Imm}}(M,N)$ there exists a
smooth path $\varphi$ in $\on{Diff}(M)$ with $\varphi(0,\;.\;)=\on{Id}_M$
depending smoothly on $f$ such that
the path $\tilde f$ given by $\tilde f(t,x)=f(t,\varphi(t,x))$ is horizontal:
$$\overline{g} \big( P_{\tilde f}(\partial_t\tilde f),T\tilde f.TM \big) =0.$$
Thus any path in shape space can be lifted to a horizontal path of immersions.
\end{lem*}
The basic idea is to write the path $\varphi$ as the integral curve of a time dependent vector field.
This method is called the Moser-trick (see \cite[Section 2.5]{Michor102}).
\begin{demo}{Proof}
Since $P$ is invariant, one has $(r^\varphi)^* P = P$ or
$P_{f\circ \varphi}(u\circ \varphi)=(P_fu)\circ\varphi$ for $\varphi\in\on{Diff}(M)$.
In the following $f\circ\varphi$ will denote the map $f(t, \varphi(t,x))$, etc.
One looks for $\varphi$ as the integral curve of a time dependent vector field $\xi(t,x)$ on
$M$, given by $\varphi_t=\xi\circ \varphi$.
The following expression must vanish for all $x \in M$ and $X_x \in T_x M$:
\begin{align*}
0&=\overline{g}\Big( P_{f\circ\varphi}\big(\partial_t(f\circ \varphi)\big)(x), T(f\circ \varphi).X_x \Big) \\&=
\overline{g}\Big( P_{f\circ\varphi}\big((\partial_t f)\circ\varphi +Tf.(\partial_t\varphi)\big)(x), T(f\circ \varphi).X_x \Big)
\\ &=
\overline{g}\Big( \big((P_{f}(\partial_tf)) +P_{f}(Tf.\xi)\big)\big(\varphi(x)\big),Tf\circ T\varphi.X_x \Big)
\end{align*}
Since $T\varphi$ is surjective, $T\varphi.X$ exhausts the tangent space $T_{\varphi(x)}M$, and one has
$$\big((P_{f}(\partial_tf)) +P_{f}(Tf.\xi)\big)\big(\varphi(x)\big)\quad \perp \quad f.$$
This holds for all $x \in M$, and by the surjectivity of $\varphi$, one also has that
$$\big((P_{f}(\partial_tf)) +P_{f}(Tf.\xi)\big)(x)\quad \perp \quad f$$
at all $x \in M$.
This means that the tangential part $\big(P_{f}(\partial_tf) + P_f(Tf.\xi)\big)^\top$ vanishes.
Using the time dependent vector field
$$\xi=-(P_f^\top )^{-1} \big((P_f \partial_t f)^\top\big)$$
and its flow $\varphi$ achieves this.
\qed\end{demo}
\subsection[Geodesic equation]{Geodesic equation on shape space}\label{so:gesh}
By the previous section and theorem~\ref{sh:sub},
geodesics in $B_i$ correspond exactly to horizontal geodesics in ${\on{Imm}}$.
The equations for horizontal geodesics in the space of immersions have been written down in
section~\ref{sh:gesh}. Here they are specialized to Sobolev-type metrics:
\begin{thm*}
The geodesic equation on shape space for a Sobolev-type metric $G^P$
is equivalent to the set of equations
\begin{equation*}\left\{\begin{aligned}
f_t &= f_t^{\on{hor}} \in {\on{Hor}}, \\
(\nabla_{\partial_t} f_t)^{\on{hor}} &=
\frac12P^{-1}\Big(\adj{\nabla P}(f_t,f_t)^\bot -\overline{g}(Pf_t,f_t).\on{Tr}^g(S)\Big)\\
&\qquad-P^{-1}\Big(\big((\nabla_{f_t}P)f_t\big)^\bot-\on{Tr}^g\big(\overline{g}(\nabla f_t,Tf)\big) Pf_t\Big),
\end{aligned}\right.\end{equation*}
where $f$ is a horizontal path of immersions.
\end{thm*}
These equations are not handable very well since taking the horizontal part
of a vector to ${\on{Imm}}$ involves inverting an elliptic pseudo-differential operator,
see section~\ref{so:ho}.
However, the formulation in the next section is much better.
\subsection[Geodesic equation]{Geodesic equation on shape space in terms of the momentum}\label{so:geshmo}
The geodesic equation in terms of the momentum has been derived in section~\ref{sh:geshmo}
for a general metric on shape space. Now it is specialized to Sobolev-type metrics
using the formula for the $H$-gradient from section~\ref{so:me}.
As in section~\ref{so:gemo} the momentum $G^P(f_t,\cdot)$ is identified with
$Pf_t \otimes {\on{vol}}(g)$. By definition, the momentum is horizontal if it annihilates all
vertical vectors. This is the case if and only if $Pf_t$ is normal to $f$.
Thus the splitting of the momentum in horizontal and vertical parts is given by
$$Pf_t \otimes {\on{vol}}(g) = (Pf_t)^\bot \otimes {\on{vol}}(g) + Tf.(Pf_t)^\top \otimes {\on{vol}}(g).$$
This is much simpler than the splitting of the velocity in horizontal and vertical parts where
a pseudo-differential operator has to be inverted, see section~\ref{so:ho}.
Thus the following version of the geodesic equation on shape space is the easiest to solve.
\begin{thm*}
The geodesic equation on shape space is equivalent to the set of equations
for a path of immersions $f$:
\begin{equation*}
\left\{\begin{aligned}
p &= Pf_t \otimes {\on{vol}}(g), \qquad Pf_t = (Pf_t)^\bot, \\
(\nabla_{\partial_t}p)^{\on{hor}} &= \frac12 \Big(\adj{\nabla P}(f_t,f_t)^\bot-\overline{g}(Pf_t,f_t).\on{Tr}^g(S)\Big) \otimes {\on{vol}}(g).
\end{aligned}\right.
\end{equation*}
\end{thm*}
The equation for geodesics on ${\on{Imm}}$ without the horizontality condition is
$$\nabla_{\partial_t}p = \frac12\big(\adj{\nabla P}(f_t,f_t)^\bot-2Tf.\overline{g}(Pf_t,\nabla f_t)^\sharp
-\overline{g}(Pf_t,f_t).\on{Tr}^g(S)\big)\otimes{\on{vol}}(g),
$$
see section~\ref{so:gemo}.
It has been proven in section~\ref{sh:gesh} that the vertical part of this equation
is satisfied automatically when the geodesic is horizontal.
Nevertheless this will be checked by hand because the proof is much simpler here
than in the general case.
If $f_t$ is horizontal then by definition $Pf_t$ is normal to $f$. Thus one has for any $X \in \mathfrak X(M)$ that
\begin{align*}
g\big((\nabla_{\partial_t}Pf_t)^\top, X \big) &=
\overline{g}(\nabla_{\partial_t}Pf_t, Tf.X ) =
0-\overline{g}(Pf_t,\nabla_{\partial_t}Tf.X)\\&=
-\overline{g}(Pf_t,\nabla_X f_t)=
-g\big(\overline{g}(Pf_t,\nabla f_t)^\sharp,X).
\end{align*}
Thus
\begin{align*}
\big(\nabla_{\partial_t} p\big)^{\on{vert}} &=
\big((\nabla_{\partial_t} Pf_t) \otimes {\on{vol}}(g) + Pf_t \otimes D_{(f,f_t)} {\on{vol}}(g) \big)^{\on{vert}} \\&=
Tf.(\nabla_{\partial_t} Pf_t)^\top \otimes {\on{vol}}(g) + Tf.(Pf_t)^\top \otimes D_{(f,f_t)} {\on{vol}}(g) \\&=
-Tf.\overline{g}(Pf_t,\nabla f_t)^\sharp \otimes {\on{vol}}(g) + 0 ,
\end{align*}
which is exactly the vertical part of the geodesic equation.
\section{Geodesic distance on shape space}\label{ge}
It came as a big surprise when it was discovered in \cite{Michor98}
that the Sobolev metric of order zero induces vanishing geodesic distance on shape space $B_i$.
It will be shown that this problem can be overcome by using higher order Sobolev metrics.
The proof of this result is based on bounding the $G^P$-length of a path from below by its area swept out.
The main result is in section~\ref{ge:no}.
The same ideas are contained in
\cite[section~2.4]{Bauer2010}, \cite[section~7]{Michor118} and \cite[section~3]{Michor102}.
\subsection{Geodesic distance on shape space}\label{ge:ge}
\emph{Geodesic distance} on $B_i$ is given by
$${\on{dist}}_{G^P}^{B_i}(F_0,F_1) = \inf_F L_{G^P}^{B_i}(F),$$
where the infimum is taken over all $F :[0,1] \to B_i$ with $F(0)=F_0$ and $F(1)=F_1$.
$L_{G^P}^{B_i}$ is the length of paths in $B_i$ given by
$$L_{G^P}^{B_i}(F) = \int_0^1 \sqrt{G^P_F(F_t,F_t)} dt \quad \text{for $F:[0,1] \to B_i$.}$$
Letting $\pi:{\on{Imm}} \to B_i$ denote the projection, one has
$$L_{G^P}^{B_i}(\pi \circ f) = L_{G^P}^{{\on{Imm}}}(f) =\int_0^1 \sqrt{G^P_f(f_t,f_t)} dt$$
when $f:[0,1]\to {\on{Imm}}$ is horizontal.
In the following sections, conditions on the metric $G^P$ ensuring that ${\on{dist}}_{G^P}^{B_i}$ separates
points in $B_i$ will be developed.
\subsection{Vanishing geodesic distance}\label{ge:va}
\begin{thm*}
The distance ${\on{dist}}_{H^0}^{B_i}$ induced by the Sobolev $L^2$ metric of order zero vanishes.
Indeed it is possible to connect any two distinct shapes by a path of arbitrarily short length.
\end{thm*}
This result was first established by Michor and Mumford
for the case of planar curves in \cite{Michor98}.
A more general version can be found in \cite{Michor102}, where the same result
is proven also on diffeomorphism groups.
\subsection{Area swept out}\label{ge:ar}
For a path of immersions $f$ seen as a mapping $f:[0,1] \times M \to N$ one has
$$(\text{area swept out by $f$})=\int_{[0,1]\times M} {\on{vol}}(f(\cdot,\cdot)^* \overline{g})
=\int_0^1 \int_M \norm{f_t^\bot} {\on{vol}}(g) dt.$$
\subsection{Area swept out bound}\label{ge:ar1}
\begin{lem*}
Let $G^P$ be a Sobolev type metric that is at least as strong as the $H^0$-metric, i.e.
there is a constant $C_1 > 0$ such that
\begin{align*}
\norm{h}_{G^P} \geq C_1 \norm{h}_{H^0} = C_1 \sqrt{\int_M \overline{g}(h,h) {\on{vol}}(g)} \qquad \text{for all $h \in T{\on{Imm}}$.}
\end{align*}
Then one has the area swept out bound for any path of immersions $f$:
\begin{align*}
C_1 \ (\text{area swept out by $f$})
\leq \max_t \sqrt{{\on{Vol}}\big(f(t)\big)} . L_{G^P}^{{\on{Imm}}}(f).
\end{align*}
\end{lem*}
The proof is an adaptation of the one given in \cite[section~7.3]{Michor118} for almost local metrics.
\begin{proof}
\begin{align*}
L_{G^P}^{{\on{Imm}}}(f)&=\int_0^1 \norm{f_t}_{G^P} dt \geq
C_1 \int_0^1 \norm{f_t}_{H^0} dt \\&\geq
C_1 \int_0^1 \norm{f_t^\bot}_{H^0} dt =
C_1 \int_0^1 \Big(\int_M \norm{f_t^\bot}^2 {\on{vol}}(g) \Big)^{\frac12} dt \\&\geq
C_1 \int_0^1 \Big(\int_M {\on{vol}}(g) \Big)^{-\frac12} \int_M 1.\norm{f_t^\bot} {\on{vol}}(g) dt\\&\geq
C_1 \min_t \Big(\int_M {\on{vol}}(g) \Big)^{-\frac12} \int_{[0,1]\times M} {\on{vol}}(f(\cdot,\cdot)^* \overline{g}) \\&=
C_1 \Big(\max_t \int_M {\on{vol}}(g) \Big)^{-\frac12}\ (\text{area swept out by $f$})
\qedhere
\end{align*}
\end{proof}
\subsection{Lipschitz continuity of $\sqrt{{\on{Vol}}}$}\label{ge:li}
\begin{lem*}
Let $G^P$ be a Sobolev type metric that is at least as strong as the $H^1$-metric, i.e.
there is a constant $C_2 > 0$ such that
\begin{align*}
\norm{h}_{G^P} \geq C_2 \norm{h}_{H^1} =
C_2 \sqrt{\int_M \overline{g}\big( (1+\Delta) h,h \big) {\on{vol}}(g)} \qquad \text{for all $h \in T{\on{Imm}}$.}
\end{align*}
Then the mapping
$$\sqrt{{\on{Vol}}}:(B_i,{\on{dist}}_{G^P}^{B_i}) \to \mathbb R_{\geq 0}$$
is Lipschitz continuous, i.e. for all $F_0$ and $F_1$ in $B_i$ one has:
$$
\sqrt{{\on{Vol}}(F_1)}-\sqrt{{\on{Vol}}(F_0)}
\leq \frac{1}{2 C_2} {\on{dist}}_{G^P}^{B_i}(F_0,F_1).
$$
\end{lem*}
For the case of planar curves, this has been proven in \cite[section~4.7]{Michor107}.
\begin{proof}
\begin{align*}
\partial_t {\on{Vol}} &= \int_M \Big(\on{div}^g(f_t^\top)-\overline{g}\big(f_t^\bot, \on{Tr}^g(S)\big)\Big) {\on{vol}}(g) \\&=
0+\int_M \overline{g}(f_t, \nabla^* Tf) {\on{vol}}(g) =
\int_M (g^0_1 \otimes \overline{g})(\nabla f_t, Tf) {\on{vol}}(g) \\&\leq
\sqrt{\int_M \norm{\nabla f_t}_{g^0_1 \otimes \overline{g}}^2 {\on{vol}}(g)}
\sqrt{\int_M \norm{Tf}_{g^0_1 \otimes \overline{g}}^2 {\on{vol}}(g)} \\&\leq
\norm{f_t}_{H^1}\ \sqrt{{\on{Vol}}} \leq \frac{1}{C_2} \norm{f_t}_{G^P}\ \sqrt{{\on{Vol}}} .
\end{align*}
Thus
\begin{align*}
\partial_t \sqrt{{\on{Vol}}(f)}=\frac{\partial_t {\on{Vol}}(f)}{2 \sqrt{{\on{Vol}}(f)}}\leq
\frac{1}{2 C_2} \norm{f_t}_{G^P}.
\end{align*}
By integration one gets
\begin{align*}
\sqrt{{\on{Vol}}(f_1)}-\sqrt{{\on{Vol}}(f_0)} &=
\int_0^1 \partial_t \sqrt{{\on{Vol}}(f)}dt \leq
\int_0^1 \frac{1}{2 C_2} \norm{f_t}_{G^P} =
\frac{1}{2 C_2}\ L_{G^P}^{{\on{Imm}}}(f).
\end{align*}
Now the infimum over all paths $f:[0,1] \rightarrow {\on{Imm}}$ with $\pi(f(0))=F_0$ and $\pi(f(1))=F_1$ is taken.
\end{proof}
\subsection{Non-vanishing geodesic distance}\label{ge:no}
Using the estimates proven above and the fact that the area swept out separates points at least on $B_e$,
one gets the following result:
\begin{thm*}
The Sobolev type metric $G^P$ induces non-vanishing geodesic distance on $B_e$ if it is
stronger or as strong as the $H^1$-metric, i.e. if there is a constant $C > 0$ such that
\begin{align*}
\norm{h}_{G^P} \geq C \norm{h}_{H^1} =
C \sqrt{\int_M \overline{g}\big( (1+\Delta) h,h \big) {\on{vol}}(g)} \qquad \text{for all $h \in T{\on{Imm}}$.}
\end{align*}
\end{thm*}
\begin{proof}
By lemma \ref{ge:ar1} we have
\begin{align*}
C_1 \ (\text{area swept out by $f$})
\leq \max_t \sqrt{{\on{Vol}}\big(f(t)\big)} . L_{G^P}^{{\on{Imm}}}(f).
\end{align*}
Now we use the Lipschitz continuity \ref{ge:li} of $\sqrt{{\on{Vol}}}$ and that area swept out separates
points on $B_e$.
\end{proof}
\section[Sobolev metrics induced by the Laplacian]{Sobolev metrics induced by the Laplace operator}\label{la}
The results on non-vanishing geodesic distance from the previous section lead us to consider
operators $P$ that are induced by the Laplacian operator:
$$P=1+A \Delta^p, \quad P \in \Gamma\big(L(T{\on{Imm}};T{\on{Imm}})\big)$$
for a constant $A>0$.
(See section~\ref{no:la} for the definition of the Laplacian that is used in this work.)
At every $f \in {\on{Imm}}$, $P_f$ is a positive, selfadjoint and bijective operator
of order $2p$ acting on $T_f{\on{Imm}} = \Gamma(f^*TN)$.
Note that $\Delta$ depends smoothly on the immersion $f$ via the pullback-metric $f^*\overline{g}$,
so that the same is true of $P$.
$P$ is invariant under the action of the reparametrization group $\on{Diff}(M)$.
It induces the Sobolev metric
$$G_f^P(h,k)=\int_M \overline{g}\big(P_f (h),k\big) {\on{vol}}(g)
=\int_M \overline{g}\Big(\big(1+A (\Delta^{f^*\overline{g}})^p \big)h,k\Big) {\on{vol}}(f^*\overline{g}). $$
When $A=1$ we write $H^p := G^{1+\Delta^p}$.
In this section we will calculate explicitly for $P=1+A \Delta^p$
the geodesic equation and conserved momenta that have been deduced
in section~\ref{so} for a general operator $P$. The hardest part will be the
partial integration needed for the adjoint of $\nabla P$.
As a result we will get explicit formulas that are ready to be implemented numerically.
\subsection{Other choices for $P$}\label{la:ot}
Other choices for $P$ are the operator $P=1+A (\nabla^*)^p \nabla^p$
corresponding to the metric
$$G_f^P(h,k)=\int_M \big(\overline{g}(h,k)+A \overline{g}(\nabla^p h,\nabla^p k) \big) {\on{vol}}(g),$$
and other operators that differ only in lower order terms. Since these operators all
have the same principal symbol, they induce equivalent metrics on each tangent space $T_f {\on{Imm}}$.
It would be interesting to know if the induced geodesic distances on $B_i$ are equivalent as well.
\subsection{Adjoint of $\nabla P$}\label{la:ad}
To find a formula for the geodesic equation one has to calculate the adjoint of $\nabla P$,
see section~\ref{so:ge}.
The following calculations at the same time show the existence of the adjoint.
It has been shown in section~\ref{so:ad}
that the invariance of the operator $P$ with respect to reparametrizations
determines the tangential part of the adjoint:
\begin{align*}
\adj{\nabla P}(h,k)\big)^\top &=\on{grad}^g \overline{g}(Ph,k)-\big(\overline{g}(Ph,\nabla k)+\overline{g}(\nabla h,Pk)\big)^\sharp.
\end{align*}
It remains to calculate its normal part using the variational formulas from section~\ref{va}.
In the following calculations there will be terms of the form $\on{Tr}(g^{-1} s_1g^{-1} s_2)$,
where $s_1,s_2$ are two-forms on $M$.
When the two-forms are seen as mappings $TM \to T^*M$, they can be composed with $g^{-1}:T^*M \to TM$.
Thus the expression under the trace is a mapping $TM \to TM$ to which the trace can be applied.
When one of the two-forms is vector valued, the same tensor components as before are contracted.
For example when $h \in \Gamma(f^*TN)$ then $s_2=\nabla^2 h$ is a two-form on $M$ with values in $f^*TN$.
Then in the expression $\on{Tr}(g^{-1}.s_1.g^{-1}.s_2)$ only $TM$ and $T^*M$ components are contracted,
whereas the $f^*TN$ component remains unaffected.
{\allowdisplaybreaks
\begin{align*}
&\int_M \overline{g}\big(m^\bot,\adj{\nabla P}(h,k)\big) {\on{vol}}(g)=
\int_M \overline{g}\big((\nabla_{m^\bot} P)h,k\big) {\on{vol}}(g)\\
&\quad=A\sum_{i=0}^{p-1}\int_M\overline{g}((\nabla_{m^\bot}\Delta)\Delta^{p-i-1}h ,\Delta^{i}k ){\on{vol}}(g)\\
&\quad =A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}\Big(g^{-1}.D_{(f,m^\bot)}g.g^{-1}\nabla^2\Delta^{p-i-1}h\Big) ,\Delta^{i}k \Big){\on{vol}}(g)\\
&\qquad\qquad -A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\nabla_{\big(\nabla^*(D_{(f,m^\bot)}g)+\frac12
d\on{Tr}^g(D_{(f,m^\bot)}g)\big)^\sharp}\Delta^{p-i-1}h ,\Delta^{i}k \Big){\on{vol}}(g)\\
&\qquad\qquad+A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\nabla^*R^{\overline{g}}(m^\bot,Tf)\Delta^{p-i-1}h ,\Delta^{i}k \Big){\on{vol}}(g)\\
&\qquad\qquad-A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(m^\bot,Tf)\nabla\Delta^{p-i-1}h\big) ,\Delta^{i}k \Big){\on{vol}}(g)\\
&=
A\sum_{i=0}^{p-1}\int_M\on{Tr}\Big(g^{-1}.D_{(f,m^\bot)}g.g^{-1} \overline{g}(\nabla^2\Delta^{p-i-1}h,\Delta^{i}k )\Big) {\on{vol}}(g)\\
&\qquad\qquad -A\sum_{i=0}^{p-1}\int_M (g^0_1\otimes \overline{g})\Big(\nabla\Delta^{p-i-1}h ,(\nabla^*D_{(f,m^\bot)}g)\otimes\Delta^{i}k \Big){\on{vol}}(g)\\
&\qquad\qquad -A\sum_{i=0}^{p-1}\int_M (g^0_1\otimes \overline{g})\Big(\nabla\Delta^{p-i-1}h ,\frac12 d\on{Tr}^g(D_{(f,m^\bot)}g)\otimes\Delta^{i}k \Big){\on{vol}}(g)\\
&\qquad\qquad+A\sum_{i=0}^{p-1}\int_M(g^0_1\otimes \overline{g})\Big(R^{\overline{g}}(m^\bot,Tf)\Delta^{p-i-1}h ,\nabla\Delta^{i}k \Big){\on{vol}}(g)\\
&\qquad\qquad-A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(m^\bot,Tf)\nabla\Delta^{p-i-1}h\big) ,\Delta^{i}k \Big){\on{vol}}(g)
\end{align*}
Using the following symmetry property of the curvature tensor (see \cite[24.4.4]{MichorH}):
$$\overline{g}(R^{\overline{g}}(X,Y)Z,U)=-\overline{g}(R^{\overline{g}}(Y,X)Z,U)=-\overline{g}(R^{\overline{g}}(Z,U)Y,X)$$
yields:
\begin{align*}
&\int_M \overline{g}\big(m^\bot,\adj{\nabla P}(h,k)\big) {\on{vol}}(g)=\\
&\qquad=
A\sum_{i=0}^{p-1}\int_Mg^0_2\Big(D_{(f,m^\bot)}g,\overline{g}(\nabla^2\Delta^{p-i-1}h,\Delta^{i}k )\Big) {\on{vol}}(g)\\
&\qquad\qquad -A\sum_{i=0}^{p-1}\int_M g^0_1\Big(\overline{g}(\nabla\Delta^{p-i-1}h,\Delta^{i}k),\nabla^*D_{(f,m^\bot)}g \Big){\on{vol}}(g)\\
&\qquad\qquad -A\sum_{i=0}^{p-1}\int_M g^0_1\Big(\overline{g}(\nabla\Delta^{p-i-1}h,\Delta^{i}k) ,\frac12 \nabla\on{Tr}^g(D_{(f,m^\bot)}g) \Big){\on{vol}}(g)\\
&\qquad\qquad+A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(\Delta^{p-i-1}h,\nabla\Delta^{i}k)Tf\big) ,m^\bot \Big){\on{vol}}(g)\\
&\qquad\qquad-A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(\nabla\Delta^{p-i-1}h,\Delta^{i}k)Tf\big) , m^\bot\Big){\on{vol}}(g)\\
&\qquad=
A\sum_{i=0}^{p-1}\int_Mg^0_2\Big(D_{(f,m^\bot)}g,\overline{g}(\nabla^2\Delta^{p-i-1}h,\Delta^{i}k )\Big) {\on{vol}}(g)\\
&\qquad\qquad -A\sum_{i=0}^{p-1}\int_M g^0_2\Big(\nabla\overline{g}(\nabla\Delta^{p-i-1}h,\Delta^{i}k),D_{(f,m^\bot)}g \Big){\on{vol}}(g)\\
&\qquad\qquad -\frac{A}{2}\sum_{i=0}^{p-1}\int_M \Big(\nabla^*\overline{g}(\nabla\Delta^{p-i-1}h,\Delta^{i}k) \Big)\on{Tr}^g(D_{(f,m^\bot)}g) {\on{vol}}(g)\\
&\qquad\qquad+A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(\Delta^{p-i-1}h,\nabla\Delta^{i}k)Tf \big),m^\bot \Big){\on{vol}}(g)\\
&\qquad\qquad-A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(\nabla\Delta^{p-i-1}h,\Delta^{i}k)Tf\big) , m^\bot\Big){\on{vol}}(g)\\
&\qquad=
A\sum_{i=0}^{p-1}\int_Mg^0_2\Big(D_{(f,m^\bot)}g,\overline{g}(\nabla^2\Delta^{p-i-1}h,\Delta^{i}k )\Big) {\on{vol}}(g)\\
&\qquad\qquad -A\sum_{i=0}^{p-1}\int_M g^0_2\Big(\overline{g}(\nabla^2\Delta^{p-i-1}h,\Delta^{i}k),D_{(f,m^\bot)}g \Big){\on{vol}}(g)\\
&\qquad\qquad -A\sum_{i=0}^{p-1}\int_M g^0_2\Big(\overline{g}(\nabla\Delta^{p-i-1}h,\nabla\Delta^{i}k),D_{(f,m^\bot)}g \Big){\on{vol}}(g)\\
&\qquad\qquad -\frac{A}{2}\sum_{i=0}^{p-1}\int_M \Big(\nabla^*\overline{g}(\nabla\Delta^{p-i-1}h,\Delta^{i}k) \Big)\on{Tr}^g(D_{(f,m^\bot)}g) {\on{vol}}(g)\\
&\qquad\qquad+A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(\Delta^{p-i-1}h,\nabla\Delta^{i}k)Tf \big), m^\bot \Big){\on{vol}}(g)\\
&\qquad\qquad-A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(\nabla\Delta^{p-i-1}h,\Delta^{i}k)Tf\big) , m^\bot\Big){\on{vol}}(g)\\
&\qquad=
-A\sum_{i=0}^{p-1}\int_Mg^0_2\Big(D_{(f,m^\bot)}g,\overline{g}(\nabla\Delta^{p-i-1}h,\nabla\Delta^{i}k )\Big) {\on{vol}}(g)\\
&\qquad\quad -\frac{A}{2}\sum_{i=0}^{p-1}\int_M
\Big(\nabla^*\overline{g}(\nabla\Delta^{p-i-1}h,\Delta^{i}k) \Big)\on{Tr}^g(D_{(f,m^\bot)}g) {\on{vol}}(g)\\
&\qquad\qquad+A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(\Delta^{p-i-1}h,\nabla\Delta^{i}k)Tf \big), m^\bot \Big){\on{vol}}(g)\\
&\qquad\qquad-A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(\nabla\Delta^{p-i-1}h,\Delta^{i}k)Tf\big) , m^\bot\Big){\on{vol}}(g)\\
&\qquad=
-A\sum_{i=0}^{p-1}\int_Mg^0_2\Big(-2.\overline{g}(m^\bot,S),\overline{g}(\nabla\Delta^{p-i-1}h,\nabla\Delta^{i}k )\Big) {\on{vol}}(g)\\
&\qquad\qquad -\frac{A}{2}\sum_{i=0}^{p-1}\int_M \Big(\nabla^*\overline{g}(\nabla\Delta^{p-i-1}h,\Delta^{i}k) \Big)\on{Tr}^g\big(-2.\overline{g}(m^\bot,S)\big) {\on{vol}}(g)\\
&\qquad\qquad+A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(\Delta^{p-i-1}h,\nabla\Delta^{i}k)Tf \big),m^\bot \Big){\on{vol}}(g)\\
&\qquad\qquad-A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(\nabla\Delta^{p-i-1}h,\Delta^{i}k)Tf\big) , m^\bot\Big){\on{vol}}(g)
\\&\qquad=
\int_M \overline{g}\Big(m^\bot,2A\sum_{i=0}^{p-1}\on{Tr}\big(g^{-1} S g^{-1} \overline{g}(\nabla\Delta^{p-i-1}h,\nabla\Delta^{i}k ) \big)\Big)\\&\qquad\qquad
+\int_M \overline{g}\Big(m^\bot,A\sum_{i=0}^{p-1} \big(\nabla^*\overline{g}(\nabla\Delta^{p-i-1}h,\Delta^{i}k) \big) \on{Tr}^g(S)\Big) {\on{vol}}(g)\\
&\qquad\qquad+A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(\Delta^{p-i-1}h,\nabla\Delta^{i}k)Tf \big), m^\bot \Big){\on{vol}}(g)\\
&\qquad\qquad-A\sum_{i=0}^{p-1}\int_M\overline{g}\Big(\on{Tr}^g\big(R^{\overline{g}}(\nabla\Delta^{p-i-1}h,\Delta^{i}k)Tf\big) , m^\bot\Big){\on{vol}}(g).
\end{align*}
}
From this, one can read off the normal part of the adjoint.
Thus one gets:
\begin{lem*}
The adjoint of $\nabla P$ defined in section~\ref{so:ad} for the operator $P=1+A\Delta^p$ is
\begin{align*}
\adj{\nabla P}(h,k)&=
2A\sum_{i=0}^{p-1}\on{Tr}\big(g^{-1} S g^{-1} \overline{g}(\nabla\Delta^{p-i-1}h,\nabla\Delta^{i}k ) \big)
\\&\qquad
+A\sum_{i=0}^{p-1} \big(\nabla^*\overline{g}(\nabla\Delta^{p-i-1}h,\Delta^{i}k) \big) \on{Tr}^g(S)\\
&\qquad+A\sum_{i=0}^{p-1}\on{Tr}^g\big(R^{\overline{g}}(\Delta^{p-i-1}h,\nabla\Delta^{i}k)Tf \big)\\
&\qquad-A\sum_{i=0}^{p-1}\on{Tr}^g\big(R^{\overline{g}}(\nabla\Delta^{p-i-1}h,\Delta^{i}k)Tf \big)
\\&\qquad
+Tf.\Big[\on{grad}^g \overline{g}(Ph,k)-\big(\overline{g}(Ph,\nabla k)+\overline{g}(\nabla h,Pk)\big)^\sharp\Big].
\end{align*}
\end{lem*}
\subsection{Geodesic equations and conserved momentum}\label{la:ge}
The shortest and most convenient formulation of the geodesic equation is in terms
of the momentum $p=(1+A\Delta^p)f_t \otimes {\on{vol}}(g)$, see sections~\ref{so:gemo} and \ref{so:geshmo}.
\begin{thm*}
The geodesic equation on ${\on{Imm}}(M,N)$ for the $G^P$-metric
with $P=1+A \Delta^p$ is given by:
$$\left\{\begin{aligned}
p &= (1+A \Delta^p)f_t \otimes {\on{vol}}(g), \\
\nabla_{\partial_t}p&=\Bigg(
A\sum_{i=0}^{p-1}\on{Tr}\big(g^{-1} S g^{-1} \overline{g}(\nabla(\Delta^{p-i-1}f_t),\nabla\Delta^{i}f_t ) \big)\\&\quad
+\frac{A}{2}\sum_{i=0}^{p-1}\big(\nabla^*\overline{g}(\nabla(\Delta^{p-i-1}f_t),\Delta^{i}f_t) \big).\on{Tr}^g(S)\\&\quad
+2A\sum_{i=0}^{p-1}\on{Tr}^g\big(R^{\overline{g}}(\Delta^{p-i-1}f_t,\nabla\Delta^{i}f_t)Tf\big)\\&\quad
-\frac12\overline{g}(Pf_t,f_t) \on{Tr}^g(S)
-Tf.\overline{g}(Pf_t,\nabla f_t)^\sharp\Bigg) \otimes {\on{vol}}(g).
\end{aligned}\right.$$
This equation is well-posed by theorem \ref{so:we} since all
conditions are satisfied.
For the special case of plane curves, this agrees with the geodesic equation calculated in
\cite[section~4.2]{Michor107}.
\end{thm*}
$P=1+A\Delta^p$ and consequently $G^P$ are invariant under the action of the reparametrization group
${\on{Diff}}(M)$. According to section~\ref{so:mo} one gets:
\begin{thm*}
The momentum mapping for the action of ${\on{Diff}}(M)$ on ${\on{Imm}}(M,N)$
$$g\Big(\big((1+A\Delta^p)f_t \big)^\top\Big) \otimes {\on{vol}}(g)\in \Gamma(T^*M\otimes_M{\on{vol}}(M))$$
is constant along any geodesic $f$ in ${\on{Imm}}(M,N)$.
\end{thm*}
The horizontal geodesic equation for a general metric on ${\on{Imm}}$ has been derived in
section~\ref{sh:geshmo}. In section~\ref{so:geshmo} it has been shown that this equation
takes a very simple form. Now it is possible to write down this equation specifically for
the operator $P=1+A\Delta^p$:
\begin{thm*}
The geodesic equation on shape space for the Sobolev-metric $G^P$ with $P=1+A\Delta^p$ is equivalent
to the set of equations
\begin{equation*}
\left\{\begin{aligned}
p &= Pf_t \otimes {\on{vol}}(g), \qquad Pf_t = (Pf_t)^\bot, \\
(\nabla_{\partial_t}p)^{\on{hor}} &=
\Bigg(A\sum_{i=0}^{p-1}\on{Tr}\big(g^{-1} S g^{-1} \overline{g}(\nabla\Delta^{p-i-1}f_t,\nabla\Delta^{i}f_t ) \big)
\\&\qquad+ \frac{A}{2}\sum_{i=0}^{p-1} \big(\nabla^*\overline{g}(\nabla\Delta^{p-i-1}f_t,\Delta^{i}f_t) \big) \on{Tr}^g(S)
\\&\qquad+2A\sum_{i=0}^{p-1}\on{Tr}^g\big(R^{\overline{g}}(\Delta^{p-i-1}f_t,\nabla\Delta^{i}f_t)Tf\big)
\\&\qquad-\frac12\overline{g}(Pf_t,f_t).\on{Tr}^g(S) \Bigg) \otimes {\on{vol}}(g),
\end{aligned}\right.
\end{equation*}
where $f$ is a path of immersions.
For the special case of plane curves, this agrees with the geodesic equation calculated in
\cite[section~4.6]{Michor107}.
\end{thm*}
\section{Surfaces in $n$-space}\label{su}
This section is about the special case where the ambient space $N$ is $\mathbb R^n$.
The flatness of $\mathbb R^n$ leads to a simplification of the geodesic equation, and the
Euclidean motion group acting on $\mathbb R^n$ induces additional conserved quantities.
The vector space structure of $\mathbb R^n$ allows to define a
Fr\'echet metric.
This metric will be compared to Sobolev metrics.
Finally in section~\ref{su:co} the space of concentric hyper-spheres in $\mathbb R^n$
is briefly investigated.
\subsection{Geodesic equation}\label{su:ge}
The covariant derivative $\nabla^{\overline{g}}$ on $\mathbb R^n$ is but the usual derivative.
Therefore the covariant derivatives $\nabla_{\partial_t} f_t$ and $\nabla_{\partial_t}p$ in the geodesic
equation can be replaced by $f_{tt}$ and $p_t$, respectively.
(Note that ${\on{Imm}}(M,\mathbb R^n)$ is an open subset of the Fr\'echet vector space
$C^\infty(M,\mathbb R^n)$.)
Also, the curvature terms disappear because $\mathbb R^n$ is flat.
Any of the formulations of the geodesic equation presented so far can thus be
adapted to the case $N=\mathbb R^n$.
We want to show how the geodesic equation simplifies further under the additional assumptions
that $\dim(M)=\dim(N)-1$ and that $M$ is orientable. Then it is possible define a unit
vector field $\nu$ to $M$. The condition that $f_t$ is horizontal then simplifies
to $Pf_t = a.\nu$ for $a \in C^\infty(M)$.
The geodesic equation can then be written as an equation for $a$.
However, the equation is slightly simpler
when it is written as an equation for $a.{\on{vol}}(g)$.
In practise, ${\on{vol}}(g)$ can be treated as a function on $M$ because
one can identify ${\on{vol}}(g)$ with its density with respect to $du^1 \wedge \ldots \wedge du^{n-1}$,
where $(u^1, \ldots, u^{n-1})$ is a chart on $M$. Thus multiplication by ${\on{vol}}(g)$
does not pose a problem.
\begin{thm*}
The geodesic equation for a Sobolev-type metric $G^P$ on shape space
$B_i(M,\mathbb R^n)$ with $\dim(M)=n-1$
is equivalent to the set of equations
\begin{equation*}
\left\{\begin{aligned}
Pf_t &= a.\nu \\
\partial_t\big(a.{\on{vol}}(g)\big) &= \frac12 \overline{g}\big(\adj{\nabla P}(f_t,f_t),\nu\big)
-\frac12 \overline{g}(Pf_t,f_t) \overline{g}\big(\on{Tr}^g(S),\nu\big),
\end{aligned}\right.
\end{equation*}
where $f$ is a path in ${\on{Imm}}(M,\mathbb R^n)$ and $a$ is a time-dependent function on $M$.
\end{thm*}
\begin{proof}
Applying $\overline{g}(\cdot,\nu)$ to the geodesic equation \ref{so:geshmo}
on shape space in terms of the momentum one gets
\begin{align*}
\partial_t\big(a.{\on{vol}}(g)\big) &=
\partial_t\ \overline{g}\big(Pf_t \otimes {\on{vol}}(g),\nu\big) \\&=
\overline{g}\Big(\nabla_{\partial_t}\big(Pf_t \otimes {\on{vol}}(g)\big),\nu\Big)
+ \overline{g}\big(Pf_t \otimes {\on{vol}}(g),\nabla_{\partial_t} \nu\big) \\&=
\frac12 \overline{g}\big(\adj{\nabla P}(f_t,f_t),\nu\big) -\frac12 \overline{g}(Pf_t,f_t) \overline{g}\big(\on{Tr}^g(S),\nu\big)+ 0.
\qedhere
\end{align*}
\end{proof}
Let us spell this equation out in even more details for the $H^1$-metric.
This is the case of interest for the numerical examples in section~\ref{nu}.
\begin{thm*}
The geodesic equation on shape space $B_i(M,\mathbb R^n)$
for the Sobolev-metric $G^P$ with $P=1+A\Delta$ is equivalent
to the set of equations
\begin{equation*}
\left\{\begin{aligned}
Pf_t &= a.\nu \\
\partial_t \big(a.{\on{vol}}(g)\big) &=
\Big(A g^0_2\big(s, \overline{g}(\nabla f_t,\nabla f_t ) \big)
-\frac{\on{Tr}(L)}{2} \big(\norm{f_t}_{\overline{g}}^2 + A \norm{\nabla f_t}_{g^0_1\otimes\overline{g}}^2 \big)
\Big) {\on{vol}}(g),
\end{aligned}\right.
\end{equation*}
where $f$ is a path of immersions,
$a$ is a time-dependent function on $M$,
$s=\overline{g}(S,\nu) \in \Gamma(T^0_2 M)$ is the shape operator,
$L=g^{-1} s \in \Gamma(T^1_1 M)$ is the Weingarten mapping,
and $\on{Tr}(L)$ is the mean curvature.
\end{thm*}
\begin{proof}
The fastest way to get to this equation is to apply $\overline{g}(\cdot,\nu)$ to
the geodesic equation on ${\on{Imm}}$ from section~\ref{la:ge}.
This yields
\begin{align*}
\partial_t \big(a.{\on{vol}}(g)\big) &=
\Big(A \on{Tr}\big(g^{-1}.s.g^{-1} \overline{g}(\nabla f_t,\nabla f_t ) \big)
+ \frac{A}{2} \big(\nabla^*\overline{g}(\nabla f_t,f_t) \big) \on{Tr}(L)\\&\qquad
-\frac12\overline{g}(Pf_t,f_t).\on{Tr}(L) \Big) {\on{vol}}(g)\\&=
\Big(A g^0_2\big(s, \overline{g}(\nabla f_t,\nabla f_t ) \big)
- \frac{A}{2} \on{Tr}^g\big(\overline{g}(\nabla^2 f_t,f_t) \big) \on{Tr}(L)\\&\qquad
- \frac{A}{2} \on{Tr}^g\big(\overline{g}(\nabla f_t,\nabla f_t) \big) \on{Tr}(L)
-\frac12\overline{g}\big((1+A\Delta)f_t,f_t\big)\on{Tr}(L) \Big) {\on{vol}}(g)\\&=
\Big(A g^0_2\big(s, \overline{g}(\nabla f_t,\nabla f_t ) \big)
- \frac{A}{2} \on{Tr}^g\big(\overline{g}(\nabla f_t,\nabla f_t) \big) \on{Tr}(L)\\&\qquad
-\frac12\overline{g}\big(f_t,f_t\big).\on{Tr}(L) \Big) {\on{vol}}(g).
\end{align*}
Notice that the second order derivatives of $f_t$ have canceled out.
\end{proof}
\subsection{Additional conserved momenta}
If $P$ is invariant under the action of the Euclidean motion group $\mathbb R^n\rtimes\on{SO}(n)$,
then also the metric $G^P$ is in invariant under this group action
and one gets additional conserved quantities as described in \cite[section~2.5]{Michor107}:
\begin{thm*}
For an operator $P$ that is invariant under the action of the
Euclidean motion group $\mathbb R^n\rtimes\on{SO}(n)$,
the linear momentum
$$ \int_M Pf_t {\on{vol}}(g)\in(\mathbb R^n)^* $$
and the angular momentum
\begin{align*}
\forall X\in \mathfrak{so}(n): \int_M \overline{g}( X.f,Pf_t ) {\on{vol}}(g) \\
\text{or equivalently } \int_M (f\wedge Pf_t )
{\on{vol}}(g)\in\textstyle{\bigwedge^2}\mathbb R^n\cong \mathfrak{so}(n)^*
\end{align*}
are constant along any geodesic $f$ in ${\on{Imm}}(M,\mathbb R^n)$.
The operator $P=1+A \Delta^p$ satisfies this property.
\end{thm*}
\subsection{Fr\'echet distance and Finsler metric}\label{su:fr}
The Fr\'echet distance on shape space $B_i(M,\mathbb R^n)$ is defined as
\begin{align*}
{\on{dist}}_\infty^{B_i}(F_0,F_1) = \inf_{f_0,f_1} \norm{f_0 - f_1}_{L^\infty},
\end{align*}
where the infimum is taken over all $f_0, f_1$ with $\pi(f_0)=F_0, \pi(f_1)=F_1$.
As before, $\pi$ denotes the projection $\pi:{\on{Imm}} \to B_i$.
Fixing $f_0$ and $f_1$, one has
\begin{align*}
{\on{dist}}_\infty^{B_i}\big(\pi(f_0),\pi(f_1)\big) = \inf_{\varphi} \norm{f_0 \circ \varphi - f_1}_{L^\infty},
\end{align*}
where the infimum is taken over all $\varphi \in \on{Diff}(M)$.
The Fr\'echet distance is related to the Finsler metric
\begin{align*}
G^\infty : T {\on{Imm}}(M,\mathbb R^n) \rightarrow \mathbb R, \qquad h \mapsto \norm{h^\bot}_{L^\infty}.
\end{align*}
\begin{lem*}
The pathlength distance induced by the Finsler metric $G^\infty$
provides an upper bound for the Fr\'echet distance:
\begin{align*}
{\on{dist}}_\infty^{B_i}(F_0,F_1) \leq {\on{dist}}_{G^\infty}^{B_i}(F_0,F_1) = \inf_f \int_0^1 \norm{f_t}_{G^\infty} dt,
\end{align*}
where the infimum is taken over all paths
$$f:[0,1] \to {\on{Imm}}(M,\mathbb R^n) \quad \text{with} \quad \pi(f(0))=F_0, \pi(f(1))=F_1.$$
\end{lem*}
\begin{proof}
Since any path $f$ can be reparametrized such that $f_t$ is normal to $f$, one has
$$\inf_f \int_0^1 \norm{f_t^\bot}_{L^\infty} dt = \inf_f \int_0^1 \norm{f_t}_{L^\infty} dt, $$
where the infimum is taken over the same class of paths $f$ as described above. Therefore
\begin{align*}
{\on{dist}}_\infty^{B_i}(F_0,F_1) &= \inf_f \norm{f(1)-f(0)}_{L^\infty}
= \inf_f \norm{ \int_0^1 f_t dt}_{L^\infty}
\leq \inf_f \int_0^1 \norm{f_t}_{L^\infty} dt \\ &
= \inf_f \int_0^1 \norm{f_t^\bot}_{L^\infty} dt
= {\on{dist}}_{G^\infty}^{B_i}(F_0,F_1). \qedhere
\end{align*}
\end{proof}
It is claimed in \cite[theorem~13]{MennucciYezzi2008} that $d_\infty={\on{dist}}_{G^\infty}$. However,
the proof given there only works on the vector space $C^\infty(M,\mathbb R^n)$ and not on $B_i(M,\mathbb R^n)$.
The reason is that convex combinations of immersions are used in the proof,
but that the space of immersions is not convex.
\subsection{Sobolev versus Fr\'echet distance}\label{su:fr2}
It is a desirable property of any distance on shape space to be stronger than the Fr\'echet
distance. Otherwise, singular points of a shape could move arbitrarily far away
without increasing the distance much.
As the following result shows, Sobolev metrics of low order do not have this property.
The authors believe that this property is true
when the order of the metric is high enough, but were not able to prove this.
\begin{lem*}
Let $G^P$ be a metric on $B_i(M,\mathbb R^n)$ that is weaker than or at least as weak as a Sobolev $H^p$-metric with
$p < \frac{\dim(M)}2+1$, i.e.
\begin{align*}
\norm{h}_{G^P} \leq C \norm{h}_{H^p} = C \sqrt{\int_M \overline{g}\big( (1+\Delta^p) h,h \big) {\on{vol}}(g)}
\qquad \text{for all $h \in T{\on{Imm}}$.}
\end{align*}
Then the Fr\'echet distance can not be bounded by the $G^P$-distance.
\end{lem*}
\begin{proof}
It is sufficient to prove the claim for $P=1+\Delta^p$.
Let $f_0$ be a fixed immersion of $M$ into $\mathbb R^n$,
and let $f_1$ be a translation of $f_0$ by a vector $h$ of length $\ell$.
It will be shown that the $H^p$-distance between $\pi(f_0)$ and $\pi(f_1)$
is bounded by a constant $2L$ that does
not depend on $\ell$, where $\pi$ denotes the projection of ${\on{Imm}}$ onto $B_i$.
Then it follows that the $H^p$-distance can not be bounded from below by the Fr\'echet distance,
and this proves the claim.
For small $r_0$, one calculates the $H^p$-length of the following path of immersions:
First scale $f_0$ by a factor $r_0$, then translate it by $h$, and then
scale it again until it has reached $f_1$.
The following calculation shows that under the assumption $p<m/2+1$
the immersion $f_0$ can be scaled down to zero in finite $H^p$-pathlength $L$.
Let $r: [0,1] \to [0,1]$ be a function of time with $r(0)=1$ and $r(1)=0$.
\begin{align*}
L_{{\on{Imm}}}^{G^P}\big(r.f_0\big)&=\int_0^1\sqrt{\int_M\overline{g}\Big(r_t.\big(1+(\Delta^{(r.f_0)^*\overline{g}})^p\big)(f_0),r_t.f_0\Big){\on{vol}}\big((r.f_0)^*\overline{g}\big)}dt\\
&=\int_0^1\sqrt{\int_M r^2_t.\overline{g}\Big(\big(1+\frac{1}{r^{2p}}(\Delta^{f_0^*\overline{g}})^p\big)(f_0),f_0\Big)r^m{\on{vol}}\big(f_0^*\overline{g}\big)}dt\\
&=\int_1^0\sqrt{\int_M\overline{g}\Big(\big(1+\frac{1}{r^{2p}}(\Delta^{f_0^*\overline{g}})^p\big)(f_0),f_0\Big)r^m{\on{vol}}\big(f_0^*\overline{g}\big)}dr =: L
\end{align*}
The last integral converges if $\frac{m-2p}{2}<-1$, which holds by assumption.
Scaling down to $r_0>0$ needs even less effort.
So one sees that the length of the shrinking and growing part of the path is bounded by $2L$.
The length of the translation is simply $\ell \sqrt{r_0^m {\on{Vol}}(f_0)}=O(r^{m/2})$
since the Laplacian of the constant vector field vanishes. Therefore
\begin{equation*}
{\on{dist}}_{B_i}^{G^P}\big(\pi(f_0),\pi(f_1)\big) \leq {\on{dist}}_{{\on{Imm}}}^{G^P}(f_0,f_1) \leq 2L. \qedhere
\end{equation*}
\end{proof}
\subsection{Concentric spheres}\label{su:co}
For a Sobolev type metric $G^P$ that is invariant under the action of the $SO(n)$ on $\mathbb R^n$,
the set of hyper-spheres in $\mathbb R^n$ with common center $0$
is a totally geodesic subspace of $B_i(S^{n-1},\mathbb R^n)$.
The reason is that it is the fixed point set
of the group $SO(n)$ acting on $B_i$ isometrically.
(One also needs uniqueness of solutions to the geodesic equation to prove that
the concentric spheres are totally geodesic.)
This section mainly deals with the case $P=1+\Delta^p$.
First we want to determine under what conditions the set of concentric spheres is geodesically complete
under the $G^P$-metric.
\begin{lem*}
The space of concentric spheres is complete with respect to the $G^P$
metric with $P=1+A\Delta^p$ iff $p\geq(n+1)/2$.
\end{lem*}
\begin{proof}
The space is complete
if and only if it is impossible to scale a sphere down to zero or up to
infinity in finite $G^P$ path-length.
So let $f$ be a path of concentric spheres.
It is uniquely described by its radius $r$. Its velocity is $f_t=r_t.\nu$, where $\nu$
designates the unit normal vector field.
One has
$$\overline{g}\big(g^{-1}.S,\nu\big)=:L=-\tfrac{1}{r}\on{Id}_{TM},
\quad \on{Tr}(L^k)=(-1)^k\tfrac{n-1}{r^k},\quad {\on{Vol}}=r^{n-1}\tfrac{n\pi^{n/2}}{\Gamma(n/2+1)}.$$
Keep in mind that $r$ and $r_t$ are constant functions on the sphere, so that all
derivatives of them vanish.
Therefore
\begin{align*}
\Delta \nu&=\nabla^*(\nabla \nu)=\nabla^*(-Tf.L)=\on{Tr}^g\Big(\nabla(Tf.L)\Big)
\\&=\on{Tr}^g\Big(\nabla(Tf).L\Big)+\on{Tr}^g\Big(Tf.(\nabla L)\Big)\\&=
\on{Tr}(L^2).\nu+ \on{Tr}^g\Big(Tf.\nabla(-\tfrac1r\on{Id}_{TM})\Big)=\frac{n-1}{r^2}.\nu+0
\end{align*}
and
\begin{align*}
Pf_t&=(1+A\Delta^p)(r_t.\nu)=r_t.\left(1+A\frac{(n-1)^p}{r^{2p}}\right).\nu.
\end{align*}
From this it is clear that the path $f$ is horizontal. Therefore its length as a path in $B_i$
is the same as its length as a path in ${\on{Imm}}$.
One calculates its length as in the proof of \ref{su:fr2}:
\begin{align*}
L^{G^P}_{B_i}(f)&=\int_0^1\sqrt{G_f^P(f_t,f_t)}dt=
\int_0^1\sqrt{\int_M r_t^2.\left(1+A\frac{(n-1)^p}{r^{2p}}\right){\on{vol}}(g)}dt\\&
=\int_0^1 |r_t|\sqrt{ \left(1+A\frac{(n-1)^p}{r^{2p}}\right)\frac{n.\pi^{n/2}}{\Gamma(n/2+1)}r^{n-1}}dt\\&
=\sqrt{\frac{n.\pi^{n/2}}{\Gamma(n/2+1)}}\int_{r_0}^{r_1} \sqrt{\left(1+A\frac{(n-1)^p}{r^{2p}}\right)r^{n-1}}dr.
\end{align*}
The integral diverges for $r_1 \to \infty$ since the integrand is greater than $r^{(n-1)/2}$.
It diverges for $r_0 \to 0$ iff $(n-1-2p)/2 \leq -1$, which is equivalent to $p \geq (n+1)/2$.
\end{proof}
The geodesic equation within the space of concentric spheres
reduces to an ODE for the radius that can be read off
the geodesic equation in section \ref{la:ge}:
\begin{align*}
r_{tt}=-r_t^2\Big(\frac{n-1}{2r}-\frac{p.A.(n-1)^p}{r\big(r^{2p}+A(n-1)^p\big)}\Big).
\end{align*}
\section{Diffeomorphism groups}\label{di}
For $M=N$ the space ${\on{Emb}}(M,M)$ equals the \emph{diffeomorphism group of $M$}.
An operator $P \in \Gamma\big(L(T{\on{Emb}};T{\on{Emb}})\big)$ that is invariant under reparametrizations
induces a right-invariant Riemannian metric on this space.
Thus one gets the geodesic equation for
right-invariant Sobolev metrics on diffeomorphism groups and well-posedness of this equation.
To the authors knowledge, well-posedness has so far only been shown for the special case
$M=S^1$ in \cite{Constantin2003}
and for the special case of Sobolev order one metrics in \cite{GayBalmaz2009}.
Theorem~\ref{so:we} establishes this result for arbitrary compact $M$ and
Sobolev metrics of arbitrary order.
In the decomposition of a vector $h \in T_f{\on{Emb}}$ into its tangential and normal components
$h=Tf.h^\top + h^\bot$, the normal part $h^\bot$ vanishes.
Also $S=\nabla Tf$ vanishes.
Thus the geodesic equation on ${\on{Diff}}(M)$ in terms of the momentum $p$ is given by (see \ref{so:gemo})
\begin{equation*}
\left\{\begin{aligned}
p &= Pf_t \otimes {\on{vol}}(g), \\
\nabla_{\partial_t}p &=-Tf.\overline{g}(Pf_t,\nabla f_t)^\sharp \otimes {\on{vol}}(g).
\end{aligned}\right.
\end{equation*}
Note that this equation is not right-trivialized, in contrast to
the equation given in \cite{Arnold1966,Michor102,Michor109}, for example.
The special case of theorem \ref{so:we} now reads as follows:
\begin{thm*} Let $p\ge 1$ and $k>\frac{\dim(M)}2+1$ and let $P$ satisfy assumptions \thetag{1--3}
of \ref{so:we}.
Then the initial value problem for the geodesic equation
has unique local solutions in the Sobolev manifold $\on{Diff}^{k+2p}$ of
$H^{k+2p}$-diffeomorphisms. The solutions depend smoothly on $t$ and on the initial
conditions $f(0,\;.\;)$ and $f_t(0,\;.\;)$.
The domain of existence (in $t$) is uniform in $k$ and thus this
also holds in $\on{Diff}(M)$.
Moreover, in each Sobolev completion $\on{Diff}^{k+2p}$, the Riemannian exponential mapping $\exp^{P}$ exists
and is smooth on a neighborhood of the zero section in the tangent bundle,
and $(\pi,\exp^{P})$ is a diffeomorphism from a (smaller) neigbourhood of the zero
section to a neighborhood of the diagonal in
$\on{Diff}^{k+2p}\times \on{Diff}^{k+2p}$.
All these neighborhoods are uniform in $k>\dim(M)/2+1$ and can be chosen
$H^{k_0+2p}$-open, for $k_0 > \dim(M)/2+1$. Thus both properties of the exponential
mapping continue to hold in $\on{Diff}(M)$.
\end{thm*}
\section{Numerical results}\label{nu}
It is of great interest for shape comparison to solve the \emph{boundary value problem}
for geodesics in shape space.
When the boundary value problem can be solved,
then any shape can be encoded as the initial momentum of a geodesic
starting at a fixed reference shape.
Since the initial momenta all lie in the same vector space,
this also opens the way to statistics on shape space.
There are two approaches to solving the boundary value problem.
In \cite{Michor118} the first approach of minimizing \emph{horizontal path energy}
over the set of curves in ${\on{Imm}}$
connecting two fixed boundary shapes has been taken.
This has been done for several almost local metrics.
For these metrics it is straightforward to calculate the horizontal energy because
the horizontal bundle equals the normal bundle.
However, in the case of Sobolev type metrics
the horizontal energy involves the inverse of a differential operator (see section~\ref{so:ho}),
which makes this approach much harder.
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth-10pt]{bump_1}
\caption{Geodesic where a bump is formed out a flat plane.
The initial momentum is $a=\sin(x)\sin(y)$.
Time increases linearly from left to right. The final time is $t=5$. }
\label{nu:bump_1}
\end{figure}
\begin{figure}[h]
\centering
\begin{psfrags}
\def\PFGstripminus-#1{#1}%
\def\PFGshift(#1,#2)#3{\raisebox{#2}[\height][\depth]{\hbox{%
\ifdim#1<0pt\kern#1 #3\kern\PFGstripminus#1\else\kern#1 #3\kern-#1\fi}}}%
\providecommand{\PFGstyle}{}%
\psfrag{a08220A}[cr][cr]{\PFGstyle $0.8220$}%
\psfrag{a08220B}[Bc][Bc]{\PFGstyle $0.8220$}%
\psfrag{a08220}[Bc][Bc][1.][0.]{\PFGstyle $0.8220$}%
\psfrag{a08225A}[cr][cr]{\PFGstyle $0.8225$}%
\psfrag{a08225B}[Bc][Bc]{\PFGstyle $0.8225$}%
\psfrag{a08225}[Bc][Bc][1.][0.]{\PFGstyle $0.8225$}%
\psfrag{a08230A}[cr][cr]{\PFGstyle $0.8230$}%
\psfrag{a08230B}[Bc][Bc]{\PFGstyle $0.8230$}%
\psfrag{a08230}[Bc][Bc][1.][0.]{\PFGstyle $0.8230$}%
\psfrag{a08235A}[cr][cr]{\PFGstyle $0.8235$}%
\psfrag{a08235B}[Bc][Bc]{\PFGstyle $0.8235$}%
\psfrag{a08235}[Bc][Bc][1.][0.]{\PFGstyle $0.8235$}%
\psfrag{a08240A}[cr][cr]{\PFGstyle $0.8240$}%
\psfrag{a08240B}[Bc][Bc]{\PFGstyle $0.8240$}%
\psfrag{a08240}[Bc][Bc][1.][0.]{\PFGstyle $0.8240$}%
\psfrag{a08245A}[cr][cr]{\PFGstyle $0.8245$}%
\psfrag{a08245B}[Bc][Bc]{\PFGstyle $0.8245$}%
\psfrag{a08245}[Bc][Bc][1.][0.]{\PFGstyle $0.8245$}%
\psfrag{a08250A}[cr][cr]{\PFGstyle $0.8250$}%
\psfrag{a08250B}[Bc][Bc]{\PFGstyle $0.8250$}%
\psfrag{a08250}[Bc][Bc][1.][0.]{\PFGstyle $0.8250$}%
\psfrag{a0A}[Bc][Bc]{\PFGstyle $0$}%
\psfrag{a0}[Bc][Bc][1.][0.]{\PFGstyle $0$}%
\psfrag{a1A}[tc][tc]{\PFGstyle $1$}%
\psfrag{a1B}[Bc][Bc]{\PFGstyle $1$}%
\psfrag{a1}[Bc][Bc][1.][0.]{\PFGstyle $1$}%
\psfrag{a2A}[tc][tc]{\PFGstyle $2$}%
\psfrag{a2B}[Bc][Bc]{\PFGstyle $2$}%
\psfrag{a2}[Bc][Bc][1.][0.]{\PFGstyle $2$}%
\psfrag{a3A}[tc][tc]{\PFGstyle $3$}%
\psfrag{a3B}[Bc][Bc]{\PFGstyle $3$}%
\psfrag{a3}[Bc][Bc][1.][0.]{\PFGstyle $3$}%
\psfrag{a4A}[tc][tc]{\PFGstyle $4$}%
\psfrag{a4B}[Bc][Bc]{\PFGstyle $4$}%
\psfrag{a4}[Bc][Bc][1.][0.]{\PFGstyle $4$}%
\psfrag{a5A}[tc][tc]{\PFGstyle $5$}%
\psfrag{a5B}[Bc][Bc]{\PFGstyle $5$}%
\psfrag{a5}[Bc][Bc][1.][0.]{\PFGstyle $5$}%
\psfrag{eA}[Bc][Bc]{\PFGstyle $G^P(f_t,f_t)$}%
\psfrag{e}[bc][bc]{\PFGstyle $G^P(f_t,f_t)$}%
\psfrag{tA}[Bc][Bc]{\PFGstyle $t$}%
\psfrag{t}[cl][cl]{\PFGstyle $t$}%
\includegraphics[width=\textwidth-10pt]{energyplot}
\end{psfrags}
\caption{Conservation of the energy $G^P(f_t,f_t)$ along the geodesic in figure~\ref{nu:bump_1}.
The true value of $G^P(f_t,f_t)$ is $\tfrac{\pi^2}{4(1+2A)} \approx 0.822467$ for $A=1$.
The maximum time step used in blue and green is 0.1. For purple and cyan it is 0.05. The number
of grid points used in blue and cyan is 100 times 100. For green and purple it is 200 times 200.}
\label{nu:energyplot}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth-10pt]{A_geodesic}
\caption{Letter A forming along a geodesic path. Time increases linearly from left to right.
The final time is $t=0.8$.
Top and bottom row are different views of the same geodesic. }
\label{nu:A_ge}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth-10pt]{A_initial}
\caption{Initial velocity $f_t(0,\cdot)$ and momentum $a(0,\cdot)$ of the geodesic in figure \ref{nu:A_ge}.
Both are shown first from above, then from the side. }
\label{nu:A_in}
\end{figure}
\begin{figure}[h]
\centering
\includegraphics[width=\textwidth-10pt]{selfintersection}
\caption{A self-intersection forming along a geodesic. Time increases linearly from left to right. }
\label{nu:se}
\end{figure}
The second approach is the method of \emph{geodesic shooting}.
This method is based on iteratively solving the initial value problem
while suitably adapting the initial conditions.
The theoretical requirements of existence of solutions to the geodesic equation and smooth dependence on
initial conditions are met for Sobolev type metrics, see section \ref{so:we}.
This makes geodesic shooting a promising approach.
{\it The first step towards this aim is to numerically solve the initial value problem for geodesics,
at least for the $H^1$-metric and the case of surfaces in $\mathbb R^3$,
and that is what will be presented in this work.
}
The geodesic equation on shape space is equivalent to the horizontal geodesic equation
on the space of immersions. For the case of surfaces in $\mathbb R^3$,
it takes the form given in section~\ref{su:ge}.
This equation can be conveniently set up
using the DifferentialGeometry package incorporated in the computer algebra system Maple
as demonstrated in figure~\ref{nu:maple}.
(The equations that have actually been solved were simplified
by multiplying intermediate terms with suitable powers of
$\sqrt{{\on{vol}}(g)}$, but for the sake of clearness this has not been included in the Maple code
in figure~\ref{nu:maple}.)
\begin{figure}[h]
\lstset{basicstyle=\small\ttfamily,
backgroundcolor=\color[gray]{0.9},
numbers=left, numberstyle=\scriptsize, stepnumber=2, numbersep=5pt,
commentstyle=\small,columns=flexible,
showstringspaces=false}
\begin{lstlisting}
with(DifferentialGeometry);with(Tensor);with(Tools);
DGsetup([u,v],[x,y,z],E);
f := evalDG(f1(t,u,v)*D_x+f2(t,u,v)*D_y+f3(t,u,v)*D_z);
G := evalDG(dx &t dx + dy &t dy + dz &t dz);
Gamma_vrt := 0 &mult Connection(dx &t D_x &t du);
Tf := CovariantDerivative(f,Gamma_vrt);
g:=ContractIndices(G &t Tf &t Tf,[[1,3],[2,5]]);
g_inv:=InverseMetric(g);
Gamma_bas:=Christoffel(g);
det:=Hook([D_u,D_u],g)*Hook([D_v,D_v],g)-Hook([D_u,D_v],g)^2;
ft := evalDG(diff(f1(t,u,v),t)*D_x+diff(f2(t,u,v),t)*D_y
+diff(f3(t,u,v),t)*D_z);
ft := convert(ft,DGtensor);
S := CovariantDerivative(Tf,Gamma_vrt,Gamma_bas);
cross:=evalDG((dy &w dz) &t D_x + (dz &w dx) &t D_y + (dx &w dy) &t D_z);
N:=Hook([ContractIndices(Tf &t D_u,[[2,3]]),
ContractIndices(Tf &t D_v,[[2,3]])],cross);
nu:=convert(N/sqrt(Hook([N,N],G)),DGvector);
s := ContractIndices(G &t S &t nu, [[1,3],[2,6]]);
L := ContractIndices(g_inv &t s,[[2,3]]);
Gftft := ContractIndices(G &t ft &t ft,[[1,3],[2,4]]);
Cft := CovariantDerivative(ft,Gamma_vrt);
CCft := CovariantDerivative(Cft,Gamma_vrt,Gamma_bas);
Dft := ContractIndices(-g_inv &t CCft,[[1,4],[2,5]]);
Pft := evalDG(ft+A*Dft);
GCftCft := ContractIndices(Cft &t Cft &t G,[[1,5],[3,6]]);
gGCftCft := ContractIndices(g_inv &t GCftCft,[[2,3]]);
TrLgGCftCft := ContractIndices(L &t gGCftCft,[[1,4],[2,3]]);
TrgGCftCft := ContractIndices(gGCftCft,[[1,2]]);
TrL := ContractIndices(L,[[1,2]]);
# b(t,u,v) := a(t,u,v)*sqrt(det);
eq1 := ContractIndices(Pft &t dx,[[1,2]])*sqrt(det) =
b(t,u,v)*ContractIndices(nu &t dx,[[1,2]]);
eq2 := ContractIndices(Pft &t dy,[[1,2]])*sqrt(det) =
b(t,u,v)*ContractIndices(nu &t dy,[[1,2]]);
eq3 := ContractIndices(Pft &t dz,[[1,2]])*sqrt(det) =
b(t,u,v)*ContractIndices(nu &t dz,[[1,2]]);
eq4 := diff(b(t,u,v),t) =
(A*TrLgGCftCft - TrL/2*(Gftft+A*TrgGCftCft))*sqrt(det);
\end{lstlisting}
\caption{Maple source code to set up the geodesic equation.}
\label{nu:maple}
\end{figure}
Unfortunately, Maple (as of version 14) is not able to solve
PDEs with more than one space variable numerically.
Thus the equations were translated into Mathematica.
The PDE was solved using the method of lines. Spatial discretization was done
using an equidistant grid, and spatial derivatives were
replaced by finite differences.
The time-derivative $f_t$ appears implicitly in the equation
$P_f(f_t)=a.\nu$, and this remains so
when the operator $P_f$ is replaced by finite differences.
The solver that has been used is
the Implicit Differential-Algebraic
(IDA) solver that is part of the SUNDIALS suite and is integrated into Mathematica.
IDA uses backward differentiation of order 1 to 5 with
variable step widths. Order 5 is standard and has also been used here.
At each time step, the new value of $f_t$ is computed using some previous values of $f$, and then
the new value of $f$ is calculated from the equation $P_f( f_t)=a.\nu$.
The dependence on $f$ in this equation is of course highly nonlinear.
A Newton method is used to solve it.
This operation is quite costly and has to be done at every step,
which is a main disadvantage of backward differentiation algorithms.
Explicit methods are probably much better adapted
to the problem. The implementation of an explicit solver
is ongoing common work of the authors with Martins Bruveris and Colin Cotter.
In the examples that follow, $f$ at time zero is a square $[0,\pi]\times[0,\pi]$ flatly embedded in $\mathbb R^3$.
This is a manifold with boundary, but it can be seen as a part of a bigger closed manifold.
Zero boundary conditions are used for both $f$ and $a$.
It remains to specify an initial condition for $a$.
As a first example, let us assume that $a$ at time zero equals $\sin(x)\sin(y)$,
where $x,y$ are the Euclidean coordinates on the square.
The resulting geodesic is depicted in figure~\ref{nu:bump_1}.
In the absence of a closed-form solution of the geodesic equation, one way to
check if the solution is correct is to see if the energy $G^P(f_t,f_t)$
is conserved. Figure~\ref{nu:energyplot} shows
this for the geodesic from figure~\ref{nu:bump_1}
with various space and time discretizations.
A more complicated example of a geodesic is shown in figure \ref{nu:A_ge} and \ref{nu:A_in}.
There, the initial velocity was chosen to be a smoothened version of a black and white
picture of the letter A. The initial momentum was computed from it using a discrete
Fourier transform.
Finally, it is shown in figure \ref{nu:se} that self-intersections of the surface can occur.
This is not due to a numerical error but part of the theory,
and can be an advantage or a disadvantage depending
on the application.
\bibliographystyle{plain}
|
{
"timestamp": "2011-06-27T02:01:59",
"yymm": "1009",
"arxiv_id": "1009.3616",
"language": "en",
"url": "https://arxiv.org/abs/1009.3616"
}
|
\section{Introduction}
The idea that the fundamental scale of gravity can be as low as TeV has drawn a lot of interest among the physics community
recently. In some brane-world models \cite{ADD98} where this TeV scale gravity is realized, one can principally expect to see some stringy
effects in the upcoming TeV colliders and in addition the signature of space-time noncommutavity. Interests in the noncommutative(NC)
field theory arose from the pioneering work by Snyder \cite{Snyder47} and has been revived recently due to developments connected
to string theories in which the noncommutativity of space-time is an important characteristic of D-brane dynamics at the low energy
limit\cite{Connes98,Douglas98,SW99}. Although Douglas {\it et al.} \cite{Douglas98} in their pioneering work have shown that noncommutative field theory is a well-defined quantum field theory, the question that remains is whether the string theory prediction and the noncommutative effect can be seen at the energy scale attainable in present or near future experiments instead of the $4$-$d$ Planck scale $M_{pl}$. A notable work by Witten {\it et al.} \cite{Witten96} suggests that one can see some stringy effects by lowering the threshold value of commutativity to \,{\ifmmode\mathrm {TeV}\else TeV\fi}, a scale which is not so far from present or future collider scale.
~ What is space-time noncommutavity? It means space and time no longer commute with each other. Now writing the space-time coordinates as operators we find
\begin{equation}
[\hat{X}_\mu,\hat{X}_\nu]=i\Theta_{\mu\nu}
\label{NCSTh}
\end{equation}
where the matrix $\Theta_{\mu\nu}$ is real and antisymmetric. The NC parameter $\Theta_{\mu\nu}$ has dimension of area and reflects the extent to which the space-time coordinates are noncommutative i.e. fuzzy. Furthermore, introducing a NC scale $\Lambda$, we rewrite Eq. \ref{NCSTh} as
\begin{equation}
[\hat{X}_\mu,\hat{X}_\nu]=\frac{i}{\Lambda^2} c_{\mu\nu}
\label{NCST}
\end{equation}
where $\Theta_{\mu\nu}(=c_{\mu \nu}/\Lambda)$ and $c_{\mu\nu}$ has the same properties as $\Theta_{\mu\nu}$. To study an ordinary field theory in such a noncommutative fuzzy space, one replaces all ordinary products among the field variables with Moyal-Weyl(MW)
\cite{Douglas} $\star$ products defined by
\begin{equation}
(f\star
g)(x)=exp\left(\frac{1}{2}\Theta_{\mu\nu}\partial_{x^\mu}\partial_{y^\nu}\right)f(x)g(y)|_{y=x}.
\label{StarP}
\end{equation}
Using this we can get the noncommutative quantum electrodynamics(NCQED) Lagrangian as
\begin{equation} \label{ncQED}
{\cal L}=\frac{1}{2}i(\bar{\psi}\star \gamma^\mu D_\mu\psi
-(D_\mu\bar{\psi})\star \gamma^\mu \psi)- m\bar{\psi}\star
\psi-\frac{1}{4}F_{\mu\nu}\star F^{\mu\nu} \label{NCL},
\end{equation}
which are invariant under the following transformations
\begin{eqnarray}
\psi(x,\Theta) \rightarrow \psi'(x,\Theta) &=& U \star \psi(x,\Theta), \\
A_{\mu}(x,\Theta) \rightarrow A_{\mu}'(x,\Theta) &=& U \star A_{\mu}(x,\Theta) \star U^{-1} + \frac{i}{e} U \star \partial_\mu U^{-1},
\end{eqnarray}
where $U = (e^{i \Lambda})_\star$. In the NCQED Lagrangian [Eq.\ref{ncQED}]
$D_\mu\psi=\partial_\mu\psi-ieA_\mu\star\psi$,$~~(D_\mu\bar{\psi})=\partial_\mu\bar{\psi}+ie\bar{\psi}\star
A_\mu$, and $F_{\mu\nu}=\partial_{\mu} A_{\nu}-\partial_{\nu}
A_{\mu}-ie(A_{\mu}\star A_{\nu}-A_{\nu}\star A_{\mu})$.
The alternative is the Seiberg-Witten(SW)\cite{SW99,Douglas98,Connes98,Jurco}
approach in which both the gauge parameter $\Lambda$ and the gauge field $A^\mu$
is expanded as
\begin{eqnarray} \label{swps}
\Lambda_\alpha (x,\Theta) &=& \alpha(x) + \Theta^{\mu\nu} \Lambda^{(1)}_{\mu\nu}(x;\alpha) + \Theta^{\mu\nu} \Theta^{\eta\sigma} \Lambda^{(2)}_{\mu\nu\eta\sigma}(x;\alpha) + \cdot \cdot \cdot \\
A_\rho (x,\Theta) &=& A_\rho(x) + \Theta^{\mu\nu} A^{(1)}_{\mu\nu\rho}(x) + \Theta^{\mu\nu} \Theta^{\eta\sigma} A^{(2)}_{\mu\nu\eta\sigma\rho}(x) + \cdot \cdot \cdot
\end{eqnarray}
and when the field theory is expanded in terms of this power series Eq. (\ref{swps}) one ends up with an infinite tower of higher dimensional operators which renders the theory nonrenormalizable. However, the advantage is that this construction can be applied to any gauge theory with arbitrary matter representation. In the WM
approach the group closure property is only found to hold for the $U(N)$ gauge theories and the matter content is found to be in the (anti)-fundamental and adjoint representations.
Using the SW-map, Calmet {\it et al.} \cite{Calmet} first constructed a model with noncommutative gauge invariance which was close to the usual commuting standard model(CSM) and is known as the {\it minimal} noncommutative standard model(mNCSM) in which they listed several Feynman rules comprising NC interaction. Intense phenomenological searches \cite{Hewett01} have been made to unravel several interesting features of this mNCSM. Hewett {\it et al.} explored several processes e.g.
$e^+ e^- \rightarrow e^+ e^-$ (Bhabha), $e^- e^- \rightarrow e^- e^-$ (M\"{o}ller),
$e^- \gamma \rightarrow e^- \gamma$, $e^+ e^- \rightarrow \gamma \gamma$ (pair annihilation), $\gamma \gamma \rightarrow e^+ e^-$ and $\gamma \gamma \rightarrow \gamma \gamma$ in the context of NCQED and NCSM.
Recently, one of us has investigated the impact of $Z$ and photon exchange in the Bhabha and the M\"{o}ller scattering, which is reported in \cite{pdas}. Conroy {\it et al.} \cite{Conroy} have investigated the process $e^+ e^- \rightarrow \gamma \rightarrow \mu^+ \mu^-$ in the context of NCQED and predicted a reach of $\Lambda = 1.7$ TeV. In addition to the photon($\gamma$) exchange, we also consider the $s$-channel exchange of the $Z$ boson. Now in a generic NCQED the triple photon vertex arises to order
${\mathcal{O}}(\Theta)$, which however is absent in this mNCSM. Another formulation of the NCSM came to the forefront through the pioneering work by Melic {\it et al.} \cite{Melic:2005ep}
where such a triple neutral gauge boson coupling \cite{Trampetic} appears naturally in the gauge sector. We will call this the nonminimal version of NCSM or simply NCSM. The Feynman rules to order $\mathcal{O}(\Theta)$ were presented in their work \cite{Melic:2005ep}. In 2007, Alboteanu {\it et al} presented the $\mathcal{O}(\Theta^2)$ Feynman rules for the first time. In the present work we will confine ourselves within this nonminimal version of the NCSM and use the Feynman rules given in Alboteanu {\it et al.} \cite{Ana}.
In Sec. II we present the cross section of $e^+ e^- \rightarrow \gamma, Z \rightarrow \mu^+ \mu^-$. The numerical analysis and the prospects of TeV scale noncommutative geometry are discussed in Sec. III. Finally, we summarize our results in Sec. IV.
\section{$ e^+ e^-\rightarrow \mu^+ \mu^-$ scattering in the NCSM}
The muon pair production process $e^- (p_1) e^+ (p_2) \rightarrow \mu^- (p_3) \mu^+ (p_4)$ proceeds via the $s$ channel exchange of $\gamma$ and $Z$ bosons in the NCSM, like the standard model. The corresponding Feynman diagrams are shown in Fig. \ref{feyn}.
\begin{figure}[htbp]
\vspace{5pt}
\centerline{\hspace{-3.3mm}
{\epsfxsize=14cm\epsfbox{feyn.eps}}}
\hspace{2.5cm}
\vspace*{-0.5in}
\caption{Feynman diagrams for $ e^+ e^-\rightarrow \gamma,Z \rightarrow \mu^+ \mu^-$ in the NCSM.}
\protect\label{feyn}
\end{figure}
\noindent In order to have the cross section to order $\mathcal{O}(\Theta^2)$, we include the order $\mathcal{O}(\Theta^2)$ Feynman rule. The scattering amplitude to order $\Theta^2$ for the photon mediated diagram can
be written as
\begin{eqnarray} \label{gamma}
{\mathcal{A}}_\gamma = \frac{4 \pi \alpha}{s} \left[{\overline v}(p_2) \gamma_\mu u(p_1)\right]
\left[{\overline u}(p_3) \gamma^\mu v(p_4)\right] \times
\left[(1 - \frac{(p_2 \Theta p_1)^2}{8}) + \frac{i}{2} (p_2 \Theta p_1)\right] \nonumber \\ \times \left[(1 - \frac{(p_4 \Theta p_3)^2}{8} ) + \frac{i}{2} (p_4 \Theta p_3)\right]
\end{eqnarray}
and the same for the $Z$ boson mediated diagram as
\begin{eqnarray} \label{Z}
{\mathcal{A}}_Z = \frac{\pi \alpha}{\sin^2(2\theta_W) s_Z} \left[{\overline v}(p_2) \gamma_\mu (a + \gamma^5) {\overline u}(p_1)\right] \times \left[{\overline u}(p_3) \gamma^\mu (a + \gamma^5) {\overline v}(p_4)\right] \nonumber \\ \times \left[(1 - \frac{(p_2 \Theta p_1)^2}{8}) + \frac{i}{2} (p_2 \Theta p_1)\right] \times \left[(1 - \frac{(p_4 \Theta p_3)^2}{8} ) + \frac{i}{2} (p_4 \Theta p_3)\right]
\end{eqnarray}
\noindent where $s=(p_1 + p_2)^2$, $\alpha = e^2/4\pi$ and $\theta_W$ is the Weinberg angle, $a = 4 \sin^2(\theta_W) - 1$. In the above
$s_Z = s - m_Z^2 - i m_Z \Gamma_Z$, where $m_Z$ and $\Gamma_Z$ are the mass and decay width of the $Z$ boson. The Feynman rules required for the above scattering process are listed in Appendix A.
\noindent The spin-averaged squared-amplitude is given by
\begin{equation} \label{Ampsq}
{\overline {|{\mathcal{A}}|^2}} = {\overline {|{\mathcal{A}}_\gamma|^2}} + {\overline {|{\mathcal{A}}_Z|^2}} + 2 {\overline {Re({\mathcal{A}}_\gamma {\mathcal{A}}_Z^{ \dagger })}}.
\end{equation}
The different terms in the spin- averaged squared-amplitude are given in Appendix C. We use the Feynman rule to order ${\mathcal{O}}(\Theta^2)$ while calculating several squared-amplitude and interestingly we found that all lower order terms, i.e. ${\mathcal{O}}(\Theta)$, ${\mathcal{O}}(\Theta^2)$, and ${\mathcal{O}}(\Theta^3)$, get canceled (see Appendixes C and D for further discussions). With these the differential cross section can be written as
\begin{equation} \label{dsigma}
\frac{d \sigma}{d \Omega} = \frac{1}{64 \pi^2 s} {\overline {|{\mathcal{A}}|^2}},
\end{equation}
where $\sigma$ = $\sigma(\sqrt{s}, \Lambda, \theta, \phi)$. From Eq. \ref{dsigma} we can obtain $\sigma$, $ d\sigma/d\cos\theta $ and $ d\sigma/d\phi $ as
\begin{eqnarray}
\label{sigma}
\sigma &=& \int_{-1}^1 d(\cos\theta) \int_0^{2 \pi} d\phi \frac{d \sigma}{d \Omega}, \\
\label{dsdcostheta}
\frac{d\sigma}{d\cos\theta} &=& \int^{2 \pi}_0 d\phi \frac{d \sigma}{d \Omega}, \\
\label{dsdphi}
\frac{d\sigma}{d\phi} &=& \int^1_{-1} d(\cos\theta) \frac{d \sigma}{d \Omega}.
\end{eqnarray}
\section{Numerical Analysis}
In this section, we analyze the total cross section and angular distributions of the differential cross section in the presence of space-time non commutativity obtained in the earlier section. Before making a detailed analysis, let us make some general remarks regarding the observation of noncommutative effects. Since we assume $c_{\mu\nu}=(\xi_i,~\epsilon_{ijk}\chi^k)$,
where $\xi_i = (\vec{E})_i$ and $\chi_k = (\vec{B})_k$ are constant vectors in a frame that is stationary with respect to fixed stars, the vectors $(\vec{E})_i$ and $(\vec{B})_k$ point in fixed directions which are the same in all frames of reference. However, as the Earth rotates around its axis and revolves around the Sun, the direction of $\vec{E}$ and $\vec{B}$ will change continuously with time dependence which is a function of the coordinates of the laboratory. The observables that are measured will thus show a characteristic time dependence. It is important to be able to measure this time dependence to verify such noncommutative theories. In our analysis, we have assumed the vectors $\vec{E}= \frac{1}{\sqrt{3}} (\hat{i} + \hat{j} + \hat{k}) $ and $\vec{B}= \frac{1}{\sqrt{3}} (\hat{i} + \hat{j} + \hat{k})$ i.e. they behave like constant vectors. This can be true only at some instant time at most.
\subsection{Production cross section vs the machine energy in the NCSM}
In Fig.2 ~we show the total cross section $\sigma(e^- e^+ \rightarrow \mu^- \mu^+)$ as a function of the center-of-mass energy $E_{com}(=\sqrt{s})$ (GeV). The lowermost solid curve(in each of the two figures) corresponds to the CSM (recovered from the NCSM in the $\Lambda \longrightarrow \infty $ limit), whereas the uppermost (long-dashed) curve, next to the uppermost (short-dashed) and next-to-next uppermost (i.e. dotted) curves, arises in the NCSM with $\Lambda = 800,~900,$ and $1000$~GeV, respectively.
We observe that although the deviation from the commuting standard model is small at relatively lower energies, it starts becoming significant at $\sim 1400$ GeV and becomes more pronounced with the increase in machine energy.
Moreover we can also see that at a given machine energy $E_{com}$, the deviations become larger with smaller values of $\Lambda$.
\begin{figure}[htbp]
\vspace{-1.15in}
\centerline{\hspace{-12.3mm}
{\epsfxsize=9cm\epsfbox{csvenergy1.ps}} \hspace{-0.25in} {\epsfxsize=9cm\epsfbox{csvenergy2.ps}} }
\vspace*{-1.25in}
\caption{The cross section $\sigma(e^- e^+ \rightarrow \mu^- \mu^+)$ (fb) as a function of the machine energy $E_{com}=\sqrt{s}$ (in GeV). The figure on the right corresponds to $\sqrt{s} \ge 1400$~GeV. }
\protect\label{sigplot}
\end{figure}
\vspace*{-0.25in}
We made an estimate of the number of events per year($yr^{-1}$) in the case of ILC (International Linear Collider).
Assuming that the ILC will run for a year with the integrated luminosity $\mathcal{L} = 100~fb^{-1}$, the number of events $N_{SM}~$($yr^{-1}$) at $\sqrt{s} = 1750$ GeV in the CSM is expected to be
$N_{SM}(=\sigma \times {\mathcal{L}} = 36 \times 100) = 3600$ $yr^{-1}$. The expected number of events(signals)in the NCSM are given in Table 1. Fixing the machine energy $E_{cm}$ at $1750$ GeV, if we lower $\Lambda$ from $900$ GeV to $800$ GeV, the number of NC events $N$ per year increases from $3800$ $yr^{-1}$ to $4200$ $yr^{-1}$, which is larger than $N_{SM}(=3600~yr^{-1})$. Note that the NC signal is always larger than the SM background.
\newpage
\begin{center}
Table 1
\end{center}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$\Lambda$ & NC signal ($\sigma$)(fb) & ${\mathcal{L}}(fb^{-1})$ & N (events per year)\\
\hline
\hline
800 & 42 & 100 & 4200 \\
\hline
900 & 38 & 100 & 3800 \\
\hline
1000 & 37 & 100 & 3700 \\
\hline
\end{tabular}
\end{center}
\noindent {\it Table 1: Progressive reduction of the NC signal and the number of events per year with the increase in the NC scale $\Lambda$. The machine energy is fixed at $E_{com} = 1750$ GeV. The integrated luminosity of the ILC is assumed to be ${\mathcal{L}}=100~fb^{-1}$ $yr^{-1}$}.
\subsection{Angular distribution of muon pair production $e^- e^+ \rightarrow \mu^- \mu^+$ in the NCSM }
The angular distribution of the final state scattered particles is a useful tool to understand the nature of new physics. We will now see how the azimuthal distribution of the final state scattered particles can be used to separate out the noncommutative geometry, the NCSM, from the other type of new physics models e.g supersymmetry, brane-world gravity, unparticle scenario, little Higgs models etc.
\begin{figure}[htbp]
\vspace{-1.15in}
\centerline{\hspace{-12.3mm}
{\epsfxsize=9cm\epsfbox{csvphi.ps}} }
\vspace{-1.25in}
\caption{{The $ \frac{d\sigma}{d\phi} $($fb/rad$) distribution as a function of $\phi$(in rad). The machine energy $E_{com}(=\sqrt{s}$) is fixed at $1.75$ TeV. The lowest horizontal curve is due to the SM, whereas the plots above the horizontal one, as we move up correspond to $\Lambda = 1.0, 0.9,$ and $0.8$ TeV, respectively.}}
\protect\label{dsdphiplot}
\end{figure}
In Fig. \ref{dsdphiplot} we show $\frac{d\sigma}{d\phi}$ as a function of the azimuthal angle $\phi$. For the angular analysis study, we fixed the machine energy $E_{com}(=\sqrt{s}$) at $1.75$ TeV. The standard model which is completely $\phi$ symmetric, predicts a flat distribution for $d\sigma/d\phi$. The lowest horizontal curve establishes this fact. Other plots above the horizontal one, as we move up, correspond to $\Lambda = 1.0, 0.9 $ and $0.8$ TeV, respectively in the NCSM. The departure from the flat behavior is due to $ p_2 \Theta p_1$ and $p_4 \Theta p_3$ terms in Eqs. \ref{gamma} and \ref{Z} that bring in the $\phi$ dependence which is thus observed in Fig. \ref{dsdphiplot}. Interestingly, the curves show several maxima and minima. The largest maxima for each of the three curves is peaked at $\phi = 0.78$ rad, whereas the second largest maxima is found to be located at $\phi \sim 4$ rad. Two minimas are found: they are located at $\phi = 2.6$ rad and $5.3$ rad, respectively. Note that in each of the above three plots, if we set $\Lambda = \infty$, the lowest horizontal SM curve is recovered. It is worthwhile to note that such an azimuthal distribution clearly reflects the exclusive nature of the noncommutative geometry which is rarely to be found in other types/classes of new physics models and hence may serve as Occam's razor- either selecting or ruling out a class of new physics model(s).
We next analyze the polar distribution. In Fig. \ref{dsdcosthetaplot}, $\frac{d\sigma}{dcos\theta}$ is plotted as a function of $cos\theta$ with the machine energy $E_{com}$ being fixed at $1.75$ TeV.
\begin{figure}[htbp]
\vspace{-1.15in}
\centerline{\hspace{-14.3mm}
{\epsfxsize=9cm\epsfbox{csvcosth.ps}}}
\vspace{-1.15in}
\caption{The $ \frac{d\sigma}{dcos\theta} $($fb$) distribution as a function of $cos\theta$ is shown. The machine energy $E_{com}(=\sqrt{s}$) is fixed at $1.75$ TeV. The lowermost curve corresponds to the CSM, whereas the plots above the horizontal one, as we move up, correspond to $\Lambda = 1.0, 0.9,$ and $0.8$ TeV, respectively in the NCSM.}
\protect\label{dsdcosthetaplot}
\end{figure}
Note the asymmetry of the distribution around the $cos\theta = 0$ line of Fig. \ref{dsdcosthetaplot}. The lowermost plot in Fig. \ref{dsdcosthetaplot} corresponds to the standard model and the plots, as we move up correspond to $\Lambda = 1.0, 0.9,$ and $0.8$ TeV, respectively.
The uppermost curve in the figure corresponding to $\Lambda = 0.8$ TeV exhibits maximal deviation from the lowermost CSM curve (obtained by setting $\Lambda \rightarrow \infty$).
\section{Conclusion}
The idea that around the TeV scale the space and time coordinates become noncommutative(i.e. no longer commutative in nature) draws a lot of attention in the physics community. We explored the impact of space-time noncommutativity in the fundamental processes $e^+ e^- \rightarrow \gamma, Z \rightarrow \mu^+ \mu^-$. Interestingly, we found that when we use the ${\mathcal{O}}(\Theta^2)$ Feynman rules, the
${\mathcal{O}}(\Theta)$, ${\mathcal{O}}(\Theta^2)$ and ${\mathcal{O}}(\Theta^3)$ contributions to the cross section simply get canceled and the lowest order contribution to the cross section appears at ${\mathcal{O}}(\Theta^4)$. We made our analyses to this order.
The plots showing the total cross section as a function of the machine energy $E_{com}$ establish the fact that at and above $E_{com} \ge 1400$ GeV, one can expect to see the effect of noncommutative geometry at a linear collider. Setting the ILC energy at $E_{com} = 1750$ GeV, the NC scale $\Lambda = 0.8$ TeV and assuming an integrated luminosity about $100~fb^{-1}$, we estimate the signal (NC event) as about $4200$ per year and the background as (SM event) $3600$ per year.
The azimuthal distribution $d\sigma/d\phi$, completely $\phi$ symmetric (flat) in the SM, deviates substantially in the NCSM. The deviation increases as the NC scale
$\Lambda$ decreases. Such a nontrivial azimuthal distribution is a unique feature of the NCSM and is quite uncommon in other classes of new physics models. We also study the $d\sigma/dcos\theta$ distribution as a function of $cos\theta$. Clearly, the asymmetry around the $cos\theta = 0$ curve persists even when space-time is noncommutative. Thus the noncommutative geometry is quite rich in terms of its phenomenological implications and it is worthwhile to explore several other interesting processes, potentially relevant for the future International Linear Collider.
\vspace*{-0.15in}
\begin{acknowledgments}
The work of P.K.Das is supported by the DST Fast Track Project No. SR/FTP/PS-11/2006.
\end{acknowledgments}
\vspace*{-0.15in}
|
{
"timestamp": "2010-09-21T02:00:48",
"yymm": "1009",
"arxiv_id": "1009.3554",
"language": "en",
"url": "https://arxiv.org/abs/1009.3554"
}
|
\section{Introduction}
\label{sec:introduction}
\subsection{Triangulations, multitriangulations and $0$-$1$-fillings}
\sloppypar
The systematic study of $0$-$1$-fillings of polyominoes with
restricted chain lengths likely originates in an article by Jakob
Jonsson~\cite{Jonsson2005}. At first, he was interested in a
generalisation of triangulations, where the objects under
consideration are maximal sets of diagonals of the $n$-gon, such that
at most $k$ diagonals are allowed to cross mutually. Thus, in the
case $k=1$ one recovers ordinary triangulations. He realised these
objects as fillings of the staircase shaped polyomino with
row-lengths $n-1, n-2,\dots,1$ with zeros and ones. The condition
that at most $k$ diagonals cross mutually then translates into the
condition that the longest north-east chain in the filling has length
$k$, see Definition~\ref{dfn:fillings-and-chains}. Instead of
studying fillings of the staircase shape only, he went on to consider
more general shapes which he called \Dfn{stack} and \Dfn{moon
polyominoes}, see Definition~\ref{dfn:moon} and
Figure~\ref{fig:moon}.
For stack polyominoes he was able to prove that the number of maximal
fillings depends only on $k$ and the multiset of heights of the
columns, not on the particular shape of the polyomino. He
conjectured that this statement holds more generally for moon
polyominoes, which was eventually proved by the
author~\cite{Rubey2006} using a technique introduced by Christian
Krattenthaler~\cite{Krattenthaler2006} based on Sergey Fomin's growth
diagrams for the Robinson-Schensted-Knuth correspondence. However,
the proof given there is not fully bijective: what one would hope for
is a correspondence between fillings of any two moon polyominoes that
differ only by a permutation of the columns. This article is a step
towards this goal.
\subsection{RC-graphs and the subword complex}
RC-graphs (for \lq reduced word compatible sequence graphs\rq,
see~\cite{MR1281474}, also known as \lq pipe dreams\rq\
see~\cite{MR2180402}) were introduced by Sergey Fomin and Anatol
Kirillov~\cite{MR1394950} to prove various properties of Schubert
polynomials. Namely, for a given permutation $w$, the Schubert
polynomial $\mathfrak S_w$ can be regarded as the generating function
of rc-graphs, see Remark~\ref{rmk:Schubert}.
A different point of view is to consider them as facets of a certain
simplicial complex. Let $w_0$ be the long permutation $n\cdots21$,
and consider its reduced factorisation
$$Q=s_{n-1}\cdots s_2 s_1\; s_{n-1}\cdots s_3 s_2\; \cdots\cdots\; s_{n-1}s_{n-2}\; s_{n-1}.$$
Then the subword complex associated to $Q$ and $w$ introduced by
Allen Knutson and Ezra Miller~\cite{MR2180402,MR2047852} has as
facets those subwords of $Q$ that are reduced factorisations of $w$.
Subword complexes enjoy beautiful topological properties, which are
transferred by the main theorem of this article to the simplicial
complex of $0$-$1$-fillings, as observed by Christian
Stump~\cite{Stump2010}, see also the article by Luis Serrano and
Christian Stump~\cite{SerranoStump2010}.
The intimate connection between maximal fillings and rc-graphs
demonstrated by the main theorem of this article,
Theorem~\ref{thm:filling-dream}, \emph{should} not have come as a
surprise. Indeed, Sergey Fomin and Anatol Kirillov \cite{MR1471891}
established a connection between reduced words and reverse plane
partitions already thirteen years ago, which is not much less than
the case of Ferrers shapes in Theorem~\ref{thm:ne-se}. They even
pointed towards the possibility of a bijective proof using the
Edelman-Greene correspondence.
More recently, the connection between Schubert polynomials and
triangulations was noticed by Alexander Woo~\cite{Woo2004}. Vincent
Pilaud and Michel Pocchiola~\cite{PilaudPocchiola2009} discovered
rc-graphs (under the name \lq beam arrangements\rq) more generally
for multitriangulations, however, they were unaware of the theory of
Schubert polynomials. In particular, Theorem 3.18 of Vincent
Pilaud's thesis~\cite{Pilaud2010} (see also Theorem~21
of~\cite{PilaudPocchiola2009}) is a variant of our
Theorem~\ref{thm:filling-dream} for multitriangulations.
Finally, Christian Stump and the author of the present article became
aware of an article by Vincent Pilaud and Francisco
Santos~\cite{MR2471876} that describes the structure of
multitriangulations in terms of so-called $k$-stars (introduced by
Harold Coxeter). We then decided to translate this concept to the
language of fillings, and discovered pipe dreams yet again.
\section{Definitions}
\label{sec:definitions}
\subsection{Polyominoes}
\label{sec:polyominoes}
\begin{figure}[h]
\begin{equation*}
\begin{array}{ccc}
\young(:::\hfil,%
::\hfil\hfil\hfil,%
::\hfil\hfil\hfil\hfil,%
\hfil\hfil\hfil\hfil\hfil\hfil\hfil,%
\hfil\hfil\hfil\hfil\hfil\hfil\hfil,%
:\hfil\hfil\hfil\hfil\hfil\hfil,%
:::\hfil\hfil)
&
\young(\hfil\hfil\hfil\hfil\hfil\hfil\hfil,%
\hfil\hfil\hfil\hfil\hfil\hfil\hfil,%
::\hfil\hfil\hfil\hfil,%
::\hfil\hfil\hfil,%
:::\hfil)
&
\young(\hfil\hfil\hfil\hfil\hfil\hfil\hfil,%
\hfil\hfil\hfil\hfil\hfil\hfil\hfil,%
\hfil\hfil\hfil\hfil,%
\hfil\hfil\hfil,%
\hfil)
\end{array}
\end{equation*}
\caption{a moon-polyomino, a stack-polyomino and a Ferrers diagram}
\label{fig:moon}
\end{figure}
\begin{dfn}\label{dfn:polyominoes}
A \Dfn{polyomino} is a finite subset of the quarter plane
$\mathbb N^2$, where we regard an element of $\mathbb N^2$ as a
cell. A \Dfn{column} of a polyomino is the set of cells along a
vertical line, a \Dfn{row} is the set of cells along a horizontal
line. We are using \lq English\rq\ (or matrix) conventions for the
indexing of the rows and columns of polyominoes: the top row and
the left-most column have index $1$.
The polyomino is \Dfn{convex}, if for any two cells in a column
(rsp. row), the elements of $\mathbb N^2$ in between are also cells
of the polyomino. It is \Dfn{intersection-free}, if any two
columns are \Dfn{comparable}, {\it i.e.}, the set of row
coordinates of cells in one column is contained in the set of row
coordinates of cells in the other. Equivalently, it is
intersection-free, if any two rows are comparable.
For example, the polyomino
\begin{equation*}
\young(::\hfil,%
::\hfil\hfil\hfil,%
\hfil\hfil\hfil\hfil\hfil,%
\hfil\hfil\hfil\hfil,%
::\hfil)
\end{equation*}
is convex, but not intersection-free, since the first and the last
columns are incomparable.
\end{dfn}
\begin{dfn}\label{dfn:moon}
A \Dfn{moon polyomino} (or L-convex polyomino) is a convex,
intersection-free polyomino. Equivalently we can require that any
two cells of the polyomino can be connected by a path consisting of
neighbouring cells in the polyomino, that changes direction at most
once. A \Dfn{stack polyomino} is a moon-polyomino where all
columns start at the same level. A \Dfn{Ferrers diagram} is a
stack-polyomino with weakly decreasing row widths
$\lambda_1,\lambda_2,\dots,\lambda_n$, reading rows from top to
bottom.
Because a moon-polyomino is intersection free, the set of rows of
maximal length in a moon polyomino must be consecutive. We call
the set of rows including these and the rows above the \Dfn{top
half} of the polyomino. Similarly, the set of columns of maximal
length, and all columns to the right of these, is the \Dfn{right
half} of the polyomino. The intersection of the top and the
right half is the \Dfn{top right quarter} of $M$.
\end{dfn}
\subsection{Fillings and Chains}
\label{sec:fillings-chains}
\begin{dfn}\label{dfn:fillings-and-chains}
A \Dfn{$0$-$1$-filling} of a polyomino is an assignment of numbers
$0$ and $1$ to the cells of the polyomino. Cells containing $0$
are also called \Dfn{empty}.
A \Dfn{north-east chain} is a sequence of non-zero entries in a
filling such that the smallest rectangle containing all its
elements is completely contained in the moon polyomino and such
that for any two of its elements one is strictly to the right and
strictly above the other.
\end{dfn}
As it turns out, it is more convenient to draw dots instead of ones
and leave cells filled with zeros empty. Two examples of (rather
special) fillings of a moon polyomino are depicted in
Figure~\ref{fig:top-bot}. In both examples the length of the longest
north-east chain is $2$.
\begin{dfn}
$\Set F_{01}^{ne}(M, k)$ is the set of $0$-$1$-fillings of the moon
polyomino $M$ whose longest north-east chain has length $k$ and
that are \Dfn{maximal}, {\it i.e.}, assigning an empty cell a $1$
would create a north-east chain of length $k+1$. For a vector
$\Mat r$ of integers, $\Set F_{01}^{ne}(M, k, \Mat r)$ is the
subset of $\Set F_{01}^{ne}(M, k)$ consisting of those fillings
that have exactly $\Mat r_i$ zero entries in row $i$.
For any filling in $\Set F_{01}^{ne}(M, k)$, and an empty cell
$\epsilon$, there must be a chain $C$ such that replacing the $0$
with $1$ in $\epsilon$, and adding $\epsilon$ to $C$, would make
$C$ into a $(k+1)$-chain. In this situation, we say that $C$ is a
\Dfn{maximal chain for} $\epsilon$.
\end{dfn}
For example, when $M$ is the moon polyomino {\tiny$\young(:\hfil\hfil,\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\hfil,:\hfil\hfil)$}, the set $\Set
F_{01}^{ne}(M, 1)$ consists of ten fillings, as can be inferred from
Figure~\ref{fig:poset}.
\begin{rmk}
Note that extending the first $k$ rows and columns of a Ferrers
diagram does not affect the set $\Set F_{01}^{ne}$, which is why we
choose to fix the number of zero entries instead of entries equal
to $1$, although the latter might seem more natural at first
glance.
\end{rmk}
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.6]
\node (v1) at (2.5cm, 5.0cm) [draw=none] {$1$};
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\node (v6) at (0.0cm,1.7845cm) [draw=none] {$6$};
\node (v5) at (1.3874cm,0.0cm) [draw=none] {$5$};
\node (v4) at (3.6126cm,0.0cm) [draw=none] {$4$};
\node (v3) at (5.0cm,1.7845cm) [draw=none] {$3$};
\node (v2) at (4.5048cm,4.0097cm) [draw=none] {$2$};
\draw [thick,grey] (v1) to (v2);
\draw [thick,grey] (v1) to (v3);
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\draw [thick,grey] (v2) to (v3);
\draw [thick,grey] (v2) to (v4);
\draw [thick] (v2) to (v5);
\draw [thick,grey] (v2) to (v7);
\draw [thick,grey] (v3) to (v4);
\draw [thick,grey] (v3) to (v5);
\draw [thick] (v3) to (v6);
\draw [thick] (v3) to (v7);
\draw [thick,grey] (v4) to (v5);
\draw [thick,grey] (v4) to (v6);
\draw [thick,grey] (v5) to (v6);
\draw [thick,grey] (v5) to (v7);
\draw [thick,grey] (v6) to (v7);
\content{0.78}{(6.7,5.5)}{%
0/0/$1$,1/0/$2$,2/0/$3$,3/0/$4$,4/0/$5$,5/0/$6$,
-1/1/$7$,-1/2/$6$,-1/3/$5$,-1/4/$4$,-1/5/$3$,-1/6/$2$}
\node at (9,2.4)
{\young(\mbox{$\color{grey}\bullet$}\g\mbox{$\bullet$}\hfil\mbox{$\color{grey}\bullet$}\g,\mbox{$\color{grey}\bullet$}\hfil\mbox{$\bullet$}\mbox{$\color{grey}\bullet$}\g,\mbox{$\bullet$}\x\mbox{$\color{grey}\bullet$}\g,\hfil\mbox{$\color{grey}\bullet$}\g,\mbox{$\color{grey}\bullet$}\g,\mbox{$\color{grey}\bullet$})};
\end{tikzpicture}
\begin{tikzpicture}[scale=0.95]
\node (v1) at (1cm, 6cm) [draw=none, grey] {$\bullet$};
\node (v2) at (1.5cm, 6cm) [draw=none, grey] {$\bullet$};
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\node (v6) at (3cm, 5.5cm) [draw=none] {$\bullet$};
\node (v7) at (3.5cm, 5.5cm) [draw=none] {$\bullet$};
\node (v8) at (3.5cm, 5cm) [draw=none] {$\bullet$};
\node (v9) at (3.5cm, 4.5cm) [draw=none] {$\bullet$};
\node (v10) at (3.5cm, 4cm) [draw=none, grey] {$\bullet$};
\node (v11) at (3.5cm, 3.5cm) [draw=none, grey] {$\bullet$};
\node (w1) at (1.5cm, 5.5cm) [draw=none] {$\bullet$};
\node (w2) at (2cm, 5.5cm) [draw=none] {$\bullet$};
\node (w3) at (2cm, 5cm) [draw=none] {$\bullet$};
\node (w4) at (2.5cm, 5cm) [draw=none] {$\bullet$};
\node (w5) at (3cm, 5cm) [draw=none] {$\bullet$};
\node (w6) at (3cm, 4.5cm) [draw=none] {$\bullet$};
\node (w7) at (3cm, 4cm) [draw=none] {$\bullet$};
\draw [thick, grey] (1cm,6cm) -- (2cm,6cm);
\draw [thick] (2cm,6cm) -- (3cm,6cm) -- (3cm,5.5cm) -- (3.5cm,5.5cm) -- (3.5cm,4.5cm);
\draw [thick] (1.5cm,5.5cm) -- (2cm,5.5cm) -- (2cm,5cm) -- (3cm,5cm) -- (3cm,4cm);
\draw [thick, grey] (3.5cm,4.5cm) -- (3.5cm,3.5cm);
\node at (6.65, 4.8)
{\young(\mbox{$\color{grey}\bullet$}\g\mbox{$\bullet$}\x\mbox{$\bullet$}\hfil,:\mbox{$\bullet$}\x\hfil\mbox{$\bullet$}\x,::\mbox{$\bullet$}\x\mbox{$\bullet$}\x,:::\hfil\mbox{$\bullet$}\x,::::\mbox{$\bullet$}\mbox{$\color{grey}\bullet$},:::::\mbox{$\color{grey}\bullet$})};
\end{tikzpicture}
\caption{a $2$-triangulation with corresponding filling of the
staircase $\lambda_0$ and a fan of two Dyck paths with
corresponding filling of the reverse staircase $\lambda_0^{rev}$.}
\label{fig:triangulation-Dyck}
\end{figure}
\begin{rmk}
For the staircase shape $\lambda_0$ with $n-1$ rows the set $\Set
F_{01}^{ne}(\lambda_0, k)$ has a particularly beautiful
interpretation, namely as the set of $k$-triangulations of the
$n$-gon. More precisely, label the vertices of the $n$-gon
clockwise from $1$ to $n$, and identify a cell of the shape in row
$i$ and column $j$ with the pair $(n-i+1, j)$ of vertices. Thus,
the entries in the filling equal to $1$ define a set of diagonals
of the $n$-gon. It is not hard to check that a north-east chain of
length $k$ in the filling corresponds to a set of $k$ mutually
crossing diagonals in the $n$-gon.
Maximal fillings of the reverse staircase shape $\lambda_0^{rev}$
for a given $k$ are in bijection with fans of $k$ Dyck paths. An
illustration of both correspondences is given in
Figure~\ref{fig:triangulation-Dyck}. These correspondences were
Jakob Jonsson's~\cite{Jonsson2005} starting point to prove (in a
quite non-bijective fashion) that there are as many
$k$-triangulations of the $n$-gon as fans of $k$ non-intersecting
Dyck paths with $n-2k$ up steps each. Luis Serrano and Christian
Stump~\cite{SerranoStump2010} provided the first completely
bijective proof of this fact, which we generalise in
Section~\ref{sec:Edelman-Greene}. Remarkably, Alex
Woo~\cite{Woo2004} used the same methods already much earlier for
the case of triangulations and Dyck paths, {\it i.e.}, $k=1$.
\end{rmk}
\subsection{Pipe dreams}
In this section we collect some results around pipe dreams and
rc-graphs. All of these statements can be found in~\cite{MR1281474}
together with precise references.
\begin{figure}
\centering
\begin{tikzpicture}
\tpipedream{0.475}{(1.95, 0.6875)}{%
0/0/black/black,1/0/black/black,3/0/black/black,4/0/black/black,5/0/black/black,6/0/black/white,%
0/1/black/black,1/1/black/black,4/1/black/black,5/1/black/white,%
1/2/black/black,3/2/black/black,4/2/black/white,%
0/3/black/black,3/3/black/white,%
0/4/black/black,1/4/black/black,2/4/black/white,%
0/5/black/black,1/5/black/white,%
0/6/black/white%
}%
\cpipedream{0.475}{(1.95, 0.6875)}{%
2/0/black/black,2/1/black/black,3/1/black/black,0/2/black/black,%
2/2/black/black,1/3/black/black,2/3/black/black}%
\content{0.475}{(1.95, 1.6375)}{%
0/0/$1$,1/0/$2$,2/0/$3$,3/0/$4$,4/0/$5$,5/0/$6$,6/0/$7$,
-1/1/$1$,-1/2/$2$,-1/3/$6$,-1/4/$4$,-1/5/$7$,-1/6/$5$,-1/7/$3$}%
%
\content{0.475}{(5.95, 1.1625)}{%
0/0/\mbox{$\bullet$},1/0/\mbox{$\bullet$},3/0/\mbox{$\bullet$},4/0/\mbox{$\bullet$},5/0/\mbox{$\bullet$},6/0/\mbox{$\bullet$},%
0/1/\mbox{$\bullet$},1/1/\mbox{$\bullet$},4/1/\mbox{$\bullet$},5/1/\mbox{$\bullet$},%
1/2/\mbox{$\bullet$},3/2/\mbox{$\bullet$},4/2/\mbox{$\bullet$},%
0/3/\mbox{$\bullet$},3/3/\mbox{$\bullet$},%
0/4/\mbox{$\bullet$},1/4/\mbox{$\bullet$},2/4/\mbox{$\bullet$},%
0/5/\mbox{$\bullet$},1/5/\mbox{$\bullet$},%
0/6/\mbox{$\bullet$},%
2/0/+,2/1/+,3/1/+,0/2/+,%
2/2/+,1/3/+,2/3/+}%
\content{0.475}{(5.95, 1.6375)}{%
0/0/$1$,1/0/$2$,2/0/$3$,3/0/$4$,4/0/$5$,5/0/$6$,6/0/$7$,
-1/1/$1$,-1/2/$2$,-1/3/$6$,-1/4/$4$,-1/5/$7$,-1/6/$5$,-1/7/$3$}%
%
\content{0.475}{(9.95, 1.1625)}{%
0/0/\mbox{$\bullet$},1/0/\mbox{$\bullet$},3/0/\mbox{$\bullet$},4/0/\mbox{$\bullet$},5/0/\mbox{$\bullet$},6/0/\mbox{$\bullet$},%
0/1/\mbox{$\bullet$},1/1/\mbox{$\bullet$},4/1/\mbox{$\bullet$},5/1/\mbox{$\bullet$},%
1/2/\mbox{$\bullet$},3/2/\mbox{$\bullet$},4/2/\mbox{$\bullet$},%
0/3/\mbox{$\bullet$},3/3/\mbox{$\bullet$},%
0/4/\mbox{$\bullet$},1/4/\mbox{$\bullet$},2/4/\mbox{$\bullet$},%
0/5/\mbox{$\bullet$},1/5/\mbox{$\bullet$},%
0/6/\mbox{$\bullet$},%
2/0/3,2/1/4,3/1/5,0/2/3,%
2/2/5,1/3/5,2/3/6}%
\content{0.475}{(9.95, 1.6375)}{%
0/0/$1$,1/0/$2$,2/0/$3$,3/0/$4$,4/0/$5$,5/0/$6$,6/0/$7$,
-1/1/$1$,-1/2/$2$,-1/3/$6$,-1/4/$4$,-1/5/$7$,-1/6/$5$,-1/7/$3$}
\end{tikzpicture}
\caption{the reduced pipe dream associated to the reduced
factorisation $s_3 s_5 s_4 s_5 s_3 s_6 s_5$ of $1,2,6,4,7,5,3$.}
\label{fig:dreams}
\end{figure}
\begin{dfn}\label{dfn:pipe}
A \Dfn{pipe dream} for a permutation $w$ is a filling of a the
quarter plane $\mathbb N^2$, regarding each element of
$\mathbb N^2$ as a cell, with \Dfn{elbow joints} $\textelbow$ and a
finite number of \Dfn{crosses} $\textcross$, such that a pipe
entering from above in column $i$ exits to the left from row
$w^{-1}(i)$. A pipe dream is \Dfn{reduced} if each pair of pipes
crosses at most once, it is then also called \Dfn{rc-graph}.
$\Set{RC}(w)$ is the set of reduced pipe dreams for $w$, and, for a
vector $\Mat r$ of integers, $\Set{RC}(w, \Mat r)$ is the subset of
$\Set{RC}(w)$ having precisely $\Mat r_i$ crosses in row $i$.
\end{dfn}
Usually it will be more convenient to draw dots instead of elbow
joints and sometimes to omit crosses. We will do so without
further notice.
\begin{rmk}
We can associate a reduced factorisation of $w$ to any pipe dream
in $\Set{RC}(w)$ as follows: replace each cross appearing in row
$i$ and column $j$ of the pipe dream with the elementary
transposition $(i+j-1, i+j)$. Then the reduced factorisation of
$w$ is given by the sequence of transpositions obtained by reading
each row of the pipe dream from right to left, and the rows from
top to bottom. An example can be found in Figure~\ref{fig:dreams},
where we write $s_i$ for the elementary transposition $(i,i+1)$.
\end{rmk}
\begin{rmk}\label{rmk:Schubert}
Using reduced pipe dreams, it is possible to define the Schubert
polynomial $\mathfrak S_w$ for the permutation $w$ in a very
concrete way. For a reduced pipe dream $D\in\Set{RC}(w)$, define
$x^D=\prod_{(i,j)\in D} x_i$, where the product runs over all
crosses in the pipe dream. Then the Schubert polynomial is just
the generating function for pipe dreams:
\begin{equation*}
\mathfrak S_w = \sum_{D\in\Set{RC}(w)} x^D.
\end{equation*}
This definition of Schubert polynomials and their evaluation by
Sergey Fomin and Anatol Kirillov~\cite{MR1471891} was used by
Christian Stump~\cite{Stump2010} to give a simple proof of the
product formula for the number of $k$-triangulations of the $n$-gon
\begin{equation*}
\prod_{1\leq i,j<n-2k} \frac{i+j+2k}{i+j}.
\end{equation*}
\end{rmk}
We now define an operation on pipe dreams which was introduced in a
slightly less general form by Nantel Bergeron and Sara
Billey~\cite{MR1281474}. It will be the main tool in the proof of
Theorem~\ref{thm:filling-dream}.
\begin{dfn}
Let $D\in\Set{RC}(w)$ be a pipe dream. Then a \Dfn{chute move} is
a modification of $D$ of the following form:
\begin{equation*}
\begin{array}{@{}c@{}}\\[-5ex]
\begin{array}{@{}r|c|c|c|c|c|l@{}}
\multicolumn{5}{c}{}&\multicolumn{1}{c}{
\phantom{+}}&
\multicolumn{1}{c}{\begin{array}{@{}c@{}}\\{.\hspace{1pt}\raisebox{2pt}{.}\hspace{1pt}\raisebox{4pt}{.}}\end{array}}
\\\cline{2-6}
&\mbox{$\bullet$}&+&\cdots&+&+\\\cline{2-3}\cline{5-6}
&+ &+&\cdots&+&+\\\cline{2-3}\cline{5-6}
&\multicolumn{5}{c|}{\vdots\hfill\vdots\hfill\vdots\hfill}&\\\cline{2-3}\cline{5-6}
&+ &+&\cdots&+&+\\\cline{2-3}\cline{5-6}
&\mbox{$\bullet$}&+&\cdots&+&\mbox{$\bullet$}\\\cline{2-6}
\multicolumn{1}{c}{\begin{array}{@{}c@{}}{.\hspace{1pt}\raisebox{2pt}{.}\hspace{1pt}\raisebox{4pt}{.}}\\ \\ \end{array}}&
\multicolumn{1}{c}{\phantom{+}}
\end{array}
\quad\stackrel{\text{chute}}\rightsquigarrow\quad
\begin{array}{@{}r|c|c|c|c|c|l@{}}
\multicolumn{5}{c}{}&\multicolumn{1}{c}{
\phantom{+}}&
\multicolumn{1}{c}{\begin{array}{@{}c@{}}\\{.\hspace{1pt}\raisebox{2pt}{.}\hspace{1pt}\raisebox{4pt}{.}}\end{array}}
\\\cline{2-6}
&\mbox{$\bullet$}&+&\cdots&+&\mbox{$\bullet$}\\\cline{2-3}\cline{5-6}
&+ &+&\cdots&+&+\\\cline{2-3}\cline{5-6}
&\multicolumn{5}{c|}{\vdots\hfill\vdots\hfill\vdots\hfill}&\\\cline{2-3}\cline{5-6}
&+ &+&\cdots&+&+\\\cline{2-3}\cline{5-6}
&+ &+&\cdots&+&\mbox{$\bullet$}\\\cline{2-6}
\multicolumn{1}{c}{\begin{array}{@{}c@{}}{.\hspace{1pt}\raisebox{2pt}{.}\hspace{1pt}\raisebox{4pt}{.}}\\ \\ \end{array}}&
\multicolumn{1}{c}{\phantom{+}}
\end{array}
\\[-3ex]
\end{array}
\end{equation*}
More formally, a \Dfn{chutable rectangle} is a rectangular region
$r$ inside a pipe dream $D$ with at least two columns and two rows
such that all but the following three locations of $r$ are crosses:
the north-west, south-west, and south-east corners. Applying a
\Dfn{chute move} to $D$ is accomplished by placing a \textcross\ in
the south-west corner of a chutable rectangle $r$ and removing the
\textcross\ from the north-east corner of $r$. We call the inverse
operation \Dfn{inverse chute move}.
\end{dfn}
The following lemma was given by Nantel Bergeron and Sara
Billey~\cite[Lemma~3.5]{MR1281474} for two rowed chute moves, the
proof is valid for our generalised chute moves without modification:
\begin{lem}\label{lem:chute-closure}%
The set $\Set{RC}(w)$ of reduced pipe dreams for~$w$ is closed
under chute moves.
\end{lem}
\begin{proof}
The pictorial description of chute moves in terms of pipes
immediately shows that the permutation associated to the pipe dream
remains unchanged. For example, here is the picture associated
with a three rowed chute move:
\begin{equation*}
\begin{tikzpicture}[scale=0.88]
\tpipedream{0.5}{(0,0)}{%
0/0/black/black,%
0/2/black/black,7/2/black/black}%
\cpipedream{0.5}{(0,0)}{%
1/0/grey/grey,2/0/grey/grey,3/0/grey/grey,4/0/grey/grey,5/0/grey/grey,6/0/grey/grey,7/0/grey/grey,%
0/1/grey/grey,1/1/grey/grey,2/1/grey/grey,3/1/grey/grey,4/1/grey/grey,5/1/grey/grey,6/1/grey/grey,7/1/grey/grey,%
1/2/grey/grey,2/2/grey/grey,3/2/grey/grey,4/2/grey/grey,5/2/grey/grey,6/2/grey/grey}%
\end{tikzpicture}
\quad\raisebox{0.5cm}{$\stackrel{\text{chute}}\rightsquigarrow$}\quad
\begin{tikzpicture}[scale=0.88]
\tpipedream{0.5}{(0,0)}{%
0/0/black/black,7/0/black/black,%
7/2/black/black}%
\cpipedream{0.5}{(0,0)}{%
1/0/grey/grey,2/0/grey/grey,3/0/grey/grey,4/0/grey/grey,5/0/grey/grey,6/0/grey/grey,%
0/1/grey/grey,1/1/grey/grey,2/1/grey/grey,3/1/grey/grey,4/1/grey/grey,5/1/grey/grey,6/1/grey/grey,7/1/grey/grey,%
0/2/grey/grey,1/2/grey/grey,2/2/grey/grey,3/2/grey/grey,4/2/grey/grey,5/2/grey/grey,6/2/grey/grey}%
\end{tikzpicture}
\end{equation*}
\end{proof}
\begin{rmk}
It follows that chute moves define a partial order on $\Set{RC}(w)$,
where $D$ is covered by $E$ if there is a chute move transforming
$E$ into $D$. Nantel Bergeron and Sara Billey restricted their
attention to two rowed chute moves. For this case, their main
theorem states that the poset defined by chute moves has a unique
maximal element, namely
$$
D_{top}(w)=\left\{(c,j): %
c\leq \#\{i: i < w^{-1}_j, w_i>j\}\right\}.
$$
It is easy to see that considering general chute moves, the poset
has also a unique minimal element, namely
$$
D_{bot}(w)=\left\{(i,c): %
c\leq \#\{j: j > i, w_j < w_i\}\right\}.
$$
In the next section we will show a statement similar in spirit to
the main theorem of Nantel Bergeron and Sara Billey for the more
general chute moves defined above.
\end{rmk}
\begin{figure}
\begin{center}
\small
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\node (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) at (626bp,495bp) [draw,draw=none] {${\addtolength{\arraycolsep}{-0.2ex}\addtolength{\arrayrulewidth}{0.2ex}{\raisebox{-.6ex}{$\begin{array}[b]{cccccc}\hhline{------}\lr{\bullet}&\lrg{\bullet}&\lrg{}&\lr{\bullet}&\lr{\bullet}&\lr{\bullet}\\\hhline{------}\lrg{\bullet}&\lrg{\bullet}&\lrg{\bullet}&\lrg{\bullet}&\lr{\bullet}\\\hhline{-----}\lrg{}&\lrg{}&\lrg{}&\lrg{\bullet}\\\hhline{----}\lr{\bullet}&\lrg{}&\lrg{\bullet}\\\hhline{---}\lr{\bullet}&\lr{\bullet}\\\hhline{--}\lr{\bullet}\\\hhline{-}\end{array}$}}}$};
\node (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) at (358bp,495bp) [draw,draw=none] {${\raisebox{-.6ex}{$\begin{array}[b]{cccccc}\hhline{------}\lr{\bullet}&\lr{\bullet}&\lr{\bullet}&\lr{\bullet}&\lr{}&\lr{\bullet}\\\hhline{------}\lr{\bullet}&\lr{}&\lr{\bullet}&\lr{\bullet}&\lr{\bullet}\\\hhline{-----}\lr{}&\lr{}&\lr{\bullet}&\lr{\bullet}\\\hhline{----}\lr{\bullet}&\lr{\bullet}&\lr{\bullet}\\\hhline{---}\lr{}&\lr{\bullet}\\\hhline{--}\lr{\bullet}\\\hhline{-}\end{array}$}}$};
\node (--bullet+--bullet++--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) at (540bp,723bp) [draw,draw=none] {${\raisebox{-.6ex}{$\begin{array}[b]{cccccc}\hhline{------}\lr{\bullet}&\lr{\bullet}&\lr{}&\lr{\bullet}&\lr{}&\lr{\bullet}\\\hhline{------}\lr{\bullet}&\lr{\bullet}&\lr{\bullet}&\lr{\bullet}&\lr{\bullet}\\\hhline{-----}\lr{}&\lr{}&\lr{\bullet}&\lr{\bullet}\\\hhline{----}\lr{\bullet}&\lr{}&\lr{\bullet}\\\hhline{---}\lr{\bullet}&\lr{\bullet}\\\hhline{--}\lr{\bullet}\\\hhline{-}\end{array}$}}$};
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\draw [black,->] (--bullet+--bullet+--bullet+++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (206bp,789bp) and (206bp,781bp) .. (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (590bp,675bp) and (582bp,665bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [very thick,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet) ..controls (496bp,916bp) and (499bp,914bp) .. (502bp,912bp) .. controls (535bp,890bp) and (549bp,897bp) .. (583bp,876bp) .. controls (583bp,876bp) and (584bp,876bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet++--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet) ..controls (434bp,1017bp) and (437bp,1008bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (120bp,789bp) and (120bp,781bp) .. (--bullet+--bullet+--bullet+++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (250bp,916bp) and (247bp,914bp) .. (244bp,912bp) .. controls (211bp,890bp) and (197bp,897bp) .. (163bp,876bp) .. controls (163bp,876bp) and (162bp,876bp) .. (--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (295bp,451bp) and (317bp,432bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet++--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (458bp,333bp) and (457bp,325bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (185bp,675bp) and (182bp,666bp) .. (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (300bp,1017bp) and (298bp,1009bp) .. (--bullet+--bullet++++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (266bp,561bp) and (263bp,552bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (301bp,786bp) and (293bp,774bp) .. (287bp,762bp) .. controls (269bp,727bp) and (273bp,714bp) .. (249bp,684bp) .. controls (235bp,666bp) and (217bp,650bp) .. (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (576bp,789bp) and (584bp,779bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet) ..controls (378bp,1131bp) and (384bp,1122bp) .. (--bullet+--bullet++--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (454bp,675bp) and (454bp,667bp) .. (--bullet+--bullet++--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [very thick,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet++--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet) ..controls (657bp,786bp) and (664bp,774bp) .. (669bp,762bp) .. controls (683bp,728bp) and (694bp,689bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (305bp,903bp) and (308bp,894bp) .. (--bullet+--bullet++--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (544bp,447bp) and (544bp,439bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (540bp,789bp) and (540bp,781bp) .. (--bullet+--bullet++--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (226bp,570bp) and (278bp,540bp) .. (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (336bp,917bp) and (333bp,914bp) .. (330bp,912bp) .. controls (311bp,895bp) and (303bp,895bp) .. (287bp,876bp) .. controls (264bp,845bp) and (268bp,832bp) .. (249bp,798bp) .. controls (244bp,789bp) and (239bp,780bp) .. (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet) ..controls (187bp,333bp) and (193bp,324bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet) ..controls (108bp,561bp) and (116bp,551bp) .. (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (540bp,561bp) and (540bp,553bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (238bp,675bp) and (244bp,666bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (490bp,675bp) and (498bp,665bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (590bp,561bp) and (582bp,551bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (84bp,789bp) and (76bp,779bp) .. (--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (265bp,689bp) and (298bp,668bp) .. (325bp,648bp) .. controls (325bp,648bp) and (326bp,647bp) .. (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet);
\draw [very thick,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (626bp,675bp) and (626bp,667bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (244bp,447bp) and (244bp,439bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet++--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (416bp,447bp) and (410bp,438bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet++--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (379bp,699bp) and (395bp,691bp) .. (411bp,684bp) .. controls (448bp,667bp) and (462bp,670bp) .. (497bp,648bp) .. controls (497bp,648bp) and (497bp,648bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (380bp,789bp) and (372bp,779bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (270bp,674bp) and (240bp,649bp) .. (239bp,648bp) .. controls (224bp,616bp) and (226bp,575bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (413bp,561bp) and (405bp,551bp) .. (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (340bp,444bp) and (336bp,432bp) .. (334bp,420bp) .. controls (326bp,386bp) and (321bp,346bp) .. (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet+--bullet++--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (136bp,675bp) and (139bp,666bp) .. (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (391bp,903bp) and (394bp,894bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (170bp,789bp) and (162bp,779bp) .. (--bullet+--bullet+--bullet+++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet+--bullet++--bullet+--bullet++--bullet+--bullet) ..controls (322bp,219bp) and (324bp,211bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+++--bullet+--bullet++--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (167bp,903bp) and (160bp,894bp) .. (--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (427bp,189bp) and (409bp,130bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet++--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (335bp,1017bp) and (341bp,1008bp) .. (--bullet+--bullet++--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (507bp,333bp) and (499bp,323bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (364bp,561bp) and (364bp,553bp) .. (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (490bp,561bp) and (498bp,551bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (379bp,792bp) and (397bp,776bp) .. (--bullet+--bullet++--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (366bp,447bp) and (368bp,439bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet++--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet) ..controls (543bp,903bp) and (542bp,895bp) .. (--bullet+--bullet++--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (594bp,447bp) and (587bp,438bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [very thick,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet) ..controls (407bp,765bp) and (396bp,718bp) .. (373bp,684bp) .. controls (357bp,662bp) and (345bp,664bp) .. (325bp,648bp) .. controls (325bp,648bp) and (324bp,647bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet) ..controls (398bp,1017bp) and (395bp,1008bp) .. (--bullet+--bullet++--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet++--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (408bp,333bp) and (414bp,324bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet) ..controls (500bp,464bp) and (493bp,460bp) .. (487bp,456bp) .. controls (458bp,437bp) and (447bp,439bp) .. (420bp,420bp) .. controls (420bp,420bp) and (419bp,420bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet++--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet);
\draw [very thick,->] (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet++--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet) ..controls (626bp,789bp) and (626bp,781bp) .. (--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet++--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet) ..controls (195bp,346bp) and (198bp,344bp) .. (201bp,342bp) .. controls (231bp,321bp) and (243bp,326bp) .. (272bp,306bp) .. controls (272bp,306bp) and (273bp,306bp) .. (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+++--bullet+--bullet++--bullet+--bullet++--bullet+--bullet);
\draw [black,->] (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet) ..controls (194bp,447bp) and (202bp,437bp) .. (--bullet+--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++++--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet);
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\draw [black,->] (--bullet+--bullet+--bullet+++--bullet+--bullet+--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet) ..controls (108bp,447bp) and (116bp,437bp) .. (--bullet+--bullet+--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet+--bullet++--bullet+--bullet++--bullet+--bullet);
\end{tikzpicture}}
\end{center}
\caption{the poset of reduced pipe dreams for the permutation $1, 2,
6, 4, 5, 3$. The interval of $0$-$1$-fillings with $k=1$ of the
moon polyomino \protect{\tiny$\young(:\hfil\hfil,\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\hfil,:\hfil\hfil)$}\ is emphasised.}
\label{fig:poset}
\end{figure}
After generating and analysing some of these posets using
\texttt{Sage}~\cite{Sage-Combinat}, see Figure~\ref{fig:poset} for an
example, we became convinced that they should have much more
structure:
\begin{cnj}\label{cnj:lattice}
The poset of reduced pipe dreams defined by (general) chute moves
is in fact a lattice.
\end{cnj}
There is another natural way to transform one reduced pipe dream into
another, originating in the concept of flipping a diagonal of a
triangulation. Namely, consider an elbow joint in the pipe dream.
Since any pair of pipes crosses at most once, there is at most one
location where the pipes originating from the given elbow joint
cross. If there is such a crossing, replace the elbow joint by a
cross and the cross by an elbow joint. Clearly, the result is again
a reduced pipe dream, associated to the same permutation.
It is believed (see Vincent Pilaud and Michel
Pocchiola~\cite{PilaudPocchiola2009}, Question~51) that the
simplicial complex of multitriangulations can be realised as a
polytope, in this case the graph of flips would be the graph of the
polytope. Note that the graph of chute moves is a subgraph of the
graph of flips. Is Conjecture~\ref{cnj:lattice} related to the
question of polytopality?
\section{Maximal Fillings of Moon Polyominoes and Pipe Dreams}
\label{sec:maximal-fillings-rc}
Consider a maximal filling in $\Set F_{01}^{ne}(M, k)$. Recall that
we regard a moon polyomino $M$ as a finite subset of $\mathbb N^2$.
Also, recall that a pipe dream is nothing but a filling of
$\mathbb N^2$ with elbow joints and a finite number of crosses.
Thus, replacing zeros in the filling of the moon polyomino with
crosses, and all cells in the filling containing ones as well as all
cells not in $M$ with elbow joints, we clearly obtain a pipe dream
for some permutation $w$. An example of this transformation is given
in Figure~\ref{fig:pipe-dream-filling}. We will see in this section
that the pipe dreams obtained in this way are in fact reduced.
\begin{figure}
\centering
\begin{tikzpicture}
\node at (0,0)
{\young(:\mbox{$\bullet$}\hfil,\mbox{$\bullet$}\x\hfil\hfil,\hfil\mbox{$\bullet$}\hfil\mbox{$\bullet$},:\hfil\hfil\mbox{$\bullet$},:\mbox{$\bullet$}\x)};
\node at (2.8975,-0.005)
{\young(:\hfil\hfil,\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\hfil,:\hfil\hfil\hfil,:\hfil\hfil)};
\content{0.475}{(1.95, 1.1625)}{%
0/0/\mbox{$\bullet$},1/0/\mbox{$\bullet$},3/0/\mbox{$\bullet$},4/0/\mbox{$\bullet$},5/0/\mbox{$\bullet$},6/0/\mbox{$\bullet$},%
0/1/\mbox{$\bullet$},1/1/\mbox{$\bullet$},4/1/\mbox{$\bullet$},5/1/\mbox{$\bullet$},%
1/2/\mbox{$\bullet$},3/2/\mbox{$\bullet$},4/2/\mbox{$\bullet$},%
0/3/\mbox{$\bullet$},3/3/\mbox{$\bullet$},%
0/4/\mbox{$\bullet$},1/4/\mbox{$\bullet$},2/4/\mbox{$\bullet$},%
0/5/\mbox{$\bullet$},1/5/\mbox{$\bullet$},%
0/6/\mbox{$\bullet$},%
2/0/+,2/1/+,3/1/+,0/2/+,%
2/2/+,1/3/+,2/3/+}%
%
\node at (6.8975,-0.005)
{\young(:\hfil\hfil,\hfil\hfil\hfil\hfil,\hfil\hfil\hfil\hfil,:\hfil\hfil\hfil,:\hfil\hfil)};
\tpipedream{0.475}{(5.95, 0.6875)}{%
0/0/black/black,1/0/black/black,3/0/black/black,4/0/black/black,5/0/black/black,6/0/black/white,%
0/1/black/black,1/1/black/black,4/1/black/black,5/1/black/white,%
1/2/black/black,3/2/black/black,4/2/black/white,%
0/3/black/black,3/3/black/white,%
0/4/black/black,1/4/black/black,2/4/black/white,%
0/5/black/black,1/5/black/white,%
0/6/black/white%
}%
\cpipedream{0.475}{(5.95, 0.6875)}{%
2/0/black/black,2/1/black/black,3/1/black/black,0/2/black/black,%
2/2/black/black,1/3/black/black,2/3/black/black}%
\content{0.475}{(5.95, 1.6375)}{%
0/0/$1$,1/0/$2$,2/0/$3$,3/0/$4$,4/0/$5$,5/0/$6$,6/0/$7$,
-1/1/$1$,-1/2/$2$,-1/3/$6$,-1/4/$4$,-1/5/$7$,-1/6/$5$,-1/7/$3$}%
\end{tikzpicture}
\caption{a maximal filling and the associated pipe dream.}
\label{fig:pipe-dream-filling}
\end{figure}
One may notice that the permutation associated with the pipe dream so
constructed depends somewhat on the embedding of the polyomino into
the quarter plane. Although one can check that this dependence is
not substantial for what is to follow, we will assume for simplicity
that the top row and the left-most column of the polyomino have index
$1$ and indices increase from top to bottom and from left to right.
Even without the knowledge that the pipe dream is reduced we can
speak of chute moves applied to fillings in $\Set F_{01}^{ne}(M, k)$.
However, a priori it is not clear under which conditions the result
of such a move is again a filling in $\Set F_{01}^{ne}(M, k)$. In
particular, we have to deal with the fact that under this
identification all cells outside $M$ are also filled with \emph{elbow
joints}, corresponding to \emph{ones}. Of course, to determine the
set of north-east chains we have to consider the original filling and
the boundary of $M$, and \emph{disregard} elbow joints outside.
Similar to the approach of Nantel Bergeron and Sara Billey we will
consider two special fillings $D_{top}(M, k)$ and $D_{bot}(M, k)$.
These will turn out to be the maximal and the minimal element in the
poset having elements $\Set F_{01}^{ne}(M, k)$, where one filling is
smaller than another if it can be obtained by applying chute moves to
the latter. Figure~\ref{fig:top-bot} displays an example of the
following construction:
\begin{dfn}
Let $M$ be a moon polyomino and $k\geq0$. Then $D_{top}(M,
k)\in\Set F_{01}^{ne}(M, k)$ is obtained by putting ones into all
cells that can be covered by any rectangle of size at most $k\times
k$, which is completely contained in the moon polyomino, and that
touches the boundary of $M$ with its lower-left corner.
Similarly, $D_{bot}(M, k)\in\Set F_{01}^{ne}(M, k)$ is obtained by
putting ones into all cells that can be covered by any rectangle of
size at most $k\times k$, which is completely contained in the moon
polyomino, and that touches the boundary of $M$ with its
upper-right corner.
\end{dfn}
\begin{figure}[h]
\begin{equation*}
\young(:::\mbox{$\bullet$}\x\hfil\hfil,%
::\mbox{$\bullet$}\x\hfil\hfil\hfil,%
::\mbox{$\bullet$}\x\hfil\hfil\hfil,%
\mbox{$\bullet$}\x\mbox{$\bullet$}\hfil\hfil\hfil\hfil\hfil,%
\mbox{$\bullet$}\x\mbox{$\bullet$}\hfil\hfil\hfil\hfil\hfil,%
:\mbox{$\bullet$}\x\mbox{$\bullet$}\x\hfil\hfil\mbox{$\bullet$},%
:\mbox{$\bullet$}\x\mbox{$\bullet$}\x\mbox{$\bullet$}\x\mbox{$\bullet$},%
:::\mbox{$\bullet$}\x\mbox{$\bullet$}\x)%
\quad
\young(:::\mbox{$\bullet$}\x\mbox{$\bullet$}\x,%
::\mbox{$\bullet$}\x\mbox{$\bullet$}\x\mbox{$\bullet$},%
::\mbox{$\bullet$}\hfil\hfil\mbox{$\bullet$}\x,%
\mbox{$\bullet$}\x\hfil\hfil\hfil\mbox{$\bullet$}\x\mbox{$\bullet$},%
\mbox{$\bullet$}\x\hfil\hfil\hfil\mbox{$\bullet$}\x\mbox{$\bullet$},%
:\hfil\hfil\hfil\hfil\hfil\mbox{$\bullet$}\x,%
:\hfil\hfil\hfil\hfil\hfil\mbox{$\bullet$}\x,%
:::\hfil\hfil\mbox{$\bullet$}\x)%
\end{equation*}
\caption{The special fillings $D_{top}(M,k)$ and $D_{bot}(M,k)$ for
$k=2$ of a moon polyomino.}
\label{fig:top-bot}
\end{figure}
We can now state the main theorem of this article:
\begin{thm}\label{thm:filling-dream}\sloppypar
Let $M$ be a moon polyomino and $k\geq 0$. The set $\Set
F_{01}^{ne}(M, k, \Mat r)$ can be identified with the set of
reduced pipe dreams $\Set{RC}\big(w(M, k), \Mat r\big)$ having all
crosses inside of $M$ for some permutation $w(M, k)$ depending only
on $M$ and $k$: replace zeros with crosses and all cells containing
ones as well as all cells not in $M$ with elbow joints.
More precisely, the set $\Set F_{01}^{ne}(M, k)$ is an interval in
the poset of reduced pipe dreams $\Set{RC}\big(w(M, k)\big)$ with
maximal element $D_{top}(M, k)$ and minimal element $D_{bot}(M,
k)$.
\end{thm}
As already remarked in the introduction various versions of this
theorem were independently proved by various authors by various
methods. The most general version is due to Luis Serrano and
Christian Stump~\cite[Theorem~2.6]{SerranoStump2010}, whose proof
employs properties of subword complexes and who thus obtain
additionally many interesting properties of the simplicial complex of
$0$-$1$-fillings.
The advantage of our approach using chute moves is the demonstration
of the property that $\Set F_{01}^{ne}(M, k)$ is in fact an interval
in the bigger poset of reduced pipe dreams. In particular, if
Conjecture~\ref{cnj:lattice} turns out to be true then $\Set
F_{01}^{ne}(M, k)$ is also a lattice. An illustration is given in
Figure~\ref{fig:poset}.
Let us first state a very basic property of chute moves as applied to
fillings:
\begin{lem}\label{lem:chute-moon-closure}
Let $M$ be a moon polyomino. Chute moves and their inverses
applied to a filling in $\Set F_{01}^{ne}(M, k)$ produce another
filling in $\Set F_{01}^{ne}(M, k)$ whenever all zero entries
remain in $M$.
\end{lem}
\begin{proof}
We only have to check that chain lengths are preserved, which is
not hard.
\end{proof}
Most of what remains of this section is devoted to prove that there
is precisely one filling in $\Set F_{01}^{ne}(M, k)$ that does not
admit a chute move such that the result is again in $\Set
F_{01}^{ne}(M, k)$, namely $D_{bot}(M, k)$, and precisely one filling
that does not admit an inverse chute move with the same property,
namely $D_{top}(M, k)$.
Although the strategy itself is actually very simple the details turn
out to be quite delicate. Thus we split the proof into a few
auxiliary lemmas. Let us fix $k$, a moon polyomino $M$, and a
maximal filling $D\in\Set F_{01}^{ne}(M, k)$ different from
$D_{bot}(M, k)$. We will then explicitly locate a chutable
rectangle. Throughout the proof maximality of the filling will play
a crucial role. The first lemma is used to show that certain cells
of the polyomino must be empty because otherwise the filling would
contain a chain of length $k+1$:
\begin{lem}[Chain induction]\label{lem:chain-induction}
Consider a maximal filling of a moon polyomino. Let $\epsilon$ be
an empty cell such that all cells below $\epsilon$ in the same
column are empty too, except possibly those that are below the
lowest cell of the column left of $\epsilon$. Assume that for
\emph{each} of these cells $\delta$ there is a maximal chain for
$\delta$ strictly north-east of $\delta$. Then there is a maximal
chain for $\epsilon$ strictly north-east of $\epsilon$.
Similarly, let $\epsilon$ be an empty cell such that all cells left
of $\epsilon$ in the same row are empty too, except possibly those
that are left of the left-most cell of the row below $\epsilon$.
Assume that for \emph{each} of these cells $\delta$ there is a
maximal chain for $\delta$ strictly north-east of $\delta$. Then
there is a maximal chain for $\epsilon$ strictly north-east of
$\epsilon$.
\end{lem}
\begin{rmk}
Note that for the conclusion of Lemma~\ref{lem:chain-induction} to
hold we really have to assume that \emph{all} cells below
$\epsilon$ are empty: in the maximal filling for $k=1$
\begin{equation*}
\young(\mbox{$\bullet$}\epsilon\mbox{$\bullet$},%
\hfil\delta\mbox{$\bullet$},%
\mbox{$\bullet$}\x)
\end{equation*}
there is a maximal chain for $\delta$ north-east of
$\delta$, but no maximal chain for $\epsilon$ north-east of
$\epsilon$. The following example demonstrates that it is equally
necessary that the filling is maximal:
\begin{equation*}
\young(\mbox{$\bullet$},\epsilon\mbox{$\bullet$},\hfil\mbox{$\bullet$})
\end{equation*}
\end{rmk}
\begin{proof}
Assume on the contrary that there is no maximal chain for
$\epsilon$ north-east of $\epsilon$. Consider a maximal chain
$C_\epsilon$ for $\epsilon$ that has as many elements north-east of
$\epsilon$ as possible. Let $\delta$ be the cell in the same
column as $\epsilon$, below $\epsilon$, in the same row as the top
entry of $C_\epsilon$ which is south-east of $\epsilon$. By
assumption, there is a maximal chain $C_\delta$ for $\delta$
north-east of $\delta$. We have to consider two cases:
If the widest rectangle containing $C_\epsilon$ is not as wide as
the smallest rectangle containing $C_\delta$, then the entry of
$C_\epsilon$ to the left of $\delta$ would extend $C_\delta$ to a
$(k+1)$-chain, which is not allowed:
\begin{center}
\setlength{\unitlength}{0.5cm}
\begin{picture}(12,11)
\put(2,0){\framebox(8,10){}}
\put(10,10){$C_\epsilon$}
\put(0,3){\framebox(12,5){}}
\put(12,8){$C_\delta$}
\put(5.5,0){\dashbox{0.3}(1,10){}}
\put(5.5,3){\framebox(1,1){$\delta$}}
\put(5.5,6){\framebox(1,1){$\epsilon$}}
\put(4,3){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(3.5,1.8){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(2.5,1){\makebox(1,1){${.\hspace{1pt}\raisebox{2pt}{.}\hspace{1pt}\raisebox{4pt}{.}}$}}
\put(7,7){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(7.6,8){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(8.5,8.5){\makebox(1,1){${.\hspace{1pt}\raisebox{2pt}{.}\hspace{1pt}\raisebox{4pt}{.}}$}}
\put(7,4){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(8.8,5){\makebox(1,1){${.\hspace{1pt}\raisebox{2pt}{.}\hspace{1pt}\raisebox{4pt}{.}}$}}
\put(11,5.8){\makebox(1,1){$\mbox{$\bullet$}$}}
\end{picture}
\end{center}
If the smallest rectangle containing $C_\epsilon$ is at least as
wide as the widest rectangle containing $C_\delta$, then we obtain
a maximal chain for $\epsilon$ north-east of $\epsilon$ by
induction. Let $c_\epsilon^1, c_\epsilon^2,\dots$ be the sequence
of elements of $C_\epsilon$ north-east of $\epsilon$, and
$c_\delta^1, c_\delta^2,\dots$ the sequence of elements of
$C_\delta$ north-east of $\delta$. We will show that
$c_\epsilon^i$ must be strictly north and weakly west of
$c_\delta^i$, for all $i$. Thus, the elements $c_\epsilon^1,
c_\epsilon^2,\dots$ together with the elements of $C_\delta$
outside the smallest rectangle containing $C_\epsilon$ form a
maximal chain for $\epsilon$ north-east of $\epsilon$.
$c_\epsilon^1$ is strictly north of $c_\delta^1$, since otherwise
$C_\delta$ would be a maximal chain for $\epsilon$. $c_\epsilon^1$
cannot be strictly east of $c_\delta^1$, since in this case
$c_\delta^1$ together with $C_\epsilon$ would be a $(k+1)$-chain.
Suppose now that $c_\epsilon^{i-1}$ is strictly north and weakly
west of $c_\delta^{i-1}$. $c_\delta^i$ cannot be strictly
north-east of $c_\epsilon^{i-1}$, since this would yield a
$k$-chain north-east of $\epsilon$. $c_\delta^i$ must be strictly
east of $c_\epsilon^{i-1}$, since $c_\delta^i$ is strictly east of
$c_\delta^{i-1}$, which in turn is weakly east of
$c_\epsilon^{i-1}$ by the induction hypothesis. Thus,
$c_\epsilon^{i-1}$ is weakly north and strictly west of
$c_\delta^i$.
$c_\epsilon^i$ cannot be strictly north-east of $c_\delta^i$, since
then the elements of $C_\epsilon$ south-west of $\epsilon$ together
with the elements $c_\delta^1,\dots,c_\delta^i$ and
$c_\epsilon^i,c_\epsilon^{i+1}, \dots$ would form a $(k+1)$-chain.
Finally, $c_\epsilon^i$ must be strictly north of $c_\delta^i$,
since $c_\epsilon^i$ is strictly north of $c_\epsilon^{i-1}$, which
in turn is weakly north of $c_\delta^i$.
\end{proof}
\begin{lem}\label{lem:chutable-rectangle}
Consider a maximal filling of a moon polyomino. Suppose that there
is a rectangle with at least two columns and at least two rows
completely contained in the polyomino, with all cells empty except
the north-west, south-east and possibly the south-west corners.
Then the south-west corner is indeed non-empty, {\it i.e.}, the
rectangle is chutable.
\end{lem}
Note that we must insist that the south-west corner of the rectangle
is part of the polyomino. Here is a maximal filling with $k=1$,
where the three cells in the south-west do not form a chutable
rectangle, since the south-west corner is missing:
\begin{equation*}
\young(:\mbox{$\bullet$}\hfil\mbox{$\bullet$},%
\mbox{$\bullet$}\hfil\hfil\hfil,%
\mbox{$\bullet$}\hfil\hfil\mbox{$\bullet$},%
:\mbox{$\bullet$}\x)%
\end{equation*}
However, we can weaken this assumption in a different way:
\begin{lem}\label{lem:chutable-rectangle-2}
Consider a maximal filling of a moon polyomino. Suppose that there
is a rectangle with at least two columns and at least two rows such
that all cells of its top row and its right column are contained in
the polyomino. Assume furthermore that all cells of the rectangle
that are in the polyomino are empty except the north-west,
south-east and possibly the south-west corners. Finally, suppose
that there is no maximal chain for the cell in the north-east
corner strictly north east of it. Then the cell in the south-west
corner is indeed in the polyomino and non-empty, {\it i.e.}, the
rectangle is chutable.
\end{lem}
\begin{proof}[Proof of Lemma~\ref{lem:chutable-rectangle}]
Suppose on the contrary that the cell in the south-west corner is
empty, too. Then, the situation is as in the following picture:
\begin{center}
\setlength{\unitlength}{0.5cm}
\begin{picture}(6,4)%
\put(0,0){\framebox(6,4){}} %
\put(0,0){\framebox(1,1){$\delta$}} %
\put(5,0){\framebox(1,1){$\mbox{$\bullet$}$}} %
\put(5,3){\framebox(1,1){$\epsilon$}}%
\put(0,3){\framebox(1,1){$\mbox{$\bullet$}$}} %
\end{picture}
\end{center}
Since the filling is maximal but the cells $\delta$ and $\epsilon$
are empty, there must be maximal chains for these cells. The
corresponding rectangles must not cover any of the two cells
containing ones, since that would imply the existence of a
$(k+1)$-chain. Thus, any maximal chain for $\delta$ must be
strictly south-west of $\delta$, and any maximal chain for
$\epsilon$ must be strictly north-east of $\epsilon$. Since the
polyomino is intersection free, the top row of the rectangle
containing the maximal chain for $\epsilon$ is either contained in
the bottom row of the rectangle containing the maximal chain for
$\delta$, or vice versa. In both cases, we have a contradiction.
\end{proof}
The next lemma parallels the main Lemma~3.6 in the article by Nantel
Bergeron and Sara Billey~\cite{MR1281474}:
\begin{lem}\label{lem:two-column-chute}
Consider a maximal filling of a moon polyomino. Suppose that there
is a cell $\gamma$ containing a $1$ with an empty cell $\epsilon$
in the neighbouring cell to its right, such that there are at least
as many cells above $\gamma$ as above $\epsilon$. Then the filling
contains a chutable rectangle.
Similarly, suppose that there is a cell $\gamma$ containing a $1$
with an empty cell $\epsilon$ in the neighbouring cell below it,
such that there are at least as many cells right of $\gamma$ as
right of $\epsilon$. Then the filling contains a chutable
rectangle.
\end{lem}
\begin{proof}
Suppose that all of the cells in the column containing $\epsilon$,
which are below $\epsilon$ and weakly above the bottom cell of the
column containing $\gamma$, are empty. Let $\delta$ be the lowest
cell in this region. There must then be a maximal chain for
$\delta$ that is north-east of $\delta$. By
Lemma~\ref{lem:chain-induction}, we conclude that there is also a
maximal chain for $\epsilon$ north-east of $\epsilon$. However,
then the $1$ in the cell left of $\epsilon$ together with this
chain yields a $(k+1)$-chain, since the rectangle containing the
maximal chain for $\epsilon$ extends by hypothesis to the column
left of $\epsilon$.
We can thus apply Lemma~\ref{lem:chutable-rectangle} to the
following rectangle: the south-east corner being the top non-empty
cell below $\epsilon$, and the north-west corner being the lowest
cell containing a $1$ in the column of $\gamma$, strictly above the
chosen south-east corner.
\end{proof}
Finally, the main statement follows from a careful analysis of
fillings different from $D_{bot}(M, k)$, repeatedly applying the
previous lemmas to exclude obstructions to the existence of a
chutable rectangle:
\begin{thm}\label{thm:maximal-fillings-chutable}
Any maximal filling other than $D_{bot}(M,k)$ admits a chute move
such that the result is again a filling of $M$. Any maximal
filling other than $D_{top}(M,k)$ admits an inverse chute move such
that the result is again a filling of $M$.
\end{thm}
\begin{proof}
Suppose that all cells in the top-right quarter of $M$ that contain
a $1$ in $D_{bot}(M,k)$ also contain a $1$ in the filling $F$ at
hand. It follows, that all cells that are empty in $D_{bot}(M,k)$
are empty in $F$, too, because there is a maximal chain for each of
them. Thus, in this case $F=D_{bot}(M,k)$.
Otherwise, consider the set of left-most cells in the top-right
quarter, that contain a $1$ in $D_{bot}(M,k)$ but are empty in $F$,
and among those the top cell, $\epsilon$. If its left or lower
neighbour contains a $1$, we can apply
Lemma~\ref{lem:two-column-chute} and are done. Otherwise, we have
to find a rectangle as in the hypothesis of
Lemma~\ref{lem:chutable-rectangle}. The difficulty in this
undertaking is to prove that the lower left corner is indeed part
of the polyomino. To ease the understanding of the argument, we
will frequently refer to the following sketch:
\begin{center}
\setlength{\unitlength}{0.5cm}
\begin{picture}(21,12)
\put(0,9){\line(1,0){2}}
\put(2,9){\line(0,1){1}}
\put(2,10){\line(1,0){1.5}}
\multiput(3.5,10)(0.5,0){35}{\line(1,0){0.1}}
\put(21,10.1){$R$}
%
\put(3.8,10.3){{.\hspace{1pt}\raisebox{2pt}{.}\hspace{1pt}\raisebox{4pt}{.}}}
\put(5,11){\line(1,0){1}}
\put(6,11){\line(0,1){1}}
\multiput(6,0)(0,0.5){24}{\line(0,1){0.25}}
\multiput(6,12)(0.5,0){8}{\line(1,0){0.25}}
\put(1,7){\framebox(1,1){$\mbox{$\bullet$}^\alpha$}}
\put(2,7){$\overbrace{\makebox(4,1){}}^\ell$}
\put(1,7){\dashbox{0.3}(15,1){}}
\put(10,7){\framebox(1,1){$\epsilon$}}
\put(10,8){\framebox(4,4){$k\times k$}}
\put(10,8){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(10,11){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(13,8){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(13,11){\makebox(1,1){$\mbox{$\bullet$}$}}
%
\put(10,0){\framebox(1,1){$\mbox{$\bullet$}^\beta$}}
\put(10,0){\dashbox{0.3}(1,7){}}
%
\put(3,5){\framebox(1,1){$\mbox{$\bullet$}^{\alpha'}$}}
\put(12,2){\framebox(1,1){$\mbox{$\bullet$}^{\beta'}$}}
\put(3,2){\framebox(1,1){$\omega$}}
\put(12,5){\framebox(1,1){$\delta$}}
%
\put(16,7){\framebox(4.7,3.7){$X$}}
\put(16,7){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(16,9.7){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(19.7,7){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(19.7,9.7){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(14,8){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(15,8){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(14,9.7){\makebox(1,1){$\mbox{$\bullet$}$}}
\put(15,9.7){\makebox(1,1){$\mbox{$\bullet$}$}}
\end{picture}
\end{center}
By construction, there is a $k\times k$ square filled with ones
just above $\epsilon$, and there is no $k$-chain strictly
north-east of $\epsilon$. This implies in particular that the top
cell in the left-most column of the polyomino must be lower than
the top row of the $k\times k$ square, because otherwise there
could not be any maximal chain for $\epsilon$.
By Lemma~\ref{lem:chain-induction} there must therefore be a
non-empty cell left of $\epsilon$, which we label $\alpha$, and a
non-empty cell below $\epsilon$, which we label $\beta$. Note that
there may be entries to the right of $\epsilon$, in the same row,
which are non-empty. However, we can assume that to the right of
the first such entry all other cells in this row are non-empty,
too, because otherwise we could apply
Lemma~\ref{lem:two-column-chute}.
We can now construct a chutable rectangle: let $\beta'$ be the top
cell containing a $1$ below an empty cell weakly to the right of
$\epsilon$, and if there are several, the left-most. Also, let
$\alpha'$ be the lowest cell among the right-most containing a $1$,
which are weakly below $\alpha$, but strictly above $\beta'$. Let
$\delta$ be the cell in the same row as $\alpha'$ and the same
column as $\beta'$. Let $\omega$ be the cell in the same column as
$\alpha'$ and the same row as $\beta'$ -- a priori we do not know
however that $\omega$ is a cell in the polyomino. We then apply
Lemma~\ref{lem:chutable-rectangle-2} to the rectangle defined by
$\alpha'$ and the first non-empty cell to the right of $\omega$, in
the same row.
To achieve our goal we show that there cannot be a maximal chain
for $\delta$ north-east of $\delta$. Suppose on the contrary that
there is such a chain. At least its top-right element must be in a
row (denoted $R$ in the sketch) strictly above the top cell of the
column containing $\alpha'$: otherwise, $\alpha'$ together with
this chain would form a $(k+1)$-chain. In the sketch, the
non-empty cells that are implied are indicated by the rectangle
denoted $X$, which must be of size $k\times k$ at least.
If the size of the region to the right of $\alpha$ indicated by
$\ell$ in the sketch is $0$ we let $\sigma$ be the bottom-left cell
of a maximal chain for $\epsilon$. Otherwise, define $\sigma$ to
be the bottom-left cell of a maximal chain for the right neighbour
of $\alpha$. In both cases, $\sigma$ must be in a column weakly
west of $\alpha$ -- in the latter case because there are fewer than
$k$ cells above the right neighbour of $\alpha$.
By intersection free-ness applied to the column containing $\sigma$
and the columns containing cells of $X$, the latter columns must
all extend at least down to the row containing $\sigma$. But in
this case the maximal chain with bottom left cell $\sigma$ can be
extended to a $(k+1)$-chain using the cells in $X$.
We have shown that a maximal chain for $\delta$ must have some
elements south-east of $\delta$. We can now apply
Lemma~\ref{lem:chutable-rectangle-2}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:filling-dream}]
All pipe dreams in $\Set{RC}(w)$ contained in $M$ are maximal
$0$-$1$ fillings of $M$, since they can be generated by applying
sequences of chute moves to $D_{top}(M,k)$.
Since we can apply chute moves to any maximal $0$-$1$-filling of
$M$ except $D_{bot}(M,k)$, all such fillings arise in this fashion.
(We have to remark here that in case the pipe dream associated to
some filling would not be reduced, applying chute moves eventually
exhibits that the filling was not maximal.) Together with
Lemma~\ref{lem:chute-moon-closure}, this implies that all fillings
$F_{01}^{ne}(M, k)$ have the same associated permutation.
Note that this procedure implies, as a by-product, that all maximal
$0$-$1$-fillings of $M$ have the same number of entries equal to
zero, {\it i.e.}, the simplicial complex of $0$-$1$-fillings is
pure.
\end{proof}
\section{Applying the Edelman-Greene correspondence}
\label{sec:Edelman-Greene}
Using the identification described in the previous section, we can
apply a correspondence due to Paul Edelman and Curtis
Greene~\cite{MR871081}, that associates pairs of tableaux to reduced
factorisations of permutations. This in turn will yield the desired
bijective proof of Jakob Jonsson's result at least for stack
polyominoes.
The main result of this section was obtained for Ferrers shapes
earlier by Luis Serrano and Christian Stump~\cite{SerranoStump2010}
using the same proof strategy. For stack polyominoes the description
of the $P$-tableau is different, thus we believe it is useful to
repeat the arguments here.
The following theorem is a collection of results from Paul Edelman
and Curtis Greene~\cite{MR871081}, Richard Stanley~\cite{MR782057}
and Alain Lascoux and Marcel-Paul Sch\"utzenberger~\cite{MR686357},
and describes properties of the \Dfn{Edelman-Greene} correspondence:
\begin{thm}\label{thm:edelman-greene}
Let $w$ be a permutation and $s_i$ be the elementary transposition
$(i, i+1)$. Consider pairs of words $(u,v)$ of the same length
$\ell$, such that $s_{v_1},s_{v_2},\dots,s_{v_\ell}$ is a reduced
factorisation of $w$ and $u_i\leq u_{i+1}$, with $u_i=u_{i+1}$ only
if $v_i>v_{i+1}$.
There is a bijection between such pairs of words and pairs $(P, Q)$
of Young tableaux of the same shape, such that $P$ is column and
row strict and whose reading word is a reduced factorisation of
$w$, and such that the transpose of $Q$ is semistandard. Moreover,
if $w$ is vexillary, {\it i.e.}, $2143$-avoiding, the tableau $P$
is the same for all reduced factorisations of $w$.
\end{thm}
This correspondence can be defined via row insertion. We insert a
letter $x$ into row $r$ of a tableau $P$ whose last letter is
different from $x$ as follows: if $x$ is (strictly) greater than all
letters in row $r$, we just append $x$ to row $r$. If row $r$
contains both the letters $x$ and $x+1$ we insert $x+1$ into row
$r+1$. Otherwise, let $y$ be the smallest letter in row $r$ that is
strictly greater than $x$, replace $y$ in row $r$ by $x$ and insert
$y$ into row $r+1$.
We can now construct the pair of tableaux $(P, Q)=(P_\ell, Q_\ell)$
from a pair of words $(u, v)$ as in the statement of
Theorem~\ref{thm:edelman-greene}: let $P_0$ and $Q_0$ be empty
tableaux. Insert the letter $v_i$ into the first row of $P_{i-1}$ to
obtain $P_i$, and place the letter $u_i$ into the cell of $Q_i$
determined by the condition that $P_i$ and $Q_i$ have the same shape.
It turns out that the permutations associated to moon polyominoes are
indeed vexillary:
\begin{prop}\label{prop:vexillary}
For any moon-polyomino $M$ and any $k$ the permutation $w(M, k)$ is
vexillary.
\end{prop}
\begin{rmk}
There are vexillary permutations which do not correspond to moon
polyominoes. For example, the only two reduced pipe dreams for the
permutation $4,2,5,1,3$ are
\begin{equation*}
\begin{tikzpicture}
\content{0.475}{(0, 0)}{%
0/0/+,1/0/+,2/0/+,3/0/\mbox{$\bullet$},4/0/\mbox{$\bullet$},%
0/1/+,1/1/\mbox{$\bullet$},2/1/+,3/1/\mbox{$\bullet$},%
0/2/+,1/2/\mbox{$\bullet$},2/2/\mbox{$\bullet$},%
0/3/\mbox{$\bullet$},1/3/\mbox{$\bullet$},%
0/4/\mbox{$\bullet$}}%
\content{0.475}{(0, 0.475)}{%
0/0/$1$,1/0/$2$,2/0/$3$,3/0/$4$,4/0/$5$,
-1/1/$4$,-1/2/$2$,-1/3/$5$,-1/4/$1$,-1/5/$3$}%
\end{tikzpicture}
\quad\raisebox{40pt}{\text{and}}\quad
\begin{tikzpicture}
\content{0.475}{(0, 0)}{%
0/0/+,1/0/+,2/0/+,3/0/\mbox{$\bullet$},4/0/\mbox{$\bullet$},%
0/1/+,1/1/\mbox{$\bullet$},2/1/\mbox{$\bullet$},3/1/\mbox{$\bullet$},%
0/2/+,1/2/+,2/2/\mbox{$\bullet$},%
0/3/\mbox{$\bullet$},1/3/\mbox{$\bullet$},%
0/4/\mbox{$\bullet$}}%
\content{0.475}{(0, 0.475)}{%
0/0/$1$,1/0/$2$,2/0/$3$,3/0/$4$,4/0/$5$,
-1/1/$4$,-1/2/$2$,-1/3/$5$,-1/4/$1$,-1/5/$3$}%
\end{tikzpicture}
\end{equation*}
\end{rmk}
\begin{proof}
It is sufficient to prove the claim for $k=0$, since the empty cells
in the filling $D_{top}(M, k)$ for any $k$ again form a moon
polyomino. Thus, suppose that the permutation associated to $M$ is
not vexillary. Then we have indices $i<j<k<\ell$ such that
$w(j)<w(i)<w(\ell)<w(k)$. It follows that the pipes entering in
columns $i$ and $j$ from above cross, and so do the two pipes
entering in columns $k$ and $\ell$, and thus correspond to cells of
the moon polyomino. Since any two cells in the moon polyomino can
be connected by a path of neighbouring cells changing direction at
most once, there is a third cell where either the pipes entering
from $i$ and $\ell$ or from $j$ and $k$ cross, which is impossible.
\end{proof}
\begin{thm}[for Ferrers shapes, Luis Serrano and Christian Stump \cite{SerranoStump2010}]\label{thm:ne-se}
Consider the set $\Set F_{01}^{ne}(S, k, \Mat r)$, where $S$ is a
stack polyomino. Let $\mu_i$ be the number of cells the
$i$\textsuperscript{th} row of $S$ is indented to the right, and
suppose that $\mu_1=\dots =\mu_k=\mu_{k+1}=0$.
Let $u$ be the word $1^{\Mat r_1}, 2^{\Mat r_2},\dots$ and let $v$
be the reduced factorisation of $w$ associated to a given pipe
dream. Then the Edelman-Greene correspondence applied to the pair
of words $(u, v)$ induces a bijection between $\Set F_{01}^{ne}(S, k,
\Mat r)$ and the set of pairs $(P, Q)$ of Young tableaux satisfying
the following conditions:
\begin{itemize}
\item the common shape of $P$ and $Q$ is the multiset of column
heights of the empty cells in $D_{top}(S, k)$,
\item the first row of $P$ equals $(k+1, k+2+\mu_{k+2},
k+3+\mu_{k+3},\dots)$, and the entries in columns are
consecutive,
\item $Q$ has type $\{1^{\Mat r_1}, 2^{\Mat r_2},\dots\}$, and
entries in column $i$ are at most $i+k$.
\end{itemize}
Thus, the common shape of $P$ and $Q$ encodes the row lengths of
$S$, the entries of the first row of $P$ encode the left border
of $S$, and the entries of $Q$ encode the filling.
\end{thm}
\begin{rmk}
In particular, this theorem implies an explicit bijection between
the sets $\Set F_{01}^{ne}(S_1, k, \Mat r)$ and $\Set
F_{01}^{ne}(S_2, k, \Mat r)$, given that the multisets of column
heights of $S_1$ and $S_2$ coincide.
Curiously, the most natural generalisation of the above theorem to
moon polyominoes is not true. Namely, one may be tempted to
replace the condition on $Q$ by requiring that the entries of $Q$
are between $Q_{top}$ and $Q_{bot}$ component-wise. However, this
fails already for $k=1$ and the shape
\begin{equation*}
\Yvcentermath0
\young(:\hfil\hfil,%
:\hfil\hfil,%
\hfil\hfil\hfil,%
\hfil\hfil\hfil)\,\,,
\end{equation*}
with $P=\young(345,5)$, $Q_{top}=\young(123,3)$ and
$Q_{bot}=\young(234,4)$. In this case, the tableau
$Q=\young(124,3)$ has preimage
\begin{equation*}
\Yvcentermath0
\young(:\mbox{$\bullet$}\hfil,%
:\mbox{$\bullet$}\x\hfil,%
\mbox{$\bullet$}\hfil\mbox{$\bullet$},%
\mbox{$\bullet$}\hfil\mbox{$\bullet$}).
\end{equation*}
\end{rmk}
\begin{rmk}
One might hope to prove Conjecture~\ref{cnj:lattice} by applying
the Edelman-Greene correspondence, and checking that the poset is a
lattice on the tableaux. However, at least for the natural
component-wise order on tableaux, the correspondence is not order
preserving, not even for the case of Ferrers shapes.
\end{rmk}
\begin{proof}
In view of Proposition~\ref{prop:vexillary}, to obtain the tableau
$P$ it is enough to insert the reduced word given by the filling
$D_{top}(S, k)$ using the Edelman-Greene correspondence, which is
not hard for stack polyominoes.
It remains to prove that the entries in column $i$ of $Q$ are at
most $i+k$ precisely if $(u,v)$ comes from a filling in $\Set
F_{01}^{ne}(S, k)$. To this end, observe that the shape of the
first $i$ columns of $P$ equals the shape of the tableau obtained
after inserting the pair of words $\left((u_1,
u_2,\dots,u_\ell),(v_1, v_2,\dots,v_\ell)\right)$, where $\ell$
is such that $u_\ell\leq k+i$ and $u_{\ell+1}>k+i$.
Namely, this is the case if and only if the first $i+k+\mu_{i+k+1}$
positions of the permutation corresponding to $(v_1,
v_2,\dots,v_\ell)$ coincide with those of the permutation $w$
corresponding to $v$ itself, as can be seen by considering
$D_{top}(w)$, whose empty cells form again a stack polyomino.
This in turn is equivalent to all letters $v_m$ being at least
$k+i+1+\mu_{k+i+1}$ for $m>\ell$, {\it i.e.}, whenever the
corresponding empty cell of the filling occurs in a row below the
$(i+k)$\textsuperscript{th} of $S$, and thus, when it is inside
$S$.
\end{proof}
\section*{Acknowledgements}
I am very grateful to my wife for encouraging me to write this note,
and for her constant support throughout. I would also like to thank
Thomas Lam and Richard Stanley for extremely fast replies concerning
questions about Theorem~\ref{thm:edelman-greene}.
I would like to acknowledge that Christian Stump provided a
preliminary version of \cite{Stump2010}. Luis Serrano and Christian
Stump informed me privately that they were able to prove that all
$k$-fillings of Ferrers shapes yield the same permutation $w$,
however, their ideas would not work for stack polyominoes. I was
thus motivated to attempt the more general case.
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{
"timestamp": "2011-05-19T02:02:56",
"yymm": "1009",
"arxiv_id": "1009.3919",
"language": "en",
"url": "https://arxiv.org/abs/1009.3919"
}
|
\section{INTRODUCTION}
Massive stars play an important role in the evolution of galaxies. They have strong winds and emit a large fraction of their radiation as UV photons; at the end of their evolution they explode as supernova, recycling enriched material into the interstellar medium. Indeed, during their short lives, they are responsible for a large amount of the momentum and kinetic energy input into the interstellar gas.
Thus, the formation of these massive stars, as well as their interaction with their natal environment, is one of the most important subjects in astrophysics. These massive stars are formed in molecular clouds, at places where local agglomerations of matter \citep{Blitz91}, made up of dense gas and appreciable concentrations of dust, may undergo quasi-static gravitational contraction \citep{McKee03}, forming the so called pre-stellar core (T $\approx$ $10$ - $30$ K). This phase presents the youngest epoch in which one can identify a high mass star in the process of formation. Due to their low temperatures, they are detectable at $\lambda$ $\approx$ $4-8$ $\mu$m as absorption sources when seen against the bright Galactic plane, and are detectable in the far infrared (FIR) and submilimeter (sub-mm) in emission \citep{Ward98}. This phase does not last more than $10^{6}$ years \citep{Ward94}.
The subsequent phase is the hot core \citep{Kurtz00} phase. Hot cores (HC) have T $>$ $100$ K and are dense ($n_{H_{2}}$ = $10^7$ $cm^{-3}$). A rapidly accreting massive star is located inside the core. The massive star acquires most of its mass in this phase and, due to this accretion, becomes sufficiently hot, and substantial UV photons are produced. The surrounding hydrogen is rapidly ionized forming a hyper compact H\,{\sc{ii}} region (HCH\,{\sc{ii}}), but this hot gas is not typically detectable in the optical. HCH\,{\sc{ii}} regions are defined as being smaller than $0.01$ pc \citep{Kurtz02} and are very faint or undetectable even at $cm$ wavelengths \citep{Churchwell02} due to their small emission measure. A few of these regions were observed in the hydrogen recombination lines H42$\alpha$-H66$\alpha$ with FWHM $\approx$ $50$-$180$ $kms^{-1}$ \citep*{Johnson98}. Little is known about HCH\,{\sc{ii}} regions, but they are treated as an intermediate stage between HCs and the ultra compact H\,{\sc{ii}} regions (UCH\,{\sc{ii}}).
UCH\,{\sc{ii}} regions represent the earliest phase in which the newly born massive star can be detected by its ionizing radiation. This detection is not direct yet. The natal dust cocoon that surrounds the ionized hydrogen radiates in the mid and far infrared. Differently from low mass stars, massive stars start to burn hydrogen well before the accretion phase finishes \citep{Bernasconi96}. Aided by its wind, the intense radiation from the massive star dissipates and evacuates the surrounding gas and dust that gradually expands \citep{Wood89}. As it does so, its optical depth diminishes and the OB-type exciting star becomes revealed, first in the near infrared and as the gas expands it becomes revealed also in the optical domain. The UCH\,{\sc{ii}} region also becomes larger, forming a compact H\,{\sc{ii}} and finally a normal H\,{\sc{ii}} region, when the OB star exhibits a naked photosphere.
A complete knowledge about the formation and evolution of the massive ionizing stars is fundamental to understanding the evolution of the H\,{\sc{ii}} regions as a whole and their influence on Galactic structure. To further this goal, we have made detailed studies of the stellar content of 35 Galactic H\,{\sc{ii}} regions, where 24 of them have been classified as giant H\,{\sc{ii}} regions (GH\,{\sc{ii}}, $N_{LyC}$ $>$ $10^{50}$ photons per second). These GH\,{\sc{ii}} regions are the best tracers of the spiral structure of the Milky Way, and we argue that some distances to these objects, derived by kinematic techniques are systematically overestimated.
In this work, we have made a study of the stellar content in the near infrared domain of each H\,{\sc{ii}} region from our sample, indicating, when it is possible, the ionizing sources as well as massive young stellar object (MYSO) candidates. The presence (or not) of a cluster of stars (which is typically defined as a clear overdensity in the stellar counts), young stellar objects, and nebular emission were used to establish an evolutionary stage for each star-forming region. Here, we have adopted an evolutionary scale from the youngest ($stage$ $A$) to the most evolved ($stage$ $D$).
In many H\,{\sc{ii}} regions \citep{Blum99,Blum00,Blum01,Figueredo05,Figueredo08}, the spectral type of the ionzing sources were identified, as well as massive objects still surrounded by disks or circumstellar envelopes, MYSOs, which typically do not yet reveal their photospheric features due to the emission from hot circumstellar dust. The disks of MYSOs may be identified by modelling the Keplerian velocities from the CO band head emission profile \citep[e.g.,][]{Blum04} seen toward some of these objects. These studies used the Spectral Atlas of Hot, Luminous Stars at 2 $\mu$m \citep*{Hanson96} to determine the spectral type of the massive stars in giant H\,{\sc{ii}} regions. In many cases significant differences from kinematic distances were found using spectroscopic parallaxes. An important kinematic discrepancy was pointed out by \citet{Xu06}. They showed that the distance to the massive star-forming region W3OH, in the Perseus spiral arm, derived from trigonometric parallax is smaller than that obtained from radio kinematic techniques by a factor of $2$. This difference from the kinematic distance to W3 (by a factor of 2) is similar to that found by Navarete et al. (in preparation) using $K$-band spectrophotometric results. Also, classical T-Tauri Stars (CTTS), objects that exhibit long-wavelength dust emission, generally atributed to a circumnstellar disk, are identified, when present, through near infrared color excess.
The procedure used to analyse the presence of MYSOs, ionizing stars and the evolutionary stage for each H\,{\sc{ii}} region is discussed in the section 4. The individual study of the stellar content of each H\,{\sc{ii}} region is given in the section 5. In the section 6, we present the MYSOs found in our sample and their classifications from near- and mid-infrared photometry. Another difficulty with such regions, is to determine their distances. The most common manner to obtain a distance of a H\,{\sc{ii}} region is using kinematic methodologies. In this work, we compare these kinematic distances with that from non-kinematic techniques. These non-kinematic distances are derived from trigonometric parallax as well as spectrophotometric parallax. In the section 7, we have collected trigonometric distances from the literature, as well as, the spectral type of the ionizing sources of some H\,{\sc{ii}} regions (when they exist in the literature) to derive spectrophotometric distances. Both distances (from trigonometric and spectrophotometric parallax), show discrepancies with kinematic distances.
\begin{table*}
\caption{H\,{\sc{ii}} regions used in the present work. Names and Galactic coordinates are given in columns 1, 2 and 3, respectively. Kinematic distances are presented in columm 4. $N(LyC)$ are presented in columm 5; most of them are Giant H\,{\sc{ii}} regions, $N(LyC)$ $>$ $10^{50}$ $s^{-1}$. The seeing for each region ($K_s$-band) is given in column 6. In column 7, we show the $evolutionary$ $stage$ derived in this work. In column 8 we indicate if the cluster is closer (CL), further away (FW), agrees (AG) with the adopted kinematic distance or if the data are not conclusive (unknown-UN).}
\begin{tabular}{cccccccc}
\hline
Name & $l$ & $b$ & $d_{Kin}^{1}$ & $N(LyC)$ & Seeing$^{2}$ & Evolutionary & Distance \\
& & & $(kpc)$ & $log(s^{-1})$ & $(")$ $K_s$-band & Stage & Classification \\ \hline
M8 & $5.97$ & $-1.18$ & $2.8$ & $50.19$ & $0.84^a$ & B & AG \\
W31-South$^{1}$ & $10.2$ & $-0.3$ & $4.5$ & $50.66$ & $0.56^b$ & B-C & CL$^{5}$\\
W31-North$^{1}$ & $10.3$ & $-0.1$ & $15.1$ & $50.90$ & $0.87^a$ & B & CL \\
W33$^{4}$ & $12.8$ & $-0.2$ & $3.9$ & $50.01$ & $0.69^c$ & A & UN \\
M17 & $15.0$ & $-0.7$ & $2.4$ & $51.22$ & $0.61^b$ & B & CL \\
(4) & $22.7$ & $-0.4$ & $10.6$ & $49.73$ & $0.77^b$ & C-D & CL \\
W42 & $25.4$ & $-0.2$ & $11.5$ & $50.93$ & $0.59^b$ & B & CL$^{5}$ \\
W43 & $30.8$ & $-0.2$ & $6.2$ & $50.83$ & $0.78^c$ & C & CL$^{5}$ \\
K47$^{4}$ & $45.5$ & $+0.1$ & $7.0$ & $49.67$ & $0.78^b$ & A & UN \\
W51 & $48.9$ & $-0.3$ & $5.5$ & $50.03$ & $1.20^a$ & B & CL \\
W51A & $49.5$ & $-0.4$ & $5.5$ & $50.94$ & $0.99^a$ & B & CL$^{5}$ \\
W3$^{3}$ & $133.7$ & $+1.2$ & $4.2$ & $50.25$ & $0.86^6$ & C & CL$^{5}$ \\
RCW42 & $274.0$ & $-1.1$ & $6.4$ & $50.36$ & $0.72^a$ & B & AG \\
RCW46 & $282.0$ & $-1.2$ & $5.9$ & $50.32$ & $1.80^a$ & B & AG \\
NGC3247 & $284.3$ & $-0.3$ & $4.7$ & $50.96$ & $0.81^a$ & B-C & CL \\
NGC3372 & $287.4$ & $-0.6$ & $2.5$ & $50.11$ & $0.69^a$ & C & CL \\
NGC3603 & $291.6$ & $-0.5$ & $7.9$ & $51.50$ & $0.75^a$ & B-C & CL \\
-- & $298.2$ & $-0.3$ & $10.4$ & $50.87$ & $0.53^b$ & B & UN \\
-- & $298.9$ & $-0.4$ & $10.4$ & $50.87$ & $0.81^a$ & A & UN \\
(4) & $305.2$ & $+0.0$ & $3.5$ & $49.53$ & $0.75^a$ & A & UN \\
(4) & $305.2$ & $+0.2$ & $3.5$ & $49.64$ & $0.81^a$ & B-C & AG \\
(4) & $308.7$ & $+0.6$ & $4.8$ & $48.59$ & $1.02^a$ & D & AG \\
RCW87$^{4}$ & $320.1$ & $+0.8$ & $2.7$ & $48.85$ & $1.17^a$ & B & UN \\
-- & $320.3$ & $-0.2$ & $12.6$ & $50.11$ & $1.41^a$ & A-B & UN \\
RCW92$^{4}$ & $322.2$ & $+0.6$ & $4.0$ & $49.52$ & $1.20^b$ & A-B & UN \\
RCW97 & $327.3$ & $-0.5$ & $3.0$ & $50.14$ & $0.81^a$ & A & AG \\
-- & $331.5$ & $-0.1$ & $10.8$ & $51.16$ & $0.84^a$ & A & CL \\
-- & $333.1$ & $-0.4$ & $3.5$ & $50.08$ & $0.59^b$ & B & CL$^{5}$ \\
-- & $333.3$ & $-0.4$ & $3.5$ & $50.04$ & $0.84^a$ & A & UN \\
-- & $333.6$ & $-0.2$ & $3.1$ & $50.43$ & $1.14^a$ & A & UN \\
RCW108$^{4}$ & $336.5$ & $-1.5$ & $1.5$ & $48.29$ & $0.99^a$ & A-B & AG \\
(4) & $336.8$ & $-0.0$ & $10.9$ & $50.48$ & $0.77^b$ & A & AG \\
RCW122$^{4}$ & $348.7$ & $-1.0$ & $2.7$ & $48.41$ & $0.88^b$ & A-B & AG \\
(4) & $351.2$ & $+0.7$ & $1.2$ & $49.67$ & $0.99^a$ & B & FW \\
RCW131$^{4}$ & $353.2$ & $+0.6$ & $1.0$ & $49.32$ & $0.99^a$ & B & FW \\ \hline
\label{table1}
\end{tabular}
\\
\footnotesize{
{\parbox{06.7in}{(1) Kinematic distances adopted here are from \citet{Russeil03}; exceptions are W31-South and W31-North for which we have used distances from \citet{Corbel04};
(2) Except for W33 and W43, which are based on CIRIM data, all the regions above have data from CTIO Blanco-4 meter telescope (ISPI or OSIRIS). Instruments used are denoted a - ISPI, b - OSIRIS and c - CIRIM;
(3) For W3 we have used 2MASS photometric data;
(4) These regions aren't in the sample of Conti \& Crowther (2004), but we have derived the $N(LyC)$ following their work.
(5) These regions have spectrophotometric distances which differ from kinematic results.
}}}
\end{table*}
\section{SELECTION OF THE SAMPLE}
In this work, we present a near infrared study of the stellar content of 35 Galactic H\,{\sc{ii}} regions (Table \ref{table1}). Our sample encompasses that of \citet{Conti04}. In that paper, they conducted a Galactic census of Galactic Giant H\,{\sc{ii}} regions, based on the all-sky 6-cm data set of \citet{Kuchar97}, in connection with the kinematic distances obtained by \citet{Russeil03}. Some H\,{\sc{ii}} regions of our sample were based on the \citet{Dutra03} and \citet{Bica03a} catalogs, who discovered new infrared clusters in the southern and northern hemispheres based on the 2MASS catalog.
In the following sections, we will present near infrared photometric data of each Galactic H\,{\sc{ii}} region of our sample. We present color-color and color-magnitude diagrams (C-C and C-M diagrams, respectively) of each of them. Also, we present false color images of $JHK_s$ ($J$ is blue, $H$ is green and $K_s$ is red), and false color images of 4.5, 5.8 and 8.0 $\mu$m IRAC-{\it Spitzer} images (blue, green and red, respectively).
\section{NEAR- AND MID-INFRARED OBSERVATIONS}
The $J$ ($\lambda$ $\approx$ 1.28 $\mu$$m$, $\Delta$$\lambda$ $\approx$ 0.3 $\mu$$m$), $H$ ($\lambda$ $\approx$ 1.63 $\mu$$m$, $\Delta$$\lambda$ $\approx$ 0.3 $\mu$$m$) and $K_s$ ($\lambda$ $\approx$ 2.19 $\mu$$m$, $\Delta$$\lambda$ $\approx$ 0.4 $\mu$$m$) band images were obtained on the nights of 1, 4 and 20 May 1999; 19 and 21 May 2000; 10 and 12 July 2001, at the Cerro Tololo Interamerican Observatory (CTIO) 4-m Blanco telescope, using the facility infrared imager OSIRIS (FOV of 93$\times$93 arcsec and pixel scale of 0.161''/pixel). On the nights of 3, 4, 5, 6 and 11 July 2005 and 3, 4, 5, 6 and 7 June 2006 we obtained images using the facility infrared imager ISPI (FOV of 10.25 $\times$ 10.25 arcmin and pixel scale of 0.3''/pixel), also at Blanco 4m telescope. Also, on the nights of 28 and 29 August 1998 we obtained images on the CTIO 4-m telescope using the infrared facility CIRIM (FOV of 102 $\times$ 102 arcsec and pixel scale of 0.40``/pixel). OSIRIS, ISPI and CIRIM are described in instrument manuals, found on the CTIO web pages (http://www.ctio.noao.edu).
The data were processed with standard methodology for near infrared images: the images were linearized and corrected for bad pixels, flatfielded and sky subtracted using a blank sky image. The fluxes were extracted using the IDL code Starfinder \citep{Diolaiti00}, except for the regions W31-South and G333.1-0.4 for which we have used the published photometry (\citet{Blum00} and \citet{Figueredo05}, respectively). The fluxes were calibrated according to the 2MASS photometric system \citep{Skrutskie06} to produce a self consistent set of magnitudes, including that from published data. Also, for the W3 H\,{\sc{ii}} region we have used 2MASS $JHK_s$ images and photometric data \citep{Skrutskie06}.
For saturated objects, we have adopted 2MASS photometric data. For non-detections in the $J-$band, we have adopted a limiting magnitude based on $90$th percentile of detected objects. This procedure was based on a test where we have added 9000 artificial stars randomly in our images. These stars had magnitudes varying from $J$ = 15.0 to $J$ = 19 mag and in intervals of 0.5 mag. In some situations, we also needed to use this procedure in $H$-band. These objcets, where we have adopted the $90\%$ limiting magnitudes, are represented by arrows insted of points. The inclination of the arrows follows the interstellar reddening.
Also, we present IRAC-{\it Spitzer} color images. IRAC (Infrared Array Camera) is the mid-infrared camera on the {\it Spitzer} Space Telescope, with four arrays observing at 3.6, 4.5, 5.8 and 8.0 $\mu$m \citep{Fazio04}. The images were obtained using the software leopard (http://archive.spitzer.caltech.edu/) and the {\it Spitzer} program ID for each H\,{\sc{ii}} region is indicated in its respective subsection. The mosaic images were constructed from bcd IRAC images using Mopex software and the IRAC photometry of MYSO candidates was realized using the IDL code Starfinder on the mosaic images following the photometric calibration manual (http://ssc.spitzer.caltech.edu/irac/iracinstrumenthandbook/).
\section{\bf ANALYSES}
\subsection{Reddening Vectors}
There are several interstellar extinction laws in the literature, e.g. \citet{Mathis90,Indebetouw05,Nishiyama06}, but the photometric system plays a very important role in such a choice when we are dealing with color-color diagrams. We have chosen the reddening vector from \citet{Straizys08b} which is derived by fitting a large number of Red Clump (RC) stars along the Galactic plane. RC stars are the metal rich equivalents of the horizontal branch stars and are assumed to have absolute luminosities weakly dependent on ages and chemical composition, and thus are used as standard candles.
The \citet{Stead09} interstellar extinction law ($A_{\lambda} \propto \lambda^{-\alpha}$) has an exponent of $\alpha$ = 2.14, which is one of the largest values derived so far. On the other hand, \citet{Mathis90} has one of the smallest values ($\alpha$ = 1.70). These laws are extreme situations and should cover the range of reddening expected in the Galaxy.
With the choice of standard candles (RC stars) and the availability of deeper datasets \citep[$e.g.$, 2MASS, UKIDSS, see:][respectively]{Skrutskie06,Hewett06}, which cover a large portion of the Galactic plane (and therefore higher values of extinction), the value of $\alpha$ has increased from \citet{Mathis90} where $\alpha$ = 1.70, \citet{Indebetouw05} with $\alpha$ = 1.86 and \citet{Nishiyama06} with $\alpha$ = 1.99. Using $\alpha$ = 2.14 and the central wavelengths for the 2MASS system we derive a slope $E_{J-H}/E_{H-K_s}$ = 2.07, which is in excellent agreement with the results from \citet{Straizys08b}; based on 2MASS data they find a slope of $E_{J-H}/E_{H-K_s}$ = 2.00 (from their slope we rederived the exponent, $\alpha$ = 2.02). \citet{Stead09} also have pointed out that their results are in agreement with the results from \citet{Indebetouw05} if one uses the same effective wavelengths of the 2MASS filters as \citet{Stead09} have used. We have used \citet{Straizys08b} reddening lines in the color-color diagrams, since their results are based on the same photometric system as ours (2MASS system), and their results (slope of $E_{J-H}/E_{H-K_s}$ and $\alpha$) lie between the two extreme interstellar laws illustrated above \citep{Mathis90,Stead09}.
\subsection{C-C and C-M Diagrams}
We use the C-C and C-M (Color-Color and Color-Magnitude, respectively) diagrams (Fig.~\ref{fig:CMD-template}, left and right, respectively) to select candidates to ionizing sources in each H\,{\sc{ii}} region, and follow up $K_s$-band spectroscopy with the aim of deriving the spectrophotometric distances. Some regions are deeply embedded in nebulosity and their stellar content is not detectable. In others, the H\,{\sc{ii}} region does not seem to be associated with a clear over-density of point sources. But, for most of our sample of H\,{\sc{ii}} regions, we find embedded stellar clusters and clear candidates for ionizing sources.
\begin{figure*}
\begin{minipage}[b]{0.49\linewidth}
\includegraphics[height=8.0cm,width=\linewidth]{fig01a.jpg}
\end{minipage}
\begin{minipage}[b]{0.49\linewidth}
\includegraphics[height=8.0cm,width=\linewidth]{fig01b.jpg}
\end{minipage}
\caption{{ Example of C-C and C-M diagrams. 1a): Color-color diagram (C-C), with reddening lines for a M-type star (upper) and for an O-type star (lower). Here, we compare two different interstellar reddening laws. Black lines have a slope of $E_{J-H}/E_{H-K_s}$ = 1.83 \citep{Mathis90}. Blue lines have a slope of 2.07 \citep{Stead09}. For both cases, we have marked the $A_K$ values. 1b): Color-magnitude diagram (C-M) with the location of the main sequence only affected by distance (left sequence) and the main sequence affected also by the interstellar reddening (line to the right). The two extreme cases of reddening vectors are also displayed.}}
\label{fig:CMD-template}
\end{figure*}
In the C-C diagram (Fig.~\ref{fig:CMD-template}a), we can see several lines in black and blue, where four are continuous and two are dashed. Each color represents an interstellar reddening slope ($E_{J-H}/E_{H-K_s}$) for color-color diagrams. The upper lines (black and blue) are the reddening lines for a M-type star, while the lower lines are for an O-type star. The intrinsic colors for the M-type star were obtained from \citet{Frogel78} and the intrinsec colors for the O-type star are from \citet{Koornneef83}. These intrinsic colors were corrected for the 2MASS photometric system using the relations from \citet{Carpenter01}. The dashed lines show the location of the CTTS sequence and the expected reddening lines. T--Tauri stars are low-mass young stellar objects stars and they can be separated in two subclasses: CTTS and weak-line T--Tauri stars (WTTS). CTTS are thought to evolve first into WTTS, where they become virtually disk-less and no longer shows signs of significant accretion, and eventually into solar-like stars on the main sequence \citep{Robrade07}.
Objects to the right of the CTTS region can be more embedded (younger) YSOs. The brighter of which are MYSOs.
It should be noted that some MYSOs will have excess emission in all the near infrared bands due to the reprocessing of their intense radiation. Thus, the CC diagram does not show a unique distinction between the effects of extinction and excess emission. Nevertheless, deeply embedded objects are often found to the right in the near infrared C-C diagram due to stronf $K-$band excess.
Finally, there are others types of young objects, such as Herbig Haro stars \citep[e.g.][]{Nishiyama07,Subramaniam06}, but we do not attempt to identify them specifically in this work.
In the C-M diagram (Fig.~\ref{fig:CMD-template}b) we have labeled the position of the brightest candidate members. In this plot we see two lines that represent the main sequence stars at the adopted (kinematic) distance for each HII region. This main sequence line is constructed using $M_V$ for O-type stars from \citet*{Vacca96} and for the others stars we have used $M_V$ from \citet{Wegner07}, for all stars the $M_{V}-M_{K}$ and $M_{H}-M_{K}$ colors used here are from \citet{Koornneef83}. All these magnitudes and colors were corrected to the 2MASS system. The first line, to the left, represents the main sequence for the foreground objects without reddening (only the inverse square law with distance was considered) and a second line, to the right, represents the main sequence for the members of a cluster (interstellar reddening also included). Two reddening vectors, with $A_{K}$ = 1.0 mag, are also plotted. They show the effect on the main sequence stars of interstellar extinction.
In the diagrams, we compare the effect of two different interstellar extinction laws discussed above. In the C-C diagram, the black lines have a slope of $E_{J-H}/E_{H-K_s}$ = 1.83 \citep{Mathis90} while the blue lines have a slope of 2.07 \citep{Stead09}. Neither slope is related to any particular photometric system, they were derived from their respective universal interstellar extinction laws \citep{Mathis90,Stead09}. However, there are various types of filters with different effective wavelengths and the reddening measured will depend on which filters are used.
In most H\,{\sc{ii}} regions two groups of objects are displayed in these (C-C and C-M) diagrams. The first one is the foreground objects, and the second group is formed by the members of the clusters themselves (e.g. Fig.~\ref{fig:G305-CMD}) projected along the same line of sight. Foreground objects can be distinguished from cluster objects by using their position in the diagrams together with qualitative information in the images. In some situations, there are different groups of objects belonging to the same H\,{\sc{ii}} region. This may occur when a bright cluster has swept away it's natal material from the central region, and triggered star formation at its periphery (or where stars are independently forming in the nearby molecular cloud) producing both main sequence cluster stars and young stellar objects. Also, differential reddening may scatter the distribution of objects in these diagrams.
In most of the C-C and C-M diagrams, objects with excess emission in the $K_{s}$-band (evidenced by large $H - K_{s}$ color) are seen. The brightest of these objects are the MYSOs, recently formed massive stars in the earliest phases of their the life. Such objects are stars in which nuclear fusion has most likely started in the core, but they have not yet begun to ionise their surroundings to form an HII region \citep{Urquhart09} and are surrounded by warm dust and/or disks and so often do not show photospheric features. Many of these objects are likely late O-type or early B-type stars, so-called OB stars of second rank whose more massive neighbors have already shed their natal envelopes and disks.
\subsection{NIR and MIR Images, Evolutionary Stages}
The present sample contains star clusters in different stages of formation. Using our $JHK_{s}$ photometric data, together with {\it Spitzer} images, we can estimate the evolutionary stage of each region by making several assumptions. An evolutionary stage can be inferred with the adoption of the following criteria: In the first stage ($stage$ $A$), the image is dominated by nebulosity in the $K_{s}$-band (mainly Br$\gamma$ at 2.167 $\micron$), in the {\it Spitzer} image the PAH emission (mainly in the 5.8 and 8.0 $\micron$, green and red, respectively) is dominant and there are few stars; In the second ($stage$ $B$), we can see a cluster of stars with some `naked' star candidates, a large number of CTTS and some MYSOs; nebulosity in both images is not so dominant. In the next stage ($stage$ $C$), we detect only minor nebulosity ($K_{s}$-band) and some emission in the {\it Spitzer} image mainly due to gas warm dust (red), with a well-defined cluster of 'naked' stars and a few CTTS and no MYSOs. In the fourth stage ($stage$ $D$), we just see a cluster of stars and no nebulosity in the region. In each region of our sample, we have used these assumptions to estimate an evolutionary stage, which goes from the younger ($A$) to older ($D$) regions.
\section{\bf NEAR- and MID-IFRARED IMAGES WITH COLOR-COLOR AND COLOR-MAGNITUDE DIAGRAMS}
\subsection{G5.97-1.18 (M8)}
A few stars possibly associated with a stellar cluster were detected at R.A.: 18h03m40s and Dec.: -24d22m40s (J2000). Nebular emission (mainly $Brackett$ $gamma$) is strong (Fig.~\ref{fig:G5-2-color}, which makes it very difficult to study the stellar content. This object is also the well known Hourglass region of the Lagoon Nebula (M8) and it is home to the O7 star Herschel 36 \citep{Thompson06}. Due to this nebular emission, we can see few objects associated with this region.
The {\it Spitzer} image ({\it Spitzer} program ID: 30570) shows a central bright region, associated with the embedded objects \#01 and \#41, and a nebulosity with main contributions from the 5.8 and 8.0 $\micron$ bands (green and red, respectively and mainly associated to PAHs) dominating all the field. The stars present in the $JHK_{s}$ color image are very embedded in the bright central nebula. Inside that nebula, we can distinguish two sources, likely the ionizing sources of this region, objects \#01, which is also called Hershel 36, and \#41 (Fig.~\ref{fig:G5-2-color}, left side). Unfortunately, object \#01 is saturated in the $JHK_{s}$ image, and we could not obtain good photometry for it. But, its coordinates, centered on the nebula, suggest it is Herschel 36. We find 2MASS $J$, $H$ and $K_{s}$-band photometry ($J$ = 7.94; $H$ = 7.45 and $K_{s}$ = 6.91 mag). However, \citet{Goto06} show, with better resolution data, that this is not a single object. Near our object \#01 we also detect Her 36 SE, which is a red extension 0".25 southeast of Herschel 36. Object \#41 is also in the center of the nebulosity. Its position in the diagrams (C-C and C-M diagram, Fig.~\ref{fig:G5-2-CMD}) show that it is also likely to be an ionizing source of this region.
Object \#01 is indicated in the C-M and C-C diagrams (Fig.~\ref{fig:G5-2-CMD}) based on its 2MASS photometry. We can see, in the C-C diagram, it displays some color excess. Object \#41 is located in the C-C diagram in a region of infrared excess. It is a region between the CTTS region and the YSO area. Other objects, \#26, \#37, \#49, \#63, \#66, \#71, \#78, \#82 and \#108 (with $H - K_{s}$ $\approx$ 1.3 mag) are located well between the reddening lines. Objects, \#151 and \#115 are to the right of the O-type reddening line, in the CTTS region, and in the {\it Spitzer} image they present small MIR emission. Object \#432 is outside the CTTS range, and is probably a YSO. In the {\it Spitzer} image we can see it as a red object. The number of CTTS is notably larger than the number of YSOs.
This information suggests an evolutionary $stage$ $B$. The size of both images is $\approx$ 3 arcmin on a side. The adopted distance is 2.8 kpc \citep{Russeil03} and its Lyman continuum flux from \citet{Conti04} is $1.55$ $\times$ $10^{50}$ photons per second. Looking at the position of the brightest objects of this region and the reddening vector, the kinematic distance seems to be in agreement with our data.
\subsection{\bf G10.2-0.3 (W31 - South)}
The Galactic GH\,{\sc{ii}} region G10.2-0.3 is part of the W31 complex \citep{Shaver70}. It is one of the largest H\,{\sc{ii}} complexes in the Galaxy with intense star-forming regions. \citet{Wilson74} show that no optical nebulosity appears to be associated with this region, and that this complex is actually formed by three H\,{\sc{ii}} regions: (i) G10.2-0.3 (W31 - South; RA=18:09:21.0, Dec=-20:19:30.9 (J2000)); (ii) G10.3-0.1 (W31 - North; RA=18:08:52.2, Dec=-20:05:53.4 (J2000)) and (iii) G10.6-0.4 (W31B; RA=18:10:28.7, Dec=-19:55:51.7 (J2000)).
Here, we discuss the GH\,{\sc{ii}} region: W31-South (G10.2-0.3), where a stellar cluster was detected. \citet{Kim02} classified this region as a shell morphological type. \citet{Wilson72} derived the kinematic distance as $d_{kin}$ $>$ 4.4 $\pm$ 0.9 kpc (corrected for $R_{\odot}$ $=$ 8.5 kpc), and \citet{Corbel04} derived 4.5 $\pm$ 0.6 kpc ($R_{\odot}$ $=$ 8.5 kpc). Using this distance, \citet{Conti04} derived a Lyman continuum flux of $4.57$ $\times$ $10^{50}$ $s^{-1}$.
\citet*{Blum01} made a detailed study in the near infrared domain of this region. In the Fig.~\ref{fig:W31-color} we reproduce their $JHK_{s}$ image and in the Fig.~\ref{fig:W31-CMD} we reproduce their photometric data transformed to the 2MASS photometric system. In their work, they identified YSOs and four O-type stars. \citet{furness09} recently observed these O-stars with the {\it Spitzer} IRS.
The {\it Spitzer} image ({\it Spitzer} program ID: 3337) at the right shows nebular emission that did not appear in the optical domain \citep{Wilson74}, and it is a little faint in the near infrared image (left hand). The O-type stars \citep[\#2, \#3, \#4 and \#5,][]{Blum01} are faint in this {\it Spitzer} image. This color image points to the presence of nebular material, mainly strong PAH (Polycyclic Aromatic Hidrocarbons) emission (shown in red and not detected at $4.5$ $\mu$m, which is more intense at $\approx$ 6 $\mu$m). The $4.5$ $\mu$m (blue), on the other hand, contains one potentially strong emission feature from H\,{\sc{ii}} regions, the free-free $Br\alpha$ recombination line at 4.05 $\mu$m. Dust is present in all bands, but is strongly present at 8.0 $\mu$m (red).
In the Fig.~\ref{fig:W31-CMD}, we see the C-C and C-M diagrams (color-color and color-magnitude diagrams, respectively). There, we can see two groups of points: (i) one at $H - K_{s}$ $\approx$ 0.4 mag representing the foreground objects and (ii) another at $H - K_{s}$ $\approx$ 1.5 mag representing the cluster members.
In the C-M diagram (Fig.~\ref{fig:W31-CMD}, right), we see some bright objects in the second group: \#2, \#3, \#4 and \#5. Since objects \#1, \#9, \#15, \#26 and \#30 are to the right of the CTTS loci, they are classified as YSOs. In this region we can identify a cluster of stars associated with a nebulosity. The main sequence plotted there is for the kinematic distance, d = 4.5 kpc. Looking at this line and the reddening vector ($A_{K}$ = 1 line), we can see the brightest cluster members seem to be brighter than a reddened O3-type star. This suggests that this kinematic distance is larger than expected, as found by \citet{Blum01} and confirmed by \citet{furness09}.
In the C-C diagram (Fig.~\ref{fig:W31-CMD}, left), we see the O-type stars \citep{Blum01}, cited above, between the lines of natural interstellar reddening. The YSO candidates are at the right of the O-type line, which indicates an excess in $K_{s}$-band emission. This excess comes from circumstellar material that does not allow us to see their photospheric features. Using these diagrams to identify the O-type stars, it was possible to select them for follow up $K$-band spectroscopy. \citet{Blum01} determined a spectrophotometric distance to this region. They showed that objects: \#2, \#3, \#4 and \#5 are, in fact, O-type stars (O5.5 V) by comparing their spectra with that of a $K$-band catalogue of hot stars \citep{Hanson96}. In this way, they found a distance of $d$ $\approx$ 3.4 $\pm$ 0.3 kpc; \citep[see also][]{furness09}. This distance is smaller than the lower limit of the kinematic distances of \citet{Wilson74} and \citet{Corbel04} cited above. Since this region has some objects with naked photospheres, several CTTS and some YSOs, we classify it as $stage$ $B-C$.
\subsection{G10.3-0.1 (W31 - North)}
A stellar cluster was detected at R.A.: 18h08m59s and Dec.: -20d04m50s (J2000). \citet{Wilson74} pointed out that this region is part of the W31 complex. As discussed above, \citet{Corbel04} have shown this region is just in the line of sight of W31, but it is much farther. They have derived a distance of 15.1 kpc. This distance may be too large for the region, as can be seen in the effect of inverse square law in the main sequence location when using this value. The main sequence indicates the ionizing sources of this region (OB-type stars) should be fainter than our data for that distance (Fig.~\ref{fig:G10-CMD},right side). The distance to this cluster might be smaller, which would provide a better fit between the apparent main sequence and the bright stellar content (i.e., the stars clustered near \#82 in Figure A6). The brightest objects may be evolved massive stars in the cluster (\#4, \#6, \#7, \#10 and \#31) given the significant gap between them and the next brightest stars. Using the distance of 15.1 kpc, \citet{Conti04} indicate that this region has $N(LyC)$ of $7.94$ $\times$ $10^{50}$ $s^{-1}$.
In the color images (mainly the $JHK_{s}$ image, Fig.~\ref{fig:G10-color}) we can see a small cluster of stars. In the {\it Spitzer} image ({\it Spitzer} program ID: 146), we see the nebulosity, mainly, in the SE direction with a bright object (\#1032, a massive YSO candidate). The images have size of $\approx$ 2.0 $\times$ 1.5 arcmin.
The white box shows the area used to obtain the photometry. Looking at the diagrams (C-C and C-M, Fig.~\ref{fig:G10-CMD}), we note that objects numbers \#4, \#6, \#7, \#10 and \#31 seem to be evolved stars and are not very close on the expected main sequence location for the cluster of stars. Object \#48 is a foreground object. Objects \#72, \#82 and \#92, are very close to the line of reddening for O-type stars in the C-C diagram (Fig.~\ref{fig:G10-CMD}). Objects \#81 and \#106 are in the CTTS $loci$, and objects \#96 and \#1032 (which is very bright in the {\it Spitzer} image) are at the right of the region for CTTS, indicating they are YSOs. Object \#1032 was not detected at $J$-band, so we have used a limiting magnitude of $J$ = 17.0 mag for $90\%$ detectability.
With this information, two YSOs, several CTTS and a well-defined cluster of stars we can put this region in the evolutionary $stage$ $B$. The kinematic distance of 15.1 kpc may be too large as can be seen in the C-M diagram. If objects \#72, \#82 and \#92 belong to the cluster and are on the main sequence, then a smaller distance is indicated. Objects \#10 and \#31 may be background giant stars since they are apparently bright but lie along the reddening line at large extinction. Alternatively, they could be luminous evolved stars in the cluster seen behind a higher column of dust.
\subsection{G12.8-0.2 (W33)}
No stellar cluster was detected at R.A.: 18h14m14s and Dec.: -17d55m47s (J2000). The distance to this region is 3.9 kpc \citep{Russeil03}. This region belongs to a more extended H\,{\sc{ii}} region, the W33 complex \citep*{Beck98}. Following the work of \citet{Conti04} we derived the Lyman continuum flux, using $T_{e}$ from \citet{Downes80} and $S_{\nu}$ from \citet{Kuchar97}. The derived $N(LyC)$ is $1.02$ $\times$ $10^{50}$ $s^{-1}$, which tells us this is a GH\,{\sc{ii}} region. \citet{Keto89} have observed an expanding shell of gas around the H\,{\sc{ii}} region with $NH_{3}$ and derived a dynamical time scale of $\sim$ $10^{5}$ years for the complex.
In the color images we can see a bipolar structure. But we can not see a well-defined cluster of stars. The image sizes are $\sim$ 1.0 arcmin on a side. In the {\it Spitzer} image ({\it Spitzer} program ID: 146) we can see PAHs emission (green and red) as well as the dark cloud also visible in the $JHK_{s}$ image.
Objects \#3, \#4 and \#6 are in the foreground. Objects \#5, \#17 and \#27 are following the interstellar reddening lines. Objects \#1, \#2, \#7, \#8 and \#10 have excess emission and are in the region of massive YSOs. The tip of the arrows indicate the positions of the objects considering the limiting magnitude of $J$ = $H$ = 17.5 mag (detections above $>$ 90$\%$ completeness) and their inclination is due to the reddening effect. Object \#8 was detected only in $K_{s}$-band, its inclination follows the interstellar law adopted. The objects with vertical lines were detected in $H$ and $K_{s}$-band, so their $H - K_{s}$ is well determined.
The absence of a cluster makes a distance determination impossible. This absence of a cluster, no CTTS and only a few MYSOs (C-C and C-M diagrams, Fig.~\ref{fig:W33-CMD}) indicate this region is at $stage$ $A$.
\subsection{G15.0-0.7 (M17)}
A stellar cluster was detected at R.A.: 18h20m30s and Dec.: -16d10m48s (J2000). \citet{Conti04} have derived a Lyman continuum flux of $N(LyC)$ = $1.66$ $\times$ $10^{51}$ $s^{-1}$ using a distance of 2.4 kpc (from \citet{Russeil03}.
\citet*{Hanson97} have performed a near infrared study of this region, and have identified nine O-type stars using a $K$-band spectral classification scheme. These stars were used to derive a spectrophotometric distance of 1.3 kpc, smaller than the results obtained from kinematic tehcniques \citep[2.4 kpc from][]{Russeil03}. \citet*{Chini80} have made a multicolor study (UBVRI) in the stellar content of M17, and in subsequent works, \citep*[e.g.][]{Chini98,Chini04} have shown the presence of YSOs in this young region.
The M17 H\,{\sc{ii}} region is larger than that shown in the color images (Fig.~\ref{fig:M17-color}). Our data are focused on the central cluster, but with better spatial resolution. Both images show a dark cloud to the East, while in the {\it Spitzer} image ({\it Spitzer} program ID: 107) we can see nebular emission in the SW direction. In the \citet{Hanson97} work, object \#189 is resolved into our objects \#100 and \#200; this effect does not explain the shorter distance obtained by them, since their spectrophotometric distance is based on many OB stars. Unfortunately, these objects are saturated, and the 2MASS images do not have sufficient spatial resolution to separate them.
Objects \#1, \#2, \#3, \#4 and \#7 are stars that belong to the M17 star cluster, since they have similar colors (Fig.~\ref{fig:M17-CMD}). Objects \#8 and \#17 follow the interstellar reddening lines, while objects \#10 and \#24 are MYSOs candidates. Object \#24 was not detected in $J$-band, so we have used a limiting magnitude of $J$ = 16.0 mag (see above for definition of limiting mag). Objects \#5, \#6 and \#23 are in the CTTS $loci$.
The presence of a cluster of stars, several CTTS and some YSOs indicate this is a region in an evolutionary $stage$ $B$. The discrepancy between the main sequence line and the bright stars shows the kinematic distance is larger than observed in our data \citep[consistent with][]{Hanson97}, since the tip of the main sequence line at this distance is fainter than some of our objects.
\subsection{G22.7-0.4}
A stellar cluster was detected at R.A.: 18h34m09s and Dec.: -09d14m26s (J2000). This region appears old, as can be seen from the images which lack strong nebulosity. In larger {\it Spitzer} images ({\it Spitzer} program ID: 146), we can see this region lies on the line of sight to the W41 complex, which has radio coordinates at $\approx$ 7 arcmin NE but has a diameter of $\theta$ = 18.93 arcmin \citep*{Smith78}. This cluster of stars ([MCM2005b]9) is included in the Glimpse catalog of new star clusters \citep{Mercer05}. The star cluster is easily seen and we can see some nebulosity in the {\it Spitzer} image (Fig.~\ref{fig:W41-color}, right side). The size of both images is $\approx$ 2.5 arcmin on a side. If this cluster belongs to the W41 complex, which is not obvious, we can assume its distance is 10.6 kpc \citep{Russeil03} and has a Lyman continuum flux of $N(LyC)$ = $5.37$ $\times$ $10^{49}$ $s^{-1}$. \citet{Leahy08} derived a distance of 4.9 kpc to the region SNR W41 (G23.3-0.3) and overlapping H\,{\sc{ii}} regions. This cluster of stars, which seems to be in projection in the line of sight, was not considered in their work. \citet{Messineo10} found a spectrophotometric distance of 4.2 $\pm$ 0.4 kpc, using two identified O9-B2 supergiants (our objects \#1 and \#6).
Looking at the diagrams (C-C and C-M, Fig.~\ref{fig:W41-CMD}), we can note two groups of objects. The first group of stars with $H - K_{s}$ $\approx$ 0.8 mag and a second with $H - K_{s}$ $\approx$ 1.0 mag. The diagrams show us that objects \#1, \#2, \#3, \#4 and \#6 are on the expected location for stars affected only by the interstellar reddening. These bright objects are saturated in our data and the adopted magnitude values are from 2MASS.
The main sequence line for this distance does not match the observed data well. This cluster of stars is likely closer than what is expected from kinematic results and, in fact, it probably does not belong to the W41 complex. It is more likely a foreground cluster of (evolved) stars. Although the cluster appears evolved, we see some nebulosity in the MIR with a few CTTS. We thus place it in an evolutionary $stage$ $C-D$
\subsection{G25.4-0.2 (W42)}
A few stars associated with an embedded stellar cluster were detected at R.A.: 18h38m15s and Dec.: -06d47m58s (J2000). This region is located in the fourth Galactic quadrant and \citet{Lester85} determined that W42 is at the `near' kinematic distance (3.7 kpc for $R_{\odot}$ = 8 kpc). \citet{Conti04} derived a $N(LyC)$ of $8.51$ $\times$ $10^{50}$ photons per second, using the adopted kinematic distance of 11.5 kpc from \citet{Russeil03}, if using the `near' distance from \citet{Lester85} it would not be giant ($0.9$ $\times$ $10^{50}$ photons). \citet*{Blum00} have made a detailed study of this region. They presented high-spatial resolution $J$, $H$ and $K_{s}$-band images of this massive star cluster. In the Fig.~\ref{fig:W42-color}, to the left, we can see a color image reproduced from \citet{Blum00}. The respective diagrams with the near infrared photometry are presented at the Fig.~\ref{fig:W42-CMD}. The {\it Spitzer} image ({\it Spitzer} program ID: 186) shows nebular emission (mainly in red, $8.0$ $\micron$), which indicates the presence of young embedded stars.
\citet{Blum00} obtained $K_{s}$-band spectra of three of the brightest four stars in the center of the cluster (objects \#1, \#2 and \#3). Object W42 No. \#1, the brightest star, was classified as kO5-O6. With these spectra, \citet{Blum00} derived a ZAMS distance of 2.2 kpc, almost half of the `near' kinematic value \citep{Lester85}. Objects, \#2 and \#3 show no stellar absorption features. This fact, combined with their position in the C-C diagram showing excess emission, lead us to classify them as MYSOs. Object \#57 is very bright in the {\it Spitzer} image but almost invisible in the near infrared image. Since it was not detected at $J$-band, we have used a magnitude limit of $J$ = 16.5 mag, and we suggest it is an excellent YSO candidate.
The presence of nebulosity, CTTS and some MYSOs indicate this region is at $stage$ $B$. The images have $\approx$ 1.5 arcmin on a side. The main sequence line also does not fit our data; as can be seen, the kinematic distance is much larger than that which would be expected to a good fit.
\subsection{G30.8-0.2 (W43)}
A stellar cluster was detected at R.A.: 18h47m37s and Dec.: -01d56m42s (J2000). \citet*{Blum99} have made a detailed study of the stellar content of this region. They have presented $J$, $H$ and $K_{s}$-band data and a new distance to this region, based on $K$-band spectrophotometric parallax.
In the near infrared color image, we can see a small and very crowded, cluster of stars. This cluster is surrounded by a dark lane with some foreground objects in the line of sight. In the {\it Spitzer} image ({\it Spitzer} program ID: 186), we can see the presence of modest nebular emission (Fig.~\ref{fig:W43-color}). \citet{Blum99} have obtained $K$-band spectra of three of the brightest stars in the center of the cluster. Objects \#1, \#2 and \#3 are in the CTTS region (Fig.~\ref{fig:W43-CMD}), but their spectra show photospheric features.
\citet{Blum99} find that W43 No. \#1, the brightest star, has a spectrum similar to the optically classified WN7 star WR 131 \citep*{Figer97} and W43 Nos. \#2 and \#3 are O-type stars. The distance to this region was determined to be 4.3 kpc, while \citet{Russeil03} derived a distance of 6.2 kpc. \citet{Conti04}, using the kinematic distance, have derived a $N(LyC)$ of $6.76$ $\times$ $10^{50}$ photons per second. Object \#9 is very bright in the {\it Spitzer} image and is very faint in the near infrared image. The limiting magnitude used for objects not detected in $J$-band is $J$ = 17.0 mag.
The presence of a cluster of stars with most objects in the CTTS $loci$, and a few YSOs, together with a Wolf-Rayet star, indicate this is a slightly evolved star-forming region, and we classify it in the evolutionary $stage$ $C$.
\subsection{G45.5+0.1 (K47)}
G45.5+0.1 (K47) is located at R.A.: 15h09m59s and Dec.: -58d17m26s (J2000) and no stellar cluster was detected. The adopted kinematic distance to this region is 7.0 kpc \citep{Russeil03}. For this distance, we have obtained a $N(LyC)$ of $4.68$ $\times$ $10^{49}$ $s^{-1}$ following the work of \citet{Conti04}. This is a small region with only a few (detected) stars associated with it. In the {\it Spitzer} image ({\it Spitzer} program ID: 187) the nebulosity dominates all the field of view (Fig.~\ref{fig:K47-color}, right side), and we can see the central bright region. The image sizes are $\approx$ 1.5 arcmin on each side.
Looking at the diagrams (C-C and C-M, Fig.~\ref{fig:K47-CMD}), we can see that objects \#1, \#10 and \#12, with $H - K_{s}$ $\approx$ 0.2 mag, are likely in the foreground. Objects \#2, \#5, \#8 and \#9 are on the expected main sequence location for O-type stars (Fig.~\ref{fig:K47-CMD}, left side). Object \#20 was not detected in $J$-band, so we assumed the value derived from the completeness limit $J$ = 16.5 mag. Its real $J - H$ color will follow the arrow. Object \#6 is in the CTTS region, but its photometry may be contaminated with nebular emission since it appears as bright as a de-reddened O3 star (C-M diagram). Several objects with larger excess are also seen.
The comparative amount of $K_{s}$-band excess objects and the color images indicates that this is a region in the evolutionary $stage$ $A$. The analyses of the kinematic distance, in this case, is inconclusive due to the absence of a cluster.
\subsection{G48.9-0.3 (W51)}
A stellar cluster was detected at R.A.: 19:22:15.0s and Dec.: +14:04:20s (J2000). This is one of the most luminous complexes of massive star-forming regions in the Galaxy \citep{Goldader94} with multiple H\,{\sc{ii}} regions \citep{Wilson70} with at least six regions hosting embedded clusters, all of them optically obscured \citep*{Kumar04}. \citet{Sato10} derived a trigonometric parallax distance of $5.41^{+0.31}_{-0.28}$ kpc to the Main/South part of this compelx, using $H_{2}O$ maser. In the $JHK_{s}$ image, we can see a well-defined cluster of stars as well as nebulosity associated with it. In the {\it Spitzer} image ({\it Spitzer} program ID: 187) the nebular pattern is easily seen. The bright red object in the central part is a contamination of object \#63 by an image artefact. The adopted distance to this region is 5.5 kpc \citep{Russeil03}. Using that distance, \citet{Conti04} derived a $N(LyC)$ of $1.07$ $\times$ $10^{50}$ photons per second (i.e., a GH\,{\sc{ii}} region).
The most prominent stars present a $H - K_{s}$ $\approx$ 0.5 mag, but we can find objects, associated with the nebulosity, with smaller values: \#7, \#15, \#49, \#62, \#63, \#65 and \#202 at $H - K_{s}$ $\approx$ 0.25 mag, as well as objects more reddened $H - K_{s}$ $\approx$ 1.0 mag (\#31, \#40 and \#144). Objects \#91, \#240 and \#540 are in the CTTS loci. In the $JHK_{s}$ color image we can see a cometary shape in the nebulosity (Fig.~\ref{fig:W51-color}, left side), while in the {\it Spitzer} image this shape is more complex (Fig.~\ref{fig:W51-color}, right side). The arrows indicate the location of the (not detected in $J$-band) YSOs: objects \#203, \#238, \#526 and \#1063. Their position in the C-C diagram follow the limiting arrows (based on a magnitude limit of $J$ = $H$ = 17.0 mag). \citet{Kang09} have made a study of embedded young stellar object candidates in the W51 complex and objects \#526 and \#1063 were also indicated as YSOs. Object \#63 is the brightest in the cluster. In the diagrams, its position suggests it may be an unobscured O-type star, while in the {\it Spitzer} image it is, still, very bright.
The presence of some CTTS and a few MYSOs, nebulosity, and a well-defined cluster suggest this region is at evolutionary $stage$ $B$. In this region, we see that the tip of the main sequence is fainter than objects \#7 and \#15 if they are assumed not to be evolved or foreground stars, which indicates the adopted kinematic distance may be larger than the real distance, i.e. W51 may be closer than what is given by kinematic results. This is consistent with the low value of reddening to the cluster too. Alternately, if \#7 and \#15 are not part of the cluster, then the kinematic distance may be accurate.
\subsection{G49.5-0.4 (W51A)}
A few stars possibly associated with an embedded stellar cluster were detected at R.A.: 19h23m42s and Dec.: +14d30m33s (J2000). This is one of the most luminous regions in the W51 complex, which is divided into eight smaller radio sources: W51A to W51H. The W51 complex is located at a kinematic distance of 5.5 kpc (near distance), adopting the value derived by \citet{Russeil03}. For this distance, \citet{Conti04} derived for W51A a $N(LyC)$ of $8.71 \times 10^{50}$ photons per second, indeed a GH\,{\sc{ii}} region. Nebulosity (Fig.~\ref{fig:W51A-color}) is well distributed through the field of view of the image with bright and dark components.
In the C-C diagram there are several objects in the CTTS region (objects \#7, \#17, \#24, \#25, \#44, \#50, and \#103). Also we see YSO candidates (objects \#45, \#61, \#62, \#73 and \#98). Objects \#21, \#32, \#57 and \#60 are quite close to the line of reddening for O-type stars. Objects \#52 and \#59 are foreground sources.
\citet{Figueredo08} have made a detailed study of the stellar content of this region. They have used spectrophotometric parallax of 4 O-type stars (\#44, \#50, \#57 and \#61; O5, O6.5, O4 and O7.5, respectively) to derive a distance of 2.0 $\pm$ 0.3 kpc. The arrows in the C-C diagram are based on the magnitude limit of $J$ = 16.5 mag, and indicate objects not detected in $J$-band. The presence of a few cluster members, associated with the color images which shows strong nebular emission (mainly $Br\gamma$) and the absence of stellar objects on the {\it Spitzer} image ({\it Spitzer} program ID: 187), indicates this region is very young.
Also, \citet{Barbosa08} have presented high spatial resolution spectroscopy of two very massive young stars in early formation stages, W51 IRS 2E and IRS 2W, (Fig.~\ref{fig:W51-color} left side). Both of them are embedded sources in the Galactic compact H\,{\sc{ii}} region W51 IRS2. Barbosa et al. find a distance of 5.1 kpc based on their spectrum of the source associated with W51d in IRS2.
Moreover, \citet{Xu09} have derived a trigonometric parallax to IRS2W using 12 GHz methanol masers and obtained a distance of $5.1^{+2.9}_{-1.4}$ kpc, close to the adopted kinematic value and that of \citet{Barbosa08}. \citep{Sato10} using $H_{2}O$ maser parallax, in the W51 Main/South region, found a distance of $5.41^{+0.31}_{-0.28}$ kpc. The discrepancy on the distances of W51A and IRS2 indicates that these two regions may not be physically connected and that the stars observed by \citet{Figueredo08} are closer along the line of site and projected onto W51A.
Objects IRS2E and IRS2W are associated with star forming regions of evolutionary $stage$ $A$, while the others objects are associated with type $B$. There are some objects brighter than the tip of the main sequence line, but they are likely foreground objects.
\subsection{G133.7+1.2 (W3)}
A stellar cluster was detected at R.A.: 02h26m34.4s and Dec.: +62d00m45s (J2000). It is at the Perseus spiral arm, and its adopted kinematic distance is 4.2 kpc \citep{Russeil03}. Included in the sample of GH\,{\sc{ii}} regions of \citet{Conti04}, it has a Lyman continuum flux of $N(LyC)$ = $1.78 \times 10^{50}$ photons per second. The $JHK_{s}$ results presented here (images and photometry) are from 2MASS.
In the $JHK_{s}$ color image, we see a cluster of stars in the center of the field, and some nebular emission to the NW and to the SE, surrounding the cluster. Each of the $J$, $H$ and $K_{s}$ images is a 18$'$ $\times$ 18$'$ mosaic, constructed from several 2MASS images. In the {\it Spitzer} image ({\it Spitzer} program ID: 127), we see the nebulosity of this field in detail, and it appears that this nebulosity belongs to a unique region, which is not so obvious in the $JHK_{s}$ image.
The brightest star of the central cluster (\#159) was used by \citet{Humphreys78} to derive a spectrophotometric distance (in the optical domain) to this region, and they found a distance of 2.2 kpc. However, the adopted kinematic distance is 4.2 kpc \citep{Russeil03}. Using trigonometric parallax, \citet{Xu06} derived a distance of 1.95 kpc to the star-forming W3OH. W3OH is a region that belongs to the W3 complex, and it is seen in the $JHK_{s}$ and {\it Spitzer} images indicated by the star \#248 (see Fig.~\ref{fig:W3-color}). The distances obtained by parallaxes (spectrophotometric and trigonometric) are in a good agreement with each other and both smaller than the kinematic result by a factor of 2.
Furthermore, Navarete et al. (in preparation), derived a distance of 1.85 $\pm$ 0.92 kpc to W3. They have used $K$-band spectrophotometric parallax (to the O-type stars \#159, \#390 and \#559) and their results are in accordance with that from \citet{Xu06} and \citet{Humphreys78}.
We can see in the C-C diagram (Fig.~\ref{fig:W3-CMD}, on the left) that objects \#252, \#347, \#390 and \#559, with $H - K_{s}$ $\approx$ 0.5 mag, are near the O-type reddening line, and objects \#444 and \#248 are in the CTTS region.
The tip of the main sequence is fainter than the brighest object of this region (\#159), as is expected since the kinematic distance appears to be too large. This region has a well-defined cluster, nebulosity is seen in both images, especially in the {\it Spitzer} image. There are several CTTS (e.g.: \#444 and \#248) and some massive YSOs. So, we can classify this region as evolutionary $stage$ $B$, while the central cluster is in a evolutionary $stage$ $C$.
\subsection{G274.0-1.1 (RCW42)}
The Galactic GH\,{\sc{ii}} region G274.0-1.1 is also known as RCW42 and a stellar cluster was detected at R.A.: 09h24m30.1s and Dec.: -51d59m07s (J2000). It belongs to a larger structure, a shell called $GSH 277+00+36$ that is at a distance of 6.5 kpc \citet{Macclureetal03}, is 600 pc in diameter, and extends above and below 1 kpc of the Galactic midplane. In the $JHK_{s}$ color image (Fig.~\ref{fig:G274-color}, left side), we see a crowded cluster of stars surrounded by a reddish nebula. We can see, in the NE part of this region, a dark cloud obscuring most of the background stars, possibly precluding the detection of other cluster members. regions like this are very difficult to analyse for cluster membership due to their crowded fields and embedded stars. The {\it Spitzer} image ({\it Spitzer} program ID: 40791) shows a field dominated by weak nebular emission, with the region surrounding the cluster emitting mostly at 8.0 $\micron$ (red).
The distance of G274.0-1.1 used by \citet{Conti04} is 6.4 kpc \citep{Russeil03}. That distance leads to a Lymann continuum luminosity of $N(LyC)$ $\approx$ 2 $\times$ $10^{50}$ photons per second. This implies, at least, a dozen early O-type stars associated within the region.
Looking at the diagrams (C-C and C-M), the objects numbers \#20, \#24, \#30 and \#32, with $H - K_{s}$ $\approx$ 0.5 mag, are on the expected main sequence location and affected only by the interstellar reddening. Moreover, these stars are close to sites of nebular emission, some of them near the center of the cluster. This suggests these objects may be the ionizing sources of the H\,{\sc{ii}} region. Object \#6 is in the foreground. Also, object \#14, which is a bright star and less reddened than the others cited above, could be an O3-O4V star. On the other hand, objects \#21, \#36, \#40 and \#42 show a color excess, objects \#31 and \#33 are bright in $K_{s}$-band and are at the right of the CTTS region with $H - K_{s}$ $\approx$ 1.8 mag (see the CCD in Fig.~\ref{fig:G274-CMD}).
The cluster members present a large range of colors, indicating they are very embedded, and we also see a large amount of CTTS as well some massive YSOs, but the nebular component does not emit strongly (it is mostly 'dark') Thus we suggest an evolutionary $stage$ $B$. In this region, it is not clear if the main sequence line (adopted kinematic distance) is in agreement with the observed data. However, the tip of this main sequence is brighter (as one would expect) than our data, which indicates the adopted kinematic distance may be correct.
\subsection{G282.0-1.2 (RCW46)}
A stellar cluster was detected at R.A.: 10h06m38.1s and Dec.: -57d12m28s (J2000) toward the GH\,{\sc{ii}} region also known as RCW46. In the $JHK_{s}$ color image (Fig.~\ref{fig:G282-color}, left side), we see a small crowded cluster of stars. Nebulosity, in the central part, is visible in both images and in the {\it Spitzer} color image ({\it Spitzer} program ID: 30734) we can see a surrounding shell nebulosity with heated dust emitting at 8.0 $\micron$ at the central part. In the southeast part of this region, there is a dark cloud obscuring most of the background stars.
The distance of G282.0-1.2 used by \citet{Conti04} was 5.9 kpc \citep{Russeil03}. That distance leads to a Lymann continuum luminosity of $N(LyC)$ $\approx$ $2.09$ $\times$ $10^{50}$ photons per second (GH\,{\sc{ii}} region).
Looking at the diagrams (C-C and C-M, both at Fig.~\ref{fig:G282-CMD}), the objects \#9, \#10, \#11, \#12, \#13, \#18 and \#20, with $H - K_{s}$ $\approx$ 1.0 mag, are near the expected main sequence location and affected only by the interstellar reddening. These stars are close to the center of the cluster. This suggests they may be the ionizing sources of this H\,{\sc{ii}} region. Also, the object \#5, which is a bright star and less reddened, is in the line of reddening of a M-type star (C-C diagram). Objects \#8, \#15 and \#16 seem to be foreground stars. On the other hand, object \#10 seems to be a highly reddened late O-type star (see the C-C diagram in Fig.~\ref{fig:G282-CMD}), possibly on the far side of the cluster. In the C-M diagram, object \#31 and \#46 appear like very reddened O-type stars, and in the C-C diagram, we can see they present a $K_{s}$-band excess, and are in the YSO area.
The {\it Spitzer} image shows a shell-like structure with some stars well inside the shell. There are 2 YSOs, several CTTS and almost no nebulosity in the near infrared image, mostly visible in the {\it Spitzer} image and a well-defined cluster. We thus put it in the evolutionary $stage$ $B$. The kinematic distance may be correct here since the tip of the main sequence line is brighter than the observed data. Except object \#5, which, if de-reddened, may be brighter than the tip of this main sequence, but it is not clear if this object belongs to the H\,{\sc{ii}} region.
\subsection{G284.3-0.3 (NGC3247)}
A stellar cluster was detected at R.A.: 10h24m17.3s and Dec.: -57d45m36s (J2000). This is a typical H\,{\sc{ii}} region. A well defined cluster with a strong nebulosity surrounding it. The distance to this region is 4.7 kpc \citep{Russeil03}. At this distance, its Lymann continuum luminosity is $N(LyC)$ $\approx$ $9.12$ $\times$ $10^{50}$ photons per second \citep{Conti04}. In the cluster we find some stars as O-type candidates. In the {\it Spitzer} image ({\it Spitzer} program ID: 195, Fig.~\ref{fig:G284-color}, right side), we also see the few brightest stars. Object \#13 is remarkable since it shows a jet above it. Emission in 5.8 $\micron$ (green) dominates the field at NW of the central cluster, while in the SE direction it is emission at 8.0 $\micron$ that dominates.
Actually, this region extends over a larger area, and in this work we depicted only the central part. The larger (not shown) image measures $\approx$ 10 arcmin, but the region reproduced at Fig.~\ref{fig:G284-color} covers only $\approx$ 3.35 arcmin on a side.
Looking at the diagrams (C-C and C-M, Fig.~\ref{fig:G284-CMD}), we can easily seen two sets of objects. The first group of stars with $H - K_{s}$ $\approx$ 0.5 mag and a second group with $H - K_{s}$ $\approx$ 1.0. In the {\it Spitzer} image (Fig.~\ref{fig:G284-color}, right side) we can see the objects with infrared excess and the nebulosity. The main candidates to be ionizing sources (\#3, \#4, \#7, \#8, \#10, \#11, \#12, \#13, \#14, \#20, \#21 and \#25) are in the first group of objects. Unfortunately, objects \#3 and \#4 are saturated in our images, so we have used 2MASS data. Objects numbers \#54, \#60 and \#70 are YSO candidates.
Since, there are many CTTS, and several MYSOs, and a well-defined cluster with nebulosity surrounding it (as can be seen in the near and mid infrared images), we can associate this region with $stage$ $B-C$. It is clear the O-type stars at the main sequence are fainter (more distant) than our data suggesting the cluster is closer to us than given by kinematic distance. This indicates the kinematic distance may be in error, and the real distance may be closer.
\subsection{G287.4-0.6 (NGC3372)}
G287.4-0.6 is also known as the Carina nebula (NGC3372) and a stellar cluster was detected at R.A.: 10h43m50.1s and Dec.: -59d32m47s (J2000). The distance of G287.4-0.6 used by \citet{Conti04} was 2.5 kpc \citep{Russeil03}. That distance leads to a Lymann continuum luminosity of $N(LyC)$ $\approx$ $1.29$ $\times$ $10^{50}$ photons per second, indicating this is a GH\,{\sc{ii}} region.
In the $JHK_{s}$ color image (Fig.~\ref{fig:G287-color}, left side), we have focused on the crowded cluster of stars. In a larger area (not shown here), we note the well-known strong nebula surrounding this cluster of stars. In the {\it Spitzer} image ({\it Spitzer} program ID: 30734) the 5.8 $\micron$ (green, and not strongly) dominates the field. Looking at the image, we can see that this region is not very distant from the Sun, since the stars can easily be distinguished and they are not strongly reddened by interstellar extinction (C-M and C-C diagrams).
Looking at the diagrams (C-C and C-M), we note objects \#100, \#1, \#2, \#4, \#5, \#8, \#10, \#14, \#17 and \#45, with $H - K_{s}$ $\approx$ 0.1 mag, are on the expected main sequence and affected only by the interstellar reddening. These stars are close to the center of the region (except objects \#1, \#2 and \#5, which may be not connected to the main cluster), indicating they may be the ionizing sources of this region. Object \#45 is remarkable due the presence of a bow shock above it.
We can see (color image, C-C and C-M diagrams) some reddened objects, but they appear to be objects behind the dust lane of this H\,{\sc{ii}} region. Object \#138, with $H - K_{s}$ $\approx$ 2.5 mag, is located at a position of an O-type star with a strong infrared excess. In the C-C diagram, this object is located to the right of the position of the CTTS, indicating it is a YSO.
The brightest objects (\#100, \#1, \#2, \#3, \#4, \#5 and \#8) are saturated in our data, so we have used 2MASS values for $J$, $H$ and $K_{s}$ for each of them, as the resolution of 2MASS is poorer than our images, these fluxes may be affected by nearby objects. But, assuming these fluxes are are well determined in the 2MASS catalog, the brightest stars of the cluster are above the tip of the main sequence line. This indicates that the kinematic distance might be too large, and the real distance could be smaller.
We classify this region as evolutionary $stage$ $C$, since we can see the presence of a central cluster cleared of nebular emission. In the central area, we see little nebulosity in both images, there are several CTTS and just one YSO, which is not located near the central cluster.
\subsection{G291.6-0.5 (NGC3603)}
A stellar cluster was detected at R.A.: 11h15m07.1s and Dec.: -61d15m37s (J2000). The adopted distance used by \citet{Conti04} was 7.9 kpc \citep{Russeil03}. That distance leads to a Lymann continuum luminosity of $N(LyC)$ $\approx$ $3.16$ $\times$ $10^{51}$ photons per second. \citet{Melena08} have derived a spectrophotometric distance of 7.6 kpc, but in the optical domain. \citet{Moffat02} have found a large number of X-ray sources, these sources were found with greater frequency toward the cluster center and with no obvious optical counterparts.
In both color images (Fig.~\ref{fig:NGC3603-color}), $JHK_{s}$ and {\it Spitzer} ({\it Spitzer} program ID: 40791), we see nebulosity surrounding a well defined cluster of stars. Except objects \#6, \#19 and \#20, which are labeled in the $JHK_{s}$ image, all the other objects are located inside the black box.
Using isochrone fitting, \citet{Stolte04} derived a distance of 6.0 kpc to this region, which is significantly smaller than that derived by kinematic technique, 7.9 kpc, \citet{Russeil03}. \citet{Melena08} have made a detailed study in the stellar content of NGC3603 and found several O-type stars.
The crowded cluster (\#6, \#10, \#25, \#29, \#34, \#49, \#51, \#55, \#68 and \#71) in the center of the image provides the ionizing sources of this region. All the objects in the central bright part are located at the tip of the C-M diagram, with $H - K_{s}$ $\approx$ 0.3 mag, and are inside the red square in the C-C diagram (Fig.~\ref{fig:NGC3603-CMD}). Object \#6 is saturated in our images, so we have used 2MASS values ($J$ = 8.60; $H$ = 8.03 and $K_{s}$ = 7.72 mag). Object \#20 is in the CTTS region at the C-C diagram, but if de-reddened (C-M diagram) it would be a `naked' O-type star. Object \#19 is in the YSO region.
The presence of this well-defined cluster, some nebulosity in the neighbourhood of the star cluster (as can be seen in both images, the near infrared and {\it Spitzer}), several CTTS and some MYSOs leads to a region between $stages$ $B$ or $C$. This is also another case of strong disagreement between the main sequence line (kinematic distance) and the observed data: the cluster appears to be closer than what is predicted by kinematic techniques. However, the tip of the observed sequence includes evolved stars, and this makes it brighter than the main sequence.
\subsection{G298.2-0.3}
A few stars associated with an embedded stellar cluster were detected at R.A.: 12h09m58.1s and Dec.: -62d50m00s (J2000). The adopted distance used by \citet{Conti04} was 10.4 kpc \citep{Russeil03}. That distance leads to a Lymann continuum luminosity of $N(LyC)$ $\approx$ $7.41$ $\times$ $10^{50}$ photons per second.
In the $JHK_{s}$ color image (Fig.~\ref{fig:G298-2-color}, left side), we see nebulosity that spans the image. We also see a very embedded cluster and, at the central part of this region, some bright stars. Since the nebulosity dominates this field, we have detected few objects (Fig.~\ref{fig:G298-2-CMD}) that might be associated with the embedded cluster. The {\it Spitzer} image ({\it Spitzer} program ID: 189) shows us that this region is very young with an embedded cluster of stars in the bright area.
The best candidates to be an ionizing source of this region is object \#4. This object seems to be less affected by the nebulosity than the others (C-C diagram, Fig.~\ref{fig:G298-2-CMD}).
Looking at the diagrams, we note objects number \#5, \#10, \#21, \#58 and \#26 are the brightest objects and have an $H - K_{s}$ around 0.5 mag. Object \#32 is near the O-type reddening line in the C-C diagram. Object \#23 is a bright object located in the CTTS region.
The presence of an embedded cluster indicates this is a region at $stage$ $B$. As can be seen in the $JHK_{s}$ color image, the cluster of stars is compact and very crowded; this may indicate the real distance to this region is large, as suggested by the kinematic results. The cluster members (\#21, \#23 and \#32), if de-reddened would be brighter than the tip of the main sequence line, but they can be foreground objects. The analyses of the kinematic distance is inconclusive.
\subsection{G298.9-0.4}
The Galactic GH\,{\sc{ii}} region G298.9-0.4 is located at R.A.: 12h15m25.1s and Dec.: -63d01m13s (J2000) and no stellar cluster was detected. The adopted distance used by \citet{Conti04} was 10.4 kpc \citep{Russeil03}. That distance leads to a Lymann continuum luminosity of $N(LyC)$ $\approx$ $7.41$ $\times$ $10^{50}$ photons per second.
In the $JHK_{s}$ color image (Fig.~\ref{fig:G298-color}, left side), we see small nebulosity, and several field stars. In the line of sight of the small nebulosity, we see only a few stars. The {\it Spitzer} image ({\it Spitzer} program ID: 189) shows a bright and red (8.0 $\micron$) area that coincides with the nebulosity and a few embedded objects in the $JHK_{s}$ image. Object \#21 is very bright in the {\it Spitzer} image.
The identification of ionizing source candidates in this region is not so obvious. In the C-M diagram (Fig.~\ref{fig:G298-CMD}), objects \#12 and \#13 are in the area expected for an ionizing source. However, looking at the C-C diagram, object number \#13 is near the reddening line for M-type stars. Object \#12 is near the line of reddening for O-type stars, without infrared excess, but its connection with the nebulosity is not so direct when we look at the images. On the other hand, object \#21, which is very bright and shows a large $K_{s}$-band excess is a MYSO.
The presence of nebular emission, and the presence of several CTTS with one MYSO indicate this region is at $stage$ $A$. In this case, it is very difficult to point to stars associated with nebulosity. The brightest objects (\#12 and \#13) may be foreground stars. So, it is not obvious if the adopted kinematic distance is correct or not.
\subsection{G305.2+0.0}
The H\,{\sc{ii}} region G305.2+0.0 is located at R.A.: 13h11m15s and Dec.: -62d45m20s (J2000), and a few stars associated with an embedded stellar cluster were detected. It is at a distance of 3.5 kpc \citep{Russeil03}, and has $N(LyC)$ = $3.39$ $\times$ $10^{49}$ $s^{-1}$. This region was divided in two parts (Fig.~\ref{fig:G305-0-color}), and the photometry was carried out on the objects within both white rectangles (Fig.~\ref{fig:G305-0-color}, on the left). Strong nebular emission can be seen in both regions, mainly at longer wavelenths. But, we can not see a cluster of stars. The nebulosity becomes more evident in the {\it Spitzer} image ({\it Spitzer} program ID: 189), where in some places the image is saturated. Also, we can see shells and a cavity in the lower left (SE direction, also evident in the near infrared).
Looking at the diagrams (Fig.~\ref{fig:G305-0-CMD}), we note that objects numbers \#134, \#266 present very similar colors ($H - K_{s}$ $\approx$ 0.8 mag), while \#126, with $H - K_{s}$ $\approx$ 0.2 mag, seems to be a foreground star in the line of sight of the nebulosity. \#86, \#125 and \#252 seem to be background stars with a large amount of nebular material in front of them. In the C-C, object \#873 is in the YSOs region.
In the diagrams (Fig.~\ref{fig:G305-0-CMD}), we can see two sets of objects. The first group of stars, with $H - K_{s}$ $\approx$ 0.30 mag, are likely foreground objects, while the sparse group of stars, with $H - K_{s}$ $\approx$ 0.75 mag, are likely members of the cluster.
Since we don't see a well defined cluster, the analyses of the kinematic distance is inconclusive. The Brackett gamma emission is strong, there is at least one YSO (object \#873) and some stars in the CTTS region; we find this region is at $stage$ $A$.
\subsection{G305.2+0.2}
A stellar cluster was detected at R.A.: 13h11m40s and Dec.: -62d33m09s (J2000). Its distance is 3.5 kpc \citep{Russeil03} and its Lyman continuum flux is $N(LyC)$ = $4.36$ $\times$ $10^{49}$ $s^{-1}$. The presence of a rich cluster of stars is evident. In the Fig.~\ref{fig:G305-color}, we show a 2'x2' portion of the ISPI image, centered on the cluster of stars. A faint nebular emission appears to surround the cluster, and is better seen in the {\it Spitzer} image ({\it Spitzer} program ID: 189). This faint nebulosity indicates this cluster is very evolved.
Looking at the diagrams (C-C and C-M, Fig.~\ref{fig:G305-CMD}), we suggest that objects numbers \#38, \#39, \#40, \#59 and \#65 are on the expected main sequence location for O-type stars and are affected only by interstellar reddening. These stars are close to the nebulosity, as seen in the {\it Spitzer} image (Fig.~\ref{fig:G305-color}, right side), which indicates they may be the ionizing sources of the H\,{\sc{ii}} region.
In the diagrams (Fig.~\ref{fig:G305-CMD}), we can see two groups of objects. The first group of stars with $H - K_{s}$ $\approx$ 0.3 mag and a second group with $H - K_{s}$ $\approx$ 0.75. The cluster members are in the second group of points, and the first group are likely foreground objects. There are some objects with $K_{s}$-band excess. Objects \#129 and \#174 follow the reddening vectors of main sequence stars, while the object \#134 has an excess emission in $K_{s}$-band and it is near the region of the CTTS.
The well-defined cluster together with surrounding nebular emission, several CTTS and some YSOs indicate that this is a region between $stages$ $B$ and $C$. The agreement between the kinematic distance and our observed data, also, seems to be valid in this region.
\subsection{G308.7+0.6}
A stellar cluster was detected at R.A.: 13h40m12.1s and Dec.: -61d43m46s (J2000). It is at a distance of 4.8 kpc \citep{Russeil03}, and we derived a $N(LyC)$ = $3.89$ $\times$ $10^{48}$ $s^{-1}$. This seems to be an evolved H\,{\sc{ii}} region, since the cluster members are well distinguished, and we can not see any nebulosity surrounding them in the $JHK_{s}$ color image. In the same way, the {\it Spitzer} image ({\it Spitzer} program ID: 190) does not show strong nebulosity, only a tiny amount of emission at 8.0 $\micron$.
In the $JHK_{s}$ color image (Fig.~\ref{fig:G308-color}, left side), we see a cluster of stars, which is evident in the diagrams at $H - K_{s}$ around 0.70 mag.
Looking at the diagrams (C-C and C-M), objects \#28, \#31, \#44 and \#51 are on the expected main sequence location and affected only by interstellar reddening. These stars are close to the center of the cluster. These stars are located in a first group of objects with $H - K_{s}$ $\approx$ 0.2 mag. A second group of stars is also seen with $H - K_{s}$ $\approx$ 0.7 mag. In this second group we find objects \#4, \#9 and \#22. Objects \#4 and \#22 are only affected by interstellar reddening, but object \#9 has an excess in $K_{s}$-band as can be seen in the C-C diagram (Fig.~\ref{fig:G308-CMD}, left size).
Also, there are no bright embedded objects. The absence of nebulosity is an indication that the winds from the massive stars have had time enough to sweep away the gas and intracluster dust. Also, we can see some objects redder than the others in the $JHK_{s}$ image. These objects may be surrounded by circumstellar material emiting strongly in the $K_{s}$-band and in the IRAC channel 1 (e.g. \#4 and \#9).
The objects \#1 and \#4 are saturated in our images. From 2MASS, their magnitudes are: \#1: $J$ = 10.36; $H$ = 7.84 and $K_{s}$ = 6.44 mag; and \#4: $J$ = 10.47; $H$ = 8.60 and $K_{s}$ = 7.83 mag. These data show object \#1 is, actually, a very bright object with infrared excess and located in the CTTS region ($H - K_{s}$ = $1.40$ mag and $J - H$ = $2.53$ mag).
In the C-M diagram, we see that the main sequence location is in good agreement with our observed data, which indicates that the adopted kinematic distance may be correct. This region has a well-defined cluster, there is no nebulosity in both images and a few CTTS. We thus assign it an evolutionary $stage$ $D$.
\subsection{G320.1+0.8 (RCW87)}
A stellar cluster was detected at R.A.: 15h05m25.1s and Dec.: -57d30m57s (J2000) toward G320.1$+$0.8, also called RCW87. Its distance is 2.7 kpc \citep{Russeil03} and has $N(LyC)$ = $3.55$ $\times$ $10^{48}$ $s^{-1}$. This seems to be a young H\,{\sc{ii}} region, since the cluster members are still surrounded by nebular emission (Fig.~\ref{fig:G320-color}).
In the $JHK_{s}$ color image (Fig.~\ref{fig:G320-color}, left side), we see a crowded cluster of stars and a bubble nebula is easily defined in the {\it Spitzer} image ({\it Spitzer} program ID: 190). In the {\it Spitzer} image, the contribution of the gas is more evident and stars \#3 and \#15 are obvious bright point sources in the IRAC channel 1 (centered at 3.5 $\mu$m).
Looking at the diagrams (C-C and C-M), objects \#9 and \#14 are on the expected main sequence location, but only object \#14 is close to the center of the H\,{\sc{ii}} region, which indicates it may be the ionizing source of the H\,{\sc{ii}} region.
There are bright objects in the $K_{s}$-band with large infrared colors: \#10 and mainly \#3, \#15 and \#106. In the C-C diagram, we see that object \#10 is close to the line of reddening of a M-type star. But if we consider the normal scatter from the hot star line, it is possible it would be an ionizing source (in the CMD it is in the position for a reddened O-type star). However, object \#10 is not close to the center of the H\,{\sc{ii}} region. Also, in the C-C diagram we see that object \#106 is close to the line of reddening of an O-type star. Object \#3 is in the CTTS region. Object \#15 is very bright in the $K_{s}$-band and presents a large infrared excess emission. Since it was not detected in the $J$-band, we have adopted the magnitude limit $J$ = 17.0 mag. Its real position in the C-C diagram follows the arrow. The presence of a cluster, nebulosity in both images, one YSO and several CTTS indicate this is a region of evolutionary $stage$ $B$.
The agreement between the kinematic distance (main sequence line) and our observed data is not obvious in this case, since the $N_{LyC}$ and the kinematic distance means that there is only a single late O-type star. This is inconsistent with the C-M diagram that show three O-type candidates (\#9, \#10 and \#14). But if we consider that objects \#9 and \#10 do not belong to this region, the kinematic distance may be correct.
\subsection{G320.3-0.2}
The Galactic GH\,{\sc{ii}} region G320.3-0.2 is located at R.A.: 15h09m59s and Dec.: -58d17m26s (J2000), and no stellar cluster is evident. There is no strong nebular emission in the $JHK_{s}$ image. Also, we do not see a well-defined cluster. However, in the {\it Spitzer} image ({\it Spitzer} program ID: 190, Fig.~\ref{fig:G320-2-color}, right side), we find nebulosity mainly at 8.0 $\micron$ (red). \citet{Conti04} derived a $N(LyC)$ of $1.29$ $\times$ $10^{50}$ photons per second using a distance of $12.6$ kpc \citep{Russeil03}.
Looking at the diagrams (C-C and C-M, Fig.~\ref{fig:G320-2-CMD}), objects numbers \#4 and \#6 seem to be on the expected main sequence location, but they don't seem to be O-type stars (C-C diagram), and are affected only by the interstellar reddening. However, both objects are saturated in our images, so we have used 2MASS photometry for them. These stars are close to the nebulosity, as seen in the {\it Spitzer} image (Fig.~\ref{fig:G320-2-color}, right side). Object \#13 presents a high reddening and is bright in the $K_{s}$-band, but looking at the C-C diagram (Fig.~\ref{fig:G320-2-CMD}, left side) this object does not show color excess. Actually, these objects (\#4, \#6, \#13, and also objects \#55 and \#142) may be in the foreground, projected in the direction of the nebulosity. Object \#90 seems to be associated with this region due the shell-like structure in the {\it Spitzer} image. Objects \#90 and \#203 are in the CTTS region and have aproximately the same $H - K_{s}$ color as object \#13.
The assignment of the evolutionary stage of this region is not easy, since we don't see a cluster, there is little nebulosity in both images and there aren't YSOs. However, the nebular emission, seen in the {\it Spitzer} image, may indicate an incipient cluster in the center of the field. We suggest this region is in a $stage$ $A-B$. The absence of an obvious cluster, together with nebular emission ({\it Spitzer} image), indicates this region may be at a larger distance, as predicted by kinematic results and the brightest objects are in the foreground.
\subsection{G322.2+0.6 (RCW92)}
G322.2+0.6 (RCW92) is located at R.A.: 15h18m39.1s and Dec.: -56d38m49s (J2000), and a few stars associated with an embedded stellar cluster were detected. \citet{Russeil03} derived a distance of 4.0 kpc and using this distance we obtained a Lyman continuum flux of $N(LyC)$ = $3.31$ $\times$ $10^{49}$ $s^{-1}$. In the near infrared color image (Fig.~\ref{fig:G322-color}), we see a cluster with embedded stars. And in the {\it Spitzer} image ({\it Spitzer} program ID: 146) the nebulosity dominates all the field, and shows that the cluster of stars seems to be in a cavity, or that a bubble of gas and dust is surrounding the cluster of stars.
We see in the $JHK_{s}$ image that the majority of the objects in this small field of view are that in the small cluster of embedded stars. Due to this strong nebulosity, outside this central cluster the stars are 'white' foreground or 'red' background objects. Objects \#1 and \#4, with $H - K_{s}$ $\approx$ 0.75 mag, seem to be associated with this region, due the near infrared color image and their location on the diagrams. Objects numbers \#2, \#3 and \#8, with $H - K_{s}$ $\approx$ 0.5 mag, are probably foreground stars projected onto this obscured region. Objects \#6 and \#9 present excess in the $K_{s}$-band and are in the CTTS region. Object \#7, also has a $K_{s}$-band excess, but more accentuated; it seems to be an YSO.
The cluster, the presence of CTTS and a massive YSO with the strong nebulosity in the {\it Spitzer} image indicate this is a region in the evolutionary $stage$ $A-B$.
\subsection{G327.3-0.5 (RCW97)}
G327.3-0.5 (RCW97) is located at R.A.: 15h53m02s and Dec.: -54d35m16s (J2000), and no stellar cluster was detected. This region does not seem to be very evolved, since we can not see an obvious cluster of stars. In the $JHK_{s}$ color image (Fig.~\ref{fig:RCW97-color}), we can see nebular emission and some foreground stars. However, this region is likely to be more complex than the near infrared data suggest. It could be a cluster with a very dark lane running through the middle, or two related ones. Indeed, the {\it Spitzer} image ({\it Spitzer} program ID: 191) shows a bubble of gas and dust to the NE and another one smaller near the center of the image, and a third at the position of object \#5. This may indicate the action of massive stars (or a cluster of massive stars) at different positions.
The kinematic distance is 3.0 kpc \citep{Russeil03}. \citet{Conti04} derived its Lyman continuum flux, $N(LyC)$ = $1.38$ $\times$ $10^{50}$ photons per second (a GH\,{\sc{ii}} region).
Looking at the diagrams (C-C and C-M, Fig.~\ref{fig:RCW97-CMD}), objects \#2 and \#5, with $H - K_{s}$ $\approx$ 0.8 mag, are on the expected main sequence location for O-type stars. Objects \#13 and \#36, with $H - K_{s}$ $<$ 0.5 mag, are bluer than objects \#2 and \#5, suggesting that these objects are in the foreground. Objects \#2 and \#5 are close to the nebulosity, and in C-C diagram they do not show excess in $K_{s}$-band. Object \#2 is near the M-type reddening line, while \#5 and \#19 are near the O-type line. These facts indicate \#5 and \#19 may be the ionizing sources of the H\,{\sc{ii}} region.
Object \#23 may be a background object, while objects \#16 and \#87 (not detected in $J$-band) present high infrared excess emission and are YSOs, though \#87 is not very bright in the $K_{s}$-band. The adopted $J$-band magnitude for a $90\%$ detectability is $J$ = 16.0 mag.
The nebulosity in the near infrared and in the {\it Spitzer} images, the absence of a cluster, some CTTS and a few YSOs indicate this is a region in the evolutionary $stage$ $A$. In this region, the adopted kinematic distance and our observed data seems to agree.
\subsection{G331.5-0.1}
The Galactic GH\,{\sc{ii}} region G331.5-0.1 is located at R.A.: 16h12m07s and Dec.: -51d27m03s (J2000), and a few stars possibly associated with a stellar cluster were detected. \citet{Russeil03} derived a distance of 10.8 kpc. At that distance, this region has a Lyman continuum luminosity of ($NLyC$) $1.45$ $\times$ $10^{51}$ photons $s^{-1}$ \citep{Conti04}.
Objects \#1, \#2 and \#3 are saturated in our data. Using 2MASS photometry we get: \#1: $J$ = 9.14; $H$ = 6.94 and $K_{s}$ = 5.88 mag. \#2: $J$ = 13.10; $H$ = 9.66 and $K_{s}$ = 7.84 mag. \#3: $J$ = 10.04; $H$ = 8.50 and $K_{s}$ = 7.96 mag. None of them has color excess. On the other hand, object number \#100 has a large excess emission in $K_{s}$-band (C-C diagram, Fig.~\ref{fig:G331-5-CMD}, left side), and it is very bright in the {\it Spitzer} image ({\it Spitzer} program ID: 191), suggesting that it is an YSO.
In the near infrared color image, we can see two regions of embedded stars. We can see a small cluster dominated by stars \#3, \#19, \#57, \#52, \#61, \#68, \#69 and \#71. In the bottom region we have objects \#1, \#2, \#99 and \#100. Most of the objects, in both regions, are foreground objects. We can see nebulosity in the near infrared and {\it Spitzer} images, but the presence of a cluster of stars is not so obvious. In the {\it Spitzer} image we can see some cavities, which may indicate the influence of massive stars over the nebular material.
This region has many CTTS, however, objects \#52 and \#100 are MYSOs. These characteristics indicate this is a region in the evolutionary stage $A$. The tip of the main sequence is fainter than the brightest suspected cluster members (e.g. \#57, \#58, and \#61), which indicates that the cluster maybe closer than the adopted kinematic distance.
\subsection{G333.1-0.4}
A stellar cluster was detected at R.A.: 16h21m03s and Dec.: -50d36m19s (J2000) and is a GH\,{\sc{ii}} region. \citet{Russeil03} derived a distance of 3.5 kpc. At that distance, this region has a Lyman continuum luminosity of $1.20$ $\times$ $10^{50}$ photons $s^{-1}$ \citep{Conti04}.
\citet{Figueredo05} have made a detailed study of the stellar content of this region. Object numbers \#1 and \#2, with $H - K_{s}$ $\approx$ 0.5 mag, were identified as O-type stars. Their $K_{s}$-band spectra were used to derive the spectroscopic parallax of this region. \citet{Figueredo05} derived a distance of $2.6$ $\pm$ $0.2$ kpc, which is smaller than that derived by the kinematic techniques.
Fig.~\ref{fig:G333-1-color} \citep[reproduced from][]{Figueredo05} shows a cluster of stars near the bottom of the image, with some objects still very embedded toward the top of the image. The {\it Spitzer} image ({\it Spitzer} program ID: 191) shows nebular emission and some bright objects (YSOs candidates).
Fig.~\ref{fig:G333-1-CMD} shows the photometric results as C-C and C-M diagrams. Objects \#10 and \#11 are in the CTTS region. Also, this region has several YSOs, for example, objects \#4, \#6, \#9, \#13, \#14, \#18, \#416, \#472, \#488 and \#598. Object \#18 (also an YSO) has a large infrared excess emission, it is very bright in the {\it Spitzer} image, and was not detected in $J$-band. Its adopted $J$-band magnitude is $J$ = 18.0 mag.
The presence of nebulosity in the near infrared and {\it Spitzer} images, the well-defined cluster, some CTTS and a large percentage of YSOs indicate this region is at evolutionary $stage$ $B$. The main sequence line seems to fit our data, but the smaller distance derived from spectrophotometric results is more reliable \citep{Figueredo05}.
\subsection{G333.3-0.4}
G333.3-0.4 is located at R.A.: 16h21m31.7s and Dec.: -50d26m23s (J2000). In this GH\,{\sc{ii}} region we can see two regions of nebular emission. A small cluster is located at the position of the upper nebulosity. The identification of individual objects is very hard due to extinction, nebuar emission, and source crowding. In the {\it Spitzer} image ({\it Spitzer} program ID: 191, Fig.~\ref{fig:G333-color}, right side), we see the nebulosity and almost no stars. The adopted distance to this region is 3.5 kpc \citep{Russeil03}, with a Lyman continuum luminosity of $1.10$ $\times$ $10^{50}$ photons $s^{-1}$ \citep{Conti04}.
Looking at the C-M diagram (Fig.~\ref{fig:G333-CMD}b), object numbers \#1, \#2 and \#4, with $H - K_{s}$ $\approx$ 0.7 mag, are on the expected main sequence location for O-type stars. These stars are close to the nebulosity, as seen also in the {\it Spitzer} image (Fig.~\ref{fig:G333-color}, right side). These facts together, indicate they may be ionizing sources of the H\,{\sc{ii}} region. Also, object \#36 presents a large reddening, with a $H - K_{s}$ $\approx$ 2.2 mag, and is bright in the $K_{s}$-band, but looking at the C-C diagram (Fig.~\ref{fig:G333-CMD}, left side) we see this object does not look like an YSO.
In this region, we do not see a rich star cluster, and in the diagrams we see some CTTS and YSOs. The region is best described as evolutionary $stage$ $A$. The projected size of the region may suggest it is at a large distance, but this analyses is inconclusive.
\subsection{G333.6-0.2}
G333.6-0.2 is located at R.A.: 16h22m11.9s and Dec.: -50d05m56s (J2000), and no stellar cluster was detected. The distance to this GH\,{\sc{ii}} region is $3.1$ kpc \citep{Russeil03} and using this distance \citet{Conti04} derived a Lyman continuum luminosity of $2.69$ $\times$ $10^{50}$ $s^{-1}$. \citet{Becklin73} noted this is the most luminous H\,{\sc{ii}} region in the wavelenght interval between 1 - 25 $\mu$m (radiating 5 $\times$ $10^{5}$ $L_{\odot}$ in this wavelength range). This region presents a very high obscuration \citep*[$A_{V}$ $\sim$ 21,][]{Rubin83} and it is difficult to associate it with a star cluster. \citet{Hyland80} have suggested that this region is, actually, a `blister' source, since it has a large intrinsic luminosity ($L \sim 3 \times 10^{6} L_{\odot})$ and presents a low degree of ionization.
The $JHK_{s}$ image exhibits many field stars and bright nebulosity in the central region with some very embedded objects. In the {\it Spitzer} image ({\it Spitzer} program ID: 191), we see mainly the nebular material with some foreground objects (Fig.~\ref{fig:G333-2-color}, right side).
Looking at the images and diagrams (C-C and C-M, Fig.~\ref{fig:G333-2-CMD}), we identify two groups of objects. The first group of stars with $H - K_{s}$ = 0.5 mag are likely members of the cluster, while the second group with $H - K_{s}$ = 2.0 mag (\#2, \#12 and \#25) seems to be composed by background objects. Also, we note that object \#4 has excess emission and together with objects \#10 and \#29 is well inside the bright region. As we can see in the near infrared image, the nebular emission is very strong in this region. The objects \#4 and \#10 seem to be YSOs. It is difficult to identify other objects in this region, due to image crowding and intense nebular emission. Objects \#7, \#16 and \#24 are near the reddening line of M-type stars, and objects \#5, \#6, \#9, \#15, \#19, and \#55 are near the reddening line of O-type stars.
The strong nebulosity in the near infrared and {\it Spitzer} images, the absence of a cluster, the presence of some CTTS and YSOs indicate this is a region at evolutionary $stage$ $A$. Also, due to the absence of a cluster, it is difficult to analyze the kinematic distance.
\subsection{G336.5-1.5 (RCW108)}
G336.5-1.5 (RCW108) is located at R.A.: 16h39m58.3s and Dec.: -48d52m38s (J2000), where a small stellar cluster was detected. We have adopted a distance of 1.5 kpc \citep{Russeil03} and the Lyman continuum flux for this distance is $N(LyC)$ = $1.95$ $\times$ $10^{48}$ photons per second, the least luminous source in our sample. This is an extremely obscured region, that shows strong nebular emission. Due to this strong nebulosity, it is difficult to identify its stellar content.
Nevertheless, a careful examination of the images and diagrams (C-C and C-M diagrams, Fig.~\ref{fig:G336-2-CMD}) suggests that objects \#1 and \#18 are near reddening line for O-type stars. Object \#8 is near the M-type reddening line, and objects \#40 and \#71 seem to be background stars. Object \#27 is a foreground star. Objects \#3 and \#4 are close to the bright central region, as seen in the images (Fig.~\ref{fig:G336-2-color}). However, \#3 presents a high excess emission, is bright in the $K_{s}$-band and is in the YSOs region of the C-C diagram (Fig.~\ref{fig:G336-2-CMD}, left side). Objects \#4 and \#15 lie in the CTTS $region$.
The presence of strong nebulosity in the near infrared and {\it Spitzer} images ({\it Spitzer} program ID: 112), the small cluster in the central area, some YSOs and several CTTS indicate this is a region at $stage$ $A-B$. If objects like source \#1 are cluster members, then the main sequence maybe in the correct position for the kinematic distance.
\subsection{G336.8-0.0}
The Galactic GH\,{\sc{ii}} region G336.8-0.0 is located at: 16h34m37s -47d36m47.8s (J2000) and is at a distance of 10.9 kpc \citep{Russeil03}. No stellar cluster was detected at these coordinates. Following the work of \citet{Conti04}, we derived a $N(LyC)$ = $3.02$ $\times$ $10^{50}$ photons per second. In the near infrared color image the nebulosity is not so obvious, while in the {\it Spitzer} color image ({\it Spitzer} program ID: 191), the nebulosity is stronger including a bright compact source which dominates the field. In neither image do we see a well defined cluster of stars (Fig.~\ref{fig:G336-color}).
Looking at the diagrams (C-C and C-M diagrams, Fig.~\ref{fig:G336-CMD}), we find that objects \#4, \#5, \#6, \#10 and \#22, with $H - K_{s}$ $\approx$ 1.0 mag, appear to be on the expected main sequence location for O-type stars and are affected only by interstellar reddening. Also, objects \#8, \#23 and \#29 are in the line of sight of the small cluster of embedded stars, and they are on the CTTS $region$.
Object number \#7, with $H - K_{s}$ $\approx$ 0.1 mag, is likely a foreground object. Objects \#55 and \#68 are to the right of the CTTS line of reddening, indicating their $K_{s}$-band excess, but they don't appear in the {\it Spitzer} image.
All the brightest objects are foreground candidates. There is not a well-defined cluster, there is some nebulosity in the images and the presence of some CTTS and two MYSO candidates indicate this is a region in an evolutionary $stage$ $A$. Due to the absence of a cluster, and a small nebulosity, it seems this region is very far away, and the adopted kinematic distance may be correct.
\subsection{G348.7-1.0 (RCW122)}
G348.7-1.0 (RCW122) is located at: 17h20m05.8s -38d57m37s (J2000), where a few stars possibly associated with an embedded stellar cluster were detected. The distance of this region is 2.7 kpc \citep{Russeil03}. Following the work of \citet{Conti04}, we derived a $N(LyC)$ = $2.57$ $\times$ $10^{48}$ photons per second.
In this region, the presence of a stellar cluster is not so obvious. The nebular emission is very strong and the ionizing sources seem to be behind the nebulosity. Most of the stars present in the $JHK_{s}$ color image are foreground as can be seen in the C-C diagram (Fig.~\ref{fig:RCW122-CMD}, left side). The background component is very difficult to see here due to the strong obscuration. In the {\it Spitzer} image ({\it Spitzer} program ID: 192, Fig.~\ref{fig:RCW122-color}, right side), we see strong nebulosity associated with the region and the sources with infrared excess. In the dark region of the near infrared color image, we see point sources in the {\it Spitzer} image suggesting the cluster is hidden by a dark cloud along our line of site.
There is a group of stars with $H - K_{s}$ $\approx$ 0.3 mag (\#1, \#4, \#6, \#9, \#10, \#16 and \#22) which are likely in the foreground. Objects \#2, \#7, \#18 and \#21, with $H - K_{s}$ $\approx$ 0.7 mag, are close to the reddening line of O-type stars. Also, objects \#3, \#8, \#11 and \#20 are in the CTTS region. Object \#5 is an YSO. Object \#1 seems to be foreground and is not in the line of sight of the nebulosity.
The diagrams show this region has several CTTS and YSOs, together with the presence of nebulosity and is lacking a well defined cluster of stars. We thus place this region in an evolutionary $stage$ $A-B$. The main sequence line seems to do a good fit in our photometric data, which may indicate the kinematic distance is correct.
\subsection{G351.2+0.7}
G351.2+0.7 is located at R.A.: 17h20m04.1s and Dec.: -35d56m10s (J2000), and a cluster of stars was detected. This is a very obscured H\,{\sc{ii}} region and is part of the large complex NGC6334 \citep[first identified by][]{Moran90}. This region is at a distance of 1.2 kpc; using this distance we have derived its Lyman continuum flux following \citet{Conti04}, $N(LyC)$ = $4.68$ $\times$ $10^{49}$ photons per second.
G351.2+0.7 is described as a ring/shell of radio emission \citep{Jackson-Kraemer99} with a ring radius of about 1 arcmin (0.5 pc). No source was detected at the ring's center or as an ionizing source of the ring \citep{Jackson-Kraemer99}. It seems to be very young, and its members are still embedded in their natal clouds. A dark molecular cloud can be easily seen at the SW. This cluster is sweeping the material away and is eroding the surrounding material (Fig.~\ref{fig:G351-color}, lower right). The {\it Spitzer} image ({\it Spitzer} program ID: 20201, Fig.~\ref{fig:G351-color}, right side) shows nebulosity and several embedded sources.
Looking at the diagrams (C-C and C-M diagrams, Fig.~\ref{fig:G351-CMD}), we identify a group of stars with $H$ - $K_{s}$ around 0.30 mag: \#16, \#18, \#20, \#23, \#24, \#27, \#28, \#32 and \#59. However, these objects are sparsely distributed in the image which may indicate these stars are in the foreground. Objects \#18, \#24, \#23 and \#27 are very close to the reddening line for O-type stars. Still close to the O-type reddening line, but with a larger infrared color, we note objects: \#7, \#12 and \#213. Some combination of these could be the ionizing sources of the H\,{\sc{ii}} region. Object \#7 is saturated in our image, so we have used 2MASS photometry. Objects \#54, \#75 and \#82 seem to be background stars. Objects \#11, \#21, \#29, \#40, \#82 and \#83 are more close to the reddening line of M-type stars. Object \#111 presents a high $K_{s}$-band excess (C-C diagram, Fig.~\ref{fig:G351-CMD}, left side) and is likely a YSO with $H - K_{s}$ $\approx$ 1.9 mag. Objects \#17, \#42, \#58 and \#78 are in the CTTS $region$.
The presence of clusters of stars, nebulosity in both images, some YSOs and several CTTS indicate this is a region in $stage$ $B$. The tip of the main sequence line is brighter than our brightest objects. This suggests that the adopted kinematic distance may be correct, or the region is a little further away than the kinematic distance.
\subsection{G353.2+0.6 (RCW131)}
G353.2+0.6 (RCW131) is located at R.A.: 17h25m37s and Dec.: -34d21m26s (J2000), where a cluster of stars was detected. This region is part of the NGC6357 star forming complex \citep*{Massi97} and its distance is 1.0 kpc \citep{Russeil03}. We have derived the Lyman continuum flux at this distance, following \citet{Conti04} we obtained $N(LyC)$ = $2.09$ $\times$ $10^{49}$ photons per second.
In the $JHK_{s}$ color image (Fig.~\ref{fig:G353-color}, left side), we see a large number of stars. Many of them are foreground objects. In the {\it Spitzer} image ({\it Spitzer} program ID: 20201, Fig.~\ref{fig:G353-color}, right side), we see, more easily, the nebulosity and some structures like pilars and filaments, a result of the action of stellar winds from massive stars.
Objects \#01, \#02, \#4 and \#7 are saturated in our images, so we have used 2MASS photometry. Looking at the diagrams (C-C and C-M diagrams, Fig.~\ref{fig:G353-CMD}), we identify two distinct group of objects. The first group of stars with $H - K_{s}$ = 0.3 mag are likely members of the cluster (\#01, \#02, \#4, \#7, \#37, \#41, \#44, \#54, \#60, \#65, \#66, \#76 and \#157) and are close to the O-type reddening line, except objects \#02 and \#54. These two objects are spatially (Fig.~\ref{fig:G353-color}) very close to each other, and their magnitudes could be affected by this proximity. The second group of stars with $H - K_{s}$ $\approx$ 1.5 mag are likely background objects. Objects \#68, \#74 and \#266 have larger infrared colors, but are close to the reddening lines of O-type stars. These stars are close to the nebulosity (Brackett gamma emission), this indicates they may be the ionizing sources of the H\,{\sc{ii}} region. Objects \#19 and \#33 are closer to the reddening line of M-type stars. Objects \#135 and \#185, among others, show reddened colors and are bright in the $K_{s}$-band, but their positions in the C-C diagram indicate they are in the CTTS region.
The presence of nebulosity in the NIR and MIR, several possibly `naked' O-stars, surrounding nebulosity, CTTS and YSOs indicate this is a region in evolutionary $stage$ $B$. The C-M diagram shows us the tip of the main sequence is brighter than the brightest stars. This may indicate that the kinematic distance may be correct, or a the region is a little further away than the kinematic distance.
\section{MYSOs in H\,{\sc{ii}} regions}
The objects selected as MYSOs candidates throughout this work are shown in Table 2. In some cases the IRAC images are crowded and do not have enough spatial resolution to resolve the cluster members. In other situations, the objects are so bright at longer wavelenghts that they become saturated, which might indicate they are MYSOs. Also, there are situations in which the nebular emission is so intense that it is not possible to determine the magnitude of the object.
In Figure 2, an IRAC-{\it Spitzer} color-color diagram is shown for the objects with measured magnitudes in the four IRAC channels (11 objects). Using the critera of \citet{Allen04} for IRAC color-color diagrams, it was possible to indicate five Class 0/I objects (green closed circles), two Class II objects (blue closed squares), one Class III object (red closed triangle) and two `naked photosphere' objects (black crosses).
Only 11 objects (from a total of 65) have the 4 IRAC-channel magnitudes determined. For the remaining objects, that present at least the [3.6] and [4.5] $\mu$m measured magnitudes, it is possible to indicate if these objects are YSOs by using the identified YSOs from the Figure 2 in a $K_{s}$-[3.6] X [3.6]-[4.5] $\mu$m diagram (Figure 3). The identified YSOs from Figure 2 are represented by green closed circles and the YSO candidates (in the vicinity of the identified ones) are represented by black open circles. The `naked photosphere' objects are around [3.6]-[4.5] = 0.5 $\mu$m (Figure 3), together with the `naked photosphere' objects identified in Figure 2. In all, we identified 14 YSO candidates, and 21 `naked photosphere' candidates.
Table 2 also shows cases in which the objects were saturated in the IRAC-{\it Spitzer} images. In the Table 2, the `naked photosphere' objects are indicated by {\it NP}.
\begin{footnotesize}
\begin{table*}
\begin{center}
{\parbox{06.7in}{ \textbf{Table 2.} Massive YSOs identified using the color-color and color-magnitude diagrams (C-C and C-M, respectively) and with a counterpart in their respective {\it Spitzer} images. Columns 1 and 2 are the identifications of each region studied in this work. Column 3 gives the candidate number. Columns 4 and 5 are the coordinates (2000). Column 6 gives the MYSO $K_{s}$-band magnitude. Columns 7, 8, 9 and 10 give the IRAC magnitudes. Column 11 gives, when it is possible, the classification of the MYSO. Column 7--10 identification for the non detected objects: (1) Undetected object; (2) Saturated object; (3) Strong nebulosity obscurating the object and (4) Crowded cluster.}}
\begin{tabular}{ccccccccccc}
\hline
region & Name & Obj.& $R.A. (J2000)$ & $Dec. (J2000)$ & $K_s (mag)$&3.6$\mu$m&4.5$\mu$m&5.8$\mu$m&8.0$\mu$m&Class. \\ \hline
G5.97-1.18 &M8 &\#01 &18:03:40.32&-24:22:42.70& $6.91$& 6.90 & 5.85 & 4.10 & (2) & YSO? \\
G5.97-1.18 &M8 &\#41 &18:03:40.37&-24:22:39.42& $9.18$& 7.10 & 6.14 & 4.66 & (2) & YSO? \\
G5.97-1.18 &M8 &\#432 &18:03:38.63&-24:22:24.20&$11.52$& 9.74 & 8.51 & 7.96 & 4.74 & YSO \\
G10.2-0.3$^a$ &W31-South&\#01 &18:09:27.64&-20:19:13.02& $9.45$& 7.42 & 6.17 & (1) & (1) & YSO? \\
G10.2-0.3$^a$ &W31-South&\#09 &18:09:26.98&-20:19:08.53&$10.67$& 7.97 & 6.65 & (1) & (1) & YSO? \\
G10.2-0.3$^a$ &W31-South&\#15 &18:09:27.28&-20:19:35.74&$11.02$& 9.70 & (1) & (1) & (1) & ? \\
G10.2-0.3$^a$ &W31-South&\#26 &18:09:26.28&-20:19:23.40&$11.49$& 8.10 & 6.70 & (1) & (1) & YSO? \\
G10.2-0.3$^a$ &W31-South&\#30 &18:09:25.80&-20:19:17.79&$11.83$& 8.83 & 7.50 & (1) & (1) & YSO? \\
G10.3-0.1 &W31-North&\#96 &18:08:58.20&-20:05:14.00&$11.49$& 9.85 & 9.74 & 9.32 & (1) & {\it NP}?\\
G12.8-0.2 &W33 &\#01 &18:14:13.46&-17:55:38.95&$12.50$& 8.23 & 6.10 & 3.99 & (1) & YSO? \\
G12.8-0.2 &W33 &\#02 &18:14:12.53&-17:55:43.09&$12.63$& (1) & (1) & (1) & (1) & ? \\
G12.8-0.2 &W33 &\#07 &18:14:13.01&-17:55:27.94&$13.74$& (1) & (1) & (1) & (1) & ? \\
G12.8-0.2 &W33 &\#08 &18:14:14.58&-17:55:50.79&$13.95$& (1) & (1) & (1) & (1) & ? \\
G12.8-0.2 &W33 &\#10 &18:14:14.32&-17:55:56.71&$14.02$& (1) & (1) & (1) & (1) & ? \\
G15.0-0.7 &M17 &\#10 &18:20:30.55&-16:11:04.71&$10.08$& 8.05 & 6.88 & 6.28 & 5.81 & YSO \\
G15.0-0.7 &M17 &\#24 &18:20:30.73&-16:10:53.42&$10.97$& (1) & (1) & (1) & (1) & ? \\
G25.4-0.2$^b$ &W42 &\#03 &18:38:15.30&-06:47:51.88&$10.42$& 8.72 & 7.81 & (2) & (3) & YSO? \\
G25.4-0.2$^b$ &W42 &\#57 &18:38:14.57&-06:48:02.34&$12.96$& 7.60 & 6.78 & (2) & (3) & YSO? \\
G30.8-0.2$^c$ &W43 &\#09 &18:47:37.12&-01:56:42.54&$11.63$& 8.83 & 6.34 & 4.57 & 4.03 & YSO \\
G30.8-0.2$^c$ &W43 &\#10 &18:47:38.51&-01:56:43.17&$11.68$& 8.02 & 6.57 & 5.21 & (1) & YSO? \\
G45.5+0.1 &K47 &\#20 &19:14:22.09&+11:08:24.35&$13.28$& 10.12 & 9.14 & (3) & (3) & YSO? \\
G48.9-0.3 &W51 &\#203 &19:22:15.26&+14:04:27.88&$11.43$& 11.64 & 11.26 & 11.36 & (1) & {\it NP}?\\
G48.9-0.3 &W51 &\#238 &19:22:11.64&+14:02:16.63&$11.61$& 9.39 & 9.10 & 8.39 & (1) & YSO? \\
G48.9-0.3 &W51 &\#526 &19:22:07.82&+14:03:13.37&$12.66$& 8.63 & 7.19 & 6.05 & 4.86 & YSO \\
G48.9-0.3 &W51 &\#1063&19:22:19.02&+14:05:07.18&$13.59$& 9.58 & 8.26 & 7.14 & 6.26 & YSO \\
G49.5-0.4$^d$ &W51A &\#45 &19:23:42.67&+14:30:27.56&$12.46$& (3) & (3) & (3) & (3) & ? \\
G49.5-0.4$^d$ &W51A &\#61 &19:23:47.21&+14:29:43.69&$12.49$& (3) & (3) & (3) & (3) & ? \\
G49.5-0.4$^d$ &W51A &\#62 &19:23:40.42&+14:29:32.22&$12.26$& (3) & (3) & (3) & (3) & ? \\
G49.5-0.4$^d$ &W51A &\#73 &19:23:52.05&+14:28:50.30&$11.55$& (3) & (3) & (3) & (3) & ? \\
G49.5-0.4$^d$ &W51A &\#98 &19:23:42.80&+14:30:29.70&$12.98$& (3) & (3) & (3) & (3) & ? \\
G274.0-1.1 &RCW42 &\#21 &09:24:25.76&-51:59:25.08&$11.01$& (1) & (1) & (3) & (3) & ? \\
G274.0-1.1 &RCW42 &\#31 &09:24:25.97&-51:59:23.59&$11.60$& 7.79 & 7.91 & (3) & (3) & {\it NP}?\\
G274.0-1.1 &RCW42 &\#33 &09:24:26.39&-51:59:19.96&$11.78$& 8.71 & 8.66 & (3) & (3) & {\it NP}?\\
G282.0-1.2 &RCW46 &\#31 &10:06:38.99&-57:11:58.35&$12.01$& 8.21 & 8.26 & (1) & (3) & {\it NP}?\\
G282.0-1.2 &RCW46 &\#46 &10:06:36.89&-57:12:31.69&$12.56$& 9.55 & 8.88 & 8.60 & (3) & {\it NP}?\\
G284.3-0.3 &NGC3247 &\#54 &10:24:01.13&-57:45:35.46&$10.25$& 10.55 & 9.87 & 9.57 & (4) & {\it NP}?\\
G284.3-0.3 &NGC3247 &\#60 &10:23:55.74&-57:45:08.48&$10.39$& (1) & (1) & (1) & (1) & ? \\
G287.4-0.6 &NGC3372 &\#138 &10:43:35.12&-59:31:48.57&$11.17$& 9.28 & 8.76 & 8.52 & 8.26 & YSO \\
G291.6-0.5 &NGC3603 &\#19 &11:15:11.38&-61:16:44.91& $8.91$& 8.56 & 7.62 & 7.00 & 6.05 & YSO \\
G298.9-0.4 &-- &\#21 &12:15:20.01&-63:01:10.49&$10.70$& 6.85 & 6.24 & (2) & (2) & YSO?\\
G305.2+0.0 &-- &\#873 &13:11:16.22&-62:46:21.57&$13.01$& 7.17 & 6.28 & 5.55 & 4.36 & YSO \\
G305.2+0.2 &-- &\#134 &13:11:31.72&-62:32:30.14&$11.25$& 10.56 & 10.05 & (1) & (1) & {\it NP}?\\
G320.1+0.8 &RCW87 &\#15 &15:05:17.21&-57:30:02.31& $8.57$& 9.30 & 8.60 & 8.41 & (2) & {\it NP}?\\
G322.2+0.6 &RCW92 &\#07 &15:18:38.78&-56:38:49.70&$11.17$& (1) & (1) & (1) & (3) & ? \\
G327.3-0.5 &RCW97 &\#16 &15:53:09.68&-54:34:31.13& $9.78$& 8.64 & 8.57 & 6.60 & (1) & {\it NP}?\\
G327.3-0.5 &RCW97 &\#87 &15:53:03.15&-54:35:24.43&$11.82$& 7.20 & 6.71 & 6.19 & 6.46 & YSO \\
G331.5-0.1 &-- &\#100 &16:12:08.98&-51:28:02.93&$10.50$& 9.03 & 8.02 & 6.59 & 5.52 & YSO \\
G331.5-0.1 &-- &\#2758&16:12:10.01&-51:28:37.84&$14.52$& 6.79 & 5.85 & 7.41 & (2) & YSO? \\
G333.1-0.4$^e$&-- &\#04 &16:21:04.56&-50:35:42.00&$10.93$& 7.62 & 5.93 & 3.76 & (3) & YSO? \\
G333.1-0.4$^e$&-- &\#09 &16:21:02.47&-50:35:38.72&$12.12$& 9.15 & 8.68 & 8.05 & (3) & {\it NP}?\\
G333.1-0.4$^e$&-- &\#06 &16:21:00.43&-50:35:08.37&$11.20$& 7.61 & 7.14 & 5.56 & (3) & {\it NP}?\\
G333.1-0.4$^e$&-- &\#13 &16:20:59.70&-50:35:14.34&$12.28$& 10.50 & 10.03 & (1) & (3) & {\it NP}?\\
G333.1-0.4$^e$&-- &\#14 &16:21:00.50&-50:35:09.37&$12.19$& 9.36 & 8.89 & 6.95 & (3) & {\it NP}?\\
G333.1-0.4$^e$&-- &\#18 &16:21:02.62&-50:35:54.85&$12.46$& (2) & (2) & (2) & (3) & YSO? \\
G333.1-0.4$^e$&-- &\#416 &16:21:02.07&-50:35:16.02&$15.72$& 6.91 & 6.44 & (1) & (3) & {\it NP}?\\
G333.1-0.4$^e$&-- &\#472 &16:21:04.02&-50:35:07.41&$15.05$& (1) & (1) & (1) & (3) & ? \\
G333.1-0.4$^e$&-- &\#488 &16:21:00.71&-50:35:05.39&$14.86$& (1) & (1) & (1) & (3) & ? \\
G333.1-0.4$^e$&-- &\#598 &16:21:06.68&-50:35:41.39&$10.50$& (1) & (1) & (1) & (3) & ? \\ \hline
\end{tabular}
\label{table2}
\end{center}
\end{table*}
\end{footnotesize}
\begin{footnotesize}
\begin{table*}
\begin{center}
{\textbf{Table 2} - Continued}
\begin{tabular}{ccccccccccc}
\hline
region & Name & Obj.& $R.A. (J2000)$ & $Dec. (J2000)$ & $K_s (mag)$&3.6$\mu$m&4.5$\mu$m&5.8$\mu$m&8.0$\mu$m&Class. \\ \hline
G333.6-0.2 &-- &\#04 &16:22:09.60&-50:05:59.13& $7.92$& (2) & (2) & (2) & (2) & YSO? \\
G333.6-0.2 &-- &\#10 &16:22:09.37&-50:06:00.59& $8.46$& (2) & (2) & (2) & (2) & YSO? \\
G336.5-1.5 &RCW108 &\#03 &16:40:01.06&-48:51:51.83& $8.93$& 6.93 & 6.00 & 4.79 & 5.10 & YSO \\
G336.8-0.0 &-- &\#55 &16:34:51.22&-47:33:16.11&$13.65$& 11.06 & 10.45 & (1) & (1) & {\it NP}?\\
G336.8-0.0 &-- &\#68 &16:34:47.51&-47:32:10.16&$13.82$& 11.93 & 11.59 & (1) & (1) & {\it NP}?\\
G348.7-1.0 &RCW122 &\#05 &17:20:06.65&-38:57:30.38&$10.83$& 10.82 & 8.61 & (1) & (1) & YSO? \\
G351.2+0.7 &-- &\#111 &17:19:57.87&-35:57:50.84&$10.42$& 7.79 & 7.12 & 6.07 & (3) & YSO? \\ \hline
\end{tabular}
\label{table2b}
\end{center}
\begin{footnotesize}
{\parbox{06.7in}{References: (a) Blum et al. (2001); (b) Blum et al. (2000); (c) Blum et al. (1999); (d) Figuer\^edo et al. (2008); (e) Figuer\^edo et al. (2005).\\
Non detection objects: \\
(1) Undetected object; (2) Saturated object; (3) Strong nebulosity obscurating the object; (4) Crowded cluster.}}
\end{footnotesize}
\end{table*}
\end{footnotesize}
\begin{figure}
\includegraphics[height=08cm,width=08.5cm]{fig02.jpg}
\caption{{ Confirmed YSOs based on the four channel color-color diagram for the IRAC magnitudes according to \citet{Allen04} color classification.}}
\end{figure}
\begin{figure}
\includegraphics[height=08cm,width=08.5cm]{fig03.jpg}
\caption{{ Color-color plot combining $K_{s}$-band and 2 IRAC-{\it Spitzer} channels for objects without measured magnitude in the four IRAC-{\it Spitzer} channels, mainly at [8.0] $\mu$m. It is possible to identify `naked photosphere' stars (open black squares) and to suggest additional YSO candidates (open black circles).}}
\end{figure}
\section{Spectrophotometric and Trigonometric Parallaxes Distances}
In previous sections, we have used kinematic distances to the H\,{\sc{ii}} regions to check them with the photometry in C-M diagrams. Now, we compare these kinematic distances with that derived from trigonometric and from $K$-band spectrophotometric parallaxes. These non-kinematic results are useful to check the kinematic distances used to map the spiral pattern of the Milky Way. This sample of non-kinematic distances of H\,{\sc{ii}} regions encompass some of our objects, as well as that from other star-forming regions, and it is useful since it shows that there is an overall discrepancy between these two methodologies, kinematic and non-kinematic.
Using VLBI several authors \citep[see:][]{Reid09a} have derived trigonometric parallax distances to star forming regions. These results can be compared with the kinematic results in Table 3. In columns 1 and 2, we show the galactic coordinates. In column 3 we list the names of the regions. In column 4 the kinematic distances are given. In column 5 the published distances from trigonometric parallaxes are shown. In column 6, we give references.
\begin{footnotesize}
\begin{table*}
\begin{center}
{\parbox{06.7in}{\textbf{Table 3.} regions with distances derived from trigonometric parallax. In columns 1 and 2 are the galactic coordinates, in the column 3 names of the regions, and in column 4 the kinematic distances. In the column 5, we show the trigonometric parallax distances. References are given in column 6.}}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$l$ & $b$ & \it{region}&$d_{Kin.}^{15}$&$d_{\pi}$ &\it{Ref.} \\
& & & $kpc$ & $kpc$ & \\ \hline
0.67 & $-0.03$ & Sgr B2 & $8.50$ & $7.90^{+0.80}_{-0.70}$ & 14 \\
23.01 & $-0.41$ & G23.0-0.4 & $4.97$ & $4.59^{+0.38}_{-0.33}$ & 5 \\
23.44 & $-0.18$ & G23.4-0.2 & $5.60$ & $5.88^{+1.37}_{-0.93}$ & 5 \\
23.66 & $-0.13$ & G23.6-0.1 & $5.04$ & $3.19^{+0.46}_{-0.35}$ & 6 \\
35.20 & $-0.74$ & G35.2-0.7 & $1.98$ & $2.19^{+0.24}_{-0.20}$ & 4 \\
35.20 & $-1.74$ & G35.2-1.7 & $2.85$ & $3.27^{+0.56}_{-0.42}$ & 4 \\
49.49 & $-0.37$ & W51 (IRS2) & $5.52$ & $5.13^{+2.90}_{-1.40}$ & 3 \\
59.78 & $+0.06$ & G59.7+0.1 & $3.07$ & $2.16^{+0.10}_{-0.09}$ & 3 \\
109.87 & $+2.11$ & Cep A & $1.09$ & $0.70^{+0.04}_{-0.04}$ & 2 \\
111.54 & $+0.78$ & NGC7538 & $5.61$ & $2.65^{+0.12}_{-0.11}$ & 2 \\
122.02 & $-7.07$ & IRAS00420 & $3.97$ & $2.17^{+0.05}_{-0.05}$ & 7 \\
123.07 & $-6.31$ & NGC281 & $2.69$ & $2.82^{+0.24}_{-0.24}$ & 8 \\
133.95 & $+1.06$ & W3(OH) & $4.28$ & $1.95^{+0.04}_{-0.04}$ & 9 \\
135.28 & $+2.80$ & WB 89-437 & $8.68$ & $6.00^{+0.02}_{-0.02}$ & 10 \\
188.95 & $+0.89$ & S252 & $4.06$ & $2.10^{+0.03}_{-0.03}$ & 1 \\
196.45 & $-1.68$ & S269 & $3.98$ & $5.28^{+0.24}_{-0.22}$ & 11 \\
209.01 & $-19.38$ & Orion & $0.99$ & $0.44^{+0.02}_{-0.02}$ & 12 \\
232.62 & $+1.00$ & G232.6+1.0 & $1.92$ & $1.68^{+0.11}_{-0.09}$ & 1 \\
239.35 & $-5.06$ & VY CMa & $1.56$ & $1.14^{+0.11}_{-0.09}$ & 13 \\ \hline
\end{tabular}
\label{table3}
\end{center}
\begin{footnotesize}
{\parbox{06.7in}{References: (1) - \citet[][a]{Reid09a}; (2) - \citet{Moscadelli09}; (3) - \citet{Xu09}; (4) - \citet{Zhang09}; (5) - \citet{Brunthaler09}; (6) - \citet{Bartkiewicz08}; (7) - \citet{Moellenbrock09}; (8) - \citet{Sato08}; (9) - \citet{Xu06,Hachisuka06}; (10) - \citet{Hachisuka09}; (11) - \citet{Honma07}; (12) - \citet{Hirota07,Menten07}; (13) - \citet{Choi08}; (14) - \citet[][c]{Reid09c}; (15) - \citet[][b]{Reid09b}.}}
\end{footnotesize}
\end{table*}
\end{footnotesize}
$K$-band spectrophotometric distances are derived from the distance modulus ({\it i.e.} $m_{K} - M_{K} = 5 \times log(d) - 5 + A_{K}$) and the adoption of an interstellar extinction law. In this section, we have used H\,{\sc{ii}} regions with identifyied ionizing O-type stars found in the literature. Most of these regions are also in our sample. For each ionizing O-type star (for the spectral types of the ionizing sources, see the references that are listed in the notes of Table 4), we have used $M_{K}$ from \citet{Hanson96}, $M_{V}$ from \citet{Vacca96} and $V - K$ \citet{Koornneef83}. Also, two extreme interstellar extinction laws were used to analyse the effect of this parameter on the result. The two laws used are from \citet{Mathis90} and \citet{Stead09}. The law from \citet{Mathis90} gives an exponent of $\alpha$ = 1.70, while the law of \citet{Stead09} has an exponent of $\alpha$ = 2.14, which represent extreme situations of high and low interstellar extinctions, respectively. Indeed, since $A_{K} \propto \lambda_{K}^{-\alpha}$, \citet{Mathis90} give the largest values for $A_{K}$ compared to the values derived using \citet{Stead09}. This implies that the spectrophotometric distances derived using \citet{Mathis90} law, column 6 of Table 4, are smaller than those derived using \citet{Stead09}, column 7 of Table 4. When a H\,{\sc{ii}} region has more than one identified ionizing sources (column 4 in Table 4 shows the number of identified ionizing sources for each region), we consider the median of the spectrophotometric distances, for each interstellar extinction law. The median distance of both interstellar extinction laws are shown in column 8. These two extreme situations were used for 26 H\,{\sc{ii}} regions we have found in the literature and the results still show discrepancies with that from kinematic techniques. As can be seen in the Table 4, most of the regions have smaller distances than the kinematic results. The kinematic distances shown in the column 5 of Table 4 are, as throughout this work, from \citet{Russeil03}. Note that our spectrophotometric distances (columns 6, 7 and their median in column 8) are not the published values. Here we used only the published spectral types of the ionizing sources and computed distances based on adopting the two extreme interstellar laws.
\begin{footnotesize}
\begin{table*}
\begin{center}
{\parbox{06.7in}{\textbf{Table 4.} Spectrophotometric distances. Columns 1 and 2 give the galactic coordinates. Column 3 gives the names of the H\,{\sc{ii}} regions. In the column 4, we list the number of stars used in each region. In the column, the kinematic distances are given. Column 6 shows the spectrophotometric distance using the \citet{Mathis90} interstellar extinction law and column 7 the distance using the \citet{Stead09} interstellar extinction law. Column 8 is the average between both spectrophotometric distances.}}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|}
\hline
$l$ &$b$ &\it{region} &$N^{\circ}$&$d_{kin}^{1}$&$d_{Mathis}$&$d_{Stead}$ & $\langle d_{spec} \rangle$\\
& & & & kpc & kpc & kpc \\ \hline
$6.0$ & $-1.2$ & M8$^{2}$ & 1 & $2.80\pm1.0$& $0.90\pm0.35$& $0.99\pm0.38$ & $0.95$\\
$10.1$ & $-0.3$ & W31-South$^{3}$ & 4 & $4.50\pm0.6$& $3.01\pm1.12$& $4.10\pm1.52$ & $3.55$\\
$10.3$ & $-0.1$ & W31-Norte$^{2}$ & 2 &$15.10\pm1.3$& $2.02\pm0.77$& $2.76\pm1.08$ & $2.39$\\
$15.0$ & $-0.7$ & M17$^{4}$ & 3 & $2.40\pm0.5$& $2.01\pm0.75$& $2.19\pm0.82$ & $2.10$\\
$25.4$ & $-0.2$ & W42$^{5}$ & 1 &$11.50\pm0.3$& $2.46\pm0.90$& $2.89\pm1.07$ & $2.67$\\
$30.8$ & $-0.0$ & W43$^{6}$ & 3 & $6.20\pm0.6$& $3.64\pm1.37$& $6.16\pm2.32$ & $4.90$\\
$31.4$ & $+0.3$ & G31.4+0.3$^{2}$ & 1 & $6.20\pm0.6$& $4.89\pm1.80$& $6.20\pm2.28$ & $5.55$\\
$34.3$ & $+0.1$ & G34.3+0.1$^{2}$ & 3 &$10.50\pm0.3$& $1.68\pm0.66$& $2.55\pm1.00$ & $2.11$\\
$43.2$ & $+0.0$ & W49A$^{2}$ & 1 &$11.80\pm0.4$&$10.13\pm3.73$&$15.40\pm5.67$ & $12.76$\\
$49.5$ & $-0.4$ & W51A$^{7}$ & 4 & $5.50\pm8.0$& $3.38\pm1.35$& $5.41\pm2.11$ & $4.39$\\
$133.7$ & $+1.2$ & W3$^{8}$ & 3 & $4.20\pm0.7$& $1.97\pm0.73$& $2.19\pm0.82$ & $2.08$\\
$189.0$ & $+0.8$ & G189.0+0.8$^{2}$ & 2 & $0.80\pm1.9$& $3.62\pm1.51$& $4.33\pm1.82$ & $3.97$\\
$217.4$ & $-0.1$ & BFS57$^{2}$ & 2 & $2.40\pm0.6$& $1.16\pm0.45$& $1.52\pm0.59$ & $1.34$\\
$265.1$ & $+1.5$ & G265.1+1.5$^{2}$ & 4 & $1.40\pm0.8$& $0.83\pm0.33$& $1.00\pm0.40$ & $0.91$\\
$267.7$ & $-1.1$ & G267.7-1.1$^{2}$ & 2 & $1.40\pm1.0$& $1.13\pm0.42$& $1.36\pm0.52$ & $1.24$\\
$268.0$ & $-1.0$ & RCW38$^{2}$ & 1 & $1.40\pm1.0$& $1.05\pm0.42$& $1.75\pm0.70$ & $1.40$\\
$282.0$ & $-1.2$ & G282.0-1.2$^{2}$ & 2 & $5.90\pm0.5$& $6.00\pm2.32$& $7.94\pm3.07$ & $6.97$\\
$291.3$ & $-0.7$ & NGC3576$^{2}$ & 1 & $3.10\pm9.8$& $0.87\pm0.32$& $1.09\pm0.41$ & $0.98$\\
$298.2$ & $-0.3$ & G298.2-0.3$^{2}$ & 2 &$10.40\pm0.5$& $4.18\pm1.56$& $5.29\pm1.97$ & $4.73$\\
$326.6$ & $+0.6$ & RCW95$^{2}$ & 2 & $2.80\pm0.3$& $1.63\pm0.63$& $2.02\pm0.78$ & $1.82$\\
$328.3$ & $+0.4$ & G328.3+0.4$^{2}$ & 2 & $6.10\pm0.6$& $5.27\pm2.18$& $6.34\pm2.59$ & $5.80$\\
$332.6$ & $-0.6$ & RCW106$^{2}$ & 4 & $3.50\pm0.3$& $3.25\pm1.34$& $4.67\pm1.88$ & $3.96$\\
$333.1$ & $-0.4$ & G333.1-0.4$^{9}$ & 2 & $3.50\pm0.3$& $3.29\pm1.23$& $3.85\pm1.43$ & $3.57$\\
$345.2$ & $+1.0$ & RCW116B$^{2}$ & 2 & $1.70\pm0.6$& $2.09\pm0.88$& $2.68\pm1.13$ & $2.38$\\
$348.2$ & $-1.0$ & RCW121$^{2}$ & 1 & $2.70\pm0.5$& $2.74\pm1.01$& $2.95\pm1.08$ & $2.84$\\
$351.2$ & $+0.7$ & NGC6334$^{2}$ & 1 & $1.20\pm1.1$& $1.30\pm0.50$& $2.34\pm0.90$ & $1.82$\\
$351.6$ & $-1.3$ & G351.6-1.3$^{2}$ & 2 &$14.30\pm0.8$& $2.28\pm0.94$& $2.90\pm1.17$ & $2.59$\\ \hline
\end{tabular}
\label{table4}
\end{center}
\begin{footnotesize}
{\parbox{06.7in}{Notes: (1) All kinematic distances are from \citet{Russeil03}, except W31-South and W31-North with distances from \citet{Corbel04}; The adopted distance moduli are from: (2) \citet{Bik05}; (3) \citet{Blum01}; (4) \citet{Hanson96}; (5) \citet{Blum00}; (6) \citet{Blum99}; (7) \citet{Figueredo08}; (8) Navarete et al., {\it in preparation}; (9) \citet{Figueredo05}.}}
\end{footnotesize}
\end{table*}
\end{footnotesize}
These effects, of discrepancies between the distances, can be seen in the Fig.~\ref{fig:gal-projec}, where the H\,{\sc{ii}} regions are displayed on the Galactic plane based on their distances. Black circles represent kinematic results, red triangles represent an average of the results using \citet{Mathis90} and \citet{Stead09} laws (Fig. 4a), and the trigonometric parallax distances (Fig. 4b). Discrepancies are clearly seen between the kinematic and non-kinematic results, where the second group has, in general, smaller distances. The arrows are proportional to the discrepancies.
\begin{figure*}
\begin{minipage}[b]{0.47\linewidth}
\includegraphics[height=09cm,width=09cm]{fig04a.jpg}
\end{minipage} \hfill
\begin{minipage}[b]{0.47\linewidth}
\includegraphics[height=09cm,width=09cm]{fig04b.jpg}
\end{minipage}
\caption{{ Distribution of the H\,{\sc{ii}} regions in the Galactic plane. In the left (Fig. 4a) are the distances derived with $K$-band spectrophotometry, these distances are an average of the distances using \citet{Mathis90} and \citet{Stead09}. In the right (Fig. 4b) are the distances derived using trigonometric parallax. In both panels the kinematic distances are also shown. }}
\label{fig:gal-projec}
\end{figure*}
\section{Conclusion}
In this work, we present a near infrared and {\it Spitzer} (mid infrared) study of 35 Galactic H\,{\sc{ii}} regions. These regions were chosen from the catalogs of \citet{Conti04}, \citet{Bica03a} and \citet{Dutra03}. \citet{Conti04} have carried out a complete census of 6-cm selected H\,{\sc{ii}} regions and identified 56 as Giant H\,{\sc{ii}} regions, based in part on mid- and far-IR fluxes from MSX and IRAS. \citet{Bica03a} and \citet{Dutra03} have carried out a 2MASS $J$, $H$, and $K_{s}$ survey of infrared star cluster across the Milky Way.
All the distances listed in the Table 1 are from kinematic techniques (rotational velocity plus a galactic rotation model) for consistency. Among our sample of 35 Galactic H\,{\sc{ii}} regions, we have defined 24 as GH\,{\sc{ii}} regions based on the kinematic distance and radio continuum luminosity.
In this paper, we have focused on the $J$, $H$, and $K_{s}$-band photometric properties of the regions. Using morphological clues in near infrared and {\it Spitzer} IRAC images along with the C-C and C-M diagrams, we place each region into a qualitative evolutionary stage labeled {\it A}, {\it B}, {\it C}, or {\it D}. In the first ($stage$ $A$) we identify regions still very embedded in gas and dust, with little evidence of their emergent stellar component, like G12.8-0.2 (W33), G333.3-0.4, G333.6-0.2, for example. regions in the second $stage$ ($B$) have a well-defined cluster of stars, but with several objects with infrared excess (T-Tauri stars and YSOs) like G5.97-1.18 (M8) and G10.3-0.1 (W31-North). The third, $stage$ $C$, are those regions that we can distinguish a well-defined cluster of stars, a few objects with infrared excess and a nebulosity surrounding the naked cluster (i.e. on its periphery), like G30.8-0.2 (W43) and G287.4-0.6 (NGC3372). The fourth $stage$ ($D$) represents regions in which we do not see nebulosity, the cluster is well-defined and stars are completely typified by colors for normal photospheres, like G308.7+0.6.
In this near and mid infrared study of the stellar content of these star-forming regions, we have also identified a sample of massive YSOs in our images and C-M and C-C diagrams, based on their large luminosities, as well as large infrared excess. As expected, the presence of YSOs, particularly the massive ones, is more prominent in the less evolved regions, where there is strong nebular emission. We present the list of the massive YSOs detected in this work in the Table 2.
Qualitatively, we have used main sequence lines in the C-M diagrams to verify if the kinematic distance is consistent with the cluster members position in these diagrams. In some regions, we have shown large discrepancies, where the tip of the main sequence (O-type stars) is fainter than the brightest objects of the cluster. This implies that the real distance is smaller than that adopted from kinematic methodology. Other distance determinations, like spectrophotometric and trigonometric parallax have verified (typically) smaller distances compared to kinematic distances. In our sample of 35 Galactic H\,{\sc{ii}} regions, we suggest that roughly a third are consistent with a closer distance than is derived from kinematic techniques. These regions are marked as CL (closer) in the Table 1. We find nine H\,{\sc{ii}} regions have kinematic distances that are qualitatively consistent with main sequence locations in our C-M diagrams. These regions are marked as AG (agree) in the Table 1. We could not speculate on the agreement between the kinematic distances and the photometric data for ten of our sample of H\,{\sc{ii}} regions, due to small number of detected objects or the presence of known evolved stars. These regions are marked as UN (unknown) in the Table 1. Finally, two H\,{\sc{ii}} regions may be further away than their kinematic distances. Interestingly they are close to the Galactic center, where the determination of kinematic distances is not so easy. Our images and C-M and C-C diagrams provide excellent candidate sources to observe spectroscopically and so expand the sample of GH\,{\sc{ii}} regions with known distances.
Quantitatively, we compared the kinematic distances with distances derived from $K$-band spectrophotometric and trigonometric parallax to 26 star forming regions found in the literature. We used two extreme interstellar extinction laws in the determination of the distance. In general, the distances derived by these two non-kinematic techniques are smaller than that derived by kinematic thecniques. The same discrepancies were found when we compare the results from trigonometric parallax with the kinematic thechniques.
There are three main conclusions in this work: 1) in most cases clusters are seen; 2) one can distinguish several evolutionary stages among these objects; and 3) the photometric distances are in many cases smaller than the kinematic values (similar to what is inferred from spectrophotometric and trigonometric parallaxes). Plans are underway to revisit the distance discrepancies among the kinematic, trigonometric and spectrophotometric determinations with the aim of better understanding our Galactic sprial structure.
\section*{Acknowledgments}
APM and AD are grateful to the Brazilian agency CNPq-MCT for continuous financial support. EF thanks L'Or\'eal-UNESCO-ABC for Brazil's 2009 For Women in Science grant. AD, EF and CLB acknowledge FAPESP for continuous financial support. PSC wishes to thank the NSF for continuous support. Based on observations obtained at the CTIO 1.5-m and Blanco 4-m telescopes, which are operated by the Association of Universities for Research in Astronomy Inc. (AURA), under a cooperative agreement with the National Science Foundation (NSF) as part of the National Optical Astronomy Observatories (NOAO).
\bibliographystyle{mn2e}
|
{
"timestamp": "2010-09-21T02:04:20",
"yymm": "1009",
"arxiv_id": "1009.3924",
"language": "en",
"url": "https://arxiv.org/abs/1009.3924"
}
|
\section{Introduction and Summary of Results} \label{sec:intro}
\subsection{The Stochastic Gravitational Wave Background}
There is great interest in detecting or constraining the strength of stochastic gravitational waves (GWs) that may have been produced by a variety of processes in the early Universe, including inflation. The strength of the waves is parameterized by their energy density per unit logarithmic frequency divided by the critical energy density, $\Omega_{\rm gw}(f)$. Current observational upper limits include (i) the constraint $\Omega_{\rm gw} \alt 10^{-13} (f / 10^{-16} \, {\rm Hz})^{-2}$ for $10^{-17}~{\rm Hz}~\alt~f \alt 10^{-16} \, {\rm Hz}$ from large angular scale fluctuations in the cosmic microwave background temperature \cite{Buonanno07}; (ii) the cosmological nucleosynthesis and cosmic microwave background constraint $\int d\ln f \: \Omega_{\rm gw}(f) \alt 10^{-5}$, where the integral is over frequencies $f \agt 10^{-15}$ Hz \cite{Smith06}; (iii) the pulsar timing limit $\Omega_{\rm gw} \alt 10^{-8}$ at $10^{-9} \, {\rm Hz} \alt f \alt 10^{-8} \, {\rm Hz}$ \cite{Jenet06}; (iv) the current LIGO/VIRGO upper limit $\Omega_{\rm gw} \alt 7 \times 10^{-6}$ at $f \sim 100 \, {\rm Hz}$ \cite{Abbott09}; and (v) the limit $\int d\ln f \: \Omega_{\rm gw} \alt 10^{-1}$ for $10^{-17} \, {\rm Hz} \alt f \alt 10^{-9} \, {\rm Hz}$ from VLBI radio astrometry of quasars.
Many new techniques also promise future measurements of these primordial GWs. Firstly, it has been shown that such a GW background would leave a detectable signature in the polarization of the cosmic microwave background \citep[CMB;][]{Kamionkowski97,Seljak97}, which will be measured by many current and future observational efforts \cite{ACTPol10,Bicep10,CAPMAP08,CBI05,Clover08,CMBPol08,Ebex04,QUaD09,PIPER10,Polarbear10}. The planned space-based interferometer LISA will also set limits on the primordial stochastic gravitational wave background (SGWB) \cite{Kudoh06}. The planned successor to LISA, the Big Bang Observer, is a space-based interferometer mission designed primarily to detect the primordial SGWB \cite{Phinney03}. Finally, Seto and Cooray have suggested that measurements of the anisotropy of time variations of redshifts of distant sources could provide constraints of order $\Omega_{\rm gw} \alt 10^{-5}$ at $f \sim 10^{-12}$ Hz \cite{Seto06}. For more details on GWs, the search for them, and the SGWB, see the review articles \cite{Buonanno07, Allen97, Maggiore00}.
\subsection{High Precision Astrometry}
The possibility of using high precision astrometry to detect GWs has been considered by many authors \cite{gaia00, Braginsky90, Damour98, Fakir94, Flanagan93, Gwinn97, Jaffe04, Kaiser97, Linder86, Makarov10, Mignard10, Pyne96, Schutz09}. There was an early suggestion by Fakir \cite{Fakir94} that GW bursts from localized sources could be detectable by the angular deflection $\Delta \theta$ to light rays that they would produce. Fakir claimed that $\Delta \theta \propto 1/b$, where $b$ is the impact parameter. This claim was shown later to be false, and in fact the deflection scales as $1/b^3$ \cite{Damour98, Flanagan93}. Therefore the prospects for using astrometry to detect waves from localized sources are not promising \cite{Schutz09}.
However, the situation is different for a SGWB, as first discussed by Braginsky et al. \cite{Braginsky90}. For a light ray propagating through a SGWB, one might expect the direction of the ray to undergo a random walk, with the deflection angle growing as the square root of distance. However, this is not the case; the deflection angle is always of order the strain amplitude $h_{\rm rms}$ of the GWs, and does not grow with distance \footnotemark[1] \cite{Braginsky90, Kaiser97, Linder86}. \footnotetext[1]{It is sometimes claimed in the literature that the deflection angle depends only on the GWs near the source and observer. In fact, this is not true, as we discuss in Appendix A. A similar claim about the frequency shift that is the target of pulsar timing searches for GWs is also false in general.} Specifically, a SGWB will cause apparent angular deflections which are correlated over the sky and which vary randomly with time, with an rms deflection $\delta_{\rm rms}(f)$ per unit logarithmic frequency interval of (see Eq. \ref{eqn:2ptnn} below)
\begin{equation} \delta_{\rm rms}(f) \sim h_{\rm rms}(f) \sim \frac{H_0}{f} \sqrt{\Omega_{\rm gw}(f)}. \label{eqn:deltrms} \end{equation}
Suppose now that we monitor the position of N sources in the sky, with an angular accuracy of $\Delta \theta$, over a time T. For a single source, one could detect an angular velocity (proper motion) of order $\sim\Delta \theta/T$, and for N sources, a correlated angular velocity of order $\sim\Delta \theta/(T \sqrt{N})$ should be detectable. The rms angular velocity from (\ref{eqn:deltrms}) is $\omega_{\rm rms}(f)~\sim~f~\delta_{\rm rms}(f)~\sim~H_0~\sqrt{\Omega_{\rm gw}(f)}$, and it follows that one should obtain an upper limit on $\Omega_{\rm gw}$ of order \cite{Pyne96}
\begin{equation} \Omega_{\rm gw}(f) \alt \frac{\Delta \theta^2}{N T^2 H_0^2}. \label{eqn:omapprox}\end{equation}
\noindent This bound will apply at a frequency of order $f\sim1/T$. It will also apply at lower frequencies \cite{Pyne96} since the angular velocity fluctuations are white (equal contributions from all frequency scales), assuming a flat GW spectrum $\Omega_{\rm gw}~=$~const. The quantity that will be constrained by observations is roughly this total $\Omega_{\rm gw}$, $\int_{f \alt T^{-1}} d \ln f \Omega_{\rm gw}(f)$.
The advent of microarcsecond astrometry has started to make the prospects for constraining GW backgrounds more interesting. The future astrometry mission GAIA (Global Astrometric Interferometer for Astrophysics) is expected to measure positions, parallaxes and annual proper motions to better than $20 \: \mu\text{as}$ for more than 50 million stars brighter than $V \sim 16$ mag and 500,000 quasars brighter than $V \sim 20$ mag \cite{gaia00}. Similarly the Space Interferometry Mission (SIM) is expected to achieve angular accuracies of order $10 \: \mu\text{as}$. Estimates of the sensitivities of these missions to a SGWB, at the $\Omega_{\rm gw} \sim 10^{-3}$ -- $10^{-6}$ level, are given in Refs. \cite{gaia00,Makarov10,Mignard10}.
VLBI radio interferometry is another method that can be used to detect the astrometric effects of a SGWB on distant sources. This method detects the same pattern as that discussed in this paper for visible astrometry, and differs from astrometry using the GAIA satellite in its longer duration (tens of years versus a few years for GAIA), and in the smaller number of sources, on the order of hundreds, that have currently been measured using this method. In the radio, the planned Square Kilometer Array (SKA) is also expected to be able to localize sources to within $\sim 10 \mu\text{as}$ \cite{Fomalont04}. Jaffe has estimated that with $10^6$ QSO sources, the SKA could achieve a sensitivity of order $\Omega_{\rm gw} \sim 10^{-6}$ \cite{Jaffe04}.
The astrometric signals due to a SGWB expected for a single object are quite small, on the order of $0.1 \: \mu$as yr$^{-1}$, much smaller than the typical intrinsic proper motion of a star in our galaxy. We therefore propose to use quasars as our sources, since their extragalactic distances cause their expected intrinsic proper motions to be smaller than those expected from a SGWB \cite{gaia00}. The construction of a non-rotating reference frame using quasars in astrometric studies will remove the $l=1$ dipole component of the measured quasar proper motions, but will leave intact the $l=2$ and higher multipoles which are expected to be excited by GWs.
Using the estimate $N \sim 10^6$ (GAIA), $\Delta \theta \sim 10 \: \mu\text{as}$, $T \sim 1$ yr gives from Eq. (\ref{eqn:omapprox}) the estimate
\begin{equation} \Omega_{\rm gw} \alt 10^{-6} \nonumber\end{equation}
\noindent at $f \alt 10^{-8}$ Hz for astrometry. This is an interesting sensitivity level, roughly comparable with that obtainable with pulsar timing \cite{Jenet06}.
Astrometry has already been applied to obtain upper limits on $\Omega_{\rm gw}$ using a number of different observations. First, Gwinn et. al analyzed limits on quasar proper motions obtained from VLBI astrometry, and obtained the upper limit $\Omega_{\rm gw} \alt 10^{-1}$ for $10^{-17} \, {\rm Hz} \alt f \alt 10^{-9} \, {\rm Hz}$ \cite{Gwinn97}. This limit was recently updated by Titov, Lambert and Gontier \cite{Titov10}. Finally, Linder analyzed observed galaxy correlation functions to obtain the limit $\Omega_{\rm gw} \alt 10^{-3}$ for $10^{-16} \, {\rm Hz} \alt f \alt 10^{-10} \, {\rm Hz}$ \cite{Linder88}.
All of these analyses used a relativly simple model of the effect of graitational waves on proper motions. In this paper we give a detailed computation of the spectrum of angular fluctuations produced by a stochastic background, including the relative strengths of E- and B-type multipoles for each order $l$. In a subsequent paper we will follow up with a derivation of the optimal data analysis method and a computation of the $\Omega_{\rm gw}$ sensitivity level, to confirm the existing crude estimates of the sensitivity of future astrometric missions such as GAIA.
\subsection{Summary of results}
For a source in the direction ${\bf n}$, the effect of the GW background is to produce an apparent angular deflection $\delta {\bf n}({\bf n},t)$. We first find a general formula for the angular deflection of a photon, for an arbitrary GW signal $h_{ij}$, emitted by a source that can be at a cosmological distance. This deflection is derived in Secs. \ref{sec:mink} and \ref{sec:FRW} below, and is given by [cf. Eq. (\ref{eqn:deflFRW})]
\begin{align} \delta n^i =& \frac{1}{2} \Bigg\{ n^j h_{ij}(0) - n^i n^j n^k h_{jk}(0)- \frac{\omega_0}{\zeta_s}\left( \delta^{ik} - n^i n^k \right)n^j \cdot \nonumber\\&\cdot \left[ -2 \int_0^{\zeta_s} d\zeta' \int_0^{\zeta'} d\zeta'' h_{jk,0}(\zeta'') + n^l \int_0^{\zeta_s} d\zeta' \int_0^{\zeta'} d\zeta'' \left( h_{jk,l}(\zeta'') + h_{kl,j}(\zeta'') - h_{jl,k}(\zeta'') \right) \right] \Bigg\}.\nonumber\end{align}
\noindent Here, $\mathbf{n}$ is the direction to the source, $\omega_0$ is the emitted frequency of the photon, $\zeta$ parameterizes the path of the photon $\tau(\zeta) = \tau_0 + \omega_0 \zeta$, $ x^i(\zeta) = -\zeta \omega_0 n^i$, $h_{ij}(\tau,\mathbf{x})$ is treated as a function of $\zeta$ through this parameterization of the photon path, $\zeta_s$ is the value of $\zeta$ at the emission event of the photon at the source, and the spacetime metric is
\begin{equation} ds^2 = a(\tau)^2 \left\{ -d\tau^2 + \left[\delta_{ij} + h_{ij}(\tau,\mathbf{x})\right] dx^i dx^j \right\}. \nonumber \end{equation}
We then specialize to the limit in which the sources are many gravitational wavelengths away and to plane waves propagating in the direction $\mathbf{p}$ to obtain a simple formula, which generalizes a previous result of Pyne et al \cite{Pyne96}. We find that the deflection, as a function of time $\tau$ and direction on the sky $\mathbf{n}$, is given by
\begin{equation} \delta n^{\hat{i}}(\tau,{\bf n}) = \frac{n^i + p^i}{2 (1 + \mathbf{p}\cdot\mathbf{n})} h_{jk}(0) n_j n_k - \frac{1}{2} h_{ij}(0) n_j,\nonumber\end{equation}
\noindent where $\mathbf{p}$ is the direction of propagation of the GW, and $h_{ij}(0)$ is the GW field evaluated at the observer, $h_{ij}(\tau,{\bf 0})$.
The main result of this paper is a computation of the statistical properties of the angular deflection resulting from a SGWB, which is carried out in Secs. \ref{sec:corr} and \ref{sec:spect}. The apparent angular deflection caused by such a GW background is a stationary, zero-mean, Gaussian random process. We compute the fluctuations in $\delta {\bf n}$ by making two different approximations: (i) The GW modes which contribute to the deflection have wavelengths $\lambda$ which are short compared to the horizon size $c \, H_0^{-1}$ today. (ii) The mode wavelengths $\lambda$ are short compared to the distances to the sources; this same approximation is made in pulsar timing searches for GWs \cite{Detweiler79}. Since our calculations are only vaild for GWs with wavelengths much smaller than the horizon, the contribution from waves with wavelengths comparable to the horizon scale will cause a small deviation from our results (on the order of a few percent for a white GW spectrum).
The total power in angular fluctuations is then
\begin{equation} \left< \delta {\bf n}({\bf n},t)^2 \right> = \theta_{\rm rms}^2 = \frac{1}{4 \pi^2} \int d\ln f \left( \frac{H_0}{f} \right)^2 \Omega_{\rm gw}(f). \label{eqn:2ptnn} \end{equation}
\noindent Taking a time derivative gives the spectrum of fluctuations of angular velocity or proper motion:
\begin{equation} \left< \delta \dot{\bf n}({\bf n},t)^2 \right> = \int d\ln f H_0^2 \Omega_{\rm gw}(f), \nonumber \end{equation}
\noindent which gives a rms angular velocity $\omega_{\rm rms}(f)$ of order
\begin{equation} \omega_{\rm rms}(f) \sim H_0 \sqrt{\Omega_{\rm gw}} \sim 10^{-2} \mu \text{as} \: \text{yr}^{-1} \left( \frac{\Omega_{\rm gw}}{10^{-6}} \right)^{1/2}. \nonumber\end{equation}
\noindent This is the signal that we hope to detect.
\bigskip
We now discuss how the angular fluctuations are distributed on different angular scales, or equivalently how the power is distributed in the spherical harmonic index $l$. The total angular fluctuations can be written as
\begin{equation} \left< \delta {\bf n}({\bf n},t)^2 \right> = \int d\ln f \sum_{l=2}^\infty \left[ \theta_{{\rm rms}, l}^{E}(f)^2 + \theta_{{\rm rms}, l}^{B}(f)^2 \right]. \label{eqn:thrms}\end{equation}
\noindent Here $\theta^{E}_{{\rm rms},l}(f)^2$ is the total electric-type power in angular fluctuations per unit logarithmic frequency in multipole sector $l$, and $\theta^{B}_{{\rm rms},l}(f)^2$ is the corresponding magnetic-type power. These quantities can be written as
\begin{equation} \theta_{{\rm rms}, l}^{Q}(f)^2 = \theta_{\rm rms}^2 \,g_{Q} \,\sigma(f) \, \alpha^{QQ}_l, \end{equation}
\noindent where $Q=E$ or $B$. The various factors in this formula are as follows. The factors $g_{E}$ and $g_{B}$ are the fractions of the total power carried by E-modes and B-modes respectively, and satisfy $g_{E} + g_{B}=1$. Their values are $g_{E} = g_{B} = 1/2$, implying that electric and magnetic type fluctuations have equal power. The function $\sigma(f)$ describes how the power is distributed in frequency, and is the same for all multipoles, both electric and magnetic. It is normalized so that $\int d\ln f \sigma(f) =1$, and is given explicitly by [cf.\ Eq.\ (\ref{eqn:2ptnn}) above]
\begin{equation} \sigma(f) = \frac{ f^{-2} \: \Omega_{\rm gw}(f)}{\int d\ln f' f^{\prime \, -2} \: \Omega_{\rm gw}(f')}. \label{eqn:sigma} \end{equation}
\noindent Finally, the angular spectra $\alpha^{EE}_l$ and $\alpha^{BB}_l$ describe how the power is distributed in different multipoles, starting with the quadrupole at $l=2$, and are normalized so that
\begin{equation} \sum_{l=2}^\infty \alpha^{QQ}_l =1 \end{equation}
\noindent for $Q=E$ and $Q=B$. We show that $\alpha^{EE}_l = \alpha^{BB}_l$, and this spectrum is plotted in Fig. \ref{fig:alphaQ} and tabulated in table \ref{tab:alphaQ}. These coefficients are well fit by the power law $\alpha^{EE}_l = 32.34 \: l^{-4.921}$. We note that the result for the quadrupole, $\alpha^{EE}_2 = 5/6$, has previously been derived using a different method in Ref. \cite{Pyne96}.
\begin{figure}[t!]
\centering
\includegraphics[width=0.6\textwidth]{fig1.eps}
\caption{Here we plot the coefficients $\alpha_l^{EE}$ as defined in Eq. (\ref{eqn:SQlf}) vs. multipole $l$.}
\label{fig:alphaQ}
\end{figure}
\begin{table*}
\centering
\caption{ First 10 nonzero multipole coefficients $\alpha_l^{EE}$ as defined in Eq. (\ref{eqn:SQlf}) and plotted in Fig. \ref{fig:alphaQ}.\label{tab:alphaQ} }
\begin{tabular}{ | l | r | }
\hline
\begin{centering} $l$ \end{centering}& \multicolumn{1}{c|}{$\alpha_l^{EE}$}\\
\hline
\hline
2 & 0.833333 \\
3 & 0.116667 \\
4 & 0.03 \\
5 & 0.0104762 \\
6 & 0.00442177 \\
7 & 0.00212585 \\
8 & 0.00112434 \\
9 & 0.000639731 \\
10 & 0.000385675 \\
11 & 0.000243696 \\
\hline
\end{tabular}
\end{table*}
\section{Calculation of Astrometric Deflection in a Minkowski Background Spacetime} \label{sec:mink}
\subsection{ Setting the stage--Minkowski Calculation }
We will first calculate the angular deflection due to a small GW perturbation on a flat background metric,
\begin{equation} ds^2 \equiv g_{\mu \nu} dx^{\mu} dx^{\nu} = -dt^2 + (\delta_{ij} + h_{ij})dx^i dx^j.\end{equation}
\noindent We are considering the effect of these GWs on a photon traveling from a source to an observer, with an unperturbed worldline $x^{\alpha}_0(\lambda) = \omega_0 ( \lambda, - \lambda {\bf n})+ (t_0,0,0,0),$ where $-{\bf n}$ is the direction of the photon's travel, $\omega_0$ is its unperturbed frequency, and the photon is observed at the origin at time $t_0$. The photon's unperturbed 4-momentum is given by $k^{\alpha}_0 = \omega_0 (1, -{\bf n}).$
To calculate the geodesics that the photon, source and observer follow, we need the connection coefficients in this metric. There are three non-zero connection coefficients:
\begin{equation} \Gamma^{k}_{0 i} = \frac{1}{2} h_{k i, 0},\quad \Gamma^{0}_{i j} = \frac{1}{2} h_{i j , 0},\quad \Gamma^{k}_{i j} = \frac{1}{2} [ h_{k i, j} + h_{k j, i} - h_{i j, k} ]. \label{eqn:gamma}\end{equation}
\noindent First, using the geodesic equation
\begin{equation} \frac{d^2x^{\alpha}}{d\tau^2} = - \Gamma^{\alpha}_{\beta \gamma} u^{\beta} u^{\gamma}, \label{eqn:geodesic}\end{equation}
\noindent it is straightforward to verify that the paths of stationary observers in these coordinates are geodesics. Therefore we can assume that both the source and observer are stationary in these coordinates, with
\begin{align} &x_{obs}^i(t) = 0 \nonumber \\
&x_{s}^i(t) = x_{s}^i = \text{constant}. \nonumber\end{align}
\noindent The affine parameter of the source is therefore
\begin{equation} \lambda_s = - \frac{\left|{\bf x}_s\right|}{\omega_0}. \nonumber\end{equation}
\subsection{ Photon Geodesic}
Next, we solve the geodesic equation (\ref{eqn:geodesic}) for the path of a photon traveling from the source to the observer in the perturbed metric. We write this path as the sum of contributions of zeroth and first order in $h$,
\begin{equation} x^{\alpha}(\lambda) = x^{\alpha}_0(\lambda) + x^{\alpha}_1(\lambda). \end{equation}
\noindent Similarly, the photon 4-momentum is
\begin{equation} k^{\alpha}(\lambda) = k^{\alpha}_0(\lambda) + k^{\alpha}_1(\lambda). \end{equation}
\noindent We note that the connection coefficients are all first order in $h$, so keeping only first order terms, we will use only the unperturbed photon 4-momentum in the geodesic equation, yielding
\begin{equation} \frac{d^2x_1^0}{d\lambda^2} = -\frac{\omega_0^2}{2} n^i n^j h_{ij,0},\end{equation}
\begin{equation} \frac{d^2x_1^k}{d\lambda^2} = - \frac{\omega_0^2}{2} [ -2 n^i h_{ki,0} + n^i n^j \left( h_{ki,j} + h_{kj,i} - h_{ij,k} \right) ]. \end{equation}
We now integrate the geodesic equation to obtain the perturbed photon 4-momentum and trajectory. The right hand sides are evaluated along the photon's unperturbed path from $\lambda=0$ at the present time back to $\lambda$, since they are already first order in $h$. We define
\begin{align} \mathcal{I}_{ij}(\lambda) &= \int_0^{\lambda} d\lambda' h_{ij,0}(\lambda'), &\mathcal{J}_{ijk}(\lambda) &= \int_0^{\lambda} d\lambda' h_{ij,k}(\lambda'),\nonumber\\
\mathcal{K}_{ij}(\lambda) &= \int_0^{\lambda} d\lambda' \int_0^{\lambda'} d\lambda'' h_{ij,0}(\lambda''), &\mathcal{L}_{ijk}(\lambda) &= \int_0^{\lambda} d\lambda' \int_0^{\lambda'} d\lambda'' h_{ij,k}(\lambda''),\label{eqn:termdef}\end{align}
\noindent where $h_{ij}(\lambda)$ means $h_{ij}(t_0 + \omega_0 \lambda, -\omega_0 \lambda {\bf n})$. We find
\begin{align} k^{0}_1(\lambda) &= -\frac{\omega_0^2}{2} n^i n^j \mathcal{I}_{ij}(\lambda) + I_0, &k^{j}_1(\lambda) &= - \frac{\omega_0^2}{2} n^i R_{ij}+ J_0^j, \nonumber\\
x^{0}_1(\lambda) &= -\frac{\omega_0^2}{2} n^i n^j \mathcal{K}_{ij}(\lambda) + I_0 \lambda + K_0, &x^{j}_1(\lambda) &= - \frac{\omega_0^2}{2} n^i S_{ij} + J_0^j \lambda + L_0^j, \label{eqn:intkx}\end{align}
\noindent where $I_0$, $J_0^j$, $K_0$ and $L_0^j$ are constants of integration, and we have defined the quantities
\begin{equation} R_{ij}(\lambda) \equiv \left[ -2 \mathcal{I}_{ij}(\lambda) + n^k \left( \mathcal{J}_{ijk}(\lambda) + \mathcal{J}_{jki}(\lambda) - \mathcal{J}_{ikj}(\lambda) \right) \right], \label{eqn:Rdef} \end{equation}
\begin{equation} S_{ij}(\lambda) \equiv \left[ -2 \mathcal{K}_{ij}(\lambda) + n^k \left( \mathcal{L}_{ijk}(\lambda) + \mathcal{L}_{jki}(\lambda) - \mathcal{L}_{ikj}(\lambda) \right) \right]. \end{equation}
\subsection{ Boundary conditions }
We determine the eight integration constants $I_0$, $J_0^j$, $K_0$ and $L_0^j$ using the boundary conditions of the problem, namely that the photon path passes through the detection event $x^{\mu}_{obs} = (t_0,0,0,0)$, that it is null, that the photon is emitted with the unperturbed frequency $\omega_0$, and that the photon path intersects the path of the source at some earlier time.
\begin{enumerate}
\item {\em Photon path must pass through detection event}
First, the perturbed photon trajectory must pass through the detection event $t = t_0$, $x^i = 0$. Therefore,
\begin{equation} x^{\mu}(0) = x^{\mu}_0(0) + x^{\mu}_1(0) = (t_0,0,0,0), \nonumber\end{equation}
\noindent giving
\begin{equation} K_0 = \frac{\omega_0^2}{2} n^i n^j \mathcal{K}_{ij}(0)=0, \quad L^j_0 = \frac{\omega_0^2}{2} n^i S_{ij}(0)=0, \end{equation}
\noindent where we have used the fact that by definition $\mathcal{K}_{ij}(0)=S_{ij}(0)=0$.
\item {\em Photon geodesic is null}
The geodesic of the photon must be null, which gives one more constraint: $g_{\mu \nu} k^{\mu} k^{\nu} = 0$. This is already true to zeroth order. To first order we get:
\begin{equation} 0 = h_{\mu \nu} k^{\mu}_0 k^{\nu}_0 + 2 \eta_{\mu \nu} k^{\mu}_1 k^{\nu}_0, \nonumber\end{equation}
\noindent where $k^{\alpha}_0 = \omega_0 (1, -{\bf n})$. Inserting the expression for the perturbed 4-momentum $k_1^{\alpha}$ given by Eqs. (\ref{eqn:termdef}) , (\ref{eqn:intkx}) and (\ref{eqn:Rdef}) , and simplifying using
\begin{equation} \frac{\text{d}}{\text{d}\lambda} h_{ij} = \omega_0 h_{ij,0} - \omega_0 n_k h_{ij,k} \label{eqn:dh}\end{equation}
\noindent shows that all of the terms involving $\lambda$ cancel out, as they must, leaving the condition
\begin{equation} I_0 + n_i J_0^i = \frac{1}{2}\omega_0 n^i n^j h_{ij}(0). \label{eqn:nullcnst}\end{equation}
\item {\em Photon is emitted with frequency $\omega_0$}
The photon is emitted at the source with the unperturbed frequency $\omega_0 = - g_{\mu \nu} k^{\mu} u_s^{\nu}$. The 4-velocity of the source is $u_s^{\mu} = (1, 0, 0, 0)$ as it has constant spatial coordinate position, so the constraint becomes $-g_{\mu 0} k^{\mu} = \omega_0$. The source emits the photon at $\lambda = \lambda_s$, so from Eq. (\ref{eqn:intkx}) this yields
\begin{equation} I_0 = \frac{\omega_0^2}{2} n^i n^j \mathcal{I}_{ij}(\lambda_s) \label{eqn:w0cnst}\end{equation}
\item {\em Perturbed photon path must hit source worldline somewhere}
The constraint that the perturbed photon trajectory must hit the source worldline somewhere can be written as
\begin{equation} x^j(\tilde{\lambda_s}) = x_s^j = x^j_0(\tilde{\lambda_s}) + x^j_1(\tilde{\lambda_s})\end{equation}
\noindent for some $\tilde{\lambda_s}$. To zeroth order we have $\tilde{\lambda_s} = \lambda_s$, but there will be a first order correction. Inserting the expression (\ref{eqn:intkx}) for the perturbation of the geodisic gives
\begin{equation} x_s^j = -\omega_0 \tilde{\lambda_s} n^j - \frac{\omega_0^2}{2} n^i S_{ij}(\tilde{\lambda_s}) + \tilde{\lambda_s} J_0^j. \label{eqn:source} \end{equation}
Projecting this equation perpendicular to ${\bf n}$ gives a formula for the perpendicular component of $J_0^i$:
\begin{equation} J^i_{0\;\perp} = \frac{\omega_0^2}{2 \lambda_s} \left( \delta^{ik} - n^i n^k \right) n^j S_{jk}(\lambda_s). \end{equation}
\noindent Here on the right hand side we have replaced $\tilde{\lambda_s}$ with $\lambda_s$, which is valid to linear order. Adding to this our earlier result for the component of $J_0^i$ parallel to ${\bf n}$ in Eqs. (\ref{eqn:nullcnst}) and (\ref{eqn:w0cnst}) gives
\begin{equation} J_0^i = \frac{\omega_0^2}{2 \lambda_s} n^j S_{jk}(\lambda_s) \left( \delta^{ik} - n^i n^k \right) - \frac{\omega_0^2}{2} n^i n^j n^k \mathcal{I}_{jk}(\lambda_s) + \frac{1}{2} \omega_0 n^i n^j n^k h_{jk}(0). \end{equation}
\end{enumerate}
\subsection{ Perturbation to Observed Frequency }\label{sec:omegacomp}
We calculate the observed photon frequency $\omega_{obs} = -g_{\mu \nu} k^{\mu} u_{obs}^{\nu}$, where $u_{obs}^{\nu} = (1,0,0,0)$, and check our result against standard formulae for the frequency shift, used in pulsar timing searches for GWs \cite{Anholm09}. The observed frequency is, from Eqs. (\ref{eqn:intkx}) and (\ref{eqn:w0cnst}),
\begin{equation} \omega_{obs} = k^0(0) = \omega_0 + I_0 = \omega_0 + \frac{\omega_0^2}{2} n^i n^j \mathcal{I}_{ij}(\lambda_s). \label{eqn:obsfreq}\end{equation}
\noindent Using the definition (\ref{eqn:termdef}), the perturbed redshift is therefore
\begin{equation} z\equiv \frac{\omega_0 - \omega_{obs}}{\omega_0} = -\frac{\omega_0}{2} n^i n^j \int_0^{\lambda_s} d\lambda' h_{ij,0}(\lambda'). \label{eqn:z}\end{equation}
\noindent For a plane wave traveling in the direction of the unit vector ${\bf p}$, we have
\begin{equation} h_{ij} = h_{ij}(t-\mathbf{p}\cdot\mathbf{x})=h_{ij}\left[\omega_0 \lambda (1+ \mathbf{p}\cdot\mathbf{n})\right], \nonumber\end{equation}
\noindent giving
\begin{equation} h_{ij,0}\equiv\frac{\partial}{\partial t} h_{ij}=\frac{1}{\omega_0 (1+\gamma)} \frac{\partial}{\partial \lambda} h_{ij}, \nonumber\end{equation}
\noindent where $\gamma = \mathbf{p}\cdot\mathbf{n}$. This gives for the redshift
\begin{equation} z = -\frac{1}{2 (1+\gamma)} n^i n^j \left[ h_{ij}(\lambda_s) - h_{ij}(0) \right], \end{equation}
\noindent which agrees with \cite{Anholm09} up to a sign, which is an error in their calculation \cite{CPC}.
\subsection{ Local Proper Reference Frame of Observer }
We must also account for the changes induced in the basis vectors of the observer's local proper reference frame due to the presence of the GW. We introduce a set of orthonormal basis vectors $\vec{e}_{\hat{\alpha}}$ which are parallel transported along the observer's worldline, with $\vec{e}_{\hat{0}} = \vec{u}$. The parallel transport equation for the spatial vectors gives
\begin{equation} u^{\alpha} e^{\beta}_{\hat{j} ; \alpha} = u^{\alpha} \left[\partial_{\alpha} e^{\beta}_{\hat{j}} + \Gamma^{\beta}_{\alpha \gamma} e^{\gamma}_{\hat{j}}\right] = 0. \label{eqn:plltrans}\end{equation}
\noindent We separate the basis vectors into two pieces, $ e^i_{\hat{j}} = \delta^i_{\hat{j}} + \delta e^i_{\hat{j}}$, where we assume that the unperturbed basis vectors are aligned with the coordinate basis directions.
Using ${\vec u} = \partial_t$, and the connection coefficients (\ref{eqn:gamma}) of the metric , Eq. (\ref{eqn:plltrans}) gives us an expression for the perturbation to the basis tetrad:
\begin{equation} \delta e^i_{\hat{j}}(t) = - \frac{1}{2} h_{i\hat{j}}(t,{\bf 0}) + \omega_{i \hat{j}}, \nonumber\end{equation}
\noindent where $\omega^i_{\hat{j}}$ is a matrix of constants. Now, we observe that $e_{\hat{j}}$ is an orthonormal set of three-vectors, which gives us six constraints on the constants $\omega^i_{\hat{j}}$:
\begin{equation} \left( \eta_{mn} + h_{mn} \right)\left( \delta^m_{\hat{j}} + \delta e^m_{\hat{j}} \right) \left( \delta^n_{\hat{k}} + \delta e^n_{\hat{k}} \right) = \delta_{\hat{j} \hat{k}}.\nonumber\end{equation}
\noindent This is identically correct to zeroth order; to first order we get $ \delta e_{j \hat{k}} + \delta e_{k \hat{j}} + h_{jk} = 0$, or, inserting our equation for $\delta e$, and assuming that $h_{ij} = h_{ji}$, we find $\omega_{ij} = -\omega_{ji}$, i.e. that the constants $\omega_{ij}$ are antisymmetric in their indices. These constants parameterize an arbitrary infinitesimal time-independent rotation. Evaluating now at the detection event gives
\begin{equation} \delta e^i_{\hat{j}} = - \frac{1}{2} h_{i\hat{j}}(0) + \omega_{i\hat{j}}. \end{equation}
\noindent For the remainder of this paper we will set to zero the term $\omega_{i\hat{j}}$, since it corresponds to a time-independent, unobservable angular deflection. The deflections caused by GWs will be observable because of their time dependence.
\subsection{ Observed Angular Deflection }
We can express the four-momentum of the incoming photon in the above reference frame as
\begin{equation} k^{\alpha}(0) = \omega_{obs} u^{\alpha} - \omega_{obs} n^{\hat{j}} e^{\alpha}_{\hat{j}}, \label{eqn:obsk}\end{equation}
\noindent where $\delta_{\hat{j} \hat{k}} n^{\hat{j}} n^{\hat{k}} = 1$, $u^{\alpha}$ is the observer's 4-velocity, and $\omega_{obs}$ is given by equation \ref{eqn:obsfreq}. Note that we evaluate all quantities at the detection event $t = t_0$, ${\bf x} = 0$. Plugging in our results for the perturbed 4-momentum and the observed frequency, we obtain an equation for the observed direction to the source $n^{\hat{j}}$
\begin{align} k^i(0) =& - \omega_0 n^i + \frac{\omega_0^2}{2 \lambda_s} n^j S_{jk}(\lambda_s) \left( \delta^{ik} - n^i n^k \right) - \frac{\omega_0^2}{2} n^i n^j n^k \mathcal{I}_{jk}(\lambda_s) + \frac{1}{2} \omega_0 n^i n^j n^k h_{jk}(0)\nonumber\\
=& - \left(\omega_0 + \frac{\omega_0^2}{2} n^k n^l \mathcal{I}_{kl}(\lambda_s) \right) n^{\hat{j}} \left( \delta^i_j - \frac{1}{2} h^i_j(0) \right). \label{eqn:obsdir}\end{align}
We decompose the direction to the source into zeroth and first order pieces as $n^{\hat{j}} = n^{\hat{j}}_0 + \delta n^{\hat{j}}$. The zeroth order terms in Eq. (\ref{eqn:obsdir}) give us $n_0^{\hat{j}} = n^j$. Plugging this into the first order terms and simplifying, we find the perturbation to the source direction
\begin{equation} \delta n^{\hat{i}} = \frac{1}{2} \left\{n^j h_{ij}(0) - \frac{\omega_0}{\lambda_s} n^j S_{jk}(\lambda_s) \left( \delta^{ik} - n^i n^k \right) - n^i n^j n^k h_{jk}(0) \right\}.\nonumber\end{equation}
\noindent Inserting our definition of $S_{jk}$, we obtain the solution to the source direction perturbation in Minkowski space
\begin{align} \delta n^{\hat{i}} =& \frac{1}{2} \Bigg\{ n^j h_{ij}(0) - n^i n^j n^k h_{jk}(0) - \frac{\omega_0}{\lambda_s}\left( \delta^{ik} - n^i n^k \right)n^j \nonumber\\&\times \left[ -2 \int_0^{\lambda_s} d\lambda' \int_0^{\lambda'} d\lambda'' h_{jk,0}(\lambda'') + n^l \int_0^{\lambda_s} d\lambda' \int_0^{\lambda'} d\lambda'' \left( h_{jk,l}(\lambda'') + h_{kl,j}(\lambda'') - h_{jl,k}(\lambda'') \right) \right] \Bigg\}. \label{eqn:defl}\end{align}
\noindent As a check of the calculation, we see that $\delta n^{\hat{i}}$ is orthogonal to $n^i$, so that $n^i + \delta n^{\hat{i}}$ is a unit vector, as expected.
We now specialize to the case of a plane wave propagating in the direction of the unit vector ${\bf p}$,
\begin{equation} h_{ij}(t,\mathbf{x}) = h_{ij}(t-\mathbf{p}\cdot\mathbf{x}). \nonumber\end{equation}
\noindent Using the identity (\ref{eqn:dh}) we can reduce the double integrals in Eq. (\ref{eqn:defl}) to single integrals, obtaining
\begin{equation} \delta n^{\hat{i}} = \left( \delta^{ik} - n^i n^k \right) n^j \Bigg\{ - \frac{1}{2} h_{jk}(0) + \frac{p_k n_l}{2 (1 + \mathbf{p}\cdot\mathbf{n})} h_{jl}(0) + \frac{1}{\lambda_s} \int_0^{\lambda_s} d\lambda \left[ h_{jk}(\lambda) - \frac{p_k n_l}{ 2 (1 + \mathbf{p}\cdot\mathbf{n}) } h_{jl}(\lambda) \right] \Bigg\}. \label{eqn:defl1}\end{equation}
\noindent Evaluating this explicitly for the plane wave
\begin{equation} h_{ij}(t,\mathbf{x}) = \text{Re}\left[ \mathcal{H}_{ij} e^{-i \Omega (t - \mathbf{p}\cdot\mathbf{x})} \right] \nonumber\end{equation}
\noindent gives
\begin{align} \delta n^{\hat{i}} =& \text{Re}\Bigg[ \Bigg( \left\{ 1 + \frac{i (2 + \mathbf{p}\cdot\mathbf{n}) }{\omega_0 \lambda_s \Omega (1 + \mathbf{p}\cdot\mathbf{n})} \left[1 - e^{-i \Omega \omega_0 (1 + \mathbf{p}\cdot\mathbf{n}) \lambda_s }\right] \right\} n^i \nonumber\\
&+ \left\{ 1 + \frac{i}{\omega_0 \lambda_s \Omega (1 + \mathbf{p}\cdot\mathbf{n})} \left[1 - e^{-i \Omega \omega_0 (1 + \mathbf{p}\cdot\mathbf{n}) \lambda_s }\right] \right\} p^i \Bigg) \frac{n^j n^k \mathcal{H}_{j k} \text{e}^{-i \Omega t_0}}{2 (1 + \mathbf{p}\cdot\mathbf{n})}\nonumber\\
&- \left\{ \frac{1}{2} + \frac{i}{\omega_0 \lambda_s \Omega (1 + \mathbf{p}\cdot\mathbf{n})} \left[1 - e^{-i \Omega \omega_0 (1 + \mathbf{p}\cdot\mathbf{n}) \lambda_s }\right] \right\} n^j \mathcal{H}^i_j \text{e}^{-i \Omega t_0}\Bigg] .
\label{eqn:m1}\end{align}
If we define the observed angles $(\theta,\phi)$ by $n^{\hat{i}} = \left(\sin\theta \cos\phi, \sin\theta \sin\phi, \cos\theta\right)$, then the observed angular deflections are
\begin{equation} \delta \theta = \text{e}^{\hat{i}}_{\hat{\theta}} \delta n^{\hat{i}}, \quad \delta \phi = \frac{\text{e}^{\hat{i}}_{\hat{\phi}} \delta n^{\hat{i}}}{\sin\theta}, \label{eqn:sphbasis}\end{equation}
\noindent where $\text{e}^{\hat{i}}_{\hat{\theta}} = \left(\cos\theta \cos\phi, \cos\theta \sin\phi, -\sin\theta\right)$ and $\text{e}^{\hat{i}}_{\hat{\phi}} = \left(-\sin\phi,\cos\phi,0\right)$.
As another check of our calculation, we now compare our result with the coordinate (gauge-dependent) angular deflection computed by Yoo et al. \cite{Yoo09}. Starting from our Eq. (\ref{eqn:defl}), we disregard the first term, which arises from the change from the coordinate basis to the parallel transported orthonormal basis. The remaining terms in Eq. (\ref{eqn:defl}) give the coordinate angular deflection $\delta n^i$. Simplifying using the identity (\ref{eqn:dh}) and the identity $\int_0^{x} dx' \int_0^{x'}dx'' f(x'') = \int_0^{x} dx' (x-x') f(x')$ gives
\begin{equation} \delta n^i = - \frac{1}{2} n^i n^j n^k h_{jk}(0) + \left( \delta^{ij} - n^i n^j \right) \int^{\lambda_s}_0 d\lambda \left\{\frac{h^{jk}(\lambda) - h^{jk}(0)}{\lambda_s} n_k + \frac{\omega_0}{2} \left( \frac{\lambda_s - \lambda}{\lambda_s} \right) \partial_j \left( n^k n^l h_{kl} \right)\right\}. \label{eqn:Yoo}\end{equation}
\noindent When combined with Eqs. (\ref{eqn:sphbasis}), this agrees with Eqs. (13) and (14) of \cite{Yoo09}, specialized to only tensor perturbations, up to an overall sign. The sign flip is due to the fact that Ref. \cite{Yoo09} uses a convention for the sign of angular deflection, explained after their Eq. (16), which is opposite to ours.
\subsection{ The Distant Source Limit }
We now specialize to the limit where the distance $\omega_0 \left|\lambda_s\right|$ to the source is large compared to the wavelength $\sim c \: \Omega^{-1}$ of the GWs. As discussed in the Introduction, astrometry is potentially sensitive to waves with a broad range of frequencies, extending from the inverse of the observation time (a few years) down to the Hubble frequency. Therefore this assumption is a nontrivial limitation on the domain of validity of our analysis. However, for sources at cosmological distances (the most interesting case), this assumption is not a significant limitation.
In this limit, we can neglect the second term in each of the three small square brackets in Eq. (\ref{eqn:m1}), giving
\begin{equation} \delta n^{\hat{i}}(t,\mathbf{n}) = \text{Re}\left[ \left( n^i + p^i \right) \frac{\mathcal{H}_{jk} n_j n_k \text{e}^{-i\Omega t}}{2 (1 + \mathbf{p}\cdot\mathbf{n})} - \frac{1}{2} \mathcal{H}_{ij} n_j \text{e}^{-i\Omega t} \right], \label{eqn:dsl3}\end{equation}
\noindent where we have written $t$ for $t_0$. This result agrees with and generalizes a calculation of Pyne et al. \cite{Pyne96}. We note that this same approximation is used in pulsar timing searches for GWs \cite{Anholm09}. In that context the approximation is essentially always valid, since pulsar distances are large compared to a few light years, and the properties of pulsar frequency noise imply that that pulsar timing is only sensitive to GWs with periods of order the observation time, and not much lower frequencies, unlike the case for astrometry.
\section{Generalization to Cosmological Spacetimes} \label{sec:FRW}
Of course, we do not live in Minkowski space. The apparent homogeneity and isotropy of the universe imply that our universe has an FRW geometry, with line element:
\begin{equation} ds^2 = g_{\alpha \beta} dx^{\alpha} dx^{\beta} = a(\tau)^2 \left\{ -d\tau^2 + \left[\delta_{ij} + h_{ij}(\tau,\mathbf{x})\right] dx^i dx^j \right\}, \label{eqn:FRW}\end{equation}
\noindent where $\tau$ is conformal time, and we specialize to the transverse traceless gauge in which $\delta^{ij} h_{ij} = \delta^{ij} \partial_i h_{jk} =0$. To translate our calculation in Minkowski spacetime to this new metric, we define an unphysical, conformally related metric ${\bar g}_{\alpha\beta} = a(\tau)^{-2}
g_{\alpha\beta}$ given by
\begin{equation} {\bar g}_{\alpha\beta} dx^\alpha dx^\beta = -d\tau^2 + \left[\delta_{ij} + h_{ij}(\tau,{\bf x}) \right] dx^i dx^j, \label{eqn:gbar}\end{equation}
\noindent which has an associated unphysical derivative operator ${\bar \nabla}_\alpha$.
\subsection{ Stationary Observers are Freely Falling }
As before, it is straightforward to check that observers who are stationary in the coordinates (\ref{eqn:FRW}) are freely falling. Therefore we assume as before that the observer and source are stationary:
\begin{equation} x_{obs}^i(t) = 0, \quad x_{s}^i(t) = x_{s}^i. \nonumber\end{equation}
\subsection{ Null Geodesic in the Conformal Metric }
Let us consider a photon traveling from a distant source to us, which follows a null geodesic in the physical metric $g_{\alpha \beta}$. Its path is also a null geodesic of the conformally related metric ${\bar g}_{\alpha\beta}$, though it it is not affinely parameterized in this metric \cite{Wald84}. Specifically, the physical 4-momentum of the photon $k^{\mu}$ must satisfy the geodesic equation $k^{\mu} \nabla_{\mu} k_{\nu} = 0$. If we define a conformally related, unphysical 4-momentum ${\bar k}_{\mu} = k_{\mu}$, whose contravariant components are then related to those of the physical 4-momentum by
\begin{equation} {\bar k}^{\mu} = {\bar g}^{\mu\nu} {\bar k}_{\nu} = a(\tau)^2 g^{\mu\nu} {\bar k}_{\nu} = a(\tau)^2 g^{\mu\nu} k_{\nu} = a(\tau)^2 k^{\mu}, \end{equation}
\noindent then we find that
\begin{equation} {\bar k}^{\mu} {\bar \nabla}_{\mu} {\bar k}_{\nu} = a(\tau)^2 k^{\mu} {\bar \nabla}_{\mu} k_{\nu}. \end{equation}
\noindent From \cite{Wald84} we know that for any vector $v^{\alpha}$, and conformally related derivatives $\nabla_{\alpha}$ and ${\bar \nabla}_{\alpha}$, we have $\nabla_{\alpha} v_{\beta} = {\bar \nabla}_{\alpha} v_{\beta} - C^{\gamma}_{\alpha \gamma} v_{\gamma}$, where $C^{\gamma}_{\alpha \gamma} = 2 \delta^{\gamma}_{ (\alpha}\nabla_{\beta)} \ln a - g_{\alpha \beta} g^{\gamma \delta} \nabla_{\delta} \ln a$. Thus, we find
\begin{align} {\bar k}^{\mu} {\bar \nabla}_{\mu} {\bar k}_{\nu} &= a(\tau)^2 k^{\mu} \nabla_{\mu} k_{\nu} + a(\tau)^2 k^{\mu} k_{\rho} \left( 2 \delta^{\rho}_{ (\mu}\nabla_{\nu)} \ln a - g_{\mu \nu} g^{\rho \sigma} \nabla_{\sigma} \ln a \right) \nonumber\\ &= a(\tau)^2 k^{\mu} \nabla_{\mu} k_{\nu} + a(\tau)^2 \left( k^{\rho} k_{\rho} \nabla_{\nu} \ln a + k^{\mu} k_{\nu} \nabla_{\mu} \ln a - k_{\nu} k^{\sigma} \nabla_{\sigma} \ln a \right) \nonumber\\&= a(\tau)^2 k^{\mu} \nabla_{\mu} k_{\nu}, \end{align}
\noindent where to get the last line we have used that the geodesic is null. Therefore, if $k^{\mu}$ is a null geodesic of the physical metric $g_{\mu\nu}$, then ${\bar k}^{\mu}$ is a null geodesic of the conformally related metric ${\bar g}_{\mu\nu}$. If $\lambda$ is an affine parameter of the geodesic, it will not be an affine parameter of the geodesic in the unphysical metric. The affine parameter ${\bar \lambda}$ in the unphysical metric is related to $\lambda$ by
\begin{equation} \frac{d{\bar \lambda}}{d\lambda} = \frac{1}{a(\tau(\lambda))^2}. \end{equation}
\subsection{ Parallel Transport of Basis Vectors in FRW Background Spacetime }
We next investigate the parallel transport of the observer's basis tetrad in a FRW background spacetime. From the form (\ref{eqn:FRW}) of the metric, we anticipate that the basis vectors must scale as $a^{-1}$ to remain normalized. Thus, we will define the basis vectors and their perturbations as
\begin{equation} e^i_{\hat{j}} = \frac{1}{a}\left(\delta^i_{\hat{j}} + \delta e^i_{\hat{j}}\right). \label{eqn:plfrw}\end{equation}
\noindent The relevant connection coefficients are
\begin{equation} \Gamma^i_{0k} = \frac{\dot{a}}{a} \delta^i_k + \frac{1}{2}\delta^{im} h_{mk,0}. \label{eqn:cc}\end{equation}
\noindent The parallel transport equation (\ref{eqn:plltrans}) for the spatial basis vectors gives us
\begin{equation} \partial_0 e^i_{\hat{j}} + \Gamma^i_{0 k} e^k_{\hat{j}} = 0. \end{equation}
\noindent Plugging in the connection coefficients (\ref{eqn:cc}) and the basis vector expansion (\ref{eqn:plfrw}), we get
\begin{equation} \partial_0 \delta e^i_{\hat{j}} + \frac{1}{2}\delta^{im} h_{m\hat{j}}=0, \end{equation}
\noindent the same equation as before. The solution, as before, will be
\begin{equation} \delta e^i_{\hat{j}}(t) = - \frac{1}{2} h^i_j(t). \end{equation}
\subsection{ Generalization of Angular Deflection Computation }
We parametrize the photon path in the background spacetime by
\begin{equation} \tau(\zeta) = \tau_0 + \omega_0 \zeta, \quad x^i(\zeta) = -\zeta \omega_0 n^i, \end{equation}
\noindent where $\zeta$ is an affine parameter of the unphysical metric (\ref{eqn:gbar}) (denoted ${\bar \lambda}$ above). From the decomposition (\ref{eqn:obsk}), the observed source direction is
\begin{equation} n^{\hat j} = \frac{g_{\alpha \beta} k^{\alpha} e^{\beta}_{\hat{j}}}{g_{\alpha \beta} k^{\alpha} u^{\beta}}. \end{equation}
\noindent We rewrite all the quantities in this expression in terms of their conformally transformed versions
\begin{equation} {\bar g}_{\alpha \beta} = a^{-2} g_{\alpha \beta}, \quad {\bar k}^{\alpha} = a^2 k^{\alpha}, \quad {\bar u}^{\alpha} = a u^{\alpha}, \quad {\bar e}^{\alpha}_{\hat{j}} = a e^{\alpha}_{\hat{j}}, \end{equation}
\noindent which are the quantities that are used in the Minkowski spacetime calculation of Sec. \ref{sec:mink}. This gives
\begin{equation} n^{\hat j} = \frac{{\bar g}_{\alpha \beta} {\bar k}^{\alpha} {\bar e}^{\beta}_{\hat{j}}}{{\bar g}_{\alpha \beta} {\bar k}^{\alpha} {\bar u}^{\beta}}, \end{equation}
\noindent the same expression as in Minkowski spacetime. Therefore, the final result is the same expression (\ref{eqn:defl}) as before, except that it is written in terms of the non-affine parameter $\zeta$:
\begin{align} \delta n^{\hat i} =& \frac{1}{2} \Bigg\{ n^j h_{ij}(0) - n^i n^j n^k h_{jk}(0)- \frac{\omega_0}{\zeta_s}\left( \delta^{ik} - n^i n^k \right)n^j \nonumber\\
&\times \left[ -2 \int_0^{\zeta_s} d\zeta' \int_0^{\zeta'} d\zeta'' h_{jk,0}(\zeta'') + n^l \int_0^{\zeta_s} d\zeta' \int_0^{\zeta'} d\zeta'' \left( h_{jk,l}(\zeta'') + h_{kl,j}(\zeta'') - h_{jl,k}(\zeta'') \right) \right] \Bigg\}. \label{eqn:deflFRW}\end{align}
\subsection{ The Distant Source Limit }
We now specialize again to the limit where the distance to the source is large compared to the wavelength $\sim c \: \Omega^{-1}$ of the GWs. We also assume that the wavelength $c \: \Omega^{-1}$ is small compared to the horizon scale, but we allow the sources to be at cosmological distances.
Starting from Eq. (\ref{eqn:deflFRW}) and paralleling the derivation of Eq. (\ref{eqn:Yoo}) we obtain
\begin{equation} \delta n^{\hat i}(\tau_0,{\bf n}) = \frac{1}{2} s_{ik} n_j h_{jk}(0) + \frac{s_{ik} n_j}{\zeta_s} \int_0^{\zeta_s} d\zeta \left[ h_{jk}(\zeta) - h_{jk}(0) \right] + \frac{\omega_0 s_{ik}}{2} \int_0^{\zeta_s} d\zeta \left( \frac{\zeta_s - \zeta}{\zeta_s} n_j n_l h_{jl,k}(\zeta) \right), \label{eqn:dsl1}\end{equation}
\noindent where $s_{ik} = \delta_{ik} - n_i n_k$. Now the wave equation satisfied by the metric perturbation is
\begin{equation} \left[ \partial^2_{\tau} + 2 \frac{a_{,\tau}}{a} \partial_{\tau} - {\bf \nabla}^2 \right] h_{ij}(\tau,{\bf x}) = 0, \nonumber\end{equation}
\noindent and plane wave solutions are of the form
\begin{equation} h_{ij}(\tau,{\bf x}) = \text{Re} \left\{ \mathcal{H}_{ij} e^{i\Omega \mathbf{p}\cdot\mathbf{x}} q_{\Omega}(\tau) \right\}, \nonumber\end{equation}
\noindent where the mode function $q_{\Omega}$ satisfies
\begin{equation} q_{\Omega}'' + 2 \frac{a'}{a} q_{\Omega}' + \Omega^2 q_{\Omega} = 0. \label{eqn:dsl2}\end{equation}
We now evaluate the angular deflection (\ref{eqn:dsl1}) for such a plane wave, in the limit where $\varepsilon \equiv a'/(\Omega a) \ll 1$, i.e. the limit where the wavelength $\sim a/\Omega$ of the GW is much smaller than the the horizon scale $\sim a^2/a'$. In the second term in (\ref{eqn:dsl1}), the term $h_{jk}(\zeta)$ is rapidly oscillating, and so its integral can be neglected compared to the integral of $h_{jk}(0)$; corrections will be suppressed by powers of $\varepsilon$. In the third term in (\ref{eqn:dsl1}), the integrand is rapidly oscillating, and so the integral will be dominated by contributions near the endpoints, up to $\mathcal{O}(\varepsilon)$ corrections. However the integrand vanishes at $\zeta=\zeta_s$, and thus the integral is dominated by the region near $\zeta=0$. In that region we can use the leading order WKB approximation to the mode function solution of (\ref{eqn:dsl2}),
\begin{equation} q_{\Omega}(\tau) = \frac{1}{a(\tau)} e^{-i \Omega \tau}, \nonumber\end{equation}
\noindent and to a good approximation we can replace $a(\tau)$ by $a(\tau_0)$. Thus we see that the same answer is obtained for distant sources as in our Minkowski spacetime calculation, even for sources at cosmological distances. From Eq. (\ref{eqn:dsl3}) we obtain
\begin{equation} \delta n^{\hat{i}}(\tau_0,{\bf n}) = \frac{n^i + p^i}{2 (1 + \mathbf{p}\cdot\mathbf{n})} h_{jk}(0) n_j n_k - \frac{1}{2} h_{ij}(0) n_j\label{eqn:dsl4}\end{equation}
\noindent for plane waves in the direction ${\bf p}$.
\section{Calculation of Angular Deflection Correlation Function} \label{sec:corr}
Now that we have calculated the deflection of the observed direction to a distant source due to an arbitrary metric perturbation $h_{ij}$, we would like to determine the properties of the deflection produced by a SGWB, such as that produced by inflation.
\subsection{ Description of SGWB as a Random Process }
In the distant source limit, the angular deflection (\ref{eqn:dsl4}) depends only on the GW field $h_{ij}$ evaluated at the location of the observer for each direction of propagation $\mathbf{p}$. Moreover, we have restricted attention to modes with wavelengths short compared to the Hubble time. Therefore, it is sufficient to use a flat spacetime mode expansion to describe the stochastic background. This expansion is (see, e.g. Ref. \cite{Flanagan93})
\begin{equation} h_{ij}(\mathbf{x}, t) = \sum_{A=+,\times} \int_0^{\infty} df \int d^2\Omega_{\mathbf{p}} \: h_{A \mathbf{p}}(f) \: e^{2 \pi i f (\mathbf{p}\cdot\mathbf{x} - t)} \: e^{A,\mathbf{p}}_{ij} + c.c., \label{eqn:hSGWB}\end{equation}
\noindent where $f$ and $\mathbf{p}$ are the frequency and direction of propagation of individual GW modes, $h_{A\mathbf{p}}$ are the stochastic amplitudes of modes with polarization $A$ and direction $\mathbf{p}$, and the polarization tensors $e^{A,\mathbf{p}}_{ij}$ are normalized such that $e^{A,\mathbf{p}}_{ij} e^{B,\mathbf{p}*}_{ij} = 2 \delta^{AB}$.
We will assume that $h_{ij}(\mathbf{x}, t)$ is a Gaussian random process, as it is likely to be the sum of a large number of random processes. We also assume that it is zero-mean and stationary. It follows that the mode amplitudes $h_{A \mathbf{p}}(f)$ satisfy
\begin{align} \langle h_{A \mathbf{p}}(f) \: h_{B \mathbf{p'}}(f') \rangle\;\: =&\: 0,\nonumber\\ \langle h_{A \mathbf{p}}(f) \: h_{B \mathbf{p'}}(f')^* \rangle =&\: \frac{3 H_0^2\Omega_{\rm gw}(f)}{32 \pi^3 f^3} \: \delta(f-f') \: \delta_{AB} \: \delta^2(\mathbf{p},\mathbf{p}') \label{eqn:h2pt}\end{align}
\noindent for $f,f'\geq 0$, where $H_0$ is the Hubble parameter and $\delta^2(\mathbf{p},\mathbf{p}')$ is the delta function on the unit sphere (see, e.g., \cite{Flanagan93}).
Since the angular deflection $\delta\mathbf{n}(\mathbf{n},t)$ depends linearly on the metric perturbation, it will also be a stationary, zero-mean, Gaussian random process, whose statistical properties are determined by its two point correlation function $\langle \delta n^i \delta n^j \rangle$. Specializing our expression (\ref{eqn:dsl4}) for the angular deflection to the form (\ref{eqn:hSGWB}) of the metric perturbation, we find
\begin{equation} \delta n^i(\mathbf{n},t) = \sum_{A=+,\times} \int_0^{\infty} df \int d^2\Omega_{\mathbf{p}} \: h_{A \mathbf{p}}(f) \: e^{-2\pi i f t} \: \mathcal{R}_{ikl}(\mathbf{n},\mathbf{p}) \: e^{A,\mathbf{p}}_{kl} + c.c. \label{eqn:dn}, \end{equation}
\noindent where
\begin{equation} \mathcal{R}_{ikl}(\mathbf{n},\mathbf{p}) = \frac{1}{2} \left( \frac{\left[n_i + p_i\right]n_k n_l}{1 + \mathbf{p}\cdot\mathbf{n}} - n_k \delta_{il} \right). \label{eqn:R}\end{equation}
\subsection{ Power Spectrum of the Astrometric Deflections of the SGWB }
So, we need only evaluate the two-point correlation function to gain full knowledge of the statistical properties of the angular deflection due to the SGWB. Writing out this quantity explicitly using Eq. (\ref{eqn:dn}),
\begin{align} \langle \delta n^i(\mathbf{n},t) \: \delta n^{j}(\mathbf{n}',t') \rangle &= \sum_{A,B=+,\times} \int_0^{\infty} df df' \int d^2\Omega_{\mathbf{p}} d^2\Omega_{\mathbf{p}'} \bigg\langle \left[h_{A \mathbf{p}}(f) \; e^{-2\pi i f t} \; \mathcal{R}_{ikl}(\mathbf{n},\mathbf{p}) \; e^{A,\mathbf{p}}_{kl} + c.c.\right] \nonumber \\
&\times \left[h_{B \mathbf{p}'}(f')^*\; e^{2\pi i f' t'} \; \mathcal{R}_{jrs}(\mathbf{n}',\mathbf{p}') \; \left(e^{B,\mathbf{p}'}_{rs}\right)^* + c.c.\right] \bigg\rangle.\label{eqn:firstcor}\end{align}
\noindent The average, which is an average over ensembles, acts only on the stochastic amplitudes $h_{A \mathbf{p}}$. Using the mode 2 point function (\ref{eqn:h2pt}) in Eq. (\ref{eqn:firstcor}), we get the simplified result
\begin{equation} \langle \delta n^i(\mathbf{n},t) \: \delta n^j(\mathbf{n}',t') \rangle = \int_0^{\infty} df \frac{3 H_0^2}{32 \pi^3} f^{-3} \Omega_{\rm gw}(f) e^{-2\pi i f (t-t')} H_{ij}(\mathbf{n},\mathbf{n}') + c. c., \label{eqn:2ptH}\end{equation}
\noindent where we have defined
\begin{equation} H_{ij}(\mathbf{n},\mathbf{n}') = \sum_{A=+,\times} \int d^2\Omega_{\mathbf{p}} \mathcal{R}_{ikl}(\mathbf{n},\mathbf{p}) \; e^{A,\mathbf{p}}_{kl} \; \mathcal{R}_{jrs}(\mathbf{n}',\mathbf{p}) \; \left(e^{A,\mathbf{p}}_{rs}\right)^*. \label{eqn:H}\end{equation}
\subsection{ Basis Tensors and their symmetries \label{sec:basis}}
We simplify the expression (\ref{eqn:H}) for $H_{ij}$ further using the identity
\begin{equation} \sum_{A=+,\times} e^{A,\mathbf{p}}_{ij} \left( e^{A,\mathbf{p}}_{kl} \right)^* = 2 P_{ijkl}, \label{eqn:ident}\end{equation}
\noindent where $P_{ijkl}$ is the projection tensor onto the space of traceless symmetric tensors orthogonal to $\mathbf{p}$, given by
\begin{equation} 2 P_{ijkl} = \delta_{ik} \delta_{jl} + \delta_{il} \delta_{jk} - \delta_{ij} \delta_{kl} + p_i p_j p_k p_l - \delta_{ik} p_j p_l - \delta_{jl} p_i p_k - \delta_{il} p_j p_k - \delta_{jk} p_i p_l + \delta_{ij} p_k p_l + \delta_{kl} p_i p_j. \label{eqn:P}\end{equation}
\noindent This gives
\begin{equation} H_{ij}(\mathbf{n},\mathbf{n}') = 2 \int d^2\Omega_{\mathbf{p}} \mathcal{R}_{ikl}(\mathbf{n},\mathbf{p}) P_{klrs} \mathcal{R}_{jrs}(\mathbf{n}',\mathbf{p}).\label{eqn:HR}\end{equation}
Noting that the correlation function (\ref{eqn:2ptH}) is perpendicular to $\mathbf{n}$ on its first index and $\mathbf{n}'$ on its second, we can decompose it onto a basis of tensors with this property:
\begin{equation} H_{ij}(\mathbf{n},\mathbf{n}') = \alpha(\mathbf{n},\mathbf{n}') A_i A_j + \beta(\mathbf{n},\mathbf{n}') A_i C_j + \gamma(\mathbf{n},\mathbf{n}') B_i A_j + \sigma(\mathbf{n},\mathbf{n}') B_i C_j, \label{eqn:decomp}\end{equation}
\noindent for some scalar functions $\alpha$, $\beta$, $\gamma$ and $\sigma$. Here we have defined
\begin{equation} \mathbf{A} = \mathbf{n} \times \mathbf{n}', \quad \mathbf{B} = \mathbf{n} \times \mathbf{A}, \quad \mathbf{C} = -\mathbf{n}' \times \mathbf{A}. \label{eqn:ABC}\end{equation}
\noindent We can deduce from Eq. (\ref{eqn:HR}) that $H_{ij}(\mathbf{n},\mathbf{n}')^* = H_{ji}(\mathbf{n}',\mathbf{n})$. Noting that $A_i(\mathbf{n}',\mathbf{n}) = -A_i(\mathbf{n},\mathbf{n}')$, and $B_i(\mathbf{n}',\mathbf{n}) = -C_i(\mathbf{n},\mathbf{n}')$, this symmetry applied to the expansion (\ref{eqn:decomp}) gives
\begin{equation} \alpha(\mathbf{n},\mathbf{n}')^* = \alpha(\mathbf{n}', \mathbf{n}), \quad \sigma(\mathbf{n},\mathbf{n}')^* = \sigma(\mathbf{n}',\mathbf{n}), \quad \beta(\mathbf{n},\mathbf{n}')^* = \gamma(\mathbf{n}',\mathbf{n}). \nonumber\end{equation}
\noindent We see from Eq. (\ref{eqn:H}) that $H_{ij}$ transforms as tensor under rotations. This implies that the functions $\alpha$, $\beta$, $\gamma$ and $\sigma$ must be invariant under rotations, and can only depend on the angle $\Theta$ between $\mathbf{n}$ and $\mathbf{n}'$. Thus, $ \alpha(\mathbf{n},\mathbf{n}') = \alpha(\mathbf{n}',\mathbf{n})=\alpha(\Theta)$ and so forth, so $\alpha$ and $\sigma$ must be real.
Next, we note that the expression (\ref{eqn:HR}) for $H_{ij}(\mathbf{n},\mathbf{n}')$ is invariant under the parity transformation $\mathbf{n}\rightarrow-\mathbf{n}$ and $\mathbf{n}'\rightarrow-\mathbf{n}'$. Looking then at the basis tensors, we see that $\mathbf{A}$ is invariant under this transformation, while $\mathbf{B}$ and $\mathbf{C}$ change sign. Thus, in order to insure that $H_{ij}$ is invariant, it can only have terms multiplying $A_i A_j$ and $B_i C_j$, so $\beta(\Theta)=0=\gamma(\Theta)$.
Having taken the symmetries of the problem into consideration, we have found $H_{ij}$ to be of the form
\begin{equation} H_{ij}(\mathbf{n},\mathbf{n}') = \alpha(\Theta) A_i A_j + \sigma(\Theta) B_i C_j. \label{eqn:decomp2}\end{equation}
\subsection{ Solving the General Integral }
We can evaluate the coefficients in the expansion (\ref{eqn:decomp2}) of $H_{ij}$ by contracting it with the basis tensors:
\begin{equation} A^i A^j H_{ij} = \sin^4(\Theta) \alpha(\Theta), \quad B^i C^j H_{ij} = \sin^4(\Theta) \sigma(\Theta). \nonumber\end{equation}
Rewriting these using Eq. (\ref{eqn:HR}), we find
\begin{equation} \alpha(\Theta) = \frac{2}{\sin^4(\Theta)} \int d^2\Omega_{\mathbf{p}} A^i \mathcal{R}_{ikl}(\mathbf{n},\mathbf{p}) P_{klrs} A^j \mathcal{R}_{jrs}(\mathbf{n}',\mathbf{p})^*, \end{equation}
\begin{equation} \sigma(\Theta) = \frac{2}{\sin^4(\Theta)} \int d^2\Omega_{\mathbf{p}} B^i \mathcal{R}_{ikl}(\mathbf{n},\mathbf{p}) P_{klrs} C^j \mathcal{R}_{jrs}(\mathbf{n}',\mathbf{p})^*. \end{equation}
To simplify the calculation, we define the quantities $\kappa = \mathbf{n}\cdot\mathbf{p}$, $\kappa' = \mathbf{n}'\cdot\mathbf{p}$, $\lambda = \mathbf{n}\cdot\mathbf{n}'$, $\mu = \mathbf{A}\cdot\mathbf{p}$, which satisfy $\mu^2 + \lambda^2 + \kappa^2 + \kappa'^2 = 1 + 2 \lambda \kappa \kappa'$. Using these definitions and the definition (\ref{eqn:R}) of $\mathcal{R}_{ikl}$, we can write
\begin{align} A^i \mathcal{R}_{ikl}(\mathbf{n},\mathbf{p}) &= \frac{1}{2} n_k \left( \frac{\mu n_l}{1 + \kappa} -A_l \right), &A^j \mathcal{R}_{jrs}(\mathbf{n}',\mathbf{p}) &= \frac{1}{2} n'_r \left( \frac{\mu n'_s}{1 + \kappa'} -A_s \right), \nonumber\\
B^i \mathcal{R}_{ikl}(\mathbf{n},\mathbf{p}) &= \frac{1}{2} n_k \left( -\frac{\kappa' + \lambda}{1 + \kappa} n_l + n'_l \right), &C^j \mathcal{R}_{jrs}(\mathbf{n}',\mathbf{p}) &= \frac{1}{2} n'_r \left( -\frac{\kappa + \lambda}{1 + \kappa'} n'_s + n_s \right). \nonumber\end{align}
\noindent We can then rewrite our expressions for $\alpha$ and $\sigma$
\begin{equation} \alpha(\Theta) = \frac{1}{4 \sin^4(\Theta)} \int d^2\Omega_{\mathbf{p}} 2P_{klrs} n_k \left( \frac{\mu n_l}{1 + \kappa} -A_l \right) n'_r \left( \frac{\mu n'_s}{1 + \kappa'} -A_s \right), \label{eqn:aP}\end{equation}
\begin{equation} \sigma(\Theta) = \frac{1}{4 \sin^4(\Theta)} \int d^2\Omega_{\mathbf{p}} 2P_{klrs} n_k \left( -\frac{\kappa' + \lambda}{1 + \kappa} n_l + n'_l \right) n'_r \left( -\frac{\kappa + \lambda}{1 + \kappa'} n'_s + n_s \right) .\label{eqn:sP}\end{equation}
\noindent Let's define two new variables $\nu^2 = (1-\kappa^2)$, $\nu'^2 = (1-\kappa'^2)$. Applying the definition (\ref{eqn:P}) of the projection tensor $P_{klrs}$, we can calculate the necessary contractions of $P_{klrs}$ for $\alpha$:
\begin{align} 2 P_{klrs} n_k A_l n'_r A_s &= \left( \lambda - \kappa \kappa' \right) \left( 1 - \lambda^2 - \mu^2 \right), &2 P_{klrs} n_k n_l n'_r A_s &= \mu \left( \kappa' \kappa^2 - 2 \lambda \kappa + \kappa' \right), \nonumber\\
2 P_{klrs} n_k A_l n'_r n'_s &= \mu \left( \kappa \kappa'^2 - 2 \lambda \kappa' + \kappa \right),
&2 P_{klrs} n_k n_l n'_r n'_s &= \nu^2 \nu'^2 - 2 \mu^2, \label{eqn:Pcontralpha}\end{align}
\noindent and for $\sigma$:
\begin{align} 2 P_{klrs} n_k n'_l n'_r n_s &= \nu^2 \nu'^2,
&2 P_{klrs} n_k n_l n'_r n_s &= \nu^2 \left( \lambda - \kappa \kappa' \right), \nonumber\\
2 P_{klrs} n_k n_l n'_r n'_s &= \nu^2 \nu'^2 - 2 \mu^2,
&2 P_{klrs} n_k n'_l n'_r n'_s &= \nu'^2 \left( \lambda - \kappa \kappa' \right). \label{eqn:Pcontrsigma}\end{align}
\noindent Plugging these back into Eqs. (\ref{eqn:aP}) and (\ref{eqn:sP}) and simplifying, we find
\begin{equation} \alpha(\Theta) = \frac{1}{4 \sin^4(\Theta)} \int d^2\Omega_{\mathbf{p}} \left[ (\lambda-\kappa \kappa')(1-\lambda^2) -\mu^2(1+\lambda) + \frac{2 \mu^2 (\lambda+\kappa)(\lambda+\kappa')}{(1+\kappa)(1+\kappa')} \right] = -\sigma(\Theta). \nonumber\end{equation}
\noindent Noticing that we can do the integrals $\int d^2\Omega_{\mathbf{p}} \mu^2 = \frac{4\pi}{3} \sin^2\Theta$ and $\int d^2\Omega_{\mathbf{p}} \kappa \kappa' = \frac{4\pi}{3} \cos\Theta$, but that the last term is more complicated, we find
\begin{equation} \alpha(\Theta) = -\sigma(\Theta) = \frac{\pi}{3} \frac{(\cos\Theta - 1)}{\sin^2\Theta} + \frac{1}{2 \sin^4\Theta} \int d^2\Omega_{\mathbf{p}} \frac{\mu^2 (\lambda+\kappa)(\lambda+\kappa')}{(1+\kappa)(1+\kappa')}. \label{eqn:alphaf}\end{equation}
We can reduce the two dimensional integral (\ref{eqn:alphaf}) to a one dimensional integral by parameterizing $\mathbf{p}$ in spherical polar coordinates $\theta_p$ and $\phi_p$, choosing $\mathbf{n} = (0, \,\sin(\Theta/2), \,\cos(\Theta/2))$ and $\mathbf{n}' = (0, \,-\sin(\Theta/2), \,\cos(\Theta/2))$ and integrating over $\phi_p$. This gives
\begin{equation} \alpha(\Theta) = -\sigma(\Theta) = \frac{\pi}{3} \frac{(\cos(\Theta) - 1)}{\sin^2(\Theta)} + \frac{\pi}{2 \sin^2\Theta} \int_0^{\pi} d\theta_p \sin\theta_p \left\{ \sin^2\theta_p + 8 \cos(\Theta/2) \left[ \cos\theta_p + \cos(\Theta/2) \right] \left[ \text{g}(\theta_p,\Theta) - 1 \right] \right\}, \label{eqn:alph1int}\end{equation}
\noindent where
\begin{equation} \text{g}(\theta_p,\Theta) = \frac{\left| \cos\theta_p + \cos(\Theta/2) \right|}{\left[ 1 + \cos\theta_p \cos(\Theta/2) \right]}. \end{equation}
\noindent We perform the integral over $\theta_p$, and find the final form of the function $\alpha(\Theta)$
\begin{equation} \alpha(\Theta) = -\sigma(\Theta) = \frac{\pi}{3 \sin^2\Theta} \left( 7 \cos\Theta - 5 \right) - \frac{32 \pi}{\sin^4\Theta} \ln\left( \sin(\Theta/2) \right) \sin^6(\Theta/2).\label{eqn:alphafin}\end{equation}
\noindent A plot of the function $\alpha(\Theta)$ is shown in Fig. \ref{fig:alpha}.
To summarize, we have now completed the calculation of the angular deflection correlation function. The final answer is given by Eq. (\ref{eqn:2ptH}), with $H_{ij}(\mathbf{n},\mathbf{n}')$ given from Eqs. (\ref{eqn:decomp2}) and (\ref{eqn:alphafin}) as
\begin{equation} H_{ij}(\mathbf{n},\mathbf{n}') = \alpha(\Theta) \left( A_i A_j - B_i C_j \right). \label{eqn:Hdecompalph}\end{equation}
\noindent Here the vectors $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ are defined by Eqs. (\ref{eqn:ABC}), and $\alpha(\Theta)$ is given
by Eq. (\ref{eqn:alphafin}).
\begin{figure}[t!]
\centering
\includegraphics[width=0.6\textwidth]{fig2.eps}
\caption{Here we plot the function $\alpha(\Theta)$, the coefficient of $H_{ij}(\mathbf{n},\mathbf{n}')$ as shown in Eq. (\ref{eqn:Hdecompalph}), as a function of the angle $\Theta$ between $\mathbf{n}$ and $\mathbf{n}'$.}
\label{fig:alpha}
\end{figure}
\subsection{ Special Case: Coincidence }
As a check of our calculation, we can solve for the two-point correlation function exactly in the case that $\mathbf{n}=\mathbf{n}'$. Using Eqs. (\ref{eqn:R}), (\ref{eqn:P}) and (\ref{eqn:HR}), the integral simplifies to
\begin{equation} H_{ij}(\mathbf{n},\mathbf{n}) = \frac{1}{4} \int d^2\Omega_{\mathbf{p}} \left[1 - \left(\mathbf{p}\cdot\mathbf{n})^2\right)\right]\left( \delta_{ij} - n_i n_j \right).\nonumber\end{equation}
\noindent We can solve this integral analytically, getting
\begin{equation} H_{ij}(\mathbf{n},\mathbf{n}) = \frac{2 \pi}{3} \left( \delta_{ij} - n_i n_j \right). \label{eqn:Hcoinc}\end{equation}
\noindent This corresponds to the limit of $\alpha(\Theta)(A^i A^j - B^i C^j)$ as $\mathbf{n}\rightarrow\mathbf{n}'$, with $\alpha(\Theta) = 2 \pi/(3 \Theta^2) + \mathcal{O}(\Theta^{-1})$ from Eq. (\ref{eqn:alphafin}). Inserting the coincidence limit (\ref{eqn:Hcoinc}) into the correlation function (\ref{eqn:2ptH}) yields the formula (\ref{eqn:2ptnn}) for the total rms angular fluctuations discussed in the introduction.
\section{ Spectrum of Angular Deflection Fluctuations \label{sec:spect}}
\subsection{ Overview }
In the previous section we computed the correlation function $\langle \delta n^i(\mathbf{n},t) \delta n^j(\mathbf{n}',t') \rangle$ as a function of the unit vectors $\mathbf{n}$ and $\mathbf{n}'$. However for many purposes it is more useful to perform a multipole decomposition of the angular deflection, and to compute the spectrum of fluctuations on different angular scales $l$, as is done with cosmic microwave background anisotropies. We decompose $\delta \mathbf{n}(\mathbf{n},t)$ as
\begin{equation} \delta \mathbf{n}(\mathbf{n},t) = \sum_{l m} \delta n_{E lm}(t) \mathbf{Y}^E_{lm}(\mathbf{n}) + \delta n_{B lm}(t) \mathbf{Y}^B_{lm}(\mathbf{n}), \label{eqn:exp}\end{equation}
\noindent where $\mathbf{Y}^E_{lm}$ and $\mathbf{Y}^B_{lm}$ are the electric- and magnetic-type transverse vector spherical harmonics defined by
\begin{equation} \mathbf{Y}^E_{lm}(\mathbf{n}) = (l(l+1))^{-1/2} \mathbf{\nabla} Y_{lm}(\mathbf{n}),\quad \mathbf{Y}^B_{lm}(\mathbf{n}) = (l(l+1))^{-1/2} (\mathbf{n}\times\mathbf{\nabla}) Y_{lm}(\mathbf{n}). \label{eqn:Ydefn}\end{equation}
\noindent We will show in this section that the statistical properties of the coefficients are given by
\begin{equation} \langle \delta n_{Q lm}(t) \: \delta n_{Q' l'm'}(t')^* \rangle = \delta_{Q Q'} \delta_{l l'} \delta_{m m'} \int_0^{\infty} df \cos[2\pi f (t - t')] S_{Q l}(f) \label{eqn:2ptfnl} \end{equation}
\noindent for $Q,Q' = E$ or $B$, for some spectrum $S_{Q l}(f)$, a function of frequency $f$ and of angular scale $l$. The formula (\ref{eqn:2ptfnl}) shows that different multipoles of the angular deflection are statistically independent, as required by spherical symmetry of the stochastic background. Also the electric-type and magnetic-type fluctuations are uncorrelated, as required by parity invariance of the stochastic background (see below).
The spectrum $S_{Q l}(f)$ is given by
\begin{equation} S_{Q l}(f) = \frac{4\pi}{2l+1} \theta_{rms}^2 \frac{\sigma(f)}{f} g_Q \alpha_l^{QQ}. \label{eqn:SQlf} \end{equation}
\noindent Here $\theta_{rms}^2$ is the total rms angular fluctuation squared, given by Eq. (\ref{eqn:2ptnn}) in the introduction. The function $\sigma(f)$ describes how the power is distributed in frequency. It is the same for all multipoles, is normalized according to $\int d(\, \ln f) \: \sigma(f) = 1$, and is given explicitly by Eq. (\ref{eqn:sigma}) in the introduction. The quantities $g_E$ and $g_B$ are the fraction of the total power in electric-type and magnetic-type fluctuations, and are $g_E = g_B = 1/2$. Finally the angular spectra $\alpha_l^{EE}$ and $\alpha_l^{BB}$ describe the dependence on angular scale, which is the same for all frequencies. They are normalized according to
\begin{equation} \sum_{l=2}^{\infty} \alpha_l^{QQ} = 1, \end{equation}
\noindent and are the same for $E$ and $B$ modes, $\alpha_l^{EE} = \alpha_l^{BB}$. This spectrum is plotted in Fig \ref{fig:alphaQ} and the first 10 values are listed in Table \ref{tab:alphaQ}. We note that these coefficients are well fit by the power law $\alpha^{EE}_l = 32.34 \: l^{-4.921}$.
Before proceeding with the derivation of the spectrum (\ref{eqn:2ptfnl}), we first derive from (\ref{eqn:2ptfnl}) the expression (\ref{eqn:thrms}) discussed in the introduction for the total fluctuation power. Squaring the expansion (\ref{eqn:exp}), taking an expected value, and then using (\ref{eqn:2ptfnl}) gives
\begin{align} \langle \delta \mathbf{n}(\mathbf{n},t)^2 \rangle &= \sum_{Q l m} \sum_{Q' l' m'} \mathbf{Y}^Q_{lm}(\mathbf{n}) \mathbf{Y}^{Q'}_{l'm'}(\mathbf{n})^* \langle \delta n_{Q lm}(t) \delta n_{Q' l'm'}(t')^* \rangle \nonumber\\ &= \sum_{Ql} \int_0^{\infty} \frac{\sigma(f)}{f} \sum_{m=-l}^l \left| \mathbf{Y}^Q_{lm}(\mathbf{n}) \right|^2 \theta_{rms}^2 \: \frac{4 \pi}{2l+1} \: g_Q \; \alpha_l^{QQ}. \end{align}
\noindent Using Uns$\ddot{\text{o}}$ld's theorem for vector spherical harmonics,
\begin{equation} \sum_{m=-l}^l \left| \mathbf{Y}^Q_{lm}(\mathbf{n}) \right|^2 = \frac{2l+1}{4\pi} ,\nonumber\end{equation}
\noindent gives
\begin{equation} \langle \delta \mathbf{n}(\mathbf{n},t)^2 \rangle = \sum_{Ql} \int_0^{\infty} \theta_{rms}^2 \: \frac{\sigma(f)}{f} \: g_Q \, \alpha_l^{QQ}, \end{equation}
\noindent which reduces to Eq. (\ref{eqn:thrms}). Note that using the normalization conventions for $\alpha_l^{QQ}$ and $\sigma(f)$ now gives $\langle \delta \mathbf{n}(\mathbf{n},t)^2 \rangle = \theta_{rms}^2 (g_E + g_B) = \theta_{rms}^2$, showing consistency of the definitions.
\subsection{ Derivation }
We now turn to a derivation of the spectrum (\ref{eqn:SQlf}). First we note that the vector spherical harmonics are transverse in the sense that $\mathbf{Y}^Q_{lm}(\mathbf{n})\cdot\mathbf{n} = 0$ for $Q=E,B$, and are orthogonal in the sense that
\begin{equation} \int d^2\Omega_{\mathbf{n}} Y^Q_{lmi}(\mathbf{n}) Y^{Q'i*}_{l'm'}(\mathbf{n}) = \delta_{Q Q'} \delta_{l l'} \delta_{m m'}. \nonumber\end{equation}
\noindent Using this orthogonality property, we can extract the coefficients of the expansion (\ref{eqn:exp})
\begin{equation} \delta n_{Q lm}(t) = \int d^2\Omega_{\mathbf{n}} \delta n_i(\mathbf{n},t) Y^{Qi*}_{lm}(\mathbf{n}). \nonumber\end{equation}
\noindent Thus we can write for the correlation function between two of these coefficients
\begin{equation} \langle \delta n_{Qlm}(t) \delta n_{Q'l'm'}(t')^* \rangle = \int d^2\Omega_{\mathbf{n}} d^2\Omega_{\mathbf{n}'} Y^{Q*}_{lmi}(\mathbf{n}) Y^{Q'}_{l'm'j}(\mathbf{n}') \langle \delta n^i(\mathbf{n},t) \delta n^j(\mathbf{n}',t') \rangle, \label{eqn:twoptlm} \end{equation}
\noindent or more explicitly, using Eq. (\ref{eqn:2ptH})
\begin{equation} \langle \delta n_{Qlm}(t) \delta n_{Q'l'm'}(t')^* \rangle = \frac{3 H_0^2}{16 \pi^3} \int_0^{\infty} df \cos[2\pi f (t - t')] \: \frac{\Omega_{\rm gw}(f)}{f^3} \: C_{Q l m Q' l' m'}, \label{eqn:twopt} \end{equation}
\noindent where
\begin{equation} C_{Q l m Q' l' m'} = \int d^2\Omega_{\mathbf{n}} d^2\Omega_{\mathbf{n}'} Y^{Q*}_{lmi}(\mathbf{n}) Y^{Q'}_{l'm'j}(\mathbf{n}') H_{ij}(\mathbf{n},\mathbf{n}'). \label{eqn:C}\end{equation}
We now argue that the EB cross-correlation vanishes. From Eq. (\ref{eqn:Ydefn}), we see that $\mathbf{Y}^E_{lm}(\mathbf{n})$ has the same parity under $\mathbf{n}\rightarrow-\mathbf{n}$ as $Y_{lm}(\mathbf{n})$, while the parity of $\mathbf{Y}^B_{lm}(\mathbf{n})$ is opposite. From section \ref{sec:basis} above $H_{ij}(\mathbf{n},\mathbf{n}')$ is invariant under both $\mathbf{n}\rightarrow-\mathbf{n}$ and $\mathbf{n}'\rightarrow-\mathbf{n}'$. Thus, if $Q=E$, $Q' = B$ in Eq. (\ref{eqn:twopt}), the integral will be symmetric under $\mathbf{n}\rightarrow-\mathbf{n}$ but antisymmetric under $\mathbf{n}'\rightarrow-\mathbf{n}'$, causing the integral over $d^2 \Omega_{\mathbf{n}'}$ to vanish. Therefore, $EB$ cross correlations vanish, and we need only calculate the $EE$ and $BB$ correlation functions.
\subsubsection{ EE correlation }
Inserting the definition (\ref{eqn:Ydefn}) of the electric vector spherical harmonics and the formula (\ref{eqn:Hdecompalph}) for $H_{ij}$ into Eq. (\ref{eqn:C}) and integrating by parts, we obtain
\begin{equation} C_{E l m E' l' m'} = \frac{1}{l(l+1)} \int d^2\Omega_{\mathbf{n}} d^2\Omega_{\mathbf{n}'} Y^*_{lm}(\mathbf{n}) Y_{l'm'}(\mathbf{n}') \beta^{EE}(\Theta), \label{eqn:C2} \end{equation}
\noindent where the function $\beta^{EE}$ is given by
\begin{equation} \beta^{EE}(\Theta) = \nabla_i \nabla'_j \left[ H_{ij}(\mathbf{n},\mathbf{n}') \right] = \nabla_i \nabla'_j \left\{\alpha(\Theta) \left[ A_i A_j - B_i C_j \right]\right\}. \label{eqn:bEEdef1}\end{equation}
\noindent Here $\nabla_i$ and $\nabla'_j$ denote normal three dimensional derivatives with respect to $\mathbf{x}$ and $\mathbf{x}'$, where $\mathbf{n} = \mathbf{x}/|\mathbf{x}|$ and $\mathbf{n}' = \mathbf{x}'/|\mathbf{x}'|$. Integration by parts on the unit sphere of this derivative operator is valid as long as the radial component of the integrand vanishes, from the identity $\nabla_i v^i = \partial_r v^r + 2 v_r/r + \nabla_A v^A$, where $\nabla_A$ denotes a covariant derivative on the unit sphere. It can be checked that the radial components do vanish in the above computation.
Next, we expand the function $\beta^{EE}$ in terms of Legendre polynomials, and use the spherical harmonic addition theorem, which gives
\begin{align} \beta^{EE}(\Theta) &= \sum_l \beta^{EE}_l P_l(\cos\Theta) \nonumber\\ &= \sum_{lm} \frac{4 \pi}{2l+1} \beta^{EE}_l Y_{lm}(\mathbf{n}) Y_{lm}(\mathbf{n}')^*\label{eqn:bEEexp}\end{align}
\noindent Inserting this into Eq. (\ref{eqn:C2}) and using the orthogonality of spherical harmonics gives
\begin{equation} C_{E l m E' l' m'} = \delta_{l l'} \delta_{m m'} \frac{1}{l (l+1)} \frac{4 \pi}{2l+1} \beta^{EE}_l. \end{equation}
\noindent Inserting this into Eq. (\ref{eqn:twopt}) now yields the correlation function given by Eqs. (\ref{eqn:2ptfnl}) and (\ref{eqn:SQlf}), and using the definitions (\ref{eqn:2ptnn}) and (\ref{eqn:sigma}) of $\theta_{rms}^2$ and $\sigma(f)$ allows us to read off the electric multipole spectrum
\begin{equation} g_E \: \alpha_l^{EE} = \frac{3}{4 \pi l (l+1)} \beta_l^{EE}. \label{eqn:bEEalph}\end{equation}
\noindent We will show below that $g_E = 1/2$.
It remains to explicitly evaluate the function $\beta^{EE}(\Theta)$ defined in Eq. (\ref{eqn:bEEdef1}) and evaluate its expansion coefficients. We have
\begin{equation} \beta^{EE}(\Theta) \equiv \nabla_i \nabla'_j \left[\alpha(\Theta)T^{ij}\right] = \left[\nabla_i \nabla'_j \alpha(\Theta)\right] T^{ij} + \left[\nabla_i \alpha(\Theta)\right] \left(\nabla'_j T^{ij}\right) + \left[\nabla'_j \alpha(\Theta)\right] \left(\nabla_i T^{ij}\right) + \alpha(\Theta)\left(\nabla_i \nabla'_j T^{ij}\right), \label{eqn:bEEdef}\end{equation}
\noindent where we have defined $T^{ij} = \left(A^i A^j(\mathbf{n},\mathbf{n}') - B^i C^j(\mathbf{n},\mathbf{n}')\right)$. Using $A^i = \epsilon^{ijk}n_j n'_k$, $B^i = (\mathbf{n}\cdot\mathbf{n'}) n^i - n'^i$, $C^i = (\mathbf{n}\cdot\mathbf{n'}) n'^i - n^i$, we can write the tensor $T^{ij}$ in Cartesian coordinates as
\begin{equation} T^{ij} = \epsilon^{ikl} \epsilon^{jrs} n_k n'_l n_r n'_s - \left( (\mathbf{n}\cdot\mathbf{n}') n^i - n'^i \right) \left( (\mathbf{n}\cdot\mathbf{n}') n'^j - n^j \right). \nonumber\end{equation}
\noindent Using $\nabla_i n_j = \delta_{ij} - n_i n_j$, $\nabla'_i n'_j = \delta_{ij} - n'_i n'_j $, $\nabla'_i n^j = \nabla_i n'^j = 0$, and $\nabla_l \epsilon^{ijk} = \nabla'_l \epsilon^{ijk} = 0$, we calculate the derivatives
\begin{align} \nabla_i T^{ij} = \left(1 - 3 (\mathbf{n}\cdot\mathbf{n}')\right) \left( (\mathbf{n}\cdot\mathbf{n}') n'^j - n^j \right),& \quad\quad \nabla'_j T^{ij} = \left(1 - 3 (\mathbf{n}\cdot\mathbf{n}')\right) \left( (\mathbf{n}\cdot\mathbf{n}') n^i - n'^i \right), \nonumber\\ \nabla_i \nabla'_j T^{ij} =&-9 (\mathbf{n}\cdot\mathbf{n}')^2 + 2 (\mathbf{n}\cdot\mathbf{n}') + 3 . \label{eqn:dT}\end{align}
For the gradients of $\alpha$, we use the fact that $\cos(\Theta) = \mathbf{n}\cdot\mathbf{n'}$, so that $- \sin(\Theta) \nabla_i \Theta = n'_i - (\mathbf{n}\cdot\mathbf{n}') n_i$, and similarly for $\nabla'_j$. Thus, we find
\begin{align} \nabla_i \alpha(\Theta)&= - \alpha'(\Theta) \frac{n'_i - (\mathbf{n}\cdot\mathbf{n}') n_i}{\sin(\Theta)}, \quad\quad \nabla'_j \alpha(\Theta)= - \alpha'(\Theta) \frac{n_j - (\mathbf{n}\cdot\mathbf{n}') n'_j}{\sin(\Theta)}\nonumber\\
\nabla_i \nabla'_j \alpha(\Theta)&= \alpha'(\Theta) \left\{ \frac{\delta_{ij} - n_i n_j - n'_i n'_j + (\mathbf{n}\cdot\mathbf{n}')n_i n'_j}{-\sin(\Theta)} + \frac{\cos(\Theta)\left[n'_i - (\mathbf{n}\cdot\mathbf{n}') n_i\right]\left[ n_j - (\mathbf{n}\cdot\mathbf{n}') n'_j \right]}{-\sin^3(\Theta)} \right\} \nonumber\\
&+ \alpha''(\Theta) \frac{\left[n'_i - (\mathbf{n}\cdot\mathbf{n}') n_i\right]\left[ n_j - (\mathbf{n}\cdot\mathbf{n}') n'_j \right]}{\sin^2(\Theta)}. \label{eqn:dalph} \end{align}
\noindent Plugging Eqs. (\ref{eqn:dT}) and (\ref{eqn:dalph}) into Eq. (\ref{eqn:bEEdef}), we get
\begin{align} \beta^{EE}(\Theta) =& \left[ -9 \cos^2(\Theta) + 2 \cos(\Theta) + 3 \right] \alpha(\Theta) - \sin^2(\Theta)\alpha''(\Theta) \nonumber \\& + \left[ 1 - 6 \cos(\Theta)\right] \sin(\Theta)\alpha'(\Theta). \label{eqn:brack2}\end{align}
\noindent Next, we insert the expression (\ref{eqn:alphafin}) for $\alpha(\Theta)$ to obtain
\begin{equation} \beta^{EE}(\Theta) = \frac{4 \pi}{3} \Big( 4 + \left( 1 - \cos\Theta \right) \left\{ 12 \ln\left[ \sin(\Theta/2) \right] - 1 \right\} \Big). \label{eqn:beefin}\end{equation}
\noindent We numerically compute the coefficients $\beta^{EE}_l$ of the Legendre polynomial expansion (\ref{eqn:bEEexp}) of $\beta^{EE}(\Theta)$, and from them compute $\alpha_l^{EE}$ using Eq. (\ref{eqn:bEEalph}). The result is plotted in Fig. \ref{fig:alphaQ} and tabulated in table \ref{tab:alphaQ}.
\subsubsection{ BB correlation }
We now calculate the $BB$ correlation in a similar manner to the $EE$ case above. Inserting into Eq. (\ref{eqn:C}) the definition (\ref{eqn:Ydefn}) of magnetic vector spherical harmonics and integrating by parts, we find
\begin{equation} C_{B l m B l' m'} = \frac{1}{l (l+1)} \int d^2\Omega_{\mathbf{n}} d^2\Omega_{\mathbf{n}'} Y^*_{lm}(\mathbf{n}) Y_{l'm'}(\mathbf{n}') \beta^{BB}(\Theta), \nonumber\end{equation}
\noindent where
\begin{equation} \beta^{BB}(\Theta) = \nabla_l \nabla'_p \left[\epsilon_{ikl}\epsilon_{jmp} n_k n'_m \alpha(\Theta) T_{ij} \right]. \label{eqn:bBB}\end{equation}
\noindent As before, we can derive from here the form (\ref{eqn:2ptfnl}) and (\ref{eqn:SQlf}) of the spectrum, with $\alpha_l^{BB}$ given by
\begin{equation} g_B \: \alpha_l^{BB} = \frac{3}{4 \pi l (l+1)} \beta_l^{BB}. \nonumber\end{equation}
We now show that $\beta^{BB}(\Theta) = \beta^{EE}(\Theta)$, from which it follows that $g_E = g_B = 1/2$ and that $\alpha_l^{EE} = \alpha_l^{BB}$. To see this we evaluate the cross products in (\ref{eqn:bBB}) using $\mathbf{n}\times\mathbf{A} = \mathbf{B}$, $\mathbf{n}\times\mathbf{B} = -\mathbf{A}$, $\mathbf{n}'\times\mathbf{C} = \mathbf{A}$. This gives
\begin{equation} \epsilon_{ikl} \epsilon_{jmp} n_k n'_m H_{ij} = H_{lp}, \nonumber\end{equation}
\noindent and using the definitions (\ref{eqn:bEEdef1}) and (\ref{eqn:bBB}) of $\beta^{EE}$ and $\beta^{BB}$, it follows that $\beta^{BB} = \beta^{EE}$.
\acknowledgments
LGB acknowledges the support of the NSF Graduate Fellowship Program. EF thanks the Theoretical Astrophysics Including Relativity Group at Caltech, and the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge, for their hospitality as this paper was being written. This research was supported in part by NSF grants PHY-0757735 and PHY-0555216.
\input{ms.bbl}
|
{
"timestamp": "2010-12-07T02:04:01",
"yymm": "1009",
"arxiv_id": "1009.4192",
"language": "en",
"url": "https://arxiv.org/abs/1009.4192"
}
|
\section{Introduction}\label{rim_intro}
Circumstellar discs result from the conservation of angular momentum
of a collapsing cloud that forms a star at its centre. Discs of
various masses and shapes are observed by direct imaging or
spectroscopically from low mass brown dwarfs (e.g.,
\citealt{Mohanty2004ApJ...609L..33M,Jayawardhana2003AJ....126.1515J,Natta2001A&A...376L..22N})
to massive O stars \citep{Cesaroni2007prpl.conf..197C}. Spectral
Energy Distributions (SED) are dominated by dust grain emission and
are influenced by disc geometry, dust mass, and grain properties.
Herbig~Ae stars (2--4 M$_{\odot}$) are among the most studied
pre-main-sequence stars surrounded by discs because of their isolated
nature and intrinsic brightness
\citep{Natta2007prpl.conf..767N}. Their SEDs can be classified into
two groups according to the ratio between the far-infrared to
near-infrared fluxes \citep{Meeus2001A&A...365..476M}. Group I objects
show much stronger far-infrared to near-infrared fluxes than their
group II counterparts. Early versions of the so-called puffed-up rim
model succeeded to explain the dichotomy
\citep{Dullemond2001ApJ...560..957D,Dominik2003A&A...398..607D}.
Group II objects are discs that absorb most of the stellar flux within
a geometrically thin but optically thick inner rim that faces directly
the star, depriving the outer disc of radiation. As a consequence the
cold outer disc is geometrically flatter and emits only weakly in the
far-infrared. On the contrary, the rim in group I objects does not
block enough stellar radiation and the outer disc flares, emitting
strongly in the far-infrared. 2D dust radiative transfer models that
include dust scattering and hydrostatic disc structure assuming that
the gas temperature is equal to the dust temperature, which is derived
from dust energy balance, underestimate the flux in the near-infrared
around 3~$\mu$m
\citep{Meijer2008A&A...492..451M,Vinkovic2006ApJ...636..348V}. \citet{Vinkovic2006ApJ...636..348V}
invoke the presence of an optically thin halo above the inner disc to
explain the excess 3$\mu$m emission. The problem arises from the
insufficient height of the inner rim: a low rim results in a small
emitting area. \citet{Acke2009A&A...502L..17A} require that their
inner rims have to be 2 to 3 times higher than the hydrostatic
solution that fits the SEDs using a Monte-Carlo radiative transfer
code. Models that incorporate detailed physics of grain evaporation
with grain size distribution change the structure of the inner rim and
help to fit simultaneously the SED and near-infrared interferometric
data (e.g.,
\citealt{Isella2006A&A...451..951I,Tannirkulam2007ApJ...661..374T,Kama2009A&A...506.1199K}). The
structure of the inner rim impacts the overall shape of the SEDs up to
a few tens of microns. Initially discovered in the SED of Herbig~Ae
discs, inner rim emission is also detected in discs around classical
T~Tauri stars \citep{Muzerolle2003ApJ...597L.149M}. Observationally,
the inner disc geometry is varied. Studies with sufficient
baseline-coverage show that either an extra optical thin component is
required
\citep{Tannirkulam2008ApJ...689..513T,Benisty2010A&A...511A..75B} or
that the rim has a skewed asymmetric geometry
\citep{Kraus2009A&A...508..787K}. A detailed modelling of these
observations is beyond the scope of this paper.
All previous studies of inner disc rims share the assumption that the
gas and dust temperatures are equal to simplify the radiative
transfer. In these studies, the temperatures are derived from the dust
energy balance. This assumption is valid in the inner optically thick
parts of disc midplanes where gas and dust are thermally coupled via
inelastic collisions, a phenomenon called thermal accommodation
\citep{Tielens2005pcim.book.....T}. In all other parts, the gas is
mostly heated by hot photoelectrons from dust grains and Polycyclic
Aromatic Hydrocarbons and attains higher temperatures than the
dust. The inner rim is naturally higher than when the gas and dust
temperature are equal and the overall disc surfaces are more
vertically extended \citep*{Woitke2009A&A...501..383W}. In discs where
the gas and dust are well mixed (i.e. without dust settling), the
change in geometry can affect the SEDs.
In this paper we explore the possibility that inner rims are much
higher when the gas and dust temperatures are computed separately and
self-consistently. A detailed study of inner and outer
thermo-chemical disc structures together with emission lines in
Herbig~Ae discs is presented in \citet{Kamp2010A&A...510A..18K}. In
this paper we focus on the inner disc structure of well-mixed discs
and the images and spectral signature of inner rims in the infrared.
\begin{figure*}
\centering
{\includegraphics[angle=0,width=18cm]{herbig_fiducial2.eps}}
\caption{The upper right and left panels show the dust and gas
temperature structures respectively for the fiducial model. The
dust temperature contours are overplotted for $T_{\mathrm{dust}}=$
50, 100, 500, and 1000~K in both panels. The location of
$A_{\mathrm{V}}=$~5 and 50 are also shown. The entire outer disc
at $R>100$~AU is located at $A_{\mathrm{V}}<$~5. The lower-right
panel show the disc density structure for the entire disc (up to
$R_{\mathrm{out}}=$~300~AU). The gas temperature over dust
temperature ratio for the fiducial model in the lower-left
panel. The gas can be more than $\sim$~5 times warmer than the
dust in the upper disc atmosphere but
$T_{\mathrm{gas}}=T_{\mathrm{dust}}$ at
$A_{\mathrm{V}}>(1-5)$. The ratio
$T_{\mathrm{gas}}/T_{\mathrm{dust}}$ is explicitly truncated at 5
in the figure.}
\label{fig_fiducial}
\end{figure*}
This paper is organized as follow. We briefly describe the {\sc
ProDiMo} code and recent additions to the code such as the SED and
image calculation in Sect.~\ref{prodimo}. We proceed with a
presentation of our fiducial disc model, the disc parameters that are
varied, and the fixed-structure disc model in
Sect.~\ref{model_description}. We then present and discuss the results
of our simulations when the gas and dust temperatures are equal
(thermal-coupled models) and when they are computed self-consistently
(thermal-decoupled models), varying several disc parameters in
Sect.~\ref{results_discussion}. Finally, we conclude about the inner
disc structures and their effects on the SEDs in
Sect.~\ref{conclusion}.
\begin{center}
\begin{table*}
\caption{Disc parameters. When a parameter has multiple entries, the values in bold correspond to the values of the fiducial model.}\label{tab_DiscParameters}
\begin{tabular}{lll}
\hline
stellar mass & $M_*$ & {\bf 2.2}~M$_\odot$ \\
stellar luminosity &$L_*$ & {\bf 32}~L$_\odot$ \\
effective temperature & $T_{\mathrm {eff}}$ & {\bf 8600}~K\\
disc mass & $M_{\mathrm{disc}}$ & {\bf 10}{\boldmath $^{-2}$}, 10$^{-3}$, 10$^{-4}$ M$_\odot$ \\
disc inner radius & $R_{\mathrm {in}}$ & {\bf 0.5}, 1 , 10~AU\\
disc outer radius & $R_{\mathrm {out}}$ & {\bf 300}~AU\\
vertical Column density power law index & $\epsilon$ & 1, 1.5, {\bf 2}\\
gas to dust mass ratio & $\delta$ & {\bf 100}\\
dust grain material mass density & $\rho_{\mathrm{dust}}$ & {\bf 2.5} g cm$^{-3}$ \\
minimum dust particle size & $a_{\mathrm{min}}$ & {\bf 0.05} $\mu$m\\
maximum dust particle size & $a_{\mathrm{max}}$ & 10, {\bf 50}, 200 $\mu$m\\
dust size distribution power law & $p$ & {\bf 3.5}\\
H$_2$ cosmic ray ionization rate & $\zeta_{\mathrm{CR}}$ & {\bf 1.7 $\times$ 10}{\boldmath $^{-17}$} s$^{-1}$\\
ISM UV field w.r.t. Draine field & $\chi$ & {\bf 0.1}\\
abundance of PAHs relative to ISM & $f_{\rm PAH}$ & {\bf 0.01}, 0.1\\
$\alpha$ viscosity parameter & $\alpha$ & {\bf 0.0}\\
\hline
\end{tabular}
\ \\
\end{table*}
\end{center}
\section{{\sc ProDiMo} code description}\label{prodimo}
{\sc ProDiMo} is designed to compute self-consistently the (1+1)D
hydrostatic disc structure in thermal balance and kinetic chemical
equilibrium. The disc can be active or passive depending on the
choice of the viscosity parameter $\alpha$. We first provide here a
brief description of {\sc ProDiMo}. Interested readers are referred to
\citet*{Woitke2009A&A...501..383W} to find detailed explanations of
the physics implemented in the code. We continue by giving the
rational for the ``soft-inner edge'' surface density profile
implemented in {\sc ProDiMo}. We finish by explaining the computation
of Spectral Energy Distributions and images, which is a new feature of
the code.
\subsection{General description}
The code iterates the computation of the disc density, gas and dust
temperature, and chemical abundance structure until successive
iterations show less than 1\% change.
The iteration starts with a given hydrostatic disc structure computed
from the previous iteration or from an assumed structure. The dust
radiative transfer module computes the dust temperatures and the disc
local mean specific intensities $J_\nu(r,z)$. The specific intensities
are used to calculate the photochemical rates. The 2D dust
radiative-transfer module of {\sc ProDiMo} has been benchmarked
against other codes that use other methods such as Monte-Carlo
\citep{Pinte2009A&A...498..967P}. The continuum radiative transfer
includes absorption, isotropic scattering, and thermal
emission. The grains are assumed spherical. The lack of anisotropic
scattering prevents us to calculate completely accurate images. In
case of anisotropic scattering, one expects the far upper part of the
disc rim to be dimmer (backward scattering) than the near lower part
(forward scattering). However, isotropic scattering is sufficiently
accurate for generating precise SEDs.
The grains have sizes between $a_{\mathrm{min}}$ and
$a_{\mathrm{max}}$ and follow a power-law size distribution with index
$p$. The absorption and scattering efficiencies are computed using Mie
theory for compact spherical grains. Grains of different sizes are
assumed to have the same temperature. This limitation in our model
does not prevent the comparison between self-consistently computed gas
temperature models (thermal-decoupled models) and models with
$T_{\mathrm{gas}}=T_{\mathrm{dust}}$ (thermo-coupled models). In this
paper, the dust composition and size distribution are constant
throughout the disc. We adopt the interstellar dust optical constants
of \citet{Laor1993ApJ...402..441L} for amorphous grains since we are
only interested in modelling featureless broad SEDs.
Once the dust temperature $T_{\mathrm{dust}}(r,z)$ and the continuum
mean specific intensities $J_\nu(r,z)$ are known, the gas temperature
and chemical concentrations are consistently calculated assuming
thermal balance and kinetic chemical equilibrium. The chemistry
network includes 71 gas and ice species. The chemical reactions
comprise photo-reactions (photoionization and photodissociation) with
rates computed using cross-sections and specific intensities $J_\nu$
calculated by the 2D radiative transfer module, gas phase reactions,
gas freeze-out and evaporation (thermal desorption, photodesorption,
and cosmic-ray induced desorption), and H$_2$ formation on grain
surfaces. The chemical rates are drawn from the {\sc UMIST} 2007
database \citep{Wooddall2007A&A...466.1197W} augmented by rates from
the {\sc NIST} chemical kinetic website. The photo-cross-sections are
taken from \citet{vanDishoeck2008}. The H$_2$ formation
on grain surfaces follows the prescription of
\citet{Cazaux2004ApJ...604..222C} with the most recent grain surface
parameters.
The gas can be heated by line absorption and cooled by line emission.
The line radiative transfer is computed via the escape probability
formalism. The gas is also heated by interactions with hot
photoelectrons from dust grains and Polycyclic Aromatic Hydrocarbons
and with cosmic rays. Thermal accommodation on grain surfaces can
either heat or cool the gas depending on the temperature difference
between the gas and the dust
\citep{Tielens2005pcim.book.....T}. Thermal accommodation dominates at
high densities and optical depths, driving the gas and dust towards
the same temperatures.
Finally, the gas temperature and molecular weight determine the local
pressure at each grid point, which is used to modify the disc structure
according to the vertical hydrostatic equilibrium. The new disc structure is
compared to the previous one to check for convergence.
\subsection{Soft Inner edge}
The radial dependence of the vertical column density is assumed to
deviate from a power-law in the innermost layers, and is implemented
according to the "soft-edge"-description of \citealp*[(see Section
3.1)]{Woitke2009A&A...501..383W}. We assume that the gas has been pushed
inwards by the radial pressure gradients around the inner edge, and
has spun up according to angular momentum conservation, until the
increased centrifugal force balances the radial pressure gradient. The
procedure results in a surface layer in radial hydrostatic equilibrium
at constant specific angular momentum where the column density
increases gradually from virtually zero to the desired value at the
point where we start to apply the power-law. The thickness of this
"soft" edge results to be typically a few 10\% of the inner
radius $R_{\mathrm{in}}$.
\begin{figure}
\centering
\resizebox{\hsize}{!}
{\includegraphics[width=18cm]{herbig_fiducial_soft_edge_thermodecoupled_photospheric_height_inner_disk.ps}}
\caption{Inner rim density structure and photospheric height
($A_{\mathrm{V}}$=1 contour) for the fiducial model (``soft-edge''
and $T_{\mathrm{gas}}$ computed by thermal balance).}
\label{inner_rim_photospheric_height}
\end{figure}
\subsection{Spectral Energy Distribution calculation}
\begin{figure}
\centering
\resizebox{\hsize}{!}{\includegraphics[]{dust_opac_3sizes.ps}}
\caption{ The dust opacity computed by Mie theory for the three values
of $a_{\mathrm{max}}$ (10, 50, 200 $\mu$m) and a power-law dust grain
size distribution (index 3.5). A line showing an opacity with a
$\lambda^{-1}$ opacity law is added for comparison.}
\label{fig_opacity}
\end{figure}
Based on the results of the continuum dust radiative transfer solution
(see \citet{Woitke2009A&A...501..383W}, Sect.\ 4) we have developed a
new {\sc ProDiMo} module for SED and image calculation based on formal
solutions of the dust continuum radiative transfer equation along a
bundle of parallel rays that cross the disc at given inclination angle
$i$ with respect to the disc rotation axis. The rays start at an
image plane put safely outside of the disc. The ray direction
$\overrightarrow{n}$ and the 3D origin of the image plane
$\overrightarrow{p_0}$ are given by
\begin{equation}
\overrightarrow{n}=\left(\begin{array}{c}
\sin i\\
0\\
\cos i\end{array}\right)
\end{equation}
and
\begin{equation}
\overrightarrow{p_{0}}=R\ \overrightarrow{n},
\end{equation}
where $R$ is a large enough distance to be sure that the plane cannot
intersect the disc. The plane is aligned along the two perpendicular
unit vectors
\begin{equation}
\overrightarrow{n_{z}}=\left(\begin{array}{c}
0\\
1\\
0\end{array}\right)
\end{equation}
and
\begin{equation}
\overrightarrow{n_{y}}=\left(\begin{array}{c}
-\cos i\\
0\\
\sin i\end{array}\right).
\end{equation}
The rays are organized in log-equidistant concentric rings in the
image plane, using polar image coordinates ($r$,$\theta$). The 3D
starting point of one ray $\overrightarrow{x_0}$, with image
coordinates $x=r \sin(\theta)$ and $y=r \cos(\theta)$, is given by
\begin{equation}
\overrightarrow{x_{0}}=\overrightarrow{p_{0}}+x\ \overrightarrow{n_{x}}+y\ \overrightarrow{n_{y}}
\end{equation}
From these ray starting points $\vec{x}_0$, the radiative transfer
equation is solved backwards along direction $-\overrightarrow{n}$,
using an error-controlled ray integration scheme as described in
Woitke, Kamp, \& Thi (2009, Sect 4.2). The source function and opacity
structure in the disc have been saved from the previous run of {\sc
ProDiMo}'s main dust radiative transfer. The resulting intensities
at the location of the image plane, $I_\nu(r,\theta)$, are used to
directly obtain monochromatic images.
\begin{figure*}
\centering
{\includegraphics[angle=0,width=18cm,height=19cm]{rim_seds_aug10.ps}}
\caption{Spectral energy distribution computed for the various
models. The upper left panel shows the fiducial model
($M_{\mathrm{disc}}$=10$^{-2}$ M$_\odot$) with decreasing surface
power index $\epsilon$ and in dot-dash purple line the SEDs
computed for the fixed disc structure models with the flaring
index $p$=1.25 and $\epsilon$=2. The upper-right panel shows the
SED for the fixed-structure model with $\epsilon=2$ and three
values of the flaring index $p$. The upper-left panel is for discs
seen face-on. All the other panels are for discs seen at 45
degree. The other panels show the effect of varying the disc total
mass (10$^{-2}$, 10$^{-3}$, and 10$^{-4}$ M$_\odot$), the grain
upper size limit ($a_{\mathrm{max}}=$~10~$\mu$m and
$a_{\mathrm{max}}=$~200~$\mu$m), the inner disc radius (0.5, 1,
and 10~AU), and the effect of adopting a ``sharp-edge'' density
profile compared to a ``soft-edge'' density profile. SEDs in solid
lines are computed in the case $T_{\mathrm{gas}}$ is computed by
thermal balance while SEDs in dash lines are computed assuming
$T_{\mathrm{gas}}=T_{\mathrm{dust}}$. The SED of the fiducial
model is shown in red. The input stellar spectrum is plotted in
light blue. The spectral sampling is much higher for the stellar
spectrum than for the SEDs. The flux in the UV and optical is the
sum of the direct stellar flux and the stellar flux scattered by
the disc.}
\label{fig_SEDs}
\end{figure*}
To retrieve the spectral flux under inclination $i$, we integrate over
the image plane as
\begin{equation}
F_\nu = \frac{1}{d^2} \int I_\nu(r,\theta) r dr d\theta
\end{equation}
where $d$ is the object's distance. We use $N_\theta=160$ and $N_r=200$
rays. Due to the log-spacing of the concentric rings, the method
safely resolves even tiny structures originating from the inner rim.
\begin{figure*}
\centering
{\includegraphics[width=18cm]{panels_fiducial_hard_edge_coupled_decoupled_fixed.eps}}
\caption{Inner rim density structure for the fiducial model but with
different disc structure prescriptions as indicated above each
panel.}
\label{inner_rim_structures}
\end{figure*}
\begin{figure*}
\centering
{\includegraphics[width=18cm]{panels_images_1.98mu_2.eps}}
\caption{Image of the inner disc at 1.98 $\mu$m for the fiducial
model ($M_{\mathrm{disc}}$=10$^{-2}$ M$_\odot$, $\epsilon=2$) for
the two possible combinations of inner edge prescriptions and two
ways to compute the gas temperature and for the fixed-structure
disc structure. In the fixed structure case, the scale-height is
0.72~AU at radius 10~AU and the flaring index is 1.25. The disc
is seen at an inclination of 45 degree. The inner radius was set
at 0.5~AU. We can see a second rim at 1--2~AU, which is the most
pronounced in the case of soft-edge and gas temperature computed
by thermal balance. The emission from the second bump contributes
significantly to the total flux. The fixed-disc model has a
flaring index of $p=$~1.25.}
\label{fig_image_1.98_rim}
\end{figure*}
\begin{figure*}
\centering
{\includegraphics[width=18cm]{panels_images_5mu_2.eps}}
\caption{Same as Fig.~\ref{fig_image_1.98_rim} but the images were
generated at 4.96 $\mu$m}
\label{fig_image_4.96_rim}
\end{figure*}
\section{Disc parameters}\label{model_description}
\subsection{Fiducial model}\label{fiducial_model_description}
\begin{figure*}
\centering
{\includegraphics[width=18cm,angle=0]{surface_density_gradient_inner_rim4.eps}}
\caption{Inner rim density structure for the fiducial model
($M_{\mathrm{disc}}$=10$^{-2}$ M$_\odot$) with surface power index
$\epsilon=$1.5 and $\epsilon=$1. The left panels are the models
with $T_{\mathrm{gas}}=T_{\mathrm{dust}}$. The right panels are
the models with $T_{\mathrm{gas}}$ computed by gas thermal
balance. These models show a much taller rim and also a secondary
rim at $\sim$~1.2 AU.}
\label{fig_surf_density_inner_rim}
\end{figure*}
\begin{figure*}
\begin{minipage}[b]{0.48\linewidth}
\centering
{\includegraphics[scale=0.48]{herbig_fiducial_DDN_density.ps}}
\end{minipage}
\begin{minipage}[b]{0.48\linewidth}
\centering
{\includegraphics[scale=0.48]{herbig_fiducial_DDN_photosphere.ps}}
\end{minipage}
\begin{minipage}[b]{0.48\linewidth}
\centering
{\includegraphics[scale=0.48]{herbig_fiducial_DDN_image.ps}}
\end{minipage}
\begin{minipage}[b]{0.48\linewidth}
\centering
{\includegraphics[scale=0.48]{rim_seds_fiducial_DDN.ps}}
\end{minipage}
\caption{\label{fig_DDN} Inner disc density structure (upper-left
panel), photospheric height (upper-right panel), near-infrared image
(lower-left panel), and comparison between the SED for the fiducial
model (lower-right panel) but with M$_*$=2.5 M$_\odot$,
$T_{\mathrm{eff}}$=10,000~K, and $L_*$=47 L$_\odot$. We show the
$L_*=$~32~$L_\odot$ stellar spectrum only. The red-dashed line in the
upper-left panel encloses the region where the gas temperature is
higher than 1000~K. Notice the large amount of scattered light in
the blue part of the visible range in the SEDs caused by the
extremely flaring inner disc atmosphere for the more luminous star.}
\end{figure*}
We modelled circumstellar discs around one typical Herbig~Ae star with
effective temperature of 8600~K, mass of 2.2~M$_{\odot}$, and
luminosity of 32~L$_{\odot}$. In this study, we did not try to fit the
SED of any particular object. The input stellar spectrum is taken from
the {\sc PHOENIX} database of stellar spectra
\citep{Brott2005ESASP.576..565B}.
The discs are illuminated from all sides by a low interstellar medium
UV flux ($\chi=0.1$) in addition to the stellar flux. We chose a low
flux because previous studies do not take interstellar UV flux into
account. The stellar flux dominates over the standard interstellar
flux in the inner disc by orders of magnitude.
The abundance of PAH ($f_{\mathrm{PAH}}$) in discs controls the gas
temperature and the disc flaring, which in turn affects the dust
temperature. The exact value of $f_{\mathrm{PAH}}$ can only be constrained by
a simultaneous fit to the broad SED and the PAH features for a specific
object. We chose an arbitrary low abundance (1\%) and tested the effects on
the SED for the fiducial model with $f_{\mathrm{PAH}}$=0.1.
The only parameters that were allowed to change are the disc total
mass ($M_{\mathrm{disc}}=$~10$^{-4}$, 10$^{-3}$, and 10$^{-2}$
M$_\odot$), the surface power-law index ($\Sigma=\Sigma_o \times
(r/r_o)^{-\epsilon}$ with $\epsilon=$~1.0, 1.5, and 2.0), the disc
inner radius ($R_{\mathrm{in}}=$~0.5, 1 , 10~AU), and the maximum
grain radius ($a_{\mathrm{max}}=$~10, 50, 200~$\mu$m).
We chose $\epsilon=2$ as our fiducial value for the surface density
profile index. Surface density profiles as steep as $r^{-2}$ for the
inner disc are a signature of self-gravitating discs
\citep{Rice2009MNRAS.396.2228R}. However, we consider discs with
maximum disc mass over star mass ratio of 4.5 $\times$ 10$^{-3}$. A
recent derivation of the Solar Nebula also gives a steep surface
density profile inside 50~AU \citep{Desch2007ApJ...671..878D}.
Shallower decreases in surface density ($\epsilon=1.5$) are more
commonly observed for the outer disc ($R>$~50~AU).
The density falls off at $R_{\mathrm{in}}$ following the ``soft-edge''
model described earlier. The minimum grain size $a_{\mathrm{min}}$ is
kept at 0.05 $\mu$m. The power-law index of the dust grain size
distribution follows the standard interstellar value of -3.5. A
power-law index of -3.5 results from grain-grain collision theory
\citep{Dohnanyi1969JGR....74.2531D}. We mimic grain growth by varying
the maximum grain radius $a_{\mathrm{max}}$ from the fiducial value of
50 to 200 $\mu$m. We also studied the effect of smaller grains by
setting the maximum grain size to 10~$\mu$m.
The choice of the inner disc radius at 0.5 AU was dictated by disc
accretion and dust grain physics. A disc is truncated at the
co-rotation radius in the absence of a magnetic field or of a low-mass
companion, either star, brown dwarf, or giant planet, orbiting close
to the star. The co-rotation radius
$r_{\mathrm{rot}}=GM_*/v_{\mathrm{rot}}^2$ is defined as the distance
from the central star where the Keplerian disc rotates at the same
speed than the star photosphere $v_{\mathrm{rot}}=v_*$ (e.g.,
\citealt{Shu1994ApJ...429..781S}). Assuming a photospheric rotation
speed $v_* = 50-150$ km s$^{-1}$, typical of young stars
\citep{vandenAncker1998A&A...330..145V}, we obtain
$r_{\mathrm{rot}}\simeq$ 0.08-0.76 AU for $M_*=2.2$ M$_\odot$. Dust
grains do not exist above their sublimation temperature, which is
around 1500~K for silicate grains. The dust sublimation radius is the
distance from the star beyond which dust exists and can be estimated by
$r_{\mathrm{d}}=\sqrt{Q_R(L_*+L_{\mathrm{acc}})/16 \pi
\sigma}/T_{\mathrm{sub}}^{2}$ where
$Q_R=Q_{\mathrm{abs}}(a,T_*)/Q_{\mathrm{abs}}(a,T_{\mathrm{sub}})$ the
ratio of the dust absorption efficiency at stellar temperature $T_*$
to its emission efficiency at the dust sublimation temperature
$T_{\mathrm{sub}}$, $a$ is the mean grain radius and $\sigma$ is the
Stefan constant. For large silicate grains ($a \geq 1\ \mu$m), $Q_R$
is relatively insensitive to the stellar effective temperature and
close to unity because most stellar radiation lies at wavelengths
shorter than the grain radius. For smaller grains, the value of $Q_R$
is significantly increased \citep{Monnier2002ApJ...579..694M}. For
one micron grains, the sublimation radius in our fiducial model is
$r_{\mathrm{d}}= 0.19$~AU assuming $L_{\mathrm{acc}}<<L_*$ (passive
disc) and $Q_R \simeq 1$ (large grains). For grains as small as 0.05
$\mu$m in radius, we obtain $Q_R \simeq 20$ and $r_{\mathrm{d}}=
0.88$~AU. The inner disc location at 0.5~AU in our models is within
the theoretical range of inner disc radii. Moreover grains as small as
0.05$\mu$m can survive at 0.5~AU. Large luminosity ($L_*>100$
L$_\odot$) fast rotators ($v \sin i>200$ km s$^{-1}$) like 51~Oph
(with $L_*$= 260$^{+60}_{-50}$ L$_\odot$ and $v_*$=267$\pm$5 km
s$^{-1}$ ) may have a inner dust-free gas-rich region between the
co-rotation radius and the dust sublimation radius
\citep{Thi2005A&A...430L..61T,Tatulli2008A&A...489.1151T}. Another
example is AB Aur where the corotation radius is smaller than the
sublimation radius for the small grains but not for the large
grains. Observational evidence of an inner gas-rich dust-poor region
has been found for this object \citep{Tannikurlam2008ApJ...677L..51T}.
We also chose to simulate discs with inner radius at 1 and 10~AU to
model typical gaps possibly created by planets. The outer radius
$R_{\mathrm{out}}$ was set at 300~AU. By default passive discs are
modelled (i.e., $\alpha=$~0). Since we focus on young discs the
standard interstellar medium value of 100 for the gas-to-dust mass
ratio was adopted for all models. The gas and dust were assumed
well-mixed with no dust settling. The common model parameters are
summarized in Table~\ref{tab_DiscParameters} with the parameters for
the fiducial model printed in boldface . For each parameter set, the
code was run in the thermal-coupled mode
($T_{\mathrm{dust}}=T_{\mathrm{gas}}$) and thermal-decoupled mode in a
50 $\times$ 50 non-regular grid. An example of the location of the
grid points is given by \citet{Kamp2010A&A...510A..18K}. All SEDs were
computed for a source located at the typical distance of 140~pc at two
inclinations: face-on (0 degree) and 45 degree.
The stellar properties of our fiducial model differs from earlier
studies ($T_{\mathrm{eff}}$= 10000~K, $M_*$=2.5 M$_\odot$, and
$L_*$=47~L$_\odot$). We plot in Fig.~\ref{fig_DDN} the density
structure, photospheric height, near-infrared image, and SEDs for two
inclinations using the earlier studies properties. The disc structures
and SEDs are different for the two sets of stellar parameters. The
hotter disc atmosphere around the 47 L$_\odot$ star flares much more
than the disc around the lower luminosity star. The infrared
luminosity is increased but the shape of the SED does not differ much
as the stellar luminosity is increased.
\subsection{Fixed-structure model}\label{mcfost_model_descrption}
\begin{center}
\begin{table}
\caption{Disc parameters for the fixed-structure model.}\label{tab_MCFOSTParameters}
\begin{tabular}{llll}
\hline
reference scale height & $H_0$ & 0.72 & AU\\
reference radius & $R_0$ & 10 & AU\\
flaring index & $p$ & 1.2, {\bf 1.25}, 1.3 & \\
\hline
\end{tabular}
\ \\
\end{table}
\end{center}
We compare the results of the hydrostatic disc models to the results
of a fixed-structure model. The disc height $H$ is parametrised by the
functional $H(R)=H_0(R/R_0)^p$, where $H_0$ is the reference scale
height at reference radius $R_0$ and $p$ is the flaring index. The
scale height is used to compute the density $n_{\mathrm{H}}(R,z)$ at
height $z$ by $n_{\mathrm{H}}(R,z)=n_0(R)exp(-(z/H)^2)$. The value of
the parameters are given in Table~\ref{tab_MCFOSTParameters} and were
chosen such that the SED matches that produced by the fiducial
model with ``soft-edge'' and $T_{\mathrm{gas}}$ computed by thermal
balance. We vary the value of the flaring index $p$ between 1.2 and
1.3 to study its effects on the SEDs. All the other parameters are the
same than for the fiducial model.
\section{Results and discussion}\label{results_discussion}
In this section, we discuss the effects on the disc gas density
structure and SED when we vary a few parameters in the case of equal
gas and dust temperature and when they are computed independently. We
first show the structure and SED of the fiducial disc in
Sect.~\ref{fiducial_model}. The SEDs and images of the fixed-structure
disc is discussed in Sect.~\ref{mcfost_model_results}. We continue by
addressing the effects on the disc structure and SED when we use the
``soft-edge'' versus the ``sharp-edge'' inner rim prescription
(Sect.~\ref{sharp_soft_edge}), when we vary the surface density
profile (Sect.~\ref{surf_density_profile}), the disc mass
(Sect.~\ref{disc_mass}), the maximum grain size
(Sect.~\ref{big_grains}), and the inner disc radius
(Sect.~\ref{inner_radius}).
\subsection{Fiducial model}\label{fiducial_model}
Figure~\ref{fig_fiducial} contains some results for the fiducial
model. The dust and gas temperature distribution are shown in the
upper two panels while the lower-right panel is the disc density
structure. The disc areas where the gas and dust temperature are
decoupled are emphasized in the lower-left panel.
Figure~\ref{fig_opacity} shows the dust opacity (scattering and
absorption) for three maximum grain sizes (10, 50, and 200~$\mu$m).
The opacity is dominated by scattering at wavelengths below 10~$\mu$m
and by absorption at longer wavelengths.
In the thermal-coupled models, the dust temperature structure is
divided into a hot inner rim, a warm upper layer, and a vertical
isothermal interior. The lower-left panel in Fig.~\ref{fig_fiducial}
shows the ratio between the gas and dust temperature. The gas and dust
temperature are equal at vertical optical depths $A_{\mathrm{V}}>$~1
for a 10$^{-2}$ M$_\odot$ disc. The 2D dust continuum
radiative-transfer results confirm the analytical two-zone
decomposition popularized by \citet{Chiang1997ApJ...490..368C}. The
SED of the fiducial model is shown in solid-red lines in
Figure~\ref{fig_SEDs}. Model images of the inner disc at 1.98 $\mu$m
and 4.96~$\mu$m are shown in Fig.~\ref{fig_image_1.98_rim} and
Fig.~\ref{fig_image_4.96_rim} respectively. The flux at 1.98~$\mu$m
comes mostly from the inner rounded rim. The region just behind the
rim is deprived of stellar photons, is cooler, and thus does not emit
strongly (shadowing effect).
The gas temperature remains above 1000~K in the disc atmosphere up to
a few AU. Together with the decrease in the vertical gravitational
pull, they explain the emergence of a secondary density bump at
$\sim$~1.3~AU. The presence of the hot ''finger'' in disc atmospheres
is typical of disc models that compute the gas and dust temperature
independently
\citep*{Ercolano2009ApJ...699.1639E,Glassgold2009ApJ...701..142G,Woitke2009A&A...501..383W,Nomura2005A&A...438..923N,Kamp2004ApJ...615..991K,Jonkheid2004A&A...428..511J}.
The shape of the disc photospheric height ($A_{\mathrm{V}}$=1
contour) is shown in Figure~\ref{inner_rim_photospheric_height}. The
height at 0.6 AU is located at $z/r \sim$~1. The curvature exhibits
the same shape than in models where grains of different sizes
sublime at different radii \citep{Isella2005A&A...438..899I} or
with density-dependent sublimation temperature
\citep{Tannirkulam2007ApJ...661..374T}. The curvature of the inner
rim is caused by gas angular momentum conservation and pressure
gradient and is independent on the dust physics. Likewise the
location of the inner rim is set by hydrodynamic constraints (disc
truncation or presence of a companion) and not by dust sublimation
physics (see Sect.~\ref{fiducial_model_description}). However we
have assumed that the gas and the dust are well-mixed. Therefore,
the gas and dust grain density structures may be independent if
grain-growth and settling occur.
\subsection{Fixed-structure model}\label{mcfost_model_results}
The inner disc structure for the $H_0=0.72$~AU and flaring index
$p=$1.25 fixed-structure model is shown in the upper-right panel of
Fig.~\ref{inner_rim_structures}. The inner rim height is
$\sim$~0.1~AU. The SEDs that correspond to the fixed-structure models
are drawn for inclination 0 ($p$=1.25) and 45 degree ($p$=1.2, 1.25,
1.3) on the two top panels of Fig.~\ref{fig_SEDs}. The flaring index
$p$ impacts on the amount of warm dust grains in the disc atmosphere
that emit beyond 100~$\mu$m. The SED of the fixed-structure model
with flaring index $p=$1.25 matches relatively well the SED of the
fiducial model. The images at 1.98 and 4.96~$\mu$m for the $p=$1.25
model are shown on the upper right panel of
Fig.~\ref{fig_image_1.98_rim} and Fig.~\ref{fig_image_4.96_rim}.
When viewed exactly face-on, the SEDs at 3$\mu$m of a fixed-structure
disc show very weak near-IR fluxes, which arise from the rim
\citep{Meijer2008A&A...492..451M}. The cause of this feature
is that the projected inner rim emission area is null when the disc
is seen face-on. In contrast, the hot-dust emitting area in the
soft-edge models with rounded inner rim show much less dependency on
the viewing angle.
At 45 degree inclination the flux at 3~$\mu$m
matches the fluxes from all the other disc-structure models. The flux
comes mostly from the edge facing the star, which cannot be seen at
null inclination. The SED between 8 and 100~$\mu$m is extremely
sensitive to the value of the flaring index $p$ because the more flare
a disc is, the more stellar photons are intercepted by the outer disc,
which predominately emits in the far infrared. The short wavelength
flux (3--8 $\mu$m) is mostly sensitive to the height of the rim, which
is constant, and changes only a little when we vary $p$.
SEDs alone cannot be used to differentiate between the inner disc
structures. On the other hand, images at short wavelengths (see
Fig.~\ref{fig_image_1.98_rim}) show differences between the models.
The emission at 1.98~$\mu$m of the fiducial model is concentrated in
the rounded inner rim with contribution from the second rim. The
emission of the fixed-disc model is spread over the first
AU. Discriminating disc models by fitting simultaneously the SED and
images has already been used successfully to study the disc around the
T~Tauri star IM~Lupi \citep{Pinte2008A&A...489..633P}.
\begin{figure*}
\centering
{\includegraphics[angle=0,width=18cm]{disc_mass_inner_rim3.eps}}
\caption{Inner rim density structure for the $M_{\mathrm{disc}}$=10$^{-3}$ M$_\odot$ (upper panels) and $M_{\mathrm{disc}}$=10$^{-4}$ M$_\odot$ (lower panels). The left panels are the models with $T_{\mathrm{gas}}=T_{\mathrm{dust}}$. The right panels are the models with $T_{\mathrm{gas}}$ computed by gas thermal balance.}
\label{fig_disc_mass_inner_rim}
\end{figure*}
\subsection{Testing the effect of ``soft'' versus ``sharp'' edge}\label{sharp_soft_edge}
Current disc models assume that the disc density decreases
monotonically from the inner edge to the outer radius (``sharp-edge''
model). We study the effect of adopting our ``soft-edge'' model
compared to ``sharp-edge'' model on the disc structure and
SEDs. Figure~\ref{inner_rim_structures} shows the density structure
for the fiducial disc parameters assuming
$T_{\mathrm{gas}}=T_{\mathrm{dust}}$ on the left panel and of
$T_{\mathrm{gas}}$ computed by gas thermal on the middle panel. Those
two density structures contrast with the two upper panels in
Fig.~\ref{fig_surf_density_inner_rim}.
First the rim is extremely narrow in the ``sharp-edge'' models. On the
other hand, the main effect of the 'soft-edge' model is to create a
rounded rim surface whatever the surface density profile (see
Fig.~\ref{inner_rim_structures}). The density gradients at the rim are
clearly seen in the density structure plots. The projected rounded rim
appears as an ellipse, similar to other theoretical studies
\citep{Isella2005A&A...438..899I,Kama2009A&A...506.1199K}.
In the ``sharp-edge'' models, the strength of the 3~$\mu$m excess
emission depends on the inclination of the disc with respect to the
observer. This dependency stems from the thinness of the rim and was
already found in previous studies of thermal-coupled models
\citep{Meijer2008A&A...492..451M}. On the other hand, the 3~$\mu$m
excess depends much less on the inclination for the ``soft-edge''
models (see upper-left panel of Fig.~\ref{fig_SEDs}).
Secondly, there is no secondary ``bump'' in the structure at
$\sim$1.2~AU, even when $T_{\mathrm{gas}}$ is computed by thermal
balance. The presence of the second bump in the ``soft-edge''
decoupled model can explain the difference in fluxes between the
models in the 4 and 40 micron region in the lower-right panel of
Figure~\ref{fig_SEDs}. The optically-thick rim in the ``sharp-edge''
models prevents a large amount of photons to reach the disc surface at
radius 1--1.5~AU (the so-called ``shadow''), therefore suppressing the
rise of the second bump. The bump exists because the gas stays at
temperatures $>$~1000~K up to 4~AU at the surface while the dust
temperatures are only $>$~300~K. The bump creates a large emitting
area for the warm dust, which translates into strong emission in the
4--40 $\mu$m region in the SEDs (See Fig.~\ref{fig_image_4.96_rim} and
upper-left panel of Fig.~\ref{fig_SEDs}).
In the ``sharp-edge'' models, the mid-infrared flux is depressed
compared to the ``soft-edge'' models because of the lower disc
height behind the rim.
\begin{figure*}
\centering
{\includegraphics[angle=0,width=18cm]{grain_amax_inner_rim2.eps}}
\caption{Inner rim density structure for the
$a_{\mathrm{max}}=$~10~$\mu$m and
$a_{\mathrm{max}}=$~200~$\mu$m. The left panels are the models
with $T_{\mathrm{gas}}=T_{\mathrm{dust}}$. The right panels are
the models with $T_{\mathrm{gas}}$ computed by gas thermal
balance.}
\label{fig_grain_size_structure}
\end{figure*}
\subsection{Effects of PAH abundance}\label{pah_effects}
The PAHs are the main heating of the agents of the gas
\citep{Kamp2010A&A...510A..18K}. A disc with $f_{\mathrm{PAH}}$=0.1 is warmer than
a disc $f_{\mathrm{PAH}}$=0.01. A warmer disc is more extended vertically and intercepts
more radiation from the star (Fig.~\ref{fig_fPAH} upper-left compared to
Fig.~\ref{fig_surf_density_inner_rim} upper-right panel) although the
shape of the photospheric height does not vary significantly
(Fig.~\ref{fig_fPAH} upper-right panel).
The overall emission in the continuum and in the gas lines is higher
(Fig.~\ref{fig_fPAH} lower-right panel). We also show an image
at 1.98~$\mu$m in the lower-left panel of Fig.~\ref{fig_fPAH}.
\begin{figure*}
\begin{minipage}[b]{0.48\linewidth}
\centering
{\includegraphics[scale=0.48]{herbig_fiducial_fPAH0.1_density.ps}}
\end{minipage}
\begin{minipage}[b]{0.48\linewidth}
\centering
{\includegraphics[scale=0.48]{herbig_fiducial_fPAH0.1_photosphere.ps}}
\end{minipage}
\begin{minipage}[b]{0.48\linewidth}
\centering
{\includegraphics[scale=0.48]{herbig_fiducial_fPAH0.1_image.ps}}
\end{minipage}
\begin{minipage}[b]{0.48\linewidth}
\centering
{\includegraphics[scale=0.48]{rim_seds_pah.ps}}
\end{minipage}
\caption{\label{fig_fPAH} Inner disc density structure (upper-left panel), photospheric height (upper-right panel), near-infrared image (lower-left panel), and comparison between the SED with $f_{\mathrm{PAH}}$=0.01 and $f_{\mathrm{PAH}}$=0.1 for the fiducial model (lower-right panel). The red-dashed line in the upper-left panel encloses the region where the gas temperature is higher than 1000~K.}
\end{figure*}
\subsection{Effects of varying the surface density profile}\label{surf_density_profile}
We show the results of varying the surface density power-law index
from $\epsilon=$~1.0 to $\epsilon=$~2.0. All other parameters are kept
constant. The density structure for the thermal-coupled
($T_{\mathrm{dust}}=T_{\mathrm{gas}}$) and thermal-decoupled
($T_{\mathrm{gas}}$ from thermal balance) models are displayed in
Fig.~\ref{fig_surf_density_inner_rim} for the inner disc on the right
and left respectively. The Spectral Energy Distribution for the models
are shown in the upper-left panel of Fig.~\ref{fig_SEDs}.
The structure of thermal-decoupled disc models differ substantially
from their thermal-coupled counterparts. The gas is heated by gaseous
and dust grain photoprocesses that convert ultraviolet photons into
fast-moving electrons, which in turn share the energy to the gas
(mostly atomic and molecular hydrogen). In the upper disc layers,
the density is too low for efficient gas-grain thermal accommodation
and the gas and dust are thermally decoupled. The
gas mostly cools by line emissions, which become quickly optically
thick, while dust grains cool by optically thinner continuum
emission. As a result, the gas remains at higher temperature than
the dust grains.
Below the disc atmosphere, the ultraviolet flux is
attenuated and thermal accommodation between gas and dust grows. The
gas there is mostly molecular and frozen onto grain surfaces. Finally,
in the highly extinct midplane thermal accommodation dominates and
drives towards equal gas and dust temperatures. The dust temperature
structure is not affected by the change in density structure as there
are still two layers: a warm upper layer and a vertical isothermal
interior.
\begin{figure*}
\centering
{\includegraphics[angle=0,width=18cm]{Rin_inner_rim4.eps}}
\caption{Inner rim density structure for the $R_{\mathrm{in}}$=1
(upper panels) and 10~AU (lower panels). The left panels are the
models with $T_{\mathrm{gas}}=T_{\mathrm{dust}}$. The right panels
are the models with $T_{\mathrm{gas}}$ computed by gas thermal
balance.}
\label{fig_inner_radius_rim}
\end{figure*}
The inner rim is much higher when the gas temperature is determined by
detailed heating and cooling balance. The SED reflects relatively
well the amount of hot grains and thermal-decoupled models show a
larger amount because of higher rims.
In the thermal-coupled model, the rim is much lower and does not block
photons to reach the area behind the
rim. \citet{Acke2009A&A...502L..17A} studied the SED from Herbig~Ae
discs using a Monte-Carlo code coupled with hydrostatic equilibrium
assuming equal temperatures. They found good matches to observational
data when they artificially rise the height of the rim, consistent
with our results when the gas thermal balance is computed.
The second bump at 1--2~AU that appears in the thermal-decoupled
models are not present in the $T_{\mathrm{gas}}=T_{\mathrm{dust}}$
models (the left panels in Fig.~\ref{fig_surf_density_inner_rim}). As
already mentioned in the discussion of ``sharp-edge'' versus
``soft-edge'' models, the second bump is caused by the gas that is
hotter than the dust at the disc surfaces. A rounded optically thinner
rim helps more scattered light to reach the disc surfaces compared to
a narrow optically thicc rim.
The existence of a puffed-up rim is a necessary but not sufficient
condition for strong 3 $\mu$m excess. The surface area of the rim has
to be large, i.e. the rim has to be high enough. In the seminal paper
on disc inner rim by \citet{Dullemond2001ApJ...560..957D}, the height
of the rim to the rim distance from the star ratios were found to be
of the order of 0.1--0.25. We find that the rim in the case of
$\epsilon=2.0$ is 0.25 AU high at 0.5~AU. Here we have defined the disc
height as the location of the disc where gas density is 10$^{6}$
cm$^{-3}$. A higher rim is needed in our modelling to produce
strong emission at 3$\mu$m because the
\citet{Dullemond2001ApJ...560..957D} model assumed a single dust
temperature at 1500~K for the rim while in our models only the dust in
the rim at the midplane reaches 1500~K. In the upper-left panel of
Fig.~\ref{fig_fiducial}, we can see a vertical decreasing dust
temperature gradient from the midplane (1500K) to the surface
($<$1000~K) at the rim radius.
The mid-infrared continuum emission at a specific wavelength reflects
the amount of dust grains at a temperature roughly given by Wien's
law. As found in previous studies, discs flare in the outer disc in
the thermal-coupled models. The rim deprives the disc atmosphere
behind the rim of UV photons. This shadowing effect has been invoked
to explain SED of disc with low 30--100 $\mu$m emission flux
\citep{Dullemond2004A&A...417..159D,Meijer2009A&A...496..741M,verhoeff2010A&A...516A..48V}.
This shadowing effect is less pronounced if dust scattering is taken
into account. Dust scattering ensures that some photons reach the
``shadowed'' area behind the rim.
The main effect of warmer upper layers in the decoupled models is
puffed-up and strongly flaring upper layers from the inner rim to the
outer radius. Although the inner rim is twice higher than in the
thermal-coupled case, the extra dust opacity in the rim is not
sufficient to attenuate the UV flux responsible for the gas heating
behind the rim. Denser gas are found higher in the disc and more
grains are raised to higher temperature, resulting in slightly
stronger 10--50 $\mu$m continuum flux. In general our hydrostatic
disc models flare strongly.
\subsection{Effects of varying the disc mass}\label{disc_mass}
In this series of models, the surface density profile is kept at 2.0
and we vary the disc mass ($M_{\mathrm{disc}}$=10$^{-4}$, 10$^{-3}$,
and 10$^{-2}$ M$_\odot$). The structure and SEDs for this series are
shown in Fig.~\ref{fig_disc_mass_inner_rim} and \ref{fig_SEDs}
respectively. The near-IR flux is weakly affected by the disc mass.
On the other hand, the flux in the 10--50 $\mu$m region is sensitive
to the total dust mass and thus decreases with decreasing total disc
mass (the gas-to-dust mass ratio is kept constant at 100), consistent
with the findings of \citet{Acke2009A&A...502L..17A}. From the SEDs in
Fig.~\ref{fig_SEDs}, the disc mass is the most important parameter
controlling the near- to far-IR flux ratio. As the opacity decreases
with wavelength (see Fig.~\ref{fig_opacity}), the flux depends less and
less on the disc geometry and more and more on the total dust mass,
which scales with the gas mass if a constant gas-to-dust mass ratio is
assumed.
\subsection{Effects of varying the maximum grain size}\label{big_grains}
The maximum grain size has a relatively weak influence on the inner
disc structure (Fig.~\ref{fig_grain_size_structure}). Therefore the
flux at 3--5~$\mu$m does not change significantly with increasing dust
grain upper size limit (see Fig.~\ref{fig_SEDs}). On the other hand,
the effects on the 10--70~$\mu$m shape of the SED are significant. The
bigger the grains are, the cooler they are and the emission is weaker
as testified by the higher opacity per unit mass for the small grains
\ref{fig_opacity}. In addition, big grains have less surface area per
volume and therefore the photoelectric heating efficiency
decreases. The disc is cooler and flares less, which results in less
intercepted stellar photons. The combined effects concur to decrease
the 10--70 flux when grains are big. Beyond 100~$\mu$m, the grain
emissivity depends strongly on the presence of larger grains.
At wavelength larger than their sizes, grains are efficient
emitters. Therefore the flux drop at wavelengths longer than 70~$\mu$m
is less pronounced for big grains. In summary, increasing the maximum
grain radius results in weaker emission at 30--70 $\mu$m but stronger
emission at longer wavelengths.
\subsection{Effects of varying the inner disc radius}\label{inner_radius}
Figure \ref{fig_SEDs} and \ref{fig_inner_radius_rim} show the SEDs and
inner disc structure for $R_{\mathrm{in}}$=~1 and 10~AU. A round rim
structure is present for all inner disc radii. Since the hottest
grains are cooler, the peak of the near-infrared bump has shifted to
$\sim$~4~$\mu$m in the $R_{\mathrm{in}}$=~1~AU model. The spectral
signature of the inner rim has disappeared for the
$R_{\mathrm{in}}$=~10~AU model. The presence of the puffed-up rim does
not depend on the actual value of the inner radius but only rims close
to the star have hot enough dust to emit strongly in the
near-infrared.
\section{Conclusions}\label{conclusion}
We modelled the structure, Spectral Energy Distribution, and infrared
images of protoplanetary discs around Herbig Ae star and analysed the
difference between the assumption of equal gas and dust temperatures
and when both temperatures are computed independently. We also
compared the hydrostatic disc model SEDs to the SEDs of a disc with
prescribed density structure.
The disc structure is governed by the gas pressure support, which in
turn depends on the gas molecular weight and temperature. The height
of the inner rim in models with calculated gas temperature exceeds
those where the gas and dust temperature are equal by a factor 2 to
3. Higher rims result in large emitting areas and thus slight larger
near-infrared excess. Our treatment of the inner disc density fall-off
(``soft-edge'') results in rounded inner rims consistent with
near-infrared interferometric studies. The discs also show a second
density bump that manifests itself as stronger emission between 3 and
30 $\mu$m. The flux beyond 30 $\mu$m is mostly sensitive to the disc
mass. The maximum grain radius affects weakly the SEDs. The effect of
gaps is to remove hot dust grains which emit predominately in the
near-IR. As a result, the flux in the near-IR is suppressed. The shape
of the SED from 3 to 100 $\mu$m cannot be used to discriminate between
the inner disc structure models. Together with the SED, images with
high spatial resolution may be used to differentiate between disc
models.
Our study stresses the importance of understanding the interplay
between the gas and the dust in protoplanetary discs. This interplay
shapes the disc structures, which in turn control the shape of the
SEDs.
\section{Acknowledgments}
WFT was supported by a Scottish Universities Physics Alliance (SUPA)
fellowship in Astrobiology at the University of Edinburgh. W.-F.\ Thi
acknowledges PNPS, CNES and ANR (contract ANR-07-BLAN-0221) for
financial support. We thank Ken Rice for discussions on inner disc
structures.
\bibliographystyle{mn2e}
|
{
"timestamp": "2010-09-23T02:02:15",
"yymm": "1009",
"arxiv_id": "1009.4374",
"language": "en",
"url": "https://arxiv.org/abs/1009.4374"
}
|
\section{Introduction}
A fundamental question in the study of strongly correlated systems concerns how a
quantum many-particle system prepared in an initial state which is not an exact eigenstate
of the Hamiltonian evolves in time, and under what conditions the system at long times
thermalizes as opposed to reaching a novel athermal state.~\cite{rev2010}
This question is particularly relevant now due to
experiments in cold-atomic gases which provide practical realizations of almost
ideal many-particle systems where the interaction between particles
and the external potentials acting on them can be changed rapidly in time.~\cite{rev2008}
Motivated by this, there has been considerable
theoretical interest in studying the time-evolution of
one-dimensional systems which are initially prepared in a spatially
inhomogeneous state by
the application of external confining potentials.
Nonequilibrium time-evolution is triggered when the external potentials are rapidly
turned off which may be accompanied with a rapid change in the interaction between particles.
For example,
the time-dependent density matrix renormalization group (TDMRG) has been used to study
the time-evolution of a domain wall in the XXZ spin-chain,~\cite{tdmrg1,tdmrg2} the conformal field
theory approach
to study domain wall time evolution in the transverse-field Ising
chain at the gapless point,~\cite{CalabreseDW}
and the Algebraic Bethe Ansatz (ABA)
to study Loschmidt echos for the XXZ chain
for an initial domain wall state.~\cite{Caux10a}
ABA has also been used to study geometric quenches {i.e.}, the
time-evolution arising after two spatially separated regions have
been coupled together.~\cite{Caux10b}
The dynamics of hard-core bosons after an
initial confining potential was switched off was
studied in Ref.~\onlinecite{Rigolqc}. Here
it was found that the initial energy of
confinement resulted in the appearance of quasi-condensates at finite momentum.
Time-evolution of an initial density inhomogeneity after an interaction quench at the
Luther-Emery point was studied in Ref.~\onlinecite{Foster10} where a power-law amplification
of the initial density profile was found.
In this paper we study how a one-dimensional ($1$D) system prepared initially in a
domain wall state corresponding to a density $\rho(x\rightarrow \pm \infty)=\pm \rho_0$,
(where $x$ is the coordinate along the chain, and $\rho_0$ is a constant)
evolves in time after a sudden interaction and potential quench.
The $1$D system is modeled using the
quantum sine-Gordon (QSG) model which captures the low energy physics of
a variety of one-dimensional systems such
as the spin-1/2 chain, interacting fermions with back-scattering interactions arising due to
Umpklapp processes, and
interacting bosons in an optical lattice.~\cite{Giamarchi}
The QSG model is integrable, its exact solution can be obtained using
Bethe-Ansatz.~\cite{BetheA} While this property has been exploited
to a great extent to understand
equilibrium properties of many $1$D systems, extending Bethe-Ansatz
to study dynamics is a daunting task,
especially for the time-evolution of two-point correlation functions.
Thus there is a necessity to develop approximate methods to study this model.
Here we investigate the
time-evolution of the QSG model semiclassically using the truncated Wigner approximation (TWA)
to which quantum corrections are added in order to set limits
on its applicability.~\cite{Polkovreview}
Moreover our parameter regime corresponds
to an interacting bose gas whose density is initially in the form of a domain wall.
We study how this initial state
evolves in time as a result of a sudden switching on of an optical lattice, which
may or may not be accompanied by an interaction quench. An optical lattice is a source of
back-scattering interactions or Umpklapp processes, that tends to localize the bosons.
Our aim is to understand how this physics affects the time-evolution of the domain wall
state. Note that domain walls like the one we study here
have been created experimentally by subjecting equal mixtures of $^{87}Rb$ atoms in two
different hyperfine states to an external magnetic field gradient.~\cite{Weld09} Studying
quantum dynamics in such systems may soon be experimentally feasible.
One consequence of
quenched dynamics in integrable models is that the system often does not thermalize,
with the long time behavior
depending non-trivially on the initial state. Here we find that an initial state in the form of
a domain wall evolves at long times into a current carrying state even in the presence of a
back-scattering interaction of moderate strength.
Moreover,
this net current flow has interesting consequences
for the behavior of two-point correlation functions.
The lack of decay of current found here is consistent with the
fact that the dc conductivity of a 1D system is infinite
even in the presence of back-scattering or Umpklapp processes.~\cite{Rosch00}
The origin of the infinite conductivity is the large number of conserved quantities
in a 1D system, where some of them have a nonzero overlap with the
current,~\cite{Rosch00,Zotos97}
thus preventing an initial current carrying state from decaying to zero.
We also justify the steady state current obtained from TWA by studying the
QSG model at the Luther-Emery point. The Luther-Emery point
is an exactly solvable point in the
gapped phase of the model. In particular we study how an initial current carrying
state evolves with time and find that a steady
state current (albeit of reduced magnitude) persists at long times. We also
study how this current affects two-point correlation functions.
Since the QSG model is
a simplified model that neglects band-curvature and higher-order back-scattering or
Umpklapp
processes, an important question concerns
to what extent it can capture quenched dynamics in realistic systems.
The nonequilibrium time-evolution of the above domain wall initial state was
studied both for the exactly-solvable lattice model of the $XX$ spin chain, and
its continuum counterpart, the Luttinger model.~\cite{Lancaster10a}
The study of the density and various two-point correlation functions
revealed that both the lattice and the continuum model reached the same nonequilibrium
steady state, but differed in the
details of the time-evolution. Continuum theories
are far easier to handle both numerically and analytically than their lattice
counterparts. Therefore to what extent they can capture the steady state behavior
after a quench for general parameters is an
open and important question which is beyond the scope of this paper.
The paper is organized as follows. In section~\ref{TWA} we study the time evolution of
an initial domain wall state after a quench employing TWA. Results for
the density, current and two-point correlation functions are presented. In section~\ref{TWAqc}
we present results for the first quantum corrections to TWA for some representative
cases and discuss the general applicability
of the TWA results. In section~\ref{LE} we present
results for a quench at the exactly solvable Luther-Emery point for an initial
current carrying state. Here results for the
steady-state current as well as two-point correlation functions are presented.
Section~\ref{Conc} contains our conclusions.
\begin{figure}
\includegraphics[
width=0.95\columnwidth
]{fig1.eps}
\caption{(Color online) Density at time $t$=$15$ after the quench
for $K$=$0.9$, $\gamma$=$1$ and several different $g$.
}
\label{szgcomp}
\end{figure}
\begin{figure}
\includegraphics[
width=0.95\columnwidth
]{newszcontours.eps}
\caption{(Color online)
Contour plots for the magnitude of the density for $K$=$0.9$, $\gamma$=$1$ and for
values of $g$
a). g$=$0.05, b). g$=$0.2, c). g$=$0.6 and d). g$=$1.0. The density
at $t$=$0$ is $\rho(x)$=$(1/4)\tanh{(x/3)}$.}
\label{szcp}
\end{figure}
\begin{figure}
\includegraphics[
width=0.95\columnwidth
]{currentcontours.eps}
\caption{(Color online)
Contour plots for the -(current) for $K$=$0.9$, $\gamma$=$1$ and for
values of $g$
a). g$=$0.05, b). g$=$0.2, c). g$=$0.6 and d). g$=$1.0. The density
at $t$=$0$ is $\rho(x)$=$(1/4)\tanh{(x/3)}$
}
\label{icp}
\end{figure}
\begin{figure}
\includegraphics[
width=0.95\columnwidth]{currentkcomp.eps}
\caption{(Color online) The current at $t$=$15$ for $g$=$0.05$, $\gamma$=$1$ and different $K$.
}
\label{fig4a}
\end{figure}
\begin{figure}
\includegraphics[
width=0.95\columnwidth]{currentgcomp.eps}
\caption{(Color online)
The current at $t$=$40\,$ for $K$=$0.9,\gamma$=$1\,$ and different $g$.
}
\label{fig4b}
\end{figure}
\begin{figure}
\includegraphics[
width=0.95\columnwidth]{averagecurrentt100.eps}
\caption{(Color online) Time evolution of the current after spatially averaging over
a strip of width $\delta x =40$ centered at $x=0$. }
\label{fig4c}
\end{figure}
\begin{figure}
\includegraphics[
width=0.95\columnwidth]{timeaveragedcurrent.eps}
\caption{(Color online)
Dependence of the steady-state average current on interaction $g$ for $\gamma=1$ and
$K=0.9$.
}
\label{fig4d}
\end{figure}
\section{Time-evolution using the Truncated Wigner Approximation} \label{TWA}
We start with an initial state which is the ground state of the Luttinger liquid,
\begin{eqnarray}
H_i =&&
\frac{v_F}{2\pi}\int dx \left[\left(\partial_x \theta(x)\right)^2
+ \left(\partial_x \phi(x)\right)^2\right. \nonumber \\
&&\left. +\frac{2}{v_F}h(x) \partial_x \phi(x)\right]
\label{Hi}
\end{eqnarray}
where in terms of bosonic creation and annihilation operators
$b_{p},b_{p}^{\dagger}$~\cite{Giamarchi},
\begin{eqnarray}
\phi(x) &=& -\frac{i\pi}{L}\sum_{p\neq0}\left(\frac{L|p|}{2\pi}\right)^{1/2}\frac{1}{p}
e^{-\alpha|p|/2-ipx}\left(b_{p}^{\dagger} + b_{-p}\right) \label{ft1}\\
\theta(x)
&=&\frac{i\pi}{L}\sum_{p\neq0}\left(\frac{L|p|}{2\pi}\right)^{1/2}
\frac{1}{|p|}e^{-\alpha|p|/2-ipx}\left(b_{p}^{\dagger} - b_{-p}\right)
\label{ft2}
\end{eqnarray}
and
$\left[\phi(x),\frac{1}{\pi}\partial_y\theta(y)\right]=i\delta(x-y)$. Above,
$v_F$ is the Fermi velocity or the velocity of the bosons,
$\alpha$ a short-distance cut-off, $p$ the momentum, $L$ the length of the
system, and $h(x)$ is an external chemical-potential
which couples to the density $\rho(x)$=$-\frac{1}{\pi}
\partial_x \phi(x)$.
In the ground state of $H_i$ the density simply follows
the external field $\langle\rho(x)\rangle $=$ \frac{1}{\pi v_F}h(x)$.
We choose $h(x)$=$h_0\tanh{(x/\xi)}$ so that the initial density is a domain wall
of width $\xi$.
We study the case where at time $t$=$0$ the external field $h(x)$
is switched off. At the same time an optical-lattice is suddenly switched on which may
be accompanied by an change in the interaction between bosons. Thus
the time evolution for $t>0$ is due to the quantum sine-Gordon model,
\begin{eqnarray}
H_f=&&\frac{u}{2\pi}\int dx \left[K\left(\partial_x\theta(x)\right)^2
+ \frac{1}{K}\left(\partial_x \phi(x)\right)^2
\right] \nonumber \\
&&- g\int dx \cos\left(\gamma \phi(x)\right)\label{Hf}
\end{eqnarray}
Here $u$=$v_F/K$, $K$ being the
Luttinger parameter and $g$ the strength of the back-scattering interaction arising due to
a periodic potential. The ground state of $H_f$ has two well known
phases,~\cite{Giamarchi} the localized
(gapped) phase characterized by $\langle\phi\rangle\neq0$,
and a delocalized (gapless) phase.
The periodic potential is a relevant parameter for $2-\frac{\gamma^2 K}{4}>0$,
implying that the ground state has a gap for infinitesimally small $g$.
On the other hand for $2-\frac{\gamma^2 K}{4}<0$,
a localized phase arises only for back-scattering strengths larger than a critical value
($g>g_c$). We will study quenched dynamics for parameters that are such
that $g$ is a relevant perturbation in equilibrium. Note that the initial domain
wall state is not an exact eigen-state of $H_f$. Neither is it
related to the classical solitonic solution of the QSG model
since the latter is a domain wall in the $\phi$ field,~\cite{Rajaraman} while our initial state
is a domain wall in $\partial_x\phi$.
When $g$=$0$, the time evolution of the system can be solved exactly.~\cite{Lancaster10a}
For this case an initial density inhomogeneity shows
typical light-cone dynamics~\cite{Calabrese} by spreading
out ballistically in either direction with
the velocity $u$, {\sl i.e.},
$\rho(x,t)$=$\frac{1}{2\pi v_F}\left[h(x+ut) + h(x-ut)\right]$.
Since the system is closed, the energy is conserved. However
during the course of the time-evolution, the energy density is
transferred from the density to the current, the latter having the form
\begin{eqnarray}
j(x,t)&=&\frac{1}{\pi}\frac{\partial \theta}{\partial x} \\
&=&\frac{1}{2\pi u K^2}\left[h(x-ut) - h(x+ut)\right]
\end{eqnarray}
In particular for
$h(x)$=$h_0 \tanh(x/\xi)$, the energy density at
$t$=$0$ is
${\cal E}$=$\frac{u\pi}{2K}\langle \rho(x)\rangle^2 \simeq h_0^2/(2\pi u K^3)$, while at long times,
and for positions within the light-cone ($ut> |x|$ ) the energy density
is ${\cal E}$=$\frac{u\pi K}{2}j^2$ where $j$=$-h_0/(\pi u K^2)$.
Note that while any initial
density profile will give rise to transient currents, the special feature of a
domain wall density profile is that for a system of infinite length, the steady state
behavior is characterized by a net current flow. In particular for any finite time the
current flows across a length $|x|=ut$ of the wire connecting the regions
of high and low densities $\pm \rho_0= \pm h_0/(\pi v_F) $ at the two ends.
We now explore how the time-evolution of the density, and the
long time behavior of the current and two-point correlation functions
is influenced by a back-scattering interaction ($g\neq 0$).
The results are obtained using TWA which involves solving the classical
equations of motion with initial conditions weighted by the Wigner distribution function of
the initial state. Thus TWA is exact when $g$=$0$ while the effect of
$g$ is the leading correction in powers of $\hbar$.~\cite{Polkovreview}
Since $H_i$ is quadratic in the fields, it can be diagonalized by a simple shift, {\sl i.e.},
$H_i$=$\sum_{p\neq 0} v_F |p| a_p^{\dagger} a_p$, where $b_p$=$a_p + h_p/(v_F \sqrt{2\pi |p| L})
$, $h_p$ being the Fourier transform of $h(x)$.
The initial Wigner distribution function for the $a_p$ fields are Gaussian and are accessed by
a Monte-Carlo sampling. This is followed by a Fourier transform defined in Eqns.~\ref{ft1} and~\ref{ft2}
which gives the
$\phi$ and $\theta$ fields at the initial time $t$=$0$.
The classical equations of motion are then solved on a lattice up to a time $t$. All the data sets
presented here are accompanied with error bars associated with the Monte-Carlo averaging.
Lengths will be measured in
units of the lattice spacing $a$ which is also set equal to the short-distance cut-off $\alpha$.
Energy scales will be in units of $v_F/a$. The results will be presented for
$h_0$=$\pi/4$ and an initial domain wall of width $\xi$=$3$.
\subsection{Time evolution of the density}
Fig.~\ref{szgcomp} shows the
density at a time $t$=$15$ after the quench for $K$=$0.9$, $\gamma$=$1$ and several different $g$.
The domain wall is found to
broaden with time with a velocity which is reduced from the velocity of
expansion $u$=$v_F/K$ when $g=0$.
Moreover, unlike purely ballistic motion, the shape of the domain wall changes during the
time-evolution.
The behavior of the density is clearer in the contour plots
in Fig.~\ref{szcp}. For small $g$, the time-evolution shows a light-cone behavior
along with the appearance of spatial oscillations within the light-cone.
The amplitude of the oscillations
increase with $g$, while the wavelength of the oscillations is set by $\rho_0$.
Increasing $g$ gradually blurs the
light-cone, and eventually for very large $g$ the domain wall mass becomes so large that it hardly
moves during the times calculated here.
\subsection{Time evolution of the current}
The current behaves in a manner complementary to the
density and consistent with the continuity equation.
Fig.~\ref{icp} shows contour plots for the current
for parameters that are identical to that for
the density shown in Fig.~\ref{szcp}.
The current, like the density,
fluctuates in space and time,
but on an average reaches a non-zero steady state within the light cone for
$g$ values that are not too large.
Fig.~\ref{fig4a}
shows the current at time $t$=$15$ for a given $g$ and different $K$. As $K$ decreases,
the current increases as one expects from the analytic result for $g$=$0$.
Fig.~\ref{fig4b} shows how the current behaves for
a fixed $K$ and different $g$. Increasing $g$ not only reduces the overall
velocity of expansion, but also reduces the magnitude of the current.
Fig.~\ref{fig4c} shows how the current spatially
averaged over a strip of width $40$ centered at the
origin evolves in time. There is a clear appearance of
a current carrying steady state
whose magnitude decreases with $g$.
Note that the spatial averaging under-estimates
the time required to reach steady state as it under-estimates the amount
of current for $ut \leq 20$.
The dependence of the steady state current on $g$ is plotted in
Fig.~\ref{fig4d} after time-averaging the current in Fig.~\ref{fig4c}
over a time window $t=40-100$ in order to eliminate the temporal fluctuations.
The steady state current is found to decrease linearly with $g$ for $g \ll 1$.
Note that when $g\neq0$, the current does not commute with $H_f$. Yet the system
reaches a current carrying steady state.
This is due to the fact that
the QSG model has a large number of other conserved quantities, some of which have a nonzero
overlap with the current operator, thus preventing the current to decay to zero. The lack
of decay of an initial current carrying state is also the origin of an infinite conductivity
in many integrable systems.~\cite{Giamarchi}
It was argued in Ref.~\onlinecite{Rosch00} that
at least two different non-commuting Umpklapp processes are needed to violate conservation
laws sufficiently so as to render the conductivity finite and thus cause the current
to decay to zero.
\subsection{Steady-state correlation functions}
In this subsection we will study the following two equal time two-point correlation functions,
\begin{eqnarray}
C_{\theta\theta}(xt;yt)&=&\langle e^{i\theta(x,t)} e^{-i\theta(y,t)}\rangle\\
C_{\phi\phi}(xt;yt)&=&\langle e^{i\phi(x,t)} e^{-i\phi(y,t)}\rangle
\end{eqnarray}
These are found to reach a
nonequilibrium steady state for a time $ut >|x|,|y|$, {\sl i.e.}
for observation points that are within the light-cone.
The result for $C_{\theta\theta}$
for $K$=$0.9$ and different $g$ is plotted in Fig.~\ref{cxxg}. The TWA result for
$g$=$0$ is in agreement with the analytic
result~\cite{Lancaster10a}
$C_{\theta\theta}(x,y;ut>|x|,|y|)$=$\exp\left[ih_0(y-x)/(v_F K)\right]
\left(\alpha/\mid x- y \mid\right)^{(1+K^{-2})/4}$. Thus when $g$=$0$, the correlation
function decays as a power-law
with a slightly larger exponent than in equilibrium
(the latter being
$C^{eq}_{\theta\theta}(x,y)$=$\left(\alpha/\mid x- y \mid\right)^{1/(2K)}$).
Moreover $C_{\theta\theta}$ shows
oscillations at wavelength $\lambda $=$\frac{2\pi v_F K}{h_0}
$=$ 2/j$, $j$ being the steady-state current within the light cone.
Note that the results for $h_0=0$ were obtained previously in Ref.~\onlinecite{Cazalilla06}
where the authors studied an interaction quench from a homogeneous initial state.
The physical reason for the spatial oscillations when $h_0\neq 0$ is the
dephasing of the variable canonically conjugate to the density as the domain wall broadens.
This implies a dephasing of transverse spin components in the $XX$ spin chain
resulting in a spin-wave pattern at wavelength $\lambda$.~\cite{Lancaster10a}
For a system of hard-core bosons, oscillations in $C_{\theta\theta}$
has the physical interpretation of the appearance of quasi-condensates at wave-vector
$k$=$2\pi/\lambda$.~\cite{Rigolqc}
The TWA results presented here show that these effects can
persist even in the presence of a back-scattering interaction, at least within
the continuum model.
In particular Fig.~\ref{cxxg} shows that when $g\neq 0$, $C_{\theta\theta}$ retains
the spatially oscillating form albeit at a wavelength that increases
with increasing $g$. Just as for $g$=$0$, one expects the current to set the
wavelength of the oscillations. To check this
Figs.~\ref{cxx0acomp} and~\ref{cxx0bcomp} show a comparison between $C_{\theta\theta}(xt,yt)$
and $C_{\theta\theta}(h_0$=$0)(xt,yt)\cos(\pi j (x-y))$
where $C_{\theta\theta}(h_0$=$0)(xt,yt)$ is the correlation function
at long times after a homogeneous quench from
$H_i$ to $H_f$ ($h_0$=$0$ in $H_i$), while
$j$ is the spatially averaged steady-state current in Fig.~\ref{fig4d}. The agreement
is found to be good at least for small $g$.
The second effect of $g$ on $C_{\theta\theta}$
is to give rise to a faster decay in position. This decay also becomes faster
for a given $g$ and on decreasing
$K$ (not shown here) which takes the system deeper into
the gapped phase of the QSG model.
Fig.~\ref{czzg} shows the behavior of $C_{\phi\phi}$ correlation function
after it has reached a steady state. The result for $g$=$0$
is~\cite{Cazalilla06,Lancaster10a}
$C_{\phi\phi}(x,y;ut >|x|,|y|)$=$\left(\alpha/\mid x- y\mid\right)^{(1+K^2)/4}$ which is also
characterized by a slightly faster power-law decay than in equilibrium (the latter being
$C^{eq}_{\phi\phi}(x,y)$=$\left(\alpha/\mid x- y\mid\right)^{K/2}$). Further unlike
$C_{\theta\theta}$, $C_{\phi\phi}$ has
no memory of the initial spatial inhomogeneity for $g=0$. However this
is no longer the case for nonzero $g$.
Fig.~\ref{czzg} shows that for small $g$, $C_{\phi\phi}$ can also show spatial oscillations.
Moreover, there is no appearance of
long-range order until $g$ is ${\cal O}(1)$, where the appearance of
the Ising gap corresponds to a nonzero asymptotic behavior of the two-point correlation
function. It is also consistent that the appearance of the gap in $C_{\phi\phi}$ coincides
with the value of $g$ for which the domain wall is almost static in Fig.~\ref{szcp}.
In equilibrium, the $\cos\gamma\phi$
interaction is a relevant perturbation for $2> \gamma^2 K/4$.~\cite{Giamarchi}
Thus the $H_f$ parameters considered here are
those for which the ground state is the gapped Ising phase for $g$ of any strength. Yet there is no
signature of the gap in the quenched dynamics for $g \ll 1$.
For this case the domain wall motion is ballistic,
and the $C_{\theta\theta}$ correlations persist over longer distances than in
the gapped Ising phase. Similar observations have also been made in the study of
an interaction quench both in the bose-Hubbard model~\cite{Kollath08} and for
a system of interacting
fermions~\cite{manmana} where it was found that the system continued to
show light-cone dynamics and gapless behavior
for parameters that correspond to the equilibrium gapped phase.
The time-evolution of the density and current in the XXZ chain for an initial domain
wall state was studied in Ref.~\onlinecite{tdmrg1}
employing TDMRG. There it was found that while a current
persists within the gapless phase, it decayed to zero in the gapped phase. This result
is different from what we find here where the current persists in the gapped
phase as long as $g$ is not too large. There could
be two reasons for this difference. Firstly the parameters $\gamma$, $g$ and $K$ that we use here, do
not correspond to the parameters of the XXZ chain. Secondly, it is
possible that the irrelevant operators that are not retained in
the continuum model modify the long-time behavior, even though this was not found to
be the case at the exactly solvable XX point ($J_z=0$, $K=1,g=0$).~\cite{Lancaster10a}
\begin{figure}
\includegraphics[
width=0.95\columnwidth
]{cxxgcomp.eps}
\caption{(Color online)
The equal time $C_{\theta\theta}(0t;nt)$ correlation function
at $t$=$45$ for $K$=$0.9$, $\gamma$=$1$ and different $g$.
}
\label{cxxg}
\end{figure}
\begin{figure}
\includegraphics[
width=0.95\columnwidth
]{cxxh0comp_g03.eps}
\caption{(Color online) The equal time $C_{\theta\theta}(0t;nt)$ correlation function
compared with correlation function for a homogeneous quench ($h_0$=$0$) and
modulated by $\cos(\pi j n)$ for $t$=$45$ and $g$=$0.03$, $K$=$0.9$, $\gamma$=$1$.}
\label{cxx0acomp}
\end{figure}
\begin{figure}
\includegraphics[
width=0.95\columnwidth
]{cxxh0comp_g07.eps}
\caption{(Color online) The equal time $C_{\theta\theta}(0t;nt)$ correlation function
compared with correlation function for a homogeneous quench ($h_0$=$0$) and
modulated by $\cos(\pi j n)$ for $t=45$,$g$=$0.07$, $K$=$0.9$, $\gamma$=$1$.}
\label{cxx0bcomp}
\end{figure}
\begin{figure}
\includegraphics[
width=0.95\columnwidth
]{czzgcomp.eps}
\caption{(Color online) The equal time $C_{\phi\phi}(0t;nt)$ correlation function
at $t$=$45$ and for $K$=$0.9$, $\gamma$=$1$ and different $g$.
}
\label{czzg}
\end{figure}
\section{Quantum corrections to TWA} \label{TWAqc}
An important question concerns the validity of TWA. It was shown in Ref.~\onlinecite{Polkovreview} that
in writing the time-evolution of an interacting system as a Keldysh path integral, TWA is the leading
correction in powers of $\hbar$. One may therefore check its validity by
expanding the path integral in higher powers of $\hbar$, and identify when these contributions
become significant. We evaluate the first quantum correction along the lines of
Ref.~\onlinecite{Polkovreview}.
Below we briefly outline the approach.
The expectation value of an observable $\hat{O}({\bf x}, {\bf p}, t)$
to leading order beyond TWA is~\cite{Polkovreview}
\begin{eqnarray}
&&\left\langle \hat{O}({\bf x},{\bf p},t)\right\rangle \approx
\int d{\bf x}_{0} d{\bf p}_{0} W_{0}({\bf x}_{0},{\bf p}_{0})\nonumber \\
&&\left[
1 - \int_{0}^{t}d\tau\frac{\hbar^{2}}{3!2!i^{2}}
\frac{\partial^{3}V}{\partial {\bf x}(\tau)^{3}}\frac{\partial^{3}}{\partial {\bf p}^{3}}
\right]
O_{W}({\bf x}, {\bf p},t)
\end{eqnarray}
where $W_{0}({\bf x}_{0}, {\bf p}_{0})\,$ is the initial Wigner distribution,
and $O_{W}({\bf x}, {\bf p}, t) = \int d{\bf y} \left\langle {\bf x} - {\bf y}/2\right
| \hat{O}({\bf x},{\bf p},t)\left|{\bf x} + {\bf y}/2\right\rangle e^{i{\bf p}\cdot{\bf y}/\hbar}\,$
is the Weyl symbol of the operator $\hat{O}$. In the QSG model, ${\bf x}\rightarrow \phi(x)$, ${\bf p}
\rightarrow \Pi(x) \equiv \frac{1}{\pi}\partial_{x}\theta(x)$, and
$\int d{\bf x}d{\bf p} \rightarrow \int \mathcal{D}\phi(x)\mathcal{D}\Pi(x)$.
In Ref.~\onlinecite{Polkovreview} the author implemented this correction by allowing
a stochastic quantum jump in the momentum variable during the time evolution.
This is done as follows: for each Monte Carlo step, we choose a set of initial conditions,
weighted by the initial Wigner distribution, in accordance with TWA. For each
set of initial conditions, we select a random position, $x_{n}$, and a random time, $\tau$.
During the classical evolution, the field $\Pi_{n}(t) = \Pi(x_{n},t)$ is given a quantum
kick at time $\tau\,$ by shifting $\Pi_{n}(\tau) \rightarrow \Pi_{n}(\tau) + \xi(\Delta\tau)^{1/3}$,
where $\xi\,$ is a random weight chosen from a Gaussian distribution of zero mean and
unit variance, and $\Delta\tau\,$ is a small time interval. Here, we take $\Delta \tau\,$
equal to the integration time step size, $\Delta t$. This process of
sampling $\tau$, $x_{n}$, and $\xi$ is repeated for a given set of
initial conditions. Thus the quantum correction to TWA is~\cite{Polkovreview}
\begin{equation}
\left\langle - \frac{tNg\gamma^{3}}{8}\sin[\gamma\phi_{n}(\tau)]
\left(\xi^{3}/3 - \xi\right)\hat{O}(\phi,\Pi,t)\right\rangle,
\end{equation}
where $N\,$ is the number of spatial points.
The results for the first quantum
correction for $\gamma$=$1$ and $\gamma$=$2$ are shown in Fig.~\ref{szqc1a} and Fig.~\ref{szqc1b}
respectively.
As expected, the larger the coefficient $\gamma$,
the larger the quantum fluctuations in the $\phi$ field,
causing TWA to break down sooner. We find TWA to work very well for
$\gamma$=$1$ up to times $t=15$. On the other hand, for the same times, the
quantum corrections for $\gamma$=$2$ are significant.
It is also important to understand whether the steady-state current is a result of the
truncation scheme. To check this we plot the current evaluated from TWA along with the
first quantum correction in Fig.~\ref{iqc} for $\gamma=1$
and $K=0.9$. The current is now spatially averaged over the
non-interacting light-cone $|x| < u t$.
The quantum correction is found to enhance the
current. This is expected on the grounds that TWA underestimates the quantum fluctuations, and
therefore underestimates the extent of gapless behavior in the dynamics.
\begin{figure}
\includegraphics[
width=0.95\columnwidth
]{qcor_gamma1.eps}
\caption{(Color online) TWA and first quantum correction for
the density (main panel) and equal time $C_{\theta\theta}$ correlation function (inset)
for $K$=$0.9, g$=$0.05, \gamma$=$1$ and $t$=$15$.
}
\label{szqc1a}
\end{figure}
\begin{figure}
\includegraphics[
width=0.95\columnwidth
]{qcor_gamma2.eps}
\caption{(Color online) TWA and first quantum correction for
the density (main panel) and equal time $C_{\theta\theta}$ correlation function (inset)
for $K$=$0.9, g$=$0.05, \gamma$=$2$ and $t$=$15$.
}
\label{szqc1b}
\end{figure}
\begin{figure}
\includegraphics[
width=0.95\columnwidth
]{oct31currentqc.eps}
\caption{(Color online) TWA and first quantum correction to the current spatially averaged
over the light-cone $u t$ for
$K$=$0.9$ and $\gamma$=$1$ and several different $g$.
}
\label{iqc}
\end{figure}
\section{Quenched dynamics at the Luther-Emery point for an initial current carrying state}
\label{LE}
The main result of TWA was that a current carrying state can persist even in the
gapped phase of a model.
In this section we will explore this physics at the Luther-Emery point of the
QSG model. In particular we will study
how an initial current carrying state evolves in time
when the back-scattering interaction is suddenly switched on. We will also
explore the long time behavior of two point correlation functions.
The Luther-Emery point is a special point of the
QSG model where the problem is rendered quadratic
after refermionization in
terms of left and right moving fermions $\psi_{L,R}$.~\cite{Giamarchi}
To see this, we rescale the fields in Eqn.~(\ref{Hf}) as $\phi' = \gamma\phi/2\,$ and
$\theta' = 2\theta/\gamma$. Further if $K = 4/\gamma^2$, then $H_f$ may be written
as
\begin{eqnarray}
H_{f}^{\prime} & = & -iu\int dx \left[\psi_{R}^{\dagger}(x)\partial_{x}\psi_{R}(x)
- \psi_{L}^{\dagger}(x)\partial_{x}\psi_{L}(x)\right]\nonumber\\
&+&
m\int dx \left[ \psi_{R}^{\dagger}(x)\psi_{L}(x) + \psi_{L}^{\dagger}(x)\psi_{R}(x)\right]
\label{Hf'}
\end{eqnarray}
where $m=g\pi\alpha$ and
\begin{eqnarray}
\psi_{R}(x) & = & \frac{\eta_{R}}{\sqrt{2\pi\alpha}}e^{-i[\phi'(x)-\theta'(x)]},\\
\psi_{L}(x) & = & \frac{\eta_{L}}{\sqrt{2\pi\alpha}}e^{i[\phi'(x)+\theta'(x)]},
\end{eqnarray}
$\eta_{R,L}\,$ are Klein factors to ensure the correct anticommutation
relations among the fermions.
We construct an initial current carrying state which is
the ground state of the Hamiltonian
\begin{eqnarray}
H_i^{\prime}=&&-iu\int dx \left[\psi_{R}^{\dagger}(x)
(\partial_x-i\frac{\mu}{u})\psi_{R}(x)\right. \nonumber \\
&&\left. - \psi_{L}^{\dagger}(x)(\partial_x -i\frac{\mu}{u})
\psi_{L}(x)\right]
\end{eqnarray}
where $2\mu$ is the chemical potential difference between right and left movers.
We then study the time-evolution of this state for $t >0$ due to the Hamiltonian $H_f^{\prime}$
(Eq.~\ref{Hf'}). This Hamiltonian has a back-scattering interaction
of strength $m$, and no applied chemical potential difference between
right and left movers ($\mu=0$).
Defining $\psi_{R/L}(x)=\int \frac{dk}{2\pi}e^{ikx}\psi_{R/L}(k)$,
the initial state is characterized by the occupations
\begin{eqnarray}
\langle \psi_R^{\dagger}(k)\psi_R(k)\rangle &=& \theta(-uk + \mu)\label{c1}\\
\langle \psi_L^{\dagger}(k)\psi_L(k)\rangle &=& \theta(uk -\mu)\label{c2}
\end{eqnarray}
The current is defined as
\begin{eqnarray}
j(x)=u\left[\psi_R^{\dagger}(x)\psi_R(x)
-\psi_L^{\dagger}(x)\psi_L(x)
\right]
\end{eqnarray}
Thus the initial state is characterized by a current density
\begin{eqnarray}
j_0 = \mu/\pi
\end{eqnarray}
Since the theory is quadratic, the time-evolution can be studied in terms of
\begin{eqnarray}
\psi_R(k,t) &=& \psi_R(k)f(k,t) + \psi_L(k) g(k,t)\label{c3}\\
\psi_L(k,t) &=& \psi_L(k)f^*(k,t) + \psi_R(k) g(k,t) \label{c4}
\end{eqnarray}
where $f(k,t) = \cos(\omega_k t)- i\sin(\omega_k t)\cos(2\theta_k),
g(k,t) = -i \sin(\omega_k t)\sin{2\theta_k}$, $\omega_k$=$\sqrt{m^2 + u^2k^2},
\tan(2\theta_k)$=$m/(uk)$.
Using the above it is straightforward to work out the current at long times after the quench.
The current reaches a steady state
\begin{eqnarray}
j=j_0 - (m/\pi)\tan^{-1}\left(j_0\pi/m\right)\label{jlt}
\end{eqnarray}
Note that this result is very similar to that obtained by TWA (Fig.~\ref{fig4d})
and predicts that the steady state current decays
linearly with $m$ for $m\ll j_0$, while it decays as $1/m^2$
for large $m$.
The persistence of an initial current carrying state
even in the gapped phase of a Hamiltonian was
also found in Ref.~\onlinecite{Klich}. Moreover in agreement with Ref.~\onlinecite{Klich}
we find that the
steady state current
in the limit of very small initial
current $j_0\ll m$ is found to scale as the cubic power
of the initial current $j \propto j_0^3$.
We now turn to the evaluation of the steady-state gap and two point correlation functions.
In terms of bosonic variables $\phi^{\prime} = \gamma \phi/2$ and $\theta^{\prime}= 2\theta/\gamma$, the
gap is
\begin{eqnarray}
\langle e^{2i\phi^{\prime}(x,t)}\rangle = -\langle \psi_R^{\dagger}(x)\psi_L(x)\rangle
\end{eqnarray}
while the basic two-point correlation functions are
\begin{eqnarray}
C_{\phi^{\prime}\phi^{\prime}}(x,t)
&=& \langle e^{2i\phi^{\prime}(x,t)}e^{-2i\phi^{\prime}(0,t)}\rangle \nonumber \\
&=& \langle \psi_R^{\dagger}(xt)\psi_L(xt)\psi_L^{\dagger}(0t)\psi_R(0t)\rangle\\
C_{\theta^{\prime}\theta^{\prime}}(x,t) &=&
\langle e^{-2i\theta^{\prime}(x,t)}e^{2i\theta^{\prime}(0,t)}\rangle \nonumber \\
&=& \langle \psi_R^{\dagger}(xt)\psi_L^{\dagger}(xt)\psi_L(0t)\psi_R(0t)\rangle
\end{eqnarray}
For long times after the quench we find
\begin{eqnarray}
\langle e^{2i\phi^{\prime}(x,t)}\rangle
= \frac{mu}{4\pi}\ln\left[\frac{u^2/\alpha^2}{m^2 +\pi^2j_0^2}\right]
= A(\alpha, m,j_0)
\label{gapLE}
\end{eqnarray}
where $\alpha$ is a short-distance cut-off. Thus the steady-state gap depends on the initial
current $j_0$.
The two point correlations at long times are
\begin{eqnarray}
C_{\phi^{\prime}\phi^{\prime}}(x,t) &=& |A(\alpha,m,j_0)|^2+ |\frac{1}{2}\delta(x) + i I_b + I_a|^2\\
C_{\theta^{\prime}\theta^{\prime}}(x,t)
&=& \left(\frac{1}{2}\delta(x) + i I_b + I_a\right)
\left(\frac{1}{2}\delta(x) - i I_b - I_a\right) \nonumber \\
&-& \left(I_d+i I_c\right)^2
\end{eqnarray}
where
\begin{eqnarray}
I_a &=& \int_0^{\mu/u}\frac{dk}{2\pi} \cos(k x)\frac{u^2k^2}{m^2 + u^2k^2}\\
I_b &=& \int_{\mu/u}^{\infty}\frac{dk}{2\pi}e^{-k\alpha}\sin{kx}\frac{u^2k^2}{m^2 + u^2k^2}\\
I_c &=& \int_0^{\mu/u}\frac{dk}{2\pi} \sin(k x)\frac{muk}{m^2 + u^2k^2}\\
I_d &=& \int_{\mu/u}^{\infty}e^{-k\alpha}\frac{dk}{2\pi}\cos{kx}\frac{muk}{m^2 + u^2k^2}
\end{eqnarray}
For $j_0=0$ ($\mu =0 $), $C_{\phi^{\prime}\phi^{\prime}}$ reduces to the expression derived
in.~\cite{Iucci}
For $j_0\neq0$ and long distances $\mu x/u \gg 1, m x/u \gg 1$ we find,
\begin{eqnarray}
C_{\phi^{\prime}\phi^{\prime}}(x,t)
= |A(\alpha,m,j_0)|^2 + \frac{1}{x^2}\left(\frac{\pi^2j_0^2}{\pi^2j_0^2 + m^2}
\right)^2
\end{eqnarray}
Thus the correlations are found to decay very slowly (as $1/x^2$) in position to
their long distance value of the square of the gap. This should be contrasted with
the equilibrium result where the decay to the long distance value is exponential.~\cite{Giamarchi}
It is also interesting to compare this
result with that of an interaction quench from an initial state which
is the ground state of $H_i^{\prime}(\mu=0)$.~\cite{Iucci}
For this case the decay to the long distance value
is a power law $\left(1/x^6\right)$, but with a larger
exponent than found here for the current carrying state.
The expression for $C_{\theta^{\prime}\theta^{\prime}}$ at long times after the quench is
\begin{eqnarray}
C_{\theta^{\prime}\theta^{\prime}}(x,t\rightarrow\infty) =
\frac{1}{x^2}\left(\frac{\pi^2 j_0^2}{m^2 + \pi^2j_0^2}\right)e^{-2i\pi j_0 x/u}
\end{eqnarray}
and shows a similar slow decay as $1/x^2$ in position (in contrast to an
exponential decay in equilibrium). Moreover, the current flow imposes spatial oscillations
at a wavelength which is determined by the current.
This quench at the Luther Emery point did not involve a change in the Luttinger parameter $K$.
Significantly different physics can occur after a similar quench that also changes
the value of $K$. In Ref.~\onlinecite{Foster10},
the authors showed that changing $K\,$ can lead to the existence of ``super solitons''
at the Luther Emery point, where initial density inhomogeneities spread
out with amplitudes that grow in time.
\section{Conclusions} \label{Conc}
In summary, we have performed a detailed study of quenched dynamics in an interacting $1$D system
prepared initially in a domain wall state. The model, being integrable,
never thermalizes with the system
reaching a nonequilibrium current carrying state which is robust even in the presence of
moderate back-scattering interactions.
The current has interesting consequences for the
correlation functions, most notably the appearance of spatial oscillations
in the $C_{\theta\theta}$ correlation function.
Our predictions for the current
can be tested
experimentally using presently available one-dimensional optical lattice
techniques.~\cite{rev2008,Weld09}
\newline
{\it Acknowledgments:} AM is particularly indebted to A. Rosch and T. Giamarchi for
helpful discussions. This work was supported by NSF-DMR (Award No. 0705584, 1004589
for JL and AM, and 0705847 for EG).
|
{
"timestamp": "2010-12-17T02:02:49",
"yymm": "1009",
"arxiv_id": "1009.3918",
"language": "en",
"url": "https://arxiv.org/abs/1009.3918"
}
|
\section{Introduction}
A fascinating series of recent experiments demonstrating signatures of quantum coherence in the energy transfer dynamics of a variety of systems~\cite{lee07,engel07, calhoun09, collini09sc,collini09,collini10,panitchayangkoona10,mercer09,womick09} has sparked renewed interest in modeling excitation energy transfer beyond standard methods.~\cite{gilmore05,nazir09,jang08,jang09,prior10,thorwart09,ishizaki09b,kimura07, roden09, nejad10}
This process, which occurs when energy absorbed at one site (the donor) is transferred to another nearby site (the acceptor) via a virtual photon,~\cite{andrews89} is often considered to be incoherent; the result of weak donor-acceptor interactions, treated perturbatively using Fermi's golden rule.~\cite{foerster59,dexter52} However, though this approach has proved to be immensely successful when applied in many situations,~\cite{scholes03,beljonne09} accounting for {\it quantum coherence} within the energy transfer dynamics requires an analysis beyond straightforward perturbation theory in the donor-acceptor interaction.
An alternative starting point for investigations into coherent energy transfer is to treat the system-environment interaction as a perturbation instead. Such weak-coupling theories, often referred to as being of Redfield or Lindblad type depending upon the approximations made in their derivation,~\cite{b+p} have been successfully applied to elucidate a number of effects that could be at play in multi-site donor-acceptor complexes. Examples include studying the interplay of coherent dynamics and dephasing in promoting efficient energy transfer in quantum aggregates,~\cite{olayacastro08,mohseni08,plenio08,caruso09,rebentrost09,rebentrost09b,chin10} exploring the role of environmental correlations in tuning the energy transfer process,~\cite{fassioli10} and extensions to assess the potential importance of non-Markovian effects.~\cite{renger02,rebentrost09NM}
Nevertheless, in order to properly understand the transition from coherent to incoherent energy transfer which occurs as the system-environment coupling or temperature is increased,~\cite{rackovsky73,leegwater96,gilmore06,nazir09} it is necessary to be able to describe the system dynamics
beyond either of these limiting cases.~\cite{beljonne09, cheng09,olayacastro10} Building on earlier work,~\cite{soules71,rackovsky73, kenkre74,abram75} a number of methods have been put forward to accomplish this. For example, modifications to both Redfield~\cite{zhang98,yang02, renger03,novoderezhkin04} and F\"orster~\cite{sumi99,scholes00,jang04} theory have extended the range of validity of both approaches. Moreover, it is possible to define a new perturbation term through the small
polaron transformation,~\cite{wurger98} which under certain conditions allows interpolation between the Redfield and F\"orster limits.~\cite{jang08,nazir09,jang09}
For particular forms of system-environment interaction, this can also be achieved through the hierarchical equations of motion technique.~\cite{ishizaki09b,tanaka10} Numerically exact calculations, based, for example, on path integral,~\cite{thorwart09,nalbach10} numerical renormalisation group~\cite{tornow08} and density matrix renormalisation group~\cite{prior10} methods, have also been applied to study energy transfer beyond perturbative approaches.
In this work, we investigate the conditions under which coherent or incoherent motion is expected to dominate the energy
transfer dynamics of a model donor-acceptor pair. Following Ref.~\onlinecite{nazir09}, we employ a Markovian master equation
derived within the polaron representation for this purpose,
since it allows for a consistent analysis of the dynamics from weak to strong system-bath coupling
(or, equivalently, low to high temperatures).~\cite{wurger98,rae02} In addition to presenting a full derivation of the theory,
we also extend it to explore in detail the important effects of donor-acceptor energy mismatch, deriving
analytical forms for the dissipative dynamics valid over a large range of parameter space.
Furthermore, we move beyond the scaling limit studied in Ref.~\onlinecite{nazir09} to consider
an environment frequency distribution of finite extent, characterised by a high-frequency cut-off in the bath spectral density. In the resonant case (no energy mismatch) we define a strict crossover temperature above which the energy transfer dynamics ceases to be coherent.~\cite{nazir09}
Of particular practical interest is the role played by correlations between the donor and acceptor environmental fluctuations,
suggested
as one mechanism by which quantum coherence may survive in the energy transfer process under otherwise adverse conditions~\cite{nazir09,collini09sc,hennebicq09,lee07,yu08,nalbach10, chen10, west10,sarovar09,womick09} (though see, for example, Refs.~\onlinecite{olbrich10} and~\onlinecite{abramavicius10} for alternatives). These correlations are also easily treated within our formalism, through position-dependent couplings between the system and the common environment. As the donor and acceptor are brought closer together, there comes a point at which their
separation becomes comparable to, or smaller than, the wavelength of relevant modes in the bath. As this happens, fluctuations at each site become ever more correlated, and dephasing effects are suppressed. We shall show, consequently, that as the level of correlation increases, so too does the crossover temperature to the incoherent regime. Hence, strong correlations lead to the survival of coherence at high temperatures.
Off-resonance, we find that it is less straightforward to define a crossover temperature.
In contrast to the resonant case, for sufficient energy mismatch between the donor and acceptor,
increasing the temperature causes the amplitude of the coherent contribution to decrease,
though not to disappear altogether. In principle, it then becomes possible for a coherent component to exist
in the dynamics at all but infinite temperatures. Although we are then unable to define a crossover in
quite the same way, we still find that bath correlations have a qualitatively similar effect to the resonant case,
protecting coherence in the transfer process.
The paper is organised as follows. In Section~{\ref{master_equation}} we introduce our model, and derive a master equation describing the donor-acceptor dynamics within the polaron representation. Section~{\ref{resonant_transfer}} considers the resonant case and the coherent-incoherent crossover.
In Section~{\ref{off_resonance}} we investigate
off-resonant energy transfer and obtain analytic expressions for the dynamics in a number of limits. Finally, in Section~{\ref{summary}} we summarise our results.
\section{Polaron transform master equation}
\label{master_equation}
\subsection{The system and polaron transformation}
We consider a donor-acceptor pair ($j=1,2$), each site of which is modeled as a two-level system
with ground state $\ket{G}_j$, excited state $\ket{X}_j$, and energy splitting $\epsilon_j$.
The pair interact via Coulombic energy transfer with strength $V$, which is responsible for the transfer of excitation
from one site to the other. We label the state corresponding to a single excitation on site $1$
as $\ket{1}\equiv\ket{XG}$, and that on site $2$ as $\ket{2}\equiv\ket{GX}$.
The environment surrounding the donor-acceptor pair is modelled as a common bath of harmonic oscillators, coupled linearly
to the excited state of each site. The total system-bath Hamiltonian in the single
excitation subspace (in which energy transfer occurs) is therefore written (where $\hbar=1$)
\begin{eqnarray}
H_{\rm SUB}&{}={}&\epsilon_1\ketbra{1}{1}+\epsilon_2\ketbra{2}{2}+V(\ketbra{1}{2}+\ketbra{2}{1})\nonumber\\
&&\:{+}\ketbra{1}{1}B_z^{(1)}+\ketbra{2}{2}B_z^{(2)}+\sum_ {\bf k} \omega_ {\bf k} b_ {\bf k} ^{\dagger}b_ {\bf k} ,\nonumber\\
\label{eqn:HSUB}
\end{eqnarray}
where the bath is described by creation (annihilation) operators $b_{\bf k}^{\dagger}$ ($b_{\bf k}$) with corresponding
angular frequency $\omega_{\bf k}$, and wavevector ${\bf k}$. The bath operators are
given by $B_z^{(j)}=\sum_ {\bf k} (g_ {\bf k} ^{(j)} b_ {\bf k} ^{\dagger}+g_ {\bf k} ^{(j)*}b_ {\bf k} )$, with coupling
constants $g_{\bf k}^{(j)}$. As in Ref.~\onlinecite{nazir09}, we shall consider the case in which each site
is coupled to the bosonic bath with the same magnitude $|g_{\bf k}|$, but make
the separation between the sites explicit through position-dependent phases in the coupling constants of the form
$g_{\bf k}^{(j)}=|g_{\bf k}|\mathrm{e}^{i{\bf k}\cdot{\bf r}_j}$, with ${\bf r}_j$ being the position of site $j$. As we shall see,
this form of coupling gives rise to correlations between the bath influences experienced at each site,
allowing a range of totally correlated, partially correlated, and completely uncorrelated fluctuations to be explored.~\cite{nazir09, nalbach10}
A standard weak-coupling approach to the system dynamics would
now be to derive a master equation for the evolution of the reduced system density operator under the assumption that the system-bath interaction terms, as written in Eq.~(\ref{eqn:HSUB}), can be treated as weak
perturbations.~\cite{b+p, ishizaki09} In this work, we shall instead derive a master equation describing the donor-acceptor energy transfer dynamics in the (now widely used) polaron representation,~\cite{abram75,nazir09,jang08,jang09,rae02,wurger98,nitzan} whereby we displace the bath oscillators depending on the system state. We may then identify alternative perturbation terms, which can be small over a much larger range of parameter space than those in the original representation. In particular, the polaron framework allows us to reliably explore from weak (single-phonon) to strong (multiphonon) coupling regimes between the system and the bath, provided that the energy transfer interaction $V$ does not become the largest energy scale in the problem (in which case the full polaron displacement is no longer appropriate~\footnote{To ensure this we keep $V<\omega_c$, where $\omega_c$ is a high-frequency cut-off in the bath spectral density (see Eq.~(\ref{spectral_density})). In fact, a rough validity criterion can be given as $(V/\omega_c)^2(1-B^4)\ll1$, see Ref.~\onlinecite{mccutcheon10}, which for small enough $V/\omega_c$ is satisfied regardless of the size of the system-bath coupling strength or temperature.}), and that there is no infra-red divergence in its bath-renormalised value $V_R$ (see Eq.~(\ref{B_integral}) below).~\cite{silbey84} In contrast, with a weak system-bath coupling treatment we would only be able to probe single-phonon bath-induced processes, and hence not be able to properly explore the crossover from coherent to incoherent dynamics in which we are primarily interested.
To proceed, we thus apply a unitary transformation which displaces the bath oscillators according to the location of the
excitation. Defining
$H_P=\mathrm{e}^{S}H_{\rm SUB}\mathrm{e}^{-S}$, where
\begin{equation}
S=\ketbra{1}{1} P(g_ {\bf k} ^{(1)}/\omega_ {\bf k} )+\ketbra{2}{2} P(g_ {\bf k} ^{(2)}/\omega_ {\bf k} ),
\end{equation}
with bath operators $P(\alpha_ {\bf k} )=\sum_ {\bf k} (\alpha_ {\bf k} b_ {\bf k} ^{\dagger}-\alpha_ {\bf k} ^* b_ {\bf k} )$,
results in the polaron transformed spin-boson Hamiltonian~\cite{wurger98} $H_P=H_0+H_I$,
with
\begin{equation}
H_0=\frac{\epsilon}{2}\sigma_z+V_R\sigma_x+\sum_ {\bf k} \omega_ {\bf k} b_ {\bf k} ^{\dagger}b_ {\bf k} ,
\label{eqn:HP}
\end{equation}
and
\begin{equation}
H_I=V(B_x\sigma_x+B_y\sigma_y).
\label{H_I}
\end{equation}
Here, the bias $\epsilon=\epsilon_1-\epsilon_2$, gives the energy difference between the donor and acceptor, while the Pauli operators are defined in a basis in which $\sigma_z=|1\rangle\langle1|-|2\rangle\langle2|=|XG\rangle\langle XG|-|GX\rangle\langle GX|$. The bath operators appearing in Eq.~({\ref{eqn:HP}}) are constructed as $B_x=(1/2)(B_++B_--2B)$ and $B_y=(i/2)(B_+-B_-)$, where
\begin{equation}
B_{\pm}=\prod_ {\bf k} D\Bigg(\pm\frac{(g_ {\bf k} ^{(1)}-g_ {\bf k} ^{(2)})}{\omega_ {\bf k} }\Bigg),
\end{equation}
with displacement operators $D(\pm \alpha_ {\bf k} )=\exp[\pm(\alpha_ {\bf k} b_ {\bf k} ^{\dagger}-\alpha_ {\bf k} ^* b_ {\bf k} )]$. Note that the interaction terms in Eq.~(\ref{H_I}) therefore depend upon the difference in donor and acceptor system-bath couplings $g_{\bf k}^{(1)}$ and $g_{\bf k}^{(2)}$, respectively.
Importantly, the term
driving coherent energy transfer in Eq.~(\ref{eqn:HP}) will not be treated perturbatively, though it does now have a bath-renormalised strength, $V_R=BV$,
where
\begin{equation}
B=\exp\bigg[-\sum_ {\bf k} \frac{|g_ {\bf k} |^2}{\omega_ {\bf k} ^2}(1-\cos( {\bf k} \cdot{\bf d}))\coth(\beta \omega_ {\bf k} /2)\bigg]
\label{B_summation}
\end{equation}
is the expectation value of the bath operators with respect to
the free Hamiltonian: $B=\langle B_{\pm} \rangle_{H_0}$. The donor-acceptor separation is given by ${\bf d}={\bf r}_1-{\bf r}_2$.
In order to calculate the renormalisation factor, we take the continuum limit to convert the summation in Eq.~({\ref{B_summation}}) into an integral.
Defining the bath spectral density $J(\omega)=\sum_ {\bf k} |g_ {\bf k} |^2\delta(\omega-\omega_ {\bf k} )$, which contains
information regarding both the density of oscillators in the bath with a given frequency, and also how strongly those
oscillators interact with the donor-acceptor pair, and assuming a linear, isotropic dispersion relation, we find
\begin{equation}
B=\exp\bigg[-\int_0^{\infty}\frac{J(\omega)}{\omega^2}(1-F_D(\omega,d))\coth(\beta \omega/2)\bigg].
\label{B_integral}
\end{equation}
Here, $\beta=1/k_BT$ is the inverse temperature, while the function $F_D(\omega,d)$ captures the degree of spatial correlation in the bath fluctuations seen at each site, and
is dependent upon the dimensionality of the system-bath interaction ($D=1,2,3$).~\cite{mccutcheon09,fassioli10,nalbach10}
In one dimension $F_1(\omega,d)=\cos(\omega d/c)$, with $c$ the bosonic excitation speed, in two dimensions $F_2(\omega,d)=J_0(\omega d/c)$, where $J_0(x)$ is a Bessel function of the first kind, and in three dimensions $F_3(\omega,d)=\mathrm{sinc}(\omega d/c)$. In all cases $F_D(\omega,d)\rightarrow 1$ as $d\rightarrow 0$, i.e. when the donor and acceptor are at the same position, bath fluctuations are perfectly correlated, and the energy transfer strength is not renormalised ($V_R\rightarrow V$). In fact, in this limit dissipative process
are entirely suppressed (provided $|g_{\bf k}^{(1)}|=|g_{\bf k}^{(2)}|$) and energy transfer remains coherent for all times and in all parameter regimes in our model (the single-excitation subspace is then decoherence-free.~\cite{lidar98, zanardi97})
In two and three dimensions, as $d\rightarrow \infty$, $F_D(\omega,d)\rightarrow 0$, and the renormalisation takes on the value that would be obtained by considering separate, completely uncorrelated baths surrounding the donor and acceptor. In the following, we shall characterise the degree of correlation in terms of the dimensionless parameter $\mu=c/\omega_0d$, where $\omega_0$ is a typical bath frequency scale (see Eq.~(\ref{spectral_density}) below). We therefore have $\mu=0$ in the absence of correlations, $\mu<1$ for weak correlations, and $\mu>1$ for strong correlations.
\subsection{Markovian master equation}
Having identified a new perturbation term by transforming our Hamiltonian to the polaron representation, we can now construct a master equation describing the evolution of the donor-acceptor pair reduced density operator $\rho$ up to second order in $H_I$. We employ a standard Born-Markov approach, which yields a polaron frame, interaction picture master equation of the form~\cite{b+p}
\begin{equation}
\frac{\partial\tilde{\rho}(t)}{\partial t}=-\int_0^{\infty}d\tau\,\mathrm{tr}_B\big[[\tilde{H}_I(t),[\tilde{H}_I(t-\tau),\tilde{\rho}(t) \otimes \rho_B]\big],
\label{master_equation_1}
\end{equation}
where tildes indicate operators in the interaction picture, $\tilde{O}(t)=\mathrm{e}^{i H_0 t}O\mathrm{e}^{-i H_0 t}$, and $\mathrm{tr}_B$ denotes a trace over the bath degrees of freedom. In deriving Eq.~({\ref{master_equation_1}}) we have assumed: (i) factorising initial conditions for the joint system-bath density operator within the polaron frame, $\chi(0)=\rho(0)\otimes \rho_B$, with $\rho_B=e^{-\beta H_B}/{\rm tr}_B(e^{-\beta H_B})$
being a thermal equilibrium state of the bath; (ii) that by construction the interaction is weak in the polaron frame so that we may factorise the joint density operator as $\tilde{\chi}(t)=\tilde{\rho}(t)\rho_B$ at all times; (iii) that the timescale on which the donor-acceptor system evolves appreciably in both the Schr\"{o}dinger and interaction pictures is large compared to the bath memory time $\tau_B$. Since, for the spectral density we shall consider below, $\tau_B\sim1/\omega_c$, where $\omega_c$ is a high-frequency cutoff (see Eq.~(\ref{spectral_density})), this is not too restrictive, as we must keep $V<\omega_c$ anyway in order for the polaron theory to work well. We note that interesting non-Markovian and non-equilibrium bath effects have been explored in the polaron formalism in Refs.~\onlinecite{jang08} and~\onlinecite{jang09}.
Inserting Eq.~({\ref{H_I}}) into Eq.~({\ref{master_equation_1}}), and moving back into the Schr\"{o}dinger picture, we arrive at our Markovian master equation describing the energy transfer dynamics within the single-excitation subspace, and written in the polaron frame as
\begin{equation}
\begin{split}
\frac{\partial\rho(t)}{\partial t}=&-i[(\epsilon/2)\sigma_z+V_R\sigma_x,\rho(t)]\\
&-V^2\int_0^\infty \mathrm{d}\tau\Bigl([\sigma_x,\tilde{\sigma}_x(-\tau)\rho(t)]\Lambda_{xx}(\tau)\\
&\hspace{1.7cm}+[\sigma_y,\tilde{\sigma}_y(-\tau)\rho(t)]\Lambda_{yy}(\tau)+\mathrm{H.c.}\Bigr),
\label{master_equation_2}
\end{split}
\end{equation}
where H.c. denotes Hermitian conjugation.
The effect
of the bath is now contained within the correlation functions
$\Lambda_{ll}(\tau)=\langle \tilde{B}_l(\tau)\tilde{B}_l(0)\rangle_{H_0}$, which are given explicitly by
\begin{align}
\Lambda_{xx}(\tau)&=(B^2/2)(\mathrm{e}^{\phi(\tau)}+\mathrm{e}^{-\phi(\tau)}-2)\label{Lambda_xx},\\
\Lambda_{yy}(\tau)&=(B^2/2)(\mathrm{e}^{\phi(\tau)}-\mathrm{e}^{-\phi(\tau)})\label{Lambda_yy},
\end{align}
where
\begin{equation}
\begin{split}
\phi(\tau)=2\int_0^{\infty}d\omega&\bigg[\frac{J(\omega)}{\omega^2}(1-F_D(\omega,d))\\
\times&\left(\cos \omega\tau\coth(\beta\omega/2)-i \sin \omega\tau\right)\bigg].
\label{phi}
\end{split}
\end{equation}
Notice that the phonon propagator, $\phi(\tau)$, is correlation-dependent due to the factor $(1-F_D(\omega,d))$, and so clearly the dissipative effect of the bath will be dependent upon the degree of correlation too. For example, as $d\rightarrow0$, $F_D(\omega,d)\rightarrow 1$, and the dissipative contribution
to Eq.~({\ref{master_equation_2}}) vanishes, as anticipated earlier.
\subsection{Evolution of the Bloch vector}
\label{evolution_of_the_bloch_vector}
We solve our master equation in terms of the Bloch vector, defined as
$\vec{\alpha}=(\alpha_x,\alpha_y,\alpha_z)^T=(\langle\sigma_x \rangle, \langle\sigma_y \rangle, \langle\sigma_z \rangle)^T$. As we are working exclusively in the single-excitation subspace, $\alpha_x$ and $\alpha_y$ describe the coherences between the states $|1\rangle\equiv|XG\rangle$ and $|2\rangle\equiv|GX\rangle$, while $\alpha_z$ captures the donor-acceptor population transfer dynamics generated by the coupling $V$.
Though Eq.~(\ref{master_equation_2}) is written
in the Sch\"{o}dinger picture, it is still in the polaron frame, and so we must determine how expectation values
in the polaron frame are related to those in the original, or ``lab" frame. We can see this by writing
${\alpha}_i=\mathrm{tr}_{S+B}(\sigma_i{\chi}_L(t))=\mathrm{tr}_{S+B}(\sigma_i \mathrm{e}^{-S}{\chi}(t)\mathrm{e}^S)
=\mathrm{tr}_{S+B}(\mathrm{e}^S \sigma_i e^{-S}{\rho}(t)\rho_{B})$,
where $\chi_L(t)=e^{-S}\chi(t)e^S$ is the lab frame total density operator, and we have made use of the Born approximation in the polaron frame to write $\chi(t)=\rho(t)\rho_{B}$. Since $\mathrm{e}^S \sigma_x \mathrm{e}^{-S}=\ketbra{2}{1} B_- +\ketbra{1}{2}B_+$,
$\mathrm{e}^S \sigma_y \mathrm{e}^{-S}=i(\ketbra{2}{1} B_- -\ketbra{1}{2}B_+)$, and $\mathrm{e}^S \sigma_z \mathrm{e}^{-S}=\sigma_z$, this implies that
the lab Bloch vector elements are ${\alpha}_i=B{\alpha}_{iP}$, for $i=x,y$, and ${\alpha}_z={\alpha}_{zP}$, where $\alpha_{iP}$ is an expectation value in the polaron frame:
${\alpha}_{iP}=\mathrm{Tr}_S(\sigma_i{\rho}(t))$. Alternatively, we can define a matrix $L$ which maps the polaron frame Bloch vector ($\vec{\alpha}_P$) to its lab frame counterpart ($\vec{\alpha}$): $\vec{\alpha}=L \cdot \vec{\alpha}_P$, where $L=\mathrm{diag}(B,B,1)$.
Working in terms of the Bloch vector, we arrive at an equation of motion of the form
\begin{equation}\label{labbloch}
\dot{\vec{\alpha}}(t)=M\cdot\vec{\alpha}(t)+\vec{b}.
\end{equation}
In the following, we shall often be interested in determining whether the energy transfer dynamics is predominantly coherent or incoherent. It is then helpful to write
Eq.~({\ref{labbloch}}) as
\begin{equation}
\dot{\vec{\alpha}}'(t)=M\cdot\vec{\alpha}'(t),
\end{equation}
with $\vec{\alpha}'(t)=\vec{\alpha}(t)-\vec{\alpha}(\infty)$, where
$\vec{\alpha}(\infty)=-M^{-1}\cdot \vec{b}$ is the steady state.
This makes clear that the nature of the energy transfer process
lies solely in the matrix $M$,
while the inhomogeneous term $\vec{b}$ is needed only in determining the steady state.
Equipped with the eigensystem of $M$, we may determine the corresponding time evolution as follows: an eigenvector of $M$, say $\vec{m}_i$, has equation
of motion $\dot{\vec{m}}_i=q_i\vec{m}_i$, where $q_i$ is the corresponding eigenvalue. Its subsequent evolution
then has the simple exponential form $\vec{m}_i(t)=\vec{m}_i\mathrm{e}^{q_i t}$. More generally, we can say that any initial state
$\vec{\alpha}'(0)$ will have subsequent evolution
\begin{equation}
\vec{\alpha}'(t)=\sum_{i=1}^{3} a_i \vec{m}_i \mathrm{e}^{q_i t},
\label{general_evolution}
\end{equation}
where the coefficients $a_i$ are determined by the initial conditions (i.e. the solutions of
$\vec{\alpha}'(0)=\sum_i a_i \vec{m}_i$).
The solution to the full inhomogeneous
equation is then found simply by addition of the steady state:
$\vec{\alpha}(t)=\vec{\alpha}'(t)+\vec{\alpha}(\infty)$.
\section{Resonant energy transfer}
\label{resonant_transfer}
We start by considering the important special case of resonant energy transfer, in which the interplay of coherent and incoherent effects is particularly pronounced. As we shall see, in this situation it is relatively straightforward to derive a strict criterion governing whether or not we expect the energy transfer dynamics to be able to display signatures of coherence.~\cite{nazir09,wurger98} Hence, resonant conditions provide a natural situation in which to begin to understand, for example, the role of bath spatial correlations~\cite{nazir09, nalbach10, chen10, fassioli10, yu08, hennebicq09,west10,sarovar09,womick09} or the range of the bath frequency distribution in determining the nature of the energy transfer process.
Setting the donor-acceptor energy mismatch to zero,
$\epsilon=0$, we find from Eq.~(\ref{master_equation_2}) dynamics generated by an expression of the form
$\dot{\vec{\alpha}}=M_R\cdot\vec{\alpha}+\vec{b}_R$, with
\begin{equation}
M_R=\left(
\begin{array}{ccc}
-(\Gamma_z-\Gamma_y) & 0 & 0 \\
0 & -\Gamma_y & -2 B V_R \\
0 & B^{-1}(2 V_R+\lambda_3) & -\Gamma_z
\end{array}
\right),
\label{resonant_M}
\end{equation}
and $\vec{b}_R=(-B \kappa_x,0,0)^T$, where
\begin{align}
\Gamma_y&=2V^2\gamma_{xx}(0),\\
\Gamma_z&=V^2(\gamma_{yy}(2V_R)+\gamma_{yy}(-2V_R))+2V^2\gamma_{xx}(0),\\
\lambda_3&=2V^2(S_{yy}(2V_R)-S_{yy}(-2V_R)),\\
\kappa_x&=V^2(\gamma_{yy}(2V_R)-\gamma_{yy}(-2V_R)).
\end{align}
The rates and energy shifts are related to the response functions
\begin{equation}
K_{ii}(\omega)=\int_0^{\infty}\mathrm{d}\tau\mathrm{e}^{i \omega \tau}\Lambda_{ii}(\tau)\mathrm=\frac{1}{2}\gamma_{ii}(\omega)+i S_{ii}(\omega),
\end{equation}
such that
\begin{equation}
\gamma_{ii}(\omega)=2\mathrm{Re}[K_{ii}(\omega)]=\int_{-\infty}^{+\infty}\mathrm{d}\tau\mathrm{e}^{i \omega \tau}\Lambda_{ii}(\tau),
\end{equation}
and $S_{ii}(\omega)=\mathrm{Im}[K_{ii}(\omega)]$.
The resonant steady-state is straightforwardly found to be
\begin{equation}
\alpha_x(\infty)=- B \tanh(\beta V_R),
\end{equation}
while $\alpha_y(\infty)=\alpha_z(\infty)=0$. Notice that while this is similar in form to the steady-state that would be obtained from a weak system-bath coupling treatment,~\cite{b+p, weissbook} $\alpha_x(\infty)$ is determined here by $V_R$, rather than the original coupling $V$, and there is also an extra factor of $B$ suppressing its magnitude.
The eigenvalues of $M_R$ are given by
$q_1=\Gamma_y-\Gamma_z$ and $q_2=q_3^*=-(1/2)(\Gamma_y+\Gamma_z+i \xi_R)$.
Thus, referring to Eq.~(\ref{general_evolution}), we see that
\begin{equation}
\xi_R=\sqrt{8V_R(2V_R+\lambda_3)-(\Gamma_z-\Gamma_y)^2}
\label{oscillation_frequency}
\end{equation}
determines whether or not any coherence exists within the energy transfer dynamics.
Considering the initial state $\vec{\alpha}(0)=(0,0,1)^T$, corresponding to excitation of the donor, $\rho(0)=|1\rangle\langle1|=|XG\rangle\langle XG|$, we find analytical forms for the evolution of the Bloch vector components:
\begin{eqnarray}
\alpha_x(t)&=&-B\tanh(\beta V_R)(1-\mathrm{e}^{-(\Gamma_y-\Gamma_z)t}),\\
\alpha_y(t)&=&-\frac{2 B V_R}{\xi_R}\mathrm{e}^{-(\Gamma_y+\Gamma_z)t/2}\sin\Big(\frac{\xi_R t}{2}\Big),\label{y_general_evolution}\\
\alpha_z(t)&=&\mathrm{e}^{-(\Gamma_y+\Gamma_z)t/2}\Bigl[\cos\Bigl(\frac{\xi_R t}{2}\Bigr)
+\frac{\Gamma_y-\Gamma_z}{\xi_R}\sin\Bigl(\frac{\xi_R t}{2}\Bigr)\Bigr].\label{z_general_evolution}\nonumber\\
\end{eqnarray}
Inspection of Eqs.~(\ref{oscillation_frequency}) and ({\ref{z_general_evolution}})
allows us to identify a crossover from coherent to incoherent motion in the energy transfer dynamics as the point
at which oscillations in the population difference vanish:~\cite{nazir09}
\begin{equation}
(\Gamma_z-\Gamma_y)^2=8V_R(2V_R+\lambda_3).
\label{critical_condition}
\end{equation}
For $(\Gamma_z-\Gamma_y)^2<8V_R(2V_R+\lambda_3)$, $\xi_R$ is real and both the
population difference and coherence $\alpha_y$ describe damped oscillations, while for $(\Gamma_z-\Gamma_y)^2\geq8V_R(2V_R+\lambda_3)$, $\xi_R$ is either zero or imaginary, with the resulting dynamics then being entirely incoherent.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.45\textwidth]{e-zero_dynamics.eps}
\caption{Population difference as a function of scaled time $\omega_0 t$ for temperatures
of $k_B T/\omega_0=1$ (blue dashed curve), $k_B T/\omega_0=5$ (green dotted curve), $k_B T/\omega_0=12$ (orange solid curve) and $k_B T/\omega_0=20$ (red dot-dashed curve). Parameters: $\alpha=0.05$, $V/\omega_0=0.5$,
$\omega_c/\omega_0=4$, $\epsilon=0$ and $\mu=c/\omega_0d=0.5$.}
\label{e-zero_dynamics}
\end{center}
\end{figure}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.45\textwidth]{resonant_dynamics.eps}
\caption{Population difference as a function of scaled time $\omega_0 t$ for temperatures
of $k_B T/\omega_0=5$ (blue dashed curves) and $k_B T/\omega_0=10$ (red dotted curves), and for separations
corresponding to no correlation, $\mu=c/\omega_0d=0$ (top), weak correlations, $\mu=0.5$ (middle),
and strong correlations $\mu=2$ (bottom). The insets show the evolution of the
corresponding coherence $\alpha_y$. Parameters: $\alpha=0.05$, $V/\omega_0=0.5$,
and $\omega_c/\omega_0=4$.}
\label{resonant_dynamics}
\end{center}
\end{figure}
To further analyse the behaviour of $\alpha_z(t)$, and the
conditions for which the boundary defined by Eq.~({\ref{critical_condition}}) is crossed, we now take a
specific form for the system-bath spectral density. For a large enough bath we may approximate
$J(\omega)=\sum_ {\bf k} |g_ {\bf k} |^2\delta(\omega_ {\bf k} -\omega)$ as a smooth function of $\omega$. In this work we consider a spectral
density of the form
\begin{equation}
J(\omega)=\alpha \frac{\omega^3}{\omega_0^2} \mathrm{e}^{-\omega/\omega_c},
\label{spectral_density}
\end{equation}
where $\alpha$ is a dimensionless quantity capturing the strength of the system-bath interaction, and
$\omega_0$ is a typical frequency of bosons in the bath, which sets an overall energy scale. The cubic frequency dependence in Eq.~({\ref{spectral_density}}) is typical, for example,
in describing dephasing due to coupling to acoustic phonons,~\cite{wurger98,ramsay10} but can also be used to elucidate the behaviour in which we are interested in general.~\cite{nalbach10} The cut-off frequency $\omega_c$
is needed to ensure that vacuum contributions remain finite, and is
related to parameters specific to the particular physical system one wishes to model. The inverse cut-off frequency also sets a typical relaxation timescale for the bath.~\cite{b+p}
To illustrate the dynamics and crossover behaviour in the resonant case, in Fig.~\ref{e-zero_dynamics} we plot the population difference ($\alpha_z$) as a function of the scaled time $ \omega_0 t$ for a range of temperatures, showing the transition from coherent
to incoherent transfer as the temperature is increased. In this plot, and all the following, we consider three-dimensional coupling, $F_3(\omega,d)={\rm sinc}(\omega d/c)$.
The role of bath spatial correlations in protecting coherence can be seen in Fig.~{\ref{resonant_dynamics}}, where we again plot the evolution of the population difference (the insets show the corresponding coherence $\alpha_y$), this time for representative intermediate and high-temperature cases. The different plots in Fig.~\ref{resonant_dynamics} correspond to zero correlations, characterised by $\mu=c/\omega_0d=0$ ($d\rightarrow\infty$, top), weak correlations, $\mu=0.5$ (middle), and strong correlations $\mu=2$ (botttom).~\footnote{Although we would usually expect $V$ to change with varying separation $d$, we keep it fixed for all plots presented here in order to isolate the role played by the environmental correlations.} Progressing from the uppermost plot to the lowest, we clearly see that an increase in correlation strength
prolongs the timescale over which oscillations in both the
population difference and coherence persist. Moreover, by looking at the curves corresponding to the
higher temperature (red, dotted), we can see that as the degree of correlation is
increased from zero, the dynamics moves from a regime showing purely incoherent relaxation,
to a regime which displays coherent oscillations {\it at the same temperature}.
The increase in correlations is thus able to extend the region of parameter space which permits coherence,~\cite{nazir09} as we shall now explore in greater detail.
\subsection{Coherent to incoherent transition}
We now return our attention to the crossover from
coherent to incoherent transfer, defined by Eq.~({\ref{critical_condition}}). Intuitively, we might expect the dynamics in the low-temperature (or weak-coupling) regime to be
coherent; for example,
in Fig.~{\ref{e-zero_dynamics}}
incoherent relaxation only occurs in the high-temperature limit.
If we therefore assume that the crossover itself occurs in the high-temperature regime, it is possible to derive
an analytic expression governing the crossover temperature by approximating the rates $\Gamma_y$ and $\Gamma_z$. Details of this approximation, and its range of validity, can be found in Appendix~{\ref{high_temperature_rates}}. Generally, for high enough temperatures and/or strong enough system-bath coupling (such that $\beta V_R\ll1$) we can approximate $\gamma_{xx}(\eta)\approx\gamma_{yy}(\eta)\approx\gamma_{yy}(0)$ in $\Gamma_y$ and $\Gamma_z$, where
\begin{equation}
\gamma_{yy}(0)\approx\frac{\beta B^2\mathrm{e}^{\phi_0 C_0(x,y)}}{2 \sqrt{\pi C_2(x,y) \phi_0}},
\label{strong_rate}
\end{equation}
with $\phi_0=2 \pi^2 \alpha/\omega_0^2\beta^2$, $x=\pi d/c \beta$ and $y=\omega_c\beta$. The functions $C_0(x,y)$
and $C_2(x,y)$ are given by Eqs.~({\ref{C_0}}) and ({\ref{C_2}}), and the renormalisation factor $B$ by the product of
Eqs.~({\ref{B_vacuum}}) and ({\ref{B_thermal}}). If we further assume that the
energy shift $\lambda_3$ vanishes in the high-temperature limit, Eq.~({\ref{critical_condition}}) reduces to
\begin{equation}
(\Gamma_z-\Gamma_y)=4 V_R,
\label{critical_condition_2}
\end{equation}
and we arrive at the expression
\begin{equation}
\bigg(\frac{k_B T}{\omega_0}\bigg)^2=\frac{V}{\omega_0}\frac{B \mathrm{e}^{\phi_0 C_0(x,y)}}{4 \sqrt{2 \pi^3 \alpha C_2(x,y)}},
\label{approx_critical_condition}
\end{equation}
with solution, $T_c$, giving the crossover temperature separating the coherent and incoherent regimes.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.45\textwidth]{newTCa.eps}
\caption{Crossover temperature separating the coherent and incoherent regimes against cut-off frequency, for
levels of correlation given by $\mu= c/\omega_0d=0$, $\mu=0.5$ and $\mu=1$, increasing as shown. The solid blue curves have been calculated from Eq.~(\ref{critical_condition}) (using the full rates), while the dashed red curves are solutions to the high-temperature approximation, Eq.~({\ref{approx_critical_condition}}). The inset shows the dependence
on the level of correlation ($1/\mu=\omega_0d/c$) for different cutoffs, $\omega_c/\omega_0=2$, $\omega_c/\omega_0=3$, and $\omega_c/\omega_0=4$, again increasing as shown. Parameters: $\alpha=0.05$ and $V/\omega_0=0.5$.}
\label{scaling_TC2}
\end{center}
\end{figure}
The dependence of $T_c$ on the various parameters involved in the problem is not straightforward, owing to the temperature dependence in the renormalisation factor $B$, in the functions $C_0$ and $C_2$, and in $\phi_0$.
In fact, there are three distinct and important temperature scales which determine when coherent or incoherent processes dominate: $T_0=\omega_0/(\sqrt{2\alpha}\pi k_B)$, which depends upon the system-bath coupling strength; $T_x=c/d\pi k_B$, which arises due to the fluctuation correlations and becomes unimportant in the uncorrelated case ($T_x\rightarrow0$ as $d\rightarrow\infty$); and $T_y=\omega_c/k_B$, dependent upon the cut-off frequency, and irrelevant in the scaling limit ($y\rightarrow\infty$). Hence, changes in any of $\alpha$, $d$, or $\omega_c$ can have an effect on the crossover temperature.
For example, the main part of Fig.~{\ref{scaling_TC2}} shows the solution to Eq.~({\ref{approx_critical_condition}}), i.e. the crosover temperature $T_c$,
as a function of the dimensionless cut-off frequency $\omega_c/\omega_0$. A calculation using Eq.~(\ref{critical_condition}) with the full rates, and including $\lambda_3$, is also shown for comparison. The three pairs of curves correspond to increasing levels of correlation, ordered as indicated. We see that, except for small $\omega_c/\omega_0$ in the case $\mu=0$,
where $\lambda_3$ becomes important, solutions to Eq.~(\ref{approx_critical_condition}) give an excellent approximation to the crossover temperature calculated using the full rates. This confirms that the coherent-incoherent crossover does indeed occur in the high-temperature (multiphonon) regime, and consequently could not be captured by a weak system-bath coupling treatment.
As the cut-off frequency is increased from its minimum value, the crossover temperature begins to decrease. This behaviour can be understood qualitatively by examining Eq.~({\ref{critical_condition_2}}), and considering the competition this condition captures between the rate $\Gamma_z-\Gamma_y$ and the coherent interaction $V_R$ in defining the nature of the dynamics. Larger values of the cut-off frequency correspond to
smaller values of the renormalised interaction strength $V_R$ (see e.g. Eq.~(\ref{B_vacuum})), while
the rates $\Gamma_y$ and $\Gamma_z$ vary less strongly with $\omega_c$ in this regime. Thus, increasing $\omega_c$ from its minimum value decreases $V_R$, and therefore reduces the range of temperatures for which $4V_R>\Gamma_z-\Gamma_y$ and coherent transfer can take place. Thus, the crossover temperature falls. Physically, this can be understood by noting that as the cut-off frequency is increased, so too is the effective frequency range and peak magnitude of the system-bath interaction, characterised by the spectral density [Eq.~(\ref{spectral_density})]. Hence, increasing from small $\omega_c/\omega_0$, the environment begins to exert an enhanced influence on the system behaviour, and so coherent dynamics no longer survives to such high temperatures. As $\omega_c$ continues to increase, however, we see the crossover temperature then begins to rise. The renormalisation factor $B$ tends to zero with increasing $\omega_c$ and here becomes the dominating quantity, thus causing the rate $\Gamma_z-\Gamma_y\sim \mathcal{O}(B^2)$ to vanish faster than the renormalised donor-acceptor coupling $V_R=BV$.
The interplay between the size of $\omega_c$ and the level of spatial correlation is best understood by considering
the inset of Fig.~{\ref{scaling_TC2}}. For all curves shown the crossover temperature increases as the distance $d$ is reduced, since the level of correlation $\mu$ increases correspondingly. As we have seen previously in Fig.~\ref{resonant_dynamics}, stronger correlations allow coherent dynamics to be observed at higher temperatures; since environmental effects are suppressed, so the crossover temperature $T_c$ must rise. This behaviour can
be attributed to an increase in the renormalised interaction strength, $V_R$, in relation to the rate $\Gamma_z-\Gamma_y$, this time with variations in the correlation level $\mu$. Interestingly, as the cut-off frequency is increased up to $\omega_c/\omega_0=4$ (lowest curve), we see that not only does the crossover temperature decrease, but also that the degree of correlation necessary to show a marked rise in $T_c$ increases. As can be seen by comparing the separation between the different curves in the main part of the figure, increasing the cut-off frequency tends to
suppress the extent to which correlations are able to protect coherence in the system. This tallies with the dynamics shown in Fig.~{\ref{resonant_dynamics}}, for which $\omega_c/\omega_0=4$, and correlations as high as $\mu=2$ were needed before a significant change in behaviour was seen. Finally, since the renormalisation factor $B$ tends to a constant non-zero value as the correlations vanish at large $d$ (as opposed to $B\rightarrow0$ as $\omega_c\rightarrow \infty$), the dependence of the crossover
temperature on $\mu$ is monotonic, in contrast to its dependence on $\omega_c$.
\section{Off resonance}
\label{off_resonance}
It is often the case in practice that the donor and acceptor will have
different excited state energies, $\epsilon_1-\epsilon_2=\epsilon\neq0$, and so we now turn our attention to energy transfer dynamics under off-resonant conditions.
Regarding the coherent to incoherent
transition, in the resonant case we were able to identify this point with a pair of conjugate eigenvalues converging on the real axis, thus changing oscillatory terms into relaxation. We might hope that in the off-resonant case we are able to
establish a similar crossover criterion, and again use this to investigate the effects of bath correlations and the cut-off frequency. However, we shall see that such an identification is less straightforward in the off-resonant regime.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.45\textwidth]{off_resonant_plots.eps}
\caption{Population difference for
an off-resonant donor-acceptor pair
as a function of scaled time $\omega_0 t$. Temperatures $k_B T/\omega_0=1$ (blue dashed curve), $k_B T/\omega_0=5$ (green dotted curve), $k_B T/\omega_0=12$ (orange solid curve) and
$k_B T/\omega_0=20$ (red dot-dashed curve) are shown. Parameters: $\alpha=0.05$, $V/\omega_0=0.5$, $\epsilon/\omega_0=1$, $\omega_c/\omega_0=4$, and $\mu=0.5$.}
\label{e-0d1/5_dynamics}
\end{center}
\end{figure}
We first present the full Bloch equations describing the evolution of our donor-acceptor pair for arbitrary energy mismatch. As in the resonant case, we have an equation of motion of the form
$\dot{\vec{\alpha}}=M\cdot\vec{\alpha}+\vec{b}$, but now the matrix $M$ is given by
\begin{equation}
M=\left(
\begin{array}{ccc}
-\Gamma_x & -(\epsilon+\lambda_1) & 0 \\
(\epsilon+\lambda_2) & -\Gamma_y & -2 B V_R \\
B^{-1} \zeta & B^{-1}(2 V_R+\lambda_3) & -\Gamma_z
\end{array}
\right),
\label{full_M}
\end{equation}
with $\vec{b}=( -B \kappa_x,-B\kappa_y,-\kappa_z)^T$. The rates become
\begin{align}
\Gamma_x&=V^2(\gamma_{yy}(\eta)+\gamma_{yy}(-\eta))\label{Gamma_x},\\
\Gamma_y&=2V^2\left(\frac{4V_R^2}{\eta^2}\gamma_{xx}(0)+\frac{\epsilon^2}{2 \eta^2}(\gamma_{xx}(\eta)+\gamma_{xx}(-\eta))\right),
\end{align}
with $\Gamma_z=\Gamma_x+\Gamma_y$, and the energy shifts
\begin{align}
\lambda_1&=\frac{2V^2\epsilon}{\eta}(S_{yy}(\eta)-S_{yy}(-\eta)),\\
\lambda_2&=\frac{2V^2\epsilon}{\eta}(S_{xx}(\eta)-S_{xx}(-\eta)),\\
\lambda_3&=\frac{4V^2V_R}{\eta}(S_{yy}(\eta)-S_{yy}(-\eta)).
\end{align}
The remaining quantities are
\begin{align}
\zeta&=\frac{4V^2V_R \epsilon}{\eta^2}\left(\gamma_{xx}(0)-\frac{1}{2}(\gamma_{xx}(\eta)+\gamma_{xx}(-\eta))\right),\\
\kappa_x&=\frac{2V^2V_R}{\eta}(\gamma_{yy}(\eta)-\gamma_{yy}(-\eta)),\\
\kappa_y&=\frac{8V^2V_R \epsilon}{\eta^2}\left(S_{xx}(0)-\frac{1}{2}(S_{xx}(\eta)+S_{xx}(-\eta))\right),\\
\kappa_z&=\frac{V^2\epsilon}{\eta}\left((\gamma_{xx}(\eta)-\gamma_{xx}(-\eta))+(\gamma_{yy}(\eta)-\gamma_{yy}(-\eta))\right).
\label{kappa_z}
\end{align}
Here, $\eta=\sqrt{\epsilon^2+4 V_R^2}$ is the system Hamiltonian eigenstate splitting in the polaron frame.
To exemplify the dynamics generated by the full off-resonant Bloch equations,
in Fig.~\ref{e-0d1/5_dynamics} we plot the evolution of the population difference in the case of
donor-acceptor energy mismatch, $\epsilon=2V$.
By comparison of Fig.~\ref{e-zero_dynamics} (plotted in the resonant case) and Fig.~\ref{e-0d1/5_dynamics}, we see that
the presence of a substantial
energy mismatch causes
the low-temperature population oscillations to increase in frequency but decrease markedly in amplitude, such that for $k_BT/\omega_0=5$ oscillations are now
almost imperceptible. We also see that the population difference tends to a non-zero steady-state
at low temperatures, as we might expect from simple thermodynamic arguments, since the states $\alpha_z=1$ and $\alpha_z=-1$ now have different energies. As the temperature is raised, however, the dynamics still looks to be approaching that shown in the resonant case of Fig.~\ref{e-zero_dynamics}.
\subsection{Correlated fluctuations}
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.45\textwidth]{off_resonant_dynamics.eps}
\caption{Population difference as a function of scaled time $\omega_0 t$ for temperatures
of $k_B T/\omega_0=5$ (blue dashed curve) and $k_B T/\omega_0=10$ (red dotted curve), and for separations
corresponding to no fluctuation correlations, $\mu=c/\omega_0d=0$ (top), weak correlations, $\mu=0.5$ (middle)
and strong correlations $\mu=2$ (botttom). The insets show the evolution of the
corresponding coherence $\alpha_y$. Parameters: $\alpha=0.05$, $V/\omega_0=0.5$, $\epsilon/\omega_0=0.5$, and $\omega_c/\omega_0=4$.}
\label{off_resonant_dynamics}
\end{center}
\end{figure}
Let us now look at the effect of correlated fluctuations in the off-resonant dynamics. Though analysis of the
full Bloch equations is now more complicated than in the resonant limit, we should still expect changes in the level of donor-acceptor fluctuation correlation
to have a qualitatively similar effect on the transfer process as outlined in Section~\ref{resonant_transfer}. To illustrate that this is indeed the case, in Fig.~{\ref{off_resonant_dynamics}} we plot the donor-acceptor population dynamics under off-resonant conditions for three different levels of fluctuation correlation (increasing from top to bottom). Just as we found in the resonant case of Fig.~\ref{resonant_dynamics}, an increase in correlations enhances the lifetime of coherence present in the energy transfer process, and can even move the dynamics from a high temperature (or strong-coupling) predominantly incoherent regime to an {\it effective} low temperature (or weak-coupling) regime displaying pronounced coherent oscillations. In addition, in the off-resonant case stronger correlations also serve to amplify the coherent contribution to the full energy transfer dynamics (made up of distinct coherent and incoherent parts, as we shall show below), since the renormalised interaction strength $V_R$ increases in relation to the energy mismatch $\epsilon$.
In an effort to put these qualitative observations on a more quantitive footing we could analyse the eigensystem of the full off-resonant $M$ [Eq.~(\ref{full_M})], in a similar manner to the resonant case.
However, finding the eigensystem of $M$
is now far less straightforward and analytical solutions to the full Bloch equations
are consequently lengthy, and therefore of little direct use in gaining an understanding of the behaviour seen in Figs.~\ref{e-0d1/5_dynamics} and~{\ref{off_resonant_dynamics}}.
The rest of this section is thus devoted to deriving simplified expressions for the energy transfer dynamics in two
important limits: (i) weak-coupling, or strong correlations, where coherent dynamics can dominate, and (ii) high temperatures (and weak correlations), where the dynamics is similar in both the resonant and off-resonant cases. These expressions not only provide insight into the off-resonant behaviour of the system and the effect of correlated fluctuations, but also serve to highlight the difficultly in now defining a simple crossover criterion, as was possible in the resonant case.
\subsection{Weak-coupling (or strong correlation) limit}
We begin by considering the weak system-bath coupling limit, which we obtain by expanding all relevant quantities to first order in $J(\omega)$. In fact, for strong enough fluctuation correlations this limit is attainable even if the system-bath coupling is not weak and/or the temperature not low, due to the factor $(1-F_D(\omega,d))$ appearing in Eqs.~(\ref{B_integral}) and~(\ref{phi}).
With reference to our expressions for the correlation functions [Eqs.~({\ref{Lambda_xx}}) and
({\ref{Lambda_yy}})], we see that within this approximation $\Lambda_{xx}(\tau)\rightarrow 0$ while
$\Lambda_{yy}(\tau)$ remains finite. We may then set to zero all rates and energy shifts which are functions of $\Lambda_{xx}(\tau)$ only in Eq.~(\ref{full_M}). This results in the far simpler form
\begin{equation}
M_{W}=\left(
\begin{array}{ccc}
-\Gamma_{W} & -(\epsilon+\lambda_1) & 0 \\
\epsilon & 0 & -2BV_R \\
0 & B^{-1}(2V_R+\lambda_3) & -\Gamma_{W}
\end{array}
\right),
\end{equation}
where the weak-coupling rate is given by~\cite{rozbicki08}
\begin{equation}
\Gamma_{W}=4\pi \bigg(\frac{V_R}{\eta}\bigg)^2 J(\eta)(1-F_D(\eta,d))\coth(\beta \eta/2),
\label{off_resonant_weak_rate}
\end{equation}
and the two energy shifts may be written
$\lambda_1=(\epsilon/\eta)\Lambda$ and $\lambda_3=(2 V_R/\eta)\Lambda$, with $\Lambda=2 V^2(S_{yy}(\eta)-S_{yy}(-\eta))$.
The inhomogeneous term becomes $\vec{b}_{W}=\{-B \kappa_x,0,-(\epsilon/2V_R)\kappa_x\}^T$ in the same limit, which leads to the weak-coupling steady state values of
\begin{align}
\alpha_x(\infty)&=-\frac{2BV_R}{\eta} \tanh(\beta \eta/2),\label{sigma_x_steady_state}\\
\alpha_z(\infty)&=-\frac{\epsilon}{\eta}\tanh(\beta \eta/2),
\label{sigma_z_steady_state}
\end{align}
and $\alpha_y(\infty)=0$.
As in the resonant case (in which there was no weak-coupling approximation), this steady-state has the same form as that expected from a standard weak-coupling approach, though with the replacement $V\rightarrow V_R$, and the extra factor of $B$ suppressing the coherence $\alpha_x(\infty)$. As the energy mismatch increases in relation to $V$, the weak-coupling steady state therefore becomes increasingly localised in the lower energy state $|2\rangle\equiv|GX\rangle$. Interestingly, this contrasts with the qualitatively incorrect form (at low temperatures at least) given by the Non-Interacting Blip Approximation (NIBA), $\alpha_z^{\rm NIBA}(\infty)=-\tanh{(\beta\epsilon/2)}$,~\cite{weissbook, leggett87} which predicts complete localisation in the lower energy state at zero temperature, regardless of the size of $\epsilon/V$. We should thus expect the present theory to fair far better than the NIBA for low-temperatures (or weak-coupling) in the off-resonant case, $\epsilon\neq0$. The rate $\Gamma_W$ given in Eq.~(\ref{off_resonant_weak_rate}) is also of the form expected from a weak-coupling treatment, though once more with the renormalisation $V\rightarrow V_R$. In fact, such a replacement is sometimes made by hand in weak-coupling theories to provide agreement with numerics over a larger range of parameters,~\cite{rozbicki08} though it arises naturally in the polaron formalism here. We can therefore conclude that, in addition to allowing for the exploration of multiphonon effects,~\cite{jang08, nazir09, wurger98,rae02} the polaron master equation provides a rigorous way to explore the (single-phonon) weak-coupling regime for spectral densities of the type in Eq.~(\ref{spectral_density}).~\cite{wurger98}
As before, to find the time evolution of $\vec{\alpha}$ we evaluate the
eigensystem of $M_W$ and use Eq.~({\ref{general_evolution}}). For the initial state $\vec{\alpha}(0)=\{0,0,1\}^T$ we find population dynamics
\begin{eqnarray}
\alpha_z(t)&{}={}&\frac{\epsilon}{\eta}\bigg(\frac{\epsilon}{\eta}\mathrm{e}^{-\Gamma_{W} t}-\bigl(1-\mathrm{e}^{-\Gamma_{W} t}\bigr)\tanh(\beta\eta/2)\bigg)\nonumber\\
&&\:{+}\frac{4 V_R^2}{\eta^2}\mathrm{e}^{-\frac{\Gamma_{W} t}{2}}\bigg(\cos\Bigl(\frac{\xi_{\mathrm{W}} t}{2}\Bigr)-\frac{\Gamma_{W}}{\xi_{\mathrm{W}}}
\sin\Bigl(\frac{\xi_{\mathrm{W}} t}{2}\Bigr)\bigg),\nonumber\\
\label{weak_alphaz_t}
\end{eqnarray}
where the weak coupling oscillation frequency is given by
\begin{equation}
\xi_{\mathrm{W}}=\sqrt{4\eta(\eta+\Lambda)-\Gamma_{W}^2},
\label{xi_OW}
\end{equation}
which we expect to be real to be consistent with our original expansion.
The first term in Eq.~({\ref{weak_alphaz_t}}), proportional to $(\epsilon/\eta)$ and present nowhere in the resonant case, describes incoherent
relaxation towards the steady state value given by Eq.~({\ref{sigma_z_steady_state}}). The second term, proportional to
$(V_R/\eta)^2$ and having a similar form to the resonant dynamics, describes damped oscillations with frequency $\xi_{W}$. Importantly, these oscillations have a temporal maximum amplitude of $4V_R^2/\eta^2\leq 1$, compared to $1$ in the resonant case. The effect of the energy mismatch in this limit is thus to suppress the amplitude of any oscillations in the population difference, while increasing their frequency due to the dependence of $\xi_W$ on $\eta$ in Eq.~(\ref{xi_OW}), exactly as seen in Fig.~\ref{e-0d1/5_dynamics}.
\subsection{High temperature (or far from resonance) limit}
At high temperatures and weak correlations, we find that
the population dynamics appears to be relatively insensitive to the size of the energy mismatch. In order to investigate this effect in more detail, we shall now make a high-temperature (or strong system-bath coupling) approximation to the full energy transfer dynamics.
Specifically, we consider the regime $V_R/\epsilon \ll 1$. This limit can in fact be achieved in two possible ways. Firstly,
recalling that $V_R=BV$, we see that $V_R$ can be made small by increasing the system-bath coupling strength
or temperature, such that $B \ll 1$. Alternatively, if the donor-acceptor pair are far from resonance, the ratio
$V/\epsilon$ will be small, and hence $V_R/\epsilon$ smaller still. Observing that the correlation functions given
by Eqs.~({\ref{Lambda_xx}}) and ({\ref{Lambda_yy}}) are both proportional to $B^2$, we can see that all
dissipative terms in the equation of motion, $\dot{\vec{\alpha}}=M\cdot\vec{\alpha}+\vec{b}$, are at least of order $V_R^2$. We proceed by
keeping only terms up to order $(V_R/\epsilon)^2$ in the full off-resonant $M$ and $\vec{b}$. This allows us to set $\lambda_3$, $\zeta$, $\kappa_x$ and $\kappa_y$ to
zero, while the remaining quantities reduce to
\begin{align}
\Gamma_y&=V^2(\gamma_{xx}(\eta)+\gamma_{xx}(-\eta)),\\
\Gamma_z&=V^2\big(\gamma_{xx}(\eta)+\gamma_{xx}(-\eta)+\gamma_{yy}(\eta)+\gamma_{yy}(-\eta)\big),\\
\lambda_1&=2V^2(S_{yy}(\eta)-S_{yy}(-\eta)),\\
\lambda_2&=2V^2(S_{xx}(\eta)-S_{xx}(-\eta)),\\
\kappa_z&=V^2\big(\gamma_{xx}(\eta)-\gamma_{xx}(-\eta)+\gamma_{yy}(\eta)-\gamma_{yy}(-\eta)\big).
\end{align}
Hence, in the high-temperature limit, Eq.~(\ref{full_M}) takes on the simpler form
\begin{equation}
M_{\mathrm{HT}}=\left(
\begin{array}{ccc}
-(\Gamma_z-\Gamma_y)& -(\epsilon+\lambda_1) & 0 \\
(\epsilon+\lambda_2) & -\Gamma_y & -2BV_R \\
0 & 2B^{-1}V_R & -\Gamma_z
\end{array}
\right),
\end{equation}
while the inhomogeneous term reduces to $\vec{b}_{\mathrm{HT}}=\{0,0,-\kappa_z\}^T$. We then find the approximate steady-state population difference
\begin{align}
\alpha_z(\infty)=-\bigg(1+\frac{4 V_R^2}{\epsilon^2}\bigg(\frac{\Gamma_y}{\Gamma_z}-1\bigg)\bigg)\tanh(\beta \eta/2),
\end{align}
valid up to second order in $V_R/\epsilon$. For $V_R\ll\epsilon$, this steady-state is strongly localised in the low energy state ($\alpha_z(\infty)\approx-1$) if $\epsilon\gg k_BT$, though for $\epsilon\ll k_BT$ thermal effects dominate and $\alpha_z(\infty)\approx0$ as in the resonant case. Again, this behaviour tallies with Fig.~\ref{e-0d1/5_dynamics}.
To obtain the corresponding population dynamics, we note in reference to Eq.~(\ref{general_evolution}) that the coefficients $a_i$, the eigenvectors $\vec{m}_i$, and the eigenvalues $q_i$ will contain powers of our expansion parameter $V_R/\epsilon$.
Expanding both $q_i$ and the products $a_i \vec{m}_i$ to second order, we find
\begin{eqnarray}
\alpha_z(t)&=&\mathrm{e}^{-\Gamma_z t}\left(1-\frac{4V_R^2}{\epsilon^2}\right)+\frac{4V_R^2}{\epsilon^2}\mathrm{e}^{-\Gamma_z t/2}\cos(\bar{\epsilon} t)\nonumber\\
&&\:{-}(1-\mathrm{e}^{-\Gamma_z t})\tanh\left(\frac{\beta\eta}{2}\right)\left[1+\frac{4V_R^2}{\epsilon^2}\bigg(\frac{\Gamma_y}{\Gamma_z}-1\bigg)\right]\nonumber\\
\label{alpha_z_HT}
\end{eqnarray}
where the shifted oscillation frequency is
\begin{equation}
\bar{\epsilon}=\epsilon+(1/2)(\lambda_1+\lambda_2)+2 \epsilon(V_R/\epsilon)^2.
\label{barepsilon}
\end{equation}
As in the weak-coupling case [Eq.~(\ref{weak_alphaz_t})] the evolution of
the donor-acceptor population difference consists of two contributions; incoherent relaxation towards the steady-state, and an oscillatory component with vanishing amplitude as $V_R/\epsilon\rightarrow 0$. The energy mismatch again serves to suppress oscillations in the population difference.
The most striking feature, however, of Eq.~({\ref{alpha_z_HT}}) is that there is an oscillatory component at frequency $\bar{\epsilon}$ at all. In the high-temperature
limit, we might expect that this frequency would reach a point where it becomes imaginary and $\alpha_z(t)$ displays
purely incoherent relaxation, as in the equivalent resonant case. However, we can see that this is not the case since $\bar{\epsilon}$ is always real by definition. Furthermore, at very high temperatures $\bar{\epsilon}\rightarrow\epsilon$, and it therefore also remains finite. Eq.~({\ref{alpha_z_HT}}) thus highlights an important difference between the energy transfer dynamics in resonant and off-resonant situations. In the resonant case, as temperature is increased, the energy transfer process becomes
less coherent through a reduction in oscillation frequency (i.e. $V_R$ becomes small in comparison to $\Gamma_z-\Gamma_y$), eventually reaching a point at which population relaxes incoherently towards the steady state. In the off-resonant case, the transfer process becomes less coherent
predominately through a reduction in oscillation amplitude. For high temperatures, an oscillatory component is still (in theory) present in the system, although it becomes ever more dominated by incoherent relaxation towards the steady-state population distribution, which depends upon the ratio $\epsilon/k_BT$. These features are clearly seen in Fig.~\ref{e-0d1/5_dynamics}.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.45\textwidth]{small_resonance.eps}
\caption{Coherence ($\alpha_y$) as a function of scaled time $\omega_0 t$ for resonant ($\epsilon=0$, solid curve) and off-resonant ($\epsilon/\omega_0=0.2$, dashed curve) cases. The temperature, $k_B T/\omega_0=13$, is chosen to be above the relevant crossover $T_c$ in the resonant case, such that the resonant dynamics is guaranteed to be incoherent. Parameters: $\alpha=0.05$, $V/\omega_0=0.5$,
$\omega_c/\omega_0=4$, and $\mu=0.5$. The inset shows the corresponding
population dynamics.}
\label{smallcoherences}
\end{center}
\end{figure}
Only to first order in $V_R/\epsilon$ do our expressions predict purely incoherent off-resonant population transfer:
\begin{equation}
\alpha_z(t)=\mathrm{e}^{-\Gamma_z t}-(1-\mathrm{e}^{-\Gamma_z t})\tanh(\beta \eta/2).
\end{equation}
Let us also consider the evolution of
$\alpha_y$ in the same limit:
\begin{equation}
\alpha_y(t)=-\frac{2 B V_R}{\epsilon}\mathrm{e}^{-(1/2)\Gamma_z t}\sin(\bar{\epsilon} t).\label{alphayhighT}
\end{equation}
Hence, although the donor-acceptor population itself evolves entirely incoherently in this limit, the coherences may still perform oscillations due to the energy mismatch. To illustrate the difference in the transition to incoherent population transfer on- and off-resonance, in the main part of Fig.~\ref{smallcoherences} we plot the evolution of the coherence $\alpha_y(t)$ in both cases. The parameters have been chosen such that the resonant dynamics is in the incoherent regime ($T>T_c$), hence the resonant $\alpha_y$ displays no oscillations [see Eq.~(\ref{y_general_evolution})]. In accordance with Eq.~(\ref{alphayhighT}), however, the introduction of an energy mismatch induces oscillations in the donor-acceptor coherence. While these oscillations have an almost negligible amplitude, this behaviour serves to illustrate the subtlety in defining a strict crossover from coherent to incoherent dynamics in the off-resonant case. In particular, despite the different forms of coherence behaviour, the corresponding (essentially incoherent) population dynamics shown in the inset is almost indistinguishable in the two cases, even though there should still be a strongly suppressed coherent contribution in the off-resonant curve.
\begin{figure}[!t]
\begin{center}
\includegraphics[width=0.45\textwidth]{coherences.eps}
\caption{Coherence ($\alpha_y$) as a function of scaled time $\omega_0 t$ for temperatures
of $k_B T/\omega_0=1$ (blue dashed curve), $k_B T/\omega_0=5$ (green dotted curve), $k_B T/\omega_0=12$ (orange solid curve) and $k_B T/\omega_0=20$ (red dot-dashed curve). Parameters: $\alpha=0.05$, $V/\omega_0=0.5$,
$\omega_c/\omega_0=4$, $\epsilon/\omega_0=2$ and $\mu=0.5$. The inset shows the corresponding
population dynamics.}
\label{coherences}
\end{center}
\end{figure}
An alternative way to obtain oscillations of the coherence $\alpha_y$ in a regime of predominantly incoherent population transfer is to introduce a large energy mismatch (i.e. make $V/\epsilon$ small) at low temperature, as shown in Fig.~{\ref{coherences}}. Here, for the lowest temperature considered the population relaxes towards its steady state value with little sign of oscillation, while the coherence performs oscillations with a significant amplitude and considerable lifetime. This behaviour is strongly suppressed, however, as temperature increases, such that $k_BT>\epsilon$.
\section{Summary}
\label{summary}
Motivated by recent experiments which suggest that quantum coherence can survive in energy transfer processes even under potentially adverse environmental conditions,~\cite{lee07,engel07, calhoun09,collini09sc,collini09,collini10,panitchayangkoona10,mercer09,womick09}
we have investigated various factors that determine the nature of the energy transfer dynamics in a model donor-acceptor pair. To do so, we used a polaron transform, Markovian master equation technique.~\cite{nazir09} This formalism is attractive as it allows for exploration of both the low-temperature (or weak-coupling) and high-temperature (or strong-coupling) regimes, as well as reliable interpolation between these two limits, provided the ratio $V/\omega_c$ does not become too large.~\cite{wurger98,jang09,rae02} We are also able to consistenly describe off-resonant effects, unlike in the NIBA,~\cite{leggett87,weissbook} and the influence of bath correlations.
In the resonant case we identified a crossover temperature separating coherent and incoherent energy transfer. We found a non-trivial dependence of this temperature on both the degree of spatial correlation within the bath-induced fluctuations, and also on the cut-off frequency of the bath spectral density. Smaller cut-off frequencies were found to enhance the extent to which bath spatial correlations are able to protect coherence in the system. The crossover generally occurs in a high-temperature limit where multiphonon effects dominate, and so could not be captured by a standard perturbative treatment of the system-bath interaction.
In the off-resonant case we found that coherent and incoherent regimes are less easily defined. In particular,
for a sufficiently large energy mismatch between the donor and acceptor, coherence
can in theory be present at all but infinite temperatures, albeit with an ever decreasing amplitude. However, using analytic expressions derived in various limits, we were able to characterise the off-resonant energy transfer process over much of the parameter space, illustrating the suppression of coherence in the population dynamics with increasing temperature or energy mismatch. We also showed that strong correlations have a qualitatively similar
effect to the resonant case, protecting coherence in the transfer process.
While we have concentrated in this work on elucidating general features of donor-acceptor energy transfer dynamics using a simple model system, the insight we have gained could be relevant to a variety of systems. In addition to those already mentioned,~\cite{lee07,engel07, calhoun09,collini09sc,collini09,collini10,panitchayangkoona10,mercer09,womick09} closely-spaced pairs of semiconductor quantum dots could provide a solid-state implementation of the model studied here.~\cite{gerardot05} In particular, our polaron master equation theory provides a bridge between the weak~\cite{rozbicki08} and strong~\cite{govorov05} system-bath coupling approximations already explored in this context. It would also be interesting to analyse the energy transfer dynamics of larger donor-acceptor complexes within the polaron formalism,~\cite{kolli10} to see if further understanding of the interplay between coherent and incoherent processes in such systems could be obtained. Finally, it would be desirable to perform a thorough investigation of the regime of validity of the polaron approach by comparison to numerically exact techniques.~\cite{makri98}
\acknowledgements
We are very grateful to Alexandra Olaya-Castro, Andrew Fisher, and Avinash Kolli for interesting discussions and useful comments. This research was supported by the EPSRC and Imperial College London.
|
{
"timestamp": "2011-04-05T02:03:10",
"yymm": "1009",
"arxiv_id": "1009.3942",
"language": "en",
"url": "https://arxiv.org/abs/1009.3942"
}
|
\section{\bf Introduction and Main Results}\label{intro}
There is a natural differential-topological invariant,
called the {\it Yamabe invariant},
which arises from a variational problem for the functional $E$ below
on a given compact smooth $n$-manifold $M$ (without boundary) of dimension $n \geq 3$.
It is well known that a Riemannian metric on $M$
is {\it Einstein} if and only if
it is a critical point of the normalized Einstein-Hilbert functional $E$
on the space $\mathcal{M}(M)$ of all Riemannian metrics on $M$
$$E : \mathcal{M}(M) \rightarrow \mathbb{R},\quad
g \mapsto E(g) := \frac{\int_MR_g d\mu_g}{\textrm{Vol}_g(M)^{(n-2)/n}}. $$
Here, $R_g, d\mu_g$ and $\textrm{Vol}_g(M)$ denote respectively
the scalar curvature, the volume element of $g$ and the volume of $(M, g)$.
Because the restriction of $E$ to any conformal class
$$C = [g] := \{ \textrm{e}^{2f}\cdot g\ |\ f \in C^{\infty}(M) \}$$
is bounded from below,
we can consider the following conformal invariant
(called the {\it Yamabe constant} of $(M, C)$)
$$
Y(M, C) := \inf_{\tilde{g} \in C} E(\tilde{g}).
$$
A remarkable theorem \cite{Yamabe, Trudinger, Aubin, Schoen-1, ScYa-3}
(cf.~\cite{Aubin-Book, BrLi, LePa, Schoen-3, ScYa-Book}) of
Yamabe, Trudinger, Aubin, and Schoen asserts that
each conformal class $C$ contains a minimizer $\hat{g}$ of $E|_C$,
called a {\it Yamabe metric} (or a {\it solution of the Yamabe problem}),
which is of constant scalar curvature
$$
R_{\hat{g}} = Y(M, C)\cdot \textrm{Vol}_{\hat{g}}(M)^{-2/n}.
$$
The study of the second variation of $E$
done in \cite{Koiso, Schoen-3} (cf.~\cite{Besse-Book})
leads naturally to the definition of
the following differential-topological invariant
$$
Y(M) := \sup_{C \in \mathcal{C}(M)} \inf_{g \in C} E(g) = \sup_{C \in \mathcal{C}(M)} Y(M, C),
$$
where $\mathcal{C}(M)$ denotes the space of all conformal classes on $M$.
This invariant is called the {\it Yamabe invariant} (or {\it $\sigma$-invariant}) of $M$
and it was introduced independently by
O.~Kobayashi~\cite{Kobayashi-1} and Schoen~\cite{Schoen-2}
(see also \cite{Kobayashi-2, Schoen-3}).
In the study of Yamabe invariant,
with certain geometric non-collapsing assumptions,
we will often encounter {\it Riemannian orbifolds} (or {\it Riemannian multi-folds} more generally)
as the limit spaces for sequences of Yamabe metrics (cf.~\cite{Ak, TiVi, Vi}).
For a compact $n$-orbifold $M$ with an orbifold metric $g$,
one can also define the corresponding Yamabe constant $Y(M, [g]_{orb})$ and Yamabe invariant $Y^{orb}(M)$
(see Section~2 or \cite{AkBo-2} for details).
Let $M_1$ and $M_2$ be compact $n$-orbifolds with same number of finite singularities
$\{ \check{p}_1, \cdots, \check{p}_{\ell} \}$
and $\{ \check{q}_1, \cdots, \check{q}_{\ell} \}$
respectively.
Assume that each corresponding singularities $\check{p}_j$ and $\check{q}_j$
have a same structure group $\Gamma_j ( < O(n) )$.
For each $j$,
let $B(\check{p}_j) ( \subset M_1 )$ and $B(\check{q}_j) ( \subset M_2 )$ denote respectively
open geodesic balls of sufficiently small radiuses centered at $\check{p}_j$ and $\check{q}_j$
with fixed reference orbifold metrics.
Then, the boundaries of these two balls can be naturally identified by a canonical diffeomorphism.
Let
$$
N := \big{(} M_1 - \sqcup_{j = 1}^{\ell} B(\check{p}_j) \big{)} \cup_Z
\big{(} M_2 - \sqcup_{j = 1}^{\ell} B(\check{q}_j) \big{)}
$$
be the sum of $M_1 - \sqcup_{j = 1}^{\ell} B(\check{p}_j)$ and
$M_2 - \sqcup_{j = 1}^{\ell} B(\check{q}_j)$
along their common boundary
$Z := \partial \big{(} \sqcup_{j = 1}^{\ell} B(\check{p}_j) \big{)}
= \partial \big{(} \sqcup_{j = 1}^{\ell} B(\check{q}_j) \big{)}$.
Note that $N$ has a canonical smooth structure as manifold.
For simplicity, in Section~4, we will abbreviate the above decomposition as
the generalized connected sum
$$
N = M_1 \#_{\sqcup_{j=1}^{\ell} (S^{n-1}/\Gamma_j)} M_2.
$$
One of main purposes of this paper is to prove the following fundamental inequality
for the estimate of the orbifold Yamabe invariant from above
and a sufficient condition for the equality in this inequality.
The inequality also includes a criterion for the non-positivity of the invariant:
\begin{thmA}
Under the above understandings,
assume that
$$
Y(N) \leq 0~({\rm resp.}~< 0)\quad {\rm and }\quad Y^{orb}(M_2) > 0~({\rm resp.}~\geq 0).
$$
Then,
$$
Y^{orb}(M_1) \leq Y(N) \leq 0.
$$
Moreover,
if $M_1$ admits an orbifold metric $\check{g}$
of constant scalar curvature satisfying $E(\check{g}) = Y(N)$,
then
$$
Y^{orb}(M_1) = Y(M_1, [\check{g}]_{orb}) = Y(N) \leq 0.
$$
\end{thmA}
On the computation of Yamabe invariants for {\it smooth} manifolds,
a first remarkable result is the following proved by Aubin~\cite{Aubin} (cf.~\cite{Aubin-Book}) :
$$
Y(M, C) \leq Y(S^n, [g_0]) = E(g_0)~\Big{(} = n(n-1) {\rm Vol}_{g_0}(S^n)^{2/n}~\Big{)}
$$
for any $C \in \mathcal{C}(M)$,
where $g_0$ is the standard metric of constant curvature one on the standard $n$-sphere $S^n$.
This implies both the universal estimate for $Y(M)$ from above and the computation of $Y(S^n)$
$$
Y(M) \leq Y(S^n) = n(n-1) {\rm Vol}_{g_0}(S^n)^{2/n}.
$$
Kobayashi~\cite{Kobayashi-1, Kobayashi-2} and Schoen~\cite{Schoen-3} proved that
$$
Y(S^{n-1} \times S^1) = Y(S^n).
$$
Kobayashi also gave two kind of proof for it (see \cite{AkFlPe} for the third one),
one \cite{Kobayashi-2} of them especially is based on the following important inequality,
called {\it Kobayashi's inequality}:
$$
Y(M_1^n \# M_2^n) \geq
\begin{cases}\
- \bigl(|Y(M_1^n)|^{n/2}+|Y(M_2^n)|^{n/2}\bigr)^{2/n}\ \cdots\
Y(M_1^n), Y(M_2^n) \leq 0, \\
\qquad \min \{ Y(M_1^n), Y(M_2^n) \}\ \cdots\ \text{otherwise}
\end{cases}
$$
for any two compact $n$-manifolds $M_1, M_2$.
This has been extended to some useful surgery theorems \cite{AmDaHu, Petean-1, PeYu}.
On the other hand,
some classification theorems for manifolds with positive scalar curvature metric
\cite{GrLa-1, GrLa-2, ScYa-1, ScYa-2, Stolz} lead to many examples of manifolds
with zero (or non-positive) Yamabe invariant, for instance, $Y(T^n) = 0$ for the $n$-torus $T^n$
(see \cite{Petean-2} for further development).
In 1995, LeBrun~\cite{Le-1} computed the Yamabe invariants of smooth compact quotients of complex-hyperbolic $2$-space,
which was the first example of manifolds with negative Yamabe invariant.
He and collaborators~\cite{GuLe, IsLe, Le-2, Le-3, Le-4} also computed the Yamabe invariants
for a large class of $4$-manifolds, including K\"ahler surfaces
$X$ with either $Y(X) < 0$ or $0 < Y(X) < Y(S^4)$
(see~\cite{AkNe, Anderson, BrNe} for $3$-manifolds $M^3$ with either $Y(M^3) < 0$ or $0 < Y(M^3) < Y(S^3)$~).
In particular, for any minimal complex surface of general type $X$,
he~\cite{Le-2} computed its Yamabe invariant $Y(X)$ to be
$$
Y(X) = - 4 \sqrt{2} \pi \sqrt{2 \chi(X) + 3 \tau(X)} < 0,
$$
where $\chi(X)$ and $\tau(X)$ are respectively the Euler characteristic and signature of $X$.
Moreover, if $X$ contains $(- 2)$-curves, there exist a sequence of metrics $\{ g_i \}_i$ on $X$
and a K\"ahler-Einstein orbifold metric $\check{g}$ on the canonical model $X_{can}$ of $X$
such that
$$
\lim_{i \to \infty} Y(X, [g_i]) = Y(X),\qquad \lim_{i \to \infty} d_{GH} \big{(} (X, g_i), (X_{can}, \check{g}) \big{)} = 0.
$$
Here, $d_{GH}$ denotes the Gromov-Hausdorff distance.
This result suggests naturally the following question :
``~Can one describe rigorously the above fact
in terms of $Y(X_{can}, [\check{g}]_{orb})$ and $Y^{orb}(X_{can})$~?~''
The other of main purposes of this paper is to answer it.
\begin{thmB}
Under the above settings,
the following holds
$$
Y^{orb}(X_{can}) = Y(X_{can}, [\check{g}]_{orb}) = Y(X).
$$
\end{thmB}
In Section~2, we recall the definition on the orbifold Yamabe invariant from \cite{AkBo-2}
and explain briefly some terminologies.
For the proof of Theorem~A,
we also recall some necessary terminologies and basic results
on the Yamabe invariant of cylindrical manifolds~\cite{AkBo-1}.
Applying these results to the orbifold Yamabe invariant,
we prove the first assertion of Theorem~A.
In Section~3, for the proof of the second assertion in Theorem~A,
we consider the existence problem of minimizers for the functional $E$
on compact conformal orbifolds, that is, the {\it orbifold Yamabe problem}.
Under a certain condition,
we solve this problem.
Using the solution, we can prove the second assertion.
In Section~4, we give two more typical exact computations of the orbifold Yamabe invariant
besides the proof of Theorem~B.
\noindent
{\bf Acknowledgements.}
The author would like to express his sincere gratitude
to Nobuhiro Honda and Jeff Viaclovsky for helpful discussions
on singularities of complex surfaces
and on the orbifold Yamabe invariant respectively.
He also would like to thank Claude LeBrun for useful comments.
\section{\bf The orbifold Yamabe invariant}\label{OYI}
For the sake of self-containedness,
we first recall the definition of orbifolds with finitely many singular points
which we discuss here \cite{AkBo-2}.
\begin{defi}
Let $M$ be a locally compact Hausdorff space.
We say that $M$ is an $n$-{\it orbifold with singularities}
$$
\Sigma_{\Gamma} = \{(\check{p}_1, \Gamma_1), \cdots, (\check{p}_{\ell}, \Gamma_{\ell})\}
$$
if the following conditions are satisfied: \\
\quad (1) $\Sigma := \{\check{p}_1, \cdots, \check{p}_{\ell}\} \subset M$,
and $M - \Sigma$ is a smooth $n$-manifold. \\
\quad (2) $\Gamma := \{\Gamma_1, \cdots, \Gamma_{\ell}\}$ is a collection of non-trivial finite subgroups $\Gamma_j$ of $O(n)$,
each of which acts freely on $\mathbb{R}^n - \{{\bf 0}\}$. \\
\quad (3) For each $\check{p}_j$, there exist its open neighborhood $U_j$
and a homeomorphism $\varphi_j : U_j \rightarrow \mathbb{B}_{\tau_j}({\bf 0})/\Gamma_j$ for some $\tau_j > 0$ such that
$$
\varphi_j : U_j - \{\check{p}_j\} \longrightarrow \big{(}\mathbb{B}_{\tau_j}({\bf 0}) - \{{\bf 0}\}\big{)}/\Gamma_j
$$
is a diffeomorphism.
Here, $\mathbb{B}_{\tau_j}({\bf 0}) := \{ x = (x^1, \cdots, x^n) \in \mathbb{R}^n~|~|x| < \tau_j \}$.
\end{defi}
We refer to the pair $(\check{p}_j, \Gamma_j)$ as a {\it singular point with the structure group} $\Gamma_j$
and the pair $(U_j, \varphi_j)$ as a {\it local uniformization}.
To simplify the presentation, we assume, without particular mention,
that an orbifold $M$ has only one singularity, i.e., $\Sigma_{\Gamma} = \{(\check{p}, \Gamma)\}$.
Let $\varphi : U \rightarrow \mathbb{B}_{\tau}({\bf 0})/\Gamma$ be a local uniformization
and $\pi : \mathbb{B}_{\tau}({\bf 0}) \rightarrow \mathbb{B}_{\tau}({\bf 0})/\Gamma$ the canonical projection.
We also always assume that $M$ is compact.
\begin{defi}
(1)\ \ A Riemannian metric $g \in \mathcal{M}(M - \{p\})$ is an {\it orbifold metric}
if there exists a $\Gamma$-invariant smooth metric $\hat{g}$ on the ball $\mathbb{B}_{\tau}({\bf 0})$
such that
$(\varphi^{-1} \circ \pi)^{\ast} g = \hat{g}$ on $\mathbb{B}_{\tau}({\bf 0}) - \{{\bf 0}\}$.
We denote by $\mathcal{M}^{orb}(M)$ the space of all orbifold metrics on $M$.
In the case when
$\Sigma_{\Gamma} = \{(\check{p}_1, \Gamma_1), \cdots, (\check{p}_{\ell}, \Gamma_{\ell})\}$,
the space of all orbifold metrics is defined similarly. \\
\quad (2)\ \ For an orbifold metric $g \in \mathcal{M}^{orb}(M)$,
its {\it orbifold conformal class} $[g]_{orb}$ is defined by
\begin{align*}
[g]_{orb} &:= [g] \cap \mathcal{M}^{orb}(M) \\
&\ = \{ e^{2f}\cdot g~|~f \in C^0(M) \cap C^{\infty}(M - \{\check{p}\}),
(\varphi^{-1}\circ \pi)^{\ast}f \in C^{\infty}(\mathbb{B}_{\tau}({\bf 0})) \}.
\end{align*}
We denote by $\mathcal{C}^{orb}(M)$ the space of all orbifold conformal classes.
\end{defi}
As in the smooth case, consider the normalized Einstein-Hilbert functional
$$
E : \mathcal{M}^{orb}(M) \rightarrow \mathbb{R},\quad
g \mapsto \frac{\int_MR_g d\mu_g}{\textrm{Vol}_g(M)^{(n-2)/n}}.
$$
Since the singularity has codimension at least three,
Stokes' theorem and Gauss' divergence theorem still hold over Riemannian orbifolds.
Hence, $\check{g}$ is a critical point of $E$ on ${\mathcal{M}^{orb}(M)}$
if and only if $\check{g}$ is an Einstein orbifold metric.
Then, one can define naturally the definition of the orbifold Yamabe invariant.
\begin{defi}
For a conformal orbifold $(M, [g]_{orb})$,
its {\it Yamabe constant} $Y(M, [g]_{orb})$ is defined by
$$
Y(M, [g]_{orb}) := \inf_{\tilde{g} \in [g]_{orb}} E(\tilde{g}).
$$
Moreover, the {\it orbifold Yamabe invariant} $Y^{orb}(M)$ of $M$ is also defined by
$$
Y^{orb}(M) := \sup_{[g]_{orb} \in \mathcal{C}^{orb}(M)}Y(M, [g]_{orb}).
$$
\end{defi}
Before we explain some necessary terminologies
on the Yamabe invariant of cylindrical manifolds,
we give two comments on orbifolds with {\it positive} orbifold Yamabe invariant.
\begin{rmk}
Let $(X, g)$ be a hyperK\"ahler asymptotically locally Euclidean (abbreviated to {\it ALE} ) $4$-manifold
constructed in \cite{Kr} (cf.~\cite{Na}),
where $X$ is the minimal resolution of the quotient space $\mathbb{C}^2/\Gamma$
for a non-trivial finite subgroup $\Gamma$ of $SU(2)$.
Then, $(X, g)$ has a smooth conformal compactification $(\hat{X} := X \sqcup \{p_{\infty}\}, \hat{g})$
with singularity $\{(p_{\infty}, \Gamma)\}$ \cite{ChLeWe, Vi},
which has a positive Yamabe constant $Y(\hat{X}, [\hat{g}]_{orb}) > 0$.
In \cite[Theorem~1.3]{Vi}, Viaclovsky has proved the following: \\
\quad (1)\ \ The orbifold Yamabe problem on $(\hat{X}, [\hat{g}]_{orb})$ has no solution.
This implies that the orbifold Yamabe problem is not always solvable (see Section~3 for the solvability),
in contrast with the case for smooth compact conformal manifolds. \\
\quad (2)\ \ He computed the orbifold Yamabe invariant of $\hat{X}$ as
$$
Y^{orb}(\hat{X}) = Y(\hat{X}, [\hat{g}]_{orb}) = Y(S^4)/|\Gamma|^{1/2}.
$$
However, similarly to the case for smooth compact manifolds,
there is not much exact computations of positive orbifold Yamabe invariants at present.
In the proof of both (1) and (2), one of key points is the following estimate,
called {\it refined Aubin's inequality} \cite[Theorem~B]{AkBo-2}
$$
Y(M, [g]_{orb}) \leq Y^{orb}(M) \leq \min_{1 \leq j \leq \ell} \frac{Y(S^n)}{|\Gamma_j|^{2/n}}
$$
for any compact Riemannian $n$-orbifold $(M, g)$ with singularities
$\{(\check{p}_1, \Gamma_1), \cdots, (\check{p}_{\ell}, \Gamma_{\ell})\}$.
This inequality is also crucial to give a sufficient condition
for the solvability of the orbifold Yamabe problem in Section~3.
\end{rmk}
\begin{defi}
Let $X$ be an open $n$-manifold {\it with tame ends}, i.e.,
it is diffeomorphic to $W \cup_Z (Z \times [0, \infty))$,
where $W (\subset X)$ is a relatively compact open submanifold with
boundary $\partial W =: Z \cong Z \times \{0\}$ (possibly finitely many connected component).
For a fixed $h \in \mathcal{M}(Z)$,
a complete Riemannian metric $\bar{g}$ on $X$ is called a {\it cylindrical metric modeled by} $(Z, h)$
if there exists a global coordinate function $t$ on $Z \times [0, \infty)$ such that
$\bar{g}|_{Z \times [1, \infty)}$ is the product metric $\bar{g}(z, t) = h(z) + dt^2~(~(z, t) \in Z \times [1, \infty)~)$
(see Figure~1).
Each pair $(X, \bar{g})$ is called a {\it cylindrical manifold} and $h$ a {\it slice metric}.
We denote by $\mathcal{M}^{h\textrm{-}cyl}(X)$ the space of all cylindrical metrics on $X$ modeled by $(Z, h)$.
\end{defi}
\quad \\
\input{fig12.tex}\\
\quad \\
\qquad \qquad \qquad Figure~1:~A cylindrical manifold $(X, \bar{g})$\\
For the definition of the Yamabe invariant on cylindrical manifolds,
we first recall the following fact.
On a compact Riemannian manifold $(M, g)$,
the value of functional $E(\tilde{g})$ for conformal metric $\tilde{g} := u^{4/(n-2)}\cdot g \in [g]$
can be rewritten by
$$
E(\widetilde{g}) = \frac{\int_M \big{(}\alpha_n|\nabla u|^2 + R_gu^2\big{)} d\mu_g}
{\Big{(}\int_M u^{2n/(n-2)} d\mu_g\Big{)}^{(n-2)/n}}\ \Big{(} =: Q_{(M, g)}(u)~\Big{)},
\qquad \alpha_n := \frac{4(n-1)}{n-2} > 0.
$$
\begin{defi}
The {\it Yamabe constant} $Y(X, [\bar{g}])$ of a cylindrical manifold $(X, \bar{g})$ is defined by
$$
Y(X, [\bar{g}]) := \inf_{u \in C^{\infty}_c(X), u \not\equiv 0} Q_{(X, \bar{g})}(u),
$$
where $C^{\infty}_c(X)$ denotes the space of all smooth functions on $X$ with compact supports.
Moreover, for a fixed $h \in \mathcal{M}(Z)$,
the $h$-{\it cylindrical Yamabe invariant} $Y^{h\textrm{-}cyl}(X)$ of the open manifold $X$ with tame ends
is also defined by
$$
Y^{h\textrm{-}cyl}(X) := \sup_{\bar{g} \in \mathcal{M}^{h\textrm{-}cyl}(X)} Y(X, [\bar{g}]).
$$
\end{defi}
To simplify the presentation, we also assume, without particular mention,
that each underlying manifold $X$ has only one connected tame end.
In contrast with the case for compact manifolds,
the constant $Y(X, [\bar{g}])$ is not always finite.
For instance, if the scalar curvature $R_h$ of slice metric $h$ is negative on $Z$,
then $Y(X, [\bar{g}]) = - \infty$.
As a complete criterion for the finiteness of $Y(X, [\bar{g}])$,
we have obtained the following \cite[Lemmas~2.7, 2.9]{AkBo-1}.
\begin{prop}
For $h \in \mathcal{M}(Z^{n-1})$, let $\mathcal{L}_h$ be the operator on $Z^{n-1}$ defined by
$$
\mathcal{L}_h := - \frac{4(n-1)}{n-2}\Delta_h + R_h,
$$
and $\lambda(\mathcal{L}_h)$ the first eigenvalue of $\mathcal{L}_h$.
Then, we have the following on the Yamabe constant of a cylindrical manifold $(X, \bar{g})$ with slice metric $h$. \\
\quad $\bullet$\ \ If $\lambda(\mathcal{L}_h) < 0$, then $Y(X, [\bar{g}]) = - \infty$. \\
\quad $\bullet$\ \ If $\lambda(\mathcal{L}_h) \geq 0$, then $Y(X, [\bar{g}]) > - \infty$. \\
\quad $\bullet$\ \ If $\lambda(\mathcal{L}_h) = 0$, then $0 \geq Y(X, [\bar{g}]) > - \infty$.
\end{prop}
We also note that the notion of the $h$-cylindrical Yamabe invariant is an natural extension of the one
of the orbifold Yamabe invariant \cite[Theorem~2.9]{AkBo-2}.
\begin{prop}
Let $M$ be a compact $n$-orbifold with singularity $\{(\check{p}, \Gamma)\}$ $($see {\rm Figure~2}$)$,
and $h_0 \in \mathcal{M}(S^{n-1}/\Gamma)$ the standard metric of constant curvature one.
Note that the open manifold $M - \{\check{p}\}$ is of one tame end
and $\mathcal{M}^{h_0\textrm{-}cyl}(M - \{\check{p}\}) \not= \emptyset$.
Then,
$$
Y^{orb}(M) = Y^{h_0\textrm{-}cyl}(M - \{\check{p}\}).
$$
\end{prop}
\quad \\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \input{fig19.tex}\\
\quad \\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad Figure~2. \\
Now, we can state the key inequality for $h$-cylindrical Yamabe invariants,
called {\it refined Kobayashi's inequality} \cite[Theorem~3.7]{AkBo-1}.
\begin{thm}
Let $N$ be a compact $n$-manifold and $Z$ a compact $(n-1)$-submanifold
with trivial normal bundle.
Assume that $M - Z$ has two connected components $W_1, W_2$.
Let $X_1 := \overline{W}_1 \cup_Z (Z \times [0, \infty)),
X_2 := \overline{W}_2 \cup_Z (Z \times [0, \infty))$
be the corresponding open $n$-manifolds with tame end $Z \times [0, \infty)$
$($see {\rm Figure~3}$)$.
For any $h \in \mathcal{M}(Z)$, we have
$$
Y(N) \geq
{\small \begin{cases}\
- \big{(}Y^{h\text{-}cyl}(X_1)^{n/2}+|Y^{h\text{-}cyl}(X_2)|^{n/2}\big{)}^{2/n}\ \cdots\
if\ \ Y^{h\text{-}cyl}(X_1), Y^{h\text{-}cyl}(X_2) \leq 0, \\
\ \min \{ Y^{h\text{-}cyl}(X_1), Y^{h\text{-}cyl}(X_2) \}\ \cdots\ \text{otherwise}. \end{cases} }
$$
\end{thm}
\qquad \qquad \qquad $N = \overline{W}_1 \cup_Z \overline{W}_2$ \\
\quad \\
\input{fig-n5.tex} \\
\quad \\
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad Figure~3. \\
Theorem~2.9 implies immediately the following.
\begin{cor}
Under the same setting as in Theorem~2.9,
assume that $Y(N) \leq 0\\ ({\rm resp.}~< 0)$ and
$Y^{h\textrm{-}cyl}(X_2) > 0~({\rm resp.}~\geq 0)$.
$($From Proposition~2.7, the positivity $Y^{h\textrm{-}cyl}(X_2) > 0$
implies automatically $\lambda(\mathcal{L}_h) > 0.)$
Then, we have
$$
Y^{h\textrm{-}cyl}(X_1) \leq Y(N) \leq 0.
$$
\end{cor}
We can now prove the first assertion in Theorem~A. \\
\quad \\
{\it Proof of the first assertion in Theorem~A}.\ \
In Corollary~2.10,
set $W_1 = M_1 - \sqcup_{j = 1}^{\ell} B(\check{p}_j)$,
$W_2 = M_2 - \sqcup_{j = 1}^{\ell} B(\check{q}_j)$
and $h = h_0$
on $Z = \partial W_1 = \partial W_2 \cong \sqcup _{j=1}^{\ell}\big{(}S^{n-1}/\Gamma_j\big{)}$.
Note that $X_1 = M_1 - \{\check{p}_1, \cdots, \check{p}_{\ell}\}$ and
$X_2 = M_2 - \{\check{q}_1, \cdots, \check{q}_{\ell}\}$.
Then, the first assertion follows directly from Proposition~2.8 and Corollary~2.10, that is,
$$
\qquad \qquad \qquad \qquad \qquad Y^{orb}(M_1) = Y^{h_0\textrm{-}cyl}(X_1) \leq Y(N) \leq 0.
\qquad \qquad \qquad \qquad \quad \ \ \square
$$
\begin{rmk}
For given compact manifolds $N_1$ and $N_2$,
we generally use Kobayashi's inequality in the case for
computing (or estimating) $Y(N_1 \# N_2)$
by using the values of both $Y(N_1)$ and $Y(N_2)$.
In contrast with this,
the generalized connected sum of compact orbifolds
is often ``prime'' as smooth manifold.
Hence, the opposite usage of (refined) Kobayashi's inequality
is also useful as Theorem~A.
\end{rmk}
\section{\bf The orbifold Yamabe problem}\label{EC}
In this section, we first prove the orbifold Yamabe problem
under a certain condition.
\begin{thm}
Let $(M, g)$ be a compact Riemannian $n$-orbifold with singularities
$\{(\check{p}_1, \Gamma_1), \cdots, (\check{p}_{\ell}, \Gamma_{\ell})\}$.
Assume the following strict inequality$:$
\begin{equation}
Y(M, [g]_{orb}) <~\min_{1 \leq j \leq \ell} \frac{Y(S^n)}{|\Gamma_j|^{2/n}}.
\end{equation}
Then, there exists a minimizer $\tilde{g} \in [g]_{orb}$
of the functional $E|_{[g]_{orb}}$ $($called an {\rm orbifold Yamabe metric}$)$
such that the orbifold metric $\tilde{g}$ is of constant scalar curvature
$R_{\tilde{g}} = Y(M, [g]_{orb})\cdot {\rm Vol}_{\tilde{g}}(M)^{-2/n}$.
\end{thm}
\noindent
{\it Proof.}\ \ We use here the same notations as those in Definition~2.1.
Without loss of generality,
we may assume that $M$ has only one singularity $\{(\check{p}, \Gamma)\}$.
The method adopting here for constructing approximate solutions
is similar to the one in \cite[Theorem~5.2]{AkBo-1}.
But, as background metric for getting both the uniform $C^0$-estimate of approximate solutions
and the regularity of a weak solution,
we will use rather the given orbifold metric $g$ itself
than an {\it asymptotically} cylindrical metric $\bar{g} \in [g|_X]$
on $X := M - \{\check{p}\}$
with $\bar{g} = r^{-2}\cdot g$ near the singularity $\check{p}$,
where $r(\cdot) := {\rm dist}_g(\cdot , \check{p})$.
First, note that
$$
Y(M, [g]_{orb}) = \inf_{u \in C^{\infty}_c(X), u \not\equiv 0} Q_{(X, g)}(u).
$$
Let $B_{\rho}$ be the open geodesic ball centered at $\check{p}$ of radius $\rho > 0$ with respect to $g$.
Set
$$
Y_i := \inf_{u \in C^{\infty}_c(X-\overline{B_{1/i}}), u \not\equiv 0} Q_{(X, g)}(u)
$$
for $i \in \mathbb{N}$.
We have that
$$
Y_i > Y_{i+1} > Y_{i+2} > \cdots, \qquad \qquad \qquad \qquad \ \
$$
$$
\lim_{i\to \infty} Y_i = \inf_{u \in C^{\infty}_c(X), u \not\equiv 0} Q_{(X, g)}(u)
= Y(M, [g]_{orb}).
$$
It then follows from the strict inequality~(1) and the above
that there exists a large integer $i_0$ such that
$$
Y_i < Y(S^n)/|\Gamma|^{2/n} < Y(S^n) \quad {\rm for\ any}\ \ i \geq i_0.
$$
Similarly to the case for compact manifolds without boundary,
this implies that there exists a non-negative $Q_{(X - B_{1/i}, g)}$-minimizer
$u_i \in C^{\infty}(X - B_{1/i})$ such that, for each $i \geq i_0$,
$$
Q_{(X - B_{1/i}, g)}(u_i) = Y_i,\qquad \int_{X - B_{1/i}} u_i^{\frac{2n}{n-2}} d\mu_g = 1,
$$
$$
u_i = 0\quad {\rm on}\ \ \partial B_{1/i},\qquad u_i > 0\quad {\rm in}\ \ X-\overline{B_{1/i}}.
$$
We denote the zero extension of each $u_i$ to $M$ by also the same symbol $u_i$.
Suppose that the sequence $\{u_i\}$ has a uniform $C^0$-bound, that is,
there exists a constant $L > 0$ such that
$$
||u_i||_{C^0(M)} \leq L\quad {\rm for}\ \ i \geq i_0.
$$
Under this uniform $C^0$-estimate,
then there exists a non-negative $Q_{(M, g)}$-minimizer $u \in W^{1,2}(M; g)$ with $||u||_{C^0(M)} \leq L$
such that (taking a subsequence if necessary)
$$
u_i \rightarrow u\quad {\rm weakly\ in}\ \ W^{1,2}(M; g),\qquad
u_i \rightarrow u\quad {\rm strongly\ in}\ \ L^2(M; g).
$$
Lebesgue's bounded convergence theorem combined with the above uniform $C^0$-estimate for $\{u_i\}$ implies that
$$
\int_M u^{\frac{2n}{n-2}} d\mu_g = 1.
$$
By this equation and the fact that $\{u_i\}$ is a $Q_{(M, g)}$-minimizing sequence,
we have
$$
u_i \rightarrow u\quad {\rm strongly\ in}\ \ W^{1,2}(M; g).
$$
Under the $C^0$-estimate $||u||_{C^0(M)} \leq L$,
applying the standard elliptic $L^p$-estimates to the Euler-Lagrange equations
for $u$ on $X$ and the lifting $(\varphi^{-1}\circ \pi)^{\ast} u$ on $\mathbb{B}_{\tau}({\bf 0})$,
we obtain that $u \in C^{\infty}(M)$.
Here, $u \in C^{\infty}(M)$ means that $u \in C^{\infty}(X)$
and the lifting $(\varphi^{-1}\circ \pi)^{\ast} u$ is smooth on $\mathbb{B}_{\tau}({\bf 0})$.
The maximum principle \cite[Proposition~3.75]{Aubin-Book} implies that $u > 0$ everywhere on $M$,
and then we get an orbifold Yamabe metric
$$
\tilde{g} := u^{4/(n-2)}\cdot g \in [g]_{orb}.
$$
To complete the proof, we need only to show a uniform $C^0$-estimate for the sequence $\{u_i\}$.
For each $u_i$, take a maximum point $q_i \in X$ of $u_i$,
and set $m_i := u_i(q_i)$.
Taking a subsequence if necessary,
we then have that there exists a point $q_{\infty} \in M$ such that
$$
\lim_{i \to \infty} q_i = q_{\infty}.
$$
Suppose that
$$
\lim_{i \to \infty} m_i = \infty.
$$
Then, we will lead to a contradiction as below. \\
\underline{Case~1.}\ \ $q_{\infty} \ne \check{p}$~:
Let $\{V, x = (x^1, \cdots, x^n)\}$ be a geodesic normal coordinate system
centered at $q_{\infty}$ satisfying $V \subset X$.
We may assume that $\{ |x| < 1 \} \subset V$.
Set
$$
v_i(x) := m_i^{-1}\cdot u_i\big{(}m_i^{-\frac{2}{n-2}}\cdot x + x(q_i)\big{)}\quad
{\rm for}\ \ x \in \{ |x| < m_i^{\frac{2}{n-2}}(1 - |x(q_i)|) \}.
$$
Similarly to the proof of Theorem~2.1 in \cite[Chapter~5]{ScYa-Book},
there exists a positive function $v \in C^{\infty}(\mathbb{R}^n)$ such that
$$
v_i \rightarrow v\quad {\rm in\ the}~C^2
\textrm{-topology\ on\ each\ relatively\ compact\ domain\ in}\ \ \mathbb{R}^n.
$$
Hence, $v$ satisfies the following:
$$
- \alpha_n \Delta_0 v = Y(M, [g]_{orb})\cdot v^{\frac{n+2}{n-2}}\quad {\rm on}\ \ \mathbb{R}^n,
$$
$$
\int_{\mathbb{R}^n} v^{\frac{2n}{n-2}} dx \leq
\liminf_{i \to \infty} \int_V u_i^{\frac{2n}{n-2}} d\mu_g \leq 1,
$$
where $\Delta_0$ denotes the Laplacian with respect to the Euclidean metric.
This implies that $Y(M, [g]_{orb}) \geq Y(S^n)$,
and then it contradicts to the assumption~(1). \\
\underline{Case~2.}\ \ $q_{\infty} = \check{p}$~:
In this case, we consider rather the liftings $\tilde{u}_i := (\varphi^{-1}\circ \pi)^{\ast} u_i$
on $\mathbb{B}_{\tau}({\bf 0})$ than $u_i$ themselves.
Similarly to the above, set
$$
\tilde{v}_i(x) := m_i^{-1}\cdot \tilde{u}_i\big{(}m_i^{-\frac{2}{n-2}}\cdot x + x(q_i)\big{)}\quad
{\rm for}\ \ x \in \{ x \in \mathbb{R}^n~|~|x| < m_i^{\frac{2}{n-2}}(\tau - |x(q_i)|) \}.
$$
Then, there exists a positive function $\tilde{v} \in C^{\infty}(\mathbb{R}^n)$ such that
$$
\tilde{v}_i \rightarrow \tilde{v}\quad {\rm in\ the}~C^2
\textrm{-topology\ on\ each\ relatively\ compact\ domain\ in}\ \ \mathbb{R}^n.
$$
Moreover, $\tilde{v}$ satisfies the following:
$$
- \alpha_n \Delta_0 \tilde{v} = Y(M, [g]_{orb})\cdot \tilde{v}^{\frac{n+2}{n-2}}\quad
{\rm on}\ \ \mathbb{R}^n,
$$
$$
\int_{\mathbb{R}^n} \tilde{v}^{\frac{2n}{n-2}} dx \leq
\liminf_{i \to \infty} \int_{\mathbb{B}_{\tau}({\bf 0})} \tilde{u}_i^{\frac{2n}{n-2}} d\mu_{\hat{g}} \leq |\Gamma|,
$$
where $\hat{g} := (\varphi^{-1}\circ \pi)^{\ast} g$.
This implies that
$$
Y(M, [g]_{orb}) \geq \frac{Y(S^n)}{|\Gamma|^{2/n}},
$$
and then it also contradicts to the assumption~(1).
\qquad \qquad \qquad \qquad \qquad \qquad \qquad $\square$ \\
We can now prove the second assertion in Theorem~A. \\
\quad \\
{\it Proof of the second assertion in Theorem~A}.\ \
First, we note that
$$
Y(M_1, [\check{g}]_{orb}) \leq Y^{orb}(M_1) \leq Y(N) \leq 0.
$$
It then follows from Theorem~3.1 and the above inequality
that there exists a constant scalar curvature orbifold metric
$\tilde{g} \in [\check{g}]_{orb}$ satisfying
$$
E(\tilde{g}) = Y(M_1, [\check{g}]_{orb}) \leq 0.
$$
Similarly to the case for smooth conformal manifolds,
the uniqueness of constant scalar curvature orbifold metrics
in a non-positive orbifold conformal class \cite[Lemma~2.3]{AkBo-2}
implies that, up to a scaling,
$$
\check{g} = \tilde{g}.
$$
Combining the above with the assumption $E(\check{g}) = Y(N)$,
we then have
$$
Y(N) = E(\check{g}) = Y(M_1, [\check{g}]_{orb}) \leq Y^{orb}(M_1) \leq Y(N).
$$
This implies that $Y^{orb}(M_1) = Y(M_1, [\check{g}]_{orb}) = Y(N)$.
\qquad \qquad \qquad \qquad \qquad \qquad \ \ $\square$\\
\section{\bf Exact computations}\label{EC}
We first prove Theorem~B.\\
\quad \\
{\it Proof of Theorem~B.}\ \
First, note that the canonical model $X_{can}$ is obtained by blowing down
each connected component of the union of the $(-2)$-curves in $X$ into a point.
The structure of an open neighborhood of each singular point in $X_{can}$
is modeled by one of A-D-E singularities,
that is, the quotient singularity $\mathbb{C}^2/\Gamma$
with a non-trivial finite subgroup $\Gamma < SU(2)$.
Then, $X_{can}$ admits a K\"ahler-Einstein orbifold metric $\check{g}$ \cite{Kor}
satisfying
$$
E(\check{g}) = - 4 \sqrt{2} \pi \sqrt{2 \chi(X) + 3 \tau(X)}.
$$
We denote the singularities of $X_{can}$ by
$\{(\check{p}_1, \Gamma_1), \cdots, (\check{p}_{\ell}, \Gamma_{\ell})\}$.
For each $\Gamma_j$, let $X_j$ denote the minimal resolution of $\mathbb{C}^2/\Gamma_j$.
Then, each $X_j$ admits a hyperK\"ahler ALE metric $h_j$ \cite{Kr},
and $(X_j, h_j)$ has a smooth conformal compactification
$(\hat{X_j} := X_j \sqcup \{\infty_j\}, \hat{h}_j)$
with singularity $\{(\infty_j, \Gamma_j)\}$ \cite{ChLeWe, Vi},
which has a positive Yamabe constant $Y(\hat{X}_j, [\hat{h}_j]_{orb}) > 0$.
With these understandings,
$X$ can be decomposed by
$$
X = X_{can} \#_{\sqcup_{j=1}^{\ell} (S^3/\Gamma_j)}
\big{(}\sqcup_{j = 1}^{\ell} \hat{X}_j \big{)}.
$$
By Theorem~A, this combined with $Y(X) < 0$ and $Y^{orb}(\hat{X}_j) > 0$ implies
$$
Y^{orb}(X_{can}) \leq Y(X) < 0.
$$
Recall that the K\"ahler-Einstein orbifold metric $\check{g}$ satisfies
$$
E(\check{g}) = - 4 \sqrt{2} \pi \sqrt{2 \chi(X) + 3 \tau(X)} = Y(X).
$$
This gives the desired conclusion:
$$
\qquad \qquad \qquad \qquad \quad
Y^{orb}(X_{can}) = Y(X_{can}, [\check{g}]_{orb}) = Y(X).
\qquad \qquad \qquad \qquad \quad \square\\
$$
Finally, we give two more typical exact computations of the orbifold Yamabe invariant. \\
\quad \\
\underline{\bf 1.}\quad Let $T$ be a complex $2$-dimensional torus
and $\check{T} := T/\langle{\rm id}, \iota\rangle$ the quotient $4$-orbifold with
$16$-singularities $\{(\check{p}_1, \langle{\rm id}, \iota\rangle), \cdots, (\check{p}_{16}, \langle{\rm id}, \iota\rangle)\}$.
Here, $\langle{\rm id}, \iota\rangle~(\cong \mathbb{Z}_2)$ denotes the group of degree $2$ generated by
$$
\iota : \mathbb{C}^2 \rightarrow \mathbb{C}^2,\ \ (z_1, z_2) \mapsto (- z_1, - z_2).
$$
Pushing down the flat metric on $T$ to $\check{T}$,
we have a flat orbifold metric $\check{g}_{flat}$ on $\check{T}$.
\begin{prop}
$$
Y^{orb}(\check{T}) = Y(\check{T}, [\check{g}_{flat}]_{orb}) = 0.
$$
\end{prop}
\noindent
{\it Proof.}\ \ Let $\mathcal{O}(-2)$ denote the complex line bundle
over the complex projective line $\mathbb{C}P^1$ of degree $- 2$.
Then, there exists a cylindrical metric $\bar{g}$ on $\mathcal{O}(-2)$
modeled by $(S^3/\langle{\rm id}, \iota\rangle, h_0)$ with positive scalar curvature $R_{\bar{g}} > 0$
(cf.~\cite[Example~4.1.27]{Ni-Book}).
Hence, $(\mathcal{O}(-2), \bar{g})$ has a smooth conformal
compactification $(\widehat{\mathcal{O}(-2)} := \mathcal{O}(-2) \sqcup \{\infty\}, \hat{g})$
with singularity $\{(\infty, \langle{\rm id}, \iota\rangle)\}$.
Note that, from the uniform positivity of $R_{\bar{g}}$ and the Sobolev embedding
$W^{1, 2}(\mathcal{O}(- 2); \bar{g})
\hookrightarrow L^4(\mathcal{O}(- 2); \bar{g})$,
$$
Y(\widehat{\mathcal{O}(-2)}, [\hat{g}]_{orb})
= Y(\mathcal{O}(-2), [\bar{g}]) > 0.
$$
Let $(N_1, H_1), \cdots, (N_{16}, H_{16})$ be
the $16$-copies of $(\widehat{\mathcal{O}(-2)}, \langle{\rm id}, \iota\rangle)$.
With these understandings,
the generalized connected sum
$$
X := \check{T} \#_{\sqcup_{j=1}^{\ell} (S^3/H_j)} (\sqcup_{j=1}^{\ell} N_j)
$$
is diffeomorphic to the Kummer surface, and hence $Y(X) = 0$.
By Theorem~A, we then have
$$
Y^{orb}(\check{T}) \leq Y(X) = 0.
$$
Note that
$$
E(\check{g}_{flat}) = 0 = Y(X),
$$
and hence
$$
\qquad \qquad \qquad \qquad \qquad
Y^{orb}(\check{T}) = Y(\check{T}, [\check{g}_{flat}]_{orb})
= Y(X) = 0. \qquad \qquad \qquad \qquad \square
$$
\quad \\
\underline{\bf 2.}\quad
Let $\Sigma$ be an exotic sphere of dimension
$n := 8k + 2 \geq 10$ with $\alpha([\Sigma]) \ne 0$,
where $\alpha$ is the $\alpha$-homomorphism
from the spin cobordism group $\Omega^{spin}_n$
to the $KO$-group $KO^{-n}(pt) \cong \mathbb{Z}_2$ (cf.~\cite[Chapter~2]{LaMi-Book}).
For any integer $\ell \geq 2$,
set
$$
G_{\ell} := \{ \zeta^j I \in GL(4k+1; \mathbb{C})~|~j = 0, \cdots, \ell -1 \},\quad
\zeta := \exp(2\pi \sqrt{-1}/\ell) \in \mathbb{C},
$$
where $I$ denotes the identity matrix.
The finite group $G_{\ell}$ acts the $n$-sphere
$S^n \subset \mathbb{R}^{n+1} = \mathbb{C}^{4k+1} \times \mathbb{R}$
by
$$
A : \mathbb{C}^{4k+1} \times \mathbb{R} \rightarrow \mathbb{C}^{4k+1} \times \mathbb{R},\ \
(z, t) \mapsto A\cdot (z, t) := (A\cdot z, t)\quad {\rm for}\ \ A \in G_{\ell}.
$$
Then, the quotient space $S^n/G_{\ell}$ is a compact $n$-orbifold
with two singularities
$\{(\check{p}_+ := [(0, \cdots, 0, 1)], G^+_{\ell} := G_{\ell}),
(\check{p}_- := [(0, \cdots, 0, - 1)], G^-_{\ell} := G_{\ell})\}$.
Pushing down the standard metric $g_0$ on $S^n$ to $S^n/G_{\ell}$,
we have an orbifold metric $\check{g}_0$ of constant curvature one on $S^n/G_{\ell}$.
Note that the space $((S^n/G_{\ell}) - \{\check{p}_+, \check{p}_-\}, \check{g}_0)$
is conformal to the product space
$((S^{n-1}/G_{\ell}) \times \mathbb{R}, \bar{g} := h_0 + dt^2)$.
Then, this combined with $R_{\bar{g}} = R_{h_0} = (n-1)(n-2) > 0$ and
the Sobolev embedding
$W^{1, 2}((S^{n-1}/G_{\ell}) \times~\mathbb{R}; \bar{g})
\hookrightarrow L^{2n/(n-2)}((S^{n-1}/G_{\ell}) \times \mathbb{R}; \bar{g})$
implies that
\begin{equation}
Y^{orb}(S^n/G_{\ell}) \geq Y(S^n/G_{\ell}, [\check{g}_0]_{orb})
= Y((S^{n-1}/G_{\ell}) \times \mathbb{R}, [\bar{g}]) > 0.
\end{equation}
\begin{prop}
$$
Y^{orb}(\Sigma \# (S^n/G_{\ell})) = 0.
$$
Here, $\Sigma \# (S^n/G_{\ell})$ stands for the connected sum
of $\Sigma$ and $S^n/G_{\ell}$ in the usual sense.
\end{prop}
\noindent
{\it Proof.}\ \
We first note the following.
By results of Lichnerowicz and Hitchin (cf.~\cite[Chapters~2, 4]{LaMi-Book})
for $\alpha : \Omega^{spin}_n \rightarrow \mathbb{Z}_2$,
$Y(\Sigma) \leq 0$.
On the other hand,
Petean \cite{Petean-2} proved that
any simply connected compact manifold of dimension greater than $4$
has a non-negative Yamabe invariant.
Hence, we have
$$
Y(\Sigma) = 0.
$$
Let
$$
N_{\ell} := (S^n/G_{\ell}) \#_{(S^{n-1}/G^+_{\ell}) \sqcup (S^{n-1}/G^-_{\ell})} (\overline{S^n/G_{\ell}})
$$
denotes the generalized connected sum.
Here, $\overline{S^n/G_{\ell}}$ is the same $n$-orbifold,
but equipped with the opposite orientation.
It turns out that
$$
N_{\ell} = (S^{n-1}/G_{\ell}) \times S^1,
$$
and then it is a compact spin $n$-manifold with positive Yamabe invariant.
Then, the positivity $Y(N_{\ell})~>~0$ implies that $\alpha([N_{\ell}]) = 0$,
and hence
$\alpha([\Sigma \# N_{\ell}]) = \alpha([\Sigma]) + \alpha([N_{\ell}]) \ne 0$.
Therefore,
\begin{equation}
Y(\Sigma \# N_{\ell}) = 0.
\end{equation}
We now decompose $\Sigma \# N_{\ell}$ as the generalized connected sum
$$
\Sigma \# N_{\ell} = \big{(}\Sigma \# (S^n/G_{\ell})\big{)}
\#_{(S^{n-1}/G^+_{\ell}) \sqcup (S^{n-1}/G^-_{\ell})} (\overline{S^n/G_{\ell}}).
$$
It then follows from Theorem~A combined with (2), (3) that
\begin{equation}
Y^{orb}(\Sigma \# (S^n/G_{\ell})) \leq Y(\Sigma \# N_{\ell}) = 0.
\end{equation}
On the other hand, Kobayashi's inequality for $Y^{orb}(\Sigma \# (S^n/G_{\ell}))$
still holds.
Hence,
\begin{equation}
0 = Y(\Sigma) = \min \{Y(\Sigma), Y^{orb}(S^n/G_{\ell}) \}
\leq Y^{orb}(\Sigma \# (S^n/G_{\ell})).
\end{equation}
The inequalities (4), (5) give the desired conclusion:
$$
\qquad \qquad \qquad \qquad \qquad \qquad \qquad
Y^{orb}(\Sigma \# (S^n/G_{\ell})) = 0.
\qquad \qquad \qquad \qquad \qquad \quad \square
$$
\quad \\
\quad \\
\bibliographystyle{amsbook}
|
{
"timestamp": "2010-09-21T02:01:05",
"yymm": "1009",
"arxiv_id": "1009.3576",
"language": "en",
"url": "https://arxiv.org/abs/1009.3576"
}
|
\section{Introduction}
Lerman and Malkin introduced a symplectic structure on a Deligne--Mumford stack. They also define a Hamiltonian group actions on a symplectic Deligne--Mumford stack \cite{0908.0903}. The motivation of their work is based on the following proposal: using stacks is preferable to using ordinary orbifold atlases when we study geometries of orbifolds.
The theory of stacks has been developed by algebraic geometers. Abramovich, Graber and Vistoli \cite{abramovich06:_gromov_witten_delig_mumfor} constructed an algebraic counterpart of the theory of Chen--Ruan: the orbifold Chow ring and Gromov--Witten theory on a smooth complex Deligne--Mumford stack. Afterwards Borisov, Chen and Smith \cite{borisov05:_chow_delig_mumfor} introduced the concept of stacky fan to construct a toric Deligne--Mumford stack efficiently and compute the orbifold Chow ring of the toric Deligne--Mumford stack in terms of the stacky fan. A stacky fan can be used to study Gromov--Witten theory and mirror symmetry. (See Iritani \cite{iritani07:_real_i} for example.)
The aim of this paper is to introduce a stacky polytope as a counterpart of a stacky fan and to establish a relation between stacky polytopes and stacky fans, applying the stack description developed by Lerman and Malkin. Starting with a stacky polytope, we discuss a construction of symplectic Deligne--Mumford stacks. The main theorem says that the Deligne--Mumford stack associated to a stacky polytope $\vec{\Delta}$ is equivalent to the Deligne--Mumford stack associated to a stacky fan $\vec{\Sigma}$ if $\vec{\Sigma}$ corresponds to $\vec{\Delta}$ (Theorem \ref{thm:equivalence_between_two_stacks}). We do not discuss the orbifold cohomology in this paper, but the terminology of stacky polytopes could be useful when we compute orbifold cohomology in a similar way to Borisov--Chen--Smith. (See also Chen--Hu \cite{chen06:_chen_ruan}.)
This paper is organised as follows. In section 2, we review briefly the theory of stacks and related geometric concepts. In section 3, we discuss symplectic quotients in terms of stacks, using the notion developed by Lerman and Malkin. This construction is well-known for smooth quotients. After that we introduce stacky polytopes and construct the compact symplectic Deligne--Mumford stack associated to a stacky polytope. Then we discuss stacky polytopes which generate torus quotients. For example we describe explicitly a stacky polytope which generates a weighted projective space. In section 4, we briefly review stacky fans and discuss a relation between stacky polytopes and stacky fans.
\section{Symplectic Deligne--Mumford stacks}
In this section, we review briefly the theory of Deligne--Mumford stacks and their differential or symplectic geometry. There are several expository articles of the theory of differentiable stacks: Behrend--Xu \cite{behrend08:_differ_stack_gerbes}, Heinloth \cite{heinloth05:_notes} and Metzler \cite{metzler03:_topol_smoot_stack}. As far as possible we follow the notation of Behrend--Xu.
\subsection{The category of stacks}
\label{subsec:stacks}
To deal with stacks, we first have to fix a category equipped with a Grothendieck topology. We will only use the category of smooth manifolds and smooth maps. This category is denoted by $\mathfrak{Diff}$ and equipped with the Grothendieck topology defined as follows. For a smooth manifold $U \in \mathfrak{Diff}$ a family $\{f_i: U_i \to U\}_i$ of smooth maps to $U$ is said to be a \emph{covering family} (or just a \emph{covering}) if each $f_i$ is an \etale{} map (i.e.\! a local diffeomorphism) and $\bigcup_i f_i(U_i) = U$. Then the collections of coverings define a Grothendieck topology on $\mathfrak{Diff}$ \cite[Section 2]{metzler03:_topol_smoot_stack}.
A \emph{category fibred in groupoids} (over $\mathfrak{Diff}$) is a category $\X$ equipped with a functor $F_\X: \X\to\mathfrak{Diff}$ which satisfies the following conditions.
\begin{enumerate}[(F1)]
\item For each smooth map $f: V \to U$ and an object $x \in \X$ with $F_\X(x) = U$, there is an arrow $a: y\to x$ in $\X$ such that $F_\X(a)=f$. The object $y$ is called the \emph{pullback} of $x$ by $f$ and denoted by $f^*x$ or $x|_V$.
\item If we have two arrows $a: y\to x$ and $b: z\to x$ in $\X$, then for any smooth map $f: F_\X(y)\to F_\X(z)$ with $F_\X(b) \circ f=F_\X(a)$ there is a unique arrow $c: y\to z$ such that $F_\X(c) = f$ and $bc=a$.
\end{enumerate}
By the definition, a category fibred in groupoids $\X$ has the following properties.
\begin{itemize}
\item An object $x \in \X$ is said to \emph{lie over} $U \in \mathfrak{Diff}$ if $F_\X(x) = U$. An arrow $a:y \to x$ is said to \emph{lie over} $U$ if $F_\X(a) = \id_U$. The subcategory consisting of objects and arrows lying over $U$ is denoted by $\X(U)$ and called the \emph{fibre} of $\X$ over $U$. The subcategory $\X(U)$ is a groupoid: a category whose arrows are all invertible.
\item Because of the condition (F2), the pullback $y$ in the condition (F1) is almost unique in the following sense: for two pullbacks $y$ and $z$ of $x$ by $f$ there is a unique invertible arrow $y \to z$ in $\X(U)$.
\item \squashup
For an arrow $a:y \to x$ lying over $U$ and a smooth map $f:V \to U$, the condition (F2) guarantees that there is a unique arrow $y|_V \to x|_V$. This arrow is called the \emph{pullback} of $a$ by $f$ and denoted by $f^*a$ or $a|V$.
\end{itemize}
\begin{remark}
In this paper, for any groupoid the source map and the target map are denoted by $\src$ and $\tgt$ respectively. Namely for an arrow $a:y \to x$, we have $\src(a) = y$ and $\tgt(a) = x$.
\end{remark}
A category fibred in groupoids $\X$ is said to be a \emph{stack} (over $\mathfrak{Diff}$) if $\X$ satisfies the glueing conditions \cite[Definition 2.4]{behrend08:_differ_stack_gerbes}. (We never use the conditions explicitly.)
The stacks which we mainly deal with in this paper are quotient stacks. If a Lie group $G$ acts on a manifold $M$, we can define the \emph{quotient stack} $[M/G]$ as follows. An object of the category $[M/G]$ is a pair of a principal $G$-bundle $\pi: P \to U$ and a $G$-equivariant map $\epsilon: P \to M$. Here the latter means $\epsilon(p \cdot g) = g^{-1} \cdot \epsilon(p)$ for any $g \in G$ and $p \in P$. The object is written as a diagram $U \stackrel{\pi}{\leftarrow} P \stackrel{\epsilon}{\rightarrow} M$ and often abbreviated to $P$ if no confusion arises. An arrow from $V \stackrel{\pi'}{\leftarrow} Q \stackrel{\epsilon'}{\rightarrow} M$ to $U \stackrel{\pi}{\leftarrow} P \stackrel{\epsilon}{\rightarrow} M$ is a pair of smooth maps $(f,\tilde{f})$ such that $\tilde{f}$ is $G$-equivariant and the following diagram commutes.
\[
\xymatrix@R=4pt@C=60pt{
U & P \ar[l]_{\pi} \ar[dr]^{\epsilon} & \\
& & M. \\
V \ar[uu]^{f} & Q \ar[l]^{\pi'} \ar[uu]^{\tilde{f}} \ar[ur]_{\epsilon'} &
}
\]
A functor $F: [M/G]\to\mathfrak{Diff}$ is defined by $F(U \leftarrow P \rightarrow M) = U$ and $F(f,\tilde{f}) = f$.
Pullbacks are given as follows. For a smooth map $f: V \to U$ and an object $U \stackrel{\pi}{\leftarrow} P \stackrel{\epsilon}{\rightarrow} M$ lying over $U$, the pullback $f^*P \in [M/G](V)$ is $V \stackrel{\pi'}{\leftarrow} f^*P \stackrel{\epsilon'}{\rightarrow} M$ in the following commutative diagram.
\[
\xymatrix@R=4pt@C=60pt{
U & P \ar[l]_{\pi} \ar[dr]^{\epsilon} & \\
& & M. \\
V \ar[uu]^{f} & f^*P \ar[l]^{\pi'} \ar[uu]^{\widetilde{f}} \ar[ur]_{\epsilon'}
&
}
\]
Here $f^*P = V \times_{U} P$ is the pullback of the $G$-bundle $P$ to $U$ and $\epsilon'(v,p) = \epsilon(p)$. The universality of the (ordinary) pullback guarantees the condition (F2). Therefore $[M/G]$ is a category fibred in groupoids. Moreover we can see that $[M/G]$ is a stack.
\begin{dfn}
For stacks $F_\X: \X\to\mathfrak{Diff}$ and $F_\Y: \Y\to\mathfrak{Diff}$, a \emph{morphism of stacks} is a functor $\phi: \X \to \Y$ satisfying $F_\Y \circ \phi = F_\X$.
The class of morphisms of stacks from $\X$ to $\Y$ is denoted by $\Mor(\X,\Y)$.
\end{dfn}
\begin{dfn}
Let $\phi: \X \to \Y$ and $\phi': \X \to \Y$ be two morphisms of stacks. A \emph{map of morphisms} is a natural isomorphism $\alpha:\phi \Rightarrow \phi'$ of functors.
\end{dfn}
The category of stacks (over $\mathfrak{Diff}$) forms a $2$-category. Namely for any two stacks $\X$ and $\Y$, the class $\Mor(\X,\Y)$ forms a category again. The class of arrows consists of maps of morphisms.
In a $2$-category, two morphisms are identified if there is an arrow between them in $\Mor(\X,\Y)$. Moreover two stacks are regarded as the same stack if there is an equivalence between them instead of an isomorphism.
\begin{dfn}
Two stacks $\X$ and $\Y$ are said to be \emph{equivalent} if there are morphisms of stacks $\phi:\X \to \Y$ and $\psi:\Y \to \X$ such that there are maps of morphisms $\id_\X \Rightarrow \psi\circ\phi$ and $\id_\Y \Rightarrow \phi\circ\psi$.
\end{dfn}
\begin{remark}
Some authors use the terminology ``isomorphic'' instead of ``equivalent''. In this paper we follow the usual terminology of the theory of $2$-categories.
\end{remark}
A manifold $M$ can naturally be considered as a stack $\X_M$ as follows: the class of objects consists of smooth maps whose target is $M$ and an arrow from $f:U \to M$ and $g:V \to M$ is a smooth map $a: U \to V$ such that $g \circ a = f$. The fibre of $\X_M$ over $U$ is given by $\mathcal{C}^\infty(U,M)$. Note that we consider the set $\mathcal{C}^\infty(U,M)$ as a discrete category.
Thanks to the Yoneda lemma, the category $\mathfrak{Diff}$ is fully embedded into the category of stacks. Therefore we identify $M$ with $\X_M$.
\begin{dfn}
A stack is said to be \emph{representable} if there is a manifold which is equivalent to the stack.
\end{dfn}
\subsection{Deligne--Mumford stacks}
\label{subsec:DM_stacks}
Note that we can always take a ($2$-)fibred product in the category of stacks.
\begin{dfn}
A morphism of stacks $p$ from a manifold $X$ to a stack $\X$ is called an \emph{atlas} (resp. \emph{\etale{} atlas}) of $\X$ if for any morphism of stacks from a manifold $Y$ to $\X$
\begin{itemize}
\item the fibred product $X \times_{\X} Y$ is representable, and
\item the projection $X \times_{\X} Y \to Y$ is a surjective submersion (resp. surjective locally diffeomorphism).
\end{itemize}
\end{dfn}
If $p: X_0 \to \X$ is an atlas of a stack $\X$, then the fibred product $X_0 \times_{\X} X_0$ is equivalent to a manifold $X_1$ and there are two surjective submersions $\src,\tgt: X_1 \to X_0$ as projections:
\[
\xymatrix@C=50pt{
X_1 \ar[r]^{\tgt} \ar[d]_{\src} & X_0 \ar[d]^{p} \\
X_0 \ar[r]_{p} & \X.
}
\]
Then $X_1 \rightrightarrows X_0$ form a Lie groupoid.
The Lie groupoid $X_1 \rightrightarrows X_0$ is said to be \emph{associated to the atlas} $p: X_0 \to \X$. Note that the above diagram is $2$-commutative, i.e.\! there is a map of morphisms from $p \circ \src$ to $p \circ \tgt$.
\begin{dfn}
Let $\X$ be a stack having an atlas $p: X_0 \to \X$. We say that the stack $\X$ is \emph{separated} if the map
\[
X_1 \to X_0 \times X_0;\ a \mapsto (\src(a),\tgt(a))
\]
is proper. This definition is independent of the choice of the atlas \cite[Section 2.4]{behrend08:_differ_stack_gerbes}.
\end{dfn}
\begin{dfn}
A stack is said to be \emph{differentiable} (resp. \emph{Deligne--Mumford}) if the stack is separated and admits an atlas (resp. \etale{} atlas).
\end{dfn}
The \emph{underlying space} of the differentiable stack $\X$ is the topological space $|\X| = X_0/\!\!\sim$, where $x \sim y$ for $x, y \in X_0$ if there is $a \in X_1$ such that $\src(a)=x$ and $\tgt(a)=y$. The topology of the underlying space is well-defined. A differentiable stack is said to be \emph{compact} if its underlying space is compact.
Each quotient stack has an atlas. For a $G$-action on $M$, the \emph{natural projection} is a morphism $p: M \to [M/G]$ defined by $p(f: U \to M) = \bigl( U \stackrel{\pi}{\leftarrow} G \times U \stackrel{\epsilon}{\rightarrow} M \bigr)$. Here $\pi$ is the projection, $\epsilon(g,u) = g \cdot f(u)$ and the right $G$-action on $G \times U$ is defined by $(g,u) \cdot h = (h^{-1}g,u)$. For every arrow $a: f' \to f$ in $M$, $p(a)$ is naturally defined. The groupoid associated to the atlas $p$ is the action groupoid $G \times M \rightrightarrows M$. Therefore if the $G$-action on $M$ is proper, then the quotient stack is differentiable. The underlying space of $[M/G]$ is the quotient topological space $M/G$.
\begin{prop}
Let $M$ be a manifold equipped with a smooth action of a Lie group $G$. If the action is proper and locally free, then there is an \etale{} atlas of the quotient stack $[M/G]$ i.e.\! the quotient stack $[M/G]$ is Deligne--Mumford.
\end{prop}
This theorem can be showed by using Theorem 1 in Crainic--Moerdijk \cite{crainic01:_foliat} (cf. Lerman--Malkin {\cite[Theorem 2.4]{0908.0903}}).
\subsection{Differential forms over a differentiable stack}
\label{subsec:forms_on_stacks}
A (global) differential form over a differentiable stack is defined as a global section of the sheaf of differential forms. A presheaf is usually defined as a contravariant functor and a sheaf can be defined over a category equipped with a Grothendieck topology. The Grothendieck topology on a stack $\X$ can be induced by the Grothendieck topology of $\mathfrak{Diff}$. Moreover for a sheaf over a differentiable stack $\X$, we can define the set of global sections. Details can be found in Behrend--Xu \cite{behrend08:_differ_stack_gerbes} and Metzler \cite{metzler03:_topol_smoot_stack}.
We define the sheaf $\Omega^k_\X$ of smooth $k$-forms on a differentiable stack $\X$ as follows. For an object $x \in \X$ lying over $U$, put $\Omega^k_\X(x) = \Omega^k(U)$. For an arrow $a:y \to x$ in $\X$ with $F_\X(a) = f:V \to U$, we assign to $\Omega^k_\X(x) \to \Omega^k_\X(y)$ the pullback $f^*:\Omega^k(U) \to \Omega^k(V)$. If $X_1 \rightrightarrows X_0$ is the groupoid associated to an atlas $p: X_0 \to \X$, then the set of global sections of $\Omega_\X^k$ is given by
\[
\Omega^k(\X) =
\bigl\{ \eta \in \Omega^k(X_0) \!\ \big|\!\ \src^*\eta = \tgt^*\eta \bigr\}.
\]
An element of $\Omega^k(\X)$ is called a (global) $k$-form on $\X$.
\begin{examples}
Suppose that a manifold $M$ is equipped with a smooth proper action of a Lie group $G$. Then the set of global $k$-forms on the quotient stack $[M/G]$ is given by
\[
\Omega^k([M/G]) = \bigl\{
\eta \in \Omega^k(M) \big| \text{$\eta$ is $G$-invariant and $\iota(\xi_M)\eta = 0$ for any $\xi \in \g$}\bigr\}.
\]
Here $\iota$ is the interior multiplication, $\g$ is the Lie algebra of the Lie group $G$ and $\xi_M$ is the infinitesimal action of $\xi$, i.e.\! $\xi_M(x) = \left.\frac{d}{d\lambda}\right|_{\lambda=0} \exp(\lambda\xi) \cdot x$.
\end{examples}
\subsection{Vector fields and symplectic forms over a Deligne--Mumford stack}
\label{subsec:vector_fields_on_stacks}
Let $p:X_0 \to \X$ be an atlas and $X_1 \rightrightarrows X_0$ the groupoids associated to the atlas. We define the groupoid of vector fields $\Vect(X_1 \rightrightarrows X_0)$ over $X_1 \rightrightarrows X_0$ as follows.
Note that we have the Lie groupoid $TX_1 \rightrightarrows TX_0$ whose structure maps are defined by derivatives of the structure maps of $X_1 \rightrightarrows X_0$. The projections define a smooth functor $\pi$ from $TX_1 \rightrightarrows TX_0$ to $X_1 \rightrightarrows X_0$. The set of objects $\Vect(X_1 \rightrightarrows X_0)$ consists of smooth functors $v$ from $X_1 \rightrightarrows X_0$ to $TX_1 \rightrightarrows TX_0$ satisfying that the composition $\pi \circ v$ is equal to the identity functor. An arrow $v \to v'$ is a natural isomorphism $\alpha:v \to v'$ satisfying that the horizontal composition $\id_{\pi}*\alpha$ is equal to the identity transformation of the identity functor for $X_1 \rightrightarrows X_0$.
\begin{thm}
[Hepworth {\cite[Theorem 3.13]{hepworth09:_vector}}] The groupoid $\Vect(X_1 \rightrightarrows X_0)$ is independent of the choice of the atlas up to category equivalences.
\end{thm}
The vector space of the equivalence classes for the groupoid $\Vect(X_1 \rightrightarrows X_0)$ is denoted by $\Vect(\X)$ and called the space of vector fields over $\X$.
\begin{thm}
[Lerman--Malkin {\cite[Proposition 2.9]{0908.0903}}]
Let $\X$ be a stack with an atlas $p:X_0 \to \X$ and $X_1 \rightrightarrows X$ the groupoid associated to the atlas. If $\X$ is Deligne--Mumford, then $\Vect(\X)$ is given by the following quotient vector space:
\[
\mathcal{V}\big/\{(v_1,v_0) \in \mathcal{V}\!\ |\!\ v_1 \in \ker(d\src) + \ker(d\tgt)\}.
\]
Here $\mathcal{V}$ is the vector space consisting of pairs $(v_1,v_0) \in \Vect(X_1) \times \Vect(X_0)$ of (ordinary) vector fields satisfying $d(\src) \circ v_1 = v_0 \circ \src$ and $d(\tgt) \circ v_1 = v_0 \circ \tgt$.
\end{thm}
Following Lerman--Malkin \cite{0908.0903}, we introduce a symplectic form on a Deligne--Mumford stack as follows. Let $\X$ be a Deligne--Mumford stack with an atlas $p: X_0 \to \X$. A vector field over $\X$ can be represented by an equivalence class $v = [v_1,v_0]$ of pair of vector fields as above. The interior multiplication $\iota(v)$ is defined by
\[
\iota(v): \Omega^2(\X) \to \Omega^1(\X);\ \omega \mapsto \iota(v_0)\omega.
\]
Here $\omega$ is represented as a $2$-form on $X_0$.
\begin{remark}
Lerman and Malkin describe a differential form over $\X$ as a pair of differential forms $(\omega_1,\omega_0)$ such that $\omega_1 = \src^*\omega_0 = \tgt^*\omega_0$ \cite{0908.0903}. We omit $\omega_1$, because it is redundant.
\end{remark}
A $2$-form $\omega$ on $\X$ is said to be \emph{nondegenerate} if the map
\[
\Vect(\X) \to \Omega^1(\X);\ v \mapsto \iota(v)\omega
\]
is a linear isomorphism. This is equivalent to the condition that $\ker \omega = \A$. Here $\A$ is the Lie algebroid of the groupoid associated to the atlas $p: X_0 \to \X$. In other words $\A$ is the pullback bundle of $\ker(d\src)$ by the unit map $X_0 \to X_1$. For a Deligne--Mumford stack we can regard the bundle $\A \to X_0$ as a subbundle of the tangent bundle $TX_0$ via $d\tgt$ \cite[Theorem 2.4]{0908.0903}. A nondegenerate closed $2$-form on $\X$ is called a \emph{symplectic form} on $\X$.
\section{The Deligne--Mumford stack associated to a stacky polytope}
\label{section:DM_stack_associated_to_stacky_polytope}
Borisov et al. define stacky fans and construct the Deligne--Mumford stack $\X_{\vec\Sigma}$ associated to a stacky fan $\vec{\Sigma}$ \cite{borisov05:_chow_delig_mumfor}. Motivated by Borisov et al., we define stacky polytopes which are symplectic counterparts and construct the symplectic Deligne--Mumford stack $\X_{\vec\Delta}$ associated to a stacky polytope $\vec{\Delta}$.
\subsection{A symplectic quotient as a stack} \label{sec:symp_toric_quot}
In this subsection we discuss a construction of symplectic quotients in terms of stacks. This is well-known as construction of symplectic orbifolds.
\begin{remark}
Lerman and Malkin construct symplectic Deligne--Mumford stacks in a different way to our construction \cite{0908.0903}. But we stick with the standard construction because it is useful when we find a stacky polytope for a given symplectic quotient in Subsection \ref{subsection:stacky_polytope_of_torus_quotients}.
\end{remark}
The $d$-dimensional torus $\T^d = \R^d/\Z^d$ linearly acts on $\C^d$:
\[
\T^d \acts \C^d; \quad
[\theta_1,\dots,\theta_d] \cdot (z_1,\dots,z_d) =
(e^{-2\pi\i \theta_1}z_1,\dots,e^{-2\pi\i \theta_d}z_d).
\]
Here $\i=\sqrt{-1}$. The action is Hamiltonian with respect to the standard symplectic structure on $\C^d$. A moment map of the action is given by
\begin{equation}
\label{eq:mu_zero}
\mu_0 : \C^d \to (\R^d)^\dual; \quad
z \mapsto \pi\sum_{\alpha=1}^d |z_\alpha|^2\vec{e}^\alpha.
\end{equation}
Here ``$\dual$'' means taking the dual space and $\vec{e}^1,\dots,\vec{e}^d$ are the dual basis of the standard basis $\vec{e}_1,\dots,\vec{e}_d$ of $\R^d$.
Let $G$ be a compact Lie group whose adjoint representation is trivial. It is easy to see that the identity component $G_0$ of $G$ is a compact torus. Given a homomorphism $\rho: G \to \T^d$, we define the smooth action of $G$ on $\C^d$ through the homomorphism. Let $\rho^\dual: (\R^d)^\dual \to \g^\dual$ be the induced linear map, where $\g$ is the Lie algebra of $G$. If we put $w^\alpha = \rho^\dual(\vec{e}^\alpha)$, then a moment map of the $G$-action is given by
\begin{equation}
\label{eq:moment_map_of_G}
\mu : \C^d \to \g^\dual; \quad
z \mapsto \pi\sum_{\alpha=1}^d |z_\alpha|^2 w^\alpha.
\end{equation}
The elements $w^1,\dots,w^d$ are called \emph{weights}.
Since the moment map is $G$-equivariant, for $\tau \in \g^\dual$ the level set $\mu^{-1}(\tau)$ is closed under the continuous action of $G$. If $\tau$ is a regular value of $\mu$, the level set $\mu^{-1}(\tau)$ is a smooth manifold. Moreover it follows from the next lemma that the $G$-action on $\mu^{-1}(\tau)$ is locally free.
\begin{lem}
\label{lem:reg-val-and-local-freeness}
For $\tau \in \g^\dual$, $\tau$ is a regular value of $\mu$ if and only if for $z \in \mu^{-1}(\tau)$ and $\xi \in \g$ the identity $\xi_{\C^d}(z)=0$ implies $\xi=0$.
\end{lem}
\begin{proof}
A covector $\tau \in \g^\dual$ is a regular value of $\mu$ if and only if for any $z \in \mu^{-1}(\tau)$ the derivative $d\mu(z): \C^d \to \g^\dual$ is surjective. The map $d\mu(z)$ is surjective if and only if $\pair{d\mu(z),\xi}|_z=0$ implies $\xi=0$ for $\xi \in \g$. Since $\pair{d\mu(z),\xi}|_z = \iota(\xi_{\C^d})\omega_0|_z$, we can conclude the lemma.
\end{proof}
If $\tau \in \g^\dual$ is a regular value of $\mu$ and $\lm$ is not empty, then the $G$-action on the level manifold $\mu^{-1}(\tau)$ is proper and locally free. Therefore the quotient stack $\C^d\sq{\tau}G = [\mu^{-1}(\tau)/G]$ is a Deligne--Mumford stack.
The restriction $\omega$ of the standard symplectic form $\omega_0$ on $\C^d$ to the level manifold $\mu^{-1}(\tau)$ is $G$-invariant. Since $\ker\omega$ coincides with the Lie algebroid of the action groupoid $G \times \lm \rightrightarrows \lm$ \cite[Proposition 5.40]{mcduff98:_introd}, $\omega$ is a symplectic form on the stack $\C^d \sq{\tau} G$. We call the quotient stack $\C^d \sq{\tau} G$ the \emph{symplectic quotient}.
The moment map $\mu: \C^d\to\g^\dual$ is often assumed to be proper because this assumption guarantees that the symplectic quotient is compact.
\begin{lem}
[Guillemin--Ginzburg--Karshon {\cite[Proposition 4.14]{guillemin02:_momen_hamil}}]
\label{lem:properness}
The moment map $\mu: \C^d \to \g^\dual$ is proper if and only if there exists a covector $\tau = \sum_{\alpha=1}^d s_\alpha w^\alpha$ with $s_\alpha \geq 0\ (\alpha=1,\dots,d)$ such that
$\{ s \in (\R^d)^\dual | \pair{s,\vec{e}_\alpha} \geq 0,\ \rho^\dual(s)=\tau \} $ is compact.
\end{lem}
This subsection can be summarised as the following theorem.
\begin{thm}
\label{thm:symp_quot_is_symp_DM_stack}
Let $(G,\rho,\tau)$ consist of
\begin{itemize}
\item \squashup
a compact Lie group $G$ whose adjoint representation is trivial,
\item \squashup
a homomorphism $\rho: G \to \T^d$ of Lie groups, and
\item \squashup
a regular value $\tau \in \g^\dual$ of the moment map \eqref{eq:moment_map_of_G}.
\end{itemize}
We assume that the triple satisfies the following conditions.
\begin{enumerate}
\item The level set $\lm$ is nonempty.
\item The moment map $\mu: \C^d\to\g^\dual$ is proper.
\end{enumerate}
Then the quotient stack $\C^d\sq{\tau}G = [\lm/G]$ is a compact symplectic Deligne--Mumford stack.
\end{thm}
\subsection{A stacky polytope}
\label{subsection:stacky_polytope}
In this subsection we define a stacky polytope and construct a symplectic Deligne--Mumford stack from a stacky polytope.
\begin{dfn
\label{dfn:stacky_polytopes}
Consider a triple $(N,\Delta,\beta)$ of
\begin{itemize}
\item \squashup
a finitely generated $\Z$-module $N$ of rank $r$,
\item \squashup
a polytope $\Delta$ with $d$ facets $F_1,\dots,F_d$ in $\t^\dual=N^\dual\otimes_\Z\R$, and
\item \squashup
a homomorphism of $\Z$-modules $\beta : \Z^d \to N$.
\end{itemize}
We call the triple $(N,\Delta,\beta)$ a \emph{stacky polytope} if the following conditions are satisfied.
\begin{enumerate}
\item \squashup
The polytope $\Delta$ is simple, i.e.\ every vertex is contained in exactly $r$ facets.
\item \squashup
Let $e_1,\dots, e_d$ be the standard $\Z$-basis of $\Z^d$. The natural map $N \to \t = N \otimes_\Z \R$ is denoted by $n \mapsto \overline{n}$. Let $\Lambda$ be the image of $N$ via the natural map. Then $\overline{\beta(e_1)},\dots, \overline{\beta(e_d)} \in \Lambda$ are vectors perpendicular to the facets $F_1,\dots, F_d$ in inward-pointing way, respectively.
\item \squashup
The cokernel of the homomorphism $\beta: \Z^d \to N$ is finite.
\end{enumerate}
\end{dfn}
The second condition implies that the polytope $\Delta$ is rational and can be described as
\begin{equation}
\label{eqn:polytope-as-intersection-of-hyperplanes}
\Delta = \left\{ \eta \in \t^\dual \!\ \middle| \!\ \pair{\eta,\overline{\beta(e_\alpha)}} \geq -c_\alpha \right\}
\end{equation}
for some $c_\alpha \in \R\ (\alpha = 1,\dots,d)$.
We construct the symplectic Deligne--Mumford stack $\X_{\vec\Delta}$ associated to a stacky polytope $\vec{\Delta}=(N,\Delta,\beta)$ as follows. First we construct a homomorphism of $\Z$-modules $\beta^\DG:(\Z^d)^\dual \to \DG(\beta)$ in the same way to Borisov et al. \cite{borisov05:_chow_delig_mumfor}.
\begin{remark}
In Borisov--Chen--Smith \cite{borisov05:_chow_delig_mumfor}, the homomorphism $(\Z^d)^\dual \to \DG(\beta)$ is denoted by $\beta^\dual$ instead of $\beta^\DG$. In this paper ``${}^\dual$'' always means ``dual'' in the usual sense. Therefore $\beta^\dual$ is the induced homomorphism $N^\dual \to (\Z^d)^\dual$ by $\beta: \Z^d \to N$, where $(\Z^d)^\dual = \Hom_\Z(\Z^d,\Z)$ and $N^\dual = \Hom_\Z(N,\Z)$.
\end{remark}
Take a projective resolution of $N$, i.e.\! an exact sequence of $\Z$-modules
\[
\xymatrix{
\cdots \ar[r]^{\rd_{\vec{F}}} &
F_2 \ar[r]^{\rd_{\vec{F}}} &
F_1 \ar[r]^{\rd_{\vec{F}}} &
F_0 \ar[r] &
N \ar[r] &
0
}
\]
with all the $F_i$'s free over $\Z$. We also take a projective resolution of $\Z^d$
\[
\xymatrix{
\cdots \ar[r]^{\rd_{\vec{E}}} &
E_2 \ar[r]^{\rd_{\vec{E}}} &
E_1 \ar[r]^{\rd_{\vec{E}}} &
E_0 \ar[r] &
\Z^d \ar[r] &
0.
}
\]
Then the homomorphism $\beta: \Z^d \to N$ lifts to a chain map $\beta: \vec{E}\to\vec{F}$ \cite[Theorem 2.2.6]{weibel94}, where $\vec{F}=\{F_i, \rd_{\vec{F}}\}_{i \geq 0}$ and $\vec{E} = \{E_i, \rd_{\vec{E}}\}_{i \geq 0}$. The mapping cone $\Cone(\beta)$ is defined as a chain complex $\{E_{i-1} \oplus F_{i}, \rd_{\Cone(\beta)} \}_{i \geq 0}$ whose boundary operator $\rd_{\Cone(\beta)}$ is given by
\[
\rd_{\Cone(\beta)} : E_{i-1} \oplus F_i \to E_{i-2} \oplus F_{i-1};\
(e,f) \mapsto \bigl(-\rd_{\vec{E}}(e),\rd_{\vec{F}}(f)-\beta(e)\bigr).
\]
Then the mapping cone naturally fits into a short exact sequence of chain complexes:
\[
\xymatrix{
0 \ar[r] & \vec{F} \ar[r] & \Cone(\beta) \ar[r] & \vec{E}[1] \ar[r] & 0,
}
\]
where $\vec{E}[1]$ is the chain complex whose $i$-th term is $E_{i+1}$. The dual sequence
\[
\xymatrix{
0 \ar[r] &
\vec{E}[1]^\dual \ar[r] &
\Cone(\beta)^\dual \ar[r] &
\vec{F}^\dual \ar[r] &
0
}
\]
is a short exact sequence of cochain complexes. It induces a long exact sequence that contains the following part:
\begin{equation}
\label{eqn:part_of_the_long_exact_sequence_containing_cohomology_of_Cone}
\xymatrix@C=16pt{
0 \ar[r] &
N^\dual \ar[r]^{\beta^\dual} &
(\Z^d)^\dual \ar[r] &
\H^1\bigl(\Cone(\beta)^\dual\bigr) \ar[r] &
\Ext_\Z^1(N,\Z) \ar[r] &
0.
}
\end{equation}
Note that the finiteness of $\coker(\beta)$ makes $\beta^\dual$ injective. Denote $\H^1\bigl(\Cone(\beta)^\dual\bigr)$ by $\DG(\beta)$ and define $\beta^\DG: (\Z^d)^\dual\to\DG(\beta)$ as the second homomorphism in the above sequence. Both $\DG(\beta)$ and $\beta^\DG$ are well-defined up to natural isomorphism \cite{borisov05:_chow_delig_mumfor}.
To construct the stack $\X_{\vec\Delta}$ as a symplectic quotient, we give a triple $(G,\rho,\tau)$ satisfying the assumptions of Theorem \ref{thm:symp_quot_is_symp_DM_stack}.
Since $N^\dual$ is naturally isomorphic to $\Lambda^\dual$, the following exact sequence forms part of the exact sequence \eqref{eqn:part_of_the_long_exact_sequence_containing_cohomology_of_Cone}:
\begin{equation}
\label{eqn:exact_sequence_including_DG}
\xymatrix@C=40pt{
0 \ar[r] &
\Lambda^\dual \ar[r]^(0.45){\beta^\dual} &
(\Z^d)^\dual \ar[r]^{\beta^\DG} &
\DG(\beta).
}
\end{equation}
Since the torus $\T$ is injective as a $\Z$-module, the functor $\Hom_\Z(-,\T)$ is exact. Applying the functor to the sequence \eqref{eqn:exact_sequence_including_DG}, we obtain the exact sequence of Lie groups:
\begin{equation}
\label{eqn:exact_seq_which_we_start_with}
\xymatrix@C=40pt{
G \ar[r]^{\rho} & \T^d \ar[r]^{\sigma} & T \ar[r] & \{\1\},
}
\end{equation}
where $G=\Hom_\Z\bigl(\DG(\beta),\T\bigr)$ and $T=\Hom_\Z\bigl({\Lambda}^\dual,\T\bigr)$. Note that we can identify $\Hom_\Z((\Z^d)^\dual,\T) = \Z^d \otimes \T$ with $\T^d = \R^d/\Z^d$. The homomorphisms $\rho$ and $\sigma$ are induced by $\beta^\DG$ and $\beta^\dual$ respectively. Since $\DG(\beta)$ is an finitely generated $\Z$-module, $G$ is a compact abelian Lie group. As Section~\ref{sec:symp_toric_quot}, the group $G$ acts on $\C^d$ with a moment map $\mu$ given by \eqref{eq:moment_map_of_G}.
\begin{lem}
Set $\tau = \sum_{\alpha=1}^d c_\alpha w^\alpha$, where $c_\alpha$ is the constant appearing Equation (\ref{eqn:polytope-as-intersection-of-hyperplanes}) and $w^\alpha = \rho^\dual(e^\alpha)$. Then $\tau$ is a regular value of the moment map $\mu$.
\end{lem}
The above lemma is proved by using Lemma \ref{lem:reg-val-and-local-freeness}. (See also the proof of Proposition 5.15 in Guillemin--Ginzburg--Karshon \cite{guillemin02:_momen_hamil}.)
\begin{lem}
\label{lem:compactness_of_stack_associated_to_stacky_polytope}
The triple $(G,\rho,\tau)$ satisfies the assumptions of Theorem \ref{thm:symp_quot_is_symp_DM_stack}.
\end{lem}
\begin{proof}
By Lemma \ref{lem:properness},
it suffices to show that
\begin{enumerate}
\item $\tau \in \bigl\{ \sum_{\alpha=1}^d s_\alpha w^\alpha \!\ \big|\!\ s_\alpha \geq 0 \bigr\}$, and
\item the set $\Delta_\tau = \bigl\{ s \in (\R^d)^\dual \ \big|\!\ \pair{s,\vec{e}_\alpha} \geq 0,\ \rho^\dual(s)=\tau \bigr\}$ is compact.
\end{enumerate}
The first condition is equivalent that $\Delta_\tau \ne \varnothing$. Putting $\tau' = \sum_{\alpha=1}^d c_\alpha \vec{e}^\alpha$, we have
\begin{align*}
\sigma^\dual(\Delta) + \tau'
&= \bigl\{
\sigma^\dual(\eta)+\tau' \in (\R^d)^\dual \big|
\pair{\eta,\sigma(\vec{e}_\alpha)} \geq -c_\alpha
\bigr\} \\
&= \bigl\{
s \in (\R^d)^\dual \big|
\pair{s,\vec{e}_\alpha} \geq 0,\
\rho^\dual(s) = \tau
\bigr\} \\
&= \Delta_\tau.
\end{align*}
Since $\Delta$ is nonempty and compact, so is $\Delta_\tau$.
\end{proof}
Applying Proposition \ref{thm:symp_quot_is_symp_DM_stack} to the triple $(G,\rho,\tau)$, we finally obtain a compact symplectic Deligne--Mumford stack $\C^d\sq{\tau}G$.
\begin{thm}
Let $\vec{\Delta}=(N,\Delta,\beta)$ be a stacky polytope. Define $(G,\rho,\tau)$ by a triple consisting of
\begin{itemize}
\item the Lie group $G = \Hom_\Z(\DG(\beta),\T)$,
\item the homomorphism $\rho:G \to \T^d$ induced by $\beta^{\DG}:(\Z^d)^\dual \to \DG(\beta)$, and
\item the covector $\tau=\sum_{\alpha=1}^d c_\alpha w^\alpha$.
\end{itemize}
Here $d$ and $c_\alpha$ are the constants appearing Definition \ref{dfn:stacky_polytopes} and $w^\alpha = \rho^\dual(e^\alpha)$. Then the triple $(G,\rho,\tau)$ satisfies the assumptions of Theorem \ref{thm:symp_quot_is_symp_DM_stack}.
\end{thm}
By Theorem \ref{thm:symp_quot_is_symp_DM_stack}, we obtain the symplectic quotient $\C^d\sq{\tau}G$ associated to the above triple $(G,\rho,\tau)$. We call the symplectic quotient the \emph{symplectic Deligne--Mumford stack $\X_{\vec\Delta}$ associated to the stacky polytope $\vec{\Delta}$}.
\begin{remark}
For the symplectic Deligne--Mumford stack associated to a stacky polytope, all stabilizer groups are abelian because $\Hom_\Z(\DG(\beta),\T)$ is abelian.
\end{remark}
\begin{remark}
Lerman and Tolman defined labelled polytopes to classify compact symplectic toric orbifolds and established a construction (Lerman--Tolman--Delzant construction) of compact symplectic toric orbifolds from labelled polytopes \cite{LT}. Their construction can also be used to produce symplectic Deligne--Mumford stacks. Ignoring a Hamiltonian structure, our construction is slightly wider than the Lerman--Tolman--Delzant construction in the following sense: Let $(N,\Delta,\beta)$ be a stacky polytope. Suppose $N$ to be free over $\Z$. Define a label $m_\alpha$ of $\alpha$-th facet $F_\alpha$ by the identity $\beta(e_\alpha) = m_\alpha \nu_\alpha$, where $\nu_\alpha$ is the primitive inward-pointing vector perpendicular to the facet $F_\alpha$. Then the rational convex polytope $\Delta$ together with the labels $m_1, \dots, m_d$ associated to the facets $F_1, \dots, F_d$ gives a labelled polytope. On the other hand every labelled polytope arises in this way.
\end{remark}
\subsection{The stacky polytope of a torus quotient}
\label{subsection:stacky_polytope_of_torus_quotients}
Given a symplectic Deligne--Mumford stack $\X$ (with abelian stabilizer groups) it is natural to ask whether there is a stacky polytope $\vec{\Delta}$ such that $\X_{\vec{\Delta}}$ is equivalent to $\X$. In this section we give a partial solution to this question: If $G$ is a torus and $\X$ is the symplectic Deligne--Mumford stack $\X$ associated to a triple $(G,\rho,\tau)$ satisfying the assumptions in Theorem \ref{thm:symp_quot_is_symp_DM_stack}, then we can find a stacky polytope $\vec{\Delta}$ in such a way that the associated stack $\X_{\vec\Delta}$ is equivalent to $\X$.
Define $\Z_G $ by $\ker(\exp:\Lie(G) \to G)$. Then the homomorphism $\rho$ induces a monomorphism of $\Z$-modules $\dot\rho: \Z_G \to \Z^d$ and we have a short exact sequence
\begin{equation}
\label{eqn:projective_resolution_of_N}
\xymatrix@C=30pt{
0 \ar[r] &
\Z_G \ar[r]^<(0.35){\dot\rho} &
\Z^d \ar[r]^<(0.35){\beta} &
N \ar[r] &
0,
}
\end{equation}
where $N$ is the cokernel of $\dot\rho$ and $\beta: \Z^d \to N$ is the natural quotient homomorphism. The image $\Lambda$ of $N$ in the vector space $\t=N\otimes\R$ is a lattice of $\t$ and the torus $T = \t/\Lambda$ can be naturally identified with the quotient torus $\T^d/\rho(G)$. The composition map of $\beta$ and the natural projection $N\to\Lambda$ gives rise to a homomorphism of tori $\sigma: \T^d \to T$.
We may take a homomorphism $s: T\to\T^d$ satisfying $\sigma \circ s = \id_T$. Consider the following map:
\[
\bar{\mu}: \lm \to \t^\dual;\ \ z \mapsto
\sum_{\alpha=1}^d \bigl(\pi|z_\alpha|^2-c_\alpha \bigr)s^\dual(\vec{e}^\alpha).\]
Here $c_1, \dots, c_d$ are real numbers satisfying $\sum_\alpha c_\alpha w^\alpha = \tau$. The image of the map is a convex polytope
\[
\Delta = \bigl\{\eta \in \t^\dual \ \big| \pair{\eta,\bar{n}_\alpha} \geq -c_\alpha\ (\alpha=1,\dots,d) \bigr\},
\]
where $n_\alpha = \beta(\vec{e}_\alpha)$ and $\bar{n}_\alpha$ is the image of $n_\alpha$ via the natural projection $N\to\Lambda$.
\begin{lem}
The triple $\vec{\Delta} = (N,\Delta,\beta)$ is a stacky polytope.
\end{lem}
\begin{proof}
The conditions in Definition~\ref{dfn:stacky_polytopes} are obviously satisfied except the simplicity of $\Delta$. For $I \subset \{1,\dots,d\}$ we define a subset $\Delta_I$ of $\Delta$ by
\[
\Delta_I = \{ \eta \in \Delta \ \big|
\pair{\eta,\bar{n}_\alpha} = -c_\alpha\ \text{ for } \alpha \in I
\}.
\]
Each face of $\Delta$ can be described as $\Delta_I$ for some $I$. According to Cieliebak--Salamon \cite[Lemma E.1]{cieliebak06:_wall}, the set $\Delta_I$ is empty or has codimension $|I|$ in $\t^\dual$. Note that $\dim \Delta = \dim \Delta_\varnothing = \dim \t^\dual$. Suppose $\Delta_I$ is a vertex of $\Delta$. If $\Delta_J$ is a nonempty facet containing the vertex $\Delta_I$, then $J \subset I$. Because $|J| = \codim \Delta_J = 1$, $J = \{j\}$ for some $j \in I$. On the other hand, for any $j \in I$, the set $\Delta_{\{j\}}$ is a nonempty facet containing the vertex $\Delta_I$.
The above discussion implies that the number of nonempty facets containing the vertex $\Delta_I$ is equal to $|I|$. Since $|I|=\codim\Delta_I=\dim\t^\dual$, the convex polytope $\Delta$ is simple.
\end{proof}
\begin{thm}
The symplectic Deligne--Mumford stack $\X_\vec{\Delta}$ associated to the stacky polytope $\vec{\Delta}$ is equivalent to $\C^d\sq{\tau}G$.
\end{thm}
\begin{proof}
Note that the short exact sequence \eqref{eqn:projective_resolution_of_N} gives a projective resolution of the $\Z$-module $N$. The dual of the mapping cone is given by
\[
\xymatrix@C=30pt{0 \ar[r] & (\Z^d)^\dual \ar[r]^(0.37){d} & (\Z^d)^\dual\oplus(\Z_G)^\dual \ar[r] & 0 \ar[r] & \cdots.}
\]
Here the differential $d$ is explicitly given by $d\vec{e}^\alpha = (-\vec{e}^\alpha,\rho^\dual(\vec{e}^\alpha))\ (\alpha=1,\dots,d)$. We can identify $\DG(\beta) = \coker(d)$ with $(\Z_G)^\dual$ by the map
\[
\DG(\beta) \to (\Z_G)^\dual;\
[\vec{e}^\alpha,w] \mapsto \rho^\dual(\vec{e}^\alpha)+w.
\]
Under this identification, the abelian Lie group $\Hom_\Z(\DG(\beta),\T)$ is the torus $G$ and the homomorphism $\Hom_Z(\DG(\beta),\T) \to \T^d$ induced by $\beta^\DG$ in the exact sequence \eqref{eqn:exact_sequence_including_DG} is the same as the homomorphism $\rho: G \to \T^d$. Therefore $\X_\vec{\Delta}$ and $\C^d\sq{\tau}G$ are both defined by the same data $(G,\rho,\tau)$.
\end{proof}
\begin{examples}
Let $\vec{w}=(w_1,\dots,w_d)$ be a $d$-tuple of positive integers. The \emph{weighted projective space} $\P(\vec{w})$ of weight $\vec{w}$ is defined as follows. Consider the homomorphism $\rho: \T \to \T^d$ defined by $\rho([\xi])=[w_1\xi,\dots,w_d\xi]$, where $\xi \in \R = \Lie(\T)$. Then the $\T$-action on $\C^d$ via $\rho$ is given by
\[
\T \acts \C^d;\
[\xi] \cdot (z_1,\dots,z_d)
= (e^{-2\pi\i w_1\xi}z_1,\dots,e^{-2\pi\i w_d\xi}z_d).
\]
The following map gives a moment map of the action:
\[
\mu: \C^d \to \R;\
(z_1,\dots,z_d) \mapsto \sum_{\alpha=1}^d \pi |z_\alpha|^2 w_\alpha.
\]
Here we identify $\Lie(\T)^\dual=\R^\dual$ with $\R$ via the dot product on $\R$. The triple $(\T,\rho,\pi)$ satisfies the assumptions of Theorem \ref{thm:symp_quot_is_symp_DM_stack}. The symplectic Deligne--Mumford stack $\C^d\sq{\pi}\T$ defined by the data $(\T,\rho,\pi)$ is called the weighted projective space $\P(\vec{w})$.
A stacky polytope $(N,\Delta,\beta)$ giving $\P(\vec{w})$ consists of the following data
\begin{itemize}
\item A $\Z$-module
$N = \coker(\dot\rho: \Z \to \Z^d) = \Z^d\big/\Z(w_1,\dots,w_d)$.
\item A convex polytope
\[
\Delta= \left\{\sum_{\alpha=1}^d s_\alpha \vec{e}^\alpha \in (\R^d)^\dual \!\ \middle| \!\ s_\alpha \geq -c_\alpha \ (\alpha=1,\dots,d),\ \sum_{\alpha=1}^d s_\alpha w_\alpha = 0 \right\}.
\]
\item The natural projection
$\beta:\Z^d \to N = \Z^d\big/\Z(w_1,\dots,w_d)$.
\end{itemize}
Here $N^\dual\otimes\R = (\R^d/\R(w_1,\dots,w_d))^\dual$ is embedded in $(\R^d)^\dual$ via the induced linear map $\dot\rho$ and $c_1,\dots,c_d$ are real constants with $\sum_{\alpha} c_\alpha w_\alpha = \pi$.
\end{examples}
\section{Stacky polytopes versus stacky fans}
We discuss the relation between stacky polytopes and stacky fans in this section. First we review briefly the Deligne--Mumford stack $\X_{\vec{\Sigma}}$ associated to a stacky fan $\vec{\Sigma}$ which is introduced by Borisov et al. \cite{borisov05:_chow_delig_mumfor}. Their construction is an extension of the quotient construction of Cox \cite{cox95}. Next we assign a stacky fan $\vec{\Sigma}$ to a stacky polytope $\vec{\Delta}$ and establish an equivalence between $\X_{\vec{\Sigma}}$ and $\X_{\vec{\Delta}}$. See Cox \cite{cox05:_lectur_toric_variet} for terminology used in the theory of toric varieties.
\begin{dfn}
Consider a triple $(N,\Sigma,\beta)$ of
\begin{itemize}
\item a finitely generated $\Z$-module $N$ of rank $r$,
\item a fan $\Sigma$ with $d$ rays $\rho_1,\dots,\rho_d$ in $\t=N\otimes_\Z\R$,
and
\item a homomorphism of $\Z$-modules $\beta : \Z^d \to N$.
\end{itemize}
We call the triple $(N,\Sigma,\beta)$ a \emph{stacky fan} if the following conditions are satisfied.
\begin{enumerate}
\item \squashup
The fan $\Sigma$ is simplicial, that is, the minimal generators of every cone $\sigma \in \Sigma$ are linearly independent in $\t$.
\item \squashup
We denote by $\bar{n}_\alpha\ (\alpha=1,\dots,d)$ the image of $n_\alpha = \beta(\vec{e}_\alpha)$ through the natural map $N \to \t$. Then $\bar{n}_\alpha$ generates the ray $\rho_\alpha$ of $\Sigma$.
\item \squashup
The cokernel of the homomorphism $\beta : \Z^d \to N$ is finite.
\end{enumerate}
\end{dfn}
Let $\C[z_1,\dots,z_d]$ be the coordinate ring of $\C^d$. The $\alpha$-th coordinate $z_\alpha$ corresponds to $\alpha$-th ray $\rho_\alpha$. For each cone $\sigma$, the monomial $\prod_{\alpha: \rho_\alpha \not\subset \sigma} z_\alpha$ is denoted by $z^{\hat{\sigma}}$. We define an ideal $J_\Sigma$ of $\C[z_1,\dots,z_d]$ as the ideal generated by the monomials $z^{\hat{\sigma}}\ (\sigma \in \Sigma)$. The Zariski open subset $\C^d\setminus\V(J_\Sigma)$ is denoted by $Z_\Sigma$.
Let $\TC$ be the complex torus $\C/\Z$. Applying the exact functor $\Hom_\Z(-,\TC)$ to the exact sequence \eqref{eqn:exact_sequence_including_DG}, we have
\[
\xymatrix@C=40pt{
G_\C \ar[r]^{\rho_\C} &
\TC^d \ar[r] &
T_\C \ar[r] &
\{\1\}.
}
\]
Here $G_\C = \Hom_\Z(\DG(\beta),\TC)$ and $T_\C = \Hom_\Z(N^\dual,\TC)$. Note that we can identify $\Hom_\Z((\Z^d)^\dual,\TC)$ with $\TC^d = \C^d/\Z^d$ naturally. The $d$-dimensional torus $\TC^d$ naturally acts on $Z_\Sigma$. Therefore the group $G_\C$ also acts on $Z_\Sigma$ through the homomorphism $\rho_\C: G_\C \to \TC^d$. Let $\X_{\vec{\Sigma}}$ be the quotient stack $\bigl[Z_\Sigma / G_\C\bigr]$.
\begin{remark}
The stack $\X_{\vec{\Sigma}}$ is usually considered as a stack over the category of schemes \cite{borisov05:_chow_delig_mumfor}. Since $Z_\Sigma$ is an open subset of $\C^d$ with respect to the usual topology and the $G_\C$-action on $Z_\Sigma$ is smooth, we consider $\X_{\vec{\Sigma}}$ as a stack over $\mathfrak{Diff}$.
\end{remark}
\begin{prop}
[Borisov--Chen--Smith \cite{borisov05:_chow_delig_mumfor}]
For each stacky fan $\vec{\Sigma} = (N,\Sigma,\beta)$, the quotient stack $\X_\vec{\Sigma}$ is a Deligne--Mumford stack. The underlying space of $\X_\vec{\Sigma}$ is the toric variety determined by the fan $\Sigma$. The stack $\X_\vec{\Sigma}$ is called the \emph{toric Deligne--Mumford stack associated to the stacky fan} $\vec{\Sigma}$.
\end{prop}
We can associate a rational fan $\Sigma \subset \t$ to each simple rational polytope $\Delta \subset \t^\dual$ \cite{cox05:_lectur_toric_variet}: each face $F$ of $\Delta$ corresponds to the cone $\sigma_F$ generated by $\bar{n}_\alpha$'s with $F_\alpha \supset F$. Then the set $\Sigma_\Delta = \bigl\{ \sigma_F\!\ \big|\!\ \text{$F$ is a face of $\Delta$} \bigr\}$ is a simplicial fan. Using this correspondence, we can associate the stacky fan $\vec{\Sigma}_\vec{\Delta} = (N,\Sigma_\Delta,\beta)$ to a stacky polytope $\vec{\Delta} = (N,\Delta,\beta)$.
\begin{thm}
\label{thm:equivalence_between_two_stacks}
Let $\vec{\Delta}$ be a stacky polytope and $\vec{\Sigma}$ the stacky fan $\vec{\Sigma}_{\vec{\Delta}}$ defined by $\vec{\Delta}$ as above. Then $\X_\vec{\Delta}$ and $\X_\vec{\Sigma}$ are equivalent as stacks over $\mathfrak{Diff}$.
\end{thm}
First of all, we note that the natural embedding $\T \to \TC$ induces the commutative diagram
\[
\xymatrix@C=50pt@R=20pt{
G_\C \ar[r]^{\rho_\C} & \TC^d \\
G \ar[r]_{\rho}\ar[u] & \T^d. \ar[u]
}
\]
Here $G = \Hom_\Z(\DG(\beta),\T)$ as Section~\ref{section:DM_stack_associated_to_stacky_polytope}. The homomorphisms $G \to G_\C$ and $\T^d \to \TC^d$ are both embeddings.
\begin{lem}
\label{lem:complexification}
Regarding $G$ as a subgroup of $G_\C$ via the above embedding, we have $G_\C = G \times \exp(\i\g)$. Here $\g = \Lie(G)$.
\end{lem}
\begin{proof}
Let $D_\mathrm{tor}$ be the torsion submodule of the finitely generated $\Z$-module $\DG(\beta)$ and $D_\mathrm{free} = \DG(\beta)/D_\mathrm{tor}$. Since $\DG(\beta) \cong D_\mathrm{free} \oplus D_\mathrm{tor}$, we have
$G = \Hom_\Z(D_\mathrm{free},\T)\times\Hom_\Z(D_\mathrm{tor},\T)$. Because the abelian group $\Hom_\Z(D_\mathrm{tor},\T)$ is finite, the Lie algebra of $\Hom_\Z(D_\mathrm{free},\T)$ is $\g$. We also have
\[
G_\C = \Hom_\Z(D_\mathrm{free},\TC)\times\Hom_\Z(D_\mathrm{tor},\TC).
\]
The lemma follows from $\Hom_\Z(D_\mathrm{tor},\T)=\Hom_\Z(D_\mathrm{tor},\T_\C)$ and $\Hom_\Z(D_\mathrm{free},\TC) = \Hom_\Z(D_\mathrm{free},\T) \times \exp(\i\g)$.
\end{proof}
\begin{lem}
\label{lem:nonzero_index}
Define $I(z) = \{ \alpha \!\ |\!\ z_\alpha = 0 \}$ for $z \in \C^d$. Suppose $F$ is a face of $\Delta$ and $\sigma_F$ the cone associated to $F$. Then $z^{\hat{\sigma}_F} \ne 0$ if and only if $F \subset \bigcap_{\alpha \in I(z)} F_\alpha$.
\end{lem}
\begin{proof}
Since $z^{\hat{\sigma}_F} = \prod_{\alpha: \rho_\alpha \not\subset \sigma}z_\alpha$, the monomial $z^{\hat{\sigma}_F}$ is not zero if and only if for all $\alpha$, $\rho_\alpha \not\subset \sigma_F$ implies $z_\alpha \ne 0$. Because of the definition of $\sigma_F$, this is equivalent to the statement that for all $\alpha$, $\alpha \in I(z)$ implies $F \subset F_\alpha$.
\end{proof}
Consider the family $\F_\tau = \bigl\{I(z) \!\ \big|\!\ z \in \lm\bigr\}$. Using the function (\ref{eq:mu_zero}), we have
\begin{align*}
\mu_0\bigl(\lm\bigr)
&= \Bigl\{ \textstyle\sum_\alpha \pi|z_\alpha|^2\vec{e}^\alpha \in (\R^d)^\dual \ \Big|\ z \in \C^d,\ \textstyle\sum_\alpha \pi|z_\alpha|^2w^\alpha = \tau \Bigr\} \\
&= \bigl\{ s \in (\R^d)^\dual \ \big|\!\ \pair{s,\vec{e}_\alpha} \geq 0 \text{ for all } \alpha,\ \rho^\dual(s)=\tau \bigr\}.
\end{align*}
We denote by $\Delta_\tau$ the above set. Then $I \in \F_\tau$ if and only if there is $s \in \Delta_\tau$ such that $I=\{ \alpha\!\ |\!\ s_\alpha=0 \}$.
\begin{lem}
\label{lem:surjectivity}
The space $Z_\Sigma$ includes $\lm$. Moreover $Z_\Sigma = \exp(\i\g) \cdot \lm$.
\end{lem}
\begin{proof}
For a set $I \subset \{1,\dots,d\}$, define $O_I = \bigl\{ z \in \C^d \!\ \big|\!\ z_\alpha=0 \text{ if and only if } \alpha \in I\bigr\}$. According to Guillemin--Ginzburg--Karshon \cite[Theorem 5.18]{guillemin02:_momen_hamil}, we have
\[
\exp(\i\g) \cdot \lm = G_\C \cdot \lm = \bigcup_{I \in \F_\tau}O_I.
\]
Therefore it suffices to show that $Z_\Sigma = \bigcup_{I \in \F_\tau}O_I$.
Suppose $z \in O_I$ for some $I \in \F_\Delta$. Then there exists $s \in \Delta_\tau$ such that $s_\alpha = 0$ if and only if $\alpha \in I(z)$. Since $s \in F = \bigcap_{\alpha \in I(z)} F_\alpha$, $F$ is a nonempty face of $\Delta$. Lemma~\ref{lem:nonzero_index} says that $z^{\hat\sigma_F} \ne 0$. Thus $z \in Z_\Sigma$.
Conversely suppose $z \in Z_\Sigma$. Then there exists a face $F$ of $\Delta$ such that $z^{\hat\sigma_F} \ne 0$. Lemma~\ref{lem:nonzero_index} says that $F \subset \bigcap_{\alpha \in I(z)} F_\alpha$. Since $\bigcap_{\alpha \in I(z)} F_\alpha$ is not empty, we have $I(z) \in \F_\Delta$ and $z \in O_{I(z)}$.
\end{proof}
\begin{lem}
\label{lem:stabilizer_blongs_to_compact_torus}
If both $z \in \lm$ and $u \cdot z \in \lm$ hold for $z \in \lm$ and $u \in G_\C$, then $u \in G$.
\end{lem}
\begin{proof}
Lemma~\ref{lem:complexification} says that there are $\xi \in \g$ and $g \in G$ with $u = g\exp(\i\xi)$. Supposing $g\exp(\i\xi) \cdot z \in \lm$, we show that $\xi=0$.
The infinitesimal action of $\xi$ at $z' = (z'_1,\dots,z'_d) \in \C^d$ is given by
\begin{align*}
\xi_{\C^d}(z')
&= \bigl(-2\pi\i\pair{w_1,\xi}z'_1,\dots,-2\pi\i\pair{w_d,\xi}z'_d\bigr)
\end{align*}
under the usual identification $T_{z'} \C^d \cong \C^d$. Since a moment map of the $G$-action on $\C^d$ is given by $\mu: \C^d \to \g^\dual$ in \eqref{eq:moment_map_of_G}, the infinitesimal action $\xi_{\C^d}$ is the same as $J\grad \pair{\mu,\xi}$, where $J$ is the complex structure on $\C^d$ and $\grad \pair{\mu,\xi}$ is the gradient vector field of $\pair{\mu,\xi}$ with respect to the standard Riemannian metric on $\C^d$.
Consider the smooth curve $c$ in $\C^d$ defined by
\[
c: \R \to \C^d;\ \lambda \mapsto g\exp(\i\lambda\xi) \cdot z.
\]
The velocity vector field of the curve is given by
\begin{align*}
\dot{c}(\lambda)
&= \bigl(2\pi\pair{w_1,\xi}c_1(\lambda),\dots,2\pi\pair{w_d,\xi}c_d(\lambda)\bigr)
= -J \xi_{\C^d}(c(\lambda))
= \grad \pair{\mu,\xi} |_{c(\lambda)}.
\end{align*}
Thus the curve $c$ is an integral curve of $\grad\pair{\mu,\xi}$ and $\pair{\mu(c(\lambda)),\xi}$ is a non-decreasing function in $\lambda$. Since both $c(0) = g \cdot z$ and $c(1) = u \cdot z$ belong to $\lm$, the $1$-form $d\pair{\mu(c(\lambda)),\xi}$ vanishes on $0 \leq \lambda \leq 1$. We have
\begin{align*}
\bigl( d\pair{\mu(c(\lambda)),\xi} \bigr) \dot{c}(\lambda)
&= \omega_0\bigl(\xi_{\C^d}(c(\lambda)),\dot{c}(\lambda)\bigr)
= \bigl( J\dot{c}(\lambda), J\dot{c}(\lambda) \bigr)_{\C^d}
= \bigl\| \dot{c}(\lambda) \bigr\|^2,
\end{align*}
where $(\cdot,\cdot)_{\C^d} = \omega_0(\cdot,J\cdot)$ is the standard Riemannian metric on $\C^d$. The above calculation implies that $\dot{c}(\lambda)=0$ if $0 \leq \lambda \leq 1$. Therefore $ \xi_{\lm}(z) = \xi_{\C^d}(z) = J\dot{c}(0) = 0$. Since the $G$-action is locally free, we can conclude that $\xi=0$.
\end{proof}
\begin{lem}
\label{lem:diffeomorphism}
The map
\[
\phi : \g \times \lm \to Z_\Sigma;\
(\xi,z) \mapsto \exp(\i\xi) \cdot z
\]
is $G_\C$-equivariant diffeomorphism. Here the $G_\C$-action on $\g \times \lm$ is defined by
\[
G_\C = G \times \exp(\i\g) \acts \g \times \lm;\
\bigl(g,\exp(\i\theta)\bigr) \cdot (\xi,z) = (\theta+\xi,g \cdot z).
\]
\end{lem}
\begin{proof}
According to Lemma~\ref{lem:surjectivity}, the map $\phi$ is well-defined and surjective. Since $G \cap \exp(\i\g) = \{\1\}$, Lemma~\ref{lem:stabilizer_blongs_to_compact_torus} implies that $\phi$ is injective. It is trivial that the map $\phi$ is $\exp(\i\g)$-equivariant. Therefore it suffices to see that the derivative $d\phi(\xi,z)$ is bijective for any $(\xi,z) \in \g \times \lm$.
For $v \in T_z\lm$ and $\theta \in \g$, take a curve $c: (-\epsilon,\epsilon) \to \lm$ satisfying $c(0)=z$ and $\dot{c}(0)=v$. According to the proof of Lemma~\ref{lem:stabilizer_blongs_to_compact_torus}, $\exp(\i\g)$-orbits transverse to $\lm$. Therefore the curve
\[
\gamma: (-\epsilon,\epsilon) \to Z_\Sigma;\
\lambda\mapsto \exp(\lambda\i\theta)c(\lambda)
\]
satisfies $\gamma(0)=z$ and $\dot{\gamma}(0)=(\theta,v) \in \t \times T_z\lm \cong T_z Z_\Sigma$. Then we have
\begin{align*}
d\phi(\xi,z)(\theta,v)
&= \left.\dfrac{d}{d\lambda}\right|_{\lambda=0} \phi(\xi+\lambda\theta,c(\lambda)) \\
&= \left.\dfrac{d}{d\lambda}\right|_{\lambda=0} \exp(\i\xi) \cdot \gamma(\lambda) \\
&= d\bigl(\exp(\i\xi)\bigr)(\theta,v).
\end{align*}
Therefore $d\phi(\xi,z)$ is bijective.
\end{proof}
Under the diffeomorphism $\phi: \g \times \lm \to Z_\Sigma$, define the $G$-equivariant map $\psi: Z_\Sigma \to \lm$ as the second projection map.
\begin{proof}%
[Proof of Theorem~\ref{thm:equivalence_between_two_stacks}]
We define a morphism of stacks $\Phi: \X_\vec{\Delta}\to\X_\vec{\Sigma}$ as follows. For an object $U \stackrel{\pi}{\longleftarrow} P \stackrel{\epsilon}{\longrightarrow} \lm$ of $\X_\vec{\Delta}$ over a manifold $U$, the morphism $\Phi$ assigns to it the object
\[
\xymatrix@C=40pt{
U &
P \times_{\lm} Z_\Sigma \ar[l]_(0.6){\Phi_\pi} \ar[r]^(0.6){\Phi_\epsilon} &
Z_\Sigma
}
\]
of $\X_\vec{\Sigma}$, where $\Phi_\epsilon: P \times_{\lm} Z_\Sigma \to Z_\Sigma$ is defined by the fibred square
\[
\xymatrix@C=50pt{
P \times_{\lm} Z_\Sigma \ar[r]^(0.6){\Phi_\epsilon} \ar[d] &
Z_\Sigma \ar[d]^{\psi} \\
P \ar[r]_(0.45){\epsilon} &
\lm,
}
\]
and $\Phi_\pi$ sends $(p,z)$ to $\pi(p)$. A free right action of $G_\C$ on $P \times_{\lm} Z_\Sigma$ is defined by
\[
P \times_{\lm} Z_\Sigma \curvearrowleft G_\C;\
(p,z) \cdot u = (p \cdot \nu(u), u^{-1} \cdot z),
\]
where $\nu: G_\C = G \times \exp(\i\g) \to G$ is the first projection. Then $\Phi_\pi$ is a principal $G_\C$-bundle.
Let $(f,\tilde{f}): P \to P'$ be an arrow from $P$ to $P'$ in $\X_\vec{\Delta}$. We assign to $\Phi(f,\tilde{f})$ the map $P \times_{\lm} Z_\Sigma \to P' \times_{\lm} Z_\Sigma$ sending $(p,z) \to (\tilde{f}(p),z)$.
It suffices to see that $\Phi$ is a monomorphism and an epimorphism \cite[Proposition 2.1]{behrend08:_differ_stack_gerbes}.
First we show that the $\Phi$ is a monomorphism. Suppose that we have two objects $P$ and $P'$ of $\X_\vec{\Delta}$ over $U$ and an arrow $\beta$ from $\Phi(P)$ to $\Phi(P')$. The arrow $\beta$ is a $G_\C$-equivariant diffeomorphism which makes the following diagram commute.
\[
\xymatrix@R=2pt{
& & P \times_{\lm} Z_\Sigma \ar@<-1ex>[lld]_(0.6){\Phi_{\pi}} \ar@<1ex>[rrd]^(0.6){\Phi_{\epsilon}} \ar@<-4pt>[dd]^{\beta} & & \\
U & & & & Z_\Sigma. \\
& & P' \times_{\lm} Z_\Sigma \ar@<1ex>[llu]^(0.6){\Phi_{\pi'}} \ar@<-1ex>[rru]_(0.6){\Phi_{\epsilon'}} & &v
}
\]
We must to show that there exists uniquely a $G$-equivariant diffeomorphism $\alpha$ from $P$ to $P'$. Since $\psi: Z_\Sigma \to \lm$ is a (trivial) principal $\exp(\i\g)$-bundle, so are $P \times_{\lm} Z_\Sigma \to P$ and $P' \times_{\lm} Z_\Sigma \to P'$. Therefore the $G_\C$-equivariant diffeomorphism $\beta$ induces a $G$-equivariant diffeomorphism $\alpha: P \to P'$. Then $\beta$ is a bundle map over $\alpha$:
\[
\xymatrix{
P \times_{\lm} Z_\Sigma \ar[r]^{\beta} \ar[d] &
P' \times_{\lm} Z_\Sigma \ar[d]
\\
P \ar[r]_{\alpha} &
P'
}
\]
Since $\Phi_{\epsilon} = \Phi_{\epsilon'} \circ \beta$, we obtain $\beta(p,z)=\bigl(\alpha(p),z\bigr)$. Taking $z \in Z_\Sigma$ with $\epsilon(p)=\psi(z)$, we have
\[
\epsilon'(\alpha(p))
= \psi\bigl(\Phi_{\epsilon'}(\beta(p,z))\bigr)
= \psi\bigl(\Phi_{\epsilon}(p,z)\bigr)
= \epsilon(p),
\]
and
\[
\pi'(\alpha(p)) = \Phi_{\pi'}(\beta(p,z)) = \Phi_{\pi}(p,z) = \pi(p).
\]
Therefore $\beta = \Phi_\alpha$. The arrow $\alpha$ is unique because $\beta$ must be a bundle map over $\alpha$.
Next we show that the $\Phi$ is an epimorphism. Suppose we have an object $\widetilde{P}$ of $\X_\vec{\Sigma}$ over a manifold $U$. Putting $P = \widetilde{P}\big/\!\exp(\i\g)$, we obtain a commutative diagram:
\[
\xymatrix@R=3pt@C=50pt{
& \widetilde{P}\ar[dd]\ar[r]^{\tilde{\epsilon}}\ar[ld]_{\tilde{\pi}}
& Z_\Sigma \ar[dd]^{\psi} \\
U & & \\
& P \ar[r]_{\epsilon} \ar[lu]^{\pi} & \lm.
}
\]
Then $P$ is an object of $\X_\vec{\Delta}$ over $U$ such that $\Phi(P)=\widetilde{P}$.
\end{proof}
\subsection*{Acknowledgements}
The author is grateful to River Chiang, Martin Guest and Reyer Sjamaar for valuable comments. This research is partially supported by NSC grant 98-2115-M-006-006-MY2, and the NCTS (South). This work was supported by National Science Council grant [98-2115-M-006-006-MY2]; and the National Center for Theoretical Sciences (South).
|
{
"timestamp": "2012-02-28T02:01:56",
"yymm": "1009",
"arxiv_id": "1009.3547",
"language": "en",
"url": "https://arxiv.org/abs/1009.3547"
}
|
\section{}
\section{Introduction}
The adiabatic theorem is important in both classical and quantum
mechanics \cite{adiabatic}. It predicts a system's dynamical
behavior subject to slowly varying system parameters. Although a
general and mathematically rigorous proof of the adiabatic theorem
is not obvious in both classical mechanics and quantum mechanics,
the adiabatic theorem has been widely used. Indeed, it is always
highly useful so long as there exist two drastically different time
scales. The adiabatic theorem has also led to the discoveries of
Berry phase \cite{Berryphase} and the classical counterpart, i.e.,
Hannay's angle \cite{Hannay}.
We focus on the classical adiabatic theorem (CAT), but as shown
below, some of our results can be applied to quantum systems as
well. Our interest here is not in a rigorous proof of the CAT, but
in dynamical fluctuations around what is predicted by CAT. As
discussed below, the possible consequences of the fluctuations
neglected by the conventional CAT can be far reaching. The
motivation of considering the fluctuations is based on a simple
observation. That is, CAT, whose proof is based on an average over
fast-varying variables, only reflects a mean dynamical behavior. As
such fluctuations on top of a mean dynamical behavior should exist
in classical adiabatic processes. Though fluctuations should be
intuitively smaller in a slower adiabatic process, their effects are
accumulated over a longer time scale and hence might not vanish even
in the adiabatic limit. For instance, in a few early
studies~\cite{Golin1, Golin2, Berry1996Non}, including the study of
``Hannay's angle of the world"~\cite{Berry1996Non,adam}, the actual
total change in canonical variables may depend on the smoothness of
the evolving adiabatic parameters. This abnormal behavior was shown
to be connected with subtle fluctuations in the action variables
from their average behavior predicted by CAT. Clearly then, a
general description of the dynamical fluctuations in adiabatically
evolving and classically integrable systems should be of importance.
We shall present in this work a general result that describes the
dynamical fluctuations inherent to classical adiabatic processes.
Roughly speaking, it establishes an interesting connection between
the actual rate of change of slowly varying system parameters and
the actual classical orbits deformed from that predicted by CAT. To
illustrate the usefulness of our general result, we design a simple
dynamical model with an adiabatically moving fixed-point solution,
from which an intriguing classical geometric phase can emerge.
We then exploit our general result to discuss the ``pollution" to
Hannay's angle in classical adiabatic processes. A mean-field model
that describes a two-mode Bose-Einstein condensate (BEC) is also
proposed to study fluctuation-induced ``pollution" to adiabatic
quantum evolution.
To tackle with dynamical fluctuations, one may quickly think of an
equation describing the time dependence of the fluctuations around
ideal adiabatic orbits. But this approach may not be fruitful
because in principle, the time dependence of any canonical variables
is already fully captured by classical canonical equations of
motion. Instead, we are concerned with how fluctuations distort
trajectories as compared with that predicted by CAT. In this sense,
our approach is somewhat in a similar spirit as an early
``multiple-time-scale-expansion" approach to corrections to
classical adiabatic invariants in chaotic systems \cite{jarzynski2}.
However, we focus on fluctuations associated with individual orbits
in integrable systems, rather than fluctuations associated with an
ensemble of chaotic trajectories in an energy shell.
This paper is organized as follows. In Sec.~II we derive a
differential equation describing the dynamical fluctuations in
classical adiabatic processes. Some related details are also
provided in Appendix. As an application, in Sec.~III we study the
case of an adiabatically moving fixed-point solution and show how an
intriguing geometric angle may emerge in a simple toy model. Based
on our general result, Sec.~IV discusses why the ``pollution" to
Hannay's angle may exist and then proposes a physical system to
study analogous fluctuation-induced pollution.
We finally give a brief summary in Sec.~V.
\section{General Description of Dynamical Fluctuations in Classical Adiabatic Processes}
Consider a classical integrable system with $N$ degrees of freedom.
Its Hamiltonian is given by $H(\mathbf{p},\mathbf{q}, \mathbf{R})$,
where canonical variables $\mathbf{p}=(p_1,p_2,\ldots,p_N)$ and
$\mathbf{q}=(q_1,q_2,\ldots,q_N)$ represent canonical momenta and
coordinates, and $\mathbf{R}$ represents a collection of system
parameters. Let $F(\mathbf{I},\mathbf{q},\mathbf{R})$ be the
generating function that induces the $\mathbf{R}$-dependent
canonical transformation from $(\mathbf{p},\mathbf{q})$ to the
action-angle variables $(\mathbf{I},\mathbf{\Theta})$, where
$I_i=\frac{1}{2\pi}\oint p_i dq_i$ and
$\mathbf{\Theta}=(\theta_1,\theta_2,\ldots,\theta_N)$.
To clearly present our derivation of a differential equation that
describes dynamical fluctuations in classical adiabatic processes,
this section is divided into four subsections representing the four
steps in our derivation. First, after expressing classical equations
of motion in the action-angle variables
$(\mathbf{I},\mathbf{\Theta})$, we define dynamical fluctuations on
top of the idealized solution given by CAT. Second, the time
dependence of the canonical variables $(\mathbf{p},\mathbf{q})$ is
expressed in terms of the dynamical fluctuations we define. Third,
directly using the canonical equations of motion and the canonical
transformation between the action-angle variables and the canonical
variables, we reexpress the time dependence of the canonical
variables in terms of the dynamical fluctuations as well as the
action-angle variables along idealized classical orbits. Finally, by
comparing results in the second and third steps a differential
equation describing the dynamical fluctuations around idealized
adiabatic orbits is obtained.
\subsection{Dynamical fluctuations}
In the $(\mathbf{I},\mathbf{\Theta})$ representation an integrable
Hamiltonian becomes $\mathcal{H}(\mathbf{I},\mathbf{R})$, which is
independent of the angle variables $\mathbf{\Theta}$. For
time-varying $\mathbf{R}=\mathbf{R}(t)$, the equations of motion for
$(\mathbf{I},\mathbf{\Theta})$ are given by~\cite{Berry1985JPA}
\begin{eqnarray} \label{new-dyna1}
\frac{d I_i}{dt}& = &-\frac{\partial \mathbf{W}}{\partial \theta_i}\cdot \frac{d\mathbf{R}}{dt}, \\
\label{new-dyna2} \frac{d\theta_i}{dt}& =
&\omega_i(\mathbf{I};\mathbf{R})+ \frac{\partial
\mathbf{W}}{\partial I_i}\cdot\frac{d\mathbf{R}}{dt} ,
\end{eqnarray}
where $\omega_i(\mathbf{I},\mathbf{R})=\partial\mathcal{H}/\partial
I_i$ is the angular frequency, and $\mathbf{W}$ is defined by
\begin{equation}
\mathbf{W} \equiv \mathbf{\nabla}_{\mathbf{R}}
F[\mathbf{I},\mathbf{q}(\mathbf{I},\mathbf{\Theta},\mathbf{R}),\mathbf{R}]-\mathbf{p}\cdot\mathbf{\nabla}_{\mathbf{R}}
\mathbf{q}(\mathbf{I},\mathbf{\Theta},\mathbf{R}).
\end{equation}
Note that $\mathbf{\nabla}_{\mathbf{R}}$ refers to the gradient in
the parameter space under fixed $(\mathbf{I},\mathbf{\Theta})$. If
\begin{eqnarray}
\epsilon\equiv \left|\frac{d\mathbf{R}}{dt}\right|
\end{eqnarray}
is much smaller than
$\omega_i|\mathbf{R}|$, one can take the average of
Eq.~(\ref{new-dyna1}) over the rapidly oscillating angle variables,
yielding $\frac{dI_i}{dt} \approx 0$ ($\mathbf{W}$ is a periodic
function of $\mathbf{\Theta}$). CAT hence identifies the action
variables as adiabatic invariants, i.e., in adiabatic processes
their values are fixed at
$\overline{\mathbf{I}}\equiv(\overline{I}_1,\overline{I}_2,\ldots,
\overline{I}_N)$. For clarity, angle variables associated with this
idealized solution are defined as $\overline{\mathbf{\Theta}}\equiv
(\overline{\theta}_1,\overline{\theta}_2,\ldots,
\overline{\theta}_N)$. We also use $\overline{\mathbf{p}}\equiv
\mathbf{p}(\overline{\mathbf{I}},\overline{\mathbf{\Theta}},\mathbf{R})$
and $\overline{\mathbf{q}}\equiv
\mathbf{q}(\overline{\mathbf{I}},\overline{\mathbf{\Theta}},\mathbf{R})$
to describe the idealized solution in terms of the (old) set of
canonical variables. With the action variables fixed at
$\overline{\mathbf{I}}$, one may then solve Eq.~(\ref{new-dyna2})
for a cyclic process from $t=0$ to $t=T$ in a straightforward
manner. One may further take the average of the idealized solution
over all possible initial angle values to obtain Hannay's angle,
which is the total mean angle change minus a dynamical angle.
The above discussion does not represent a complete description of
classical adiabatic processes. Clearly, Eq. (\ref{new-dyna1}) tells
us that $\frac{dI_i}{dt}$ is not mathematically zero:
it may possess fluctuations of the order $O(\epsilon)$ (i.e., to the first order of $\epsilon$).
As such, in performing an averaging procedure as is done in CAT
one neglects the dynamical correlation between $\mathbf{\Theta}$ and $\mathbf{I}$.
It is hence necessary to reconsider Eq.~(\ref{new-dyna1}) in order
to consider any possible real-orbit fluctuations on top of CAT. On a
real orbit we assume we have
$I_i=\overline{I}_i+\delta{I}_i$, where we have used $\delta$ to represent fluctuations from
the behavior predicted by CAT.
Equivalent to that, one can describe the same fluctuations
from the idealized orbit in terms of $\delta q_j$ and $\delta p_j$.
There are now both idealized adiabatic orbits without considering
fluctuations and true orbits with fluctuations: the geometry of an
idealized orbit can be characterized by
$\mathbf{I}=\overline{\mathbf{I}}$ and
$\overline{\mathbf{\Theta}}\in[0,2\pi)$; and that of a true orbit
with fluctuations is slightly deformed to
\begin{eqnarray}
\mathbf{I}&=&\overline{\mathbf{I}}+\delta \mathbf{I}, \\
\mathbf{\Theta}&=&\overline{\mathbf{\Theta}}+\delta\mathbf{\Theta},
\end{eqnarray}
where $\delta \mathbf{I}$ and $\delta\mathbf{\Theta}$ are assumed to
be at most of the order $O(\epsilon)$. By our definitions above, we
have
\begin{equation} \label{deltadefine}
\left(\begin{array}{c}\delta \mathbf{I} \\
\delta \mathbf{\Theta}\end{array}\right)
=\left(\begin{array}{c}\frac{\partial \overline{\mathbf{I}}}{\partial
\overline{p}_j}\delta p_j+\frac{\partial \overline{\mathbf{I}}}{\partial \overline{q}_j} \delta q_j \\
\frac{\partial \overline{\mathbf{\Theta}}}{\partial
\overline{p}_j}\delta p_j+\frac{\partial
\overline{\mathbf{\Theta}}}{\partial \overline{q}_j}\delta
q_j\end{array}\right)\equiv \left(\begin{array}{c}K \\ M
\end{array}\right) \left( \begin{array} {c} \delta \mathbf{p} \\
\delta\mathbf{q}\end{array}\right).
\end{equation}
Here and in the following the summation convention for repeated
indices is adopted. Equation~(\ref{deltadefine}) also defines two
$N\times 2N$ matrices $K$ and $M$, corresponding to the upper and
lower halves of a Jacobi matrix. Note that throughout we use
$\frac{\partial\bar{f}}{\partial\bar{x}}$ to indicate
$\frac{\partial f}{\partial x}$ evaluated at $x=\bar{x}$.
As will be seen below, it suffices to consider fluctuations of the
first order of $\epsilon$ because higher-order effects cannot be
accumulated with time. We stress that the fluctuations are
intrinsic: they are nonzero so long as $\epsilon$ is not identically
zero. In other words, fluctuations considered here exist in any
classical adiabatic process and should not be thought of an effect
arising from a too-large $\epsilon$. It should be also noted that in
principle, all the dynamical information is contained in
Eqs.~(\ref{new-dyna1}) and (\ref{new-dyna2}). However, we are
interested in developing a framework to describe how fluctuations
might behave along an idealized classical orbit.
\subsection{Canonical equations of motion in terms of fluctuations}
In terms of the fluctuations $\delta q_j$ and $\delta p_j$, we next
expand $H(\mathbf{p},\mathbf{q},\mathbf{R})$ around
$\overline{H}\equiv
H(\overline{\mathbf{p}},\overline{\mathbf{q}},\mathbf{R})$ to the
order $O(\epsilon)$, yielding the following canonical equations of
motion for $(\mathbf{q},\mathbf{p})$:
\begin{eqnarray} \label{deltapq} \nonumber
\frac{d p_i}{dt}&=&-\frac{\partial \overline{H}}{\partial
\overline{q}_i}-\frac{\partial^2 \overline{H}}{\partial
\overline{q}_i \partial \overline{p}_j}\delta
p_j-\frac{\partial^2\overline{H}}{\partial \overline{q}_i \partial
\overline{q}_j} \delta q_j \\
&=& \frac{\partial \overline{p}_i}{\partial
\overline{\theta}_j}\omega_{j}(\mathbf{I},\mathbf{R})-\frac{\partial^2
\overline{H}}{\partial \overline{q}_i \partial \overline{p}_j}\delta
p_j-\frac{\partial^2\overline{H}}{\partial \overline{q}_i \partial
\overline{q}_j} \delta
q_j; \nonumber \\
\frac{d q_i}{dt}&=&\frac{\partial \overline{H}}{\partial
\overline{p}_i}+\frac{\partial^2\overline{H}}{\partial
\overline{p}_i \partial \overline{p}_j} \delta
p_j+\frac{\partial^2\overline{H}}{\partial \overline{p}_i \partial
\overline{q}_j}\delta q_j \nonumber \\
&=& \frac{\partial \overline{q}_i}{\partial \overline{\theta}_j}
\omega_{j}(\mathbf{I},\mathbf{R})+
\frac{\partial^2\overline{H}}{\partial \overline{p}_i \partial
\overline{p}_j} \delta p_j+\frac{\partial^2\overline{H}}{\partial
\overline{p}_i \partial \overline{q}_j}\delta q_j, \label{dqdp1}
\end{eqnarray}
where we have used the following two canonical relations
\begin{eqnarray}
\frac{\partial \overline{I}_{j}}{\partial \overline{q}_i}&=&-\frac{\partial \overline{p}_i}{\partial \overline{\theta_j}};
\nonumber \\
\frac{\partial \overline{I}_{j}}{\partial \overline{p}_i}&=&\frac{\partial \overline{q}_i}{\partial \overline{\theta}_j}.
\end{eqnarray}
Through Eq.~(\ref{dqdp1}) it is seen that the time dependence of the
canonical variables $(\mathbf{p},\mathbf{q})$ is connected to the
dynamical fluctuations $\delta q_j$ and $\delta p_j$, to the first
order of $\epsilon$.
\subsection{Time-dependence of canonical variables from action-angle variables}
The time evolution of the canonical variables
$(\mathbf{p},\mathbf{q})$ may be also directly obtained from the
canonical transformation from the action-angle variables to
$(\mathbf{p},\mathbf{q})$ and from the equations of motion given by
Eqs.~(\ref{new-dyna1}) and (\ref{new-dyna2}). In particular, using
\begin{eqnarray}
\frac{d p_i}{dt}=\frac{\partial
p_i}{\partial \mathbf{R}}\frac{d\mathbf{R}}{dt}+ \frac{\partial
p_i}{\partial I_j}\frac{d I_j}{dt}+\frac{\partial p_i}{\partial
\theta_j}\frac{d\theta_j}{dt}
\end{eqnarray}
and the analogous expression for
$\frac{d q_i}{dt}$, rewriting the derivatives in
Eqs.~(\ref{new-dyna1}) and (\ref{new-dyna2}) at
$(\mathbf{I},\mathbf{\Theta})$ in terms of those at
$(\overline{\mathbf{I}},\overline{\mathbf{\Theta}})$, and
neglecting all terms that are at least $O(\epsilon^2)$, one arrives
at (see Appendix for details)
\begin{eqnarray} \nonumber
\frac{d p_i}{dt}&=&\frac{\partial \overline{p}_i}{\partial
\mathbf{R}}\frac{d\mathbf{R}}{dt}-\frac{\partial
\overline{p}_i}{\partial \overline{I}_j}\frac{\partial \mathbf{W}
}{\partial \overline{\theta}_j}\cdot\frac{d\mathbf{R}}{dt}+
\frac{\partial \delta p_i}{\partial
\overline{\theta}_j}\omega_j(\bar{\mathbf{I}}, \mathbf{R}) \\
\nonumber & &+\ \frac{\partial \overline{p}_i}{\partial
\overline{\theta}_j}\left[\frac{\partial \mathbf{W}}{\partial
\overline{I}_j}\cdot
\frac{d\mathbf{R}}{dt}+\omega_j(\bar{\mathbf{I}},
\mathbf{R})+\frac{\partial \omega_j}{\partial \overline{I}_k}\delta
I_k\right]\\ \nonumber \frac{dq_i}{dt}&=& \frac{\partial
\overline{q}_i}{\partial \mathbf{R}}\frac{d\mathbf{R}}{dt}
-\frac{\partial \overline{q}_i}{\partial
\overline{I}_j}\frac{\partial \mathbf{W}}{\partial\overline{\theta}_j}\cdot\frac{d\mathbf{R}}{dt} + \frac{\partial \delta q_i}{\partial \overline{\theta}_j}\omega_j(\bar{\mathbf{I}}, \mathbf{R}) \\
& &+\ \frac{\partial \overline{q}_i}{\partial
\overline{\theta}_j}\left[\frac{\partial \mathbf{W}}{\partial
\overline{I}_j}
\cdot\frac{d\mathbf{R}}{dt}+\omega_j(\bar{\mathbf{I}},\mathbf{R})+\frac{\partial
\omega_j}{\partial \overline{I}_k}\delta I_k\right]. \nonumber \\
\label{dqdp2}
\end{eqnarray}
Interestingly, due to the direct connection between
$(\mathbf{p},\mathbf{q})$ and $(\mathbf{I},\mathbf{\Theta})$, the
full time dependence of $(\mathbf{p},\mathbf{q})$ is connected with
dynamical fluctuations in a highly nontrivial manner. In particular,
the terms $\frac{\partial \delta p_i}{\partial \overline{\theta}_j}$
and $\frac{\partial \delta q_i}{\partial \overline{\theta}_j}$ in
Eq.~(\ref{dqdp2}) indicate that it is important to account for how
dynamical fluctuations change with $\bar{\theta}_j$. This is a
crucial piece of information regarding the overall feature of the
dynamical fluctuations.
\subsection{A differential equation describing dynamical fluctuations}
Both Eq~(\ref{dqdp1}) and Eq.~(\ref{dqdp2}) deal with the same time
dependence of $(\mathbf{p},\mathbf{q})$ and hence they should be
consistent with each other. Comparing these two equations term by
term, we arrive at the following equation,
\begin{equation} \label{ZZ1}
\Gamma \left(\begin{array}{c}\delta\mathbf{p}\\
\delta\mathbf{q}\end{array}\right)=\mathbf
{\Sigma}\cdot\frac{d\mathbf{R}}{dt}+\Pi \left(\begin{array}{c}\delta I_1\\
\delta I_2 \\ \vdots \\ \delta I_N
\end{array}\right)+\left(\begin{array}{c}\frac{\partial
\delta \mathbf{p}}{\partial \overline{\theta}_j}\omega_j\\
\frac{\partial \delta \mathbf{q}}{\partial
\overline{\theta}_j}\omega_j
\end{array}\right),
\end{equation}
where
\begin{equation}
\Gamma=\left(\begin{array}{cc}-\frac{\partial^2\overline{H}}{\partial
\overline{\mathbf{q}}\partial\overline{\mathbf{p}}}&-\frac{\partial^2\overline{H}}{\partial
\overline{\mathbf{q}}\partial\overline{\mathbf{q}}}
\\\frac{\partial^2\overline{H}}{\partial \overline{\mathbf{p}}
\partial \overline{\mathbf{p}}}&\frac{\partial^2\overline{H}}{\partial
\overline{\mathbf{p}} \partial \overline{\mathbf{q}}}
\end{array}\right)
\end{equation}
is a $2N\times2N$ matrix;
\begin{equation}
\mathbf{\Sigma}=\left(\begin{array}{c}-\frac{\partial
\overline{\mathbf{p}}}{\partial
\overline{I}_j}\frac{\partial\mathbf{W}}{\partial\overline{\theta}_j}+\frac{\partial
\overline{\mathbf{p}}}{\partial
\overline{\theta}_j}\frac{\partial\mathbf{W}}{\partial
\overline{I}_j}+\frac{\partial \overline{\mathbf{p}}}{\partial
\mathbf{R}}\\
-\frac{\partial \overline{\mathbf{q}}}{\partial
\overline{I}_j}\frac{\partial\mathbf{W}}{\partial\overline{\theta}_j}+\frac{\partial
\overline{\mathbf{q}}}{\partial
\overline{\theta}_j}\frac{\partial\mathbf{W}}{\partial
\overline{I}_j}+\frac{\partial \overline{\mathbf{q}}}{\partial
\mathbf{R}}\end{array}\right)
\end{equation}
is a $2N\times1$ vector along each direction of $\mathbf{R}$; and
\begin{equation}
\Pi=\left(\begin{array}{ccc}\frac{\partial
\overline{\mathbf{p}}}{\partial \overline{\theta}_j}\frac{\partial
\omega_j}{\partial \overline{I}_1}&\frac{\partial
\overline{\mathbf{p}}}{\partial
\overline{\theta}_j}\frac{\partial \omega_j}{\partial \overline{I}_2}&\cdots\\
\frac{\partial \overline{\mathbf{q}}}{\partial
\overline{\theta}_j}\frac{\partial \omega_j}{\partial
\overline{I}_1}&\frac{\partial \overline{\mathbf{q}}}{\partial
\overline{\theta}_j}\frac{\partial \omega_j}{\partial
\overline{I}_2}&\cdots
\end{array}\right)
\end{equation}
is a $2N\times N$ matrix. Substituting Eq.~(\ref{deltadefine}) into
Eq.~(\ref{ZZ1}), one finally obtains an equation for $\delta
\mathbf{p}$ and $\delta \mathbf{q}$ only:
\begin{equation} \label{differential}
\left(\begin{array}{c}\frac{\partial
\delta \mathbf{p}}{\partial \overline{\theta}_j}\omega_j\\
\frac{\partial \delta \mathbf{q}}{\partial
\overline{\theta}_j}\omega_j\end{array}\right)+(\Pi
K-\Gamma)\left(\begin{array}{c}\delta\mathbf{p}\\
\delta\mathbf{q}\end{array}\right)+\mathbf{\Sigma}\cdot\frac{d\mathbf{R}}{dt}=0.
\end{equation}
For a given integrable Hamiltonian, except for those related to
$\delta \mathbf{p}$ and $\delta \mathbf{q}$ and their derivatives,
all the matrices contained in Eq.~(\ref{differential}) are evaluated
at an idealized orbit and hence can be explicitly obtained.
Some remarks are in order. First, Eq.~(\ref{differential}) is not
about evolving the fluctuations
$(\delta\mathbf{q},\delta\mathbf{p})$ at one moment to the next
moment. Instead, it describes, when the system parameters reach the
current configuration $\mathbf{R}$ with a small but nonzero rate
$\frac{d\mathbf{R}}{dt}$, the deviation of the overall shape of one
true orbit from the idealized orbit without dynamical fluctuations,
i.e., the overall deformed orbit in phase space. To our knowledge,
this result is obtained for the first time here. This detailed
description of the dynamical fluctuations can be very useful for
both quantitative and qualitative considerations. The derivation
here is somewhat lengthy because the physical meaning of
$(\overline{\mathbf{I}},\overline{\mathbf{\Theta}})$ in terms of
$({\bf q},{\bf p})$ and hence the idealized orbit itself is changing
as $\mathbf{R}$ varies. Second, consistent with our treatment to the
first order of $\epsilon$, $(\delta\mathbf{p},\delta\mathbf{q})$ is
seen to depend on $\frac{d\mathbf{R}}{dt}$. If
$\frac{d\mathbf{R}}{dt}$ were identically zero, then
$\delta\mathbf{p}=\delta\mathbf{q}=0$ is one possible solution (If
$\delta\mathbf{p}\ne 0$ and $\delta\mathbf{q}\ne 0$ is still the
solution for $\frac{d\mathbf{R}}{dt}=0$, then this solution
describes the relationship between two infinitely close orbits).
Third, in the absence of the detailed information of
$(\delta \mathbf{p},\delta\mathbf{q})$ for at least one phase space
location, Eq.~(\ref{differential}) alone does not suffice to predict
$(\delta\mathbf{p},\delta\mathbf{q})$ because of its differential
form. As will be discussed later, this implies that in general,
detailed information of the time-dependence of ${\mathbf R}$, e.g.,
its smoothness, can be important for determining the dynamical
fluctuations.
Finally, because the linear Schr\"{o}dinger equation and nonlinear
Gross-Pitaeviskii (GP) equation have an exact canonical structure of
Hamiltonian dynamics \cite{Weinberg,HeslotPRD}, our results here can
be also relevant to quantum adiabatic processes.
If we now consider the mean behavior of
$(\delta\mathbf{p},\delta\mathbf{q})$ along an ideal orbit (denoted
by $\langle\cdot\rangle$), then using the fact that
$(\delta\mathbf{p},\delta\mathbf{q})$ are periodic functions of
$\overline{\mathbf{\Theta}}$, we reduce Eq.~(\ref{differential}) to
\begin{equation} \label{mean}
\left\langle(\Pi
K-\Gamma)\left(\begin{array}{c}\delta\mathbf{p}\\
\delta\mathbf{q}\end{array}\right)\right\rangle+\langle\mathbf{\Sigma}\rangle\cdot\frac{d\mathbf{R}}{dt}=0.
\end{equation}
Because the matrices $\Pi$, $K$, $\Gamma$ vary along the orbit, one
may infer from Eq.~(\ref{mean}) the statistical correlations
$\langle(\Pi K-\Gamma)\delta\mathbf{p}\rangle$ and $\langle(\Pi
K-\Gamma)\delta\mathbf{q}\rangle$, but the mean fluctuations
$\langle \delta\mathbf{p}\rangle$, $\langle
\delta\mathbf{q}\rangle$, or $\langle \delta\mathbf{I}\rangle$
remain unknown.
\section{Emergence of a geometric angle from an adiabatically moving fixed-point solution}
As a direct application of our central result in
Eq.~(\ref{differential}), here we focus on a rather simple case,
where the solution to Hamilton's equation of motion is a fixed point
in phase space if the system parameters are not changing. We denote
the fixed-point solution as $(\overline{\mathbf{p}},
\overline{\mathbf{q}})$, which are of course functions of
$\mathbf{R}$. Consider now an adiabatic process in which
$\mathbf{R}$ is changing slowly. Then the idealized orbit according
to CAT is just one adiabatically moving fixed point. In addition, at
this fixed point all functions of $(\overline{\mathbf{p}},
\overline{\mathbf{q}})$ are independent of $\overline{\bf \Theta}$
(otherwise they would be time-dependent), thus forcing their
derivatives with respect to $\overline{\bf \Theta}$ to vanish and
making an averaging over $\overline{\bf \Theta}$ [e.g., in
Eq.~(\ref{mean})] unnecessary.
We therefore obtain
\begin{eqnarray}
{\Pi}&=&0, \\
{\mathbf{\Sigma}}&=&(\frac{\partial \overline{\mathbf{p}} }{\partial
\mathbf{R}}, \frac{\partial \overline{\mathbf{q}}}{\partial
\mathbf{R}})^{T}.
\end{eqnarray}
Using these results we have the following relation from
Eq.~(\ref{differential}):
\begin{equation} \label{mean3}
\left(\begin{array}{c} \delta{\mathbf{p}}\\
\delta {\mathbf{q}}\end{array}\right)= {\Gamma}^{-1}
\left(\begin{array}{c}\frac{\partial \overline{\mathbf{p}}}{\partial \mathbf{R}}\\
\frac{\partial \overline{\mathbf{q}}}{\partial \mathbf{R}}\end{array}\right) \cdot \frac{d\mathbf{R}}{dt}.
\end{equation}
Note that the values of $\theta_i$ at a fixed point can be taken as
arbitrary. Hence the fluctuations obtained in Eq.~(\ref{mean3}) do
not have any interesting consequence for the evolution of
$\theta_i$. Furthermore, since the $K$ matrix vanishes at fixed
points (where the action reaches its minimum), one would also arrive
at $\delta{\mathbf{I}}=0$ to the first order of $\epsilon$ even
though $\delta\mathbf{q} \ne 0$ and $\delta\mathbf{p}\ne 0$.
Consider then the coupling of this system with another degree of
freedom, whose canonical coordinates are denoted by $(J,\phi)$. The
total Hamiltonian is assumed to be independent of $\phi$, denoted
$H^{\text{tot}}({\bf p}, {\bf q}, J)$. Because $J$ is a strict
constant of motion and can be regarded as a fixed system parameter
for the motion of $(\mathbf{p},\mathbf{q})$, the expression for
$\delta \mathbf{p}$ and $\delta \mathbf{q}$ in Eq.~(\ref{mean3})
still applies to fixed points in the phase space of
$(\mathbf{p},\mathbf{q})$. To seek how fluctuations predicted by
Eq.~(\ref{mean3}) may affect the motion in $\phi$, let us now
examine the angular frequency associated with $\phi$, i.e.,
\begin{eqnarray}
\omega_J({\bf p}, {\bf q}, J)\equiv \frac{\partial
H^{\text{tot}}}{\partial J}.
\end{eqnarray} Clearly, the fluctuations $\delta\mathbf{p}$ and $\delta\mathbf{q}$ will lead to
\begin{eqnarray}
\delta \omega_J(\overline{{\bf p}}, \overline{{\bf q}}, J) =
\frac{\partial \omega_J(\overline{{\bf p}}, \overline{{\bf q}},
J)}{\partial {\bf \overline{p}}}\cdot \delta {\bf p} +
\frac{\partial \omega_J(\overline{{\bf p}}, \overline{{\bf q}}, J)}{\partial {\bf \overline{q}}}\cdot \delta {\bf q}. \nonumber \\
\end{eqnarray} This fluctuation in $\omega_J({\bf p}, {\bf q}, J)$ induces an correction to the evolution of $\phi$.
Using Eq.~(\ref{mean3}), one finds an explicit expression for this
correction as follows,
\begin{eqnarray}
\phi^{\text{corr}} &=&
\int_0^T \left[ \frac{\partial \omega_{J}}{\partial \overline{\mathbf{p}}}\cdot \delta\mathbf{p}
+\frac{\partial \omega_{J}}{\partial \overline{\mathbf{q}}}\cdot \delta\mathbf{q} \right] \ dt \nonumber \\
& = & \oint \left(\begin{array}{cc}\frac{\partial \omega_J}{\partial \overline{\mathbf{p}}}, &\frac{\partial
\omega_{J}}{\partial \overline{\mathbf{q}}}
\end{array}\right){\Gamma}^{-1} \left(\begin{array}{c}\frac{\partial
\overline{\mathbf{p}}}{\partial \mathbf{R}}\\ \frac{\partial
\overline{\mathbf{q}}}{\partial \mathbf{R}}\end{array}\right) \cdot
d\mathbf{R}.
\label{thetaresult}
\end{eqnarray}
As seen from Eq.~(\ref{thetaresult}), $ \phi^{\text{corr}}$ obtained
above no longer depends on $T$ (so it will not vanish even in the
$\epsilon\rightarrow 0$ or $T\rightarrow +\infty$ limit). Rather, it
depends on the geometry in the parameter space only. $
\phi^{\text{corr}}$ is hence identified as a geometric angle that
arises from the fluctuations in a classical adiabatic process. This
is particularly interesting because here $\delta\mathbf{I}=0$, i.e.,
even when the fluctuations in the original action variables are
vanishing, there can still be a physical effect on another degree of
freedom due to the dynamical fluctuations.
To illustrate the result in Eq.~(\ref{thetaresult}) we have designed
a simple toy model with two degrees of freedom in total.
Specifically, the total Hamiltonian is given by
\begin{equation} \label{Hexample}
H^{\text{tot}}(p_1,q_1; J)=\alpha J+
\frac{1}{2}\left[\left(\frac{p_1^2}{X^2}-J\right)^2+\left(\frac{q_1^2}{Y^2}-J\right)^2\right],
\end{equation}
with $\mathbf{R}=(X>0,Y>0)$, $\phi$ being a cyclic angular
coordinate that forms a canonical pair with $J$, and $\alpha$ being
a free parameter. For the $(p_1,q_1)$ degree of freedom, this system
has a $\mathbf{R}$-dependent fixed point
\begin{eqnarray}
\overline{q}_1&=&\sqrt{\bar{J}}Y; \\ \nonumber
\overline{p}_1&=&\sqrt{\bar{J}}X,
\end{eqnarray} where $\bar{J}$ represents a
conserved value of the variable $J$.
To calculate the fluctuation-induced geometric angle seen in the
evolution of $\phi$, note first
\begin{equation}
\omega_J = \frac{\partial H^{\text{tot}}}{\partial J}=\alpha+
2J-\frac{p_1^2}{X^2}-\frac{q_1^2}{Y^2},
\end{equation} and
\begin{equation}
\frac{\partial\omega_J}{\partial
\overline{p}_1}=-\frac{2\overline{p}_1}{X^2},\
\frac{\partial\omega_J}{\partial
\overline{q}_1}=-\frac{2\overline{q}_1}{Y^2}.
\end{equation}
One may also easily obtain that the matrix ${\Gamma}$ here is just a
$2\times 2$ matrix, i.e.,
\begin{equation}
{\Gamma}_{2\times
2}=\left(\begin{array}{cc}-\frac{\partial^2\overline{H}}{\partial
\overline q_1\partial
\overline{p}_1}&-\frac{\partial^2\overline{H}}{\partial
\overline{q}_1\partial \overline{q}_1}
\\ \frac{\partial^2\overline{H}}{\partial \overline{p}_1 \partial
\overline{p}_1}&\frac{\partial^2\overline{H}}{\partial
\overline{p}_1 \partial \overline{q}_1}
\end{array}\right)=\left(\begin{array}{cc}0&-\frac{4\bar{J}}{Y^2}
\\ \frac{4\bar{J}}{X^2}&0
\end{array}\right);
\end{equation} and
\begin{equation}
\left(\begin{array}{c}\frac{\partial \overline{p}_1}{\partial \mathbf{R}}\\
\frac{\partial \overline{q}_1}{\partial
\mathbf{R}}\end{array}\right)=\left(\begin{array}{c}\sqrt{\bar{J}}\\
0\end{array}\right)\hat{X}+\left(\begin{array}{c}0\\
\sqrt{\bar{J}}\end{array}\right)\hat{Y},
\end{equation}
where $\hat{X}$ and $\hat{Y}$ are unit vectors along the $X$ and $Y$
coordinates. Finally, substituting these intermediate results into
Eq.~(\ref{thetaresult}), one finds the fluctuation-induced geometric
angle
\begin{eqnarray}
\phi^{\text{corr}}&=&
\oint_C \left(\begin{array}{cc}\frac{\partial \omega_J}{\partial \overline{p}_1}&\frac{\partial \omega_J}{\partial
\overline{q}_1} \end{array}\right)
{\Gamma}_{2\times 2}^{-1} \left(\begin{array}{c}\frac{\partial \overline{p}_{1}}{\partial \mathbf{R}}\\ \frac{\partial
\overline{q}_{1}}{\partial \mathbf{R}}\end{array}\right)
\cdot d\mathbf{R} \nonumber\\
&=&\oint_C \left(\frac{\partial\omega_J}{\partial \overline{p}_1}
\frac{X^2}{4\bar{J}}\sqrt{\bar{J}}\hat{Y},\
-\frac{\partial\omega_J}{\partial
\overline{q}_1}\frac{Y^2}{4\bar{J}}\sqrt{\bar{J}}\hat{X} \right)
\cdot d\mathbf{R} \nonumber \\ &=&\frac{1}{2}\oint_C (Y dX-X dY)
=-\iint\limits_{\partial S=C} dS. \label{phiresult}
\end{eqnarray}
As seen from the above result, here the geometric angle induced by
the fluctuations in the first degree of freedom may be interpreted
as the flux of an effective ``magnetic charge" uniformly distributed
on the $(X,Y)$ plane. The emergence of such a new classical
geometric angle from our simple calculations is hence intriguing.
It should be emphasized that in obtaining $\phi^{\text{corr}}$ in
Eq.~(\ref{phiresult}), we did not seek new action-angle variables
$(\tilde{I}_1,\tilde{\theta}_1)$ and
$(\tilde{I}_2,\tilde{\theta}_2)$ such that $H^{\text{tot}}$ becomes
a function of $\tilde{I}_1$ and $\tilde{I}_2$ only. Indeed it can be
highly complicated in general to find such a new representation due
to the coupling between the two degrees of freedom. This indicates
that $\phi^{\text{corr}}$ here has a different meaning than Hannay's
angle, because it represents a geometrical correction to the $\phi$
evolution, not to the evolution of the yet-to-be-found new angle
variables $\tilde{\theta}_1$ or $\tilde{\theta}_2$.
We also note that our result here is consistent with one of the
found terms in the previous study of the so-called ``nonlinear Berry
phase" based on GP equation~\cite{liu}. In particular, the GP
equation considered in Ref.~\cite{liu} can be mapped to that of a
classical Hamiltonian with two degrees of freedom, with the
nonlinear eigenstates mapped to classical fixed points (see also
Sec.~IV-B). Adopting our perspective here, the geometric phase
contributed by deviations from nonlinear eigenstates as analyzed in
Ref.~\cite{liu} may be understood as a classical geometric angle due
to intrinsic fluctuations in classical adiabatic processes. Indeed,
we have checked that if we apply Eq.~(\ref{mean3}) to the model
considered in Ref.~\cite{liu}, then we can obtain a
fluctuation-induced geometric phase term that is identical with a
Berry-phase correction term discovered in Ref.~\cite{liu}. Note
however, the focus of our perspective is on a general description of
the important dynamical fluctuations in a broad class of classical
adiabatic processes. In our fully classical considerations here, a
totally classical geometry angle is shown to arise in a second
degree of freedom that is coupled with the first degree of freedom
(with one adiabatically moving fixed point solution); whereas in
Ref.~\cite{liu}, the emphasis was placed on a quantum adiabatic
evolution context and the main concern is with the sum of one
familiar Berry phase and a fluctuation-induced geometric phase as a
correction.
\section{Discussion}
\subsection{Pollution to Hannay's angle}
As mentioned above, in some early studies about Hannay's angle in
some Hamiltonian systems \cite{Golin1, Golin2, Berry1996Non,adam},
it was numerically found that during an adiabatic process the total
angle change minus the dynamical angle may not be Hannay's angle.
This subtle behavior was connected with dynamical fluctuations in
classical adiabatic processes. Here we exploit our general result of
Eq.~(\ref{differential}) to shed more light on possible pollution to
Hannay's angle.
According to Eq.~(\ref{new-dyna2}) and CAT, the total change in
angle variables in a cyclic adiabatic process is given by
\begin{equation}
\triangle\theta_{i}^{\text{ideal}}(T)=\int_0^{T}\omega_{i}(\overline{\mathbf{I}},\mathbf{R})\
dt - \frac{\partial}{\partial \overline{I}_i} \oint
(\overline{\mathbf{p}}\cdot
\mathbf{\nabla}_{\mathbf{R}}\overline{\mathbf{q}}) \cdot
d\mathbf{R}. \label{standard}
\end{equation}
On the right hand side of Eq.~(\ref{standard}), the first term is
often called the dynamical angle, and the second term gives Hannay's
angle (upon an average over initial angle variables). We have also
used the notation $\theta_{i}^{\text{ideal}}$ to emphasize that it
is for idealized cases without considering any dynamical
fluctuations. Indeed, the angular frequency $\omega_{i}$ in
Eq.~(\ref{standard}) is naively assumed to be the one determined by
the idealized and constant action $\overline{\mathbf{I}}$.
However, as suggested by Eq.~(\ref{new-dyna2}), fluctuations in the
action variables $\delta\mathbf{I}$ can then correct the angular
frequency from $\omega_{i}(\overline{\mathbf{I}},\mathbf{R})$ to
$\omega_{i}(\overline{\mathbf{I}},\mathbf{R})+\frac{\partial
\omega_i(\overline{\mathbf{I}},\mathbf{R})}{\partial
\overline{\mathbf{I}}} \cdot\delta\mathbf{I}$. In terms of the
canonical variables $(\mathbf{p},\mathbf{q})$, fluctuations in
$\overline{\mathbf{p}}$ and $\overline{\mathbf{q}}$ will lead to
fluctuations in the angular frequency
\begin{eqnarray}
\delta \omega(\overline{{\bf I}}, {\bf R}) = \frac{\partial
\omega(\overline{{\bf I}}, {\bf R})}{\partial {\bf
\overline{p}}}\cdot \delta {\bf p} +
\frac{\partial \omega(\overline{{\bf I}}, {\bf R})}{\partial {\bf \overline{q}}}\cdot \delta {\bf q}. \label{deltaw}
\end{eqnarray}
For this reason, the dynamical angle obtained by a time-integral of
the idealized frequency
$\omega_{i}(\overline{\mathbf{I}},\mathbf{R})$, [see
Eq.~(\ref{standard})] should be re-examined with care. In terms of
$\delta\mathbf{p}$ and $\delta\mathbf{q}$, the real change in the
angular variables should be given by
\begin{eqnarray}
\triangle\theta_{i}^{\text{real}}(T)& = &
\triangle\theta_{i}^{\text{ideal}}(T)+\int_0^{T}
\delta\omega_i(\mathbf{I}, \mathbf{R})\ dt \nonumber \\
&=& \triangle\theta_{i}^{\text{ideal}}(T)+\int_0^{T}\frac{\partial
\omega(\overline{{\bf I}}, {\bf R})}{\partial {\bf
\overline{p}}}\cdot \delta
{\bf p}\ dt \nonumber \\
&& +\ \int_0^{T} \frac{\partial \omega(\overline{{\bf I}}, {\bf
R})}{\partial {\bf \overline{q}}}\cdot \delta {\bf q} \ dt .
\label{standardreal}
\end{eqnarray}
Because $\delta \mathbf{p}$, $\delta \mathbf{q}$ and hence $\delta
\omega$ are of the same order with $\epsilon=|d\mathbf{R}/dt|$, just
like the above fixed-point solution case, the term $\int_0^{T}
\delta\omega_i \ dt$ may not be negligible as it accumulates the
fluctuations $\delta\omega_i(\mathbf{I}, \mathbf{R})$ over an entire
adiabatic process. So the term $\int_0^{T} \delta\omega_i \ dt$
should not be neglected without a clear understanding of the
dynamics. At this point it is also clearer why we only consider
$\delta \mathbf{p}$ and $\delta \mathbf{q}$ to the first order of
$\epsilon$: including higher-order terms are unnecessary because
they will vanish in the $\epsilon\rightarrow 0$ limit.
The correction term $\int_0^{T} \delta\omega_i \ dt$ can hence give
the difference between two objects: the standard Hannay's angle, and
a numerical calculation of a geometric angle based on the expression
of
($\triangle\theta_{i}^{\text{real}}-\int_0^{T}\omega_{i}(\overline{\mathbf{I}},\mathbf{R})\
dt$). Unfortunately, unless for special fixed-point solution cases
analyzed above, we in general cannot determine the fluctuations
$\delta \mathbf{p}$ and $\delta\mathbf{q}$ from the differential
equation in Eq.~(\ref{differential}). In particular, $\delta
\mathbf{p}$ and $\delta\mathbf{q}$ can only be determined if we have
information about them for at least one given $\mathbf{\Theta}$ (as
the input). Therefore, without some detailed information of an
adiabatic process, e.g., the detailed dependence of adiabatic
parameter $\mathbf{R}(t)$ on time, information about $\delta
\mathbf{p}$ and $\delta \mathbf{q}$ is not available in general.
To see more clearly, let us discretize the adiabatic process by
dividing one adiabatic process into many time intervals
$t_1,t_2,\cdots$, during each of which $\mathbf{R}=\mathbf{R}_j$,
followed by a jump onto the next value $\mathbf{R}_{j+1}$ after the
temporal interval $t_j$ (different time intervals and different
choices for $\mathbf{R}_j$ define different adiabatic processes with
different details). Note that even for a continuous adiabatic
process, this discretized version is rather typical in numerical
simulations (as the discretized time steps decrease, the simulated
dynamics approaches a continuous process). Now for each point
$\mathbf{R}_j$, we may use Eq.~(\ref{differential}) to describe the
dynamical fluctuations, but Eq.~(\ref{differential}) is dependent on
$\mathbf{R}_j$. For a particular segment where
$\mathbf{R}=\mathbf{R}_j$, the angle variable $\mathbf{\Theta}$
changes rapidly. Obviously, different timing for the next jump will
result in different initial values of $\mathbf{\Theta}$ for next
segment $\mathbf{R}=\mathbf{R}_{j+1}$, leading to another initial
condition for the differential equation (\ref{differential})
associated with ${\bf R}=\mathbf{R}_{j+1}$. This process then
continues.
According to Eq.~(\ref{deltaw}), $\delta\omega$ and thus the
correction term $\int_0^{T} \delta\omega_i \ dt$ will then depend on
great details of a particular adiabatic process. It is for this
reason that the correction term $\int_0^{T} \delta\omega_i \ dt$ is
identified as ``pollution" to Hannay's angle, with the latter
independent of how an adiabatic process is implemented. Analysis
here also makes it clearer that the fixed-point solution case in
Sec.~III is special because a definite prediction about fluctuations
can be made therein.
It is also worth noting that, according to Eq.~(\ref{deltaw}), the
pollution vanishes if the angular frequency $\omega_i$ does not
depend on the action $\mathbf{I}$. This is the case in a linear
system such as a harmonic oscillator.
\subsection{``Pollution" to adiabatic phase evolution in a two-mode BEC model}
Finally, we propose to use a two-mode GP equation to study pollution
to a geometric phase associated with quantum adiabatic cycles, thus
making a connection between our theoretical considerations here and
a reachable experimental context. In particular, there are a number
of possibilities to experimentally realize a two-mode BEC. For
example, one may consider a BEC in a double-well potential, or a BEC
in an optical lattice occupying two bands \cite{BECexpe}. On the
mean-field level, a two-mode BEC can be described by the following
GP equation ($\hbar=1$)
\begin{eqnarray} \label{GP} \nonumber
&i\frac{d}{dt}\left(\begin{array}{c}a\\b\end{array}\right)={H}_{\text{GP}}\left(\begin{array}{c}a\\b\end{array}\right)\\
&=\frac{1}{2}\left(\begin{array}{cc}\gamma+c(|b|^2-|a|^2)&\Delta\\
\Delta&-\gamma-c(|b|^2-|a|^2)\end{array}\right)\left(\begin{array}{c}a\\b\end{array}\right), \nonumber \\
\end{eqnarray}
where $\gamma$ denotes an energy bias between the two modes, $|a|^2$
and $|b|^2$ (with $|a|^2+|b|^2=1$) represent occupation
probabilities of the two modes, $c$ gives the self-interaction
strength, and $\Delta$ denotes the coupling between the two modes.
We can consider, for example, the two parameters $\gamma$ and
$\Delta$ to implement an adiabatic cyclic process.
The dynamics described by the above GP equation can be translated
into Hamiltonian dynamics. In particular, let $p=\phi_a-\phi_b$,
$q=|a|^2$, $a=|a|e^{i\phi_a}$, $b =|b|e^{i\phi_b}$, then apart from
an overall phase parameter $\phi_b$, Eq.~(\ref{GP}) leads to
\begin{eqnarray} \label{classical} \nonumber
&\frac{dp}{dt}=-\frac{\partial H}{\partial q}, \nonumber \\
&\frac{dq}{dt}=\frac{\partial H}{\partial p}.
\end{eqnarray}
where
\begin{eqnarray} \label{CHamiltonian} \nonumber
H&=&\Delta\sqrt{q(1-q)}+\frac{\gamma}{2}(2q-1)-\frac{c}{4}(2q-1)^2.
\end{eqnarray}
It is also straightforward to find that the evolution of $\phi_b$
obeys
\begin{eqnarray}
\frac{d\phi_b}{dt}=i(\sqrt{q}e^{-ip},\sqrt{1-q})\frac{d}{dt}\left(\begin{array}{c}\sqrt{q}e^{ip}\\
\sqrt{1-q}\end{array}\right)-H-\Lambda, \label{phaseevo}
\end{eqnarray}
where
\begin{eqnarray}
\Lambda&=&-\frac{c}{4}(2q-1)^2.
\end{eqnarray}
It is seen that the evolution of the overall phase $\phi_b$ is
determined by, but will not have a back action on, the classical
trajectories determined by $H$ in Eq.~(\ref{CHamiltonian}). In this
sense, the $\phi_b$ parameter plays a similar role as the $\phi$
parameter in Sec.~III.
It is now clear that our general result of dynamical fluctuations in
classical adiabatic processes can be directly relevant to
understanding the adiabatic evolution of a two-mode BEC system. If
the adiabatic process starts from a stationary state of the GP
equation, then the dynamics is just about
an adiabatically evolving fixed-point solution of the Hamiltonian in
Eq.~(\ref{CHamiltonian}). As shown earlier (see also
Ref.~\cite{liu}), in this case a definite prediction can be made
about how accumulation of dynamical fluctuations can eventually lead
to a geometry-like correction to $\phi_b$. Consider now a
superposition state of two stationary states of the above two-mode
GP equation as the initial state of an adiabatic process. This case
then corresponds to a classical adiabatic process with
non-fixed-point solutions. As indicated by Eq.~(\ref{phaseevo}),
dynamical fluctuations can now affect the evolution of the
adiabatically evolving phase $\phi_b$, in an unpredictable way if we
do not know the details of the adiabatic process. Pollution to the
quantum phase $\phi_b$ hence emerges. Interestingly, in the same
context, how $\phi_b$ may develop an adiabatic geometric phase for
general superpositions of stationary states
was already considered in Ref.~\cite{WuPRL2005} without considering dynamical fluctuations.
It is hence of interest to numerically or even experimentally
examine the actual pollution due to the accumulation of dynamical
fluctuation effects
in such type of quantum adiabatic processes.
\section{Summary}
To summarize, we have obtained a general description of the
intrinsic dynamical fluctuations in classical adiabatic processes
associated with integrable systems. These fluctuations are typically
neglected by the conventional classical adiabatic theorem. The
dynamical fluctuations are described in this work in terms of
deviations from idealized adiabatic trajectories. As an
application, we have shown how a new kind of classical geometric
phase may emerge using an explicit example with an adiabatically
evolving fixed-point solution. We then discussed the origin of the
pollution to Hannay's angle and proposed to use a two-mode BEC
system to further study possible fluctuation-induced pollution to
one type of quantum adiabatic evolution described on a mean-field
level.
{\bf Acknowledgement} The work of Q.Z. and C.H was supported by
National Research Foundation and Ministry of Education, Singapore
(Grant No. WBS: R-710-000-008-271) and by the National Natural
Science Foundation of China (Grant No. 11105123).
\vspace{0.5cm}
|
{
"timestamp": "2012-02-10T02:01:37",
"yymm": "1009",
"arxiv_id": "1009.3623",
"language": "en",
"url": "https://arxiv.org/abs/1009.3623"
}
|
\section{introduction}\label{intro}
One of the most remarkable and fundamentally important result of
the field quantization is the Casimir effect which is a force
arising from the change of the zero point energy caused by
imposing the boundary conditions (BC) \cite{Casimir}. This force
is the macroscopic aspect of the quantum electrodynamics that
provides a direct line between quantum field theory and the
macroscopic world. The original calculation of the Casimir force
between two perfectly conducting parallel plates immersed in the
quantum electromagnetic vacuum is based on the definition of the
Casimir energy in the presence and the absence of boundary
surfaces \cite{Casimir} that leads to an attractive observable
force
\begin{equation}
F=-\frac{\hbar c}{240}\frac{1}{H^4},
\end{equation}
between the plates, where $\hbar$ is the Planck constant, $c$ is
the speed of light and $H$ is the distance between the plates. In
other way one can consider this effect by evaluating the radiation
pressure on macroscopic objects \cite{Milonni}. As the magnitude
of the Casimir force is substantial at $H<100 ~{\text{nm}}$ this
effect is relevant in nano-technology
\cite{sriva,nanoscale1,nanoscale1b,s2} and should be take into
account to design and actuate microelectromechanical (MEMS)
systems. Moreover the possibility of transducing the energy from
the vacuum is investigated by means of MEMS
\cite{pinto,Cole,Forward}.
Many attempts have been focused on observing the Casimir force and
performing high-precision measurements during last few years
\cite{Lamoreaux,Mohideen,Harris,Bressi,Decca1,Decca2}. All these
experiments are in agree with the prediction of the Casimir
\cite{Casimir} within a few percents. These deviation from the
ideal force may due to temperature, roughness of surfaces and
finite dielectric constants that have been covered in a very
recent book entitled by Advanced in the Casimir Effect
\cite{Bordag_Book}.
In 90th decade, Golestanian and Kardar developed a path integral
approach to investigate the dynamic Casimir effect in the system
of two corrugated conducting plates surrounded by the quantum
vacuum \cite{Golestanian-PRL-1997,Golestanian-PRA-1998}. Emig and
his colleagues also used the path integral formalism to obtain
normal and lateral Casimir force between two sinusoidal corrugated
perfect conductor surfaces \cite{Emig-PRL-2001,Emig-PRA-2003}.
Later on, the exact mechanical response of the quantum vacuum to
the dynamic deformations of a cavity and the rate of dissipation
have been calculated by using path integral scheme \cite{jalal}.
This motivated us to investigate the Casimir effect in the
presence of a magnetodielectric medium by quantizing the
electromagnetic (EM) field using path integral formalism. Our
system contains of a magnetodielectric medium with permitivity
$\varepsilon$ and permeability $\mu$, enclosed by two
semi-infinite ideal metals ($ \varepsilon_{\text L} \rightarrow
\infty$ for the ideal metal in left-hand side and
$\varepsilon_{\text R} \rightarrow \infty$ for right-hand side
one) as depicted in Fig.~\!(\ref{schematic-fig}). We model the
magnetodielectric medium by a continuum of harmonic oscillators
(Hopfield Model) \cite{Fardin1,Fardin2,Fardin3}.
\begin{figure}[b!]
\includegraphics[width=0.70\columnwidth]{Fig_New.pdf}
\caption{This picture illustrates the schematic figure of the
system under consideration. A magnetodielectric medium
($\varepsilon, ~\! \mu$) is enclosed between two perfect parallel
conductors ($\varepsilon_{\text R}$ and $\varepsilon_{\text L}
\rightarrow \infty$). The distance between conductors is $H$, and
the $z$ direction is perpendicular to the surfaces of the media.}
\label{schematic-fig}
\end{figure}
The outline of this paper is as follows: In Sec.\
\ref{path-scalar}, in order to introduce the scheme, we first
quantize the simple case of a scaler Klein-Gordon field in the
presence of a medium. Sec.\ \ref{force-scalar} is devoted to
obtain the Casimir force for the scalar filed in the presence of a
medium for different kinds of boundary conditions. In Sec.\
\ref{path-electromagnetic}, we develop our formalism to the case
of EM field in the presence of a magnetodielectric medium. Sec.\
\ref{force-electromagnetic} gives the Casimir force for the case
of EM field. Finally, the conclusions and outlooks are in the
Sec.\ \ref{conclusion}.
\section{Field quantization using path integrals}\label{path-scalar}
To illustrate the method and also for later convenience, before
considering the EM field in the presence of a medium, we consider
the simplest case i.e. a scaler massless field. In the next
section we will show that the Klein-Gordon field can be
corresponded to each polarization of the EM field. Therefore, let
us consider the following Lagrangian for the total system
\begin{equation}\label{2}
{\cal {L}} = {\cal{L}} _{sys}+{\cal{L}} _{mat}+{\cal{L}}_ {int},
\end{equation}
where
\begin{equation}\label{3}
{\cal{L}}_{sys} = \frac{1}{2}\,\partial ^\mu \varphi\,\partial
_\mu \varphi,
\end{equation}
is the Lagrangian density of the massless Klein$-$Gordon filed.
The medium is modeled by a continuum of harmonic oscillators as
\cite{Huttner}
\begin{equation}\label{4}
{\cal{L}}_{mat} = \int_0^\infty d\omega\,(\frac{1}{2}\rho \dot
Y_\omega^2 - \frac{1}{2}\rho \omega ^2 Y_\omega^2),
\end{equation}
where $Y_\omega$ is an oscillator's field, $\rho$ is the density
of matter field and the interaction between the system and its
medium is defined by
\begin{equation}\label{5}
{\cal{L}}_{int}=\varphi \dot P,
\end{equation}
where
\begin{equation}\label{6}
P =\int d\omega \nu(\omega)Y_\omega.
\end{equation}
In the next section we will show that the quantity $P$ is in fact
the polarization field corresponding to the medium and the
interaction (\ref{5}) will become the electric-dipole interaction.
Generally a generating function is defined by \cite{Ryder}
\begin{equation}\label{7}
Z[J] = \int {\cal {D}} [\psi] \exp{\bigg\{\imath\int d^{n + 1}
x[{\cal L}\big(\psi (x)\big) + J(x)\psi (x)]\bigg\}},
\end{equation}
where $\psi$ is the scalar field and the different correlation
functions can be found by taking the repeated functional
derivatives with respect to the source field $J(x)$. The above
partition function is Gaussian since the integrand has quadratic
form with respect to the fields. To obtain the generating function
for the interacting fields, we first calculate the generating
function for the free fields
\begin{eqnarray}\label{8}
Z_0 \!\! && \!\! [J_{\varphi} ,J_\omega] = \int
{\cal{D}}[\varphi]\prod_{\omega}{\cal{D}}[Y_\omega] \times
\nonumber\\
&& \hspace{-0.5cm} \times \exp \bigg\{ \imath \int d^{n+1} x \big[
{\cal{L}}_{sys} + {\cal{L}}_{mat} + J_{\varphi} \varphi + \int
d\omega J_\omega
Y_\omega \big ] \bigg\}. \nonumber\\
\end{eqnarray}
Using the $n$-dimensional version of Gauss's theorem we find
\begin{equation}\label{9}
\int d^{n+1} x\,\partial _\mu \varphi\, \partial ^\mu \varphi =
-\int d^{n+1} x\,\varphi\,\Box\, \varphi,
\end{equation}
where $\Box$ is the d'Alemberian in $(n+1)$-dimensional space-time
and the integration by part
\begin{equation}\label{10}
\int d^{n+1} x\,\,\dot Y_{\omega}\,\dot Y_{\omega} = -\int d^{n+1}
x\,Y_{\omega}\,\frac{{\partial ^2 }}{{\partial t^2 }}\,Y_{\omega}.
\end{equation}
the free generating function (\ref{8}) can be written as
\begin{eqnarray}\label{11}
Z_0 \!\! && \!\! [J_{\phi} ,J_\omega] = \int\prod_{\omega}{\cal
D}[Y_\omega]
{\cal D}[\varphi] \exp \bigg\{ - \frac{\imath}{2} \int d^{n + 1} x \nonumber\\
&& \times \bigg[ \varphi (x)\Box\varphi (x) + \int d\omega
Y_\omega (x)(\frac{{\partial ^2 }}{{\partial t^2 }}+\rho
\omega^2)Y_\omega (x)
\nonumber\\
&& ~~~ + J_{\varphi} (x)\varphi (x) +\int d\omega J_\omega
(x)Y_\omega (x)\bigg]\bigg\}.
\end{eqnarray}
The integral in the equation (\ref{11}) can be easily calculated
from the field version of the quadratic integrals and the result
is
\begin{eqnarray}\label{12}
Z_0 \!\! && \!\! [J_{\varphi}, J_\omega] = \nonumber\\
&& \!\!\!\!\!\!\! = \exp \bigg\{ -\frac{\imath}{2} \int d^{n+1}x
\int d^{n+1} x' \!
\bigg[ J_{\varphi} (x) G_0 (x - x')J_{\varphi} (x') \nonumber\\
&& \hspace{+0.9cm} + \int \!\! d\omega J_\omega (x) G_\omega (x
- x') J_\omega(x') \bigg] \! \bigg\},
\end{eqnarray} where
$G_\omega (x - x')$ and $G_0 (x - x')$ are the propagators for
free fields and satisfy the following equations
\begin{equation}\label{13}
\Box G_0 (x - x') = \delta (x - x'),
\end{equation}
\begin{equation}\label{14}
\{ \rho \frac{{\partial ^2 }}{{\partial t^2 }} + \rho \omega ^2 \}
G_\omega (x - x') = \delta (x - x').
\end{equation}
We employ the Fourier transformation to solve the equations
(\ref{13}) and (\ref{14}). The solutions are
\begin{equation}\label{15}
G_0 (x - x') = \frac{1}{(2\pi)^{n+1}}\int d^n {\bf k}\,d\omega\,
\frac{{e^{\imath{\bf k}\cdot({\bf x} -{\bf x}') - \imath \omega (t
- t')} }}{{\omega ^2 - {\bf k}^2 }},
\end{equation}
and
\begin{equation}\label{16}
G_\omega (x - x') = \frac{1}{(2\pi\rho)}\int d\omega'\,
\frac{{e^{ - \imath\omega' (t - t')} }}{{\omega^2-\omega'^2
}}\,\delta({\bf x}-{\bf x'}).
\end{equation}
Here the space component of the point $x\in {\cal R}^{n+1}$ is
indicated by the bold face ${\bf {x}}\in {\cal R}^{n}$ and the
time component by $t$ or $x_0\in {\cal R}$.
For further use we define
\begin{equation}\label{17}
J_p(z)=\int d\omega \nu(\omega) J_\omega(z),
\end{equation}
the generating function of the interacting fields can be written
in terms of the free generating function as \cite{Ryder}
\begin{eqnarray}\label{18}
Z[J_{\varphi},J_P] &=& Z^{ - 1} [0]e^{\imath \int d^{n+1}
z{\cal{L}}_{int} (\frac{\delta } {\delta J_\varphi
(z)},\frac{\delta }{\delta J_P(z) })} Z_0 [J_{\varphi} ,J_\omega
] \nonumber\\
&& \hspace{-1.8cm} = Z^{ - 1} [0] \sum\limits_{n = 0}^\infty
\frac{1}{n!}\bigg[ \imath\int d^{n+1}z\frac{\delta } {{\delta
J_\varphi(z) }} \cdot \frac{\partial }{{\partial z_0}}\frac{\delta
}{{\delta J_{\varphi} (z) }} \bigg]^n \times \nonumber\\
&& \hspace{+0.6cm} \times Z_0 [J_{\varphi} ,J_\omega ],
\end{eqnarray}
where $Z [0]$ is the partition function of the free space. Thus
the Green's function of Klein-Gordon filed can be obtained via
\begin{equation}\label{19}
G_{\varphi\varphi}(x - y) = \imath \frac{{\delta ^2
Z[J_{\varphi},J_P]}}{{\delta J_{\varphi} (x)\delta J_{\varphi}
(y)}}\big |_{J_{\varphi},J_{\omega} = 0}.
\end{equation}
Combining the generating function (\ref{18}), the Green's function
(\ref{19}) and the definition of ${\cal L}_{int}$ (\ref{5}) yield
the following series for the Green's function
\begin{eqnarray}\label{20}
G_{\varphi\varphi}(x - x')& =& G_0 (x - x')\nonumber\\
&& \hspace{-2.4cm} + \!\! \int \!\! d\omega \!\! \int \!\! dx_1
dx_2 G_0 (x \! - \! x_1 \!) \nu ^2 (\omega )\frac{{\partial ^2
}}{{\partial t^2 }}G_\omega (x_1 \! - \! x_2 \! )G_0 (x_2 \! -
\! x')
\nonumber\\
&& \hspace{-2.4cm} + \!\!\! \int \!\!\! d\omega \!\!\! \int \!\!\!
d\omega ' \!\!\!\! \int \!\!\! dx_1 dx_2 \!\!\! \int \!\! \! dx_3
dx_4 G_0 (x \! - \! x_1 \!)\nu ^2 (\omega )\frac{{\partial ^2
}}{{\partial t^2 }}G_\omega (x_1 \!\! - \! x_2 \! ) \!\! \times\nonumber\\
&& \hspace{-2.4cm} \times G_0 (x_2 \! - \! x_3 \!)\nu ^2 (\omega
')\frac{{\partial ^2 }} {{\partial t^2 }}G_{\omega '} (x_3 \!-
\!x_4 \!)G_0 (x_4 \! - \! x') \! + \! \cdots .
\end{eqnarray}
It is appropriate to use a general Green's function in
$(n+1)$-dimensional Fourier space that is
\begin{equation}\label{21}
G({\bf k},\omega )
=\int e^{\imath {\bf k}\cdot{\bf x} - \imath \omega t}\, G(x)\,dt\,d^{n}\,{\bf x}.
\end{equation}
Therefore the Green's functions of the free fields $\varphi$ and
$Y_\omega$ in the Fourier space might be given by
\begin{equation}\label{22}
G_0 ({\bf k},\omega ) = \frac{1}{{{\bf k}^2 - \omega ^2 }},
\end{equation}
and
\begin{eqnarray}\label{23}
G_\omega ({\bf k},\omega ') &=& \frac{1}{\rho}\int \frac{
d\omega'' dt\,d^3 {\bf x} ~{e^{-\imath\omega'' t} }}{{\omega^2 -
\omega ^{''2} - \imath0^ + }}\delta ({\bf x})e^{\imath(\omega' t
- {\bf k}\cdot{\bf x})}
\nonumber \\
&=& \frac{1}{\rho}\frac{1}{\omega^2 - \omega^{'2} -\imath 0^{+}}
=: G_\omega (\omega '),
\end{eqnarray}
respectively. Since we are interested in retarded Green's
functions we have added $ - \imath0^+$ to the denominator of the
equation (\ref{23}). Since the reservoir field is assumed to be
homogeneous, the Green's function of the reservoir does not depend
on ${\bf k}$ in the above equation. Using the equations (\ref{22})
and (\ref{23}), $G_{\varphi\varphi}(x - x')$
can be written in the Fourier space as
\begin{eqnarray}\label{24}
G_{\varphi,\varphi}({\bf k},\omega )&& \nonumber\\
&& \hspace{-1.8cm} = G_0 ({\bf k},\omega ) \{1+\sum\limits_{n =
0}^\infty[\int d\omega'\omega^2\nu ^2 (\omega' )G_{\omega'}(\omega
) G_0({\bf k},\omega )]^n\}\nonumber\\
&& \hspace{-1.8cm} = \frac{G_0 ({\bf k},\omega )} {1 - \int
d\omega'\omega^2\nu ^2 (\omega' )G_{\omega'}(\omega)G_0 ({\bf
k},\omega )} \nonumber\\
&& \hspace{-1.8cm} = \frac{1}{{{\bf k}^2-\omega ^2-
\frac{1}{\rho}\int d\omega'\frac{\nu ^2 (\omega')\omega^2 }{\omega
'^2 -\omega ^2+ \imath 0^+ }}}.
\end{eqnarray}
This Green's function can also be obtained directly from the
Heisenberg equations of motion. By direct substitution we can show
that the Green's function $G_{\varphi\varphi}(x-y)$ satisfies the
equation
\begin{eqnarray}\label{25}
&& \!\!\!\!\! \Box G_{\varphi\varphi} ({\bf x} - {\bf x}',t - t') \nonumber\\
&& \!\!\!\!\! - \frac{\partial }{{\partial t}} \!\!
\int_{-\infty}^{t} \!\!\!\!\! dt'' \! \chi (t \! - \! t''
)\frac{\partial }{{\partial t''}}G_{\varphi\varphi} ({\bf x} \!-\!
{\bf x}',t'' \! \!- \! t') \!=\! \delta ({\bf x} \!-\! {\bf x}',t
\! - \! t'),
\nonumber\\
\end{eqnarray}
which is the motion equation of dissipation field with
susceptibility of the medium $\chi(t)$, with the following Fourier
transform
\begin{equation}\label{26}
\chi(\omega ) = \frac{1}{\rho}\int d\omega '\frac{{\nu ^2 (\omega
')}}{{\omega ^{'2} - \omega ^2 + \imath 0^+ }}.
\end{equation}
From the equations (\ref{24}) and (\ref{26}) it is clear that the
modified Green's function $G_{\varphi,\varphi}({\bf k},\omega )$
can be obtained from the free field Green's function $G_0 (x-x')$
in Eq.~\!(\ref{15}) simply by replacing $\omega^2$ with
$\varepsilon(\omega)\omega^2$, where
$\varepsilon(\omega):=1+\tilde{\chi}(\omega)$. It can be easily
shown that this susceptibility satisfies the Kramers-Kronig
relations as expected.
By the same technique we can obtain correlation between the
polarization field and the Klein-Gordon field. To this end we
define $G_{\varphi,P} $ as
\begin{equation}\label{27}
G_{\varphi,P}=\frac{\delta^2 Z[J_\varphi, J_P]}{\delta J_\varphi
\delta J_P},
\end{equation}
that can be obtained via direct calculation similar to the
procedure that ended to $G_{\varphi,\varphi}({\bf k},\omega )$ as
\begin{eqnarray}\label{28}
&&G_{\varphi,P}(x - x') = \int dx_1G_0 (x - x_1)\frac{\partial}{\partial z_0}G_{0P}(x_1-x')\nonumber\\
&& + \int dx_1 dx_2 G_{0P} (x - x_1 ) \frac{{\partial ^2
}}{{\partial t^2 }}G_0 (x_1 - x_2 )G_{0P} (x_2 - x')
\nonumber\\
&& + \int dx_1 dx_2 \int dx_3 dx_4 G_{0P} (x - x_1
)\frac{{\partial ^2 }}{{\partial t^2 }}G_0 (x_1 - x_2
) \times \nonumber\\
&& \hspace{+0.0cm} \times G_{0P}(x_2 - x_3 )\frac{{\partial ^2 }}
{{\partial t^2 }}G_0 (x_3 \! - \! x_4 )G_{0P} (x_4 \! - \! x') +
\cdots,
\end{eqnarray}
where
\begin{equation}\label{29}
G_{0P}(x-x')=\int d\omega \nu^2(\omega)G_\omega(x-x').
\end{equation}
If we again write $G_{\varphi,P}(x - x')$ in the Fourier space we
find
\begin{eqnarray}\label{30}
G_{\varphi,P}({\bf k},\omega )&=& \imath \omega G_0 ({\bf
k},\omega )
G_{0P}({\bf k},\omega ) \nonumber\\
&& \hspace{-0.6cm} \times \big\{1+\sum\limits_{n = 0}^\infty[\int
d\omega'\omega^2\nu ^2 (\omega' )G_{\omega'}(\omega ) G_0({\bf
k},\omega )]^n \big\}\nonumber\\
&=& \frac{G_{0P} ({\bf k},\omega ) G_0 ({\bf k},\omega )} {1 -
\int d\omega'\omega^2\nu ^2 (\omega' )G_{\omega'}(\omega)G_0
({\bf k},\omega )}\nonumber\\
&=& \frac{G_{0P} ({\bf k},\omega )}{{{\bf k}^2-\omega ^2-
\frac{1}{\rho}\int d\omega'\frac{\nu ^2 (\omega')\omega^2 }{\omega
'^2 -\omega ^2+ \imath 0^+ }}}.
\end{eqnarray}
Comparing Eqs.~\!(\ref{16}) and (\ref{29}), yields $G_{0P} ({\bf
k}, \omega)=\int d\omega\frac{\nu^2(\omega')}{\omega'^2-\omega^2+
\imath 0^+}=\tilde{\chi}(\omega)$, consequently
$G_{\varphi,P}({\bf k},\omega )$ can be rewritten as
\begin{equation}\label{31}
G_{\varphi,P} ({\bf k},\omega )=\imath
\omega\chi(\omega)G_{\varphi,\varphi} ({\bf k},\omega).
\end{equation}
The other important correlation function is $G_{P,P}({x-x'})$
which is defined via generating function $Z[J_\varphi,J_P]$ as
\begin{equation}\label{32}
G_{P,P}({x-x'})=\frac{\delta^2Z[J_\varphi,J_P]}{\delta J_P\delta
J_P}.
\end{equation}
By straightforward calculations one can obtain $G_{P,P}({\bf
k},\omega)$ as
\begin{equation}\label{33}
G_{P,P}({\bf
k},\omega)=\frac{\nu^2(\omega)}{\omega}+\omega^2\chi^2(\omega)G_{\varphi\varphi}({\bf
k},\omega).
\end{equation}
The imaginary part of the response function can be read from
Eq.~\!(\ref{26}) as $\frac{\nu^2(\omega)}{\omega}=Im\chi(\omega)$.
Here to illustrate the validity of our results, i.e.
Eq.~\!(\ref{31}-\ref{33}), we compare them with the results of the
other methods of field quantization. In other conventional methods
of phenomenological field quantization \cite{Matloob}, the fields
can be divided into positive ($+$) and negative ($-$) frequencies
parts which satisfy the constitutive relation
\begin{equation}\label{34}
\hat{P}^{\pm}({\bf k},\omega)= \pm \imath
\omega\chi(\omega)\hat{\varphi}^{\pm}({\bf
k},\omega)+\hat{P}_N({\bf k}^{\pm},\omega),
\end{equation}
after quantization of the fields, where by $\hat{}$ we mean
operator and the operator with positive frequency is the Hermitian
conjugate of the negative one. $\hat{P}_N^{\pm}$ is the noise part
of the polarization field that according to the
fluctuation-dissipation theorem \cite{Matloob, Fardin3} satisfies
\begin{equation}\label{35}
[ \hat{P}_N^+ ({{\bf x},\omega}) , \hat{P}_N^{-} ({\bf
x},\omega)]= \pi Im\chi(\omega)\delta({\bf x}-{\bf x}'),
\end{equation}
which $[\cdots,\cdots]$ denotes the commutator of two operators.
If we use the Eqs.~\!(\ref{34}) and (\ref{35}) to obtain the
Green's functions, we achieve the same results as the
Eqs.~\!(\ref{31}-\ref{33}). This shows the validity of our path
integral quantization.
According to the definition of the Green's functions, after
integrating over gaussian fields one can read the generating
function (\ref{18}) as
\begin{eqnarray}\label{36}
Z[J_\varphi,J_P]&&=exp\bigg\{ \imath \int dx \int dx' \big[
J_\varphi G_{\varphi,\varphi}(x-x')J_\varphi \nonumber\\
&& +J_\varphi G_{P,\varphi}(x-x')J_P+J_PG_{P,P}(x-x')J_P \big]
\bigg\}. \nonumber\\
\end{eqnarray}
\section{Calculating the Casimir force}\label{force-scalar}
\subsection{General formalism}
In this section we briefly review the path integral technique to
calculate the Casimir force. Let us consider two conducting plates
faced each other at the distance $H$ and embedded in an arbitrary
medium. The field $\varphi$ satisfies the Dirichlet
\begin{equation}\label{37}
\varphi (X_\alpha) = 0,
\end{equation}
or Neumann
\begin{equation}\label{38}
{\partial_n}\varphi (X_\alpha) = 0,
\end{equation}
boundary conditions on surface, where $X_\alpha, (\alpha=1,2)$ is an
arbitrary point on the $\alpha$th conducting plate. To obtain the
partition function from the Lagrangian we use the Wick's rotation,
$(t\rightarrow \imath\tau)$ and change the signature of the
space-time from Minkowski to Euclidean. The Diriclet or Neumann
boundary conditions can be taken into account using the auxiliary
fields $\psi _\alpha (X_\alpha )$ \cite{jalal}
\begin{equation}\label{38}
\delta \big( \varphi (X_\alpha ) \big) = \int {\cal
{D}}[\psi_\alpha (X_\alpha )]e^{\imath\int d X_\alpha \psi
(X_\alpha )\varphi (X_\alpha )}.
\end{equation}
and
\begin{equation}\label{39}
\delta \big({\partial_n}\varphi (X_\alpha ) \big) = \int {\cal
{D}}[\psi_\alpha (X_\alpha )]e^{- \imath\int d X_\alpha
{\partial_n}\psi (X_\alpha )\varphi (X_\alpha )}.
\end{equation}
After Wick's rotation the Dirichlet and Neumann partition
functions can be cast into the form
\begin{equation}\label{jalal}
Z_D = Z_0^{ - 1} \int {\cal {D}}[\varphi] \prod\limits_{^{a = 1} }^2
{\cal D}[\psi_\alpha (X_\alpha )])e^{S_D[\varphi ]},
\end{equation}
and
\begin{equation}\label{41}
Z_N = Z_0^{ - 1} \int {\cal {D}}[\varphi] \prod\limits_{^{a = 1} }^2
{\cal D}[\psi_\alpha (X_\alpha )])e^{S_N[\varphi ]}
\end{equation}
respectively, where $Z_0$ is the partition function of the free
space, and
\begin{eqnarray}\label{42}
&&S_D [\varphi ] = \int d^{(n+1)} x \nonumber\\
&& \times \big\{ {\cal L} \big( \varphi(x) \big)+ \varphi (x)\!\!
\sum\limits_{\alpha = 1}^2 \! \int \!\! d^{(n)} X\delta (X \!\! -
\!\! X_\alpha )\psi _\alpha (x) \big\}.
\end{eqnarray}
and
\begin{eqnarray}\label{43}
S_N [\varphi ] = \int d^{(n+1)} x && \nonumber\\
&& \hspace{-3.8cm} \times \{{\cal L} (\varphi(x))+ \varphi
(x)\sum\limits_{\alpha = 1}^2 \int d^{(n)} X\delta (X - X_\alpha
)\partial_n\psi _\alpha (x)\}.
\end{eqnarray}
Using the same procedure of Ref.~\!(\cite{jalal}) the parttion
functions for the Dirichlet and Neumann BC can be read
\begin{equation}\label{52}
Z_D = \frac{1}{{\sqrt {\det \Gamma_D (x,y,H)} }},
\end{equation}
and
\begin{equation}\label{53}
Z_N = \frac{1}{{\sqrt {\det \Gamma_N (x,y,H)} }},
\end{equation}
where
\begin{equation}\label{54}
\Gamma_D (x,y,H) =\bigg[\begin{array}{*{20}c}
{{\cal G}(x - y,0)} & {{\cal G}(x - y,H)} \\
{{\cal G}(x - y,H)} & {{\cal G}(x - y,0)} \\
\end{array}\bigg],
\end{equation}
and
\begin{equation}\label{55}
\Gamma_N (x,y,H) =\bigg[\begin{array}{*{20}c}
-\partial^2_z{{\cal G}(x - y,0)} & -\partial^2_z{{\cal G}(x - y,H)} \\
-\partial^2_z{{\cal G}(x - y,H)} & -\partial^2_z{{\cal G}(x - y,0)} \\
\end{array}\bigg],
\end{equation}
where ${\cal G}$ is the Green's function of the fields after wick
rotation. We define the effective action as
\begin{equation}\label{56}
S_{eff} =- \imath \ln Z (H),
\end{equation}
where $\ln Z (H)$ can be either for the Dirichlet or Neumann BC,
in order to calculate the Casimir force by applying derivative
with respect to the distance between the plates
\begin{equation}\label{57}
F = \frac{{\partial S_{eff} (H)}}{{\partial H}}.
\end{equation}
It is easy to show that the contribution of the Dirichlet is the
same as that of the Neumann BC to the Casimir energy and hence the
Casimir force in the presence of a isotropic and homogenous medium
like \cite{Emig-PRA-2003}. So that in the next sections we treat
only the Dirichlet BC.
\subsection{Casimir force for different boundary
conditions}\label{Casimir-Different-BC-Scalar}
In this section we would like to obtain the Casimir force in the
presence of an absorptive medium. Before we obtain the Casimir
force for interacting fields, we investigate the possibility of
the existence of the Casimir force due to the matter field alone.
Using the Lagrangian (\ref{4}) and expression for the effective
action (\ref{56}) we find the $\Gamma_\omega $ tensor as
\begin{equation}\label{58}
\Gamma_{\omega} (x,y,H) =\bigg[\begin{array}{*{20}c}
{{\cal G}_\omega(x - y,0)} & {{\cal G}_\omega(x - y,H)} \\
{{\cal G}_\omega(x - y,H)} & {{\cal G}_\omega(x - y,0)} \\
\end{array}\bigg].
\end{equation}
But since ${\cal G}_\omega(x-y,H)=0$ for this situation, the
noninteracting matter field alone, does not lead to any modified
Casimir force. This result is clear since we model the matter
field by the Hopfield model ******[HOPFIELD's REFERENCE]******.
This model is based on an independent set of harmonic oscillators
and imposing any condition on one of these oscillators does not
affect the others, and hence we do not expect any Casimir effect.
For a dissipative field $\varphi$ (\ref{25}) we may consider three
different boundary conditions,
{\bf i}) Imposing the boundary condition on the Klein-Gordon
field: for this case the $\Gamma$ tensor can be read
\begin{equation}\label{59}
\Gamma_{\varphi\varphi} (x,y,H) =\bigg[\begin{array}{*{20}c}
{{\cal G}_{\varphi,\varphi}(x - y,0)} & {{\cal G}_{\varphi,\varphi}(x - y,H)} \\
{{\cal G}_{\varphi,\varphi}(x - y,H)} & {{\cal G}_{\varphi,\varphi}(x - y,0)} \\
\end{array}\bigg].
\end{equation}
Since $\Gamma_{\varphi \varphi}$ is diagonal in the Fourier space,
to obtain the Casimir force we proceed in this space. The Fourier
transformation of $G_{\varphi \varphi} (x -y,H)$ is
\begin{eqnarray}\label{60}
{\cal G}_{\varphi \varphi}(p,q,H) &=& \int dx d y e^{\imath p.x
+\imath q.y} G_{\varphi \varphi} (x -y,H)\nonumber\\
&=&\frac{{e^{ - n(p_0 )|p_0| h} }}{2n(p_0 )|p_0|} (2
\pi)^3\delta(p+q),
\nonumber\\
\end{eqnarray}
where $p= (p_0, {\bf p})$, $\bf p$ is a vector parallel to the
conductor, $p_0$ the temporal component of the $p$, $n(p_0)=
\sqrt{\bar{\varepsilon}(p_0)}$ and
$\bar{\varepsilon}(p_0)=\varepsilon (\imath \omega)$. Thus for the
case {\bf i} the Casimir force is
\begin{equation}\label{61}
F_{\bf i} = - \int \frac{d^3 p }{(2\pi )^3
}[\frac{E(p)}{e^{2E(p)h} - 1}],
\end{equation}
where $E(p)=[n^2 (p_0 )p_0^2 + {\bf p}^2]^{1/2}$. In the absence
of the medium between the conductors, $n(p_0)=1$, we recover the
original Casimir force between two plates immersed in the quantum
vacuum of a scalar field
\begin{equation}\label{62}
F_{\bf i}= - \int\frac{d^3p}{(2\pi)^3}\frac{p^2}{e^{2|p| H}-1}= -
\frac{\pi^2}{480 H^4},
\end{equation}
and for a non absorptive medium with the susceptibility
$\chi(t)=\chi_0 \delta(t)$, we find the modified Casimir force as
\begin{equation}\label{63}
F_{\bf i}=\frac{1}{n}F_{\text{Vac}}.
\end{equation}
The above relation is fully in agree with the result of the
Lifshitz theory of fluctuation-induced force bewteen media
\cite{Lifshitz},\cite{Milonni-Book}. The equation (\ref{61}) is
interesting since it is the reminiscent of the Bose-Einstein
distribution. In fact, $E(p)$ can be interpreted as the force
density due to the bosons in the state $p$.
{\bf ii}) Imposing the boundary condition on the polarization
field: in this case $\Gamma_{PP}$ tensor is
\begin{equation}\label{64}
\Gamma_{PP} (x,y,H) =\bigg[\begin{array}{*{20}c}
{{\cal G}_{P,P}(x - y,0)} & {{\cal G}_{P,P}(x - y,H)} \\
{{\cal G}_{P,P}(x - y,H)} & {{\cal G}_{P,P}(x - y,0)} \\
\end{array}\bigg].
\end{equation}
Here the Casimir force is
\begin{equation}\label{65}
F_{\bf ii} = - \int \frac{d^3 p }{(2\pi )^3
}[\bar{\chi}^2(p_0)\frac{E(p)}{\alpha e^{2E(p)H} - 1}],
\end{equation}
where $\alpha =E(p)Im\bar{\chi}(p_0)+\bar{\chi}^2(p_0)$. Although
the noise operators do not have any spatial correlation but their
presence on the surface can affect the Casimir force and decrease
it due to polarization.
{\bf iii}) Imposing the boundary condition on the both of
polarization and Klein-Gordon fields: we can easily show that
$\Gamma_{\varphi\varphi,PP} (x,y,H)$ is a $8 \times 8$ tensor with
the form of
\begin{eqnarray}\label{65}
\Gamma_{\varphi\varphi,PP} (p,q,H) &&
\nonumber\\
&& \hspace{-2.6cm} =\bigg[\begin{array}{*{20}c}
{\!\Gamma_{\!\varphi\varphi} (p,q,\!H\!) }&{q_0 \bar{\chi}(q_0)\Gamma_{\! \varphi\varphi} (p,q,\!H\!) \!} \\
{\!\!q_0 \bar{\chi}(q_0)\Gamma_{\!\varphi\varphi} (p,q,\!H\!)} &
{~\!q_0^2 \chi^2(q_0) \Gamma_{\! \varphi\varphi} (p,q,\!H\!) \!+\! Im \bar{\chi} (q_0) {\bf I}\!\!} \\
\end{array}\bigg], \nonumber\\
\end{eqnarray}
where ${\bf I}$ is the $2\times 2$ unit matrix multiplied by $(2
\pi)^3 \delta (p+q)$.
It can be easily shown in this case that the Casimir force will be
the same as that of imposing the boundary condition only on the
Klein-Gordon field i.e. case {\bf i}. We can interpret this
situation with the aid of Eq.~\!(\ref{34}). According to this
relation if the BC is imposed only on the Klein-Gordon field then
$P_N^{\pm}$ is not zero, and if the BCs are imposed on the both of
polarization and Klein-Gordon fields then the BC will be imposed
on $P_N^{\pm}$ automatically which according to the begining of
the Sec.\ \ref{Casimir-Different-BC-Scalar} does not lead to aany
Casimir effect.
The physical interpretation of the first and third condition is
obvious. These conditions arise when we want to calculate the
Casimir force between two perfect conductors that enclose a
medium. But the second BC can happen when we want to consider the
system of containing two dielectric slabs. This kind of boundary
condition leads to the Casimir force between two dielectric slabs
which is under consideration. Here we only consider the boundary
conditions in cases {\bf i} and {\bf iii}, and we shall treat the
case {\bf ii} in elsewhere \cite{Fardin4}.
\section{Electromagnetic field quantization in the presence of a magnetodielectric medium}\label{path-electromagnetic}
In this section we develop the formalism to EM field in the
presence of a magnetodielectric medium \cite{Fardin3}. The
Lagrangian of EM field in the presence of a medium can be written
as
\begin{equation}\label{67}
{\cal L}={\cal L}_{EM}+{\cal L}_{1mat}+{\cal L}_{2mat}+{\cal
L}_{int}
\end{equation}
where ${\cal L}_{EM}$ is
\begin{equation}\label{68}
{\cal L}_{EM}=\frac{\varepsilon_0{\bf E}^2}{2}-\frac{{\bf
B}^2}{2\mu_0},
\end{equation}
where ${\bf E}$ and ${\bf B}$ are the electric and magnetic
fields. They can be written in terms of scalar and vector
potentials $U$ and ${\bf A}$ respectively as ${\bf E}=\dot {\bf
A}-\nabla U$ and ${\bf B}=\nabla\times{\bf A}$. In this work we
use the Coulomb gauge $\nabla\cdot {\bf A}=0$, i.e. ${\bf A}$ is a
transverse field. ${\cal L}_{1mat}$ and ${\cal L}_{2mat}$ reffer
to the polarization and magnetization of the medium respectively
and can be written as
\begin{equation}\label{69}
{\cal L}_{{\bf i}mat}=\int_0^\infty d\omega \big(
\frac{1}{2}{\dot{\bf X}_{{\bf
i}\omega}^2}+\frac{1}{2}\omega^2{{\bf X}_{{\bf i}\omega}^2} \big),
\end{equation}
where ${\bf X}_{{\bf i}\omega}$ is an oscillator's vector field.
The interaction part of the Lagrangian is
\begin{equation}\label{71}
{\cal L}_{int}={\bf A}\cdot \dot{\bf P}- \nabla U . {\bf P}
+\nabla\times{\bf A}\cdot{\bf M},
\end{equation}
where ${\bf P}$ and ${\bf M}$ are polarization and magnetization
of the medium defined by
\begin{eqnarray}\label{70}
{\bf P}&=&\int_0^\infty\nu_1 (\omega){\bf X}_{1\omega},\nonumber\\
{\bf M}&=&\int_0^\infty\nu_2 (\omega){\bf X}_{2\omega}.
\end{eqnarray}
The Euler-Lagrange equations lead us to the fact that $\dot U$ is
not an independent dynamical variable. To obtain $U$ in terms of
the other independent dynamical variables, it is appropriate to
write the fields in Fourier space where the longitudinal and
transverse parts of a field can be separated using the unit
vectors ${\bf e}_3({\bf k})=\hat{\bf k}$ and ${\bf e}_\lambda({\bf
k})$ ($\lambda=1,2$) respectively. ${\bf e}_1({\bf k})$ and ${\bf
e}_2({\bf k})$ are perpendicular to each other and $\hat{{\bf
k}}$. Using these unit vectors the scalar potential can be written
in terms of the longitudinal part of the matter field as (In what
follows we indicate the fields in Fourier space by a $\tilde{}$
over them.)
\begin{equation}\label{72}
\tilde{U}= \imath \frac{\tilde{X}^\parallel_1}{\varepsilon_0 |{\bf
k}|}.
\end{equation}
By applying this recent relation to the Eq.~\!(\ref{67}), we can
easily show that the longitudinal part of the electromagnetic
field only changes the longitudinal component of ${\cal L}_{1mat}$
as
\begin{equation}\label{73}
\tilde{{\cal L}}^\parallel_{1mat}=\int_0^\infty d\omega \big(
\frac{1}{2}{\dot{\tilde{ X}}^{\parallel 2}_{1
\omega}}+\frac{1}{2}\omega'^2{\tilde{ X}^{\parallel 2}_{1 \omega}}
\big)
\end{equation}
where $\omega'=\sqrt{\omega^2+\omega^2_c}$ and
$\omega^2_c=\frac{\nu^2(\omega)}{\varepsilon_0}$. It is worth
noting that only the transverse parts of Lagrangians have
contribution to the Casimir effect and the longitudinal part does
not lead to any Casimir force. Therefore we just consider the
transverse part of the Lagrangian which in the Fourier space is
\begin{equation}\label{74}
L^\perp= \int'{d^3{\bf k}\tilde{\cal L}^\perp_{em}+\sum_{{\bf
i}=1,2}(\tilde{\cal L}^\perp_{{\bf i},mat}+ \tilde{\cal
L}^\perp_{{\bf i},int})},
\end{equation}
where
\begin{equation}\label{75}
\tilde{{\cal L}}^\perp_{em}=\varepsilon_0( \dot{\tilde{{\bf A}}
}^2-c^2{\bf \tilde B}^2),
\end{equation}
\begin{equation}\label{76}
\tilde{{\cal L}}^\perp_{{\bf i},mat}=\int d \omega \big( \rho
\dot{\tilde{\textbf{X}}}_{\bf i}^{\perp2}-\rho \omega^{2}{\bf
\tilde X}_{\bf i}^{\perp2} \big) ,
\end{equation}
and $\tilde{\cal L}^\perp_{1,int}$ and $\tilde{\cal
L}^\perp_{2,int}$ refer to the interaction of the polarization and
magnetization fields with the electromagnetic field respectively
which are
\begin{equation}\label{77}
\tilde{\cal L}^\perp_{1,int}= \int_0^\infty \!\! d\omega \nu_1 (
\omega )
\tilde{\textbf{A}}\cdot\dot{\tilde{\textbf{X}}}_1^\perp=\sum_{\lambda=1}^2\int_0^\infty
\!\! d\omega \nu_1 ( \omega ) \tilde{A}_\lambda \dot{\tilde{X}}_{1
\lambda},
\end{equation}
and
\begin{eqnarray}\label{78}
\tilde{\cal L}^\perp_{2,int} && = \int_0^\infty d\omega \nu_2 (
\omega)
\textbf{k}\times\tilde{\bf{A}}\cdot\dot{\tilde{\textbf{X}}}_2^\perp
\nonumber\\
&&=\sum_{\lambda,\lambda'=1}^2\int_0^\infty d\omega \nu_2 ( \omega
) |{\bf k}| \tilde{A}_{\lambda}
{\tilde{\text{X}}}_{2\lambda'}\epsilon_{\lambda\lambda'},
\end{eqnarray}
where $\epsilon_{\lambda\lambda'}$ is the antisymmetric tensor.
To obtain the Casimir force between two plates with axial symmetry
due to the transverse part of the Lagrangian, the filed modes can
be divided into TM and TE modes
\cite{Emig-PRL-2001,Emig-PRA-2003}. It can be shown that for our
case as we consider the electromagnetic field in the presence of a
homogenous, isotropic and flat magnetodielectric medium, similar
to \cite{Emig-PRL-2001,Emig-PRA-2003} again the field can be
divided into TM and TE modes which refer to the Dirichlet and
Neumann BC respectively. As for the situation under study here for
TM and TE modes the Casimir force is the same, therefore here we
consider only the TM mode. According to the above discussion the
full Lagrangian can be rewritten as
\begin{equation}\label{79}
{\cal {L}} = {\cal{L}} _{sys}+{\cal{L}} _{mat}+{\cal{L}}_ {int},
\end{equation}
where
\begin{equation}\label{80}
{\cal{L}}_{sys} = \frac{1}{2}\,\partial ^\mu \varphi\,\partial
_\mu \varphi,
\end{equation}
is the Lagrangian density of a massless Klein$-$Gordon filed and
\begin{equation}\label{81}
{\cal{L}}_{mat} =\sum_{{\bf i}=1}^2 \int_0^\infty
d\omega\,(\frac{1}{2}\rho \dot X_{{\bf i}\omega}^2 -
\frac{1}{2}\rho \omega ^2 X_{ {\bf i}\omega}^2).
\end{equation}
The interaction term is defined by
\begin{equation}\label{82}
{\cal{L}}_{int}=\varphi \dot P+|\nabla \varphi|M,
\end{equation}
where $P=\int d\omega\nu_1(\omega)X_{1\omega}$ and $M=\int
d\omega\nu_2(\omega)X_{2\omega}$. It can be seen that the
Lagrangian (\ref{79}) is similar to the Lagrangian (\ref{2}). In
fact these Lagrangians are the same if we take $\nu_2(\omega)=0$.
This is the reason why we called $P$ as a polarization field and
the results obtained in the Sec.\ \ref{path-scalar} can be used
for a polarizable medium.
By mixing (\ref{18}) and (\ref{82}), and use the same procedure in
the Sec.\ \ref{path-scalar}, after some manipulations we obtain
the Green's function as
\begin{eqnarray}\label{83}
G_{\varphi,\varphi}({\bf k},\omega)=\frac{1}{{\bf k} ^2 (1 - \chi
_m (\omega )) - \omega ^2 (1 + \chi _e (\omega))},
\end{eqnarray}
which is the Green's function of EM field in the presence of
magnetodielectric medium with the electric and magnetic
susceptibilities
\begin{equation}\label{84}
\chi_e (\omega ) \equiv\int_{ - \infty }^{+\infty}d\omega
'\frac{\nu_1^2(\omega ')}{{\omega - \omega ' +\imath 0^+}},
\end{equation}
and
\begin{equation}\label{85}
\chi_m (\omega )\equiv \int_{ - \infty }^{+\infty}d\omega
'\frac{\nu^2_2(\omega ')}{{\omega - \omega ' +\imath 0^+}},
\end{equation}
respectively. These susceptibilities satisfy the Kramers -Kronig
relations as expected. The other Green's functions or correlation
functions are
\begin{equation}\label{86}
G_{\varphi,P}({\bf k},\omega)=\imath \omega
\chi_e(\omega)G_{\varphi,\varphi}({\bf k},\omega),
\end{equation}
\begin{equation}\label{87}
G_{\varphi,M}({\bf k},\omega)=\imath |{\bf k}|
\omega\chi_m(\omega)G_{\varphi,\varphi}({\bf k},\omega),
\end{equation}
\begin{equation}\label{88}
G_{P,P}({\bf k},\omega)= \frac{\nu^2(\omega)}{\omega}+\omega^2
\chi_e^2(\omega)G_{\varphi \varphi}({\bf k},\omega),
\end{equation}
\begin{equation}\label{89}
G_{M,M}({\bf k},\omega)=\frac{\nu^2(\omega)}{\omega}+|{\bf
k}|^2\chi_m^2(\omega)G_{\varphi \varphi}({\bf k},\omega).
\end{equation}
The first terms in the right hand side of Eqs.~\!(\ref{88}) and
(\ref{89}) are related to the noise operators of the system which
satisfy the fluctuation-dissipation theorem, like (\ref{35}).
Consequently the generating function is
\begin{eqnarray}\label{90}
Z[J_\varphi,J_P]&=&\exp\bigg\{\imath \int dx \int dx' \bigg[
J_\varphi(x)
G_{\varphi,\varphi}(x-x')J_\varphi(x)\nonumber\\
&& \hspace{-2cm} + J_\varphi(x)
G_{P,\varphi}(x-x')J_P(x)+J_\varphi(x)
G_{P,\varphi}(x-x')J_P(x)\bigg]\bigg\}. \nonumber\\
\end{eqnarray}
This relation can be used to obtain the Casimir force in the next
section.
\section{Calculating the Casimir force for EM field in the presence of
a magnetodielectric medium}\label{force-electromagnetic}
Here again we consider three different boundary conditions similar
to the Sec.\ \ref{force-scalar} varies kinds of the fields.
{\bf i}) Imposing the boundary condition on EM field: if we impose
the Dirichlet BC on EM field the partition function becomes
\begin{equation}\label{91}
Z_D = \frac{1}{{\sqrt {\det \Gamma_{\varphi \varphi} (x,y,H)} }},
\end{equation}
where
\begin{equation}\label{92}
\Gamma_{\varphi \varphi} (x,y,h H) =\bigg[\begin{array}{*{20}c}
{{\cal G}_{\varphi,\varphi}(x - y,0)} & {{\cal G}_{\varphi,\varphi}(x - y,H)} \\
{{\cal G}_{\varphi,\varphi}(x - y,H)} & {{\cal G}_{\varphi,\varphi}(x - y,0)} \\
\end{array}\bigg],
\end{equation}
and ${{\cal G}_{\varphi,\varphi}(x - y,h)}$ is the Green's
function (\ref{83}) with imaginary time.
To calculate $Z_D$, we invoke the Fourier space where
$\Gamma_{\varphi \varphi} (x,y,h H)$ is diagonal and its elements
have the form
\begin{eqnarray}\label{93}
&&{\cal G}_{\varphi , \varphi}(p,q,H) \nonumber\\
&& = \bar{\mu}(p_0)\frac{e^{-\sqrt{n^2(p_0 )p_0^2 + {\bf p}^2}
~\!H}}{2\sqrt{n^2(p_0)
p_0^2+ {\bf p}^2}} (2 \pi)^3 \delta(p+q)
\end{eqnarray}
where here $n (p_0)=\sqrt{\bar{\mu}(p_0)\bar{\varepsilon}( p_0)}$
and $\mu(\omega)=\frac{1}{1-\chi_m(\omega)}$. Finally the Casimir
force is obtained as
\begin{equation}\label{94}
F = \int \frac{d^3 p }{(2\pi )^3 }[\frac{E(p)}{e^{2E(p)h} - 1}],
\end{equation}
where $E(p)=\sqrt{n^2(p_0 )p_0^2+{\bf p}^2}$. This equation is
like the equation (\ref{61}) the only difference is in the
definition of $n(p_0)$. If we impose the Neumann BC on the EM
field we will achieve the same result as that of the Dirichlet BC.
{\bf ii}) We can impose the BC on polarization and magnetization
fields. As we mentioned after the equation (\ref{65}), this
situation does not appear in our problem since we consider a
magnetodielectric medium surrounded by two perfect conductors
which lead to the BC on EM field. But for two magnetodielectric
slabs this kind of BC may appear which may be dealt with elsewhere
\cite{Fardin4}.
{\bf iii}) We can impose the boundary condition on both matter and
EM fields. In this case if we redo the calculations, we conclude
that this situation leads to the same result as case {\bf i}. This
shows that for a magnetodielectric medium the conditions {\bf i}
and {\bf iii} lead to the same result for the Casimir force.
\section{Conclusion}\label{conclusion}
In this article we quantized the electromagnetic field in the
presence of a magnetodielectric medium in the frame work of path
integrals. For a medium with a given susceptibility, the modified
Casimir force is obtained for different boundary conditions. The
present approach can be generalized to the case of rough perfect
conductors in the presence of a general medium straightforwardly
\cite{Fardin4}.
\vspace{-.7cm}
|
{
"timestamp": "2010-09-21T02:00:41",
"yymm": "1009",
"arxiv_id": "1009.3537",
"language": "en",
"url": "https://arxiv.org/abs/1009.3537"
}
|
\section{Introduction}
Photometric monitoring of $\eta$~Carinae (\cite{feinstein67,feinsteinetal74,sterkenetal96,sterkenetal99,vangenderenetal06,frew04,fernandezlajusetal09}) revealed that an increase in brightness at variable rates, since 1950. The mechanism behind such long-term variations are still unclear, however. In this regard, spectroscopic monitoring can put several important constraints to the diagnosis. Unfortunately, frequent spectroscopic observations of this object began just about 2 decades ago, which is not sufficient yet to draw a clear picture of what is happening to the central source.
\section{Results and discussion}
In order to verify whether or not the central source in $\eta$~Car is passing through changes, we analized ground-based spectroscopic data taken at the same phase ($\phi\approx0.3$) of the spectroscopic event, but in different cycles (\#9, \#10, \#11 and \#12).
Our analysis revealed that the lines formed in the wind of the primary star -- such as the hydrogen lines -- do not present any evidence of systematic or significant changes in line profile, as shown in Fig.\,\ref{fig1}a, for example. In that figure, the H$\delta$ line profile was normalized by the local continuum.
On the other hand, lines with high-ionization potential -- such as [Fe\,{\sc iii}]~$\lambda4657$ -- do show systematic variations, namely, the intensity of the peak of the line's narrow component is decreasing with time, relative to the local continuum and, thus, the equivalent width of the narrow component is \textit{decreasing} with time, as indicated in Fig.\,\ref{fig1}b (dotted line).
However, since forbidden lines are formed in a more extended region, and we do not know where exactly the increase in brightness is coming from, we converted the equivalent width measurements into line flux by using the $B$-band magnitudes for each epoch. After that, we normalized the line fluxes by the line flux observed in 1994 Feb 25th ($\phi=9.31$). The result is shown in Fig.\,\ref{fig1}b (dashed line).
After correcting the equivalent width of the narrow component of the [Fe\,{\sc iii}]~$\lambda4657$ emission line by the flux in the local continuum, the trend changed completely: the line flux is \textit{increasing} with time. From 1994 to 2010, the narrow component line flux increased by about 60 per cent (in the same period, the continuum flux increased by a factor of 2.5).
We know that the narrow component is formed in the Weigelt's blobs. If the equivalent width of such component is decreasing while the line flux is increasing, then we can conclude that either (1) the total extinction around the central source is decreasing in all directions, not only in our line-of-sight or (2) the effective temperature of the secondary star is increasing (or the wind opacity is decreasing).
Unfortunately, based only on our preliminary results shown in this proceedings, we can only conclude that the wind of the primary star did not change during the last 15 years. At this moment, we cannot point for sure which of the possibilities presented above is the correct one to explain the behavior of the [Fe\,{\sc iii}]~$\lambda4657$ line (although we favor the decreasing of extinction in all directions). However, further analysis of other spectral features with high-ionization potential will eventually provide us with more indications on what is happening in the central source of $\eta$~Car. That will be the subject of a more complete, forthcoming paper.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.8\textwidth]{fig1.eps}
\caption{(a) Line profile of H$\delta$ at the epochs indicated in the legend. (b) Equivalent width and relative line flux. The hydrogen line shows no significant variations throughout the last 15 years. On the other hand, the equivalent width of the [Fe\,{\sc iii}] emision line is systematically decreasing with time, but the relative line flux is increasing.}
\label{fig1}
\end{center}
\end{figure}
|
{
"timestamp": "2010-09-23T02:02:26",
"yymm": "1009",
"arxiv_id": "1009.4399",
"language": "en",
"url": "https://arxiv.org/abs/1009.4399"
}
|
\section*{Introduction}
In the structure of the matter field equation given by the Dirac fermionic field equation, the most general spinorial derivative contains torsion; because torsion is a tensor then the torsional contribution can be separated apart without spoiling the covariance of the whole derivative: after torsion has been separated away what remains is the simplest spinorial derivative plus terms given by torsion and representing additional interactions. Eventually field equations coupling torsion to the spin distribution of the spinorial field are taken; when these field equations are plugged into the matter field equation, the additional interactions will turn out to be spinorial autointeractions.
In the case in which many spinorial fields are considered, then the spin distribution is the total spin distribution given by the sum of the spin distributions of each and every single spinor involved; in each matter field equations, the additional interactions turns out to be spinorial autointeraction of the spinor with itself as well as spinorial interactions with the spinor with all other spinors that take place in the dynamics of the process. Then it is possible to consider these interactions in order to study the form they may possibly have.
In a previous paper \cite{f} we have considered the simplest case given by two spinors of which one is a spinor and the other is a single-handed spinor showing that the matter field equation in the torsional free case is formally equivalent to the matter field equation without torsion but with the electroweak gauge interactions; in the final remarks that paper raised the question about the possible extension to more general situations in which the torsional interactions may take place for two spinors with both projections although one does not interact. In this paper we consider this extension by studying the case in which torsional interactions take place for two spinors with both projections with the two left-handed projections able to mix but with the two right-handed projection unable to mix showing that the matter field equation in the free case is formally equivalent to the matter field equation in the torsionless case with the electroweak gauge interaction.
\section{Torsional interaction}
As in the previous paper, we consider a set of $k$ matter fields labeled with the indices in parentheses each of which governed by the matter field equation
\begin{eqnarray}
&i\gamma^{\mu}D_{\mu}\phi^{a}=0,\ \ \ \ a=1\hdots k
\label{matter}
\end{eqnarray}
given in the massless case, and these equations come along with the background field equations that are given for the combination of the Ricci tensor and scalar and for the Cartan tensor in terms of the energy and the spin distribution of the matter field as
\begin{eqnarray}
&G_{\alpha\beta}-\frac{1}{2}g_{\alpha\beta}G
=\frac{i}{4}\sum_{a}\left[\bar{\psi}^{a}\gamma_{\alpha}D_{\beta}\psi^{a}
-D_{\beta}\bar{\psi}^{a}\gamma_{\alpha}\psi^{a}\right]
\label{metric}
\end{eqnarray}
and
\begin{eqnarray}
&Q_{\mu\alpha\beta}
=-\frac{i}{4}\sum_{a}\bar{\psi}^{a}\{\gamma_{\mu},\sigma_{\alpha\beta}\}\psi^{a}
\label{torsion}
\end{eqnarray}
according to the prescription of the Einstein--Sciama-Kibble scheme. Then it is possible to use the torsion as given by the field equations (\ref{torsion}) in order to substitute torsion with the spin of the spinor fields as
\begin{eqnarray}
&i\gamma^{\mu}\nabla_{\mu}\psi^{a}
+\frac{3}{16}\sum_{b}\bar{\psi}^{b}\gamma_{\mu}\gamma\psi^{b}
\gamma^{\mu}\gamma\psi^{a}=0,\ \ \ \ a=1\hdots k
\label{matterfield}
\end{eqnarray}
in which we see that spinorial bilinears appear.
\subsection{Torsional interaction:\\
spin coupling of spinor and single-handed spinor}
In this paper we will consider the case given when only two spinor fields are present and although both are taken to be the full spinor their two right-handed projections are not allowed to mix.
So in the case we have two spinors the matter field equations are given by the matter field equations (\ref{matterfield}) with $k=2$ and they can be explicitly written as
\begin{eqnarray}
&i\gamma^{\mu}\nabla_{\mu}\psi^{1}
+\frac{3}{16}\bar{\psi}^{1}\gamma_{\mu}\gamma\psi^{1}\gamma^{\mu}\gamma\psi^{1}
+\frac{3}{16}\bar{\psi}^{2}\gamma_{\mu}\gamma\psi^{2}\gamma^{\mu}\gamma\psi^{1}=0\\
&i\gamma^{\mu}\nabla_{\mu}\psi^{2}
+\frac{3}{16}\bar{\psi}^{1}\gamma_{\mu}\gamma\psi^{1}\gamma^{\mu}\gamma\psi^{2}
+\frac{3}{16}\bar{\psi}^{2}\gamma_{\mu}\gamma\psi^{2}\gamma^{\mu}\gamma\psi^{2}=0
\label{matterfields}
\end{eqnarray}
as it can be seen by separating the fields.
Moreover, we can separate the right-handed and left-handed projections as
\begin{eqnarray}
\nonumber
&i\gamma^{\mu}\nabla_{\mu}\psi^{1}_{L}
+\frac{3}{16}\bar{\psi}^{1}_{L}\gamma_{\mu}\psi^{1}_{L}\gamma^{\mu}\psi^{1}_{L}
+\frac{3}{16}\bar{\psi}^{2}_{L}\gamma_{\mu}\psi^{2}_{L}\gamma^{\mu}\psi^{1}_{L}-\\
&-\frac{3}{16}\bar{\psi}^{1}_{R}\gamma_{\mu}\psi^{1}_{R}\gamma^{\mu}\psi^{1}_{L}
-\frac{3}{16}\bar{\psi}^{2}_{R}\gamma_{\mu}\psi^{2}_{R}\gamma^{\mu}\psi^{1}_{L}=0\\
\nonumber
&i\gamma^{\mu}\nabla_{\mu}\psi^{1}_{R}
-\frac{3}{16}\bar{\psi}^{1}_{L}\gamma_{\mu}\psi^{1}_{L}\gamma^{\mu}\psi^{1}_{R}
-\frac{3}{16}\bar{\psi}^{2}_{L}\gamma_{\mu}\psi^{2}_{L}\gamma^{\mu}\psi^{1}_{R}+\\
&+\frac{3}{16}\bar{\psi}^{1}_{R}\gamma_{\mu}\psi^{1}_{R}\gamma^{\mu}\psi^{1}_{R}
+\frac{3}{16}\bar{\psi}^{2}_{R}\gamma_{\mu}\psi^{2}_{R}\gamma^{\mu}\psi^{1}_{R}=0\\
\nonumber
&i\gamma^{\mu}\nabla_{\mu}\psi^{2}_{L}
+\frac{3}{16}\bar{\psi}^{1}_{L}\gamma_{\mu}\psi^{1}_{L}\gamma^{\mu}\psi^{2}_{L}
+\frac{3}{16}\bar{\psi}^{2}_{L}\gamma_{\mu}\psi^{2}_{L}\gamma^{\mu}\psi^{2}_{L}-\\
&-\frac{3}{16}\bar{\psi}^{1}_{R}\gamma_{\mu}\psi^{1}_{R}\gamma^{\mu}\psi^{2}_{L}
-\frac{3}{16}\bar{\psi}^{2}_{R}\gamma_{\mu}\psi^{2}_{R}\gamma^{\mu}\psi^{2}_{L}=0\\
\nonumber
&i\gamma^{\mu}\nabla_{\mu}\psi^{2}_{R}
-\frac{3}{16}\bar{\psi}^{1}_{L}\gamma_{\mu}\psi^{1}_{L}\gamma^{\mu}\psi^{2}_{R}
-\frac{3}{16}\bar{\psi}^{2}_{L}\gamma_{\mu}\psi^{2}_{L}\gamma^{\mu}\psi^{2}_{R}+\\
&+\frac{3}{16}\bar{\psi}^{1}_{R}\gamma_{\mu}\psi^{1}_{R}\gamma^{\mu}\psi^{2}_{R}
+\frac{3}{16}\bar{\psi}^{2}_{R}\gamma_{\mu}\psi^{2}_{R}\gamma^{\mu}\psi^{2}_{R}=0
\label{fundamentalmatterfields}
\end{eqnarray}
in which all spinors are semi-spinors in single-handed irreducible representation.
By using the Fierz identities we can write these field equations in formally equivalent ways that will be more suited for the task we want to pursue: the first insight will be to try to follow \cite{f} in order to reproduce the field equations for the hadron fields before the symmetry breaking occurring in the standard model and in doing so we will see that we will obtain two different composite scalar fields corresponding to the two different fermions in the family; yet another possibility is look for a different way in which supplementary interactions will appear in the field equations for the hadron fields before the symmetry breaking occurring in the standard model but for which there will only be one composite scalar field written as a combination of all fermion fields. However we also know that in the first case each fermion undergoes mass generation provoked by the corresponding Higgs in such a way that the fermion mass is equal to the corresponding Higgs mass, with the consequence that for a general family of hadrons the Linde-Weinberg bound is not achieved and the stability of the vacuum configuration is not accomplished; thus we have to turn to the second case in which all fermions undergo mass generation provoked by a single Higgs in such a way that it is in terms of a combination of all fermion masses that the Higgs mass is written, so that if only one family has only one hadron whose mass is higher than the Linde-Weinberg bound then the stability of the vacuum configuration is ensured for the entire system under consideration.
\paragraph{Massless fundamental hadrons and composite scalar and vector fields.} In order to see that what we have discussed above may be obtained we see that the field equations may be rearranged in a form that is the same as
\begin{eqnarray}
\nonumber
&i\gamma^{\mu}\nabla_{\mu}L
-\frac{1}{2}g\vec{\sigma}\cdot\vec{A}_{\mu}\gamma^{\mu}L
-\frac{1}{6}g'B_{\mu}\gamma^{\mu}L-G_{u}i\sigma^{2}\phi^{\ast}u-G_{d}\phi d=0\\
\nonumber
&i\gamma^{\mu}\nabla_{\mu}u-\frac{2}{3}g'B_{\mu}\gamma^{\mu}u+G_{u}i\phi^{T}\sigma^{2}L=0\\
\nonumber
&i\gamma^{\mu}\nabla_{\mu}d+\frac{1}{3}g'B_{\mu}\gamma^{\mu}d-G_{d}\phi^{\dagger}L=0
\label{equivalentfundamentalmatterfields2}
\end{eqnarray}
in which all supplementary interactions appeared as three-field interactions and therefore extra terms are present but negligible leaving a system of equations in a known form. In fact in the leading order of approximation we have that this form is that of the field equations for the hadron fields before the symmetry breaking occurring within the standard model of elementary fields.
In order to be able to get this form we have to rename the spinor fields
\begin{eqnarray}
&\left(\psi^{1}_{R}\right)=u\ \ \ \ \left(\psi^{2}_{R}\right)=d\ \ \ \ \ \ \ \ \ \ \ \
\left(\begin{tabular}{c}$\psi^{1}_{L}$\\ $\psi^{2}_{L}$\end{tabular}\right)=L
\end{eqnarray}
as new hadron fields: then we have to consider their bilinear fields defining
\begin{eqnarray}
\frac{3}{32G_{d}G_{u}}\left(\begin{tabular}{c}
$5G_{u}\bar{\psi}^{2}_{R}\psi^{1}_{L}-2G_{d}\bar{\psi}^{2}_{L}\psi^{1}_{R}$\\
$5G_{u}\bar{\psi}^{2}_{R}\psi^{2}_{L}+2G_{d}\bar{\psi}^{1}_{L}\psi^{1}_{R}$
\end{tabular}\right)=\phi
\end{eqnarray}
for the scalar field; and finally we have
\begin{eqnarray}
&\frac{9}{32}\left[\frac{1}{2}\left(\bar{\psi}^{1}_{L}\gamma_{\mu}\psi^{1}_{L}
+\bar{\psi}^{2}_{L}\gamma_{\mu}\psi^{2}_{L}\right)
+2\bar{\psi}^{1}_{R}\gamma_{\mu}\psi^{1}_{R}
-\bar{\psi}^{2}_{R}\gamma_{\mu}\psi^{2}_{R}\right]=g'B_{\mu}\\
&\frac{9}{32}
\left[\frac{1}{2}\left(\bar{\psi}^{1}_{L}\gamma_{\mu}\psi^{1}_{L}
-\bar{\psi}^{2}_{L}\gamma_{\mu}\psi^{2}_{L}\right)\right]=gA^{3}_{\mu}\\
&\frac{9}{32}
\left[\frac{i}{2}\left(\bar{\psi}^{2}_{L}\gamma_{\mu}\psi^{1}_{L}
-\bar{\psi}^{1}_{L}\gamma_{\mu}\psi^{2}_{L}\right)\right]=gA^{2}_{\mu}\\
&\frac{9}{32}
\left[\frac{1}{2}\left(\bar{\psi}^{2}_{L}\gamma_{\mu}\psi^{1}_{L}
+\bar{\psi}^{1}_{L}\gamma_{\mu}\psi^{2}_{L}\right)\right]=gA^{1}_{\mu}
\end{eqnarray}
for the vector fields. Transformation laws for the hadrons and the scalar and vector fields are assigned as before the symmetry breaking occurring within the standard model of elementary fields.
\subparagraph{Massless fundamental hadrons and composite scalar and vector fields: structure of $U(1)\times SU(2)_{L}$ local electroweak gauge interaction.} Finally we consider the field equations written as
\begin{eqnarray}
&i\gamma^{\mu}\mathbb{D}_{\mu}L-G_{u}i\sigma^{2}\phi^{\ast}u-G_{d}\phi d=0\\
&i\gamma^{\mu}\mathbb{D}_{\mu}u+G_{u}i\phi^{T}\sigma^{2}L=0\\
&i\gamma^{\mu}\mathbb{D}_{\mu}d-G_{d}\phi^{\dagger}L=0
\label{invariantequivalentfundamentalmatterfields2}
\end{eqnarray}
in which the derivatives are in compact form.
This form is obtained by defining the derivatives
\begin{eqnarray}
&\mathbb{D}_{\mu}L=\nabla_{\mu}L
+\frac{i}{2}(g\vec{\sigma}\cdot\vec{A}_{\mu}+\frac{1}{3}g'B_{\mu})L\\
&\mathbb{D}_{\mu}u=\nabla_{\mu}u+\frac{2i}{3}g'B_{\mu}u\\
&\mathbb{D}_{\mu}d=\nabla_{\mu}d-\frac{i}{3}g'B_{\mu}d
\end{eqnarray}
covariant for general $U(1)\times SU(2)_{L}$ local transformations. This generalization is possible since the massless fundamental hadrons $u$ and $d$ and also $L$ are functions of the spacetime position and so their mixing may take place with coefficients depending on the spacetime position themselves.
By following the same procedure we have followed in \cite{f} it is possible to see that a stable vacuum configuration is assumed in which in the same order of approximation discussed above it is in terms of the sum of the squared fermion masses that the Higgs mass is given, and so the mass of the Higgs depends on the masses of all leptons and hadrons within a specific family; if we speculate that the procedure in \cite{f} applied here can be fully extended then it would be possible to have a stable vacuum configuration in which in the same order of approximation discussed above it might be in terms of the sum of squared fermion masses that the Higgs mass would be given, and so the mass of the Higgs would depend on the masses of all leptons and hadrons in all families: therefore if such generalization were possible in the leading order of approximation such a mass relation would read
\begin{eqnarray}
&m_{H}^{2}\approx
m_{t}^{2}+m_{b}^{2}+m_{c}^{2}+m_{s}^{2}+m_{u}^{2}+m_{d}^{2}+m_{\tau}^{2}+m_{\mu}^{2}+m_{e}^{2}
\end{eqnarray}
in which we see that because the heaviest hadron in the last family is more massive than the Linde-Weinberg bound then the stability of the vacuum configuration would be ensured. Under these assumptions in the case here presented it would be possible to obtain an approximated mass of
\begin{eqnarray}
&m_{H}\approx 173\ \mathrm{Gev}
\end{eqnarray}
and it might be possible to reproduce the correct dynamics for the Higgs field.
\section*{Conclusion}
In this paper we have proved that the matter field equations in the most general torsional case for spinors with both projections although one does not interact are formally equivalent to the matter field equations in the simplest torsionless case plus the electroweak gauge interactions for massless quarks; to proceed in logical order next step would be to know whether this derivation can be extended to more general situations in which the torsional interactions take place for both spinors having both projections in interaction thus formally obtaining the strong gauge interaction of massless quarks.
Should all this be done the result would be that matter field equations with torsion for spinors are formally equivalent to matter field equations for gauge invariant massless fermions implying a connection between torsion and gauge interactions; as already mentioned in the previous paper the gauge interactions arise from gauging internal transformations whereas torsion generates spacetime translations (as discussed in \cite{h-h-k-n}): so spacetime translations are related to internal transformations like for supersymmetric transformations and a link between the torsion tensor and supersymmetry may be established whenever the hypotheses of the Coleman-Mandula theorem are satisfied.
Differently from the previous paper however we have here more drastic issue related to the fact that what we have obtained are matter field equations for gauge invariant massless fermions, that is before the breakdown of the gauge symmetry generating the mass of fermions themselves; in fact a symmetry breaking for the mass generation may be assigned as usual by the introduction of the Higgs field and, although in the previous paper this could be done by defining the Higgs field to be a composite state of fermion fields, in this paper a single Higgs field has to be a combination of all fermion fields: this is after all to be expected, since the initial field equations are formally equivalent to the well-known Nambu-Jona--Lasinio field equations, which produce spontaneous chiral symmetry breaking with the consequent mass generation for the fields (as discussed for nuclear interactions with mesons in \cite{n-j--l}).
In the case in which these extensions were to be achieved then we would have that the initial work started in with the Nambu-Jona--Lasinio model of nuclear interactions mediated by mesons here extended to the model of electroweak interactions mediated by bosonic states of bound fermions will be enlarged to all nuclear interactions mediated by bosonic states of bound fermions; the symmetry breaking for the mass generation would be accomplished again by bosonic states of bound fermions. Therefore this would mean that the idea put forward by Hehl, Von Der Heyde, Kerlick and Nesterin for which the nuclear interactions arise from the torsional interactions might be justified.
|
{
"timestamp": "2010-10-15T02:02:51",
"yymm": "1009",
"arxiv_id": "1009.4423",
"language": "en",
"url": "https://arxiv.org/abs/1009.4423"
}
|
\section{Introduction}
For stars in clusters the evolutionary stage, age, and luminosity
can be determined more reliably, whereas they are rather uncertain
for field stars. It is especially important to study group members
that are rare, such as LBV--stars. From this point of view, young
Cyg\,OB2 association is of special interest. Many unevolved
O/Of--stars have been identified there as well as an LBV candidate
-- the variable star No.\,12. Its luminosity is
log\,L/L$\odot$\,=\,6.26 (\cite[de Jager 1998]{deJager98}) at the
association distance of 1.7 kpc.
\begin{figure}[htb]
\begin{center}
\begin{tabular}{ll}
\includegraphics[width=2.48in,bb=100 140 540 780,clip]{Fig1a.ps} &
\includegraphics[width=2.82in,bb=30 130 540 770,clip]{Fig1b.ps} \\
\end{tabular}
\caption{H$\alpha$ (left) and He\,{\sc I}\,5876\,\AA\ (right)
profiles in the spectra of Cyg\,OB2~No.\,12 obtained on
June~12,~2001 (dotted line) and April~12,~2003 (solid line). The
absorption features seen within the H$\alpha$ core correspond to
transitions in the line rather than to the telluric spectrum, whose
contribution was carefully removed. The vertical dashes show the
positions of telluric lines, with the dash lengths proportional to
the line strength.} \label{s4-11_fig1}
\end{center}
\end{figure}
\section{Observations and results}
Optical spectra of Cyg\,OB2 No.\,12 were taken using the \'echelle
spectrographs of the 6-meter telescope of the Special Astrophysical
Observatory. On June 12, 2001, we used the PFES spectrograph
(\cite[Panchuk et al. 1998]{Panchuk98}) with a 1040$\times$1170-pixel CCD
at the prime focus and got a spectrum with a resolution of $R
=\lambda/\Delta\lambda \sim$ 15000 (20 km\,s$^{-1}$). Later we used the
NES spectrograph~(\cite[Panchuk et al. 2009]{Panchuk09}) equipped with a
2048$\times$2048-pixel CCD and an image slicer and obtained spectra with
$R \sim$\,60000 (5\,km\,s$^{-1}$) on April 12, 2003 and on
December~8,~2006.
The spectral types we derived for three dates were the same within the
errors: B5.0$\pm$0.5, B4.8$\pm$0.5 and B4.0$\pm$0.5. The
luminosity type is Ia$^+$. The high luminosity is supported by the strong
O\,{\sc I}\,7773\,\AA\ IR--triplet whose equivalent width of 1.14\,\AA\
corresponds to an absolute visual magnitude of M$_{V} <$\,8\,mag.
The radial velocities (V$_{\rm r}$) measured from the absorption line
cores vary with time and with the line intensity. The weakest lines give
V$_{\rm r}$ lower than V$_{\rm sys}$\,=\,$-$11\,km\,s$^{-1}$
(\cite[Klochkova \& Chentsov 2004]{KlochkovaChentsov04}) by 5, 14 and
15\,km\,s$^{-1}$ in 2001, 2003, 2006, respectively, suggesting a variable
expansion rate of the layers where they form. The left panel of
Fig.\,\ref{s4-11_fig1} shows that the H$\alpha$ profile varies with time,
but its principal features are preserved: a strong bell-shaped emission,
with a dip at the short-wavelength slope, a sheared peak, and extended
Thompson wings. The blue-shifted absorption is barely visible in June~2001
and is more pronounced in April~2003, but can be traced at least to
V$_{\rm r}$\,=\,$-$160\,km\,s$^{-1}$ in both cases; i.e., to the same
limit that is reached by the blue wings of the absorption lines of
Si\,{\sc II} and He\,{\sc I} (the latter shown in the right panel of
Fig.\,\ref{s4-11_fig1}).
The wind terminal velocity is $\sim$ 150\,km\,s$^{-1}$. The
intensity inversions in the upper part of the H$\alpha$ profile
indicate that the wind is not uniform. In addition to the high
velocity material mentioned above, it contains a fair amount of
material that is nearly stationary relative to the star or is even
falling onto the stellar surface. Coexistence of lines with direct
and inverse P\,Cygni profiles in the same spectrum, and even
combinations of such features in the profile of the same line leads
us to reject a spherical symmetry wind. It is possible that the slow
part of the wind also contributes to the absorption profiles. So
far, this possibility is supported by the coincident velocities for
the central dips of the H$\alpha$ line and the well-formed cores of
strong absorption lines (He\,{\sc I}\,5876\,\AA\ in 2001 and
Si\,{\sc II}\,6347\,\AA\ in 2003), as well as by the fact that the
blue shift of all the absorption lines in the 2003 spectrum relative
to their positions in 2001 was accompanied by a similar shift of the
central dip in H$\alpha$. At any rate, both the hydrogen lines and
the strongest absorption lines in the visual spectrum
Cyg\,OB2~No.\,12 are partially formed in the wind.
{\bf Acknowledgements}. This research was supported by the Russian
Foundation for Basic Research (project no.~08--02--00072\,a).
|
{
"timestamp": "2010-09-22T02:01:02",
"yymm": "1009",
"arxiv_id": "1009.3995",
"language": "en",
"url": "https://arxiv.org/abs/1009.3995"
}
|
\section{Introduction.}
\mpb
This paper has two parts. In the first one we give a result
bounding the topological complexity of metric balls in terms of
the geometry. The bound we obtain is quite precise, and as an
application we show in the second part some useful criteria for
Gromov hyperbolicity of the Poincar\'e metric on a Riemann
surface. In particular (Theorem~\ref{t:finitegenus}), if $S$ is a
surface with finite genus and we remove from $S$ any `uniformly
separated' closed set $E$, then the Poincar\'e metric on the
deleted surface $S\setminus E$ is hyperbolic if and only if $S$
was hyperbolic with its own Poincar\'e metric.
Bounding the topology in terms of the geometry is a natural topic of
research; to mention a few examples see \cite{G2}, \cite{GP},
\cite{GPW}. More concretely, it is shown in \cite{G2} that the
fundamental group of a compact $n$-manifold $M$ with sectional
curvature verifying $K \ge -k^2$ can be generated with less than $C$
elements, where $C$ is a constant which just depends on $n$, $k$ and
the diameter of $M$. Theorem~\ref{t:balls} below is the noncompact
analogue for surfaces; it bounds the number of generators of the
fundamental group of a metric ball $B(p,r')$ by a constant times the
gap between the two sides of the classical comparison inequality:
\begin{equation}\label{ball-length}
\mbox{\rm length}\,\partial B(p,r)\;\leq\;\frac{2\pi}{k}\sinh (kr)\,
,
\end{equation} where $r$ is slightly larger than~$r'$. The result
is sharp: when the bound in Theorem~\ref{t:balls} is an
equality the metric ball is a topological disk and its curvature
is constant. An essential ingredient in the proof of this Theorem
is a second order differential inequality (\ref{fundamental})
relating the area and Euler characteristic of a metric ball, as
functions of the radius.
In Sections $5$ and $6$ we apply this result to the theory of Gromov
hyperbolicity. A geodesic metric space is called hyperbolic (in the
Gromov sense) if \emph{geodesic triangles are thin.\/} This means
that there exists an upper bound (the hyperbolicity constant) for
the distance from every point in a side of any geodesic triangle to
the union of the other two sides (see Definition \ref{def:Rips}).
Gromov hyperbolic spaces are a useful tool for understanding the
connections between graphs and Potential Theory on Riemannian
manifolds (see e.g. \cite{ARY}, \cite{CFPR}, \cite{FR2}, \cite{HS},
\cite{K1}, \cite{So}). Besides, the concept of Gromov hyperbolicity
grasps the essence of negatively curved spaces, and has been
successfully used in the theory of groups (see e.g. \cite{GH},
\cite{G3} and the references therein).
In recent years many researchers have been interested in studying
Gromov hyperbolicity of the metric spaces which appear in Geometric
Function Theory. In \cite{Be} and \cite{KN} it is shown that the
Klein-Hilbert metric is Gromov hyperbolic (under particular
conditions on the domain of definition); in \cite{Ha} it is proved
that the Gehring-Osgood metric is Gromov hyperbolic, and that the
Vuorinen metric is not Gromov hyperbolic except for a particular
case. In \cite{BB} significant progress is made about the
hyperbolicity of Euclidean bounded domains with their
quasihyperbolic metric (see also \cite{BHK} and the references
therein).
The study of Gromov hyperbolicity of a Riemann surface with its
Poincar\'e metric is non-trivial. An obvious reason is that
homological `holes' may be surrounded by geodesic triangles which
are not thin. For example in the `infinite grille', a
$\ZZ^2$-covering of the genus-2 surface, triangles engulfing many
holes are quite `fat'. An even stronger reason is the result, proved
in \cite{RT3}, that the usual classes: $O_G$, $O_{HP}$, $O_{HB}$,
$O_{HD}$, and surfaces with linear isoperimetric inequality, are
logically independent of the Gromov hyperbolic class. More
precisely, in each of these classes, as well as in its complement,
some surfaces are Gromov hyperbolic and some are not (even in the
case of plane domains). This has stimulated a good number of works
on the subject, e.g. \cite{APRT}, \cite{HLPRT}, \cite{HPRT1},
\cite{HPRT2}, \cite{PRT2}, \cite{PRT3}, \cite{PT}, \cite{RT3}.
A characterization of Gromov hyperbolicity for a surface $S^*$ with
cusps and/or funnels was obtained in \cite{PRT2} and \cite{PRT3}.
The idea there was to identify the cusps and funnels of $S^*$ with
pairwise disjoint compact sets $\{E_n\}$ removed from an original
surface $S$, so that the conformal structure of $S^*$ equals that of
$S\setminus \cup_n E_n$. Of course the Poincar\'e metric changes
when removing the sets $E_n$, but control of the resulting metric in
$S^*$ was achieved in terms of local information in $S$ near
each~$E_n$. Those two works use the idea of \emph{uniform
separation:\/} the $E_n$ are placed inside compact neighborhoods
$V_n\supset E_n$ having controlled topology each, and subject to
conditions such as $\p V_n$ being neither too long, nor too close to
$E_n$ or to the other $V_m$'s. The criterion obtained in \cite{PRT3}
additionally requires a `uniform hyperbolicity' condition, namely
that a single constant is valid for the hyperbolicity of all sets
$V_n\setminus E_n$ with the metric induced from $S^*$. This
condition can be hard to ensure in practice. In this paper we give
two criteria for hyperbolicity: Theorem~\ref{t:main}, based on
uniform separation but without the uniform hyperbolicity condition
in their hypotheses, and Theorem~\ref{t:infinite}, based on
`surrounding' the $E_n$'s by curves of controlled length (each $E_n$
is thus placed inside a ball, but with less constraints than in
uniform separation). In a nutshell, Theorem~\ref{t:main} states that
$S^*$ is hyperbolic if and only if $S$ is hyperbolic and a
reasonable metric condition on the handles of $S$ is held. There
follows an even simpler characterization when $S$ has either no
genus or finite genus, since in this case $S^*$ is hyperbolic if and
only if $S$ is hyperbolic (see Corollary \ref{c:main} and
Theorem~\ref{t:finitegenus}). This criterion has already received
important use in~\cite{HPRT2}.
Two ingredients have proved essential in the proofs of these
criteria. One is the above mentioned bound on the topology of balls.
Another ingredient consists on results about stability of
hyperbolicity; this means that we can set free some apparently
important quantities and still have uniform hyperbolicity. In this
direction, Theorem~\ref{t:finite} and Corollary~\ref{c:finite0}
guarantee uniform hyperbolicity even in situations where the
`punctures' $E_n$ approach one another. Likewise
Theorem~\ref{t:clasef} gives uniform hyperbolicity of all surfaces
with a fixed bound on the lengths of all of their funnel borders
\emph{except one} (see Definition~\ref{d:clasef}); this is a
remarkable improvement of a previous result \cite[Theorem
5.3]{PRT3}, where \emph{all} such borders had to be controlled.
It is also a remarkable fact that almost every constant appearing in
the results of this paper depends just on a small number of
parameters. This is a common place in the theory of hyperbolic
spaces (see e.g. \cite{GH}) and is also typical of surfaces with
curvature $-1$ (see e.g. the Collar Lemma in \cite{R} and \cite{S},
and Theorem~\ref{t:balls}). In fact, this simple dependence is a
crucial fact in the proof of Theorem~\ref{t:main}.
\mpb
\noindent {\bf Notations and conventions.} Every surface in this
paper is connected and orientable. In Section~2 we denote by $\p
B_r$ the \emph{extrinsic} boundary $\overline{B}_r\setminus B_r$ of
a ball $B_r$ as open set of an ambient surface $S$. In Sections 4,
5, and 6 we shall consider bordered $2$-dimensional manifolds, and
then the symbol $\p$, followed by the manifold's name, will indicate
the \emph{intrinsic} boundary of its bordered structure, e.g. $\p
(S^1\times (0,1])=S^1\times\{ 1\}$. If $(X,d_X)$ is a geodesic
metric space, we shall denote by $L_X$ the induced length, and given
$Y\subset X$ we shall write $d_X|_Y$, or simply $d_Y$, for the
geodesic distance induced on $Y$ by $d_X$. When there is no possible
confusion, we will not write the subindex $X$. Finally, we denote by
$c$, $k$, $c_j$, and $k_j$, positive constants which can assume
different values in different theorems.
\spb
\section{The topology of balls.}
\spb
In this Section we give upper bounds, in terms of the radius, for
the growth of the topological complexity of distance balls in a
surface endowed with a Riemann metric. Unlike the rest of the paper,
we allow the Gaussian curvature to be zero or positive somewhere.
Given a surface $S$, the topological complexity within $S$ of each
distance ball $B(p,r)$ will be measured using the integer $n(r)$
defined as follows:
\begin{equation}\label{n1n2}
n(r) := \;\hbox{\rm minimal number of generators for }\;\pi_1\big(\,
B(p,r)\, ,\, p\,\big)\; .
\end{equation}
Given $r_0$, we are going to bound $n(r')$ for some $r'>r_0$ not far
from $r_0$. This seems unavoidable because $n(r)$ is not always a
monotonic function of~$r$. Figure~\ref{figure:balls3} describes
metric balls $B_r$ such that $n(r)$ goes up and down as $r$ takes on
three values $r_1<r_2<r_3$. The starting ball $B_{r_1}$ (leftmost in
the figure) is diffeomorphic with a disk but its frontier in $S$ is
a curve with three points of self-tangency; it has $n(r_1)=0$. The
ball $B_{r_2}$ has $n(r_2)=3$ but one of its boundary components (a
`spurious hole') bounds a triangular disk in $S$. When $r=r_3$ the
triangular hole has disappeared and then~$n(r_3)=2$. In general, we
need to go from $B_{r_0}$ to some larger $B_{r'}$ with fewer
spurious holes.
\begin{figure}[h]
\includegraphics[scale=0.5]{balls3}
\caption{}
\label{figure:balls3}
\end{figure}
\begin{obs}\label{con-borde}
{\rm In this section we always work within some closed ball
$\overline{B}(p,R)$ satisfying the following conditions:
\begin{list}{}{}
\item[--] The open ball $B(p,R)$ is not all of $S$. Thus for each $r<R$
the boundary $\partial B(p,r)$ has positive length.
\item[--] Every geodesic issuing from $p\,$ continues up to length
$R$. In particular, $\overline{B}(p,R)$ is compact.
\end{list}
}
\end{obs}
\begin{definicion}\label{ell}
For each $r\geq 0$ let $\ell (r)$ denote the length of the boundary
$\partial B(p,r)$.
\end{definicion}
\begin{teo}
\label{t:balls} Let $k,c$ be positive constants and assume $r_0>0$
is such that the ball $\overline{B}\big(\, p\, ,\,
r_0+\frac{c}{k}\,\big)$ is in the hypotheses of
Remark~\ref{con-borde}.
If the metric is real analytic and satisfies $K\ge -k^2$, or if it
is smooth and satisfies $0\ge K\ge -k^2$, then there is a radius
$r'$, strictly between $r_0$ and $r_0+\frac{c}{k}$, such that
\begin{equation}\label{top-estimate}
n(r')\;\leq\; \frac{1}{\sinh c}\left( \sinh
(kr_0+c)-\frac{k\,\ell\big(\, r_0+\frac{c}{k}\,\big)}{2\pi}\right)
\; .
\end{equation}
This inequality is also valid if $n(r')$ is defined using the
fundamental group of the closed metric $r'$-ball, and it is strict
unless the ball $B\big(\, p\, ,\, r_0+\frac{c}{k}\,\big)$ is an
injective image of the exponential map (hence a disk) and has
$K\equiv -k^2$.
\end{teo}
In particular $n(r')\leq\displaystyle\frac{\sinh (kr_0+c)}{\sinh
c}<\frac{1}{1-e^{-2c}}e^{kr_0}$. Any general bound for the
fundamental group must grow exponentially with the radius: consider
copies of a fixed $Y$-piece with $K=-k^2$; if we paste them together
following the combinatorial design of a binary tree, we obtain an
example where $n(r)$ is asymptotically equal to $c_0e^{c_1r}$ for
some constants $c_0,c_1>0$.
Using Theorem~\ref{t:balls} one can improve the constant in
\cite[Theorem 3.1]{PRT2}, a result which says that Balls of small
radius (depending on the Gauss curvature bound) are simply or doubly
connected.
\begin{obs}\label{open-vs-closed}
{\rm For analytic metrics, and for those satisfying $K\leq 0$, we
are going to see that, as $r$ increases from $0$ to $R$, the
topology of the distance ball changes only at values of $r$ which
make up a discrete set in $[0,R)$. For all other values of $r$ the
inclusion $B(p,r)\hookrightarrow\overline{B}(p,r)$ is a homotopy
equivalence. The function $\,d(p\, ,\cdot )$ thus behaves like a
Morse function.}
\end{obs}
The present Section is organized as follows. We first examine in
depth the pertinent properties of the cut and conjugate loci. Then
we establish the regularity of the function $\ell (r)$ from
Definition~\ref{ell}, and give a formula for its derivative. After
these preliminaries we prove Theorem \ref{t:balls}, for which we
shall use a differential inequality (\ref{fundamental}) which
relates area and Euler characteristic of metric balls.
\mpb
Let $\mbox{\rm Exp}_p:T_pS\to S$ be the exponential map. The
boundary $\partial B(p,r)$ is some closed subset of the following
image
\[ \mbox{\rm Exp}_p\big(\, \{\; {\bf v}\in T_pS\; ;\; \|{\bf v}\|
=r\; \}\big)\, \] which is usually a complicated curve on $S$ with
many self-intersections. In particular, some parts of this image
will lie interior to the ball $B(p,r)$, not on its boundary.
\begin{definicion}
The {\em tangential cut locus} of $p$ is the set of vectors ${\bf
v}\in T_pS$ such that $\mbox{\rm Exp}_p(t{\bf v})$ defines a
minimizing segment for $t\in [0,1]$ and not for $t\in [0,T]$ if
$T>1$. The {\em cut locus} of $p$ in $S$ is the image of the
tangential cut locus under $\mbox{\rm Exp}_p$, and its points are
called {\em cut points.}
The {\em tangential first conjugate locus} of $p$ is the set of
vectors ${\bf v}\in T_pS$ such that $\mbox{\rm Exp}_p$ has nonzero
jacobian at each $t{\bf v}$ with $t\in [0,1)$ and zero jacobian
at~$\bf v$. We then say that $\mbox{\rm Exp}_p({\bf v})$ is the
first conjugate point of $p$ along the geodesic with initial data
$p,{\bf v}$. The set of all such points, equal to the image of the
tangential first conjugate locus under $\mbox{\rm Exp}_p$, is
called {\em first conjugate locus} of $p$ in~$S$.
\end{definicion}
We work in a ball $\overline{B}(p,R)$ which is the exponential image
of the tangential ball $\overline{B}_R^{\mbox{\rm\scriptsize
T}}=\{\,{\bf v}\; ;\; \|{\bf v}\|\leq R\,\}$. Denote by $\mbox{\rm
Cut}^{\mbox{\rm\scriptsize T}}_p$ the part contained in
$\overline{B}_R^{\mbox{\rm\scriptsize T}}$ of the tangential cut
locus. Denote by $\mbox{\rm Cut}_p$ the part contained in
$\overline{B}(p,R)$ of the cut locus. Denote by $\mbox{\rm
Conj}^{\mbox{\rm\scriptsize T}}_p$ the part contained in
$\overline{B}_R^{\mbox{\rm\scriptsize T}}$ of the tangential first
conjugate locus. Denote by $\mbox{\rm Conj}_p$ the part contained in
$\overline{B}(p,R)$ of the first conjugate locus.
We base our discussion of these sets on the work of Myers
\cite{Myers1} and \cite{Myers2}, the reader may also see
\cite{Kobayashi} and \cite{Petersen}. If non-empty, the tangential
loci are described inside $T_pS$ as polar graphs:
\[
\mbox{\rm Conj}^{\mbox{\rm\scriptsize T}}_p = \{\, \|{\bf v}\|
=R_1(\theta )\,\} \quad ,\quad \mbox{\rm Cut}^{\mbox{\rm\scriptsize
T}}_p = \{\, \|{\bf v}\| =R_2(\theta )\,\} \; ,
\]
where $R_1(\theta )$ is smooth and $R_2(\theta )$ is continuous. For
$i=1,2$ the domain of $R_i(\theta )$ is either the whole unit circle
in $T_pS$, in which case the polar graph is a closed curve, or a
finite union of closed arcs in the unit circle, in which case the
polar graph is a finite union of embedded arcs with all the
endpoints on the outer circle
$\partial\overline{B}^{\mbox{\rm\scriptsize T}}_R$. These polar
graphs are compact and so are their exponential images $\mbox{\rm
Cut}_p$ and $\mbox{\rm Conj}_p$.
\begin{lema}\label{finite}
If the metric is real analytic and $\overline{B}(p,r)\neq S$, then
$\mbox{\rm Cut}_p\cap \mbox{\rm Conj}_p\cap \overline{B}(p,r)$ is a
finite set.
\end{lema}
\begin{proof} It is proved in \cite[Lemma 10]{Myers1} that any cut
point for $p$ which is also a conjugate point must be the
exponential image of a vector which is a local minimum for the norm
$\|\cdot\|$ in $\mbox{\rm Conj}^{\mbox{\rm\scriptsize T}}_p$. We
claim that the norm has finitely many local minima in $\mbox{\rm
Conj}^{\mbox{\rm\scriptsize
T}}_p\cap\overline{B}_r^{\mbox{\rm\scriptsize T}}$. Assume the
contrary, i.e. that there are infinitely many local minima for the
norm in $\mbox{\rm Conj}^{\mbox{\rm\scriptsize T}}_p\cap
\overline{B}_r^{\mbox{\rm\scriptsize T}}$; then infinitely many of
them belong to a single connected component $\mathcal C$ of
$\mbox{\rm Conj}^{\mbox{\rm\scriptsize T}}_p\cap
\overline{B}_r^{\mbox{\rm\scriptsize T}}$ and thus accumulate to
some vector ${\bf v}_0\in{\mathcal C}$ with $\|{\bf v}_0\|\leq r$.
The real analytic curve ${\mathcal C}$ and the circle centered at
$\bf 0$ and passing through ${\bf v}_0$ have a contact of infinite
order at ${\bf v}_0$, hence they coincide. But then $\mbox{\rm
Exp}_p$ sends that circle to a single point and
$\overline{B}(p,\|{\bf v}_0\| )$ is all of $S$, thereby forcing
$\overline{B}(p,r)$ to also be all of~$S$.\end{proof}
The results in \cite{Myers1} and \cite{Myers2} describe the cut
locus of a point on a surface and how it is reached by minimizing
geodesic arcs starting at such point (this second part is what most
interests us here). Under our hypothesis (analytic metric or $K\leq
0$) the set $\mbox{\rm Cut}_p$ is an embedded graph in~$S$ with
finitely many vertices and finitely many edges in each ball
$\overline{B}(p,r)$ not equal to~$S$. The points on this graph can
be of three kinds:
\begin{itemize}
\item Vertices of multiplicity $1$, i.e. points at which only one
edge arrives. These are conjugate points for $p\,$, and so they do
not exist if $K\leq 0$ and they are finite in number if the metric
is analytic. Each of these vertices is joined to $p$ by only one
minimizing geodesic arc.
\item Points of multiplicity $2$. These are the points on the
interior of the edges.
\item Vertices of multiplicity $m\geq 3$, i.e. points at which
three or more edges arrive.
\end{itemize}
Each edge is an embedded arc in $S$, and it follows from \cite[Lemma
11]{Myers1} that it is smooth except perhaps at the conjugate points
that it may contain. Thus each edge is smooth except perhaps at
finitely many points. It is proved in \cite[page 97]{Myers2} that
every interior point of an edge, smooth or non-smooth, is joined to
$p$ by exactly two minimizing geodesic arcs (of course, having the
same length). The same argument proves that if infinitely many edges
arrived at some cut point $q$ then there would exist infinitely many
minimizing geodesic arcs, all of the same length $r$, joining $p$
to~$q$ (which must then be conjugate to~$p$). If $K\leq 0$ this does
not happen because there are no conjugate points. If the metric is
real analytic then $\mbox{\rm Exp}_p$ would be a real analytic map
taking an infinity of tangent vectors at $p\,$, all with norm equal
to $r$, to the single point $q$. This would imply
$S=\overline{B}(p,r)$. Therefore a ball $\overline{B}(p,r)$ not
equal to $S$ does not contain any vertex of infinite multiplicity.
Once vertices of multiplicity $1$ are finite in number and vertices
of infinite multiplicity do not exist, the total number of vertices
and edges is finite due to topological reasons.
Let $\gamma (t)$, $1\leq t\leq 1$, be a geodesic with $\gamma (0)=p$
and ${\bf v}=\gamma'(0)$ a vector which is a local minimum for $r$
in the tangential first conjugate locus, then $q=\gamma
(1)=\mbox{\rm Exp}_p({\bf v})$ is a first conjugate point of $p$
along $\gamma$. Figure~\ref{figure:costura} shows the behavior near
$q$ of the geodesics which start at $p$ with initial velocity close
to~$\bf v$.
\begin{figure}[h]
\includegraphics[scale=0.8]{costura}
\caption{}
\label{figure:costura}
\end{figure}
Consider a circular arc $C_0\subset T_pS$ centered at $\bf 0$ and
containing $\bf v$ as midpoint. Figure~\ref{figure:ortogonales}
shows the exponential image $\Gamma_q$ of $C_0$ as well as
orthogonal trajectories of the geodesics displayed in
Figure~\ref{figure:costura}, which trajectories are subsets of the
exponential images of circular arcs centered at $\bf 0$. If
$\gamma$ is the only minimizing path from $p$ to~$q$ then the
orthogonal trajectories shown in Figure~\ref{figure:ortogonales}
lie on the boundaries of balls centered at~$p\,$; in this case the
part of the cut locus on Figures~\ref{figure:costura}
and~\ref{figure:ortogonales} will be the dotted line. If there are
more minimizing paths from $p$ to~$q$ then some part of the
orthogonal trajectories will lie on the boundary and another part
will lie in the interior of the corresponding ball; in this case
the cut locus will have, in addition to the dotted line shown,
other branches ending at~$q$.
\begin{figure}[h]
\includegraphics[scale=0.5]{ortogonales}
\caption{}
\label{figure:ortogonales}
\end{figure}
\begin{obs}\label{inf-curv}
{\rm Since the geodesics in Figure~\ref{figure:costura} meet in
pairs making an angle which tends to zero as the cut point tends to
$q$, the image $\Gamma_q=\mbox{\rm Exp}_p(C_0)$ is of class
${\mathcal C}^1$ at the point $q$, but not of class~${\mathcal
C}^2$. In fact the geodesic curvature of $\Gamma_q$ at $q$ (defined
as limit of the curvatures at points close to $q$) is a positive
infinite multiple of~$\gamma'(1)$ because small arcs of $\Gamma_q$
around $q$ are supported by distance circles of arbitrarily small
radius centered at points of the dotted line.}
\end{obs}
We next define a type of point which is of great importance in our
context.
\begin{definicion}
A {\em middle point} is a cut point $q$ at which two minimizing
geodesic arcs (of equal length) issued from $p$ meet `head on', i.e.
the velocities of the two geodesic arcs at $q$ are each a negative
multiple of the other.
\end{definicion}
If $\gamma_1$ and $\gamma_2$ are those two minimizing arcs, then
$\gamma_1$ followed by reversed $\gamma_2$ defines a geodesic loop
based at~$p$ and having $q$ as middle point, hence the name.
\begin{lema}\label{coll-finite}
Suppose the metric is real analytic or it satisfies $K\leq 0$. If
the ball $\overline{B}(p,r)$ is not all of $S$, then there are only
finitely many middle points inside it.
\end{lema}
\begin{proof}
Suppose that, on the contrary, there is an infinity of such points.
Then there is an infinite sequence $\{\, ({\bf v}_n,L_n)\,\}$ where
${\bf v}_1,{\bf v}_2,{\bf v}_3,\dots $ are unit vectors in $T_pS$
and $L_1,L_2,L_3,\dots $ are lengths bounded by the number $r$, so
that the points $\mbox{\rm Exp}_p(L_n{\bf v}_n)$ are all middle
points of loops based at~$p$. Therefore we have $\mbox{\rm
Exp}_p(2L_n{\bf v}_n)=p\,$ for all~$n$. Since $\{ L_n\}$ is bounded
we extract a subsequence, again denoted $\{\, ({\bf v}_n,L_n)\,\}$,
that converges to some pair $({\bf v}_0,L)$ with $L\leq r$.
We prove first that this cannot happen for an analytic metric. There
is an $\varepsilon >0$ such that for every unit vector ${\bf v}$
close enough to ${\bf v}_0$ the geodesic segment
\[ \Gamma ({\bf v}):=\{\,\mbox{\rm Exp}_p(t{\bf v})\; ;\;
2L-\varepsilon\leq t\leq 2L+\varepsilon\,\} \] is small and very
close to $p$. Fix one such $\varepsilon >0$ and consider the
function:
\[ f({\bf v}):=d\,\big(\, \Gamma ({\bf v})\, ,\, p\,\big)\; . \] Once
$\varepsilon$ is fixed, for $\bf v$ close enough to ${\bf v}_0$ this
distance is attained at a point interior to the segment $\Gamma
({\bf v})$, hence f is analytic in a small enough neighborhood $C$
of ${\bf v}_0$ in the unit circle. At the same time $f$ vanishes on
an infinite sequence of points of $C$ converging towards ${\bf
v}_0$, which forces $f\equiv 0$. The circular arc $C$ thus
determines a $1$-parameter family of geodesic loops based at $p$,
which must all have the same length by Gauss' Lemma. It follows that
for all ${\bf v}\in C$ we have $\mbox{\rm Exp}_p(2L{\bf v})=p$ while
the entire loop of length $2L$ with initial data $p,{\bf v}$ is
contained in $\overline{B}(p,L)$. By analytic prolongation, we
obtain that the exponential image $B$ of the tangential disk $\{
\|{\bf v}\|\leq 2L\}$ is contained in
$\overline{B}(p,L)\subseteq\overline{B}(p,R)$ and that $\mbox{\rm
Exp}_p$ maps the tangential circle $\{ \|{\bf v}\| =2L\}$ to~$p\,$.
But then we would have $B=S=\overline{B}(p,L)$, and
$\overline{B}(p,r)$ would be all of $S$ because $r\ge L$.
We now do the proof for a metric with $K\leq 0$. Assuming the middle
points $\mbox{\rm Exp}_p(L_n{\bf v}_n)$ to be pairwise distinct, the
vectors $L_n{\bf v}_n$ are pairwise distinct and may be assumed to
be all different from their limit. Then the sequence of vectors
${\bf w}_n=2L_n{\bf v}_n$ has a subsequence $\{{\bf w}_{n_k}\}$
which converges {\em tangentially} to ${\bf w}=2L{\bf v}_0$. This
means that not only is $\bf w$ the limit of $\{{\bf w}_{n_k}\}$, but
the unit vectors $({\bf w}_{n_k}-{\bf w})/\| {\bf w}_{n_k}-{\bf
w}\|$ also have a limit ${\bf u}\in T_pS$. Then $\bf u$ is a unit
vector whose image under the differential of $\mbox{\rm Exp}_p$ at
$\bf w$ is zero, thus causing $p=\mbox{\rm Exp}_p({\bf w})$ to be
conjugate to itself which is impossible if~$K\leq 0$.
\end{proof}
We want to describe the geometry of the boundaries of metric balls
centered at $p$, for a metric which is analytic or satisfies $K\le
0$. Any $q\in\partial B(p,r)$ which is not a cut point is joined
to $p$ by a unique geodesic arc of length~$r$, along which $q$ is
not conjugate to~$p$, and the boundary is smooth (analytic)
near~$q$; also $q$ is not a self-intersection point of the image
$\mbox{\rm Exp}_p\big(\{\,{\bf v}\in T_pS\, ;\, \|{\bf
v}\|=r\}\big)$. Thus the only special points the boundary can have
are the points it shares with the cut locus. We see in
Figure~\ref{figure:ortogonales} that the boundary develops a
corner when it hits an endpoint of the cut locus graph, but its
topology does not change. For a short while after that moment, the
corner angle varies but otherwise the geometry of the boundary
remains unchanged.
We are going to see that, as $r$ increases, the geometry of the
boundary $\partial B(p,r)$ changes only when said boundary hits a
cut point which is either a conjugate point, a middle point of
multiplicity~$2$, or a vertex of higher multiplicity in the cut
locus graph. For the moment let us see what happens when the
boundary hits a middle point of multiplicity~$2$. Let
$\gamma_1,\gamma_2$ be the two minimizing geodesic arcs with
$\gamma_1(0)=\gamma_2(0)=p$, $\gamma_1 (1)=\gamma_2(1)=q$, and
$\gamma_1'(1)=-\gamma_2'(1)$. For $i=1,2$ let $C_i$ be a small
circular arc centered at $\bf 0$ in $T_pS$ and having $\gamma_i'(0)$
as midpoint. The exponential images of $C_1,C_2$ are ${\mathcal
C}^1$ curves $\Gamma_1,\Gamma_2$ meeting tangentially at~$q$.
If $\Gamma_1$ and $\Gamma_2$ curve away from each other toward
$\gamma_2$ and $\gamma_1$ respectively, then as $r$ increases the
boundaries $\partial B(p,r)$ evolve near $q$ as in
Figure~\ref{figure:ojales}.
\begin{figure}[h]
\includegraphics[scale=0.3]{ojales}
\caption{}
\label{figure:ojales}
\end{figure}
If $\Gamma_1$ and $\Gamma_2$ curve away from each other toward
$\gamma_1$ and $\gamma_2$ respectively, then as $r$ increases the
boundaries $\partial B(p,r)$ evolve near $q$ as in
Figure~\ref{figure:beso}.
\begin{figure}[h]
\includegraphics[scale=0.3]{beso}
\caption{}
\label{figure:beso}
\end{figure}
\begin{prop}\label{solo-beso}
If $K\leq 0$, then the only possible local geometry when the ball
boundary hits a middle point of multiplicity~$2$ is the one shown in
Figure~\ref{figure:beso}.
\end{prop}
\begin{proof}
In this case there are no conjugate points. The exponential map at
$p\,$ is a local diffeomorphism $T_pS\to S$ which pulls the metric
on $S$ back to a metric $\overline{g}$ on the tangent space. If
$(r,\theta )$ are orthonormal polar coordinates in
$T_pS\setminus\{{\bf 0}\}$, then $\overline{g}=dr^2+\lambda(r,\theta
)^2\, d\theta^2$ where $\lambda$ is a positive smooth function with
$\lim_{r\to 0}\lambda =0$, $\lim_{r\to 0}\lambda_r =1$, and
$-\lambda_{rr}/\lambda$ equal to the Gaussian curvature
of~$\overline{g}$. This yields $\lambda_{rr}\ge 0$ and $\lambda_r\ge
1$. The unit tangent vector to any Euclidean circle in $T_pS$
centered at $\bf 0$ is ${\bf t}=(1/\lambda)\partial_{\theta}$ and
one easily computes $\overline{g}\big(\,\nabla_{\bf t}{\bf t}\, ,\,
\partial_r\,\big) =-\lambda_r/\lambda <0$, hence said circle
curves strictly inward with respect to $\overline{g}$. Therefore the
boundary of any ball centered at $p\,$ in $S$ also curves strictly
inward at every non-corner point, and when it hits a middle point of
multiplicity~$2$ the only possible local geometry is the one shown
in Figure~\ref{figure:beso}, with the two colliding fronts having
finite non-zero curvature.
\end{proof}
The situation in Figure~\ref{figure:ojales} occurs in particular
when $q$ is conjugate to $p$ along both $\gamma_1$ and $\gamma_2$.
If $q$ is conjugate to $p$ along only one of these arcs, say
$\gamma_1$ to fix ideas, then $\Gamma_1$ curves towards $\gamma_2$
with infinite curvature at~$q$, while $\Gamma_2$ has finite
curvature at~$q$. The topology is then as in
Figure~\ref{figure:ojales} but there are several possibilities for
the geometry. One possibility is Figure~\ref{figure:ojales}. Another
possibility is the first image in Figure~\ref{figure:varios} (where
$\Gamma_2$ curves toward $\gamma_2$) with the shrinking hole now in
the shape of a crescent moon. Other possibilities (not depicted)
correspond to $\Gamma_2$ having zero curvature at~$q$.
\begin{figure}[h]
\includegraphics[scale=0.3]{varios}
\caption{}
\label{figure:varios}
\end{figure}
If $q$ is neither conjugate to $p$ along $\gamma_1$ nor along
$\gamma_2$, and the metric is real analytic, then
$\Gamma_1,\Gamma_2$ are real analytic arcs tangent at~$q$. If they
had a contact of infinite order at $q$, then they would coincide and
all of their points would be middle points. Thus
Lemma~\ref{coll-finite} implies that $\Gamma_1$ and $\Gamma_2$ have
a contact of finite order at~$q$. Therefore either $\Gamma_1$ stays
on one side of $\Gamma_2$, or these two arcs cross each other
(tangentially) at~$q$. If $\Gamma_1$ stays on one side of $\Gamma_2$
then the topology is as in Figures \ref{figure:ojales}
or~\ref{figure:beso}, but the geometry has more possibilities than
the ones shown in Figures \ref{figure:ojales} and~\ref{figure:beso}.
Some (not all) of the extra possibilities are shown in
Figure~\ref{figure:varios}. If $\Gamma_1$ crosses $\Gamma_2$
tangentially at~$q$ then the geometry is equal or very similar to
that in Figure~\ref{figure:uf}: the boundary containing $q$ has a
cusp at $q$, while the nearby boundaries contain a corner whose
branches make a nonzero angle.
\begin{figure}[h]
\includegraphics[scale=0.3]{uf}
\caption{}
\label{figure:uf}
\end{figure}
We briefly discuss now higher multiplicity vertices of the cut locus
graph. Myers shows in \cite{Myers1} and \cite{Myers2} that a vertex
of multiplicity $m$ is joined to $p$ by exactly $m$ minimizing
geodesic arcs. For example, Figure~\ref{figure:Y} describes how the
minimizing geodesic arcs issued from $p$ reach a $Y$-shaped part of
the cut locus; we see three geodesic arcs ending at the triple
point.
\begin{figure}[h]
\includegraphics[scale=0.6]{Y2}
\caption{}
\label{figure:Y}
\end{figure}
If two of the minimizing geodesics reaching the triple point make
flat angles when they meet (thereby making the triple point a middle
point as well) then the situations described in the second and third
images in Figure~\ref{figure:Y} are the only possible ones, because
middle points are isolated.
If the situation is as in the first or second image in
Figure~\ref{figure:Y} then the boundary evolves toward the triple
point as shown in Figure~\ref{figure:nablas}: we see a triangular
hole decreasing in size until it disappears.
\begin{figure}[h]
\includegraphics[scale=0.4]{nablas}
\caption{}
\label{figure:nablas}
\end{figure}
In the case represented by the third image in Figure~\ref{figure:Y}
the boundary evolves as shown in Figure~\ref{figure:T}: it has
$3-1=2$ corners before hitting the triple point, one cusp when
hitting said point, and a single corner afterwards.
\begin{figure}[h]
\includegraphics[scale=0.3]{nuevaT}
\caption{}
\label{figure:T}
\end{figure}
The last image in Figure~\ref{figure:Y} occurs when two minimizing
geodesics make a concave angle at the triple point (measured without
going through the other geodesic). In this case the boundary evolves
almost like in Figure~\ref{figure:T}, the only difference being that
it also has a corner when hitting the triple point and so no cusp is
created in this case.
\begin{definicion}
The {\em interior angle} at a corner point of $\partial B(p,r)$ is
the angle between the two boundary branches ending at that point,
measured through the interior of~$B(p,r)$.
\end{definicion}
We now list together all possibilities when the boundary $\partial
B(p,r)$ touches an interior point $q$ of an edge of the cut locus,
i.e. a point of multiplicity~$2$. We then have exactly two
minimizing geodesic arcs joining $p$ to $q$, both of length $r$. In
particular, this is a self-intersection point of the image
$\mbox{\rm Exp}_p\big(\{\,{\bf v}\in T_pS\, ;\, \|{\bf
v}\|=r\}\big)$ and out of the four branches of this image that reach
$q$ only two branches are part of the boundary. We split this case
into two subcases:
\begin{itemize}
\item If $q$ is not a middle point then the part of $B(p,r)$
inside a small neighborhood of $q$ is the union $B'\cup B''$ of two
pieces whose boundaries near $q$ are at least ${\mathcal C}^1$
(recall the phenomenon in Figure~\ref{figure:ortogonales}) and
non-tangent at $q$. It follows that in such a case $\partial B(p,r)$
has a corner at $q$ and the interior angle at this corner is a
concave angle $\alpha\in (\pi ,2\pi )$. By Remark~\ref{inf-curv}, if
$q$ is also a conjugate point then at least one of the two branches
of the boundary has infinite curvature at~$q$. \item If $q$ is a
middle point then the boundaries of the two pieces $B',B''$ are
tangent at $q$, and one of the three phenomena described in Figures
\ref{figure:ojales}, \ref{figure:beso}, \ref{figure:varios},
\ref{figure:uf} occurs at~$q$. One phenomenon (see
Figure~\ref{figure:ojales} and the first image in
Figure~\ref{figure:varios}) consists on the boundary losing a small
connected component with two corners; the interior angles at the
corners remain inside $(\pi ,2\pi )$ during this process but both
tend to $2\pi$ as the hole's size tends to~$0$. Another phenomenon
(see Figure~\ref{figure:beso} and the last two images in
figure~\ref{figure:varios}) is an increase in the connectivity of
the ball; the interior angles at the created corners are both in
$(\pi ,2\pi)$ except at the instant when they are created. The third
possible phenomenon (see Figure~\ref{figure:uf}) is a cusp point on
the boundary when it touches $q$ and a single corner before and
after that instant; the topology remains unchanged during this
process.
\end{itemize}
Next we describe in detail what happens when the boundary
$\partial B(p,r)$ touches a point $q$ of multiplicity $m\geq 3$ in
the cut locus. This happens only for a finite number of values of
the radius~$r$ when the metric is analytic or with $K\leq0$,
because there is only a finite number of multiple points in such
cases. The minimizing geodesics from $p$ to $q$ make up a family
$\mathcal G$ with $m$ elements, all with the same length. Two
cases are possible:
\begin{itemize}
\item The angles at $q$ between consecutive geodesics in the
family $\mathcal G$ are all convex angles. Then $\partial B(p,r)$
has a $m$-sided polygonal component which shrinks down to the point
$q$ and then disappears. See Figure~\ref{figure:nablas} for the
$m=3$ case. (This multiple point will be a middle point if two {\em
non-consecutive} geodesics in $\mathcal G$ make flat angles).
\item There is a consecutive pair of geodesics in the family
$\mathcal G$ making a flat angle (in which case $q$ will be a middle
point) or a concave angle at~$q$. Then all other consecutive pairs
must make convex angles at~$q$. In this case the boundary $\partial
B(p,r)$ either has an $m$-sided polygonal component shrinking to~$q$
(Figure~\ref{figure:nablas} shows this for $m=3$) or its topology
remains unchanged during the process: $m-1$ corners before hitting
$q$, one cusp or one corner at $q$ when hitting it, and a single
corner after hitting~$q$ (Figure~\ref{figure:T} shows this for
$m=3$).
\end{itemize}
\begin{obs}\label{corners}
A careful examination of the above study shows that the boundary
$\partial B(p,r)$ has corners, in each connected component of
positive length, forever after hitting the cut locus of~$p\,$.
The right-hand side of (\ref{ball-length}) is the obvious estimate
for the length of the exponential image of the $r$-circle on $T_pS$.
If $\partial B(p,r)$ has corners then said exponential image has
parts lying interior to $B(p,r)$. If the inequality $K\geq -k^2$ is
strict somewhere in $B(p,r)$ then the exponential image will have
length smaller that the right-hand side of~(\ref{ball-length}).
Therefore (\ref{ball-length}) can be an equality only if $B(p,r)$ is
disjoint from the cut locus of $p\,$ and $K\equiv -k^2$ inside
$B(p,r)$.
\end{obs}
Let $R$ be the radius in Remark~\ref{con-borde}. In $B(p,R)$ we
consider the set $\mathcal N$ which comprises all conjugate points
in the cut locus, all middle points of multiplicity~$2$ in the cut
locus, and all vertices of multiplicity $3$ or greater of the cut
locus. By the above discussion, if the metric is analytic or
satisfies $K\leq 0$ then the set $\mathcal N$ is finite inside each
ball $\overline{B}(p,r)$ with $r<R$, and as $r$ increases the
boundary $\partial B(p,r)$ gains (or loses) corners only by touching
this set. This implies finiteness of the number of corner points on
each ball boundary. Since the set $\mathcal N$ is finite in each
$\overline{B}(p,r)$ with $r<R$, the distances from the points in
$\mathcal N$ to $p\,$ can be arranged into an increasing sequence:
$r_1<r_2<r_3<\cdots$ which either is finite or converges to $R$ (the
latter can only occur if $\overline{B}(p,R)=S$). If $r<R$ is
different from these values then $\partial B(p,r)$ is a finite
disjoint union of simple closed curves, each having the corner
geometry just described, and the interior angle $\alpha$ at each
corner lies in the open interval $(\pi ,2\pi )$ and can thus be
written as $\alpha =\pi+2\beta$ for some $\beta\in
(0,\frac{\pi}{2})$. Also, for such $r$ the inclusion
$B(p,r)\hookrightarrow\overline{B}(p,r)$ is a homotopy equivalence
as claimed in Remark~\ref{open-vs-closed}.
\begin{lema}\label{finitos}
If the metric is real analytic or satisfies $K\leq 0$, then the
function $\ell (r)$ from Definition~\ref{ell} is continuous for all
$r\in [0,R)$ and smooth at $r\in [0,R)\setminus\{\,
r_1,r_2,r_3\dots\,\}$.
\end{lema}
\begin{proof}
In the case of a finite sequence $r_1,\dots ,r_s$, let $I$ be one of
the following intervals
\[ (0,r_1)\; ,\; (r_1,r_2)\; ,\; \cdots \; ,\; (r_{s-1},r_s)\; ,\;
(r_s,R)\; .\] In the case of an infinite sequence, let $I$ be
$(0,r_1)$ or any interval $(r_j,r_{j+1})$. In either case the number
of connected components and the corner geometry of the boundary
$\partial B(p,r)$ do not change while $r$ ranges over~$I$. Let $N_I$
be the number of maximal smooth segments in the boundary for $r\in
I$. These segments are the exponential images of $r\cdot
C_1(r),\dots ,r\cdot C_{N_I}(r)$ where $C_1(r),\dots ,C_{N_I}(r)$
are disjoint closed circular arcs in the tangential unit circle in
$T_pS$ (the rest of the tangential circle of radius $r$ is mapped by
$\mbox{\rm Exp}_p$ into the interior of the metric ball of
radius~$r$). The endpoints of the $C_i(r)$ are functions ${\bf
v}_j(r)$, $j=1,\dots ,2N_I$, with domain~$I$. For $r\in I$ the
number $\ell (r)$ is the integral of a smooth integrand over
$C_1(r)\cup\cdots\cup C_{N_I}(r)$, hence the smoothness of $\ell
(r)$ is equivalent to the smoothness of the endpoints ${\bf v}_j(r)$
of those circular arcs. Since $r\cdot C_1(r)\cup\cdots\cup r\cdot
C_{N_I}(r)$ is disjoint with the tangential first conjugate locus,
the functions ${\bf v}_j(r)$ are smooth if and only if the boundary
corner points $\mbox{\rm Exp}_p\big(r\cdot{\bf v}_j(r)\big)$ depend
smoothly on~$r$, which we next prove to be the case.
For each $C_i(r)$ let $C'_i(r)$ be an open circular arc containing
$C_i(r)$ such that $r\cdot C'_i(r)$ is still disjoint with the
tangential first conjugate locus. Then $b_i(r)=\mbox{\rm Exp}_p\big(
r\cdot C'_i(r)\big)$ is a smooth embedded arc in $S$ such that the
boundary corner points corresponding to $C_i(r)$ are the two
intersection points defined by $b_i(r)\cap\mbox{\rm Cut}_p\,$. Since
no boundary corner point is in $\mathcal N$, the cut locus is a
smooth embedded curve near them and so $b_i(r)\cap\mbox{\rm Cut}_p$
will depend smoothly on $r$ if $b_i(r)$ meets $\mbox{\rm Cut}_p$
transversally at these two corner points. The formula for first
variation of arc length implies that if $q$ is any smooth point of
the cut locus then said locus bisects the directions of the two
minimizing geodesic segments joining $p$ to $q$. The boundary
$\partial B(p,r)$ that goes through $q$ has two corner directions at
$q$ which are the orthogonal directions to those two minimizing
geodesic segments, hence the boundary has a direction tangent to the
cut locus at $q$ if and only if $q$ is a middle point (of
multiplicity~$2$). Since for $r\in I$ no corner point of $\partial
B(p,r)$ is a middle point, the smooth segment $b_i(r)$ meets the cut
locus transversally. This implies that the boundary corner points
are smooth functions of $r$ for $r\in I$ and, as explained above,
that $\ell (r)$ is smooth in~$I$.
The continuity of $\ell (r)$ at the special values $r_1,r_2,\dots$
follows by examination of Figures \ref{figure:ojales} to
\ref{figure:T} and the analysis that we made for each of them. For
example, in Figure~\ref{figure:ojales} we see a contribution to
$\ell (r)$ which has negative derivative for $r<r_k$, has derivative
equal to $-\infty$ at $r=r_k$, and equals $0$ for $r\geq r_k$.
Moreover this defines a H\"older continuous function of $r$ near
$r=r_k$ because we proved that the two fronts whose motion gives
rise to Figure~\ref{figure:ojales} either have different curvatures
at the special point (one of them infinite) or have only a finite
order contact at such point. Similar arguments apply to the other
Figures.
\end{proof}
We shall now give a formula for $\ell'(r)$. Let $k_g$ denote the
geodesic curvature of the boundary, taken with positive sign where
the boundary is curving towards the metric ball and with negative
sign where the boundary is curving away from the metric ball. The
integral $\int_{\partial B(p,r)}k_g\, ds$ is the contribution to the
derivative $\ell '(r)$ by the smooth segments of the boundary. To
determine the contribution from the corners it is sufficient to
consider the case of two straight segments lying on the Euclidean
plane and making an angle $\alpha =\pi+2\beta$. We see in
Figure~\ref{figure:a} that this corner contributes $\,
-2\tan\beta\,$ to $\ell'(r)$. The formula for the derivative of
$\ell (r)$ is:
\begin{equation}\label{ele-prima}
\ell'(r)\; =\; \sum_i -2\tan\beta_i +\int_{\partial B(p,r)}k_g\,
ds\quad ,\quad \mbox{\rm for }\; r\notin\{\,
0,r_1,r_2,r_3\dots\,\}\; ,
\end{equation}
where the index $i$ runs over the corners of the boundary
$\partial B(p,r)$ and $\alpha_i=\pi+2\beta_i$ are the respective
interior angles at those corners.
\begin{figure}[h]
\includegraphics[scale=0.5]{a}
\caption{}
\label{figure:a}
\end{figure}
\begin{definicion}\label{chi}
A surface is of \emph{finite type} if its fundamental group is
finitely generated.
Let $S$ be a connected surface of finite type, either non-compact or
compact with non-empty boundary; for such surfaces the
\emph{Euler-Poincar\'e characteristic} is the number $\chi
(S)=1-\mbox{\rm rank } H_1(S)$. Assume further that $\partial S$ is
either empty or a disjoint union of simple closed curves; then $\chi
(S)$ coincides with $2-2g-n$, where $g$ is the genus of $S$ and $n$
is the sum of the number of connected components of $\partial S$
plus the number of ends of $S$ that are homeomorphic with~$S^1\times
[0,\infty )$ (with the curve corresponding to $S^1\times\{ 0\}$
lying interior to~$S$).
\end{definicion}
\vspace{3mm}
\begin{proof}[Proof of Theorem~\ref{t:balls}.]
Define the integer-valued function $\chi (r)$ as follows: \[ \chi
(r)\; =\; \chi\big(\, B(p,r)\,\big)\qquad \mbox{\rm for}\qquad 0<
r< r_0+\frac{1}{k}\; .\] We have three reasons for assuming that
$\overline{B}\big(\, p\, ,\, r_0+\frac{c}{k}\,\big)$ meets the
conditions of Remark~\ref{con-borde}. The first reason is that
formula~(\ref{ele-prima}) can then be used for
$0<r<r_0+\frac{c}{k}$. The second reason is that, once the balls
$B(p,r)$ have non-empty boundary for that range of values of $r$, we
can apply Definition~\ref{chi} to these balls. The third is to
ensure that the group $\pi_1\big(\, B(p,r)\, ,\, p\,\big)$ is free
in $n(r)$ generators; then $H_1\big(\, B(p,r)\,\big)$, being the
abelianization of $\pi_1\big(\, B(p,r)\, ,\, p\,\big)$, is isomorphic
with ${\mathbb Z}^{n(r)}$ and we have:
\begin{equation}\label{ji}
n(r)\; =\; \mbox{\rm rank } H_1\big(\, B(p,r)\,\big)\; =\; 1-\chi
(r)\, .
\end{equation}
In view of this, we seek an estimate for $1-\chi (r)$.
For $r\notin\{\, 0,r_1,r_2,r_3\dots\,\}$ we can use the Gauss-Bonnet
formula:
\[ 2\pi\,\chi (r) \; =\; \int_{B(p,r)}K\, d\,\hbox{area}
+\sum_i (\pi -\alpha_i)+\int_{\partial B(p,r)}k_g\, ds\, ,\] which
we rewrite as follows: \begin{equation}\label{Gau} -\sum_i
2\beta_i+\int_{\partial B(p,r)} k_g\, ds + \int_{B(p,r)}K\,
d\,\hbox{area} \; =\; 2\pi\,\chi (r)\, .\end{equation} From
$\beta_i\in\big( 0,\frac{\pi}{2}\big)$ we infer
$\beta_i<\tan\beta_i$, which together with formulas
(\ref{ele-prima}) and (\ref{Gau}) leads to
\begin{equation}\label{ji-deriv}
\ell'(r)+\int_{B(p,r)}K\, d\,\hbox{area}\,\leq\; 2\pi\,\chi (r)
\quad ,\quad \mbox{\rm for }\; r\notin\{\, 0,r_1,r_2,r_3\dots\,\}\,
,
\end{equation} the inequality being strict whenever $\partial
B(p,r)$ has at least one corner. By Remark~\ref{corners}, this is
the case whenever $B(p,r)$ contains a cut point of~$p\,$.
Define now the function:
\[ a(r):\, =\, \mbox{\rm area}\,\big(\, B(p,r)\,\big)\; ,\]
which satisfies $a'(r)=\ell (r)$ for all $r\in \big[\, 0\, ,\,
r_0+\frac{c}{k}\,\big)$, thus:
\[ a(r)\in{\mathcal C}^1\big[\, 0\, ,\,
r_0+\mbox{$\frac{c}{k}$}\,\big)
\qquad\mbox{\rm and}\qquad a(r)\in{\mathcal C}^\infty\left(\,\big[\,
0\, ,\, r_0+\mbox{$\frac{c}{k}$}\,\big)\setminus\{\,
r_1,r_2,r_3\dots\,\}\,\right)\, .\]
Introduce now the hypothesis $K\geq -k^2$. The first consequence is
that formula~\ref{ji-deriv} yields the following differential
inequality:
\begin{equation}\label{fundamental}
a''(r)-k^2a(r)\; \leq 2\pi\,\chi (r) \quad ,\quad \mbox{\rm for }\;
r\notin\{\, r_1,r_2,r_3\dots\,\}\; ,
\end{equation}
which is strict whenever $B(r,p)$ contains a cut point of $p\,$. The
second consequence is the well-known bound (\ref{ball-length}) for
boundary length in terms of the corresponding length in a hyperbolic
plane with curvature~$-k^2$. The third consequence is the bound for
area:
\begin{equation}\label{ball-area}
a (r) \;\leq\; \frac{2\pi}{k^2}\,\big(\cosh (kr)-1\big)\, ,
\end{equation}
obtained by integrating (\ref{ball-length}).
\begin{lema}\label{comparison}
Let $u(r),\overline{u}(r)$ be functions on an interval $r_0\leq r<
R$, smooth in the complement of a discrete set $Z\subset [r_0,R)$,
which satisfy:
\[ \left\{\begin{array}{ll}u(r) \in {\mathcal C}^1[r_0,R) & \\
u''(r)-k^2 u(r) = f(r) & r\notin Z\end{array}\right. \qquad\quad
\left\{\begin{array}{ll}\overline{u}(r) \in {\mathcal C}^1[r_0,R) & \\
\overline{u}''(r)-k^2 \overline{u}(r) = \overline{f}(r) & r\notin Z
\end{array}\right. \] If the following inequalities hold
\begin{eqnarray}\label{ineq-1} f(r) &\leq & \overline{f}(r)
\;\;\mbox{\rm for all }\; r\notin F\, ,\\
\label{ineq-2} u(r_0) &\leq & \overline{u}(r_0)\, ,\\
\label{ineq-3} u'(r_0)-k\, u(r_0) &\leq &
\overline{u}'(r_0)-k\,\overline{u}(r_0)\, ,
\end{eqnarray}
then $\; u\leq\overline{u}\;$ and $\; u'-k
u\leq\overline{u}'-k\overline{u}\;$ (hence also
$u'\leq\overline{u}'$) everywhere on~$[r_0,R)$.
\end{lema}
\begin{proof}
Make the ansatz $u(r)\equiv e^{kr}c(r)$. Then the conditions imposed
on $u(r)$ are equivalent to the following:
\[ c(r)\in{\mathcal C}^1[r_0,R)\qquad\mbox{\rm and}\qquad
r\notin Z\Longrightarrow\frac{d}{dr}\big(\, e^{2kr}c'(r)\,\big)\;
=\; e^{kr}f(r) \, .\] The function $c'(r)\equiv e^{-kr}\big(\,
u'(r)-ku(r)\,\big)$ is continuous on $[r_0,R)$ and smooth for
$r\notin Z$. Since $Z$ is discrete, it follows that $c'(r)$ is given
at every $r\in [r_0,R)$ by the formula:
\[ c'(r)\; =\; e^{-2kr}\,\left( e^{2kr_0}c'(r_0)+\int_{r_0}^r
e^{kt}f(t)\, dt\right) .\] We have the analogous formula for the
derivative of the function $\overline{c}(r)$ given by
$\overline{u}(r)\equiv e^{kr}\,\overline{c}(r)$. Then, in view of
(\ref{ineq-1}), the inequality $c'(r)\leq\overline{c}'(r)$ holds on
all of $[r_0,R)$ if it holds at $r=r_0$, which is the case thanks
to~(\ref{ineq-3}). Now $c(r_0)\leq\overline{c}(r_0)$ is equivalent
to~(\ref{ineq-2}), and $c(r)\leq\overline{c}(r)$ follows by
integration.
The claimed inequalities follow by multiplying
$c(r)\leq\overline{c}(r)$ and $c'(r)\leq\overline{c}'(r)$
by~$e^{kr}$.
\end{proof}
\begin{obs}
The above proof gives $u'(r)<\overline{u}'(r)$ if we have
$f<\overline{f}$ in some non-trivial interval contained in
$[r_0,r]$.
\end{obs}
Consider the interval $I_0=\big[\, r_0\, ,\, r_0+\frac{c}{k}\,\big)$
and the constant:
\[ \chi_0:=\max_{r\in I_0}\chi (r)=1-\min_{r\in I_0}n(r)\, .\]
There is an $r'\in\big(\, r_0\, ,\, r_0+\frac{c}{k}\,\big)$ such
that $\chi_0$ equals the Euler characteristic of both $B(p,r')$ and
$\overline{B}(p,r')$. Inequality~(\ref{top-estimate}) is equivalent
to the inequality:
\begin{equation}\label{conclusion}
1-\chi_0\; <\: \frac{1}{\sinh c}\left( \sinh (kr_0+c)-
\frac{k\,\ell\big(\, r_0+\frac{c}{k}\,\big)}{2\pi}\right) .
\end{equation} Another property that the constant $\chi_0$ has is
that if we define the following two functions on $I_0$:
\begin{eqnarray*}
f(r) &=& a''(r)-k^2 a(r) \, ,\\ \overline{a}(r) &=& \left(\,
a(r_0)+\frac{2\pi\chi_0}{k^2}\,\right)\cosh\big(\, k(r-r_0)\,\big)
+\frac{1}{k}\,\ell (r_0)\sinh\big(\,
k(r-r_0)\,\big)-\frac{2\pi\chi_0}{k^2}\, ,
\end{eqnarray*}
then $\overline{a}$ is everywhere smooth, and by (\ref{fundamental})
it satisfies:
\[
\overline{a}''(r)-k^2\overline{a}(r)\; =\; 2\pi\chi_0\;\geq\;
f(r)\qquad\mbox{for all }\; r\in I_0\setminus\{ r_1,r_2,\dots \}\, ,
\] the inequality being strict if $B(p,r)$ contains some cut point
of~$p\,$. We have adjusted $\overline{a}$ to satisfy
$\overline{a}(r_0)=a(r_0)\,$ and $\,\overline{a}'(r_0)=a'(r_0)$;
then Lemma~\ref{comparison} tells us that:
\begin{equation}\label{desig-1} \ell (r)\; =\; a'(r)\;\leq\;
\overline{a}'(r)\qquad \mbox{\rm for all }\; r\in I_0\,
,\end{equation} with strict inequality unless $B(p,r)$ contains no
cut point of~$p\,$.
Computing $\overline{a}'(r)$ from the explicit formula that defines
$\overline{a}$, and using (\ref{ball-length}) and (\ref{ball-area}),
one finds:
\begin{equation}\label{desig-2}
\overline{a}'(r)\;\leq\;\frac{2\pi}{k}\left[\,\vphantom{\frac{a}{a}}\big(\,
\cosh (kr_0)-1+\chi_0\,\big)\sinh\big(\, k(r-r_0)\,\big)+\sinh
(kr_0)\cosh\big(\, k(r-r_0)\,\big)\, \right]\;\;\mbox{\rm for all
}r\in I_0\, .\end{equation}
Combining (\ref{desig-1}) and (\ref{desig-2}), we get for all $r\in
I_0$:
\[ 1-\chi_0 \; \leq\; \frac{1}{\sinh \big(\, k(r-r_0)\,\big)}\left(
\sinh (kr) -\frac{k\,\ell (r)}{2\pi}\right) .\] Taking the limit as
$r\to r_0+\frac{c}{k}$, and using the following fact:
\[ \lim_{r\to
r_0+(c/k)}\ell (r)\;\geq\; \ell\big(\, r_0+\frac{c}{k}\,\big)\, ,
\]
we deduce (\ref{conclusion}).
Suppose (\ref{conclusion}) is an equality. Since $\chi_0=\chi\big(\,
B(p,r')\,\big)$, the ball $B(p,r')$ must then be disjoint from the
cut locus of $p\,$, hence diffeomorphic to a disk, and so $n(r')=0$.
But then (\ref{ball-length}) is an equality at $r= r_0+\frac{c}{k}$,
and by Remark~\ref{corners} the ball $B\big(\, p\, ,\,
r_0+\frac{c}{k}\,\big)$ is disjoint from the cut locus of $p\,$ and
has $K\equiv -k^2$.
\end{proof}
\section{Background on Gromov spaces.}
\spb
In our study of hyperbolic Gromov spaces we use the notations of
\cite{GH}. We give now the basic facts about these spaces. We refer
to \cite{GH} for more background and further results.
\spb
\begin{definicion}\label{hype}
Let us fix a point $w$ in a metric space $(X,d)$. Define the
\emph{Gromov product} of $x,y\in X$ with respect to the point $w$ as
$$
(x|y)_w:=\frac12\,\big( d(x,w)+d(y,w)-d(x,y) \big)\ge 0\,.
$$
We say that the metric space $(X,d)$ is $\d$-\emph{hyperbolic}
$(\d\ge 0)$ if
$$
(x|z)_w\ge\min\big\{ (x|y)_w, (y|z)_w \big\}-\d\,,
$$
for every $x,y,z,w\in X$. When we do not want to specify the value
of $\delta$, we say that $X$ is \emph{Gromov hyperbolic}.
\end{definicion}
\spb
It is convenient to remark that this definition of hyperbolicity is
not universally accepted, since sometimes the word `hyperbolic'
refers to negative curvature or to the existence of a Green
function. However, in this paper we only use the word {\it
hyperbolic} in the sense of Definition~\ref{hype}.
\spb
\noindent{\it Examples:}
\begin{list}{}{}
\item[(1)] Every bounded metric space $X$ is $(\diam X)$-hyperbolic.
\item[(2)] Every complete simply connected Riemannian manifold with
sectional curvature bounded from above by $-k^2$, with $k>0$, is
hyperbolic.
\item[(3)] Every tree with edges of arbitrary length is $0$-hyperbolic.
\end{list}
We refer the reader to \cite{BHK}, \cite{GH} and \cite{CDP} for
further examples.
\begin{definicion}
A metric space $X$ is a \emph{ geodesic metric space } if any two
points $x,y\in X$ can be joined by a path whose length equals
$d(x,y)$.
\end{definicion}
In general metric spaces, the \emph{length} $L(\g )$ of a path $\g
:[a,b]\to X$ is defined as $\,\sup\sum_{i=1}^n
d(\g(t_{i-1}),\g(t_{i})),$ taken over all partitions $\,
a=t_0<t_1<\cdots <t_n=b$.
\begin{definicion}
\label{def:Rips} In a general metric space $X$ a \emph{\/metric
geodesic\/} is a path $\g (t)$ such that
$d(\g(t),\g(s))=L(\g|_{[t,s]})=|t-s|$ for every $s,t\in [a,b]$, i.e.
$\g$ is minimizing and parametrized by arclength. We relax this
condition for a closed path: it only has to minimize length in its
free homotopy class.
If $T$ is a \emph{metric geodesic triangle} (i.e. its sides
$J_1,J_2,J_3,$ are metric geodesics) we say that $T$ is
$\d$-\emph{thin} if for every $x\in J_i$ we have that
$d(x,\cup_{j\neq i}J_{j})\le \d$. The space $X$ is $\d$-\emph{thin}
(or satisfies the \emph{Rips condition} with constant $\d$) if every
geodesic triangle in $X$ is $\d$-thin.
\end{definicion}
A basic result is that hyperbolicity is equivalent to the Rips
condition:
\begin{teo}
\emph{(\cite[p. 41]{GH})} Let us consider a geodesic metric space
$X.$
$(1)$ If $X$ is $\d$-hyperbolic, then it is $4\d$-thin.
$(2)$ If $X$ is $\d$-thin, then it is $4\d$-hyperbolic.
\end{teo}
From the next Section onwards, all spaces will be $2$-dimensional
Riemannian manifolds (with or without boundary) and length will be
defined as the obvious integral. Given such a surface $S$, its
distance function $d_S$ is defined by minimizing length of paths in
$S$. This turns $S$ into a geodesic metric space.
\begin{definicion}
\label{d:innermetric} For a sub-surface $X\subset S$ we have two
choices:
\begin{list}{}{}
\item[--] The \emph{extrinsic distance}, which is just $d_S$
acting only on pairs $(x,y)\in X\times X$. \item[--] The
\emph{intrinsic distance} $d_S|_X$, defined by minimizing
$d_S$-length of paths contained in~$X$. When there is no risk of
confusion we shall denote it~$d_X$.
\end{list} Likewise we have the \emph{extrinsic diameter}
$\,\diam_S(X)$ and the \emph{intrinsic diameter} $\,\diam_X(X)$.
\end{definicion}
Obviously $d_S\le d_X$ and $\,\diam_S(X)\le\diam_X(X)$. Notice also
that $X$ is always a geodesic metric space with the intrinsic
distance, not always with the extrinsic one.
\spb
Next we introduce a useful notion and use it to state
Theorem~\ref{t:treedecomp}, which will be important for the proof of
Theorem~\ref{t:clasef} below and was also used in the proofs of two
results, Theorems \ref{t:clases} and \ref{t:rsn}, which are quotes
from the previous work~\cite{PRT3}.
\begin{definicion}
\label{d:treedecomp} Let $(X,d)$ be a metric space, and let
$X=\cup_nX_n$ where $\{X_n\}_n$ is a family of connected geodesic
metric spaces such that $\eta_{nm}:=X_n\cap X_m$ are compact sets.
Further, assume that for any $n\neq m$ with
$\eta_{nm}\neq\varnothing$ the set $X\setminus\eta_{nm}$ is not
connected, and that the connected components of
$X\setminus\eta_{nm}$ containing $X_n\setminus\eta_{nm}$ are all
different from those containing $X_m\setminus\eta_{nm}$. We say that
$\{X_n\}_n$ is a $k$-\emph{tree decomposition} of $X$ if for each
$n$ we have $\sum_{m}\diam_{X_n} (\eta_{nm})\le k$.
\end{definicion}
If we define a graph with one vertex $v_n$ for each piece $X_n$, and
one edge $e_{nm}$ joining $v_n$ to $v_m$ if
$\eta_{nm}\neq\varnothing$, we obtain a tree. Hence the name.
\begin{teo}
\label{t:treedecomp} \emph{(Compare \cite[Theorem 2.9]{PRT1})} Let
us consider a metric space $X$ and a family of geodesic metric
spaces $\{X_n\}_n \subseteq X$ which is a $k$-tree
decomposition of $X$. Then $X$ is $\d$-hyperbolic if and only if
there exists a constant $c$ such that $X_n$ is $c$-hyperbolic for
every $n$. Furthermore, $\d$ (respectively $c$) is a universal
constant which only depends on $k$ and $c$ (respectively
$k$ and $\d$).
\end{teo}
\bpb
\section{Definitions and previous results on Riemann surfaces.}
\spb
In the following, Gaussian curvature is the constant $-1$. In this
Section we collect some definitions and facts concerning Riemann
surfaces which will be referred to afterwards.
\spb
An \emph{open non-exceptional} Riemann surface $S$ is the following
two things:
\begin{enumerate}
\item Conformally, it is a Riemann
surface whose universal covering space is the unit disk
$\DD=\{z\in\CC:\; |z|<1\}$. \item As a Riemannian manifold, it is
endowed with its own Poincar\'e metric, i.e. the metric obtained by
projecting the Poincar\'e metric of the unit disk $ds =2
|dz|/(1-|z|^2)$ down to $S$ by the covering map.
\end{enumerate}
\begin{obs}
{\rm (1) Note that, with this definition, every compact
non-exceptional Riemann surface without border is open.
(2) There are infinitely many metrics in the conformal class of $S$
with constant curvature $-1$, but the Poincar\'e metric is the only
complete one. In fact, a surface with a Riemann metric satisfying
$K\equiv -1$ has the unit disk as universal cover if and only if the
Riemann metric is complete.
(3) The only Riemann surfaces which are left out are the sphere, the
plane, the punctured plane and the tori. It is easy to study the
hyperbolicity of these particular cases.}
\end{obs}
\spb
A \emph{bordered non-exceptional Riemann surface} is a connected
$2$-dimensional Riemannian manifold $S$ with boundary, subject to
the following restrictions:
\begin{enumerate}
\item It is the complement $S=R\setminus U$ of an open set $U$ in an
open non-exceptional Riemann surface $R$, and the Riemann metric on
$S$ is the one induced from the Poincar\'e metric of~$R$.
\item Mild border regularity conditions: the border of $S$ is locally
Lipschitz and any ball in $R$ intersects at most a finite number of
connected components of $U\!$.
\end{enumerate}
We say that $R$ contains $S$ isometrically. Since $S$ is a closed
subset of $R$, it is geodesically complete.
\begin{obs}{\rm If instead
of removing an open set we delete a closed set $E$ from an open
non-exceptional Riemann surface $R$, then we consider $R\setminus E$
also as an open non-exceptional Riemann surface, with its own
Poincar\'e metric which has also constant curvature $-1$ but is
longer than the (incomplete) Riemannian metric induced from~$R$.}
\end{obs}
Not every $2$-dimensional Riemannian manifold $S$ with $K\equiv -1$
embeds isometrically into an open non-exceptional Riemann surface.
For such isometric embedding to exist, the following necessary
condition must be satisfied: if $\widetilde{S}$ is the universal
cover of $S$, then any Riemann metric $g$ on $S$ with $K\equiv -1$
induces a local isometry $\Phi :\widetilde{S}\to{\mathbb D}$ that is
unique up to isometries of $\mathbb D$, and if $g$ is induced by an
embedding into an open non-exceptional Riemann surface then $\Phi$
must be injective. Let $S$ be an abstract closed disk, and take any
non-injective immersion $f:S\to{\mathbb D}$; then $f$ pulls the
Poincar\'e metric back to a metric $g$ on $S$ that has $K\equiv -1$;
but $(S,g)$ does not embed isometrically into any open
non-exceptional Riemann surface, because for this $g$ one has $\Phi
=f$. On the other hand $\,\inte S$ is an open non-exceptional
Riemann surface (with the conformal structure defined by $g$) and
the corresponding Poincar\'e metric is isometric with~$\mathbb D$,
quite different from the incomplete metric~$g|_{\inte S}$.
\begin{obs}
In this paper we only consider bordered non-exceptional Riemann
surfaces whose boundary components are (simple) closed curves.
\end{obs}
\begin{lema}\label{embedding}
If a doubly connected $2$-dimensional Riemannian manifold $S$ embeds
isometrically into some open non-exceptional Riemann surface, then
it embeds isometrically into either an annulus, or a cusp, or the
unit disk.
\end{lema}
\begin{proof}
Take any open non-exceptional Riemann surface $R$ containing $S$
isometrically. Consider the fundamental group $\pi_1(R)$ and the
subgroup ${\mathcal G}\subseteq\pi_1(R)$ defined by loops contained
in $S$. The group $\mathcal G$ is either trivial or infinite cyclic.
Consider also the covering map ${\mathbb D}\to R$.
If $\mathcal G$ is trivial then $S$ has a lift $S'\subset{\mathbb
D}$ which projects homeomorphically to $S$ under the covering map.
This $S'$ is isometric with $S$ and thus provides an isometric
embedding of $S$ into the unit disk.
If $\mathcal G$ is an infinite cyclic group $\langle\mu\rangle$,
then $S$ lifts to some region $S'\subset{\mathbb D}$ and the
restriction $S'\to S$ of the covering projection is equivalent to
the quotient map $S'\to S'/\langle\phi\rangle$, where $\phi$ is the
isometric action of $\mu\in\pi_1(R)$ on $\mathbb D$. The composition
$S\approx S'/\langle\phi\rangle\hookrightarrow{\mathbb
D}/\langle\phi\rangle$ is an isometric embedding.
The isometry $\phi$ cannot be elliptic, as it is induced by
$\mu\in\pi_1(R)$. If $\phi$ is a hyperbolic isometry, then ${\mathbb
D}/\langle\phi\rangle$ is an annulus. If $\phi$ is parabolic, then
${\mathbb D}/\langle\phi\rangle$ is a cusp.
\end{proof}
Lemma~\ref{embedding} enables us to classify, as Riemannian
manifolds, the bordered non-exceptional Riemann surfaces
homeomorphic with $S^1\times [0,\infty )$.
If $R$ is an annulus and $\g\subset R$ is an essential simple
closed curve, then the closure of each connected component of
$R\setminus\g$ is a doubly connected non-exceptional bordered
Riemann surface. Such bordered surfaces are called
\emph{generalized funnels,} as well as any bordered
non-exceptional Riemann surface isometric to any of these. A
\emph{funnel} is a generalized funnel whose boundary curve is a
geodesic.
If $R$ is a complete cusp and $\g\subset R$ is an essential simple
closed curve, then the closures $S_1,S_2$ of the connected
components of $R\setminus\g$ behave as follows:
\begin{enumerate}
\item On $S_1$ the curves freely homotopic to $\g$
become shorter as they run away from $\g$. We call $S_1$
\emph{narrow cusp end,} as well as any surface isometric with it. We
associate with $S_1$ an ideal point $q$ (the \emph{puncture}) such
that the conformal structure of $S_1$ extends to $S_1\cup\{ q\}$.
\item On $S_2$ the curves freely homotopic to $\g$ get
longer as they run away from $\g$. We call $S_2$, and any isometric
surface, \emph{wide cusp end.}
\end{enumerate}
A Jordan curve $\g\subset{\mathbb D}$ bounds two closed subsets in
the unit disk: a simply connected one and a doubly connected one.
We call the latter a \emph{disk end,} as well as any bordered
non-exceptional Riemann surface isometric to it.
Lemma~\ref{embedding} implies that if a bordered non-exceptional
Riemann surface $S$ is homeomorphic with $S^1\times [0,\infty )$
then it is isometric with one of the following: a generalized
funnel, a narrow cusp end, a wide cusp end, or a disk end (note that
$S$ is geodesically complete by definition).
\spb
Given a simple closed geodesic $\g$, we call the domain $\{p\in
S:\,d_S(p,\g)<d\}$ the \emph{collar of width $d$ about} $\g$ if it
is a tubular neighborhood of $\g$ with fibres the geodesics
orthogonal to $\g$. The Collar Lemma \cite{R}, \cite[Chapter 4]{Bu}
says that there exists a collar about $\g$ of width $d_0$, where
$\cosh d_0 = \coth (L_S(\gamma)/2)$ (the shorter the curve, the
thicker the collar). We shall use it in the proof of
Theorem~\ref{t:infinite}.
\begin{obs}\label{disjuntas}
Let $S$ be a non-exceptional Riemann surface, open or with compact
border. If $g_1$ is a closed curve neither freely homotopic to a
point, nor to a puncture, then the free homotopy class of $g_1$
contains a unique closed geodesic $\g_1$ which in fact minimizes
length in said class. If $g_1$ is simple then so is the geodesic
$\g_1$. If $g_1 ,g_2$ are simple, disjoint, and not freely
homotopic, then so are the corresponding geodesics $\g_1$
and~$\g_2$. In fact, if $L(g_1),L(g_2)\le l$ and $\cosh d_0=\coth
(l/2)$ then the collars of width $d_0$ about $\g_1$ and $\g_2$ are
disjoint (see \cite[Chapter 4]{Bu}).
\end{obs}
We next show that wide cusp ends and disk ends happen very rarely.
\begin{lema}\label{wide}
Let $S$ be a non-exceptional Riemann surface, open or bordered. If
$S$ has a wide cusp end, then it can only embed isometrically into a
cusp. If $S$ has a disk end, then it can only embed isometrically
into the unit disk.
\end{lema}
\begin{proof}
Suppose $R$ is an open non-exceptional Riemann surface containing
$S$ isometrically. Then $R$ shares with $S$ an end $S_0$ which is
either a wide cusp end or a disk end. The essential simple closed
curves in $S_0$ do not give rise to a non-constant geodesic in $R$,
because then $S_0$ would be part of a funnel end. By
Remark~\ref{disjuntas}, these curves must be freely homotopic in $R$
either to a puncture, which forces $R$ to be a cusp, or to a point,
which forces $R$ to be the unit disk.
\end{proof}
We call \emph{compact annulus} any non-exceptional bordered
Riemann surface homeomorphic with $S^1\times [0,1]$. Applying
Lemma~\ref{embedding} to these surfaces we obtain the following
result, which will be essential in the proof of
Theorem~\ref{t:clasef}.
\begin{prop}\label{RS}
Let $S$ be any bordered non-exceptional Riemann surface. There is a
canonical choice $R_S$ of an open non-exceptional Riemann surface
with the following properties: (1) $S$ embeds isometrically into
$R_S$, (2) $R_S$ has the same genus as $S$, and (3) $\chi
(R_S)\geq\chi (S)$.
\end{prop}
\begin{proof}
Let $\g$ be any connected component of $\partial S$ and take a
compact annulus $A\subset S$ bounded by $\g$ and another simple
closed curve~$\eta\subset S$. When one applies Lemma~\ref{embedding}
to $A$, there are five possibilities:
\begin{enumerate}
\item We find an isometric embedding $f:A\to{\mathbb D}$ and $f(\g
)$ lies in the interior of $f(\eta )$. In this case we can glue the
bounded component of ${\mathbb D}\setminus f(\eta )$ to $S$ by
overlapping $A\setminus\eta$ with itself, thereby producing a larger
Riemannian surface that has $S$ isometrically embedded inside it and
has one less boundary component than~$S$.
\item We find an isometric embedding $f:A\to{\mathbb D}$ and $f(\eta
)$ lies in the interior of $f(\g )$. Then we glue, with overlapping,
the doubly connected component of ${\mathbb D}\setminus f(\eta )$ to $S$
and produce a surface (open or bordered) which contains $S$
isometrically and has a disk end. By Lemma~\ref{wide} the latter
surface, and hence also $S$, embeds isometrically into the unit
disk. \item $A$ embeds isometrically into a cusp $R_0$, so that $\g$
is the boundary of the narrow end of $R_0\setminus\inte S$ and
$\eta$ is the boundary of the wide end. Then we can glue, with
overlapping, the narrow end of $R_0$ bounded by $\eta$ to $S$; this
produces a surface that contains $S$ isometrically and has the
boundary curve $\g$ replaced with a puncture. \item $A$ embeds
isometrically into a cusp $R_0$, so that $\g$ is the boundary of the
wide end of $R_0\setminus\inte S$ and $\eta$ is the boundary of the
narrow end. Then we glue, with overlapping, the wide end of $R_0$
bounded by $\eta$ to $S$; we thus obtain a surface that contains $S$
isometrically and has a wide cusp end. This and Lemma~\ref{wide}
imply that $S$ embeds isometrically into a cusp, so that $\g$ is an
essential curve in that cusp. This forces $\pi_1(S)$ to be
non-trivial. \item $A$ embeds isometrically into an annulus. Then we
find a generalized funnel which we can glue to $S$ with overlapping,
and thus produce a surface that contains $S$ isometrically and has
the boundary curve $\g$ replaced with a funnel end.
\end{enumerate}
If one boundary curve of $S$ is in case (2) we take $R_S={\mathbb
D}$. If one boundary curve of $S$ is in case (4) we make $R_S$ equal
to the cusp that contains $S$ isometrically; in this case $\chi
(R_S)=0\geq\chi (S)$ because $\pi_1(S)$ is non-trivial.
Suppose now that cases (2) and (4) do not occur for any connected
component of $\partial S$. The boundary $\p S$ consists of a
sequence $\g_1,\g_2,\dots$ of simple closed curves, and we can
choose the corresponding compact annuli $A_1,A_2,\dots$ pairwise
disjoint. Then we do, simultaneously for all $\g_i$, the gluing that
corresponds to each of them (described in the odd-numbered cases),
and we obtain a surface $R_S$ which is geodesically complete, with
$K\equiv -1$, with empty boundary, and with the same genus as $S$.
Moreover, $R_S$ contains $S$ isometrically inside it and satisfies
$\chi (R_S)\geq \chi (S)$, with strict inequality if case (1) has
occurred at least once. It is easy to see that $R_S$ does not depend
on the choice of the pairwise disjoint sequence $\{ A_i\}_i$.
\end{proof}
\begin{definicion}
Let us consider a non-exceptional Riemann surface $S$ of finite type
(open or with compact border) with $\chi (S)\leq 0$. An \emph{ outer
loop } in $S$ is either the boundary geodesic of a funnel or the
minimizing curve in the free homotopy class of some connected
component of~$\p S$. We consider punctures as outer loops of zero
length.
\end{definicion}
\begin{definicion}\label{d:clasef}
Fix a non-negative integer $a$ and a positive real number $l$. We
denote by $\F(a,l)$ the set of non-exceptional Riemann surfaces of
finite type $S$ verifying the following properties:
\begin{enumerate}
\item
S has no genus and $0\geq\chi (S)\geq -a$, equivalently $0\leq
n-2\leq a$, where $n$ is the number from Definition~\ref{chi}.
\item
If $\chi (S)=0$, then the unique outer loop has length less than or
equal to $l$. If $\chi (S)<0$ then every outer loop, except perhaps
one of them, has length less than or equal to $l$.
\item If $\p S$ is non-empty, then $L_S(\p S) \le l$.
\end{enumerate}
We denote by $\S(a,l)$ the set of Riemann surfaces $S\in \F(a,l)$
verifying that every outer loop has length less than or equal to
$l$. Notice that $\S (a,l)$ and $\F (a,l)$ coincide only for~$a=0$.
\end{definicion}
\begin{teo}
\emph{(\cite[Theorem 5.3]{PRT3})} \label{t:clases} For each $l\ge 0$
and each non-negative integer $a$, there exists a constant $\d$,
which only depends on $a$ and $l$, such that every surface in
$\S(a,l)$ is $\d$-hyperbolic.
\end{teo}
\begin{definicion}
An $N$-\emph{normal neighborhood} of a subset $F$ of a Riemann
surface $S$ is a compact, connected, bordered Riemann surface
without genus $V$ such that $F\subset V\subset S$, and $\p V$ is the
union of at most $N$ simple closed curves, i.e. $\chi (V)\geq 2-N$.
A set $E=\cup_n E_n$ in an open non-exceptional Riemann surface $S$,
with each $E_n$ compact, is called $(r,s,N)$-\emph{uniformly
separated} in $S$ if for every $n$ we can choose an $N$-normal
neighborhood $V_n$ of $E_n$ such that $V_n\setminus E_n$ is
connected, $d_S (\p V_n, E_n)\ge r$, and $L_S (\p V_n)\le s$, and
the whole sequence $\{ V_n\}_n$ can be chosen so that $d_S (V_n,
V_m)\ge r$ for every $n\neq m$.
\end{definicion}
\begin{obs}\label{separated}
As each $V_n$ has zero genus by definition, it is $V_n\in
\S(N-2,s)\subset \F(N-2,s)$ independently of~$r$. Also, Alexander
duality implies that if $E_n$ and $V_n\setminus E_n$ are connected,
then $E_n$ is simply connected.
\end{obs}
\spb
The uniformly separated sets play a central role in many topics in
Complex Analysis, such as interpolation in the unit disk $\DD$ (see
\cite{Ca}), harmonic measure (see \cite{OS}) and the study of linear
isoperimetric inequalities in open Riemann surfaces (see
\cite[Theorem 1]{APR} and \cite[Theorems 3 and 4]{FR1}).
\begin{definicion}\label{def:D}
Let $S$ be an open non-exceptional Riemann surface, $E=\cup_nE_n$ an
$(r,s,N)$-uniformly separated set in $S$ and $S^*:=S \setminus E$.
For each choice of $\{V_n\}_n$ we define
$$
\aligned D_{S^*}=D_{S^*}(\{V_n\}_n):=\sup_{n,i,j}\big\{\,
& d_{S^*}|_{V_n\setminus E_n}(\eta^n_i,\eta^n_j)\; : \;
\eta^n_i,\eta^n_j
\text{ are different connected components of $\p V_n$} \\
& \; \text{and } \; \eta^n_i,\eta^n_j \text{ are in the same
connected component of } S\setminus\inte V_n\,
\big\}.
\endaligned $$
\end{definicion}
\mpb
\begin{obs}
(1) Note that if $\eta^n_i,\eta^n_j$ are in the same connected
component of $S\setminus\inte V_n$, then $S \setminus \eta_i^n$ is
connected.
(2) Recall that $d_{S^*}\neq d_S|_{S^*}$, since $(S^*,d_{S^*})$ is a
geodesically complete Riemannian manifold (the points of $E$ are at
infinite $d_{S^*}$-distance of the points of $S^*$; in fact, $S^*$
is an open non-exceptional Riemann surface).
\end{obs}
\mpb
The following results show the relevance of $D_{S^*}(\{V_n\}_n)$
(see also Theorem \ref{t:main}).
\begin{prop}
\emph{(\cite[Proposition 5.1]{PRT3})} \label{p:prt}
Let $S$ be an open
non-exceptional Riemann surface, $E=\cup_nE_n$ an
$(r,s,N)$-uniformly separated set in $S$ and $S^*:=S \setminus E$.
Let us assume also that we can choose the sets $\{V_n\}_n$ such
that $D_{S^*}(\{V_n\}_n)=\infty$. Then $S^*$ is not hyperbolic.
\end{prop}
\begin{teo}
\emph{(\cite[Theorem 5.4]{PRT3})} \label{t:rsn} Let $S$ be an open
non-exceptional Riemann surface and $E=\cup_nE_n$ an
$(r,s,N)$-uniformly separated set in $S$. Then, $S^*:=S\setminus E$
is $\d^*$-hyperbolic if and only if $S$ is $\d$-hyperbolic,
$D_{S^*}(\{V_n\}_n)$ is finite and $V_n\setminus E_n$ is
$k$-hyperbolic for every $n$ (with $d_{S^*}|_{V_n\setminus E_n}$).
Furthermore, if $D_{S^*}(\{V_n\}_n)$ is finite and $V_n\setminus
E_n$ is $k$-hyperbolic for every $n$, then $\d^*$ (respectively
$\d$) is a universal constant which only depends on
$r,s,N,k,D_{S^*}(\{V_n\}_n)$ and $\d$ (respectively
$r,s,N,D_{S^*}(\{V_n\}_n)$ and $\d^*$).
\end{teo}
In the above theorem $\{ V_n\setminus E_n\}_n$ is a family of
bordered non-exceptional Riemann surfaces, all isometrically
embedded into $S^*$, and this family is required to be uniformly hyperbolic.
This uniform hyperbolicity condition will be removed in
Section~\ref{section5}.
\spb
If $S$ has no genus, then the set in which we take the supremum that
defines $D_{S^*}$ is the empty set. Hence, we deduce the following
direct consequence.
\begin{coro}
\label{c:rsn} Let $S$ be an open non-exceptional Riemann surface
with no genus, and $E=\cup_nE_n$ an $(r,s,N)$-uniformly separated
set in $S$. Then, $S^*:=S\setminus E$ is $\d^*$-hyperbolic if and
only if $S$ is $\d$-hyperbolic and $V_n\setminus E_n$ is
$k$-hyperbolic for every $n$ (with $d_{S^*}|_{V_n\setminus E_n}$).
Furthermore, if $V_n\setminus E_n$ is $k$-hyperbolic for every $n$,
then $\d^*$ (respectively $\d$) is a universal constant which only
depends on $r,s,N,k$ and $\d$ (respectively $r,s,N$ and $\d^*$).
\end{coro}
Finally we include a technical result about the Poincar\'e metric.
\begin{lema}
\label{l:cociente} \emph{(\cite[Lemma 3.1]{APR})} Let us consider an
open non-exceptional Riemann surface $S$, a closed non-empty subset
$C$ of $S$, and a positive number $\e$. If $S^*:=S\setminus C$, then
we have that $1 < L_{S^*}(\g)/L_S(\g) < \coth (\e/2)$, for every
curve $\g\subset S$ with finite length in $S$ such that
$d_S(\g,C)\ge\e$.
\end{lema}
\spb
\section{Stability of hyperbolicity.}\label{section5}
The leading idea in this Section is that some quantitative
information that seems to influence hyperbolicity of a surface
actually is irrelevant, let us see an example. If $S$ is an open
Riemann surface and $p_1,p_2\in S$, then several conformal
invariants of $S^*=S\setminus\{ p_1,p_2\}$ (e.g. the exponent of
convergence, the first eigenvalue of the Laplace-Beltrami operator
and the isoperimetric constant) degenerate when $p_2$ tends to
$p_1$; in contrast, the hyperbolicity constant stays bounded
(stable) as $p_1$ approaches~$p_2$.
\spb
In this Section we only consider surfaces without genus, so that in
particular the number $D_{S^*}$ from Definition~\ref{def:D} is zero.
We begin by proving Theorem~\ref{t:finite} as a surprising
consequence of Theorem~\ref{t:balls} (on the topology of balls).
Then Corollary~\ref{c:finite}, an immediate consequence of
Theorem~\ref{t:finite}, is used to prove Theorem~\ref{t:clasef}.
In its turn, Theorem~\ref{t:clasef} is fundamental for the proof of
the main Theorem in Section~\ref{section:6}.
\begin{teo}
\label{t:finite} Let us consider a $\d$-hyperbolic non-exceptional
Riemann surface $S$ with no genus, and pairwise disjoint simply
connected compact sets $\{E_n\}_{n=1}^N$ in $S$. We define
$S^*:=S\setminus \cup_{n=1}^N E_n$. Assume that for each
$n=1,\dots,N,$ there exists a simple closed curve $g_n$ `surrounding
just $E_n$' with $L_{S^*}(g_n) \le l$. Then there exists a constant
$\d^*$, which only depends on $\d$, $N$ and $l$, such that $S^*$ is
$\d^*$-hyperbolic.
\end{teo}
\begin{obs}
By $g_n$ `surrounding just $E_n$' we mean that $g_n$ is
homotopically trivial in $S$, $g_n$ surrounds $E_n$ and $g_n$ does
not surround $E_k$ for $k\neq n$ ($g_n$ is `freely homotopic' to
$E_n$). Note also that $g_n \cap (\cup_k E_k)=\varnothing$, since
$L_{S^*}(g_n) < \infty$.
\end{obs}
\begin{proof}
First we prove the case $N=1$.
We have $L_{S}(g_1) < L_{S^*}(g_1) \le l$.
The curve $g_1$ surrounds a simply connected open set
$D\subset S$ with $E_1 \subset D$; then we can lift $D$ to $\tilde
D \subset \DD$ and given $z,w\in \tilde{D}$, we consider the
infinite geodesic $\eta$ in $\DD$ joining $z,w$; the geodesic
$\eta$ meets $\p \tilde D$ in $z',w'$ with $[z,w]\subset [z',w']$;
therefore, $d_{\DD}(z,w) \le
d_{\DD}(z',w') \le L_{\DD}(\tilde{g_1})/2 = L_{S}(g_1)/2 \le l/2$,
and $\diam_{S}(D) \le \diam_{\DD}(\tilde{D}) \le l/2$.
Hence, $\diam_{S}(E_1) \le l/2$.
Let us fix any $p\in E_1$; then $E_1 \subset D \subset \overline{B_S(p,l/2)}$.
Since $K\equiv -1$, by Theorem \ref{t:balls} we know that there
exists $l'\in [l,l+1]$ such that $\p B_S(p,l')$ is a union of simple
closed curves and
$$
\hbox{rank}\,H_1 \big(B_S(p,l')\big) \le \frac{\sinh(l+1)}{\sinh 1}
< e^{l}/(1-e^{-2}) < 2 \, e^l .
$$
We define $V_1$ as the closure of the ball $B_S(p,l')$; then $g_1$
is contained in $V_1$. Consequently, $d_S(E_1, \p V_1) \ge l/2$. We
also have $L_S(\p V_1) \le L_\DD(B_\DD(0,l')) = 2 \pi \sinh l'\le 2
\pi \sinh (l+1)$. The boundary $\partial V_1$ has at most $1+2e^l$
connected components, moreover $V_1$ has no genus because the
ambient surface $S$ has zero genus by hypothesis. All this implies
that $V_1$ is an $(1+2e^l)$-normal neighborhood of $E_1$ in $S$, and
$E_1$ is therefore an $(l/2, 2 \pi \sinh (l+1), 1+2e^l)$-uniformly
separated set in $S$.
We check now that $V_1^*:=V_1 \setminus E_1$ is hyperbolic with the
intrinsic distance $d_{S^*}|_{V_1^*}$ induced on $V_1^*$ by the Poincar\'e
metric of $S^*=S \setminus E_1$. By Lemma \ref{l:cociente}, since
$d_S(E_1, \p V_1) \ge l/2$,
$$
L_{S^*}(\p V_1) \le L_{S}(\p V_1) \coth (l/4) \le 2 \pi \sinh (l+1)
\coth (l/4) \,.
$$
Each component of $\partial V_1 =\partial V_1^*$ gives rise to an
outer loop in $V_1^*$. If $E_1$ is a single point then $V_1^*$ has a
puncture at $E_1$; otherwise $V_1^*$ has one additional outer loop
freely homotopic to $g_1$. In any case the sum of the number of
outer loops plus the number of punctures in $V_1^*$ is at most
$2+2e^l$. The $S^*$-length of the outer loops coming from $\partial
V_1$ is less than or equal to $2 \pi \sinh (l+1) \coth (l/4)$.
Since $L_{S^*}(g_1) \le l$, if $E_1$ is not a puncture then the
outer loop in $V_1^*$ homotopic to $g_1$ has length less than or
equal to $l<2 \pi \sinh (l+1) \coth (l/4)$
(since $g_1$ is contained in $V_1^*$).
Consequently $V_1^* \in
\S(2e^l ,2 \pi \sinh (l+1) \coth (l/4))$, and Theorem \ref{t:clases}
says that there exists a constant $\d_1$, which only depends on $l$,
such that $V_1^*$ is $\d_1$-hyperbolic.
The hypothesis of $S$ having genus zero allows us to use Corollary
\ref{c:rsn}, hence there exists a constant $\d_1^*$, which only
depends on $\d_1$ and $l$, such that $S^*$ is $\d_1^*$-hyperbolic.
This finishes the proof in the case $N=1$.
\spb
Now we prove the result by induction on $N$. We have proved it for
$N=1$. Assume that it holds for $N-1$ (note that we also have
$L_{S\setminus (E_1\cup \cdots\cup E_{N-1})}(g_n) < L_{S\setminus
(E_1\cup \cdots\cup E_{N})}(g_n) \le l$ for $n=1,\dots,N-1$).
Consequently, $S\setminus (E_1\cup \cdots\cup E_{N-1})$ is
$\d_{N-1}$-hyperbolic, where $\d_{N-1}$ only depends on $\d$, $N$
and $l$.
Since $g_N$ is a simple closed curve surrounding just $E_N$, with
$L_{S^*}(g_N) \le l$, the result for $N=1$ gives that $S^*$ is
$\d^*$-hyperbolic, with $\d^*$ a constant which only depends on
$\d$, $N$ and $l$.
\end{proof}
In Theorem~\ref{t:infinite} we shall extend Theorem~\ref{t:finite}
to the case of infinitely many sets $E_n$. It needs an extra
hypothesis: that the $E_n$'s get neither too small nor too large as
$n\to\infty$.
\vspace{3mm}
Since any puncture can be surrounded by arbitrarily short closed
curves, we deduce the following result:
\begin{coro}
\label{c:finite0} Let us consider a $\d$-hyperbolic non-exceptional
Riemann surface $S$ with no genus, and points $\{p_n\}_{n=1}^N$ in
$S$. We define $S^*:=S\setminus \{p_1,\dots,p_N\}$. Then there
exists a constant $\d^*$, which only depends on $\d$ and $N$, such
that $S^*$ is $\d^*$-hyperbolic.
\end{coro}
Corollary \ref{c:finite0} can be viewed as a result on stability
of hyperbolicity: $S^*$ is $\d^*$-hyperbolic independently of how
close or far apart the points $\{p_1,\dots, p_N\}$ are from one
another.
\spb
Since $\DD$ is hyperbolic, Theorem~\ref{t:finite} also implies the
following.
\begin{coro}
\label{c:finite} Let us consider pairwise disjoint simply connected
compact sets $\{E_n\}_{n=1}^N$ in $\DD$ and $\DD^*:=\DD \setminus
\cup_{n=1}^N E_n$. Assume that for each $n=1,\dots,N,$ there exists
a simple closed curve $g_n$ surrounding just $E_n$ with
$L_{\DD^*}(g_n) \le l$. Then there exists a constant $\d^*$, which
only depends on $N$ and $l$, such that $\DD^*$ is $\d^*$-hyperbolic.
\end{coro}
Finally we prove the following improvement of Theorem
\ref{t:clases}. It is surprising since we do not require anything
about one of the outer loops.
\begin{teo}
\label{t:clasef} For each $a$ and $l$, there exists a constant $\d$,
which just depends on $a$ and $l$, such that every $S\in \F(a,l)$ is
$\d$-hyperbolic.
\end{teo}
\begin{obs}
It is interesting to note that it is not possible to obtain a similar result to
Theorem \ref{t:clasef} if all the outer loops except two have
bounded length, as the following example shows:
if $Y_t$ is the $Y$-piece with simple closed geodesics
$\g_1\cup \g_2\cup \g_3 = \p Y_t$
such that $L(\g_1)=1$ and $L(\g_2)=L(\g_3)=t$,
then $\lim_{t\to\infty} \d(Y_t)=\infty$.
\end{obs}
\begin{proof}
We first prove the result for open surfaces.
If $S\in \S(a,l)$, then Theorem \ref{t:clases} gives the result;
this happens in particular when $a=0$. Therefore, we can assume that
$\chi (S)<0$, that an outer loop $\g_0$ satisfies $L_S(\g_0)>l$, and
that any other outer loop $\g_j$ $(j=1,\dots,N)$ verifies
$L_S(\g_j)\le l$. From $-a\leq \chi(S)=2-(N+1)<0$ we infer $2\leq
N\leq a+1$.
For open surfaces the conformal structure and the Riemann metric
determine each other, so we can consider one structure or the other
to our convenience. Having zero genus, $S$ can be represented as a
plane domain $S\subset \CC$ with $S=\O \setminus E_1\cup \cdots \cup
E_N$, $\O$ a simply connected open set, $E_1, \dots, E_N$ simply
connected compact sets, such that $\g_0$ surrounds $E_1\cup \cdots
\cup E_N$ and $\g_j$ surrounds just $E_j$ ($j=1, \dots ,N$). The
hypothesis $L_S(\g_0)>l$ implies that $\g_0$ is not a puncture and
that $\O\neq \CC$; then, by the Riemann mapping Theorem, we can
assume that $S=\DD \setminus E_1\cup \cdots \cup E_N$. Since we have
$N \le a+1$ and $L_S(\g_j)\le l$ ($j=1, \cdots , N$), by Corollary
\ref{c:finite} there exists a constant $\d$, which just depends on
$a$ and $l$, such that $S$ is $\d$-hyperbolic.
\spb
We now prove the result for bordered surfaces.
The idea of the proof is to see a bordered surface in $\F(a,l)$ as a
subset of an open surface in $\F(a,l)$, and then make use of Theorem
\ref{t:treedecomp}.
Given the bordered surface $S$, consider the open surface $R_S$ from
Proposition~\ref{RS}. If $R_S$ is the unit disk then it is
$\log(1+\sqrt2\,)$-thin (see e.g. \cite[p.130]{An}).
Assume now that $R_S$ is not the unit disk. Outer loops in $S$ are
metric geodesics (recall Definition~\ref{def:Rips}), perhaps not
Riemannian geodesics, but they give rise in $R_S$ to Riemannian
outer loops of no greater length (including punctures), or they just
shrink to points in $R_S$. Hence $R_S\in\F (a,l)$, and by the open
case there is a constant $\d_1$, just depending on $a$ and $l$, such
that $R_S$ is $\d_1$-hyperbolic.
The closure of $R_S\setminus S$ is the union of simply or doubly
connected bordered surfaces $R^1,\dots,R^s,$ with $s\le a+2$, and
the condition $L_S(\p S)\le l$ implies that $\{S,R^1,\dots,R^s\}$ is
an $l$-tree decomposition of $R$. Then, by Theorem
\ref{t:treedecomp}, there exists a constant $\d$ that depends only
on $a$ and $l$ and such that $S$ is $\d$-hyperbolic.
\end{proof}
\section{Main results on hyperbolicity}\label{section:6}
Now, taking advantage of all the tools developed in the previous
Sections, we present the main results on hyperbolicity of the paper.
The first Theorem we present improves Theorem \ref{t:rsn} by
removing the uniform hyperbolicity hypothesis, which is usually the
hardest one to check.
\begin{teo}
\label{t:main} Let $S$ be an open non-exceptional Riemann surface
and $E=\cup_n E_n$ a $(r,s,N)$-uniformly separated set in $S$, with
$E_n$ simply connected for every $n$. Then, $S^*:=S \setminus E$ is
$\d^*$-hyperbolic if and only if $S$ is $\d$-hyperbolic and the
number $D_{S^*}(\{V_n\}_n)$ from Definition~\ref{def:D} is finite.
Furthermore, $\d^*$ (respectively $\d$) is a universal constant
which only depends on $r,s,N,D_{S^*}(\{V_n\}_n)$ and $\d$
(respectively $\d^*$).
\end{teo}
\begin{obs}
\begin{enumerate}
\item Recall that if $E_n$ is simply connected, then it gives rise
to either a puncture (if $E_n$ is a single point) or a funnel (if
$E_n$ is not a single point) in $S^*$. \item Note that we do not
require anything about $\diam_S E_n$; in particular, we allow the
case $\sup_n \diam_S E_n= \infty$; in this case the funnels $F_n$
in $S^*$ corresponding to $E_n$ verify $\sup_n L_{S^*}(\p F_n) \ge
\sup_n L_{S}(\p F_n) \ge \sup_n \diam_S E_n = \infty$, which makes
the study of the hyperbolicity of $S^*$ more difficult. \item
Theorem \ref{t:main} is a known result in the particular case when
every $E_n$ is a single point (see \cite[Theorem 3.1]{PRT2}).
\end{enumerate}
\end{obs}
\begin{proof}
In order to apply Theorem \ref{t:rsn}, we just need to prove that
$V_n^*:=V_n\setminus E_n$ is $k$-hyperbolic for every $n$, where $k$
is a constant which only depends on $r,s$ and $N$.
Recall that $V_n$ is compact and belongs to $\S(N-2,s)\subset
\F(N-2,s)$ for any $n$.
If $\p V_n$ is a single closed curve (i.e. $V_n$ is a topological
disk), then there is just one outer loop in $V_n^*$ and, by Lemma
\ref{l:cociente}, $L_{S^*}(\p V_n) < L_{S}(\p V_n) \coth (r/2)\le s
\coth (r/2)$. Hence, $V_n^*\in\S(0,s \coth (r/2))=\F(0,s \coth
(r/2))$. In this case Theorem \ref{t:clases} suffices to ensure that
$V_n^*$ is $k_1$-hyperbolic, with a constant $k_1$ which only
depends on $r$ and $s$.
If $\p V_n$ is not connected, let us denote by $\g_n$ the simple
closed geodesic in $V_n^*$ which surrounds just $E_n$ (if $E_n$ is a
single point, as usual, we see $\g_n$ as a puncture and
$L_{S^*}(\g_n)=0$).
Note that any outer loop $\g$ distinct from $\g_n$ in $V_n^*$ is
freely homotopic to some closed curve in $\p V_n$.
Since Lemma \ref{l:cociente} implies $L_{S^*}(\p V_n)< L_{S}(\p V_n)
\coth (r/2)\le s \coth (r/2)$, we deduce that $V_n^*\in\F(N-1,s
\coth (r/2))$ (recall that $V_n$ has at most $N$ outer loops and
$E_n$ is simply connected for every $n$; we do not need to bound the
length of the outer loop corresponding to $E_n$). Theorem
\ref{t:clasef} guarantees that $V_n^*$ is $k_2$-hyperbolic, with a
constant $k_2$ which only depends on $r,s$ and $N$.
Now Theorem \ref{t:rsn} gives the result.
\end{proof}
We would like not to have to check the hypothesis
$D_{S^*}(\{V_n\}_n)< \infty$. The two following results allow to
remove this hypothesis if $S$ has either no genus or finite genus.
\medskip
If $S$ has no genus, then the set in which we take the supremum in
order to define $D_{S^*}$ is the empty set. Hence, we deduce the
following direct consequence.
\begin{coro}
\label{c:main} Let $S$ be an open non-exceptional Riemann surface
with no genus and $E=\cup_nE_n$ an $(r,s,N)$-uniformly separated
set in $S$, with $E_n$ simply connected for every $n$. Then,
$S^*:=S \setminus E$ is $\d^*$-hyperbolic if and only if $S$ is
$\d$-hyperbolic.
Furthermore, $\d^*$ (respectively $\d$) is a universal constant
which only depends on $r,s,N$ and $\d$ (respectively~$\d^*$).
\end{coro}
\begin{teo}
\label{t:finitegenus} Let $S$ be an open non-exceptional Riemann
surface with finite genus and $E=\cup_nE_n$ an $(r,s,N)$-uniformly
separated set in $S$, with $E_n$ simply connected for every $n$.
Then, $S^*:=S \setminus E$ is hyperbolic if and only if $S$ is
hyperbolic.
\end{teo}
\begin{proof}
If $S$ has no genus, then Corollary \ref{c:main} gives the result.
Therefore, we can assume that $S$ has genus.
Given a choice of $\{V_n\}_n$, we define the subset of indices
$\Lambda$ as the set of all $n$ such that there are different
connected components $\eta^n_i,\eta^n_j$ of $\p V_n$ in the same
connected component of $S\setminus\inte V_n$. We are going to prove
that $\Lambda$ is finite. This will imply that $D_{S^*}(\{V_n\}_n)$
is the maximum of at most $\frac{N(N-1)}{2}\cdot\card\Lambda\,$
finite distances, hence finite, and then Theorem \ref{t:main} will
finish the proof.
Since $S$ has finite genus, there exists a domain $G\subset S$
verifying the following facts: $\overline{G}$ is a compact set whose
boundary is a finite collection $g_1,\dots ,g_h$ of simple closed
curves, and $S\setminus G$ is a disjoint union $S_1\cup\cdots\cup
S_h$ where each $S_j$ is a bordered surface with no genus, and $\p
S_j=S_j\cap\overline{G}=g_j$ for $j=1,\dots ,h$.
Only finitely many of the $V_n$ intersect $\overline{G}$; otherwise
the condition $d(V_n,V_m)\ge r$ could not be satisfied for all
$n\neq m$. We next prove that if $V_n\cap\overline{G}=\varnothing$
then $n\notin\Lambda$, and finiteness of $\Lambda$ follows.
Let us suppose $V_n$ is disjoint from $\overline{G}$, that two
connected components $\eta_{ni},\eta_{ni'}$ of $\p V_n$ can be
connected in $S\setminus \inte V_n$, and derive a contradiction.
Since $V_n$ is connected, it is contained in one of the $S_j$, say
$S_{j_0}$. The path $\gamma$ connecting $\eta_{ni}$ to $\eta_{ni'}$
in $S\setminus \inte V_n$ cannot be all inside $S_{j_0}$, for that
would force $S_{j_0}$ to have genus. Therefore $\gamma$ exits
$S_{j_0}$, which it can do only by crossing $g_{j_0}$. Any time
$\gamma$ re-enters $S_{j_0}$, it must do so by crossing $g_{j_0}$.
One concludes that the parts of $\gamma$ outside $S_{j_0}$ can be
replaced with arcs of $g_{j_0}$, but this yields a continuous path
in $S_{j_0}\setminus\inte V_n$ which joins $\eta_{ni}$ to
$\eta_{ni'}$, once again forcing $S_{j_0}$ to have genus.
We conclude that $\Lambda\subseteq\{\, n\; :\; V_n\cap
G\neq\varnothing\,\}$; hence $\Lambda$ and $D_{S^*}(\{V_n\}_n)$ are
finite, as was to be proved
\end{proof}
Finally we shall prove Theorem~\ref{t:infinite}, a complementary
result to Corollary \ref{c:main} with very different hypotheses. It
is also the $N=\infty$ analogue of Theorem \ref{t:finite}. Removing
an infinity of sets $E_n$ from the initial surface $S$ can ruin
hyperbolicity if the $E_n$ become too small or too large as
$n\to\infty$. One idea is to reduce to the case $S={\mathbb D}$ and
then use annuli, instead of curves, to `surround' the sets $E_n$.
More concretely, the condition that the domain ${\mathbb D}\setminus
E$ must satisfy is having \emph{uniformly perfect boundary}, which
we define below. We also quote some results, about the Poincar\'e
and quasihyperbolic metrics of a domain, that are used in the final
proof.
\begin{definicion}
A \emph{generalized annulus} $\O$ is a doubly connected open subset
of the complex plane which is not the plane minus a point; then its
complement (in the Riemann sphere) has two connected components.
Given any generalized annulus $\O$, there exists a conformal mapping
of $\O$ onto $\{z\in\CC: \, 1< |z-a|<R \}$, for some $1<R \le
\infty$. We define the \emph{modulus} of $\O$ as
$$
\modulus \O := \frac1{2\pi} \log R \,.
$$
We say that a generalized annulus $\O$ \emph{separates} a closed set
$E$ if $\O$ does not intersect $E$ and each connected component of
the complement of $\O$ intersects $E$.
We say that $E$ is \emph{uniformly perfect} if there exists a
constant $c_1$ such that $\modulus \O \le c_1$ for every generalized
annulus separating $E$ (see \cite{BP}).
\end{definicion}
Two useful properties of the modulus are the following:
(A) If $\g$ is the simple closed geodesic for the Poincar\'e metric
in $\O$, then $\modulus \O =\pi/L_\O(\g)$ (if $\O$ has a puncture we
can see $\g$ as the puncture and then $L_{\O}(\g)=0$ and $\modulus
\O = \infty$).
(B) If $\O_1 \subseteq \O_2$, then $\modulus \O_1 \le \modulus
\O_2$.
\spb
A domain with one or more punctures is never uniformly perfect. If
we remove from the unit disk $\mathbb D$ a sequence of straight
segments whose lengths converge to zero, then the resulting domain
is not uniformly perfect. This example leads to the hypothesis
$\diam_SE_n\ge c$ in the statement of Theorem~\ref{t:infinite}.
\spb
Uniformly perfect sets verify the following interesting property:
\begin{teo}
\emph{(\cite[Corollary 1]{BP})} \label{BPthm2} Let $\O\subset\CC$ be
a non-exceptional domain and $ds=\lambda_\O (z)\, |dz|$ its
Poincar\'e metric. Define also $\d_\O (z):=\min\{|z-a|: \, a \in \p
\O \}$. The following conditions are equivalent:
\begin{enumerate}
\item There exists a positive constant $c_2$ with $\,\displaystyle
\frac{c_2}{\d_\O (z)} \le \l_\O (z) \le \frac2{\d_\O (z)}\, $ for
every $z\in\O$ \item $\p\O$ is uniformly perfect.
\end{enumerate}
\noindent Furthermore, if $\p\O$ is uniformly perfect then the
constant $c_2$ just depends on the uniformly perfect constant of
$\p\O$.
\end{teo}
If we define, as usual, the \emph{quasihyperbolic length} of a curve
$\gamma$ as
$$
k_\O (\g) := \int_\g \frac{|dz|}{\d_\O (z)} \;,
$$
then Theorem \ref{BPthm2} says that $\p\O$ is uniformly perfect if
and only if $L_\O (\g) \ge c_2 k_\O (\g)$ for every curve $\g
\subset \O$.
\spb
We need one more technical result:
\begin{lema}
\label{minLenLem} \emph{(\cite[Lemma 3.3]{HLPRT})} Let $\gamma$ be a
curve in a domain $\O\subset \RR^n$ starting at a point $x$ and with
Euclidean length~$s$. Then
$$
k_\O(\gamma) \ge \log \Big( 1 + \frac{s}{\d_\O(x)} \Big)\,.
$$
\end{lema}
\begin{teo}
\label{t:infinite} Let us consider an open non-exceptional Riemann
surface $S$ with no genus, and pairwise disjoint simply connected
compact sets $\{E_n\}_{n=1}^\infty$ in $S$ with $\diam_S E_n \ge c$
for every $n$. We define $S^*:=S\setminus \cup_{n=1}^\infty E_n$.
Assume that for each $n$ there exists a simple closed curve $g_n$
surrounding just $E_n$ with $L_{S^*}(g_n) \le l$. Then $S$ is
$\d$-hyperbolic if and only if $S^*$ is $\d^*$-hyperbolic.
Furthermore, $\d^*$ (respectively $\d$) is a universal constant
which only depends on $c$, $l$ and $\d$ (respectively $\d^*$).
\end{teo}
\begin{obs}
The conclusion of Theorem \ref{t:infinite} does not hold if we
remove either the hypothesis $\diam_S E_n \ge c$ or $L_{S^*}(g_n)
\le l$.
\end{obs}
\begin{proof}
For each $n$ there exists a simple closed geodesic $\g_n$
surrounding just $E_n$ (freely homotopic to $g_n$) with
$L_{S^*}(\g_n) \le L_{S^*}(g_n) \le l$ ($\g_n$ can not be a puncture
since $\diam_S E_n \ge c$ implies that $E_n$ is not an isolated
point).
\vspace{3mm}
\noindent{\bf Claim:} there exists a positive constant $\e_0$,
which just depends on $c$ and $l$, such that $d_S(E_n,\g_n) \ge
\e_0$ for every $n$.
\vspace{3mm}
We prove the Theorem assuming this claim. Let $V_n$ be the closure
of the simply connected open subset of $S$ surrounded by $\g_n$.
Then $V_n$ is a $1$-normal neighborhood of $E_n$. Furthermore,
$$
\begin{aligned}
d_S(E_n,\p V_n) & = d_S(E_n,\g_n) \ge \e_0 \,,
\\
L_S(\p V_n)=L_S(\g_n) & < L_{S^*}(\g_n) \le L_{S^*}(g_n) \le l \,.
\end{aligned}
$$
Given $n\neq m$, we have $L_{S^*}(\g_n), L_{S^*}(\g_m) \le l$, and
by the Collar Lemma (see \cite{R}) there are collars in $S^*$ around
$\g_n$ and $\g_m$ of width $\Arccosh \coth (l/2)$. These collars are
pairwise disjoint, as explained in Remark~\ref{disjuntas}. Then we
deduce that $d_{S^*}(\g_n,\g_m) \ge 2 \Arccosh \coth (l/2)$. Now
Lemma \ref{l:cociente} gives:
$$
d_{S}(V_n,V_m) = d_{S}(\g_n,\g_m) > \tanh(\e_0/2) d_{S^*}(\g_n,\g_m)
\ge 2 \tanh(\e_0/2) \Arccosh \coth (l/2)\,.
$$
If we define $r := \min\{\e_0, 2 \tanh(\e_0/2) \Arccosh \coth
(l/2)\}$, then $\{E_n\}_n$ is a $(r,l,1)$-uniformly separated set
in~$S$.
Corollary \ref{c:main} states that $S$ is $\d$-hyperbolic if and
only if $S^*$ is $\d^*$-hyperbolic, with the appropriate behaviour
of the constants. This finishes the proof if the claim holds.
\vspace{3mm}
\noindent Now we are going to prove the claim.
We prove first that without loss of generality we can assume
$S=\DD$: Let $\pi:\DD\longrightarrow S$ be a universal covering map.
We consider $F:=\pi^{-1}(E)$ and the connected components $\{F_n\}$
of $F$. If $\DD^*:=\DD\setminus F$, then $\pi:\DD^*\longrightarrow
S^*$ is also a covering map. Consequently, $\pi$ defines two local
isometries: $\DD\to S$ and $\DD^*\to S^*$. Given any fixed $F_n$
then $E_m:=\pi(F_n)$ verifies that $\pi:F_n\to E_m$ is a bijection.
Hence, $\diam_\DD(F_n) \ge \diam_S(E_m) \ge c$.
Let $W_n$ be the connected component of $\pi^{-1}(V_m)$ containing
$F_n$. Then $\g'_n=\partial W_n$ is the simple closed geodesic in $\DD^*$
surrounding just $F_n$ and $L_{\DD^*}(\g'_n) = L_{S^*}(\g_m) \le l$.
Consequently, $\{F_n\}_{n=1}^\infty$ verifies the hypotheses in
Theorem \ref{t:infinite}. Since $\pi$ defines bijections $W_n\to
V_m$ and $\g'_n\to\g_m$, we have $d_\DD(F_n,\g_n') =d_S(E_m,\g_m)$.
In order to prove the claim we can thus assume without loss of
generality that $S=\DD$, and $\DD^*=\DD\setminus \cup_{n=1}^\infty
E_n$.
\spb
We prove now that $\p \DD^*$ is a uniformly perfect set: Let us
consider a generalized annulus $A$ separating $\p \DD^*$; then
$A\subset \DD^*\subset \DD$ and the bounded connected component of
the complement of $A$ contains some $E_{n_0}$. Hence, $A\subset \DD
\setminus E_{n_0}$ and consequently $\modulus A\le \modulus (\DD
\setminus E_{n_0})$.
We are looking for a lower bound of $L_{\DD \setminus
E_{n_0}}(\eta_{n_0})$, where $\eta_{n_0}$ is the simple closed
geodesic for the Poincar\'e metric in $\DD \setminus E_{n_0}$.
Consider $a,b \in E_{n_0}$ with $d_\DD(a,b)=\diam_\DD E_{n_0}$, and
the simple closed geodesic $\eta$ for the Poincar\'e metric in $\DD
\setminus \{a,b\}$. We have that
$$
L_{\DD \setminus E_{n_0}}(\eta_{n_0})
> L_{\DD \setminus \{a,b\}}(\eta_{n_0})
\ge L_{\DD \setminus \{a,b\}}(\eta) \,.
$$
Since $\DD \setminus \{a,b\}$ and $\DD \setminus \{a',b'\}$ are
isometric if and only if $d_\DD(a,b)=d_\DD(a',b')$, then $L_{\DD
\setminus \{a,b\}}(\eta)=f(d_\DD(a,b))$, for some function $f:
(0,\infty) \longrightarrow (0,\infty)$. Since $\eta$ surrounds
$\{a,b\}$,
$$
f(d_\DD(a,b)) = L_{\DD \setminus \{a,b\}}(\eta)
> L_{\DD}(\eta) > d_\DD(a,b) \,;
$$
consequently, $f(t)>t$ and since $d_{\DD} (a,b)=\diam_\DD E_n \ge
c$,
$$
L_{\DD \setminus E_{n_0}}(\eta_{n_0}) > L_{\DD \setminus
\{a,b\}}(\eta) = f(d_{\DD} (a,b)) > d_{\DD} (a,b) \ge c \,,
$$
and hence $\modulus A \le \modulus (\DD \setminus E_{n_0}) \le
\pi/c$. This shows that $\p \DD^*$ is a uniformly perfect set.
\medskip
Now, by Theorem \ref{BPthm2}, there exists a constant $c_1$, which
just depends on $c$, such that $L_{\DD^*} (\g) \ge c_1 k_{\DD^*}
(\g)$ for every curve $\g \subset {\DD^*}$.
Let us consider a fixed $n$ and the simple closed geodesic $\g_n$ in
$\DD^*$ surrounding just $E_n$ (freely homotopic to $g_n$) with
$L_{S^*}(\g_n) \le L_{S^*}(g_n) \le l$. Take $p \in E_{n}$ and $q
\in \g_{n}$ with $\e:=d_\DD(E_n,\g_n)=d_\DD(p,q)$. Since
$d_\DD(0,a)=2\Arctanh a$, using a M\"{o}bius map if it is necessary,
we can assume without loss of generality that $p=0$ and $q=\tanh
(\e/2)$. Since $L_{\DD}(\g_n)> \diam_\DD E_{n} \ge c$, we can
consider a subcurve $\g_n^0 \subset \g_n$ starting at $q$ and with
$L_{\DD}(\g_n^0) = c$; then,
$$
\g_n^0 \subset \overline{B_\DD(0,\e+c)} = \overline{B_{Eucl}\big(\,
0\, ,\,\tanh ((\e+c)/2)\,\big)} \,.
$$
Then
$$
L_{Eucl}(\g_n^0) = \int_{\g_n^0} |dz| \ge \frac{1-\tanh\!^2
((\e+c)/2)}2 \int_{\g_n^0} \frac{2\,|dz|}{1-|z|^2} =
\frac{L_{\DD}(\g_n^0)}{2 \cosh\!^2((\e+c)/2)} = \frac{c}{2 \cosh\!^2
((\e+c)/2)} \;.
$$
Therefore, applying Lemma \ref{minLenLem}, we obtain
$$
\begin{aligned}
l & \ge L_{\DD^*}(g_n) \ge L_{\DD^*}(\g_n) \ge L_{\DD^*}(\g_n^0) \ge
c_1 k_{\DD^*}(\g_n^0)
\\
& \ge c_1 \log \Big( 1 + \frac{L_{Eucl}(\g_n^0)}{\d_{\DD^*}(q)}
\Big) \ge c_1 \log \Big( 1 + \frac{L_{Eucl}(\g_n^0)}{q} \Big)
\\
& \ge c_1 \log \Big( 1 + \frac{c}{2 \tanh (\e/2) \cosh\!^2
((\e+c)/2)} \Big)\,.
\end{aligned}
$$
Hence,
$$
2 \tanh (\e/2) \cosh\!^2 ((\e+c)/2) \ge \frac{c}{e^{l/c_1}-1} \;.
$$
Note that, for each fixed $c$, the function $f_c: (0,\infty)
\longrightarrow (0,\infty)$ given by $f_c(\e)=2 \tanh (\e/2)
\cosh\!^2 ((\e+c)/2)$ is positive and increasing in $\e \in
(0,\infty)$. If we define
$$
\e_0:= f_c^{-1} \Big( \frac{c}{e^{l/c_1}-1} \Big) >0 \,,
$$
then $\e_0$ just depends on $c$ and $l$, and $d_\DD(E_n,\g_n)=\e \ge
\e_0$ for every $n$. This finishes the proof of both the claim and
the Theorem.
\end{proof}
\
|
{
"timestamp": "2010-09-21T02:03:55",
"yymm": "1009",
"arxiv_id": "1009.3881",
"language": "en",
"url": "https://arxiv.org/abs/1009.3881"
}
|
\section{Introduction}
The reconstruction of an unknown quantum state from a suitable set
of measurements is called quantum tomography~\cite{Paris}, which is
a particularly important method in the study of quantum mechanics
and its various applications. Since the characterization of the
states is the central tasks in quantum-state engineerings and
controls, this technique is of great importance in the current
quantum information processing. Recently, many theoretical analysis
and experimental demonstrations have been devoted to implement the
desirable quantum-state tomographies for, e.g., the polarization
states of photons~\cite{White,Langford}, the electronic states of
trapped ions~\cite{H1}, and the solid-state qubits~\cite{Liu}, etc.
However, all these tomographic reconstructions are based on the
destructively-projective (DP) measurements, and thus are very
operational-complicated. This is because that each kind of DP
measurements, e.g., $\hat{P}_k=|k\rangle\langle k|$, is required to
be performed many times on many copies of the reconstructed state
for determining just one of the elements (e.g., $|c_k|^2$ in the
state $|\psi\rangle=\sum_kc_k|k\rangle$) in the density matrix
$\rho=|\psi\rangle\langle\psi|$.
Besides the usual DP measurements, quantum state could also be
detected by other strategies, typically such as the so-called
quantum nondemolition (QND) measurements. Basically, QND measurement
is a nondestructive detection, as the measurement-induced
back-action noises could be effectively suppressed by repeatedly
hiding them in certain observables which are not of interests. The
basic criteria for a QND measurement is that the repeated
measurements of an observable $\hat{o}$ of the same system should
yield the identical result. This means that the measured observable
must be commutative with the Hamiltonian $\hat{H}_{int}$ describing
the interaction between the measured system and the detector, i.e.,
$[\hat{H}_{int},\hat{o}]=0$. Historically, QND measurement is
proposed to explore the fundamental limitations of measurements, and
has been demonstrated in various branches of physics, such as in the
detection of gravitational waves~\cite{Braginsky}, quantum
optics~\cite{Pereira,Bencheikh,Grangier},
telecommunications~\cite{Levenson2}, and quantum
control~\cite{Wiseman}, etc. In recent years, the QND measurement
has also been successfully applied to probe the atomic qubits in the
cavity quantum electrodynamics (QED)~\cite{Nogues,Turchette}.
Furthermore, this technique was extensively used to the circuit QED
systems~\cite{Blais,Wallraff,Wallraff2,Wei,Gambetta,Bianchetti,Filipp}
for nondestructively reading out the superconducting qubits. This
QND measurement is implemented by measuring the transmission of the
driven microwave signals through a transmission line resonator,
which is dispersively coupled to the detected qubits. This is
because that the detected qubits can cause sufficiently large
state-dependent shifts of the resonator frequency. Thus by detecting
the signals of the shifted frequency of the resonator, the qubit
state will be read out. However, the QND measurements in the above
works~\cite{Blais,Wallraff,Wallraff2,Wei,Gambetta,Bianchetti,Filipp}
are only utilized to effectively distinguish the different logic
states of the detected qubit(s), which is (are) not prepared
initially at their superposed state.
Motivated by the above experiments, recently we proposed a new
scheme to nondestructively detect the superposition of these logic
states by the QND measurements~\cite{arXiv}. By taking account of
the full quantum correlations between the resonator and
dispersively-coupled qubit(s), our proposal shows that each detected
peak marks one of the logic states and the relative height of such a
peak is related to its corresponding superposed probability. This
means that one kind of the QND measurements can determine all the
diagonal elements of the density matrix of the measured quantum
state $\rho=|\psi\rangle\langle\psi|$. Similarly, the non-diagonal
elements of $\rho$ could be determined by other kinds of QND
measurements by performing the suitable unitary operations to
transfer them into the measurable diagonal locations. Therefore, the
proposed tomographic reconstructing approach is high efficient for
$N$ ($N>1$) qubits, as the number of the kinds of the QND
measurements required is significantly decreased. For example, to
tomographically reconstruct a two-qubit state, the proposed $6$-kind
QND measurements are sufficient. This is significantly simpler than
the previous schemes (requiring $15$-kind measurements) based on
either the DPs~\cite{Liu} or the individual dispersive readouts of
the logic states~\cite{Filipp}.
The paper is organized as follows: Sec.~II gives our generic model
of the transmissions of a driven resonator. In Sec.~III, we provide
a detailed analysis of the QND measurement of a single-qubit state
by probing the transmissions of the driven resonator. Next, we show
how to use these QND measurements to tomographically reconstruct an
unknown single-qubit state in the experimental circuit-QED system.
The extensions to the two-qubit case are given in Sec.~IV, where the
advantage of our proposal (compared with the previous approach based
on the DP measurements) will be explicitly revealed. The possible
generalization to the $N$-qubit situation and summarizations of our
main results are finally given in Sec.~V.
\section{Transmission of a driven empty cavity}
For the detection of the states of the qubits, we investigate the
photon transmission of a driven resonator by studying the
steady-state properties of the resonator-qubits dynamics. For
generality, we consider a cavity-QED system consisting of $N$
qubits. The Hamiltonian reads
\begin{eqnarray}
H_{}=\hbar\omega_r\hat{a}^\dagger\hat{a}+\sum_{j=1}^N[\frac{\hbar\omega_{j}}{2}\sigma_{z_j}
+\hbar g_j(\sigma_{+_j}\hat{a}+\sigma_{-_j}\hat{a}^\dagger)],
\end{eqnarray}
where $a^{(\dagger)}$ and $\sigma_{\pm_ j}$ are ladder operators for
the photon field and the $j$th qubit respectively. Also, $\omega_r$
is the cavity frequency, $\omega_{j}$ the $j$th qubit transition
frequency, and $g_j$ the coupling strength between the $j$th qubit
and the resonator. Suppose that the cavity is coherently driven by
\begin{eqnarray}
H_d=\hbar\epsilon(\hat{a}^\dagger
e^{-i\omega_dt}+\hat{a}e^{i\omega_dt}),
\end{eqnarray}
where $\epsilon$ is the real amplitude and $\omega_d$ the frequency
of the applied external driving.
Under the Born-Markov approximation, the dynamics of the whole
system with the dissipations and dephasings is described by the
following master equation~\cite{Walls}
\begin{eqnarray}
\dot{\varrho}_N&=&-\frac{i}{\hbar}[{H_N},\varrho_N]+\kappa \mathcal
{D}[a]\varrho_N+\sum_{j=1}^N\gamma_{1,j} \mathcal
{D}[\sigma_{-_j}]\varrho_N\nonumber\\
&&+\sum_{j=1}^N\frac{\gamma_{\phi,j}}{2}
\mathcal{D}[\sigma_{z_j}]\varrho_N,\\
&&H_N=H+H_d.\nonumber
\end{eqnarray}
Here, $\varrho_N$ is the density operator and the dissipation
superoperator is defined by $\mathcal
{D}[A]{\varrho_N}=A{\varrho_N}A^\dagger-A^\dagger
A{\varrho_N}/2-{\varrho_N}A^\dagger A/2$, which describes the
effects of the environment on the system. The parameters of the last
three terms in Eq.~(3) correspond to photon decay rate $\kappa$, the
$j$th qubit decay rate $\gamma_{1,j}$, and the $j$th qubit pure
dephasing rate $\gamma_{\phi,j}$, respectively.
In what follows, we begin with the master equation~(3) to calculate
the frequency-dependent transmission of the cavity, which is
proportional to the steady-state mean photon number
$\langle\hat{a}^\dagger\hat{a}\rangle$ in the cavity. Technically,
to satisfy the basic criteria for the desirable QND measurements of
the $N$-qubit system, we assume that the conditions
\begin{equation}
0<\frac{g_j}{\Delta_j},\,\frac{g_jg_{j'}}{\Delta_j\Delta_{jj'}},\,\frac{g_jg_{j'}}{\Delta_{j'}\Delta_{jj'}}\ll
1,\,\,j\neq j'=1,2,...,N,
\end{equation}
should be satisfied for assuring the effective dispersive coupling
$\sigma_{z_j}\hat{a}^\dagger\hat{a}$ between the $j$th qubit and the
cavity. These conditions assure also that the inter-bit interactions
are negligible. Above, $\Delta_j=\omega_j-\omega_r$ denotes the
detuning between the $j$th qubit and the cavity, and
$\Delta_{jj'}=\omega_j-\omega_j'$ the detuning between the $j$th and
$j'$th qubits.
For contrast, we first calculate the transmission spectrum of a
driven empty cavity. The Hamiltonian of the simplified system
reduces to ($\hbar=1$ throughout the paper)
\begin{eqnarray}
H_0=\omega_r\hat{a}^\dagger\hat{a}+\epsilon(\hat{a}^\dagger
e^{-i\omega_dt}+\hat{a} e^{i\omega_dt}).
\end{eqnarray}
After the time-dependent unitary transformation defined by the
operator $R=\exp(-i\omega_dt\hat{a}^\dagger\hat{a})$, we get the
effective Hamiltonian
\begin{eqnarray}
\tilde{H}_0=R^\dagger H_0R-iR^\dagger\partial{R}/\partial{t}
=-\Delta_{dr}\hat{a}^\dagger\hat{a}+\epsilon(\hat{a}^\dagger+\hat{a}),
\end{eqnarray}
where $\Delta_{dr}=\omega_d-\omega_r$ is the detuning of the cavity
from the driving. Consequently, we get the master equation for such
a driven empty cavity
\begin{eqnarray}
\dot{\varrho}_0&=&-i[\tilde{H}_0,\varrho_0]
+\kappa(\hat{a}\varrho_0\hat{a}^\dagger-\hat{a}^\dagger\hat{a}\varrho_0/2-\varrho_0\hat{a}^\dagger\hat{a}/2),
\end{eqnarray}
where $\varrho_0$ is the density matrix of the empty cavity.
From the above master equation, one can easily obtain the equations
of motion for the expectation values of the relevant operators, such
as mean photon number inside the cavity
$\langle\hat{a}^\dagger\hat{a}\rangle=Tr(\hat{a}^\dagger\hat{a}\varrho_0)$:
\begin{subequations}
\label{eq:whole}
\begin{eqnarray}
\frac{d\langle\hat{a}^\dagger\hat{a}\rangle}{dt}&=
&-\kappa{\langle\hat{a}^\dagger\hat{a}\rangle}-2\epsilon\mathrm{Im}{\langle\hat{a}\rangle},\label{subeq:1}
\end{eqnarray}
with
\begin{eqnarray}
\frac{d\langle\hat{a}\rangle}{dt}&=&(i\Delta_{dr}-\frac{\kappa}{2}){\langle\hat{a}\rangle}-i\epsilon.\label{subeq:2}
\end{eqnarray}
\end{subequations}
The steady-state solution of Eq.~(8) gives
\begin{eqnarray}
\frac{\langle\hat{a}^\dagger\hat{a}\rangle_{ss}}{\epsilon^2}
=\frac{1}{(\omega_d-\omega_r)^2+(\frac{\kappa}{2})^2}.
\end{eqnarray}
Obviously, the transmission spectrum of an empty cavity, which is
proportional to $\langle\hat{a}^\dagger\hat{a}\rangle$, is
well-known Lorentzian: centered at $\omega_d=\omega_r$ with the
half-width $\kappa$. Certainly, when $\omega_d$ does not
sufficiently match the cavity frequency, no photon penetrates the
cavity and thus no transmission is recorded.
\section{Tomographic reconstruction of a single-qubit state by
QND measurements}
\subsection{Nondestructive detection of a single qubit by cavity
transmissions}
Now we investigate the case, in which a single qubit with transition
frequency $\omega_1$ is dispersively coupled to the cavity mode. In
the frame rotating at drive frequency $\omega_d$ characterized by
the transformation $R$, the Hamiltonian of the system reads
\begin{eqnarray}
\tilde{H}_{1}=\frac{\tilde{\omega}_1}{2}\sigma_{z_1}+(-\Delta_{dr}+\Gamma_1\sigma_{z_1})\hat{a}^\dagger\hat{a}
+\epsilon(\hat{a}^\dagger+\hat{a}),
\end{eqnarray}
with $\tilde{\omega}_1=\omega_1+\Gamma_1$ and
$\Gamma_1={g_1^2}/{\Delta_1}$.
Under the Born-Markov approximation, the master equation for the
single-qubit plus the driven resonator is
\begin{eqnarray}
\dot{\varrho}_1&=&-i[\tilde{H}_1,\varrho_1]+\kappa \mathcal
{D}[a]\varrho_1+\gamma_{1,1} \mathcal {D}[\sigma_{-_1}]\varrho_1
\nonumber\\&&+ \frac{\gamma_{\phi,1}}{2}
\mathcal{D}[\sigma_{z_1}]\varrho_1.
\end{eqnarray}
The desirable quantity $\langle\hat{a}^\dagger\hat{a}\rangle$ can be
determined by solving the following coupled equations of motion:
\begin{subequations}
\label{eq:whole}
\begin{eqnarray}
\frac{d\langle\hat{a}^\dagger\hat{a}\rangle}{dt}&
=&-\kappa{\langle\hat{a}^\dagger\hat{a}\rangle}-2\epsilon\mathrm{Im}{\langle\hat{a}\rangle},\label{subeq:1}
\end{eqnarray}
\begin{eqnarray}
\frac{d\langle\hat{a}\rangle}{dt}&
=&(i\Delta_{dr}-\frac{\kappa}{2}){\langle\hat{a}\rangle}-i\Gamma_1{\langle\hat{a}\sigma_{z_1}\rangle}-i\epsilon,\label{subeq:2}
\end{eqnarray}
\begin{eqnarray}
\frac{d\langle\hat{a}\sigma_{z_1}\rangle}{dt}
&=&(i\Delta_{dr}-\frac{\kappa}{2}-\gamma_{1,1}){\langle\hat{a}\sigma_{z_1}\rangle}-(i\Gamma_1+\gamma_{1,1}){\langle\hat{a}\rangle}
\nonumber\\&&-i\epsilon{\langle\sigma_{z_1}\rangle},\label{subeq:3}
\end{eqnarray}
and
\begin{eqnarray}
\frac{d\langle\sigma_{z_1}\rangle}{dt}&=&-\gamma_{1,1}({\langle\sigma_{z_1}\rangle}+1).\label{subeq:4}
\end{eqnarray}
\end{subequations}
It is obvious that the additional measurement-induced dephasing rate
$\gamma_{\phi,1}$ does not influence the solution of the equations.
As the decay $\gamma_{1,1}$ of the qubit is significantly less than
the decay rate $\kappa$ of the driven cavity, the average of
$\sigma_{z_1}$ could be safely assumed to be unchanged during the
measurement. In fact, the characterized time of the detection is
determined mainly by the decay of the cavity $\kappa$. Such a
quantity is about $2\pi\times1.69$ MHZ~\cite{Bianchetti}, which is
obviously larger than $\gamma_{1,1}=2\pi\times0.02$
MHZ~\cite{Wallraff2}. Experimentally, the time interval of
completing a single QND measurement is about
$T_e=40$ns~~\cite{Bianchetti}, this is significantly shorter than
the lifetime: $T_1\sim7.3$$\mu$s and the decoherence time:
$T_2\sim500$ns~\cite{Wallraff2}. Therefore, during such a readout
the decay of the qubit is negligible, i.e.,
$\langle\sigma_{z1}(T_e)\rangle=\exp(-\gamma_{1,1}T_e)(\langle\sigma_{z1}(0)\rangle+1)-1\approx
\langle\sigma_{z1}(0)\rangle$.
Under the steady-state condition, we obtain
\begin{eqnarray}
&&\frac{{\langle\hat{a}^\dagger\hat{a}\rangle}_{ss}}{\epsilon^2}\nonumber\\
&=&\frac{2}{\kappa}\times[(\frac{\kappa}{2}+\gamma_{1,1})(\frac{\kappa^2}{4}+\frac{\gamma_{1,1}\kappa}{2}+\Gamma_1^2-\Delta_{dr}^2)+\nonumber\\
&&(\Delta_{dr}+\Gamma_1\langle\sigma_{z_1}(0)\rangle)(\kappa\Delta_{dr}+\gamma_{1,1}\Delta_{dr}+\gamma_{1,1}\Gamma_1)]\nonumber\\
&&\times[(\frac{\kappa^2}{4}+\frac{\gamma_{1,1}\kappa}{2}+\Gamma_1^2-\Delta_{dr}^2)^2\nonumber\\
&&+(\kappa\Delta_{dr}+\gamma_{1,1}\Delta_{dr}+\gamma_{1,1}\Gamma_1)^2]^{-1},
\end{eqnarray}
which is strongly related to the the initial state of qubit, thus
the qubit state could be determined by the cavity transmission.
\begin{figure}[htbp]
\includegraphics[width=8cm,height=6cm]{hf1.eps}
\caption{(Color online) Cavity transmission for the single-qubit
states versus the probe detuning $\omega_d-\omega_r$. Five cases of
the qubit states for $|\beta_1|^2=0, 0.2, 0.4, 0.5$, and $1$ are
shown. For comparison, the empty cavity (EMC) transmission is also
plotted in black line.
The peak shifts by
$-\Gamma_1$ or $\Gamma_1$ correspond to single logic state
$|0\rangle$ or $|1\rangle$. For the superposition states, the
double-peak relative heights
(in contrast to the peak height of the empty cavity
transmission) present clearly the superposed probabilities of the
two logic states. Here, the parameters are selected as:
$(\Gamma_1,\kappa,\gamma_{1,1})=2\pi\times(-7.38,1.69,0.02)$MHz
~\cite{Bianchetti,Wallraff2}.}
\end{figure}
The measured cavity transmission (normalized to the peak height of
the empty cavity transmission) versus the probe frequency detuning
are plotted in Fig.~1. Generally, the qubit is assumed to be
prepared initially in the state
$|\psi\rangle=\beta_0|0\rangle+\beta_1|1\rangle$. Obviously, when
$\beta_0=0$, or $1$, it reduces to the single logic state
$|1\rangle$ or $|0\rangle$. Compared with the empty cavity
transmission (plotted as the dark line in Fig.~1), one observes that
qubit-resonator coupling leads to a right (left) shift of the single
peak in the transmission spectrum by a quantity $-\Gamma$
($\Gamma$), which is dependent of the logic states for
$\langle\sigma_{z_1}(0)\rangle=-1$
($\langle\sigma_{z_1}(0)\rangle=1$). Thus, the shifts of the peaks
can be used to mark the logic states of the qubit. However, when the
qubit is in the superposition of the two logic states, e.g.,
$|\beta_1|^2=0.2, 0.4$, and $0.5$, respectively, we see that the
situation is very different from the case for the single logic
states. In this case, the spectrum shows two peaks whose locations
coincide with that for the single logic states, but the relative
heights of these two peaks correspond clearly to the superposed
probabilities, i.e., $|\beta_0|^2$ and $|\beta_1|^2$, respectively.
This provides an effective approach to directly measure the
superposed probabilities of a superposed state.
\subsection{Tomographic reconstruction of a single-qubit state}
Above investigation indicates that, partial information of the qubit
state, i.e., the diagonal elements of the relevant density matrix,
can be directly obtained by only one kind of the QND measurements.
While, to extract the full information of an unknown qubit state,
one should tomographically reconstruct all the elements of its
density matrix. Basically, to completely define a $d$-dimensional
density matrix $\rho$, one needs to determine $d^2-1$ real
parameters. Therefore, to determine an unknown qubit state, the key
point is to identify these parameters by virtue of the tomographic
technique.
Now we demonstrate how to perform the tomographic construction of an
arbitrary single-qubit state
$|\psi\rangle_1=\beta_0|0\rangle+\beta_1|1\rangle$, whose density
matrix operator reads
\begin{eqnarray}
\rho_1=\left(
\begin{array}{cccc}
\rho_{00} & \rho_{01} \\
\rho_{10} & \rho_{11} \\
\end{array}\right).
\end{eqnarray}
A more efficient and widely used technique is to parameterize the
density matrix $\rho_1$ on a Bloch sphere~\cite{Liu},
\begin{eqnarray}
\rho_1=\frac{1}{2}(I
+\sum_{i=x,y,z}r_i\sigma_i) =\frac{1}{2}\left(
\begin{array}{cccc}
1+r_{z} & r_{x}-ir_{y} \\
r_{x}+ir_{y} & 1-r_{z} \\
\end{array}\right).
\end{eqnarray}
Here, $I$ denotes the identity matrix, $\sigma_i$ the Pauli
matrices, and $r_i$ real parameters. Therefore, in order to
determine the single-qubit state, we must identify the three
components $(r_x, r_y, r_z)$ of the Bloch vector $\vec{r}$. As
discussed in the previous section, two diagonal elements $\rho_{00}$
and $\rho_{11}$ of the density matrix $\rho_{1}$ can be directly
determined by the two measured occupation probabilities
$|\beta_0|^2$, $|\beta_1|^2$ in the direct QND measurements. This
means that the parameter $r_z$ can be determined by the relation
$r_z=\rho_{00} -\rho_{11}$ $=|\beta_0|^2-|\beta_1|^2$. To obtain the
other two parameters $r_x$ and $r_y$, we need to determine the
non-diagonal elements. To this end, we perform the single-qubit
operations: $U_{x_1}=\exp{(i\pi\sigma_{x_1}/4)}$ and
$U_{y_1}=\exp{(i\pi\sigma_{y_1}/4)}$, to transfer them to the the
relevant diagonal locations, respectively. For example, after the
operation $U_{x_1}$, the density matrix $\rho_1$ is changed to
\begin{eqnarray}
\rho_1'=U_{x_1}\rho_1 U_{x_1}^\dagger
=\frac{1}{2}\left(
\begin{array}{cccc}
1-r_{y} & r_{x}-ir_{z} \\
r_{x}+ir_{z} & 1+r_{y} \\
\end{array}\right).
\end{eqnarray}
Now, performing another kind of QND measurements the parameters
$|\beta'_0|^2$ and $|\beta'_1|^2$ can be measured. Consequently, the
coefficient $r_y$ can be determined via the relation
$r_y=|\beta'_1|^2-|\beta'_0|^2$. Similarly, by performing the
quantum operation $U_{y_1}$ on the original density matrix $\rho_1$,
another new density matrix
\begin{eqnarray}
\rho_1''=U_{y_1}\rho_1 U_{y_1}^\dagger
=\frac{1}{2}\left(
\begin{array}{cccc}
1+r_{x} & -r_{z}-ir_{y} \\
-r_{z}+ir_{y} & 1-r_{x} \\
\end{array}\right),
\end{eqnarray}
can be obtained and the coefficient $r_x$ can be similarly
determined. Note that here the number of the unitary operations
required for implementing the quantum state tomography (based on the
QND measurements) is the same as the previous approach (based on the
usual DP measurements). Thus, for the single-qubit case the
complexity of the present approach is the same as that in the
previous one. Note that here, as the same as that in the previous
approach based on the DP measurements, three kinds of QND
measurements are still required for the present reconstructions. One
is directly applied, another is applied after the $U_{x_1}$
operation, and the final one is applied after the $U_{y_1}$
operation, thus the efficiency is not enhanced.
The remaining task is to implement the single-qubit operations
required above for transferring the non-diagonal elements to the
diagonal locations. We work with a circuit-QED system wherein a
superconducting charge qubit is coupled to the fundamental mode of a
transmission line resonator~\cite{Makhlin}. Let the qubit work at
its degeneracy point and neglect the fast oscillating terms under
the rotating-wave approximation (RWA). Following Ref.~\cite{Blais2},
under one displacement transformation, the effective Hamiltonian of
the resonator plus qubit system can be written as
\begin{eqnarray}
\tilde{H}=-\Delta_{dr}\hat{a}^\dagger\hat{a}+
\frac{\Delta_a}{2}\sigma_{z_1}
+g_1(\hat{a}^\dagger\sigma_{-_1}+\hat{a}\sigma_{+_1})
+\frac{\Omega}{2}\sigma_{x_1},\nonumber\\
\end{eqnarray}
with the detuning of the qubit transition frequency from the drive
$\Delta_a=\omega_1-\omega_d$ and the Rabi frequency
$\Omega=2\epsilon g_1/(-\Delta_{dr})$. Next, supposing that this
system works in the dispersive regime, i.e., $|g_1/\Delta_1|\ll 1$,
after the transformation
$U_1=\exp{[{-g_1}(\hat{a}^\dagger\sigma_{-_1}-\hat{a}\sigma_{+_1})/{\Delta_1}]}$,
then the above Hamiltonian becomes
\begin{eqnarray}
H_x =-\Delta_{dr}\hat{a}^\dagger\hat{a}
+\frac{\tilde{\Delta}_a}{2}\sigma_{z_1}+\frac{\Omega}{2}\sigma_{x_1},\,
\tilde{\Delta}_a=\Delta_a+\Gamma_1.\nonumber\\
\end{eqnarray}
First, if the condition $\tilde{\Delta}_a=0$ is satisfied, then the
Hamiltonian (19) produces a rotation of the qubit about the $x$
axis, i.e., $U_{x_1}$ could be generated by choosing the evolution
time $t_x=\pi/(2\Omega)$. Second, if the driving is sufficiently
detuned from the qubit and its amplitude is also sufficiently large
enough, then another approximate Hamiltonian
\begin{eqnarray}
&&H_z =-\Delta_{dr}\hat{a}^\dagger\hat{a}+
\frac{1}{2}(\tilde{\Delta}_a+\frac{1}{2}\frac{\Omega^2}{\Delta_a})\sigma_{z_1},
\end{eqnarray}
can be obtained by further performing a transformations
$U_2=\exp{({\beta}^*\sigma_{+_1}-{\beta}\sigma_{-_1})}$, with the
coefficient $\beta=\Omega/(2\Delta_a)$, on the Hamiltonian (19).
Obviously, the desirable operation $U_{z_1}$ can be implemented by
the evolution under the Hamiltonian (20) with the duration
$t_z=\pi\Delta_a/(2\Delta_a\tilde{\Delta}_a+\Omega^2)$. Third, the
desirable operation $U_{y_1}$ could be constructed as:
$U_{y_1}=\exp(i\pi\sigma_{y_1}/4)=\exp(i\pi\sigma_{z_1}/4)\exp(i3\pi\sigma_{x_1}/4)
\exp(i3\pi\sigma_{z_1}/4)$. It should be pointed out that the
durations $t_x$ (or $t_y$) of the single-qubit operations required
above for implementing the desirable tomographies is estimated as
$\sim100$ps using the experimental parameters:
$\epsilon\sim2\pi\times 20$MHz~\cite{Gambetta}, and $\Delta_{dr}\sim
\kappa/2$~\cite{Blais2}. This is significantly less by at least two
orders than the qubit decoherence time, which is measured as $\sim
500$ns~\cite{Blais2}. Therefore, the required gate operations are
accessible and the proposed tomographic reconstructions are
experimentally feasible.
As an example, we assume that the three parameters $r_x=0.6$,
$r_y=0.5$, $r_z=0.6$ are obtained through the above reconstructions,
then the reconstructed state $\rho_1$ can be written as
$\rho_1=0.8|0\rangle\langle0|
+(0.3-0.25i)|0\rangle\langle1|+(0.3+0.25i)|1\rangle\langle0|+0.2|1\rangle\langle1|$,
whose real $\rho_{ij}^{(R)}$ and imaginary $\rho_{ij}^{(I)}$ parts
(i, j=0, 1) are graphically represented in Fig.~2.
\begin{figure}[htbp]
\includegraphics[width=8cm,height=4cm]{hf2.eps}
\caption{(Color online) Graphic representations of the density
matrix $\rho_1$ for a single-qubit state. The real $\rho_{ij}^{(R)}$
and imaginary $\rho_{ij}^{(I)}$ parts of the density matrix elements
$\rho_{ij}=\langle i|\rho|j\rangle$ (i, j=0, 1) are plotted in (a)
and (b), respectively.}
\end{figure}
\section{Tomographic reconstruction of a two-qubit state by
QND measurements}
\subsection{Nondestructive detection of an unknown two-qubit state by cavity transmissions}
We extend the above sing-qubit QND measurements to the two-qubit
case. The transition frequencies of the two qubits are represented
as $\omega_1$ and $\omega_2$, respectively. In the above dispersive
condition (4) and in a framework rotating at $\omega_{d}$, the
effective Hamiltonian of the present complete system is
\begin{eqnarray}
\tilde{H}_2&=&(-\Delta_{dr}+\Gamma_1\sigma_{z_1}+\Gamma_2\sigma_{z_2})\hat{a}^\dagger\hat{a}\nonumber\\
&&+\frac{\tilde{\omega}_1}{2}\sigma_{z_1}+\frac{\tilde{\omega}_2}{2}\sigma_{z_2}
+\epsilon(\hat{a}^\dagger+\hat{a}),
\end{eqnarray}
where $\Gamma_j=g_j^2/\Delta_{j}$ and
$\tilde{\omega}_j=\omega_j+\Gamma_j$, $j=1,2$.
Similarly, the relevant master equation reads
\begin{eqnarray}
\dot{\varrho}_2&=&-i[\tilde{H}_2,\varrho_2]+\kappa \mathcal
{D}[a]\varrho_2+\sum_{j=1,2}\gamma_{1,j} \mathcal {D}[\sigma_{-_j}]\varrho_2\nonumber\\
&&+ \sum_{j=1,2}\frac{\gamma_{\phi,j}}{2}
\mathcal{D}[\sigma_{z_j}]\varrho_2.
\end{eqnarray}
and the equations of motion for the mean values of various
expectable operators are
\begin{subequations}
\label{eq:whole}
\begin{eqnarray}
\frac{d\langle\hat{a}^\dagger\hat{a}\rangle}{dt}&
=&-\kappa{\langle\hat{a}^\dagger\hat{a}\rangle}-2\epsilon\mathrm{Im}{\langle\hat{a}\rangle},\label{subeq:1}
\end{eqnarray}
\begin{eqnarray}
\frac{d\langle\hat{a}\rangle}{dt}&
=&(i\Delta_{dr}-\frac{\kappa}{2}){\langle\hat{a}\rangle}-i\Gamma_1{\langle\hat{a}\sigma_{z_1}\rangle}-i\Gamma_{2}{\langle\hat{a}\sigma_{z_2}\rangle}-i\epsilon,\nonumber\\\label{subeq:2}
\end{eqnarray}
\begin{eqnarray}
\frac{d\langle\hat{a}\sigma_{z_1}\rangle}{dt}&=&(i\Delta_{dr}-\frac{\kappa}{2}-\gamma_{1,1}){\langle\hat{a}\sigma_{z_1}\rangle}-(i\Gamma_1+\gamma_{1,1}){\langle\hat{a}\rangle}\nonumber\\
&&-i\Gamma_2{\langle\hat{a}\sigma_{z_1}\sigma_{z_2}\rangle}-i\epsilon{\langle\sigma_{z_1}\rangle},\label{subeq:3}
\end{eqnarray}
\begin{eqnarray}
\frac{d\langle\hat{a}\sigma_{z_2}\rangle}{dt}&=&(i\Delta_{dr}-\frac{\kappa}{2}-\gamma_{1,2}){\langle\hat{a}\sigma_{z_2}\rangle}-(i\Gamma_2+\gamma_{1,2}){\langle\hat{a}\rangle}\nonumber\\
&&-i\Gamma_{1}{\langle\hat{a}\sigma_{z1}\sigma_{z_2}\rangle}-i\epsilon{\langle\sigma_{z_2}\rangle},\label{subeq:4}
\end{eqnarray}
\begin{eqnarray}
\frac{d\langle\hat{a}\sigma_{z_1}\sigma_{z_2}\rangle}{dt}&=&(i\Delta_{dr}-\frac{\kappa}{2}-\gamma_{1,1}-\gamma_{1,2}){\langle\hat{a}\sigma_{z_1}\sigma_{z_2}\rangle}\nonumber\\
&&-i\epsilon{\langle\sigma_{z_1}\sigma_{z_2}\rangle}-(i\Gamma_2+\gamma_{1,2}){\langle\hat{a}\sigma_{z_1}\rangle}\nonumber\\
&&-(i\Gamma_1+\gamma_{1,1}){\langle\hat{a}\sigma_{z_2}\rangle},\label{subeq:5}
\end{eqnarray}
\begin{eqnarray}
\frac{d\langle\sigma_{z_1}\rangle}{dt}&=&-\gamma_{1,1}({\langle\sigma_{z_1}\rangle}+1),\label{subeq:6}
\end{eqnarray}
\begin{eqnarray}
\frac{d\langle\sigma_{z_2}\rangle}{dt}&=&-\gamma_{1,2}({\langle\sigma_{z_2}\rangle}+1),\label{subeq:7}
\end{eqnarray}
\begin{eqnarray}
\frac{d\langle\sigma_{z_1}\sigma_{z_2}\rangle}{dt}&=&-(\gamma_{1,1}+\gamma_{1,2})\langle\sigma_{z_1}\sigma_{z_2}\rangle-\gamma_{1,1}
\langle\sigma_{z_2}\rangle\nonumber\\
&&-\gamma_{1,2}\langle\sigma_{z_1}\rangle.\label{subeq:8}
\end{eqnarray}
\end{subequations}
Again, due to the relatively-long decoherence times of the qubits
and their sufficiently short measured times, the additional
measurement-induced dephasing and decay rates of the qubits are also
unimportant. Thus, the expectable values of the qubit operators can
still be regarded as unchanged, i.e.,
$\langle\sigma_{z_j}(t)\rangle\approx\langle\sigma_{z_j}(0)\rangle$
and
$\langle\sigma_{z_1}(t)\sigma_{z_2}(t)\rangle\approx\langle\sigma_{z_1}(0)\sigma_{z_2}(0)\rangle$,
during the desirable QND measurements. As a consequence, one can
easily solve the above Eqs. (23a-e) and finally obtain the exact
steady-state distribution of the intracavity photon number
\begin{widetext}
\begin{eqnarray}
\frac{\langle\hat{a}^\dagger\hat{a}\rangle_{ss}}{\epsilon^2}=\frac{2}{\kappa}\mathrm{Re}\left\{\frac{
F(\sum_{j,j'}{B_jD_{j'}G_j}+D_1D_2) +B_1B_2[G_{12}(D_1+D_2)
+\sum_{j,j'}{E_jG_{j'}}]-\sum_{j}{B_jE_j(D_j+B_jG_j)}}
{\sum_{j,j'}B_jE_j(D_{j'}F+D_jA)-{(B_1E_1-B_2E_2)^2}
-AD_1D_2F}\right\},\nonumber\\
j,j'=1,2,
j\neq{j'}.
\end{eqnarray}
\end{widetext}
Here,
$A=i\Delta_{dr}-\frac{\kappa}{2},B_j=i\Gamma_j,D_j=i\Delta_{dr}-\frac{\kappa}{2}-\gamma_{1,j},
E_j=i\Gamma_j+\gamma_{1,j},F=i\Delta_{dr}-\frac{\kappa}{2}-\gamma_{1,j}-\gamma_{1,j'}$,
$G_j=\langle\sigma_{z_j}(0)\rangle$, and
$G_{12}=\langle\sigma_{z_1}(0)\sigma_{z_2}(0)\rangle$.
\begin{figure}[htbp]
\includegraphics[width=8cm,height=6cm]{hf3.eps}
\caption{(Color online) (a). Cavity transmission of the cavity
versus the probe detuning $\omega_d-\omega_r$ for certain slected
two-qubit states with $|\alpha_0|^2=1$, $|\alpha_1|^2=1$,
$|\alpha_2|^2=1$, $|\alpha_3|^2=1$, $(|\alpha_0|^2$, $|\alpha_1|^2$,
$|\alpha_2|^2$, $|\alpha_3|^2)=$ $(0.5, 0, 0, 0.5)$, and
$(|\alpha_0|^2$, $|\alpha_1|^2$, $|\alpha_2|^2$, $|\alpha_3|^2)=$
$(0.1, 0.2, 0.3, 0.4)$, respectively. For comparison, the empty
cavity (EMC) transmission is also plotted in black line. Here,
The parameters are
$(\Gamma_1,\Gamma_2,\kappa,\gamma_{1,1},\gamma_{1,2})
=2\pi\times(-11.11,-9.11,1.7,0.02,0.022,)$MHz
~\cite{Filipp,Wallraff2}. In (b) only the parameters $\Gamma_1$ and
$\Gamma_2$ are modified as $\Gamma_1'=1.05 \Gamma_1$ and
$\Gamma_2'=0.85\Gamma_2$. In this case, the relative heights of the
peaks are exactly equivalent to the corresponding probabilities of
the single logic states superposed in the measured superposition
state.}
\end{figure}
We now investigate the above distributions schematically for various
typically selected two-qubit initial states. First, we assume that
the two-qubit is initially prepared at only one of the four logic
states, i.e., in the generic expression
$|\psi\rangle_2=\alpha_1|00\rangle+\alpha_2|01\rangle+\alpha_3|10\rangle+\alpha_4|11\rangle$
only one of the four probability amplitudes equals $1$, e.g.,
$\alpha_1=1$, $\alpha_2=\alpha_3$ $=\alpha_4$ $=0$. Fig.~3 shows
clearly that single peaks reveal the inputs of these four single
logic states, and they can also be distinguished by the shifts of
the central frequencies of the transmission spectrum. The peaks with
frequency shifts: $-\Gamma_1-\Gamma_2$, $-\Gamma_1+\Gamma_2$,
$\Gamma_1-\Gamma_2$, and $\Gamma_1+\Gamma_2$ mark the state
$|00\rangle$, $|01\rangle$, $|10\rangle$, and $|11\rangle$,
respectively. Thus the pulls of the cavity are strongly dependent of
the states of the qubits. For these single logic states the heights
of the single peaks are exactly equivalent and of unity value, which
is the same as that for the EMC. Next, for the superposition of the
four single logic states the situations are quite different. For
example, Fig.~3 (a) also shows that, if the two-qubit is prepared
initially as one of the Bell states: $(|\alpha_1|^2$,
$|\alpha_2|^2$, $|\alpha_3|^2$, $|\alpha_4|^2)=(0.5, 0, 0, 0.5)$,
then the transmitted spectrum of the cavity reveals two peaks; their
locations are respectively at the positions for the single states
$|00\rangle$ and $|11\rangle$, but have the same relative heights.
Moreover, for a more generic superposed state $(|\alpha_1|^2$,
$|\alpha_2|^2$, $|\alpha_3|^2$, $|\alpha_4|^2)=(0.1, 0.2, 0.3, 0.4)$
one can see that four peaks are exhibited simultaneously. The
central positions of these peaks locate at the corresponding
positions of single logic states $|00\rangle, |01\rangle,
|10\rangle$, and $|11\rangle$, respectively. The relative heights of
them read $0.1, 0.212, 0.308$, and $0.4$, respectively. Here, the
relative heights of the peaks marking the states $|00\rangle$ and
$|11\rangle$ are exactly equivalent to the superposed probabilities
$|\alpha_1|^2$ and $|\alpha_4|^2$. However, the relative heights of
the peaks marking the states $|01\rangle$ and $|10\rangle$ deviate
from the corresponding superposed probabilities $|\alpha_2|^2$ and
$|\alpha_3|^2$. This is because these two peaks are not well
distinguished due to the contributions from these two logic states'
overlap. As a consequence, each peak is higher a little than the
expected one, i.e., the superposed probability of the relevant logic
state. While, such a situation does not exist for the $|00\rangle$
and $|11\rangle$ peaks (the relative heights of them equal to the
expected ones), as they are separated sufficiently far from the
others. In Fig.~3 (b) we modify the relevant parameters such as
$\Gamma_1'=1.05 \Gamma_1$ and $\Gamma_2'=0.85\Gamma_2$. Then we find
that each peak of the transmission of the cavity is well separated
from the others, and thus its relative height is exactly equal to
the expectable superposed probability of the corresponding logic
state in the measured two-qubit state.
\subsection{High efficiency tomographic reconstruction of a two-qubit state}
The two-qubit state tomography is done in the same way as that for
the single-qubit state. The only difference is that now there are
$15$ real parameters to be determined in the 4-dimensional density
matrix operator $\rho_2$, and thus more operations are required to
transfer the nondiagonal elements in $\rho_2$ to the diagonal
locations. Generally, the 4-dimensional density matrix for a
two-qubit state
$|\psi\rangle_2=\alpha_1|00\rangle+\alpha_2|01\rangle+\alpha_3|10\rangle+\alpha_4|11\rangle$
in a complete basis $\{|1\rangle_2=|00\rangle$,
$|2\rangle_2=|01\rangle$, $|3\rangle_2=|10\rangle$,
$|4\rangle_2=|11\rangle\}$ can be represented as
\begin{eqnarray}
\rho_2=\left(
\begin{array}{cccc}
\rho_{11} & \rho_{12}&\rho_{13} & \rho_{14} \\
\rho_{21} & \rho
_{22}&\rho_{23} & \rho_{24} \\
\rho_{31} & \rho
_{32}&\rho_{33} & \rho_{34} \\
\rho_{41} & \rho_{42}&\rho_{43} & \rho_{44} \\
\end{array}\right),
\end{eqnarray}
which can also be rewritten as $\rho_2=\bar{\rho}_2/4$
with~\cite{Liu}
\begin{widetext}
\begin{eqnarray}
&& \bar{\rho}_2=\sum_{m,n=0,x,y,z}r_{mn}\sigma_{m_1}\otimes\sigma_{n_2}\nonumber\\
&&=\left(
\begin{array}{cccc}
r_{00}+r_{0z}+r_{z0}+r_{zz} &r_{0x}+r_{zx}-ir_{0y}-ir_{zy} &r_{x0}+r_{xz}-ir_{y0}-ir_{yz} &r_{xx}-r_{yy}-ir_{xy}-ir_{yx} \\
r_{0x}+r_{zx}+ir_{0y}+ir_{zy} &r_{00}-r_{0z}+r_{z0}-r_{zz} &r_{xx}+r_{yy}+ir_{xy}-ir_{yx} &r_{x0}-r_{xz}-ir_{y0}+ir_{yz} \\
r_{x0}+r_{xz}+ir_{y0}+ir_{yz} &r_{xx}+r_{yy}-ir_{xy}+ir_{yx} &r_{00}+r_{0z}-r_{z0}-r_{zz} &r_{0x}-r_{zx}-ir_{0y}+ir_{zy} \\
r_{xx}-r_{yy}+ir_{xy}+ir_{yx} &r_{x0}-r_{xz}+ir_{y0}-ir_{yz} &r_{0x}-r_{zx}+ir_{0y}+ir_{zy} &r_{00}-r_{0z}-r_{z0}+r_{zz}\\
\end{array}\right).\nonumber\\
\end{eqnarray}
\end{widetext}
Here, $\sigma_{m=x,y,z}$ are the Pauli operators and $\sigma_{0}$
identity matrix, and what we want to determine is sixteen real
parameters $r_{mn}$. Note that the first and second subscripts of
the matrix elements $\rho_{ij}$ (i, j=1, 2, 3, 4) in Eq.~(25) and
$r_{mn}$ in Eq.~(26) is labeled for the first and second qubits,
respectively.
As in the above discussion, performing the QND measurements on the
two-qubit state $\rho_2$ can directly determine all the four
diagonal elements: $\rho_{11}$, $\rho_{22}$, $\rho_{33}$ and
$\rho_{44}$, respectively, by the measured results $|\alpha_1|^2$,
$|\alpha_2|^2$, $|\alpha_3|^2$ and $|\alpha_4|^2$. As a consequence,
the parameters $r_{00}$, $r_{0z}$, $r_{z0}$ and $r_{zz}$ can be
determined by
\begin{eqnarray}
&&r_{00}=|\alpha_1|^2+|\alpha_2|^2+|\alpha_3|^2+|\alpha_4|^2=1,\nonumber\\
&&r_{0z}=|\alpha_1|^2-|\alpha_2|^2+|\alpha_3|^2-|\alpha_4|^2 ,\nonumber\\
&&r_{z0}=|\alpha_1|^2+|\alpha_2|^2-|\alpha_3|^2-|\alpha_4|^2,\nonumber\\
&&r_{zz}=|\alpha_1|^2-|\alpha_2|^2-|\alpha_3|^2+|\alpha_4|^2.
\end{eqnarray}
To determine the other $12$ parameters, we need to perform certain
unitary operations to transfer them to the diagonal locations for
other QND measurements.
It is well-known that, arbitrary two-qubit operation assisted by
arbitrary rotations of the single qubits generate an universal set
of quantum gates. So the key to implement the above required
operations for tomographies is to realize a two-qubit gate. Again,
for the experimental circuit QED system with two superconducting
charge qubits, such a gate could be implemented by using the
so-called FLICFORQ protocol~\cite{Blais2}. In fact, if the cavity is
driven by two external fields satisfying the sideband matching
condition: $\omega_{d_2}-\omega_{d_1}=\Omega_1+\Omega_2$, then an
effective Hamiltonian
\begin{eqnarray}
\tilde{H}_{\rm{FF}}=\omega_r\hat{a}^\dagger\hat{a}+
\frac{g_1g_2(\Delta'_1+\Delta'_2)}{16\Delta'_1\Delta'_2}(\sigma_{y_1}\otimes\sigma_{y_2}+
\sigma_{z_1}\otimes\sigma_{z_2}),\nonumber\\
\end{eqnarray}
can be induced in a quadruply rotating framework. Here,
$\Delta'_j={\omega}_{_j}+2{\Omega^2_{jj'}}/{\Delta_{_{jdj'}}}-\omega_r$
with $\Omega_{jj'}=2g_j\epsilon_{j'}/(\omega_{d_{j'}}-\omega_r)$,
and $\Delta_{_{jdj'}}=\omega_{_j}-\omega_{d_{j'}}$, $j, j'=1, 2$, $
j\neq j'$. Obviously, the evolution under the above Hamiltonian with
the duration, e.g., around $\sim100$ps, for the experimental
parameters~\cite{Blais2}, can produce a two-qubit operation:
\begin{equation}
U_{\rm{FF}}=\exp[i{\pi}(\sigma_{y_1}\otimes\sigma_{y_2}+
\sigma_{z_1}\otimes\sigma_{z_2})/{4}].
\end{equation}
On the other hand, the typical single-qubit gates, e.g.,
$U_{x_j},\,U_{y_j}$, and $U_{z_j}$ (j=1, 2) can be relatively easy
to produce using the similar approaches presented in Sec. III. With
such a two-qubit operation and these single-qubit gates, we show in
Table I how to perform the desirable unitary operations for
transferring the non-diagonal elements to the diagonal locations.
For example, by performing a selected operational sequence
$W=U_{\rm{FF}}U_{x_1}$ on the original density matrix $\rho_2$, we
have a new density matrix $\rho_2'=W \rho_2 W^\dagger$, and the new
diagonal elements are
\begin{eqnarray}
&&\rho_{11}'=\frac{1}{4}(r_{00}+r_{xy}-r_{yz}+r_{zx}),\nonumber\\
&&\rho_{22}'=\frac{1}{4}(r_{00}+r_{xy}+r_{yz}-r_{zx}),\nonumber\\
&&\rho_{33}'= \frac{1}{4}( r_{00}-r_{xy}+r_{yz}+r_{zx}),\nonumber\\
&&\rho_{44}'=\frac{1}{4}( r_{00}-r_{xy}-r_{yz}-r_{zx}).
\end{eqnarray}
Then, by the QND measurements the values of
$|\alpha'_1|^2,|\alpha'_2|^2$, $|\alpha'_3|^2$, and $|\alpha'_4|^2$
are given directly. As a consequence, the desirable parameters
$r_{00}$, $r_{xy}$, $r_{yz}$ and $r_{zx}$ can be obtained by the
relations:
\begin{eqnarray}
&&r_{00}=|\alpha'_1|^2+|\alpha'_2|^2+|\alpha'_3|^2+
|\alpha'_4|^2=1,\nonumber\\
&&r_{xy}=|\alpha'_1|^2+|\alpha'_2|^2-|\alpha'_3|^2-|\alpha'_4|^2,\nonumber\\
&&r_{yz}=-|\alpha'_1|^2+|\alpha'_2|^2+|\alpha'_3|^2-|\alpha'_4|^2,\nonumber\\
&&r_{zx}=|\alpha'_1|^2-|\alpha'_2|^2+|\alpha'_3|^2-|\alpha'_4|^2.
\end{eqnarray}
Similarly, other non-diagonal elements can also be determined.
Clearly, here only six kinds of QND measurements are sufficient to
tomographically reconstruct a two-qubit state. This is obviously
simpler than the previous tomographies based on the usual DP
measurements, wherein $15$ kinds of measurements are probably
required~\cite{Liu,Filipp}. Thus, the present tomographies is
essentially high efficient.
After performing all the QND measurements listed in the table, a
two-qubit state can be completely reconstructed. For example, a
two-qubit state $\rho_2$ having the following representation:
\begin{eqnarray}
\rho_2=\left(
\begin{array}{cccc}
r_{00}&r_{0x}&r_{0y}&r_{0z}\\
r_{x0}&r_{xx}&r_{xy}&r_{xz}\\
r_{y0}&r_{yx}&r_{yy}&r_{yz}\\
r_{z0}&r_{zx}&r_{zy}&r_{zz}\\
\end{array}\right)=
\left(
\begin{array}{cccc}
1 & 0 & 0 & -\frac{1}{5} \\
0 & \frac{1}{4} & 0 & \frac{3}{5} \\
0 & 0 & -\frac{1}{4} & 0 \\
-\frac{2}{5} & \frac{1}{8} & 0 & 0 \\
\end{array}\right),
\end{eqnarray}
which can be effectively reconstructed by these parameters
\begin{widetext}
\begin{eqnarray}
\rho_2=\left(
\begin{array}{cccc}
\rho_{11} & \rho_{12}&\rho_{13} & \rho_{14} \\
\rho_{21} & \rho
_{22}&\rho_{23} & \rho_{24} \\
\rho_{31} & \rho
_{32}&\rho_{33} & \rho_{34} \\
\rho_{41} & \rho_{42}&\rho_{43} & \rho_{44} \\
\end{array}\right)
=\left(
\begin{array}{cccc}
0.1 & 0.0313-0.0313i& 0.15-0.15i & -0.125i \\
0.0313+0.0313i& 0.2 & 0.125 & -0.15+0.15i\\
0.15+ 0.15i & 0.125 & 0.3 & -0.0313+0.0313i\\
0.125i& -0.15-0.15i & -0.0313+0.0313i & 0.4\\
\end{array}\right),
\end{eqnarray}
\end{widetext}
determined by six kinds of QND measurements. The simulated
reconstructions are graphically shown in Fig.~4, where
$\rho_{ij}^{(R)}$ and $\rho_{ij}^{(I)}$ are the real and imaginary
parts of the reconstructed state $\rho_2$ in the bases
$|1\rangle_2=|00\rangle$, $|2\rangle_2=|01\rangle$,
$|3\rangle_2=|10\rangle$, $|4\rangle_2=|11\rangle$, with $i, j=1, 2,
3, 4$.
\begin{figure}[htbp]
\includegraphics[width=8cm,height=4cm]{hf4.eps}
\caption{(Color online) Schematic representations of the density
matrix $\rho_2$ for a two-qubit state. The real $\rho_{ij}^{(R)}$
and imaginary $\rho_{ij}^{(I)}$ parts $(i, j=1, 2, 3, 4)$ of the
density matrix elements
in the complete bases are plotted in (a) and
(b), respectively.}
\end{figure}
\begin{table}
\caption{The operational combinations before the QND measurements to
determine the parameters for tomographically reconstructing a
two-qubit state. The subscript $"1(2)"$ of $U$ is labeled for the
operation of qubit $1(2)$.}
\begin{ruledtabular}
\begin{tabular}{lcr}
quantum operation $W$ & determined parameters\\
\hline
no & $r_{00}$, $r_{zz}$, $r_{0z}$, $r_{z0}$\\
$U_{\rm{FF}}U_{x_1}$ & $r_{00}$, $r_{xy}$, $r_{yz}$, $r_{zx}$\\
$U_{\rm{FF}}U_{y_1}$ & $r_{00}$, $r_{yx}$, $r_{zy}$, $r_{xz}$\\
$U_{\rm{FF}}U_{z_1}$ & $r_{00}$, $r_{xx}$, $r_{yy}$, $r_{zz}$\\
$U_{y_1}U_{z_1}U_{\rm{FF}}$ & $r_{00}$, $r_{0x}$, $r_{y0}$, $r_{yx}$\\
$U_{y_2}U_{z_2}U_{\rm{FF}}$ & $r_{00}$, $r_{x0}$, $r_{0y}$, $r_{xy}$\\
\end{tabular}
\end{ruledtabular}
\end{table}
\section{Discussions and Conclusions}
Generally, the quantum state tomographic constructions demonstrated
above can be extended for $N$ (with $N>2$) qubits in a
straightforward manner. This is because that the proposed QND
measurements can be directly applied to determine all the diagonal
elements of the arbitrary $N$-qubit state; the individual superposed
logic states can be inferred from the relevant positions of the
measured peaks, and the probabilities of the corresponding
computational bases superposed in the measured state could be
extracted from the relative heights of the peaks (when they separate
sufficiently from the others). Moreover, all the required operations
for the tomographic reconstructions can be implemented from the
universal set of the logic gates demonstrated.
In summary, we have proposed a scheme to perform the quantum state
tomographies by QND measurements. Differing from the usual
tomographies based on the DP measurements, here the QND measurements
are utilized. Since all the diagonal elements of the density matrix
of an unknown quantum state can be simultaneously determined by a
single kind of QND measurements, the efficiency of the present
tomographic reconstruction is definitely better for more qubits.
Specifically, our proposal is demonstrated with the current circuit
QED setup with a few charge qubits, and could be generalized to
other systems, in principle.
\section*{Acknowledgments}
This work was supported in part by the National Science Foundation
grant No. 10874142, 90921010, and the National Fundamental Research
Program of China through Grant No. 2010CB923104, and the Fundamental
Research Funds for the Central Universities No. SWJTU09CX078, and
A*STAR of Singapore under research grant No. WBS: R-144-000-189-305.
|
{
"timestamp": "2010-09-23T02:00:50",
"yymm": "1009",
"arxiv_id": "1009.4252",
"language": "en",
"url": "https://arxiv.org/abs/1009.4252"
}
|
\section{Introduction}
DGMRES is a new iterative method for computing the Drazin-inverse
solution of linear systems \cite{sid2}. Consider the linear system
\begin{equation}\label{1}
Ax=b
\end{equation}
where $A\in\Bbb{C}^{n\times n}$ is a singular matrix. We recall that
the Drazin-inverse solution of (\ref{1}) is the vector $A^Db$, where
$A^D$ is the Drazin-inverse of the singular matrix $A$.
In \cite{sid1}, A.Sidi proposed a general approach to Krylov
subspace methods for computing Drazin-inverse solution. In that
paper the authors do not put any restriction on $A$, that is, A is
non-hermitian or hermitian, index of $A$ is arbitrary and the
spectrum of A can have any shape. In \cite{sid2}, A.Sidi gave one of
the Krylov subspace method named DGMRES, which is a GMRES-like
algorithm. Like GMRES, in practical use, we often propose restarted
DGMRES which denoted by DGMRES(m). DGMRES(m) is an economical
computing and storgewise method for the Drazin-inverse solution. But
restarting slows down the convergence and often stagnates (see
\cite{zhou1},\cite{zhou2}). In the present paper we propose a DGMRES
method augmented with eigenvectors, which can accelerate the
convergence and overcome the stagnation. Classical GMRES augmented
with eigenvectors was studied by R.Morgan (see \cite{mor}). Now we
derive a DGMRES augmented with eigenvectors similarly. We give the
convergence analysis of DGMRES augmented with eigenvectors which
shows this method is more effective than DGMRES without augment.
Then some numerical experiments are presented to show the
convergence rate of DGMRES augmented with eigenvectors is
remarkably improved, especially when the matrix has a very large or
small nonzero eigenvalue.
The paper is organized as follows. In section 2, we will give a
brief review of DGMRES. In section 3, we obtain the convergence
analysis of DGMRES augmented with eigenvectors and derive the
algorithm. In section 4, we present some numerical examples to
compare the DGMRES with the DGMRES augmented with eigenvectors.
\section{DGMRES}
Throughout this paper, we suppose the index of $A$ is known. The
index of $A$, denoted by $ind(A)$, is the size of the largest Jordan
block corresponding to the zero eigenvalue of $A$. DGMRES is a
Krylov subspace methods for computing the Drazin-inverse solution
$A^Db$. For more details we refer the readers to
\cite{sid1},\cite{sid2}.
We start with a initial vector $x_0$ and the method we interested in
is to generate a sequence of vectors $x_1,x_2,\cdots,$ which
satisfies
$$x_m=x_0+q_{m-1}(A)r_0, \ \ r_0=:b-Ax_0,$$
where $q_{m-1}(\lambda)=\sum\limits_{i=1}^{m-a}c_i\lambda^{a+i-1}$
$(a:=ind(A))$. Then
\begin{equation}\label{2}
r_m:=b-Ax_m=p_m(A)r_0
\end{equation}
where $p_m(\lambda)=1-\lambda
q_{m-1}(\lambda)=1-\sum\limits_{i=1}^{m-a}c_i\lambda^{a+i}$.
The Krylov subspace we will use is
$$K_m(A;A^ar_0)=span\{A^ar_0,A^{a+1}r_0,\cdots,A^{m-1}r_0\}$$.
The vector $x_m$ produced by DGMRES satisfies
\begin{equation}\label{4}
\|A^ar_m\|=\min_{x\in x_0+K_m(A;A^ar_0)} \|A^a(b-Ax)\|.
\end{equation}
Now we give restarting DGMRES Algorithm DGMRES(m).
\begin{alg} DGMRES(m)\\
1. Choose initial guess $x_0$, and compute $r_0=b-Ax_0$ and $A^ar_0$;\\
2. Compute $\beta=\|A^ar_0\|$ and set $v_1=\beta^{-1}A^ar_0$;\\
3.Orthogonalize the vectors: $A^ar_0,A^{a+1}r_0,\cdots,A^mr_0$ by
Arnoldi-Gram-Schmidt process.\\
For $j=1,2,\cdots$ do
For $i=1,2,\cdots$ do
Compute $h_{i,j}=\langle AV_j,V_i\rangle$.
Compute $\widehat{v}_j=Av_j-\sum\limits_{i=1}^j h_{i,j}v_i$.
Set $h_{j+1,j}=\|\widehat{v}_i\|$ and
$v_{j+1}=\widehat{v}_j/h_{j+1,j}$.\\
4. For $k=1,2,\cdots,$, form the matrices $\widehat{V}_k\in
\Bbb{C}^{n\times k}$ and $\overline{H}_k\in \Bbb{C}^{(k+1)\times
k}$.
$$\widehat{V}_k=[v_1,v_2,\cdots,v_k];$$
$$\overline{H}_k=\left(\begin{array}{cccc}
h_{11} & h_{12} &\cdots &h_{1k}\\
h_{21} & h_{22} &\cdots &h_{2k}\\
0 & h_{32} &\cdots &h_{3k}\\
& &\cdots & \\
\vdots & \ddots &\ddots &\vdots \\
0 & \cdots &0 &h_{k+1,k}
\end{array}
\right)$$\\
5. For $m=a+1,\cdots$, form the matrix
$\widehat{H}_m=\overline{H}_m\overline{H}_{m-1}\cdots\overline{H}_{m-a}$.\\
6. Compute the QR factorization of $\widehat{H}_m$:
$\widehat{H}_m=Q_mR_m$, where $R_m$ is upper triangular.\\
7. Solve the system $R_my_m=\beta(Q_m^\ast e_1)$, where
$e_1=(1,0,\cdots,0)^T$.\\
8. Compute $x_m=x_0+\widehat{V}_{m-a}y_m$.\\
9. Restart: if $r_m=b-Ax_m$ satisfied with the residual norm then
stop, else let $x_0=x_m$ and go to 2.
\end{alg}
In \cite{sid1,sid2}, the convergence analysis of DGMRES is given as
follows.
\begin{lem}\cite{sid1,sid2}
Denote the spectrum of $A$ by $\sigma(A)$ and choose $\Omega$ to be
a closed domain in the $\lambda-$plane that contains
$\sigma(A)\backslash\{0\}$ but not $\lambda=0$, such that its
boundary is twice differentiable with respect to arc-length. Denote
by $\Phi(\lambda)$ the conformal mapping of the exterior of $\Omega$
onto the exterior of the unit disk $\{w:|w|\geq 1\}$. Then the
vector $x_m$ extracted from $K_m(A;A^ar_0)$ satisfies
\begin{equation}\label{5}
\|A^ar_m\|\leq Km^{a+2(\widehat{k}-1)}\rho^m,
\end{equation}
for all $m$, where $K$ is a positive constant independent of $m$,
$\widehat{k}=max\{k_j:k_j=ind(A-\lambda_j),\lambda_j\in\sigma(A)\
\backslash \{0\}\}$ and $\rho=|1/\Phi(0)|<1$.
\end{lem}
We know the $x_m's$ produced by DGMRES is
$$x_m=x_0+\sum\limits_{i=1}^{m-a}c_iA^{a+i-1}r_0=x_0+p(A)r_0.$$
Let $z_1,z_2,\cdots,z_l$ is a set of linear independent eigenvectors
corresponding to eigenvalues $\lambda_1,\lambda_2,\cdots,\lambda_l$.
We add some new vectors
$\widehat{z}_{l+1},\widehat{z}_{l+2},\cdots,\widehat{z}_n$ such that
$z_1,\cdots z_l,\widehat{z}_{l+1},\cdots,\widehat{z}_n$ is a basis
in $\Bbb{C}^n$. Then
$r_0=b-Ax_0=\sum\limits_{i=1}^l\beta_iz_i+\sum\limits_{i=1+1}^n\beta_i\widehat{z}_i$
and
\begin{eqnarray*}
&&r_m=b-Ax_m=b-A(x_0+p(A)r_0)\\
&=&r_0-Ap(A)[\sum\limits_{i=1}^l\beta_iz_i+\sum\limits_{i=l+1}^n\beta_i\widehat{z}_i]\\
&=&r_0-\sum\limits_{i=1}^l\lambda_ip(\lambda_i)z_i-\sum\limits_{i=l+1}^n\beta_iAp(A)\widehat{z}_i\\
&=&\sum\limits_{i=1}^l\beta_iz_i+\sum\limits_{i=l+1}^n\beta_i\widehat{z}_i-\sum\limits_{i=1}^l
\lambda_ip(\lambda_i)z_i-\sum\limits_{i=l+1}^n\beta_iAp(A)\widehat{z}_i\\
&=&\sum\limits_{i=1}^l(\beta_iz_i-\lambda_ip(\lambda_i)z_i)+
\sum\limits_{i=l+1}^n(\beta_i-\beta_iAp(A))\widehat{z}_i.
\end{eqnarray*}
Thus
\begin{eqnarray*}
A^ar_m&=&\sum\limits_{i=1}^l[\beta_i\lambda_i^az_i-\lambda_i^{a+1}p(\lambda_i)z_i]
+\sum\limits_{i=l+1}^n[\beta_i\lambda_i^a\widehat{z}_i-\beta_ip(A)A^{a+1}\widehat{z}_i].
\end{eqnarray*}
From Lemma 2, we know
\begin{eqnarray}\label{6}
&&\|\sum\limits_{i=1}^l[\beta_i\lambda_i^az_i-\lambda_i^{a+1}p(\lambda_i)z_i]+
\sum\limits_{i=l+1}^n[\beta_i\lambda_i^a\widehat{z}_i-\beta_ip(A)A^{a+1}\widehat{z}_i]\|\\
\nonumber
&\leq&Km^{a+2(\widehat{k}-1)}\rho^m.
\end{eqnarray}
\section{DGMRES augmented with eigenvectors}
In order to accelerate the convergence of restarting GMRES, Morgan
suggested some eigenvectors corresponding to a few of the smallest
eigenvalues add to the Krylov subspace for GMRES. Then the
convergence can be much faster. This method can be used in the
process of DGMRES.
Let $k$ be the number of the eigenvectors added to the subspace. Let
$r_0$ be the initial residual vector and $a=ind(A)$.
$K_m:=span\{A^ar_0,A^{a+1}r_0,\cdots,A^{m-1}r_0\}$ and
$K_{m,k}=span\{A^ar_0,A^{a+1}r_0,\cdots,A^{m-1}r_0,z_1,z_2,\cdots,z_k\}$,
where $z_1,z_2,\cdots,z_k$ are the eigenvectors added to $K_m$. The
approximated Drazin-inverse solution will be extracted from
$K_{m,k}$. That is
$$x_m=x_0+\sum\limits_{i=1}^k\alpha_iz_i+\sum\limits_{i=1}^{m-a}c_iA^{a+i-1}r_0$$.
which satisfies
\begin{equation}\label{11}
\|A^ar_m\|=\min_{x\in x_0+K_{m,k}(A;A^ar_0)} \|A^a(b-Ax)\|.
\end{equation}
Suppose $z_1,z_2,\cdots,z_k,\cdots,z_l$ is the set of linear
independent eigenvectors and
$z_1,\cdots,z_k,\cdots,z_l,\widehat{z}_{l+1},\cdots,\widehat{z}_n$
is a basis of $\Bbb{C}^n$. So we can write
$$r_0=\sum\limits_{i=1}^k\beta_iz_i+\sum\limits_{i=k+1}^l\beta_iz_i+\sum\limits_{i=l+1}^n\beta_i\widehat{z}_i,$$
and
\begin{eqnarray*}
&&x_m=x_0+\sum\limits_{i=1}^k\alpha_iz_i+p(A)r_0\\
&=&x_0+\sum\limits_{i=1}^k\alpha_iz_i+p(A)[\sum\limits_{i=1}^l\beta_iz_i+\sum\limits_{i=l+1}^n\beta_i\widehat{z}_i].
\end{eqnarray*}
Thus
\begin{eqnarray*}
&&r_m=b-Ax_m\\
&=&b-Ax_0-\sum\limits_{i=1}^k\alpha_i\lambda_iz_i-\sum\limits_{i=1}^l\lambda_ip(\lambda_i)\beta_iz_i-\sum\limits_{i=l+1}^nAp(A)\beta_i\widehat{z}_i\\
&=&r_0-\sum\limits_{i=1}^k\alpha_i\lambda_iz_i-\sum\limits_{i=1}^k\lambda_ip(\lambda_i)\beta_iz_i-\sum\limits_{i=k+1}^l\lambda_ip(\lambda_i)\beta_iz_i
-\sum\limits_{i=l+1}^n\beta_iAp(A)\widehat{z}_i
\end{eqnarray*}
and
\begin{eqnarray*}
&&A^ar_m=A^ar_0-\sum\limits_{i=1}^k\alpha_i\lambda_i^{a+1}z_i-\sum\limits_{i=1}^k\lambda_i^{a+1}p(\lambda_i)\beta_iz_i\\
&&\ \
-\sum\limits_{i=k+1}^l\lambda_i^{a+1}p(\lambda_i)\beta_iz_i-\sum\limits_{i=l+1}^n\beta_iAp(A)\widehat{z}_i\\
&=&\sum\limits_{i=1}^k\beta_i\lambda_i^az_i+\sum\limits_{i+1}^l\beta_i\lambda_i^az_i+\sum\limits_{i=l+1}^n\beta_iA^a\widehat{z}_i\\
&&\ \
-\sum\limits_{i=1}^k\alpha_i\lambda_i^{a+1}z_i-\sum\limits_{i=1}^k\lambda_i^{a+1}p(\lambda_i)\beta_iz_i\\
&&\ \
-\sum\limits_{i=k+1}^l\lambda_i^{a+1}p(\lambda_i)\beta_iz_i-\sum\limits_{i=l+1}^n\beta_iAp(A)\widehat{z}_i\\
&=&\sum\limits_{i=1}^k[\beta_i\lambda_i^a-\alpha_i\lambda_i^{a+1}-\lambda_i^{a+1}p(\lambda_i)\beta_i]z_i\\
&&+\sum\limits_{i=k+1}^l(\beta_i\lambda_i^a-\lambda_i^{a+1}p(\lambda_i)\beta_i)z_i+\sum\limits_{i=l+1}^n
[\beta_iA^a-\beta_iAp(A)]\widehat{z}_i\\
\end{eqnarray*}
Since DGMRES augmented with eigenvectors makes $\|A^ar_m\|$
minimized, we have
$\alpha_i=\frac{\beta_i\lambda_i^a-\lambda_i^{a+1}p(\lambda_i)\beta_i}{\lambda_i^{a+1}}$
and
\begin{equation}\label{7}
A^ar_m=\sum\limits_{i=k+1}^l(\beta_i\lambda_i^a-\lambda_i^{a+1}p(\lambda_i)\beta_i)z_i
+\sum\limits_{i=l+1}^n[\beta_iA^a-\beta_iAp(A)]\widehat{z}_i
\end{equation}
From Lemma 1, (\ref{5}), and (\ref{7}), we get the follow theorem.
\begin{thm}
Denote the spectrum of $A$ by $\sigma
(A)=\{\lambda_1,\lambda_2,\cdots,\lambda_l\}$. $\sigma_a
(A):=\{\lambda_{k+1},\lambda_{k+2},\cdots,\lambda_l\}$ $(k\leq l)$.
Choose $\Omega_a$ to be
a closed domain in the $\lambda-$plan that contains
$\sigma_a(A)\backslash\{0\}$ but not $\lambda=0$, such that its
boundary is twice differentiable with respect to arc-length. Denote
by $\Phi_a(\lambda)$ the conformal mapping of the exterior of
$\Omega_a$ onto the exterior of the unit disk $\{w:|w|\geq 1\}$.
Then the vector $x_m$ generated by $DGMRES$ augmented with
eigenvectors satisfies
\begin{equation}\label{8}
\|A^ar_m\|\leq Km^{a+2(\widehat{k}-1)}\rho_a^m,
\end{equation}
for all $m$, where $K$ is a positive constant independent of $m$,
$\widehat{k}=max\{k_j:k_j=ind(A-\lambda_j),\lambda_j\in\sigma_a(A)\
\backslash \{0\}\}$ and $\rho_a=|1/\Phi_a(0)|<1$.
\end{thm}
Compared (\ref{8}) with (\ref{5}), we know the convergence of DGMRES
augmented with eigenvectors is faster than that of DGMRES.
The implementation of DGMRES augmented with eigenvectors is
different from that of GMRES. We derive the algorithm step by step.
1. Let the initial vector $x_0=0$. Compute $\beta=\|A^ar_0\|$ and
set $v_1=(A^ar_0)/\beta$.
2. Orthogonalize the vectors $A^ar_0,\cdots,A^{m-1}r_0$ via
Arnoldi-Gram-Schmidt process.
For $j=1,2,...,m-a$.
For $i=1,2,\cdots,j$
Compute $h_{ij}=\langle Av_j,v_i\rangle$.
$\widehat{v_{j+1}}=Av_j-h_{ij}v_i$
end i
Let $h_{j+1,j}=\|\widehat{v}_j\|$
Set $v_{j+1}=\widehat{v}_j/h_{j+1,j}$
end j\\
Consequently, we get a orthogonal matrix $V_{m-a}=[v_1,v_2,\cdots,
v_{m-a}]$ and a $(m-a+1)\times (m-a)$ Hessenberg matrix
$\overline{H}^{\sharp}$.
3. Compute k approximated smallest magnitude nonzero eigenvalues of $A$ and add their corresponding eigenvectors $z_1,\cdots,z_k$ to the
subspace. Let $H^{\sharp}=[h_{ij}]_{(m-a)\times (m-a)}$. Obviously
$V_{m-a}^TAV_{m-a}=H^{\sharp}$. When $m\leq q-1$, where $q$ is the
degree of the minimal polynomial of $A$, the $\overline{H}^{\sharp}$
has full rank. We suppose $\lambda_1,\lambda_2,\cdots,\lambda_k$
are k smallest eigenvalues of $H^{\sharp}$ and $y_1,y_2,\cdots,y_k$
are the eigenvectors corresponding to them. Then $z_i:=V_{m-a}y_i$
$(i=1,2,\cdots,k)$ are the approximated eigenvectors of $A$. We add
them to the subspace and orthogonalize.
For $j=m-a+1,\cdots,m-a+k$
For $i=1,2,\cdots,j$
Compute $h_{ij}=\langle Az_{j-m+a},v_i\rangle$.
Compute $\widehat{v}_{j+1}=Az_{j-m+a}-h_{ij}v_i$
end i
Set $h_{j+1,j}=\|\widehat{v}_{j+1}$
Set $v_{j+1}=\widehat{v}_{j+1}/h_{j+1,j}$
end j.
Denote $W=[v_1,v_2,\cdots,v_{m-a},z_1,\cdots,z_k]$,
$\overline{H}^{(0)}=[h_{ij}]_{m-a+k+1,m-a+k}$, and
$V^{(0)}=[v_1,v_2,\cdots,v_{m-a},v_{m-a+1},\cdots,v_{m-a+k+1}]$. It
is easy to see $AW=V^{(0)}\overline{H}^{(0)}$ and $x_m=x_0+Wy_m$ for
some $y_m\in \Bbb{C}^{n\times n}$. It follows that
$r_m=r_0-AWy_m=r_0-V^{(0)}\overline{H}^{(0)}y_m$. So we get
\begin{equation}\label{8}
A^ar_m=A^ar_0-A^{a+1}Wy_m=\beta v_1-A^{a+1}Wy_m.
\end{equation}
From this, we should continue to deal with $A^{a+1}W$.
\begin{eqnarray*}\label{9}
A^{a+1}W&=&A^a(AW)=A^aV^{(0)}\overline{H}^{(0)} \\
&=&A^{a-1}(AV^{(0)})\overline{H}^{(0)} \\
&=&A^{a-1}[v_1,v_2,\cdots,v_{m-a},v_{m-a+1},v_{m-a+2},\cdots,v_{m-a+k+2}]\overline{H}^{(1)}\overline{H}^{(0)}\\
&=&A^{a-1}V^{(1)}\overline{H}^{(1)}\overline{H}^{(0)}\\
&\vdots&\\
&=&A^{a-(a-1)}V^{(a-1)}\overline{H}^{(a-1)}\cdots
\overline{H}^{(0)}\\
&=&V^{(a)}\overline{H}^{(a)}\overline{H}^{(a-1)}\cdots
\overline{H}^{(0)}\\
\end{eqnarray*}
Denote $\overline{H}^{(a)}\cdots \overline{H}^{(0)}$ by
$\overline{H}$. From the above, we get
\begin{equation}\label{10}
A^{a+1}W=V^{(a)}\overline{H}.
\end{equation}
Since $r_m$ is the residual vector produced by DGMRES augmented with
eigenvectros, from (\ref{8}) and (\ref{10}), we have
\begin{eqnarray*}
\|A^ar_m\|&=&\beta v_1-A^{a+1}Wy_m\|=\|\beta
v_1-V^{(a)}\overline(H)y_m\| \\
&=&\|\beta e_1-\overline{H}y_m\|=\min\limits_{y}\|\beta
e_1-\overline{H}y\|
\end{eqnarray*}
Apply the QR factorization of $\overline{H}$: $\overline{H}=QR$,
where $R$ is upper triangular. Thus $y_m$ satisfies
$Ry_m=\beta(Q^Te_1)$ and $x_m$ follows.
4. The practical computation of $\overline{H}$. From the above we
know
$$A[v_1,\cdots,v_{m-a},z_1,\cdots,z_k]=[v_1,\cdots,v_{m-a},v_{m-a+1},\cdots,v_{m-a+k+1}]\overline{H}^{(0)},$$
and
\begin{eqnarray*}
&&A[v_1,v_2,\cdots,v_{m-a},v_{m-a+1},\cdots,v_{m-a+k+1}\\
&=&[v_1,v-2,\cdots,v_{m-a},v_{m-a+1},v_{m-a+2}^{(1)},\cdot,v_{m-a+k+1}^{(1)}]\overline{H}^{(1)}
\end{eqnarray*}
where $$ \overline{H}^{(1)}=\left(
\begin{array}{ccccccc}
h_{11} & h_{12} & \cdots & h_{1,m-a} &
\widehat{h}_{1,m-a+1} & \cdots &
\widehat{h}_{1,m-a+k+1}\\
h_{21} & h_{22} & \cdots &h_{2,m-a} &\widehat{h}_{2,m-a+1}
&\cdots &\widehat{h}_{1,m-a+k+1}\\
0 &h_{32} &\cdots &h_{3,m-a} &\widehat{h}_{3,m-a+1} &\cdots
&\widehat{h}_{3,m-a+k+1}\\
& \cdots &\cdots & & & & \\
0 & 0 &\cdots &h_{m-a+1,m-a} &\widehat{h}_{m-a+1,m-a+1} &\cdots
&\widehat{h}_{m-a+1,m-a+k+1}\\
0 & 0 &\cdots &0 &\widehat{h}_{m-a+2,m-a+1} &\cdots
&\widehat{h}_{m-a+2,m-a+k+1}\\
& \cdots &\cdots &&&& \\
0 & 0 &\cdots &0 &0 &\cdots &\widehat{h}_{m-a+k+2,m-a+k+1}
\end{array}
\right) $$
$\widehat{h}_{ij}$ is the new entries we need to compute when from
$\overline{H}^{(0)}$ to $\overline{H}^{(1)}$, that is, we need to
compute $\frac{[2(m-a+2)+k]\times (k+1)}{2}$ entries additional.
Next
\begin{eqnarray*}
&&A[v_1,v_2,\cdots, v_{m-a}, v_{m-a+1},
v^{(1)}_{m-a+2},\cdots,v^{(1)}_{m-a+k+1}]\\
&=&[v_1,\cdots,v_{m-a+1},v_{m-a+2}^{(1)},v_{m-a+3}^{(2)},\cdots,v_{m-a+k+2}^{(2)}]\overline{H}^{(2)}.
\end{eqnarray*}
where $$ \overline{H}^{(1)}=\left(
\begin{array}{ccccccc}
h_{11} &h_{12} &\cdots &h_{1,m-a+1} &\widehat{h}_{1,m-a+2}
&\cdots &\widehat{h}_{1,m-a+k+2}\\
h_{21} &h_{22} &\cdots &h_{2,m-a+1} &\widehat{h}_{2,m-a+2}
&\cdots &\widehat{h}_{2,m-a+k+2}\\
0 & h_{32} &\cdots &h_{3,m-a+1} &\widehat{h}_{3,m-a+2} &\cdots
&\widehat{h}_{3,m-a+k+2}\\
& &\vdots &\vdots &&&\\
0 & 0 &\cdots &h_{m-a+2,m-a+1} &\widehat{h}_{m-a+2,m-a+2} &\cdots
&\widehat{h}_{m-a+2,m-a+k+2}\\
& & \vdots &\vdots &&&\\
0 &0 &\cdots &0&0& \cdots& \widehat{h}_{m-a+k+2,m-a+k+2}\\
\end{array}
\right) $$
Similarly, from $\overline{H}^{(1)}$ to $\overline{H}^{(2)}$, there
are $\frac{[2(m-a)+5+k]\times (k+1)}{2}$ need to be computed.
Following this way, we can continue to get
$\overline{H}^{(3)},\cdots, \overline{H}^{(a)}$ and $\overline{H}$
can be computed.
We summarize the above as the following algorithm.
\begin{alg} DGMRES augmented with eigenvectors\\
1. Pick initial vector $x_0$ and compute $r_0=b-Ax_0$ and $\beta=\|A^ar_0\|$.\\
2. Apply Arnoldi process to $A^ar_0,\cdots, A^{m-1}r_0$.
$v_1=r_0/\|r_0\|$
For $j=1,\cdots,m-a$
For $i=1,\cdots,j$
Compute $h_{ij}=\langle Av_j,v_i\rangle$
Compute $\widehat{v}_{j+1}=Av_j-h_{ij}v_i$
end i
Set $h_{j+1,j}=\|v_j\|$
$v_{j+1}=\widehat{v}_{j+1}/h_{j+1,j}$
end j\\
4. Denote $[h_{ij}]_{(m-a)\times (m-a)}$ by $H$ and compute its k
eigenvectors $y_1,\cdots,y_k$ corresponding to k smallest magnitude eigenvalues.
Compute $z_i=Vy_i$. We add $z_i's$ to the subspace and denote
$W=[v_1,\cdots,v_{m-a},z_1,\cdots,z_k]$. We apply Arnoldi process
to $AW$ and get the matrix $V^{(0)}$ and $\overline{H}^{(0)}$.
For $j=m-a+1,\cdots,m-a+k$
For $i=1,2\cdots,j$
Compute $h_{ij}=\langle Az_{j-(m-a)},v_i\rangle$
Compute $\widehat{v}_{j+1}=Az_{j-(m-a)}-h_{ij}v_i$
end i
Set $h_{j+1,j}=\|v_j\|$
Set $v_{j+1}=\widehat{v}_{j+1}/h_{j+1,j}$
end j\\
5. Compute $\overline{H}^{(1)},\cdots,\overline{H}^{(a)}$.
For $t=1,2,\cdots,a$
For $j=m-a+t,\cdots,m-a+k+t$
For $i=1,2,\cdots,j$
Compute $h_{ij}=\langle Av_j,v_i\rangle$
$\widehat{v}_{j+1}=Av_j-h_{ij}v_i$
end i
$h_{j+1,j}=\|v_j\|$
$v_{j+1}=\widehat{v}_{j+1}/h_{j+1,j}$
end j
Compute $\overline{H}=\overline{H}^{(a)}\cdots
\overline{H}^{(0)}$.\\
6. Compute the QR factorization of $\overline{H}$:
$\overline{H}=QR$, where $R$ is upper triangular. Set $c=\beta Q^T
e_1$. Solve the least-square problem $R_my_m=\beta (Q^Te_1)$ and
$x_m=x_0+Wy_m$. If $r_m:=b-Ax_m$ satisfies the residual norm then
stop; else set $x_0=x_m$ and go to 2.
\end{alg}
\section{Numerical examples}
For convenience, We denote the DGMRES augmented with eigenvectors by
ADGMRES. In this section we present some numerical examples to
compare ADGMRES to DGMRES. All the experiments were performed in
$MATLAB^\circledR$ 7.5 on an Inter Core 2 Duo 2000MHz PC with main
memory 1000M.
{\bf Example 1} We take $A$ to be a 12 by 12 singular matrix which
has the following Jordan canonical form
$$\left(
\begin{array}{cccccccccccc}
1 &1 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0\\
0 &1 &1 &0 &0 &0 &0 &0 &0 &0 &0 &0\\
0 &0 &1 &0 &0 &0 &0 &0 &0 &0 &0 &0\\
0 &0 &0 &3 &1 &0 &0 &0 &0 &0 &0 &0\\
0 &0 &0 &0 &3 &1 &0 &0 &0 &0 &0 &0\\
0 &0 &0 &0 &0 &3 &0 &0 &0 &0 &0 &0\\
0 &0 &0 &0 &0 &0 &7 &0 &0 &0 &0 &0\\
0 &0 &0 &0 &0 &0 &0 &8 &0 &0 &0 &0\\
0 &0 &0 &0 &0 &0 &0 &0 &9 &1 &0 &0\\
0 &0 &0 &0 &0 &0 &0 &0 &0 &9 &0 &0\\
0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &1\\
0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0 &0
\end{array}
\right)
$$
The right side $b=(1,1,\cdots,1)^T$. The ADGMRES uses $m=6,k=1$ and
DGMRES uses $m=7$ and denote them by DGMRES(6,1), DGMRES(7)
respectively. So they use the same size subspace. When
$\frac{\|A^ar_m\|}{\|A^ab\|}<\epsilon$, we stop. The convergence
curve for ADGMRES and DGMRES are indicated in Fig 1. It is easy to
see ADGMRES convergence faster than DGMRES
\begin{center}
\scalebox{0.6}{\includegraphics{fig1}}
{\small Fig1. The convergence curves for ADGMRES(6,1) and DGMRES(7)}
\end{center}
{\bf Example 2.} In this example the matrix $A$ is the same as the
above but let $a_{7,7}=1000$. Obviously such $A$ has a larger
eigenvalue 1000. We expect ADGMRES(4,1) convergence much faster
than $DGMRES(5)$. As indicated in Fig 2., it is the case.
\begin{center}
\scalebox{0.6}{\includegraphics{fig2}}
{\small Fig2. The convergence curves for ADGMRES(4,1) and DGMRES(5)}
when $A$ has a very lager eigenvalue.
\end{center}
From Fig 2. we can see after 70 iterations the DGMRES(5) stagnates
but after 5000 iterations the curve of ADGMRES(4,1) decreases
steeply.
{\bf Example 3.} In this example we take $a_{77}=0.001$, that is,
$A$ has a very small eigenvalue. From Fig 3. we see DGMRES(5)
stagnates after 50 iterations but ADGMRES(4,1) still works well.
\begin{center}
\scalebox{0.6}{\includegraphics{fig3}}
{\small Fig3. The convergence curves for ADGMRES(4,1) and DGMRES(5)}
when $A$ has a very small eigenvalue.
\end{center}
{\bf Example 4.} This example comes from \cite{zhou2}.
$$A=\left(
\begin{array}{cccc}
1 & 1 &1 &2\\
0 &1 &3 &4\\
0 &0 &1 &1\\
0 &0 &0 &0
\end{array}
\right), b=\left(\begin{array}{c}
-4\\
7\\
1\\
0\end{array}\right)$$
In \cite{zhou2} The author found that DGMRES(2) converges faster
than DGMRES(3). Since $A$ has only one nonzero eigenvalue $1$, we
infer from Theorem that the convergence of ADGMRES(2,1) is as the
same as that of DGMRES(2). From Table 1 we observe it is the case.
At the same time we also see DGMRES(3) stagnates. The residual norm
produced by the three methods is indicated in the following table.
\begin{center}
\begin{tabular}{c| c c c}
runs &ADGMRES(2,1) &DGMRES(3) &DGMRES(2)\\
\hline
50 &0.0038 & 0.00279 &0.0038\\
100 &$1.23\times 10^{-5}$ &0.00276 &$1.23\times 10^{-5}$\\
200 &$1.71\times 10^{-9}$ &0.00276 &$1.72\times 10^{-9}$\\
300 &$6.155\times 10^{-14}$ &0.00276 &$6.145\times 10^{-14}$\\
\end{tabular}
{\small Table 1. The residual norm produced by ADGMRES(2,1),
DGMRES(3), DGMRES(2) after 50, 100, 200, 300 runs respectively.}
\end{center}
\section{Conclusion.} The DGMRES method augmented with eigenvectors
can improve the convergence, especially when the matrix has small or
large eigenvalues. The DGMRES method often stagnates (see
\cite{zhou1,zhou2}). The DGMRES augmented with eigenvectors is a
good choice to overcome the stagnation. When $k$, the number of the
eigenvectors added to the subspace, is large, the method is
expensive. So in practical use we usually choose a small $k$.
\bibliographystyle{amsplain}
|
{
"timestamp": "2010-09-23T02:02:27",
"yymm": "1009",
"arxiv_id": "1009.4406",
"language": "en",
"url": "https://arxiv.org/abs/1009.4406"
}
|
\section{Introduction}
The phenomenon of superconductivity appearing in compounds having magnetic
elements, has received noticeable attention during the last three decades
due to the great variety of exotic electronic and magnetic correlations it
involves. Particularly interesting for this topic, has been the discovery of
the quaternary intermetallic compounds; \textit{R}Ni$_{2}$B$_{2}$C, (\textit
R} = rare earths, Y, Sc, Th) \cite{Nagaraj,Cava1} where coexistence of
antiferromagnetism and superconductivity has been observed as for example in
\textit{R} = (Tm, Er, Ho, Dy, Lu) \cite{Cava1,Cava2,Tomy,Cho,Muller,Eisaki}.
As far as we know several \textit{R}T$_{2}$B$_{2}$C intermetallics
compounds, with different rare earths (\textit{R}) and transition metal (T)
combinations, have been synthesized \cit
{Cava2,Muller,Cava3,Carter,Sampath,Massalami,Gupta2006,Anand2007,Massalami2009}
and most of them, such as HoNi$_{2}$B$_{2}$C \cite{LinPRBHolmioSolo}, show
superconductivity in spite of the presence of the rare earth magnetic
element.
The influence of the transition metal magnetism in the magnetic properties
of these compounds, seem to be of minor importance compared to that of the
rare earth ions, whose magnetic moments apparently impose the magnetic
ordering at all. Thus in some borocarbides with 3\textit{d} transition
elements such as Ni and Co, neutron-diffraction measurements \cite{LynnPRB55
, and electronic transport measurements \cite{Massalami,Schmidt}, have
revealed that no significant magnetic moment develops in the T sites. Local
structure studies at the Ni site, using Mossbauer spectroscopy on $^{57}$Fe
doped (1 at \%) samples, also support this fact \cite{Sanchez}.
Interestingly, the T elements play an indirect role in the magnetism of
magnetic \textit{R}T$_{2}$B$_{2}$C systems through the spatial dependent
indirect RKKY exchange interactions \cite{Eisaki,Gratz}, that govern the
magnetic ordering in these compounds. On the other hand, the electronic
influence of the T elements in the superconductivity of \textit{R}T$_{2}$B
_{2}$C is more relevant than that for the \textit{R} , B and C elements.
This is particularly true for the Ni based borocarbide superconductors,
where the density of state at the Fermi level is mainly due to the Ni
\textit{3d} bands \cite{Loureiro}. On the contrary, in a comparative study
of the structure and superconducting properties of \textit{R}Ni$_{2}$B$_{2}
C, Loureiro \textit{et al. }\cite{Loureiro}, showed that the superconducting
state is more strongly affected by the magnetism of the \textit{R} ion than
for the \textit{R}-ion size, at least for \textit{R} between Dy and Tm.
However, the role of the magnetism and ion-size of T elements in the
superconductivity of \textit{R}T$_{2}$B$_{2}$C when magnetic \textit{R} ions
are presents, is not clear yet. In this work we aboard the study of a
particularly interesting case: the PrT$_{2}$B$_{2}$C compounds, with T = Ni,
Co, and Pt which have revealed many peculiarities. PrNi$_{2}$B$_{2}$C, and
PrCo$_{2}$B$_{2}$C do not superconduct as measured down to 0.3 K \cit
{Narozhnyi}, however, PrPt$_{2}$B$_{2}$C does superconduct at 6 K, even in
the magnetic Pr$^{+3}$ ion presence \cite{Cava4,Dhar}. Noticeably, PrPt$_{2}
B$_{2}$C does not show any magnetic ordering at low temperatures \cite{Dhar}
but in contrast, Pr-Ni and Pr-Co based borocarbides, develop
antiferromagnetic ordering at about 4 and 8.5 K, respectively \cit
{Duran1PRB,Duran2PRB}. Recently, magnetoresistance and specific heat studies
in Pr(Co, Pt)$_{2}$B$_{2}$C \cite{Duran2PRB,Morales} have pointed out that
spin fluctuation mechanism is involved in the electronic behavior of these
two compounds. However, although evidence for spin fluctuations can be
deduced from certain features in the electronic transport measurements, the
interpretation of those properties could be not so clear. High pressure
experiments in spin fluctuators such as RCo$_{2}$ \cite{HauserRCo2pressure},
CeNi$_{5}$ \cite{JMMMCeNi5spinFlucPres}, and UPt$_{3}$ \cit
{BrodaleUPt3pressure} have proven to be a useful tool in order to make clear
if a spin fluctuation mechanism is occurring in such systems. The aim of
this paper is to enlighten the influence of the chemical and external
applied pressure on the superconducting state and magnetic scattering at low
temperature for the three Pr-based borocarbides: Pr(Ni, Co, Pt)$_{2}$B$_{2}
C. We analyzed the changes of the resistivity as a function of pressure and
temperature. We assume that interactions between itinerant electrons plays
an important role in the low temperature resistivity characteristic, and
those can be modified by applied external or internal chemical pressure.
\section{Experimental details}
Three compounds were prepared: samples of PrCo$_{2}$B$_{2}$C, PrNi$_{2}$B
_{2}$C, and PrPt$_{2}$B$_{2}$C. The single crystals were grown by cold
copper crucible method as described by Dur\'{a}{}n \textit{et al} \cit
{Duran2PRB}. All samples were characterized by X-ray diffraction using a
Bruker P4 diffractometer, with monochromatized Mo-K$\alpha $ radiation. The
cell parameters were: \textit{a }$=3.6156$ \AA , and \textit{c} $=10.3507$
\AA \thinspace\ for PrCo$_{2}$B$_{2}$C,\ \textit{a} $=3.6996$ \AA , and
\textit{c} $=9.9885$ \AA\ for PrNi$_{2}$B$_{2}$C, and \textit{a }$=3.8373$
\AA , \textit{c} $=10.761$ \AA\ for PrPt$_{2}$B$_{2}$C samples. Resistivity
measurements in the \textit{a-b} plane were performed by the four-probe
technique using gold wires of $10$ $\mu m$ diameter as electrical contacts.
Pressure experiments were performed by using a micro-cryogenic diamond anvil
cell MCDAC (cell piston-cylinder type Be-Cu cell) consisting of two
diamonds, each of $0.5$ $mm$ culet size. A Cu-Be gasket was preindented and
a $150$ $\mu m$ diameter hole was drilled at the center. The samples used
have dimensions of about $80\times 15\times 40$ $\mu m^{3}$ and were placed
in the gasket hole. The transmitting pressure medium was MgO powder. The
metallic gasket was electrically insulated, pressing over it Al$_{2}$O$_{3}$
powder of $1$ $\mu m$ grain size. As the MCDAC pressure increases, the wires
used to measure the electrical resistance may be cut off at the edge of the
diamonds because of the diamond indentation. To reduce this problem, we used
a thin aluminum foil placed under the four gold wires, with this set up
frequently we reached high quasi-hydrostatic pressures in the range about 6
GPa. Also, in order to prevent motion of the sample and of the electrical
leads, at the initial compressing, a thin mylar film was placed over them.
Additional pressure experiments in polycrystalline Pr(Ni, Pt)$_{2}$B$_{2}$C
compounds up to about 21 GPa were made using a sintered diamond Bridgman
anvil apparatus, with a pyrophyllite gasket and two steatite disks as the
pressure medium. For determination of the pressures a Pb manometer were used
\FRAME{ftbpFU}{3.269in}{3.7446in}{0pt}{\Qcb{(Color online) Variation of the
T-T shortest length between 3\textit{d} ions for the transition metal T and
PrT$_{2}$B$_{2}$C compounds, with T = (Co, Ni, Pt). It also shows the
behavior of the cell parameters as a function of the T-size.}}{\Qlb{Fig.1}}
fig1_falconi.eps}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "ICON";valid_file "F";width
3.269in;height 3.7446in;depth 0pt;original-width 8.2858in;original-height
9.5008in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'Fig1_Falconi.eps';file-properties "XNPEU";}}
\section{Results and discussion}
It is known that the \textit{R}Ni$_{2}$B$_{2}$C compounds crystallize in the
tetragonal body-centered structure (space group \textit{I4}/mmm) and when
the rare earth atom radii is increased (\textit{R} goes from La to Lu), the
\textit{c}-parameter becomes larger whereas the \textit{a}-parameter
decreases \cite{Siegrist,LynnPRB55}. This structural behavior can be
accounted by for the rigidity of the B-C and Ni-B bonds and the variable
tetrahedral angle in the NiB$_{4}$ unit. Distortions of this tetrahedral
unit are claimed to be a decisive parameter for Tc in non magnetic or
antiferromagnetic \textit{R}Ni$_{2}$B$_{2}$C and \textit{R}NiBC compounds
\cite{SanchezTetrahedro}. In the case when the size of the transition
element is increased, maintaining the same rare earth element, the
structural behavior of the unit cell seems to be slightly different. Fig. 1
displays the T-T shortest length between 3\textit{d }ions, particularly in
the Pr(Ni, Co, Pt)$_{2}$B$_{2}$C compounds and that for the metal T. Also it
shows the behavior of the \textit{a} and \textit{c} parameters for each
compound. We note that increasing the ionic radii T-size causes an
increasing of the \textit{c}-parameter and an anomaly behavior of the
\textit{a}-parameter for the PrCo$_{2}$B$_{2}$C compound. At first glance
and according to the figure, this anomaly is correlated with the variation
of the T-T shortest bond in the framework of the PrT$_{2}$B$_{2}$C
structure, and not with the ionic radii size of the T element.\FRAME{ftbpFU}
3.3572in}{2.5884in}{0pt}{\Qcb{(Color online) Normalized resistivity at 295 K
$\protect\rho _{ab}(T)/\protect\rho _{ab}(295$\ K$)$) of PrNi $_{2}$B$_{2}
C, PrCo$_{2}$B$_{2}$C, and PrPt$_{2}$B$_{2}$C compounds at room pressure.
The three systems present metallic conductivity from room temperature to
down 25 K. Inset shows the low temperature variation of the normalized
resistivity.}}{\Qlb{Fig2}}{fig2_falconi.eps}{\special{language "Scientific
Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display "ICON";valid_file
"F";width 3.3572in;height 2.5884in;depth 0pt;original-width
10.0024in;original-height 7.7012in;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Fig2_Falconi.eps';file-properties "XNPEU";}}
Fig. 2 shows the normalized electrical resistivity in the \textit{a-b} plane
as a function of temperature for PrNi$_{2}$B$_{2}$C, PrCo$_{2}$B$_{2}$C, and
PrPt$_{2}$B$_{2}$C single crystals at room pressure. The three compounds
present metallic characteristic. At low temperature Ni and Co based
compounds show notable similarities, but are not superconductors, whereas
PrPt$_{2}$B$_{2}$C has a sharp superconducting temperature at about 6 K. The
residual resistivity ratios RRR for the first two compounds are $9$ and $33$
respectively, whereas for PrPt$_{2}$B$_{2}$C is $5.5$. The residual
resistivity\ $\rho _{0}$ of all three is sample dependent, varying between 5
and 25 $\mu \Omega cm.$ The main panel of Fig. 2 shows some interesting
characteristics: Ni and Co based compounds present a notable positive
curvature from about 150 to 50 K, whereas Pt compound presents a wide bump
from about 250 to 20 K. These notable differences may signal a clearly
distinctive influence of the crystalline field at high temperatures. A
gradual but pronounced drop in resistivity comes to disturb the linear
variation to about 8 and 20 K for PrNi$_{2}$B$_{2}$C, and PrCo$_{2}$B$_{2}
C, respectively. Such resistivity behavior at relatively low temperature is
typical for magnetic elements of the \textit{R}Ni$_{2}$B$_{2}$C series, and
it has been associated with a decrease of the magnetic scattering of the
conduction electrons by rare earth ions (\cite{MiprimerPRB} and reference
therein). However, according to the results present below, it is possible
that other mechanism involving conduction electrons could also develop at
low temperature. The case for PrPt$_{2}$B$_{2}$C is quite different, after
following an upward curvature it becomes superconducting at about 6 K.
Magnetic and heat capacity measurements in this compound \cite{Dhar}, have
revealed a nonmagnetic ground state for Pr ions due to CEF effects, which is
claimed to be the reason for superconductivity. The inset of the Fig. 2
shows the resistivity behavior from 60 to 2 K for the three compounds at
room pressure. At first glance, increasing the transition metal radius\
corresponds to a major resistivity droop at low temperature.\FRAME{ftbpFU}
3.7109in}{2.9585in}{0pt}{\Qcb{(Color online) Linear fit to $\protect\rho $(T
^{2}$) from 2 to 8 K for PrCo$_{{\protect\small 2}}$B$_{{\protect\small 2}}
C. The low temperature behavior of $\protect\rho $(T) follows a T$^{2}$\ low
with a cuadratic coefficient A = 0.08 $\protect\mu \Omega cm/K^{2}.$ Inset
shows the $\protect\rho (T)$\ behavior at low temperatures.}}{\Qlb{Fig3}}
fig3_falconi.eps}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "ICON";valid_file "F";width
3.7109in;height 2.9585in;depth 0pt;original-width 6.2881in;original-height
5.0055in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'Fig3_Falconi.eps';file-properties "XNPEU";}}
The $\rho _{ab}(T)$ curve for PrCo$_{2}$B$_{2}$C from about 2 to 8 K shows a
clear T$^{2}$-law dependence with a cuadratic coefficient A, equal to 0.08
\mu \Omega $cm/K$^{2}$, see Fig. 3. This low-temperature resistivity
behavior is similar to the observed in heavy fermion systems, as for
example; YbNi$_{2}$B$_{2}$C and UPt$_{3}$ compounds \cite{Yatskar,Stewart},
and could be attributed to spin fluctuations \cite{Morales,Moriya}. Thus, in
a similar compound but simpler, RCo$_{2}$, a T$^{2}$ dependence has been
found at low temperatures, which is due to spin fluctuating characteristics
\cite{HauserRCo2pressure}. The fact that the magnitude of the cuadratic
coefficient \textit{A} of $\rho (T)$ for PrCo$_{2}$B$_{2}$C is of the order
of that for RCo$_{2}$ (\cite{MassalammiJMMM2004}, found this coefficient as
big as three orders of magnitude but in polycrystalline PrCo$_{2}$B$_{2}$C)
suggests that spin fluctuating could be the responsible mechanism for the
low temperature $\rho (T)$ behavior in this compound.\ Using the universal
relation for heavy fermion compounds; $A/\gamma ^{2}=1.0x10^{-5}\mu \Omega $
cm$(mole$K$)^{2}/mJ^{2}$\ \cite{Kadowaki} the resulted Sommerfeld
coefficient is $\gamma =89.4$\ $mJ/mol-$K$^{2}$ which is a low value
compared with that for PrNi$_{2}$B$_{2}$C ($211mJ/mol-$K$^{\mathbf{2}}$ \cit
{Duran2PRB} by specific heat measurements), but enhanced value as compared
to the normal metal Co and to other borocarbides as YCo$_{2}$B$_{2}$C \cit
{Massalami}, (Gd, Tb, Dy, Ho, Er, Tm)Ni$_{2}$B$_{2}$C \cite{MassalamiMagnon
, whose $\gamma $ is about $17$\ $mJ/mol-$K$^{2}$. \FRAME{ftbpFU}{3.4947in}
2.9758in}{0pt}{\Qcb{(Color online) The graph shows pressure effects on the
\protect\rho _{ab}(T)$\ for a PrNi$_{2}$B$_{2}$C single crystal up to 5.3
GPa. Vertical line indicates the increasing pressure. Insert is a zoom of
\protect\rho _{ab}(T,P)$\ at low temperature.}}{\Qlb{Fig4}}{fig4_falconi.ep
}{\special{language "Scientific Word";type "GRAPHIC";maintain-aspect-ratio
TRUE;display "ICON";valid_file "F";width 3.4947in;height 2.9758in;depth
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"1";cropright "1";cropbottom "0";filename 'Fig4_Falconi.eps';file-properties
"XNPEU";}}
\bigskip
Figs. 4 and 5 shown the $\rho _{ab}(T)$\ curves as a function of pressure
for PrNi$_{2}$B$_{2}$C and PrCo$_{2}$B$_{2}$C single crystals, respectively.
As we can see, these compounds reveal different pressure behaviors. The
overall trend of $\rho _{ab}(T)$ for the Pr-Ni based compound does not
change as the pressure increases up to 5.3 GPa. The linear behavior of $\rho
_{ab}(T)$ (extending from about 100 K to room temperature) is attributed to
electron-phonon scattering and under the applied pressures, it shows a slope
decreasing from 0.089 $\mu \Omega $cm/K to 0.069 $\mu \Omega $cm/K.
According to the inset of Fig. 4, the smooth drop of $\rho _{ab}(T)$ at low
temperature, which has been related to the decrease of magnetic scattering
\cite{MiprimerPRB}, is reduced with applied pressures increasing up to 5.3
GPa. In the case of PrCo$_{2}$B$_{2}$C, as can be seen in panel \textbf{b}
of Fig. 5, the pressure effects are more stronger than in PrNi$_{2}$B$_{2}
C, mainly in the low temperature regime. From room pressure to about 1.7 GPa
the high temperature behavior of $\rho _{ab}(T),$ from 300 K to about 75 K,
remains without appreciable changes and with almost a constant slope of 0.30
$\mu \Omega $cm/K. In an opposite way to PrNi$_{2}$B$_{2}$C, the low
temperature curvature of $\rho _{ab}(T)$, which is also associated with
magnetic correlations, tends strongly to be suppressed by pressure. This
tendency has also been observed by Massalami \textit{et. al.} \cit
{MassalammiJMMM2004} applying pressures up to about 1.2 GPa, the maximum
pressure value they applied. Similar to their results, we also observed that
the cuadratic behavior of $\rho _{ab}(T)$ at low temperatures is maintained
under 1.2 GPa. However, we found, applying pressures higher than 1.7 GPa on
this compound, a distinctive characteristic, namely the change of the low
temperature curvature from concave to convex and the complete disappearance
of the resistivity drop above 2.9 GPa (see panel \textbf{b} in Fig.5).
Interestingly, at this pressure the \textit{T}$^{2}$ behavior disappeared
and instead there is an appearing of a type plateau zone in $\rho _{ab}(T)$
which start about 15 K and extend down to 1.8 K, the lowest temperature
available in our experiments. Increasing the pressure up to 4.4 GPa, this
zone of constant resistivity is extended from 1.8 K up to 20 K. At this
pressure, the overall high temperature behavior of $\rho _{ab}(T)$ remains
almost with the same slope of about 0.30 $\mu \Omega $cm/K.\
The above experimental facts reveal that the electronic properties of PrCo
_{2}$B$_{2}$C are more pressure sensitive than that for PrNi$_{2}$B$_{2}$C,
mainly at the low temperature regime. Interestingly, we note that the shape
of $\rho _{ab}(T)$ for PrCo$_{2}$B$_{2}$C under pressure tends to be
qualitatively similar to that for nonmagnetic YCo$_{2}$B$_{2}$C at room
pressure \cite{Massalami}. Thus, it seems the effect of pressure in this
compound is to suppress the magnetic correlations which originate the low
temperature scattering behavior.\FRAME{ftbpFU}{3.2638in}{3.6227in}{0pt}{\Qcb
(Color online) In a) it is presented $\protect\rho _{ab}(T)$\ for PrCo$_{2}$
$_{2}$C single crystal under several pressures up to 4.4 GPa. Panel b) shows
a view of $\protect\rho _{ab}(T)$\ at low temperatures and for different
pressure values.}}{\Qlb{Fig5}}{fig5_falconi.eps}{\special{language
"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display
"ICON";valid_file "F";width 3.2638in;height 3.6227in;depth
0pt;original-width 6.3131in;original-height 7.0119in;cropleft "0";croptop
"1";cropright "1";cropbottom "0";filename 'Fig5_Falconi.eps';file-properties
"XNPEU";}}
According to Fig 5, the resistivity can be fitted to a $T^{2}$ behavior only
for a region of relatively low temperatures, interestingly under pressure
the fitting range is extended. As already we pointed out, at atmospheric
pressure the fitting goes from 2 up to about 8 K with a cuadratic
coefficient \textit{A}, equal to 0.08 $\mu \Omega $cm/K$^{2}$. At 1.7 GPa
the interval extends from 2 to about 19.5 K, with a value of \textit{A}
decreased to 0.018 $\mu \Omega $cm/K$^{2}$. Once the applied pressure
reaches the value of 2.9 GPa, it was not more possible to fit a cuadratic
function to the curvature of $\rho _{ab}(T)$ because the like plateau zone,
also presented for the curves at 3.7 and 4.4 GPa. This $\rho _{ab}(T)$
behavior at low temperature and pressures is accounted by the spin
fluctuation scenario \cite{Rossiter}, which also take in account the
decreasing of the \textit{A} parameter with the applied pressure.
Additionally, relatively low magnetic fields decreases the \textit{A}
parameter in a linear form in PrCo$_{2}$B$_{2}$C \cite{Morales}, which has
been claimed to be due to quenching of the spin fluctuation. Similar results
have been found in other systems as for example, Ce$_{0.8}$(Pr, Nd)Ni$_{5}$,
indicating that spin fluctuations tends to be suppressed both by pressure
and magnetic fields \cite{Marian,Willis}. The microscopic character of the
state resulting from applying a magnetic field is completely different from
that obtained with applying pressure; the first remains magnetic, whereas
the last tends to be a real non-magnetic state; one where no microscopic
magnetic moments exists. This is important because in the spin fluctuations
model the state above $T_{sf}$ (where the spin fluctuation appears and which
coincide with the temperature below which a $T^{2}$ law in resistivity is
valid) is non the non magnetic state (like in the stoner model) but a
magnetic state, where local moments still exist, but long range order tend
to be destroyed by the fluctuations.
The collective modes described by the spin fluctuations can readily be
excited at relatively low temperature, where the stoner excitations are
still very small but we assume they can be suppressed by two factors:
intense magnetic field and/or external applied pressure. Pressure increases
the correlation there exist between \textit{f} ions and promotes the
itinerance of \textit{f} electrons. As a result the \textit{f}-density of
state near the Fermi level is lowered modifying the electron structure and
influencing thus the prevailing long range order between the band electrons
\FRAME{ftbpFU}{3.4177in}{2.7882in}{0pt}{\Qcb{(Color online) Normalized
R(T)/R(40 K) curves at several pressures up to 2.5 GPa for polycrystalline
PrNi$_{2}$B$_{2}$C. Insert shows the behavior of $\protect\rho (P)$\ at 260
K.}}{\Qlb{Fig6}}{fig6_falconi.eps}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "ICON";valid_file "F";width
3.4177in;height 2.7882in;depth 0pt;original-width 6.3157in;original-height
5.1448in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'Fig6_Falconi.eps';file-properties "XNPEU";}}
In order to know if the $\rho (T)$ of PrNi$_{2}$B$_{2}$C could follow a
similar behavior to that for its isomorphs PrCo$_{2}$B$_{2}$C at higher
pressures, we carried out measurements for polycrystalline sample at several
pressures up to 21.5 GPa. Interestingly, we found there is a marked tendency
of $\rho (T)$ at low temperature to behave similar to that for PrCo$_{2}$B
_{2}$C (see Fig. 6). At about 13.5 GPa there is a change from negative to
positive curvature of $\rho (T)$\ at temperatures lower than about 15 K.
This curvature change was also found in Pr-Co system but at about 1.7 GPa.
It is important to mention that from the $\rho _{ab}(T=260K,P)$ curves for
PrNi$_{2}$B$_{2}$C (see inset of Fig. 6), and PrCo$_{2}$B$_{2}$C (not
shown), we discarded some structural phase changes that could be related to
these effects. The fact that Pr-Ni system requires more pressure to behaves
almost in the same way to Pr-Co system at low temperature, could be related
to changes in the \textit{c- }parameter.\FRAME{ftbpFU}{3.3114in}{2.5365in}
0pt}{\Qcb{(Color online) $R(T)$ curves for polycrystalline PrPt$_{2}$B$_{2}
C, measured up to 21.5 GPa. High pressure tends to destruct
superconductivity. Insert shows the low temperature behavior of $R(T).$}}
\Qlb{Fig7}}{fig7_falconi.eps}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "ICON";valid_file "F";width
3.3114in;height 2.5365in;depth 0pt;original-width 10.9624in;original-height
8.3835in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'Fig7_Falconi.eps';file-properties "XNPEU";}} The main difference in the
unit cell of these two compounds stands from this parameter, which is
biggest in Pr-Co system and related to the modifiable tetrahedral B-T-B
angle. On the other hand, it is known that for the spin fluctuator YMn$_{2}
, the existence of a magnetic moment on Mn sites depends largely on the
interatomic Mn-Mn distance \cite{Hauser}. Above a critical distance there
exists a magnetic moment.\ Such distance plays a key-role in determining the
magnetic properties and is sensitive to external or internal perturbations.
The case for Pt in PrPt$_{2}$B$_{2}$C could be similar. As we already point
out this compound shows an upward curvature in $\rho (T)$\ at high
temperature which has been related to crystalline electric field (CEF)
effects \cite{Duran2PRB}. We make resistivity measurements for
polycrystalline PrPt$_{2}$B$_{2}$C under several pressures up to 21.5GPa
(see fig. 7). As it can be observed, the negative curvature of $\rho (T)$\
at high temperature is not appreciably modified under pressure and the main
changes are at low temperatures. The superconducting transition temperature,
T$c$, decreases at the rate dT$c$/dP = -0.34 \ K/ GPa (see Fig. 8). It seems
there is not correlation with the decreasing of T$c$ and the unmodified
curvature related to CEF effects. A positive magnetoresistance at low
temperature in this compound has been associated with spin fluctuation \cit
{Duran2PRB}, however although this picture follows the same trends of
Pr(Ni,Co)$_{2}$B$_{2}$C, further investigations are inquired in order to
clarify this matter.\FRAME{ftbpFU}{3.5751in}{2.9845in}{0pt}{\Qcb{(Color
online) Decreasing of the superconducting transition temperature for PrPt
_{2}$B$_{2}$C, as function of pressure. The rate of decreasing of the
transition looks normal for a \textit{d} electronic compound.}}{\Qlb{Fig8}}
fig8_falconi.eps}{\special{language "Scientific Word";type
"GRAPHIC";maintain-aspect-ratio TRUE;display "ICON";valid_file "F";width
3.5751in;height 2.9845in;depth 0pt;original-width 8.3083in;original-height
6.9272in;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'Fig8_Falconi.eps';file-properties "XNPEU";}}
\section{Conclusions}
High pressure resistivity measurements in Pr(Co,Ni,Pt)$_{2}$B$_{2}$C has
been made. The first and foremost fact we found is that applied pressures of
about 4.0 GPa are able to change drastically the low temperature resistivity
behavior of PrCo$_{2}$B$_{2}$C, but it requires $\sim $ 13.0 GPa in order to
attain similar changes for PrNi$_{2}$B$_{2}$C. This means that the low
temperature electronic transport properties of PrCo$_{2}$B$_{2}$C are more
pressure sensitive than that for the isomorphous PrNi$_{2}$B$_{2}$C.
Evidence for spin fluctuation in PrCo$_{2}$B$_{2}$C is reported from the
cuadratic behavior of its resistivity at low temperature, and from the
decreasing of the cuadratic coefficient as a function of pressure. For\ PrCo
_{2}$B$_{2}$C, the magnetic scattering related to spin disorder is
suppressed at 2.9 GPa, but it remained observable at less under 5.3 GPa for
PrNi$_{2}$B$_{2}$C. For the case of the superconductor PrPt$_{2}$B$_{2}$C,
pressure does not modify the $\rho (T)$ curvature related to CEF effects,
but decreases Tc at the rate dTc/dP = -0.34 K/GPa. Finally, although these
conclusions are no decisive, we believe they would stimulate further
experimental and theoretical studies for a better understanding of the
pressure effects in the \textit{R}T$_{2}$B$_{2}$C compounds, which is far
for complete.
The authors acknowledge the MCBT, Institut N\'{e}el, (CNRS) \& UJ for the
time granted to perform some high pressure experiments. R. F. thanks to
SEP-PROMEP, UJATAB-CA175 for support. R. E. thanks UNAM-DGAPA, project No
IN-101107. We thank to S. Bern\`{e}s for the crystallographic studies and F.
Silvar for liquid He supply.
|
{
"timestamp": "2010-09-23T02:02:45",
"yymm": "1009",
"arxiv_id": "1009.4437",
"language": "en",
"url": "https://arxiv.org/abs/1009.4437"
}
|
\section{Introduction}
Let $M$ be a Riemannian manifold and $f: M \to M^{\prime}\subset M$ a
diffeomorphism. An $f$-invariant set $\Lambda$ has a dominated splitting if
if its tangent bundle $T_\Lambda M=U\oplus V$, where $U$ and $V$ are non trivial invariant continuous subbundles
such that, for constants $C>0$ and $0<\gamma<1$:
\begin{equation}
\|Df^{n}{|_{U(x)}}\|\ \|Df^{-n}{|_{V(f^{n}({x}))}}\|^{}\le C \gamma^n,
\mbox{ for
all } x\in\Lambda, n\ge 0. \label{fds}
\end{equation}
Clearly, any hyperbolic splitting is a dominated one.
Some important consequences of this property on the dynamics were given by Pujals and Sambarino in \cite{PS09}. A spectral decomposition theorem was obtained for $C^{2}$ compact surface
diffeomorphisms having dominated splitting over the limit set $\displaystyle L(f) = \overline{\bigcup _{x\in M}\left(\omega(x) \cup \alpha(x)\right)}$
where $\omega(x)$ and $\alpha(x)$ are the $\omega$ and $\alpha$-limit sets of $x$, respectively:
\textbf{Theorem (Pujals-Sambarino \cite{PS09}):} \emph{Let $M$ be a compact 2-manifold and $f:M \to M^{\prime}\subset M$ a $C^2$-diffeomorphism. Assume that $L(f)$ has a
dominated splitting. Then $L(f)$ can be decomposed into $L(f)=\mathcal{I}\cup{\mathcal{R}}\cup
{\tilde{\mathcal{L}}}$ such that:}
\begin{enumerate}
\item \emph{$\mathcal{I}$ is a set of periodic points with bounded periods
contained in a disjoint union of finitely many normally hyperbolic periodic
arcs or simple closed curves. }
\item \emph{$\mathcal{R}$ is a finite union of normally hyperbolic periodic
simple closed curves supporting an irrational rotation. }
\item \emph{$\tilde{\mathcal{L}}$ can be decomposed into a disjoint union
of finitely many compact invariant and transitive sets (called basic sets).
The periodic points are dense in $\tilde{\mathcal{L}}$ and at most
finitely many of them are non-hyperbolic periodic points. The (basic) sets
above are the union of finitely many (nontrivial) homoclinic classes.
Furthermore $f|_{\tilde{\mathcal{L}}}$ is expansive. }
\end{enumerate}
Our purpose, in this work, is to construct simple examples of dynamical systems with attractors admitting a dominated splitting and where the dynamics is of type $\mathcal{I}$ or $\mathcal{R}$. We will follow the ideas developed in \cite{MPS09} and \cite{AMS09}. In \cite{MPS09}, non conservative billiards were introduced by a modification of the reflection rule and the existence of attractors was demonstrated for a wide class of dispersing and semi-dispersing billiards and of billiards with focusing components. In \cite{AMS09} models were studied numerically and different attractors, periodic and chaotic are presented.
We concentrate on strictly convex billiard tables, with boundary formed by a unique sufficiently differentiable focusing component. They present structures like KAM stability islands and invariant rotational curves non homotopic to a point. We will investigate how, in the presence of non conservative perturbations, an invariant curve will give rise to an attractor.
The maps we consider here are more general than the pinball billiards introduced in \cite{MPS09}, as the perturbation of the angle is not necessarily biased to the normal direction.
The contents of the present paper is outlined in four sections that follow this introduction.
{In} section~\ref{sec:cones} we present the main tools needed to work with dominated splitting while section~\ref{sec:classic} deals with the basic properties of classical billiards on ovals.
In section~\ref{sec:nonela} we introduce our {\sl non elastic billiards}. They are defined as a composition of a classical billiard followed by a change of the reflection angle, corresponding to a contraction in the vertical fibers of an invariant rotational curve.
We will prove that under some differentiability hypotheses and some bounds on the contraction, there exists a compact strip in the phase space, such that the non elastic billiard map is a $C^2-$diffeomorphism from that strip onto its image. Its limit set contains the invariant curve and has a dominated splitting. Moreover, the non elastic dynamics on the invariant curve is determined by its rotation number with respect to the original classical billiard map.
This result will guide us to construct our examples of non elastic billiards on ovals with dominated splitting and attractors supporting a rational or an irrational rotation. They are presented in section \ref{sec:exa}, where we explore their properties theoretically and numerically.
Our result (Theorem \ref{nonelastic}) could clearly be established in the more general setting of $C^2$ conservative twist maps. Once we have a $C^2$ invariant rotational curve and assuming the necessary bounds, everything would work likewise in the proof. The main problem is to build specific examples, other than the obvious twist integrable case (which corresponds to the circular billiard) or the oval billiards presented here, and check, for instance and at least numerically, the size of the basin of attraction.
\section{Dominated Splitting, Cone Fields and Quadratic Forms}
\label{sec:cones}
The existence of a dominated splitting follows from the existence of an eventually strictly invariant cone field and of an uniform control of expansions and/or contractions as showed in \cite{Wo01}. The criteria presented there in Proposition~4.1 can be translated, in dimension 2, into the operational lemma below.
Let $u,v:M\mapsto TM$ be two vector fields such that $u(x)=u_x$ and $v(x)=v_x$ are linearly independent vectors in $ T_xM$.
They induce a nondegenerate quadratic form $Q$ on $ TM$ by $Q_x (a u_x + b v_x) = ab$ and a cone field given at each $x$ by $\displaystyle {\cal C}(x) = \left\{ w \in {T}_xM \, : \, Q_x (w) > 0 \right\} \cup \{ 0 \}$ and whose boundaries, at each point, are given by ${\cal C}_{0}(x) = \left\{ w \in {T}_xM \, : \, Q_x (w) = 0 \right\}$.
If the vector fields are continuous, the quadratic form and the cone field are also continuous.
Given $x \in M$ and a vector $w = a u_x + b v_x \in {T}_xM$, let
$Df_x w = a_1 u_{{f(x)}} + b_1 v_{{f(x)}} \in {T}_{{f(x)}}M$ denote the image of $w$ under the
derivative $Df_x$. Then we have
\begin{equation}
\left(
\begin{array}{c}
a_1 \\ b_1
\end{array}
\right)
=
[Df_x]_U
\left(
\begin{array}{c}
a \\ b
\end{array}
\right)
\label{eq:ab}
\end{equation}
where $[Df_x]_U$ is the matrix representation of the derivative at $x$, with the choice of $\{u_x,v_x\}$ and $\{u_{{f(x)}},v_{{f(x)}}\}$ as bases of ${T}_xM$ and ${T}_{{f(x)}}M$ respectively.
\begin{lemma}\label{lema1}
Let $\Lambda$ be a compact $f$-invariant subset of $M$. If
there is a choice of vector fields $u,v$ such that the entries of $[Df_x]_U$ are strictly positive for every $x\in \Lambda$, then $\Lambda$ has a dominated splitting.
\end{lemma}
Proof: If the entries of $[Df_x]_U$ are strictly positive for every $x\in \Lambda$ then for every $w = a u_x + b v_x $, $x\in \Lambda$, $ a b \ge 0$, $a^2 + b^2 >0$ we have $a_1 b_1 > 0$ where $Df_x w = a_1 u_{{f(x)}} + b_1 v_{{f(x)}}$.
This implies that for every $x\in \Lambda$ $Df_x ({\cal C}(x) \cup {\cal C}_{0}) \subset {\cal C} ({f(x)})$, i.e., $\cal C$ is strictly $Df$-invariant ($f$ is strictly $Q$-separated).
It follows from Proposition 4.1 in \cite{Wo01} that $\Lambda$ has a dominated splitting. \hfill $\blacksquare$
\section{Classical Billiards on Ovals}\label{sec:classic}
Let $\Gamma$ be an oval, i.e., a plane, simple, closed, $C^k, k
\ge 3$, curve, with strictly positive curvature, parameterized counterclockwise by
$\phi$, the angle between the tangent vector and an horizontal
axis. Let $R(\varphi)$ be its radius of curvature at $\varphi$.
The classical billiard problem on $\Gamma$ consists of the free
motion of a point particle inside $\Gamma$, making elastic reflections at the impacts with the boundary. The
motion is then determined by the point of reflection at $\Gamma$ and the direction of motion immediately after each reflection.
They can be given by the parameter $\varphi\in [0,2\pi)$, that
will locate the point of reflection and by the angle $\alpha \in (0,\pi)$ between the tangent vector and the outgoing trajectory, measured counterclockwise.
The classical billiard defines a map $B$ from the open cylinder $[0,2\pi)\times(0,\pi)$ into itself, which has some very well known properties (see, for instance, \cite{kat} and \cite{kato} for the properties of billiards and twist maps listed in this section).
Denoting $B(\varphi_0,\alpha_0)= (\varphi_1,\alpha_1)$, the derivative of $B$ at $(\varphi_0,\alpha_0)$ is
\begin{equation}\label{eq:deriv}
DB_{(\phi_0,\alpha_0)} =
\frac{1}{R_1 \sin \alpha_1}
\left(
\begin{array}{cc}
L - R_0 \sin \alpha_0 & L \\
L - R_0 \sin \alpha_0 - R_1 \sin \alpha_1 & L - R_1 \sin \alpha_1
\end{array}
\right)
\end{equation}
where $R_i = R(\phi_i)$ and $L$ is the distance between $\Gamma(\varphi_0)$ and $\Gamma(\varphi_1)$.
If $\Gamma $ is $C^k$, then $B$ is a $C^{k-1}$-diffeomorphism. It preserves the measure $d\nu=R(\varphi)\sin\alpha\, d\alpha d\varphi$.
It is reversible with respect to the reversing symmetry $S(\varphi,\alpha)=(\varphi,\pi-\alpha)$, as $S\circ B= B^{-1}\circ S$, meaning that the phase space is symmetric with respect to the line $\alpha = \pi/2$.
Moreover, as $\Gamma$ is an oval, $B$ has the monotone Twist property with rotation interval $(0,1)$. For each $\rho\in(0,1)$ there exists a closed, invariant, minimal set ${\cal O}_\rho\subset [0,2\pi)\times(0,\pi)$, which can be injectively projected on $(0,2\pi)$ and such that the induced dynamics preserves the order of $[0,2\pi)\sim S^1$.
This invariant set ${\cal O}_\rho$ can be a periodic orbit, an Aubry Mather set or a rotational curve , i.e., a continuous closed curve which is the graph of a Lipschitz function.
We will concentrate on billiards having an invariant rotational curve $\gamma=$ graph$(g)$, $g: [0,2\pi)\mapsto(0,\pi)$ a Lipschitz function. In general, for a given boundary $\Gamma$, the set of rotation numbers $\rho$ such that ${\cal O}_\rho$ is an invariant rotational curve is nowhere dense in $(0,1)$, but nevertheless invariant rotational curves exist on a large class of classical oval billiards.
For instance, the circular and the elliptical billiards have invariant rotational curves with any rotation number $\rho$. The billiard on a curve of constant width have the horizontal curve $g(\varphi)\equiv \pi/2$ as an invariant rotational curve with rotation number $1/2$. But, apart from special examples like these, it is difficult to find billiards with invariant rotational curves of rational rotation number. In fact, having an invariant rotational curve with rational rotation number is not a generic property for billiards on ovals. The generic dynamics is to have, for each rational rotation number, a finite number of periodic orbits with this rotation number and at least one of them is hyperbolic, with transverse homoclinic orbits \cite{dia}.
From the other side, having an invariant rotational curve with irrational rotation number is quite general. If the oval boundary $\Gamma $ is $C^k, k\geq 5$ then Lazutkin Theorem guarantees a whole family of invariant rotational curves with diophantine $\rho$ near the boundaries of the cylinder $ [0,2\pi)\times(0,\pi)$ (\cite{laz}, \cite{dou}).
And sufficiently small perturbations of the circular billiard present, in addition to the diophantine invariant rotational curves, given by the KAM theorem, uncountably many invariant rotational curves with liouvillian rotation number.
For our purposes, we will need the curve $\gamma$, or equivalentely the function $g$, to be $C^2$ or more. This is the case of the circular, the elliptical and the constant width billiards, for instance, where the rotational invariant curves are, in fact, analytic. Also, the curves guaranteed by Lazutkin Theorem are as differentiable as the billiard map, so $C^4$ or more. But, as showed in \cite{arn}, invariant rotational curves can be just a little bit more regular than Lipschitz, and so we have to impose the existence of a $C^2$ rotational invariant curve as a condition to the billiards we will use.
In the next paragraphs we present some bounds that will be necessary to prove our results.
Let us assume that the invariant set ${\cal O}_\rho$ is a closed curve $\gamma$= graph$(g)$. As $B|_\gamma$ preserves the order of $[0,2\pi)\approx S^1$ then either $g(\varphi)\equiv \pi/2$ or there exist constants $b$ and $B$ such that $0<b\leq g(\varphi)\leq B< \frac{\pi}{2}$ or
$\frac{\pi}{2} <B\leq g(\varphi) \leq b< \pi$.
The $C^2$ (or more) character of $\gamma$ also implies that a tangent vector $(1,g'(\varphi_0))$ is sent by $DB_{(\varphi_0,\alpha_0)}$ on a tangent vector to $\gamma$ at $(\varphi_1,\alpha_1)$. The preservation of orientation implies that the first coordinate of $DB_{(\varphi_0,\alpha_0)}(1,g'(\varphi_0))$ must be strictly positive.
So $$L[1+g'(\varphi_0)]-R_0\sin\alpha_0>0 \ \ \hbox{ and } \ \ 1+g'(\varphi_0)>\frac{R_0\sin\alpha_0}{L}>0$$
This implies that $g'(\varphi)>-1$ for every $\varphi$. As the billiard is reversible, the graph of $\tilde g(\varphi)=\pi-g(\varphi)$ is also a rotational invariant curve and then $\tilde g'(\varphi)=-g'(\varphi) >-1$
for every $\varphi$. So, for any $C^2$ invariant $\gamma=$graph$(g)$, $-1<g'(\varphi)<1$.
To each invariant rotational curve $\gamma$ is associated a caustic \cite{Tab95}, a curve lying inside the billiard table and
tangent to every segment of the billiard trajectory between two consecutive impacts. If $(\varphi_0,\alpha_0)$ and so
$(\phi_1,\alpha_1)$ belong to $\gamma$, then $\alpha_0=g(\varphi_0)$, $\alpha_1=g(\varphi_1)$ and we have
\begin{equation}\label{eq:caustica}
\frac{R_0 \sin \alpha_0}{1+g'(\varphi_0)} + \frac{R_1\sin \alpha_1}{1-g'(\varphi_1)} = L
\end{equation}
where the two terms on the left hand side are strictly positive. The first one measures the distance to the tangency
point (on the caustic) from the initial point $\Gamma(\varphi_0)$ and the second one, from the final point
$\Gamma(\varphi_1)$ (see \cite{cmp}).
\section{Non Elastic Billiards}\label{sec:nonela}
Let $B(\varphi_0,\alpha_0)=(\varphi_1,\alpha_1)$ be a $C^2$ classical billiard map on an oval, with a $C^2$ invariant rotational curve $\gamma_0$, given by the graph of $\alpha=g(\varphi)$.
A compact subset of the phase space $[0,2\pi)\times (0,\pi)$ with non-empty interior and whose boundaries are two distinct rotational curves (not necessarily invariant nor graphs) will be called a compact strip.
Let $I \in \mbox {${\rm {I\!R}}$}$ be a closed interval containing $0$. Given $h:I\to I$, a $C^2$ strictly increasing contraction with $h(0)=0$, we can define a non elastic billiard map $P$ on a compact strip $\Sigma$ containing $\gamma_0$ by
$$P(\varphi_0,\alpha_0) = \left(\varphi_1,\alpha_1-h(\alpha_1-g(\phi_1))\right)$$
with $\Sigma$ chosen such that if $(\varphi,\alpha)\in\Sigma$ then $\alpha-g(\varphi)\in I$.
$P$ is the composition of a classical billiard followed by a change at the reflection angle, corresponding to a contraction in the vertical fibers of the invariant rotational curve $\gamma_0$.
Observe that $h(t) \equiv 0$ corresponds to the classical unperturbed billiard and that $h(t) = t$ corresponds to a map that
sends all the points in $\Sigma$ onto the invariant curve (called {\em slap billiard} in \cite{MPS09}).
The derivative of $P$ is given by
\begin{equation}
DP_{(\varphi_0,\alpha_0)} =
\frac{1}{ R_1 \sin \alpha_1}
\left(\begin{array}{cc}
1 & 0 \\
h'_1 g'_1 & 1-h'_1
\end{array}
\right)
\left(
\begin{array}{cc}
L - R_0 \sin \alpha_0 & L \\
L - R_0 \sin \alpha_0 - R_1 \sin \alpha_1 & L - R_1 \sin \alpha_1
\end{array}
\right) \label{eqn:p}
\end{equation}
where $g'_i=g'(\varphi_i)$ and $h'_i=h'(\alpha_i-g(\varphi_i))$
Our main result is:
\begin{theorem}\label{nonelastic}
Given a classical oval billiard map $B$, with a $C^2$ invariant rotational curve $\gamma_0=\{(\varphi,g(\varphi))\}$, consider a compact strip $\Sigma$ containing $\gamma_0$ and a closed interval
$I \subset \mbox {${\rm {I\!R}}$}$, such that $\alpha-g(\varphi)\in I$ if $(\varphi,\alpha)\in\Sigma$.
If $h:I\mapsto \mbox {${\rm {I\!R}}$}$ is a $C^2$ function satisfying $h(0)=0$ and $0\leq 1-\underline l<h'(0)<1$
(with $\underline l$ depending only on $\gamma_0$), then there exists a
compact strip $S \subset \Sigma$ such that the non elastic billiard map $P$ defined by $B$, $g$ and $h$ is a
$C^2$-diffeomorphism from $S$ onto $P(S)$. Its limit set $L(P)$ contains $\gamma_0$ and has a dominated splitting. Moreover, the non elastic perturbation does not change the dynamics on $\gamma_0$.
\end{theorem}
Proof:
The non elastic billiard $P:\Sigma \to P(\Sigma)\subset
[0,2\pi)\times(0,\pi)$ is the composition of the $C^2$ classical
billiard map $B$ with the $C^2$ perturbation of the identity
$(\varphi,\alpha) \mapsto (\varphi,\alpha)-(0, h(\alpha-g(\phi)))$, where
$h$ is a $C^2$ contraction. Then $P$ is a $C^2$ diffeomorphism.
Given
$\delta> 0 $, let $u_{(\varphi,\alpha)} = (1, g'(\varphi) - \delta) $
and $v_{(\varphi,\alpha)} = (1, g'(\varphi) + \delta)$ be two
linearly independent vector fields defining the cone field ${\cal
C}(\varphi,\alpha)$ and the associated quadratic form
$Q_{(\varphi,\alpha)}$ (as in section \ref{sec:cones}).
Using the change of bases matrices, we have
{\begin{equation}
\hskip -0.115cm [DP_{(\varphi_0,\alpha_0)}]_U =
{\displaystyle \frac{1}{2\delta R_1 \sin \alpha_1}}
\left(
\begin{array}{cc}
\delta (l_{0} - \delta L) +(1- h'_1) ( \delta l_{1} - l_{01}) &
\delta (l_{0} + \delta L) -(1- h'_1) ( \delta l_{1} + l_{01}) \\
\delta (l_{0} - \delta L) - (1- h'_1)( \delta l_{1} - l_{01}) &
\delta (l_{0} + \delta L) + (1- h'_1) ( \delta l_{1} + l_{01})
\end{array}
\right) \label{eqn:dp}
\end{equation}}
where
\begin{eqnarray*}
l_{0} &=& L(1+g'_0) - R_0 \sin \alpha_0 \ \ , \ \
l_{1} = L(1-g'_1) - R_1 \sin \alpha_1 \\
l_{01} &=& L(1+g'_0)(1-g'_1) - (1-g'_1) R_0 \sin \alpha_0 - (1+g'_0) R_1 \sin \alpha_1 \ .
\end{eqnarray*}
Relation (\ref{eq:caustica}) implies that for every $(\varphi_0,\alpha_0)$ and $(\varphi_1,\alpha_1)$ on
$\gamma_0$ we have
$$l_{01} =0 \ \ , \ \
l_{0}=\frac{1+g'_0 }{1-g'_1 } R_1 \sin \alpha_1 \ \ , \ \
l_{1}=\frac{1-g'_1 }{1+g'_0 } R_0 \sin \alpha_0 .$$
The billiard boundary $\Gamma$ is an oval and as it is compact there exist
constants $a$ and $A$ and a width $D$ such that $0<a\leq
R(\varphi)\leq A$ and $0< L\leq D$. As the invariant curve is also compact,
for every $(\varphi_0,\alpha_0)$ and
$(\varphi_1,\alpha_1)$
on $\gamma_0$, there exist constants $c$ and $C$ such
that $\displaystyle C\geq l_{0}\geq c>0$ and $\displaystyle C \geq
l_{1} \geq c>0$. So for points on $\gamma_0$ there are constants $0<\underline l\leq 1$ and $0<\overline L$
such that
$$\underline l\leq \frac{l_0}{l_1} \hskip1cm \mbox{and}\hskip1cm 0<\frac{L}{l_1}\leq \overline L
\ .$$
By formula (\ref{eqn:dp}), each entry of the matrix
$[DP_{(\varphi_0,\alpha_0)}]_ U$ for $(\varphi_0,\alpha_0)\in
\gamma_0$ is of the form
$$ \delta
\left(
l_{0} \pm \delta \, L \pm \left(1- h'(0)\right)
\, l_{1}
\right)
\geq
\delta l_1
\left( \frac{l_0}{l_1} - \delta\frac{ L}{l_1} - (1- h'(0)) \right) \geq
\delta c \left(\underline l -\delta\overline L-(1- h'(0)) \right). $$
Now, if $0\leq 1-\underline l<h'(0)<1$, we can choose $\delta
>0$ such that $\delta c(\underline l -\delta\overline L-(1-
h'(0)))>0$. As $P$ is a $C^2$ diffeomorphism and remembering that
$\gamma_0$ is compact, we can find a strip $S\subset \Sigma$,
containing $\gamma_0$, where $P$ is well defined and all the
entries of $[DP_{(\varphi_0,\alpha_0)}]_ U$ are strictly positive.
Then, by lemma \ref{lema1}, $L(P)\subset S$ has a dominated splitting and contains $\gamma_0$, since $h(0)=0$.
Moreover, $P|_{\gamma_0}=B|_{\gamma_0}$ and the dynamics under $P$, on $\gamma_0$, is the same as under $B$. As $\gamma_0$ and $P$ are $C^2$, for our billiard dynamics, $\gamma_0$ is either a set of periodic points of same period linked by homo/heteroclinic arcs or supports a rational or an irrational rotation.
\hskip .4cm $\blacksquare$
This result will guide us to construct examples of non elastic billiards on ovals with limit set having a dominated splitting and supporting a rational or an irrational rotation (pieces of type $\mathcal I$ or $\mathcal R$ of Pujals-Sambarino's Theorem). This will be done in the next section.
We will pay attention to the maximal possible size of the strip $S$ and will try to see if there are other attractors on $L(P)$ than $\gamma_0$.
\section{Examples}\label{sec:exa}
\subsection{The circle}
The simplest example of a classical billiard with invariant rotational curves is the circular one. This billiard map is linear and is given by $B(\varphi_0,\alpha_0)=(\varphi_0+2\alpha_0,\alpha_0)$. The phase space $[0,2\pi)\times(0,\pi)$ is foliated by invariant horizontal curves and the dynamics on each one of them is simply a rotation of $2\alpha_0$.
We pick one invariant curve $\gamma_0$, defined by $\alpha=g(\varphi)=\overline\beta_0$. At $\gamma_0$, $g'\equiv 0$, $R_i=R$, the radius of the circle, and $\sin\alpha_i=\sin\overline\beta_0$, implying $l_0=l_1$ and $\underline l=1$.
Fix $I$, a closed interval with $0\in I\subset (-\overline\beta_0,\pi- \overline\beta_0)$ and $h:I\mapsto\mbox {${\rm {I\!R}}$}$, any $C^2$ strictly increasing contraction such that $h(0)=0$ and $0< h'(0)<1$. The non elastic billiard $P$ is defined on the strip
$[0,2\pi)\times \{I+\overline\beta_0\}$ and is given by
$P(\varphi_0,\alpha_0)=(\varphi_0+2\alpha_0,\alpha_0-h(\alpha_0-\overline\beta_0))$.
By Theorem~\ref{nonelastic}, there exists a compact strip $S$ such that $P|_S$ is a $C^2$ diffeomorphism and $L(P|_S)$ contains $\gamma_0$ and has a dominated splitting.
Now, we choose $\overline\beta_-$ and $\overline\beta_+$ such that $W=[0,2\pi)\times[\overline\beta_{-},\overline\beta_+]$ is the biggest horizontal straight strip contained in $S$. As each boundary $\gamma_\pm=\{(\varphi, \overline\beta_\pm)\}$ is invariant under $B$, we have that $P(W)\subset W$.
The map $P$ is a horizontal rotation followed by a vertical contraction. Denoting $(\varphi_n,\alpha_n)=P^n(\varphi_0,\alpha_0)$, it is easy to see that $\alpha_n\to \overline\beta_0$, as $n\to+\infty$ and so the horizontal circle $\gamma_0$ is the unique attractor of $P$ in $W$.
Moreover, the restricted map $P|_{\gamma_0}$ is just a rotation of angle $2\overline\beta_0$.
If $\overline\beta_0/\pi$ is rational, $\gamma_0$ is a normally hyperbolic simple closed curve composed by periodic points of the same period.
If $\overline\beta_0/\pi$ is irrational $\gamma_0$ is a normally hyperbolic closed curve supporting an irrational rotation.
This yields an example of a diffeomorphism, defined on a strip $W$, whose limit set has dominated splitting and is composed by a unique piece of type $\mathcal I$ or $\mathcal R$ of Pujals-Sambarino's Theorem.
Clearly, the size of the strip $W$ depends on the choice of the contraction $h$. Taking for instance $h(x)=\mu x$, with $0<\mu<1$, the non elastic billiard is given by
$P(\varphi_0,\alpha_0)=(\varphi_0+2\alpha_0,\alpha_0-\mu(\alpha_0-\overline\beta_0))$ and the basin of attraction of
$\gamma_0$ contains any straight strip $W=[0,2\pi)\times[\overline\beta_{-},\overline\beta_+]$.
\subsection{The ellipse}
A similar example is given by the classical elliptical billiard. We consider an ellipse $\Gamma$ with eccentricity $e$ and minor axis 1. Its radius of curvature $R$ satisfies $\displaystyle\sqrt{1-e^2}\leq R\leq \frac{1}{1-e^2}$.
The associated classical billiard map is denoted by $B:[0,2\pi)\times (0,\pi)\mapsto[0,2\pi)\times (0,\pi)$.
This billiard system is integrable: the function
$\displaystyle F(\varphi, \alpha)= \frac{\cos^2\alpha - e^2\cos^2\varphi}{1-e^2\cos^2\varphi}$
is a first integral (see, for instance \cite{berry}) and $[0,2\pi)\times (0,\pi)$ is foliated by the levels of $F=F_0$, with $-\frac{e^2}{1-e^2}<F_0<1$. If $0<F_0<1$, the level set consists of two invariant, analytic and symmetric rotational curves, the lower one contained in $[0,2\pi)\times (0,\frac{\pi}{2})$ and the upper one in $[0,2\pi)\times (\frac{\pi}{2} ,\pi)$. But, unlike the circular case, $B$ has two elliptic islands of period 2, obstructing the rotational invariant curves to foliate the whole phase-space, as can be seen on Figure 1(left). It also has a hyperbolic 2-periodic orbit, with a saddle connection, corresponding to the level $F_0=0$.
For a fixed $0<F_0<1$ let $\displaystyle\gamma_0$ be the lower invariant rotational curve in $ F(\phi,\alpha) = F_0 $ (the upper case is analogous). It is the graph of $\alpha=g(\varphi)$ given implicitly by $\cos \alpha = \sqrt{F_0+(1-F_0)e^2\cos^2\phi}.$
We have then that, for any $(\varphi,\alpha)\in \gamma_0$, $\sqrt{(1-F_0)(1-e^2)}\leq\sin\alpha\leq\sqrt{1-F_0}$.
Differentiating twice with respect to $\varphi$ we get $(1-F_0)e^2\sin 2\phi = g'(\phi) \sin 2\alpha$ and $2(1-F_0)e^2\cos 2\phi = 2 g'(\phi) \cos 2\alpha + g''(\phi) \sin 2\alpha$.
The extremal points of $g'$ must satisfy $\tan 2 \phi = \tan 2\alpha$ which implies $\max \{g'(\phi)\} = (1-F_0) e^2 = - \min \{g'(\phi)\}$.
We can then take
\begin{equation}\label{underline}
\underline{l} =
(1-e^2)^2 \left( \frac{1-(1-F_0) e^2}{1+(1-F_0) e^2} \right)^2 \le
\frac{(1+g'_0)^2 R_1 \sin \alpha_1}{(1-g'_1 )^2 R_0 \sin \alpha_0}
= \frac{l_0}{l_1}
\end{equation}
The associated non elastic billiard map is given by $P(\varphi_0,\alpha_0)=(\varphi_1,\alpha_1-h(\alpha_1-g(\varphi_1))$
where the contraction $h: I\mapsto \mbox {${\rm {I\!R}}$}$ is an arbitrary $C^2$ function satisfying $h(0)=0$ and $0<1-\underline l\leq h'(0)$, and $I$ is a closed interval containing $0$. Then, by Theorem~\ref{nonelastic}, there exists a compact strip $S$, containing $\gamma_0$, such that $P|_S$ is a $C^2$ diffeomorphism, $L(P|_S)$ has a dominated splitting and contains $\gamma_0$.
Let $F_\pm$ be two constants of motion satisfying $1> F_- > F_0 > F_+ > 0 $, and $\gamma_\pm$=graph$(g_\pm)$ be the lowest invariant rotational curves associated to $F_\pm$, respectively.
We also suppose that
$W=\{(\varphi,\alpha), g_{-}(\varphi)\leq\alpha\leq g_{+}(\varphi)\}\subset S$ is the biggest strip of this type contained on $S$. As $\gamma_\pm$ are invariant under $B$ we have $P(W)\subset W$.
For $(\varphi_0,\alpha_0)\in W$, we denote $(\varphi_n,\alpha_n)=P^n(\varphi_0,\alpha_0)$. Let us suppose, for instance, that $F(\varphi_0,\alpha_0)>F_0$, the other case being analogous. As $P$ is a translation on a $B$-invariant rotational curve followed by a contraction on the vertical direction toward $\gamma_0$, then $F(\varphi_n,\alpha_n)\to F_0$ and $\gamma_0$ is the $\omega$-limit of any $(\varphi_0,\alpha_0)\in W$. As in the circular case, $\gamma_0$ is the unique attractor of $P|_W$.
Using action-angle variables, Chang and Friedberg \cite{cha} have shown how to decide if the rotation number associated to the level $F_0=F(\varphi_0,\alpha_0)$ of a given initial condition $(\varphi_0,\alpha_0)$, is rational or irrational and then determine the dynamics on each level curve.
As $P|_{\gamma_0}=B |_{\gamma_0}$, this allows us to choose $F_0$ in order to have, as the unique attractor, a normally hyperbolic closed curve supporting an irrational rotation or a normally hyperbolic closed curve composed by periodic points of same period.
Although theoretically promising as a result, depending on the choice of $\gamma_0$ and $h$, the strip $W$ can be very thin. In particular it will never contain points of the elliptical islands.
In trying to go beyond the theoretical predictions and find examples of non elastic elliptical billiards defined on a bigger part of the phase space, we shall remember that weak contractions will never destroy the rotation carried by the linear ellipticity of the 2-periodic orbit. So we can not expect the associated non elastic billiard to be defined on a strip containing the islands. However, it is an interesting question whether this can be achieved by taking a sufficiently strong contraction.
We present some numerical simulations where this can be done. We choose an ellipse with $e= 0.35$ and fix the invariant curve $\gamma_0$ at $F_0=0.25$. Our choices of the eccentricity and $\gamma_0$ are rather arbitrary.
Figure~\ref{fig:elipse1} (left) below diplays the phase space of the classical elliptical billiard. The invariant curve $\gamma_0$ is enhanced. The horizontal axis corresponds to $\phi \in [0,2\pi)$ and the vertical to $\alpha \in (0,\pi)$, the left bottom corner being the origin.
We consider linear perturbations $h(x) = \mu x$, with $0<\mu<1$. $\mu =0$ implies that there is no perturbation (classical billiard) and $\mu = 1$ implies that all the points in the phase space land on the invariant curve after one iteration (slap billiard). The non elastic billiard $P(\varphi_0,\alpha_0)=(\varphi_1,\alpha_1-\mu(\alpha_1-g(\phi_1)))$
is a $C^2$ diffeomorphism on any compact strip contained on $[0,2\pi)\times(0,\pi)$. Calculating $\underline l$ by formula (\ref{underline}) we get that if $\mu>1-\underline l \approx 0.47$
there is a strip $W$ such that $\gamma_0$ is the unique attractor of $P$ on $W$ and has a dominated splitting.
Figure~\ref{fig:elipse1} (right) illustrates the basin of
attraction of the curve $\gamma_0$ for $\mu = 0.5$: black points
correspond to initial conditions which approach $\gamma_0$ under
iteration. This simulation thus indicates that the basin of
attraction of $\gamma_0$ is the whole phase space, ie, that $W$
can be any compact strip contained in $[0,2\pi)\times(0,\pi)$ and
$\gamma_0$ is the unique attractor of $P$. Note that, as we have
mentioned before, either $\gamma_0$ is composed of periodic points
of same period or supports an irrational rotation.
\begin{figure}[h]
\begin{center}
\includegraphics[viewport=0 0 570 570,width=.4\hsize]{mkp-lfig1a.eps
\hskip 0.5cm
\includegraphics[viewport=0 0 570 570,width=.4\hsize]{mkp-lfig1b.eps
\end{center}
\caption{Classical and non elastic elliptical billiards}
\label{fig:elipse1}
\end{figure}
\subsection{Non integrable billiards}
There are no other known $C^2$ ovals such that the classical billiard map is integrable, other than the circle and the ellipse. But it is well known that there are oval billiards with $C^2$ invariant rotational curves.
Taking a sufficiently differentiable, non integrable, classical oval billiard map $B$ with a $C^2$ invariant rotational curve $\gamma_0$=graph$(g)$ we can, as before, take a contraction $h$ satisfying the hypothesis of Theorem~\ref{nonelastic} and define a non elastic billiard map $P$.
Then there exists a strip $S$ on which $P$ is a $C^2$ diffeomorphism on $S$, $L(P)$ contains $\gamma_0$ and has a dominated splitting. Depending on the rotation number of $\gamma_0$ we can have, as in the previous examples, a normally hyperbolic closed curve supporting an irrational rotation or a normally hyperbolic closed curve composed by periodic points of same period. But we can no longer guarantee that the only attractor of $P$ is $\gamma_0$.
\subsubsection{Invariant straight line}
In order to explore numerically what happens in those more general
cases, we have to pick a concrete example. Although in any example one can prove
the existence of whole families of invariant curves, it is almost
impossible for any fixed curve, to write down the function $g$ for
which it is the graph.
In general this can only be achieved in very specific examples as, for instance, the constant width curves, where $g(\varphi)=\pi/2$. We will deal with a much richer example, given by symmetric perturbations of the circle.
Let $\Gamma_n$ be the oval parameterized by the angle $\varphi$ and
which radius of curvature is of the form $R(\phi)= 1 + a \cos n\phi$, $|a|<1$ and $n\geq 4$.
Tabachnikov (Section 2.11, \cite{Tab95}) showed that its associated classical billiard map have an invariant
straight line given by $g(\varphi)=\beta_0$ if $\beta_0$ satisfies $n\tan \beta_0 = \tan n\beta_0$. It is not difficult to show that the dynamics on $\gamma_0$ is an irrational rotation \cite{geraldo}.
For $n=6$, $ \beta_0 = \tan^{-1} \sqrt{7+4\sqrt{21}/3} \approx 0.41\pi$ satisfies this condition and the line $\gamma_0$ given by $\alpha = g(\phi) = \beta_0$ is invariant.
Figure \ref{fig:mesa} displays the billiard table and the corresponding phase space of $\Gamma_6$ defined by $R(\phi) = 1+ 0.01 \cos 6\phi$. As in the previous section, the horizontal axis corresponds to $\phi \in [0,2\pi)$ and the vertical to $\alpha \in (0,\pi)$, the left bottom corner being the origin. Note the invariant straight line $\gamma_0$.
\begin{figure}[h]
\begin{center}
\includegraphics[viewport=48 0 236 290,width=.15\hsize]{mkp-lfig2a.eps
\hskip 2cm
\includegraphics[viewport=0 0 570 570,width=.4\hsize]{mkp-lfig2b.eps
\end{center}
\caption{Billiard table and classical phase space for $\Gamma_6$}
\label{fig:mesa}
\end{figure}
As in the other numerical examples, we consider linear perturbations $h(x) = \mu x$, $0<\mu<1$.
On $\gamma_0$, we have $l_0 = R_1 \sin \beta_0$ and
$l_1 = R_0 \sin \beta_0$ and we can take $\underline{l} = \frac{\min R(\phi)}{\max R(\phi)} =
\frac{0.99}{1.01}$.
Then, if $\mu>1-\underline l \approx 0.02$, there exists a strip $S$ on which $P$ is a $C^2$ diffeomorphism, $L(P)$ contains $\gamma_0$ and has a dominated splitting.
Figure \ref{fig:4bacias} illustrates the basin of attraction of $\gamma_0$ for $\mu=0.1,\, 0.35,\, 0.37$ and $ 0.4$: black points correspond to initial conditions which approach $\gamma_0$ under iteration. The white tadpoles correspond to points attracted to the 6-periodic orbits, linearly elliptic for the original classical billiard.
We observe that as the contraction factor $\mu$ is increased, the basin of attraction of $\gamma_0$ grows until it eventually occupies the whole phase space.
Thus, for strong contractions, we may have $P$ defined on any compact strip $S$ and having $\gamma_0$ as its unique attractor.
\begin{figure}[h]
\begin{center}
\includegraphics[viewport=0 0 570 570,width=.4\hsize]{mkp-lfig3a.eps
\hskip 0.5cm
\includegraphics[viewport=0 0 570 570,width=.4\hsize]{mkp-lfig3b.eps
\vskip 0.5cm
\includegraphics[viewport=0 0 570 570,width=.4\hsize]{mkp-lfig3c.eps
\hskip 0.5cm
\includegraphics[viewport=0 0 570 570,width=.4\hsize]{mkp-lfig3d.eps
\end{center}
\caption{$\Gamma_6$: the basin of attraction of $\gamma_0$ for $\mu=0.1, 0.35, 0.37$ and $ 0.4$}
\label{fig:4bacias}
\end{figure}
\subsection{An example that is not one}
Any strictly convex classical billiard map is a monotone twist map
with rotation interval $(0,1)$. To each $\rho\in (0,1)$ is
associated a set ${\cal O}_\rho$. If $\rho$ is irrational, ${\cal
O}_\rho$ is either a rotational invariant curve or an Aubry-Mather
set. An Aubry Mather set is a closed, invariant, minimal set,
projecting injectively on a Cantor set of $S^1\sim[0,2\pi)$ and
such that the dynamics preserves the order of $S^1$. It is
contained in a non invariant graph of a continuous piecewise
linear Lipschitz function $\alpha=g(\varphi)$ (see, for instance \cite{kat}, Section 13.2).
Let us take a classical billiard map $B$ with two rotational invariant curves $\gamma_{-}$ and $\gamma_+$ and an Aubry-Mather set ${\cal A}$ contained in the strip bounded by $\gamma_{-}$ and
$\gamma_+$ and a perturbation $h$ on this strip.
The non elastic billiard map can be then defined as before, as $P(\varphi_0,\alpha_0) =(\varphi_1, \alpha_1 -h(\alpha_1 - g(\varphi_1))$, where $(\varphi_1, \alpha_1)=B(\varphi_0, \alpha_0)$.
As $h$ is a contraction, the limit set of $P$ will contain the Aubry-Mather set ${\cal A}$. However, we must remark that $P$ is not $C^2$ because $g$ is not even a differentiable function. Then, we can not apply Pujal-Sambarino's theorem.
\noindent {\bf Acknowledgment.}
{We thank M\'ario Jorge Dias Carneiro for his precious ideas. We thank the
Brazilian agencies FAPEMIG and CNPq for financial support.
}
|
{
"timestamp": "2011-04-20T02:01:46",
"yymm": "1009",
"arxiv_id": "1009.4187",
"language": "en",
"url": "https://arxiv.org/abs/1009.4187"
}
|
\section{Introduction}
There is a well-known correspondence between points of the affine Grassmannian for $\Gl_{n}$ and vector bundles on a projective curve together with certain trivializations. Let us recall this correspondence, as Beauville and Laszlo describe it in \cite{beauville-laszlo}.
Let $X$ be a smooth projective curve over $k$, $p\in X$ be a closed point, and choose a uniformizer $z\in \O_{X,p}$. We fix these data for the rest of these notes. For every $k$-algebra $R$ we set
\begin{equation}
\begin{split}
&X_{R} := X\otimes_{\Spec k}\Spec R, \quad X_{R}^{*} := \Spec (\O_{X}(X-\lbrace p\rbrace)\otimes_{k}R),\\
&D_{R} := \Spec R[[z]],\quad D_{R}^{*} := R((z)).
\end{split}
\end{equation}
These data determine a cartesian diagram of schemes
\hspace{\fill}
\begin{equation}\label{diagFormal}
\begin{xy}
\xymatrix{
D_{R}^{*} \ar^{\psi}[r]\ar_{i}[d] & X_{R}^{*} \ar^{j}[d] \\
D_{R} \ar^{f}[r] & X_{R}.
}
\end{xy}
\end{equation}
\hspace{\fill}
Beauville and Laszlo prove the following
\begin{proposition}[\cite{beauville-laszlo}, Proposition 1.4]\label{thmBL1}
The functor
$$
\Loop\Gl_{n}: R \mapsto \Gl_{n}(R((z)))
$$
on the category of $k$-algebras is isomorphic to the functor which associates to $R$ the set of isomorphism classes of triples $(E,\rho,\sigma)$, where $E$ is a vector bundle of rank $n$ over $X_{R}$, and $\rho$ and $\sigma$ are trivializations of $E$ over $X_{R}^{*}$ and $D_{R}$, respectively.
\end{proposition}
As a consequence they obtain
\begin{proposition}[\cite{beauville-laszlo}, Proposition 2.1 and Remark 2.2]\label{thmBL2}
The affine Grassmannian for $\Gl_{n}$, which is by definition the fpqc-sheafification of the functor $R\mapsto \Gl_{n}(R((z)))/\Gl_{n}(R[[z]])$, is isomorphic to the functor which associates to $R$ the set of isomorphism classes of pairs $(E,\rho)$, where $E$ is a vector bundle of rank $n$ over $X_{R}$, and $\rho$ is a trivialization of $E$ over $X_{R}^{*}$.
\end{proposition}
The interesting part in the proof of Proposition \ref{thmBL1} is to see why the data of trivial vector bundles of rank $n$ on $D_{R}$ and $X_{R}^{*}$, respectively, together with a transition function over $X_{R}^{*}$, determine a vector bundle on $X_{R}$. This is not a classical descent situation, since if $R$ is not Noetherian, $D_{R}$ is in general not flat over $X_{R}$. In \cite{bl-descente} Beauville and Laszlo prove that descent holds nontheless.
In the present notes we present an alternative proof of Proposition \ref{thmBL1} using the following strategy. We define the subring $A_{R}\subset R[[z]]$ as a certain localization of $\O_{X,p}\otimes_{k}R$, which depends functorially on $R$ and determines a flat neighborhood of the locus $z=0$ in $X_{R}$. Let us write $\Delta_{R} = \Spec A_{R}$ and $\Delta_{R}^{*}=\Spec A_{R}[1/z]$.
Then $\Delta_{R}\coprod X_{R}^{*}\to X_{R}$ is an fppf-covering, and if we could replace $D_{R}$ by $\Delta_{R}$ and $D_{R}^{*}$ by $\Delta_{R}^{*}$ in the formulation of Proposition \ref{thmBL1}, then this proposition would immediately follow by faithfully flat descent.
Indeed, we will show below how to arrive at this situation using a simple approximation argument. Moreover, the concrete situation will turn out to be not only fppf-local, but even Zariski-local, so that descent of vector bundles holds trivially.
\section{Vector bundles on a smooth curve}\label{sectionAlgebraic}
Note that the choice of a uniformizer $z\in \O_{X,p}$ determines an inclusion $(R\otimes_{k}\O_{X,p}) \subset R[[z]]$, $R[[z]]$ being the completion with respect to the $z$-adic valuation.
For each $f\in (R\otimes_{k}\O_{X,p})\cap R[[z]]^{\times}$
we define $S_{R,f} := (R\otimes_{k}\O_{X,p})_{f} \subset R[[z]]$. The union of all these rings, for varying $f$, will be denoted $A_{R}$. Writing $\Delta_{R} := \Spec A_{R}$ and $\Delta_{R}^{*} := \Spec A_{R}[1/z]$ we have a cartesian diagram
\hspace{\fill}
\begin{xy}
\xymatrix{
\Delta_{R}^{*} \ar^{\psi}[r]\ar^{\iota}[d] & X_{R}^{*} \ar^{j}[d] \\
\Delta_{R} \ar^{\varphi}[r] & X_{R}.
}
\end{xy}
\hspace{\fill}
Moreover we set $U_{R,f} := \Spec S_{R,f}$.
\begin{lemma}\label{lemCoverings}
The morphism $D_{R} \coprod X_{R}^{*}\to X_{R}$
is surjective. Thus $\Delta_{R} \coprod X_{R}^{*} \to X_{R}$ is an fppf-, and $U_{R,f} \coprod X_{R}^{*} \to X_{R}$ is a Zariski-covering for each $f\in (R\otimes_{k}\O_{X,p})\cap R[[z]]^{\times}$.
\end{lemma}
\begin{proof}
Let $P$ be a point of $X_{R}$ and let $A=(\O_{X}\otimes R)_{P}$ be the local ring at $P$. Either $z$ is invertible in $A$ -- then $P\in X_{R}^{*}$ -- or $z$ is in the maximal ideal $\mathfrak{p} \subset A$. In the latter case we consider $can: A\to\hat{A}=\plim A/z^{N}$ and the ideal $\hat{\mathfrak{p}} = \plim \mathfrak{p}/z^{N}$. Passing to the inverse limit over the short exact sequences
$$
0 \to \mathfrak{p}/(z^{N}) \to A/(z^{N}) \to A/\mathfrak{p} \to 0
$$
we obtain $can^{-1}(\hat{\mathfrak{p}}) = \mathfrak{p}$, and the commutative square
\hspace{\fill}
\begin{xy}
\xymatrix{
\Spec \hat{A} \ar[r]\ar[d] & \Spec R[[z]]=D_{R} \ar[d] \\
\Spec A \ar[r] & X_{R}.
}
\end{xy}
\hspace{\fill}
shows that $\hat{\mathfrak{p}}\cap R[[z]] \subset R[[z]]$ is a preimage of $P$ in $D_{R}$.
\end{proof}
Let $T$ be the functor on the category of $k$-algebras, which associates to a $k$-algebra $R$ the set of isomorphisms classes of triples $(E,\rho,\sigma)$, where $E$ is a vector bundle of rank $n$ on $X_{R}$, and
\begin{align*}
\rho: \mathcal{O}_{X_{R}^{*}}^{n} \xrightarrow{\simeq} E_{\rvert X^{*}_{R}},\\
\sigma: \mathcal{O}_{\Delta_{R}}^{n} \xrightarrow{\simeq} E_{\rvert \Delta_{R}}
\end{align*}
are trivializations. To each isomorphism class $[(E,\rho,\sigma)]\in T(R)$ we may assign the respective `transition matrix over $\Delta_{R}^{*}$'. This is independent of the actual representative of $[(E,\rho,\sigma)]$ and hence determines a morphism of functors
$$
\Phi(R): T(R) \to \Gl_{n}(A_{R}[1/z]); \quad (E,\rho,\sigma) \mapsto \Gamma(X_{R}, (\rho\rvert_{\Delta_{R}^{*}})\circ(\sigma^{-1}\rvert_{\Delta_{R}^{*}})).
$$
\begin{proposition}\label{propAlgebraic}
The morphism $\Phi(R)$ defined above is an isomorphism of functors.
\end{proposition}
\begin{proof}
We have to construct an inverse for $\Phi(R)$. To this end, we choose a matrix $g \in \Gl_{n}(A_{R}[1/z])$ and consider the following diagram of quasi-coherent sheaves on $X_{R}$,
\hspace{\fill}
\begin{xy}
\xymatrix{
E \ar[rr]\ar[d] & & \mathcal{O}_{X_{R}^{*}}^{n} \ar^{can}[d] \\
\mathcal{O}_{\Delta_{R}}^{n} \ar^{can}[r] & \mathcal{O}_{\Delta_{R}^{*}}^{n} \ar^{g}[r] & \mathcal{O}_{\Delta_{R}^{*}}^{n},
}
\end{xy}
\hspace{\fill}
where $E$ is uniquely determined up to isomorphism by requiring that the diagram be cartesian. (By abuse of notation we do not indicate the obvious push-forwards to $X_{R}$ in this diagram.) It is easy to check (by pullback to $\Delta_{R}$ and $X_{R}^{*}$, respectively) that this diagram determines trivializations of $E$ over $\Delta_{R}$ and $X_{R}^{*}$. The transition function for these two trivializations is equal to $g$ by construction.
To see that this construction indeed gives an inverse for $\Phi(R)$ it remains to check that $E$ is a vector bundle. This is immediate by Lemma \ref{lemCoverings} together with faithfully flat descent, or by the following elementary argument: the matrix $g$ involves only finitely many elements of $A_{R}[1/z]$, whence in fact $g\in S_{R,f}[1/z]$ for some $f\in (R\otimes_{k}\O_{X,p})\cap R[[z]]^{\times}$. This shows that $E$ can as well be obtained by gluing trivial bundles over $U_{R,f}$ and over $X_{R}^{*}$, respectively. Now, since $U_{R,f} \subset X_{R}$ is Zariski-open, this shows that $E$ is a vector bundle.
\end{proof}
\section{`Formal' descent of vector bundles}\label{sectionFormal}
Let us now consider the situation introduced at the beginning in diagram \eqref{diagFormal}, where we consider the formal neighborhood $D_{R} = \Spec R[[z]]$ of $\Spec R\times\lbrace p\rbrace \subset X_{R}$.
By $\hat{T}$ we denote the functor, which associates to every $k$-algebra $R$ the set of isomorphism classes of triples $(E,\rho,\sigma)$, where $E$ is a vector bundle of rank $n$ over $X_{R}$ and
\begin{align*}
\rho: \mathcal{O}_{X_{R}^{*}}^{n} \xrightarrow{\simeq} E_{\rvert X_{R}^{*}},\\
\sigma: \mathcal{O}_{D_{R}}^{n} \xrightarrow{\simeq} E_{\rvert D_{R}}
\end{align*}
are trivializations.
As in the previous section, we obtain a functorial morphism $\hat{\Phi}(R): \hat{T}(R) \to \Gl_{n}(R((z)))$ by assigning to each triple $(E,\rho,\sigma)$ the corresponding transition function over $D_{R}^{*}$.
\begin{theorem}[\cite{beauville-laszlo}, Proposition 1.4]\label{thmFormal}
The morphism $\hat{\Phi}$ is an isomorphism of functors.
\end{theorem}
\begin{proof}
In order to construct an inverse for $\hat{\Phi}$, i.e. to construct a triple $(E,\rho,\sigma)$ from a given $\gamma \in \Gl_{n}(R((z)))$, we proceed exactly as in the proof of Proposition \ref{propAlgebraic}. The only non-trivial thing to check is that the quasi-coherent sheaf $E$, defined so to make the diagram
\hspace{\fill}
\begin{equation}\label{diagX}
\begin{xy}
\xymatrix{
E \ar[rr]\ar[d] & & \mathcal{O}_{X_{R}^{*}}^{n} \ar^{can}[d] \\
\mathcal{O}_{D_{R}}^{n} \ar^{can}[r] & \mathcal{O}_{D^{*}_{R}}^{n} \ar^{\gamma}[r] & \mathcal{O}_{D^{*}_{R}}^{n},
}
\end{xy}
\end{equation}
\hspace{\fill}
cartesian, is a vector bundle over $X_{R}$. We do this by reducing to a situation where Proposition \ref{propAlgebraic} applies.
More precisely, Lemma \ref{lemDensity} below shows that every $\gamma \in \Gl_{n}(R((z)))$ can be written as a product $\gamma = g\cdot \delta$, where $g\in \Gl_{n}(A_{R}[1/z])$ and $\delta\in \Gl_{n}(R[[z]])$.
Thus diagram \eqref{diagX} `decomposes' likewise, and yields the big diagram
\hspace{\fill}
\begin{xy}
\xymatrix{
E \ar@{=}[r]\ar[dd] & E \ar[rr]\ar[d] & & \mathcal{O}_{X_{R}^{*}}^{n} \ar^{can}[d] \\
& \mathcal{O}_{\Delta_{R}}^{n} \ar^{can}[r]\ar[d] & \mathcal{O}_{\Delta_{R}^{*}}^{n} \ar^{g}[r]\ar[d] & \mathcal{O}_{\Delta_{R}^{*}}^{n}\ar[d] \\
\mathcal{O}_{D_{R}}^{n} \ar^{\simeq}_{\delta}[r] & \mathcal{O}_{D_{R}}^{n} \ar^{can}[r] & \mathcal{O}_{D^{*}_{R}}^{n} \ar_{g}[r] & \mathcal{O}_{D^{*}_{R}}^{n}.
}
\end{xy}
\hspace{\fill}
The two small squares in this diagram are trivially cartesian, while the big rectangle coincides with the square \eqref{diagX}, and is thus cartesian by definition of $E$. Consequently, the upper rectangle is cartesian, which proves that $E$ is nothing but the vector bundle corresponding to the transition matrix $g\in \Gl_{n}(A_{R}[1/z])$ under the correspondence of Proposition \ref{propAlgebraic}.
\end{proof}
\begin{lemma}\label{lemDensity}
We have $\Gl_{n}(R((z))) = \Gl_{n}(A_{R}[1/z])\cdot \Gl_{n}(R[[z]])$.
\end{lemma}
\begin{proof}
We set $B := \displaystyle\cup_{P\in R[z]\cap R[[z]]^{\times}} R[z,z^{-1},P^{-1}] \subset R((z))$ (Note that the ring $B\cap R[[z]]$ is equal to the ring $A_{R}$ in the case $X=\mathbb{P}^{1}_{k}$.). Since $B\subset A_{R}[1/z]$, it suffices to check that
$\Gl_{n}(R((z))) = \Gl_{n}(B)\cdot \Gl_{n}(R[[z]])$. First we note that $\Gl_{n}(R[[z]])\subset \Gl_{n}(R((z)))$ is open: Namely, $\det: \Mat_{n}(R[[z]]) \to R[[z]]$ is continuous and $R$ carries the discrete topology, and thus $R^{\times} \subset R$ is open. This shows that $\Gl_{n}(R[[z]]) \subset \Mat_{n}(R[[z]]) \subset \Mat_{n}(R((z)))$ are two open inclusions, so $\Gl_{n}(R[[z]])\subset \Gl_{n}(R((z)))$ is as well open. As a second step we deduce from Lemma \ref{lemUnits} below that $\Gl_{n}(B) = \Gl_{n}(R((z)))\cap \Mat_{n}(B)$. Since $\Mat_{n}(B)\subset \Mat_{n}(R((z)))$ is dense and $\Gl_{n}(R((z)))\subset \Mat_{n}(R((z)))$ is open, we conclude that $\Gl_{n}(B) \subset \Gl_{n}(R((z)))$ is dense.
These two statements together imply that $\Gl_{n}(B)\cdot \Gl_{n}(R[[z]])$ is dense and closed in $\Gl_{n}(R((z)))$, whence the lemma.
\end{proof}
\begin{lemma}\label{lemUnits}
The subring $B\subset R((z))$ defined above satisfies
$B^{\times} = R((z))^{\times} \cap B.$
\end{lemma}
\begin{proof}
We consider $f\in R((z))^{\times} \cap B$. By multiplying with a suitable $P\in R[z]\cap R[[z]]^{\times}$, we may reduce to the case $f\in R((z))^{\times} \cap R[z,z^{-1}]$. Such an $f$ has the form $f = -N + Q$, where $N\in R[z,z^{-1}]$ is a nilpotent Laurent polynomial and the leading coefficient of $Q\in R((z))^{\times}$ is a unit in $R$. Using the formula $(-N+Q)(N^{i}+N^{i-1}Q+\dotsb+Q^{i})=(-N^{i}+Q^{i})$ we may assume that $f=Q^{i}$, i.e. has a leading coefficient in $R^{\times}$. Multiplying with $z^{m}$ for a suitable $m\in\mathbb{Z}$ we obtain $z^{m}f \in R[z]\cap R[[z]]^{\times}$, which is invertible in $B$ by construction.
\end{proof}
The property of the ring $B$ which is exhibited in the last lemma is crucial for our strategy of approximation to work. This is what forces us to consider the, at first glance, rather artificial rings $A_{R}$ instead of for example just $\O_{X,p}\otimes R$. The latter would not contain the ring $B$, and in particular would not have the property of Lemma \ref{lemUnits}.
|
{
"timestamp": "2010-09-22T02:01:43",
"yymm": "1009",
"arxiv_id": "1009.4055",
"language": "en",
"url": "https://arxiv.org/abs/1009.4055"
}
|
\section{Introduction}
One of the main open problems in Cosmology is to determine the physical mechanism behind the current cosmic acceleration. This phenomenon has been evidenced by a combination of observational data~\cite{data} and, in the context of the general relativity theory, can be explained only if we admit the existence of an exotic field, the so-called dark energy. The origin and nature of this exotic component constitute a complete mystery and represents one of the major challenges not only to cosmology but also to our understanding of fundamental physics~(see, e.g., \cite{review} for more on this subject). By assuming a spatially flat geometry, this mysterious component accounts for (in units of the critical density) $\simeq 0.7$ of the cosmic composition, a value that is of the same order of magnitude of the relative density of the cold dark matter, $\simeq 0.3$. However, since these dark components are usually assumed to be independent and, therefore, scale in different ways, this would require an unbelievable coincidence, the so-called coincidence problem (CP).
A phenomenological attempt at alleviating the CP is allowing the dark matter and dark energy to interact. This phenomenology in turn gave origin to the so-called models of coupled quintessence, which have been largely explored in the literature \cite{cq, jesus, ernandes, ernandes1}. These scenarios are based on the premise that, unless some special and unknown symmetry in nature prevents or suppresses a non-minimal coupling between these components (see \cite{carroll} for a discussion), a small interaction cannot be ruled out.
The usual critique to coupled quintessence scenarios is that, in the absence of a natural guidance from fundamental physics, one needs to specify a possible interacting or coupling term between the two dark components in order to establish a model and investigate their observational and theoretical implications. In this concern, a still phenomenological but very interesting step toward a more realistic interacting or coupling law was recently discussed in Ref.~\cite{wm} (see also~\cite{alclim05}) in the context of models with vacuum decay $(\omega =-1)$. Instead of the traditional approach, Ref.~\cite{wm} deduced the new interaction law from a simple argument about the effect of the dark energy on the CDM expansion rate. Such a coupling is similar to the one obtained in Ref.~\cite{shapiro} from arguments based on renormalization group and seems to be very general, having many of the previous attempts as a particular case.
An important aspect worth emphasizing is that in the above analyses the interacting parameter $\epsilon$ has been considered constant over the cosmic evolution whereas in a more realistic case it must be a time-dependent quantity. In Ref.~\cite{ernandes2} the above arguments were extend for the case in which the interacting parameter $\epsilon$ is a function of the scale factor $a$. The analysis of Ref.~\cite{ernandes2}, however, was restricted to the case in which $\omega = -1$, which is mathematically equivalent to dynamical $\Lambda$ scenarios.
In this paper, we extend the arguments of Ref.~\cite{ernandes2} to a dark energy/dark matter interaction, where the dark energy component is described by an equation of state $p_{x}=\omega \rho_{x}$ with $\omega < 0$. We explore the dynamical behavior of this class of models and find viable cosmological solutions for two parameterizations of $\epsilon(a)$. In particular, the solutions with transient acceleration, as consequence of the interaction in the dark sector, are investigated in more detail. In order to check the observational viability of this general class of coupled quintessence scenarios, we also carry out a statistical analysis with recent observations of type Ia supernovae (SNe Ia) along with recent estimates of the CMB/BAO ratio at two different redshifts $z = 0.20$ and $z = 0.35$.
\section{Interaction in the dark sector}
\begin{figure*}
\centerline{\psfig{figure=gra0902.ps,width=2.0truein,height=1.9truein,angle=-90}
\psfig{figure=gra0901.ps,width=2.0truein,height=1.9truein,angle=-90}
\psfig{figure=gra09.ps,width=2.0truein,height=1.9truein,angle=-90}}
\label{fig:qzw}
\end{figure*}
\begin{figure*}
\centerline{\psfig{figure=gra1102.ps,width=2.0truein,height=1.9truein,angle=-90}
\psfig{figure=gra1101.ps,width=2.0truein,height=1.9truein,angle=-90}
\psfig{figure=gra11.ps,width=2.0truein,height=1.9truein,angle=-90}}
\caption{Evolution of the density parameters $\Omega_i$ ($i = b, dm, x$) as a function of $\log(a)$ for some selected combinations of $\epsilon_{0}$ and $\xi$, [Eq. (\ref{para1})] and two characteristic values of the equation-of-state parameter, $w = -0.9$ and $w = -1.1$, corresponding to quintessence and phantom behaviors, respectively. The case $\omega = -1.0$ was discussed in Ref.~\cite{ernandes2}.}
\label{fig:qzw}
\end{figure*}
According to current observations, the main contributions to the total energy-momentum tensor of the cosmic fluid are non-relativistic matter (baryonic plus dark) and a negative-pressure dark energy component. By assuming that a possible interaction occurs in the dark sector, the energy conservation equation for the two interacting components can be written as
\begin{equation}\label{coup}
\dot{\rho}_{dm} + 3 \frac{\dot{a}}{a}\rho_{dm} = -\dot{\rho}_x -
3\frac{\dot{a}}{a}(\rho_x + p_x)\;,
\end{equation}
where $\rho_{dm}$ and $\rho_x$, are the energy densities of the dark matter and dark energy, respectively, whereas $p_{x}$ is the dark energy pressure. As the dark components are exchanging energy, dark matter density will dilute in a rate whose deviation from standard case, $\rho_{dm} \propto a^{-3}$, may be characterized by the function $\epsilon(a)$, i.e.,
\begin{equation}
\label{dm}
\rho_{dm}=\rho_{dm,o} a^{-3 + \epsilon(a)},
\end{equation}
where the subscript 0 denotes current values and we have set $a_{0}=1$. Note that, contrarily to most analyses available in the literature, we consider the interaction parameter as a function of the cosmological scale factor, $\epsilon = \epsilon(a)$. In what follows we also consider that the dark energy is described by an equation of state $p_x=\omega \rho_x$, with $\omega = \rm{constant} < 0$.
By substituting the above evolution law into Eq. (\ref{coup}), we find
\begin{equation}\label{de}
\rho_{x} = \left[\rho_{x,0} + \rho_{dm,0} \int_{a}^{1}\frac{[\epsilon(\tilde{a}) + \tilde{a} \epsilon^{'} ln \tilde{a}]}{\tilde{a}^{1 -3\omega - \epsilon(\tilde{a})}} d\tilde{a} \right]a^{-3(1+\omega)},
\end{equation}
where a prime denotes derivative with respect to scale factor $a$. For $\omega=-1$ the above equation reduces to the vacuum decaying scenario recently discussed in Ref. \cite{ernandes2}, whereas for $\omega \neq - 1$ and $\epsilon = \rm{const.}$, the above expressions reduce to the scenario recently discussed in Refs.~\cite{jesus, ernandes1}.
\begin{figure*}
\centerline{\psfig{figure=grae0901.ps,width=1.7truein,height=1.8truein,angle=-90}
\psfig{figure=grae09.ps,width=1.7truein,height=1.8truein,angle=-90}
\psfig{figure=grae1101.ps,width=1.7truein,height=1.8truein,angle=-90}
\psfig{figure=grae11.ps,width=1.7truein,height=1.8truein,angle=-90}}
\caption{The same as in Fig. 1 for some selected values of $\epsilon_{0}$, [Eq. (\ref{para2})]. Note that for large positive values of $\epsilon_{0} > 1.2$ (Panel 2b and 2d), the interaction between dark matter and dark energy will drive the Universe to a new matter-dominated era in the future, when $a \rightarrow \infty$.}
\label{fig:qzw}
\end{figure*}
We assume from now on vanishing spatial curvature (and neglect the radiation contribution), so that the Friedmann equation for this interacting dark matter-dark energy scenario can be written as
\begin{equation}\label{friedmann}
{\cal{H}} = \left[ \Omega_{b,0} a^{-3} + \Omega_{dm,0}a^{-3 + \epsilon(a)} + \Omega_{x,0}f(a) \right]^{1/2}\;,
\end{equation}
where ${\cal{H}}={{H}}/{H_o}$, and $\Omega_{b,0}$, $\Omega_{dm,0}$ and $\Omega_{x,0}$ stand for, respectively, the current baryon, dark matter and dark energy density parameters. In the above equation, the dimensionless function $f(a)$ takes the following form:
\begin{equation}\label{fdea}
f(a) = \left[1+ \frac{\Omega_{dm,0}}{\Omega_{x,0}}\int_{a}^{1}\frac{[\epsilon(\tilde{a}) + \tilde{a} \epsilon^{'} \ln \tilde{a}]}{\tilde{a}^{1 -3\omega - \epsilon(\tilde{a})}} d\tilde{a} \right]a^{-3(1+\omega)}.
\end{equation}
\subsection{$\epsilon(a)$ parameterization}
In order to proceed further and study some cosmological consequences of the class of coupled quintessence scenarios discussed above, we must assume an appropriated relation for $\epsilon(a)$. In our analysis, we consider two different parameterizations for the interacting parameter:
\begin{equation}\label{para1}
\epsilon(a) = \epsilon_0a^\xi = \epsilon_0(1+z)^{-\xi}, \quad \quad \quad \quad \quad \rm{(P1)}\\\;
\end{equation}
and
\begin{equation}
\label{para2}
\epsilon (a) = \epsilon_{0} \exp{(1 -a^{-1})} = \epsilon_0\exp{(-z)}, \quad \rm{(P2)}\\\;
\end{equation}
where $\epsilon_0$ and $\xi$ may, in principle, take negative and positive values. P1 is certainly a very simple choice among some physically possible functional forms. Note, however, that for negative values of $\xi$, P1 blows up in the past, when $a \rightarrow 0$. Differently, P2 is a one-parameter, well-behaved function during the entire evolution of the Universe. Note also that P2 implies a weaker dark matter/dark energy interaction in the past, as $z$ increases.
\section{Dynamical behavior}
The time evolution of the density parameters $\Omega_b(a)$, $\Omega_{dm}(a)$ and $\Omega_{x}(a)$ can be derived by combining Eqs. (\ref{dm})-(\ref{friedmann}). They read:
\begin{subequations}
\begin{equation} \label{8a}
\Omega_{b}(a) = \frac{a^{-3}}{a^{-3} + {\rm{A}}a^{-3 +\epsilon(a)} + Bf(a)}\;,
\end{equation}
\begin{equation} \label{8b}
\Omega_{dm}(a) = \frac{a^{-3 +\epsilon(a)}}{A^{-1}a^{-3} + a^{-3 +\epsilon(a)} + Cf(a)}\;,
\end{equation}
\begin{equation} \label{8c}
\Omega_{x}(a) = \frac{f(a)} {B^{-1}a^{-3} + C^{-1}a^{-3 + \epsilon(a)} + f(a)}\;,
\end{equation}
\end{subequations}
where $A = \Omega_{dm,0}/{\Omega_{b,0}}$, ${\rm{B}} = {\Omega_{x,0}}/{\Omega_{b,0}}$ and ${\rm{C}} = {\Omega_{x,0}}/{\Omega_{dm,0}}$.
Figure 1 shows the evolution of the density parameters as function of $\log(a)$ for P1 [Eq. (\ref{para1})]. For simplicity, we consider two characteristic values of the equation-of-state parameter, $w = -0.9$ and $w = -1.1$, corresponding to quintessence and phantom behaviors, respectively. In agreement with current WMAP results~\cite{cmbnew}, we assume $\Omega_{b,0} = 0.0416$ and $\Omega_{dm,0} = 0.24$. Note that, although currently accelerated (and, therefore, possibly in agreement with SNe Ia data), models with $\epsilon_{0} > 0$ and negative values of $\xi$ (Figs. 1a and 1d) fail to reproduce the past dark matter-dominated epoch, whose existence is fundamental for the structure formation process to take place. In both cases, the dark energy and dark matter densities vanish at high-$z$ and the Universe is fully dominated by the baryons (for a CMB analysis in a baryon-dominated universe, see~\cite{silk}).
\begin{figure*}
\centerline{\psfig{figure=qz09.ps,width=2.1truein,height=1.9truein,angle=-90}
\psfig{figure=qz10.ps,width=2.1truein,height=1.9truein,angle=-90}
\psfig{figure=qz11.ps,width=2.1truein,height=1.9truein,angle=-90}}
\label{fig:qzw}
\end{figure*}
\begin{figure*}
\centerline{\psfig{figure=qze09.ps,width=2.1truein,height=1.9truein,angle=-90}
\psfig{figure=qze10.ps,width=2.1truein,height=1.9truein,angle=-90}
\psfig{figure=qze11.ps,width=2.1truein,height=1.9truein,angle=-90}}
\caption{Deceleration parameter as a function of $\log(a)$ for some selected values of $\epsilon_{0}$ and $\xi$ for P1 and P2, respectively. Note that for $\epsilon_{0} > 0$ and large positive values of $\xi$ (P1) and also when $\epsilon_{0} > 1.2$ (P2) the Universe will experience a new matter-dominated era in the future, when $a \rightarrow \infty$.}
\label{fig:qzw}
\end{figure*}
Regardless of the sign of $\epsilon_0$ and the values of $w$, well-behaved scenarios are obtained when $\xi$ takes positive values (Figs. 1b and 1e). In these cases a mix of baryons ($\lesssim 20\%$) and dark matter ($\gtrsim 80\%$) dominates the past evolution of the Universe whereas the dark energy is always the dominant component from a value of $a \lesssim 1$ on. An interesting and completely different future cosmic evolution is obtained when $\epsilon_0 > 0$ and the parameter $\xi$ takes large positive values ($\gtrsim 0.8$). This is shown in (Figs. 1c and 1f) for $\xi = 1.0$ and $\epsilon_0 = 0.1$. Note that, besides having a well-behaved past evolution and being currently accelerating, the cosmic acceleration will eventually stop at some value of $a>>1$ (when the dark energy becomes sub-dominant) and the Universe will experience a new matter-dominated era in the future, when $a \rightarrow \infty$. This kind of dynamic behavior is not found in most coupled quintessence models discussed in the literature, being essentially a feature of the so-called thawing~\cite{thaw} and hybrid~\cite{cqg} potentials, which in turn seems to be in good agreement with some requirements of String or M theories, as discussed in Ref.~\cite{fischler} (see also \cite{ed})~\footnote{The argument presented in Ref.~\cite{fischler} is that an eternally accelerating universe, a rather generic feature of many quintessence scenarios (including the standard $\Lambda$CDM model), seems not to be in agreement with String/M-theory predictions, since it is endowed with a cosmological event horizon which prevents the construction of a conventional S-matrix describing particle interactions.}.
In Fig. 2 it is shown the same as Fig. 1 for P2 [Eq. (\ref{para2})]. Note that, independently of the values of $\omega$, all values of $\epsilon_{0} > 0$ give rise to well-behaved scenarios in which the Universe had a last evolution dominated by a mix of baryons and dark matter and it is currently accelerating (dominated by dark energy). Note also that for large positive values $\epsilon_{0}$, e.g., $\epsilon_{0} \gtrsim 1.2$ (Panels 2b and 2d) the Universe will evolve to an eternal deceleration phase instead of the usual de Sitter phase.
The deceleration parameter, defined as $q=-a\ddot{a}/\dot{a}^2$, is given by
\begin{equation}\label{acela}
q= \frac{3}{2}{\frac{\Omega_{b,0}a^{-3} + \Omega_{dm,0}a^{-3+\epsilon(a)} + (1+\omega) \Omega_{x,0}f(a)} {\Omega_{b,0}a^{-3} + \Omega_{dm,0}a^{-3+\epsilon(a)} + \Omega_{x,0}f(a)}} -1,
\end{equation}
and shown in Fig. 3 as a function of $\log(a)$ for P1 and P2, respectively. In agreement with our previous discussion, we clearly see a transient acceleration phenomenon for some selected values of $\epsilon_0$ and $w$.
\begin{figure*}
\centerline{\psfig{figure=planoew.eps,width=3.2truein,height=2.0truein,angle=-90}
\psfig{figure=planoexi.eps,width=3.2truein,height=2.0truein,angle=-90}}
\caption{The results of our statistical analyses. Contours of $\chi^2$ in the plane $\omega - \epsilon_0$ (left panel) and $\epsilon_0 - \xi$, with $\omega = -1$, (right panel) for P1. These contours are drawn for $\Delta \chi^2 = 2.30$ and $6.17$. The best fit values are $\epsilon_{0}= -0.11$ and $\omega = -1.04$ (left panel), while in the case $\Lambda(t)$CDM (right panel) we have found $\epsilon_{0}= 0.49$ and $\xi = 0.41$.}
\label{fig:qzw}
\end{figure*}
\begin{figure*}
\centerline{\psfig{figure=planoeew.eps,width=3.2truein,height=2.0truein,angle=-90}
\psfig{figure=planoem.eps,width=3.2truein,height=2.0truein,angle=-90}}
\caption{Contours of $\chi^2$ in the plane $\omega - \epsilon_{0}$ (left panel) and $\Omega_{dm,0} - \epsilon_{0}$, with $\omega = -1$, (right panel) for P2. The best fit values are $\epsilon_{0} = -1.19$ and $\omega = -1.16$ (left panel), while in the case $\Lambda(t)$CDM (right panel) we have found $\Omega_{dm,0} = 0.245$ and $\epsilon_{0}= 0.035$.}
\label{fig:qzw}
\end{figure*}
\section{Observational analysis}
As we have seen, the model here discussed comprises a multitude of cosmological solutions for different combinations of its parameters. In this section, we will discuss more quantitatively the observational aspects of this class interacting scenarios. To this end we perform a joint analysis involving current SNe Ia, CMB/BAO data. In our analysis, we fix $\Omega_{b,0} = 0.0416$ from WMAP results~\cite{cmbnew} (which is also in good agreement with the bounds on the baryonic component derived from primordial nucleosynthesis~\cite{nucleo}) and consider the recent determination of the Hubble parameter $H_0 = 74.2 \pm 4.8$~\cite{hubble} in conjunction with the CMB constraint $\Omega_{dm,0}h^2 = 0.109 \pm 0.006$~\cite{cmbnew}. We use a recent SNe Ia compilation, the so-called Union sample compiled in Ref.~\cite{union} which includes recent large samples from SNLS~\cite{snls} and ESSENCE~\cite{essence} surveys, older data sets and the recently extended data set of distant supernovae observed with the Hubble Space Telescope. The total compilation amounts to 414 SNe Ia events, which was reduced to 307 data points after selection cuts.
Following Ref.~\cite{sollerman} we use constraints derived from the product of the CMB acoustic scale
\begin{equation}
\ell_{A} = \pi d_A (z_*)/r_s(z_*)\;,
\end{equation}
and the measurement of the ratio of the sound horizon scale at the drag epoch to the BAO dilation scale,
\begin{equation}
r_s(z_d )/D_V(z_{\rm{BAO}})\;.
\end{equation}
In the previous expressions, $d_A (z_*)$ is the comoving angular-diameter distance to recombination $z_* = 1089$ and $r_s(z_*)$ is the comoving sound horizon at photon decoupling given by $r_s(z_*) = \int_{z_*}^{\infty} \frac{c_s}{H(z)} dz$, which depends upon the speed of sound before recombination $(c_s)$. $z_d \simeq 1020$ is the redshift of the drag epoch (at which the acoustic oscillations are frozen in) and the so-called dilation scale, $D_V$, is given by $D_V(z) = [czr^{2}(z)/H(z)]^{1/3}$.
By combining the ratio $r_s (z_d = 1020)/r_s (z_*=1090) = 1.044 \pm 0.019$ ~\cite{Komatsu} with the measurements of $r_s(z_d )/D_V(z_{\rm{BAO}})$ at $z_{\rm{BAO}} = 0.20$ and 0.35 from Ref.~\cite{Percival}, Sollerman {\it et al.} (2009) found
$$
f_{0.20} = d_A (z_*)/D_V (0.2) = 17.55 \pm0.65
$$
$$
f_{0.35} = d_A (z_*)/D_V (0.35) = 10.10 \pm 0.38\;.
$$
In our analysis, we minimize the function $\chi^2_{\rm{T}} = \chi^2_{\rm{SNe}} + \chi^2_{\rm{CMB/BAO}}$, where $\chi_{\rm{CMB/BAO}}^{2} = \left[f_{0.2}(z|\mathbf{s}) -
f_{0.2}\right]^2/\sigma_{0.2}^2 + \left[f_{0.35}(z|\mathbf{s}) - f_{0.35}\right]^2/\sigma_{0.35}^2$ and $\mathbf{s}$ stands for the model parameters. This total $\chi^2_{\rm{T}}$ function, therefore, takes into account all the observational data discussed above.
The results of our statistical analyses are shown in Figs. 4 and 5. We show $1$ and $2\sigma$ confidence regions in the parametric spaces: $\omega - \epsilon_{0}$ and $\epsilon_{0} - \xi$ for P1 and $\omega - \epsilon_{0}$ and $\Omega_{dm,0} - \epsilon_{0}$ for P2 that arise from the joint analysis described above. Note that in all panels both negative and positive values for the interacting parameter are allowed by these analyses. Physically, this amounts to saying that not only an energy flow from dark energy to dark matter ($\epsilon_{0} > 0$) is observationally allowed but also a flow from dark matter to dark energy ($\epsilon_{0} < 0$) [see Eq. (\ref{dm})]. In right panel of Fig. 4 we show the analysis for $\epsilon_{0}$ and $\xi$ by fixing the dark energy EoS at $\omega = -1$, which is fully equivalent to the vacuum decay scenario proposed in Ref.~\cite{ernandes2}. As expected, we note that the current observational bounds on $\xi$ are quite weak since it appears as a power of the scale factor in the energy density [Eqs. (\ref{dm}) and (\ref{de})]. When $\xi$ takes more negative values $\epsilon_0 \rightarrow 0$, i.e., this scenario behaves very similarly to the standard $\Lambda$CDM model. Note also that the $\Omega_{dm,0}$ parameter (Fig. 5 - right panel) is very well bounded by observations. In the case of P1, the best-fit found are $\epsilon_{0}= -0.11$ and $\omega = -1.04$ (left panel), whereas for $\Lambda(t)$CDM model (right panel) we have found $\epsilon_{0}= 0.49$ and $\xi = 0.41$. For P2 we have found the following best-fit values $\epsilon_{0} = -1.19$ and $\omega = -1.16$ (left panel), and $\Omega_{dm,0} = 0.245$ and $\epsilon_{0}= 0.035$ (right panel - $\Lambda(t)$CDM scenario).
\section{Final Remarks}
In this paper, we have investigated a general class of models with interaction in the dark sector whose evolution law of the dark energy is deduced from the effect of the same on the CDM expansion rate. Contrary to most similar analyses available in literature, we consider a more general case in which (i) the interaction term $\epsilon$ is a function of the scale factor and (ii) the EoS parameter may take any value ($w < 0$). We have shown that many previous phenomenological approaches discussed in the literature are particular cases of our approach.
We have investigated the dynamical behaviour of this scenario and found a number viable of cosmological solutions for two parametrizations of $\epsilon(a)$ (Figs. 1, 2 and 3). In the first case (P1), when $\epsilon_{0} > 0$ and the $\xi$ parameter takes large positive values ($\gtrsim 0.8$) we have found solutions with transient acceleration, in which the dark matter-dark energy interaction will drive the Universe to a new matter-dominated era in the future. P2, although depends only on the $\epsilon_0$ parameter, also supplies solutions with transient acceleration, when $\epsilon_{0} > 1.2$. As mentioned earlier, this kind of solution seems to be in agreement with theoretical constraints from String/M theories on the quintessence potential $V(\phi)$ or, equivalently, on the dark energy equation-of-state $w$, as discussed in Ref.~\cite{fischler}.
By combining recent data of SNe Ia (Union sample) with the so-called CMB/BAO ratio at two redshifts, $z = 0.2$ and $z = 0.35$, we have shown that that both an energy flow from dark energy to dark matter as well as a flow from dark matter to dark energy are possible. For the two different parameterizations discussed here, we have also investigated $\Lambda(t)$CDM scenarios for which $\omega = -1$. We have also found that positive values of $\xi$ are largely favored over negative ones.
\begin{acknowledgments}
This work was supported by CAPES (Brazilian Research Agency). The author thanks J. S. Alcaniz for valuable discussions.
\end{acknowledgments}
|
{
"timestamp": "2010-09-21T02:03:35",
"yymm": "1009",
"arxiv_id": "1009.3841",
"language": "en",
"url": "https://arxiv.org/abs/1009.3841"
}
|
\wt{\partial}{\wt{\partial}}
\def\part_a{\wt{\partial}_a}
\def\part_b{\wt{\partial}_b}
\def\part_c{\wt{\partial}_c}
\def\part_v{\wt{\partial}_v}
\def\part_w{\wt{\partial}_w}
\def{_\alpha}{{_\alpha}}
\def{_\beta}{{_\beta}}
\def_{\alpha\beta}{_{\alpha\beta}}
\def_{\beta\alpha}{_{\beta\alpha}}
\def\partial_{k_x}{\partial_{k_x}}
\def\partial_{k_y}{\partial_{k_y}}
\def\partial_a{\partial_a}
\def\partial_b{\partial_b}
\def\partial_c{\partial_c}
\def\partial_{\rm k}{\partial_{\rm k}}
\def\wt{\partial}_{k_x}{\wt{\partial}_{k_x}}
\def\wt{\partial}_{k_y}{\wt{\partial}_{k_y}}
\def{\wt{M}}{{\wt{M}}}
\newcommand{\equ}[1]{Eq.~(\ref{eq:#1})}
\newcommand{\eqs}[2]{Eqs.~(\ref{eq:#1}) and (\ref{eq:#2})}
\begin{document}
\title{Band theory of spatial dispersion in magnetoelectrics}
\author{Andrei Malashevich}
\email{andreim@berkeley.edu}
\author{Ivo Souza}
\affiliation{
Department of Physics, University of California, Berkeley, California 94720, USA
}
\date{\today}
\begin{abstract}
Working in the crystal-momentum representation, we calculate the
optical conductivity of noncentrosymmetric insulating crystals
at first order in the wave vector of light. The time-even part
of this tensor describes natural optical activity and the
time-odd part describes nonreciprocal effects such as
gyrotropic birefringence. The time-odd part can be uniquely
decomposed into magnetoelectriclike and purely quadrupolar
contributions. The magnetoelectriclike component reduces in the
static limit to the traceless part of the frozen-ion static
magnetoelectric polarizability while at finite frequencies it
acquires some quadrupolar character in order to remain
translationally invariant.
The expression for the orbital contribution to the conductivity
at transparent frequencies
is validated by comparing numerical tight-binding calculations for
finite and periodic samples.
\end{abstract}
\pacs{78.20.Ek,75.85.+t,78.20.Bh}
\maketitle
\marginparwidth 2.7in
\marginparsep 0.5in
\def\amm#1{\marginpar{\small AM: #1}}
\def\ism#1{\marginpar{\small IS: #1}}
\def\scriptsize{\scriptsize}
\section{Introduction}
Electric and magnetic effects are closely coupled in magnetoelectric
(ME) materials. These are insulators with broken spatial-inversion
($\mathcal{P}$) and time-reversal ($\mathcal{T}$) symmetries, in which
an applied electric field $\bm{\mathcal{E}}$ induces a first-order magnetization
${\bf M}$, and conversely a magnetic field ${\bf B}$ induces a first-order
electric polarization ${\bf P}$. This cross response is described in the
static limit by a single magnetoelectric polarizability tensor
\begin{equation}
\label{eq:ME}
\alpha_{ab}\equiv\frac{\partial M_b}{\partial \mathcal{E}_a}=
\frac{\partial P_a}{\partial B_b},
\end{equation}
where the equality follows from changing the order of the mixed derivatives
of the free energy.
The ME effect has been intensively studied in recent years. While the
focus has been mostly on the static response, ME effects in the
optical range have also been observed.\cite{arima08} For oscillating
fields the thermodynamic argument leading to the second equality in
\equ{ME} does not hold because the system is not in equilibrium, and
two separate frequency-dependent polarizabilities are needed to
describe the dynamical ME coupling
\begin{equation}
\label{eq:dynME}
\chi^{\mathrm{me}}_{ab}=\frac{\partial M_a}{\partial \mathcal{E}_b},\mbox{~~~}
\chi^{\mathrm{em}}_{ab}=\frac{\partial P_a}{\partial B_b}.
\end{equation}
It was recognized already in the 1960s that the coupling,
Eq.~(\ref{eq:dynME}), leads to new optical effects in ME media, such as
gyrotropic birefringence.\cite{brown63} Since the lattice-mediated
response is frozen out at optical frequencies, the purely electronic
contribution can be isolated. The first successful measurements, on
Cr$_2$O$_3$, found that the strength of the optical ME coupling is
comparable to that of the static one.\cite{krichevtsov93}
The phenomenology of optical ME effects has been studied in
detail in the literature, starting with the work of Hornreich and
Shtrikman on gyrotropic birefringence.\cite{hornreich68} These
authors showed that this effect is a consequence of spatial
dispersion, appearing at first order in the expansion of the
effective optical conductivity tensor (defined by
Eq.~(\ref{eq:sigma_ab}) below) in powers of the wave vector ${\bf q}$ of
light
\begin{equation}
\label{eq:sigma-taylor}
\sigma_{ab}({\bf q},\omega)=\sigma^{(0)}_{ab}(\omega)
+\sigma_{abc}(\omega)q_c
+\cdots
\end{equation}
It is well known that the phenomenon of natural optical activity is
also a manifestation of spatial dispersion.\cite{landau} While natural
optical activity is associated with the $\mathcal{T}$-even part of
$\sigma_{abc}(\omega)$, optical ME effects arise from the
$\mathcal{T}$-odd part, which can be nonzero only in magnetically
ordered systems, where $\mathcal{T}$ symmetry is spontaneously broken.
A careful consideration of all response tensors which contribute to
the conductivity at linear order in ${\bf q}$ shows that these include, in
addition to the dynamic ME polarizabilities, Eq.~(\ref{eq:dynME}), the
electric-quadrupole response of the medium.
Regarding the microscopic theories needed for quantitative
calculations, there are well-established {\it molecular} theories of
spatial dispersion,\cite{barron2004,raab2005} but the corresponding
theory for crystals is not equally developed. A band theory of
natural optical activity was put forth by Natori\cite{natori75} but
has not been used in first-principles calculations. To our knowledge,
only one group has reported calculations of natural optical activity
in solids at optical wavelengths, based on a
somewhat different formulation.\cite{zhong92,zhong93} As for the optical
ME effects, quantitative estimates of their magnitude have so
far relied on cluster models to mimic the crystalline
environment.\cite{muthkumar-prb96,igarashi-prb09}
In this work, we develop a formalism for calculating
spatial-dispersion effects in the framework of band theory. One
difference with respect to previous works is that
we give a unified treatment of both ${\cal T}$-even and ${\cal
T}$-odd parts of this tensor. More importantly, we express the
transition matrix elements in the crystal momentum
representation.\cite{blount62} This choice has both practical and
formal advantages. The practical advantage is that it leads to
expressions which can be easily implemented
using localized Wannier orbitals.
On the theoretical side, the
crystal-momentum representation is the language in which the modern
theories of electric polarization,\cite{King-Smith,resta-review07}
orbital magnetization,\cite{timo05,xiao05,ceresoli06,shi07} and
orbital magnetoelectric response \cite{malash2010,essin2010} are
formulated.
As we shall see, our expression for the orbital contribution to
the $\mathcal{T}$-odd part of $\sigma_{abc}(\omega)$ generalizes to finite
frequencies the traceless part of the orbital ME polarizability
formula of Refs.~\onlinecite{malash2010,essin2010}.
The manuscript is organized as follows. In Sec.~\ref{sec:phenom}
we give a self-contained account of the phenomenology of
spatial-dispersion optics. The effective conductivity is defined and
related to the magnetoelectric and quadrupolar polarizabilities. We
then reformulate the phenomenological relations, originally obtained
for finite systems, in terms of translationally invariant quantities
which remain well defined in the thermodynamic limit. The main
results of the paper are contained in Sec.~\ref{sec:bulk}, where we
obtain a microscopic expression for the $\sigma_{abc}(\omega)$ in
periodic insulators. We then consider the
$\omega\rightarrow 0$ limit of that expression and discuss its
relation to the theory of static ME response. In
Sec.~\ref{sec:results} we implement the bulk $\sigma_{abc}(\omega)$
expression for a tight-binding model
and compare the results with calculations on finite samples
cut from the bulk crystal. We conclude in Sec.~\ref{sec:sum} with
a brief summary and outlook.
\section{Phenomenology of spatial dispersion}
\label{sec:phenom}
In this section we discuss spatial dispersion from a phenomenological
perspective. Besides introducing basic definitions and setting the
notation, the main purpose here is to arrive at
Eqs.~(\ref{eq:alpha_gamma_def})--(\ref{eq:beta-tilde}) relating the
spatially dispersive optical conductivity to
{\it translationally invariant} renormalized multipole
polarizabilities. Those relations will allow us to identify the
magnetoelectriclike and purely quadrupolar parts of the optical
response of crystals, to be calculated in Sec.~\ref{sec:bulk}.
\subsection{Effective conductivity tensor}
Consider a crystal with broken $\mathcal{P}$ and possibly broken
$\mathcal{T}$ symmetries. We are mainly interested in materials where
those symmetries are broken spontaneously, rather than by static
electric and magnetic fields, and wish to study their current response
${\bf J}({\bf q},\omega)$ to an electromagnetic plane wave
\begin{equation}
\label{eq:waveE}
\bm{\mathcal{E}}({\bf r},t)=\bm{\mathcal{E}}({\bf q},\omega)e^{i({\bf q}\cdot{\bf r}-\omega t)},
\end{equation}
\begin{equation}
\label{eq:waveB}
{\bf B}({\bf r},t)
=\frac{c}{\omega}\big[{\bf q}\times\bm{\mathcal{E}}({\bf q},\omega)\big]e^{i({\bf q}\cdot{\bf r}-\omega t)}.
\end{equation}
Because the oscillating electric and magnetic fields $\bm{\mathcal{E}}$ and ${\bm{\mathcal{\epsilon}}}$ and ${\bf B}$
are interdependent,
the linear (in the field strengths) response can be described by a
single {\it effective conductivity} tensor\cite{hornreich68,melrose}
\begin{equation}
\label{eq:sigma_ab}
J_a({\bf q},\omega)=\sigma_{ab}({\bf q},\omega)\mathcal{E}_b({\bf q},\omega).
\end{equation}
Alternatively, one may choose to work with the dielectric function
$\epsilon_{ab}({\bf q},\omega)$.\cite{landau,melrose} To first order
in ${\bf q}$ the two are related (in Gaussian cgs units) by
\begin{equation}
\label{eq:epsilon}
\epsilon_{ab}({\bf q},\omega)=\delta_{ab}+\frac{4\pi i}{\omega}\sigma_{ab}({\bf q},\omega).
\end{equation}
The leading term in the expansion of
$\sigma_{ab}({\bf q},\omega)$ in powers of ${\bf q}$, Eq.~(\ref{eq:sigma-taylor}),
is the optical conductivity in
the electric-dipole approximation. We shall focus on the next term in
the expansion, $\sigma_{abc}$, which is chiefly responsible for
spatial dispersion. Because spatial inversion takes ${\bf q}$ into $-{\bf q}$, the tensor
$\sigma_{abc}(\omega)$ necessarily vanishes in centrosymmetric systems.
Its symmetric ($\sigma^{\mathrm{S}}_{abc}$) and antisymmetric
($\sigma^{\mathrm{A}}_{abc}$) parts under the interchange of the first two indices
are, respectively, odd and even under
$\mathcal{T}$.\cite{explan-onsager} The $\mathcal{T}$-even piece
describes natural optical activity, and the $\mathcal{T}$-odd
piece describes non-reciprocal optical effects.
These include, in addition to gyrotropic
birefringence, directional dichroism\cite{arima08} and magnetochiral
effects in chiral ferromagnets.\cite{train08}
Unlike the spontaneous magneto-optical effects coming from the
$\mathcal{T}$-odd part of $\sigma_{ab}^{(0)}$ (magnetic circular
dichroism and birefringence), which require ferromagnetic or ferrimagnetic
order, gyrotropic birefringence can also occur in antiferromagnets
such as Cr$_2$O$_3$.
This is a well-known magnetoelectric material, and indeed the
physical basis for spatial dispersion rests in part on the
magnetoelectric effect.
\subsection{Multipole theory for finite systems}
The connection between spatial dispersion and the magnetoelectric
effect can be readily established by expressing ${\bf J}({\bf q},\omega)$ in terms of
the multipole moments of the charge and current distributions. We
begin by taking the spatial Fourier transform of the current density,
\begin{equation}
{\bf J}({\bf q},t)=\frac1V\int d{\bf r} e^{-i{\bf q}\cdot{\bf r}} {\bf J}({\bf r},t)
\end{equation}
and expanding in powers of ${\bf q}$,
\begin{equation}
{\bf J}({\bf q},t)={\bf J}^{(0)}(t)+{\bf J}^{(1)}({\bf q},t)+{\cal O}(q^2).
\end{equation}
Standard multipole-expansion manipulations\cite{melrose} involving the
continuity equation and integrations by parts show that
$J_a^{(0)}(t)=\partial_tP_a(t)$ and
\begin{equation}
\label{eq:J1}
J_a^{(1)}({\bf q},t)=-\frac{iq_b}{2}\partial_tQ_{ab}(t)+i\epsilon_{abc}cq_bM_c(t),
\end{equation}
where $\epsilon_{abc}$ is the antisymmetric tensor of rank three and
${\bf P}$, ${\bf Q}$, and ${\bf M}$ are the electric dipole,
electric quadrupole, and magnetic dipole moments of the sample divided
by its volume
\begin{equation}
\label{eq:P}
P_a(t)=\frac1V\int d{\bf r}\, r_a\rho(t,{\bf r}),
\end{equation}
\begin{equation}
\label{eq:Q}
Q_{ab}(t)=\frac1V\int d{\bf r}\, r_ar_b\rho(t,{\bf r}),
\end{equation}
\begin{equation}
\label{eq:M}
M_a(t)=\frac{1}{2cV}\epsilon_{abc}\int d{\bf r}\, r_bJ_c(t,{\bf r}).
\end{equation}
Fourier transforming in time we arrive at
\begin{equation}
\label{eq:J_PQM}
J_a({\bf q},\omega)=-i\omega P_a(\omega)-\frac{\omega}{2}q_bQ_{ab}(\omega)
+i\epsilon_{abc}cq_bM_c(\omega)+{\cal O}(q^2).
\end{equation}
The current induced by the monochromatic wave,
Eqs.~(\ref{eq:waveE}) and (\ref{eq:waveB}), can now be calculated from the
oscillating induced moments, which are the real parts of the following
expressions:\cite{barron2004,raab2005}
\begin{equation}
\label{eq:inducedP}
P_a=\chi^{\rm e}_{ab}\mathcal{E}_b+\frac12\chi^{\rm q}_{abc}\nabla_c\mathcal{E}_b
+\cdots+\chi^{\rm em}_{ab}B_b+\cdots,
\end{equation}
\begin{equation}
\label{eq:inducedQ}
Q_{ab}=\wt{{\chi}^{\rm q}}_{abc}\mathcal{E}_c+\cdots,
\end{equation}
\begin{equation}
\label{eq:inducedM}
M_a=\chi^{\rm me}_{ab}\mathcal{E}_b+\cdots,
\end{equation}
where the fields and their gradients are evaluated at the location of
the sample. $\chi^{\rm e}$ is the electric polarizability per unit volume,
and quantum-mechanical expressions for the remaining response
tensors are listed in Appendix~\ref{app:multipol}. $\chi^{\rm em}$ and
$\chi^{\rm me}$ are the dynamic ME polarizabilities introduced in
Eq.~(\ref{eq:dynME}); they involve matrix elements of the
electric-dipole ($E1$) and magnetic-dipole ($M1$) operators, and for
this reason are known as the $E1.M1$ terms. $\chi^{\rm q}$ and $\wt{{\chi}^{\rm q}}$ are
the $E1.E2$ terms, as they mix electric-dipole and electric-quadrupole
transitions.
In Eqs.~(\ref{eq:inducedP})--(\ref{eq:inducedM}) only those terms
which contribute to the effective conductivity up to first order in ${\bf q}$
were kept. Combining Eqs.~(\ref{eq:J_PQM})--(\ref{eq:inducedM}) with
Eqs.~(\ref{eq:waveE}) and (\ref{eq:waveB}) and comparing with
Eqs.~(\ref{eq:sigma_ab}) and (\ref{eq:sigma-taylor}) we find, upon
collecting terms linear in ${\bf q}$,
\begin{equation}
\label{eq:sigma-abc}
\sigma_{abc}=
ic(\chi^{\rm em}_{ad}\epsilon_{dbc}+\epsilon_{acd}\chi^{\rm me}_{db})
+\frac{\omega}{2}(\chi^{\rm q}_{abc}-\wt{{\chi}^{\rm q}}_{acb})
\end{equation}
Spatial dispersion is thus governed by the magnetoelectric
and quadrupolar
responses of the medium.\cite{hornreich68}
The need to include the quadrupolar terms in order to properly describe
the optical activity of oriented molecules and uniaxial crystals
was emphasized in Ref.~\onlinecite{buckingham71}.
Dividing Eq.~(\ref{eq:sigma-abc}) into symmetric (magnetic) and
antisymmetric (natural) parts under $a\leftrightarrow b$ yields
\begin{equation}
\label{eq:sigma-S}
\sigma^{\mathrm{S}}_{abc}=ic
\left(
\epsilon_{bcd}\alpha_{ad}
+\epsilon_{acd}\alpha_{bd}
\right)
+
\omega\gamma_{abc},
\end{equation}
\begin{equation}
\label{eq:sigma-A}
\sigma^{\mathrm{A}}_{abc}=ic
\left(
\epsilon_{bcd}\beta_{ad}
-\epsilon_{acd}\beta_{bd}
\right)+
\omega\xi_{abc},
\end{equation}
where we have defined
\begin{equation}
\label{eq:alpha}
\alpha_{ab}=\frac{\chi^{\rm em}_{ab}+\chi^{\rm me}_{ba}}{2}\doteq\mathrm{Re}\,\chi^{\rm em}_{ab},
\end{equation}
which reduces to Eq.~(\ref{eq:ME}) in the static limit, and
\begin{equation}
\label{eq:beta}
\beta_{ab}=\frac{\chi^{\rm em}_{ab}-\chi^{\rm me}_{ba}}{2}\doteq i\mathrm{Im}\,\chi^{\rm em}_{ab},
\end{equation}
\begin{equation}
\label{eq:gamma}
\gamma_{abc}
=\frac{\chi^{\rm q}_{abc}+\chi^{\rm q}_{bac}-\wt{{\chi}^{\rm q}}_{acb}-\wt{{\chi}^{\rm q}}_{bca}}{4}
\doteq \frac{i}{2}\mathrm{Im}\left[\chi^{\rm q}_{abc}+\chi^{\rm q}_{bac}\right],
\end{equation}
\begin{equation}
\label{eq:xi}
\xi_{abc}
=\frac{\chi^{\rm q}_{abc}-\chi^{\rm q}_{bac}-\wt{{\chi}^{\rm q}}_{acb}+\wt{{\chi}^{\rm q}}_{bca}}{4}
\doteq \frac12\mathrm{Re}\left[\chi^{\rm q}_{abc}-\chi^{\rm q}_{bac}\right].
\end{equation}
In each of these equations the second equality, denoted by the symbol
$\dot=$, only holds at nonabsorbing frequencies, for which
$\chi^{\rm em}_{ab}\dot=(\chi^{\rm me}_{ba})^*$ and
$\wt{{\chi}^{\rm q}}_{abc}\dot=(\chi^{\rm q}_{cab})^*$ (see
Appendix~\ref{app:multipol}). In this lossless regime $\sigma_{abc}$
becomes anti-Hermitian in the first two indices.
The above multipole formulation leads to a practical scheme for
calculating spatial dispersion effects, by computing the
polarizabilities $\chi^{\rm em}$, $\chi^{\rm me}$, $\chi^{\rm q}$, and $\wt{{\chi}^{\rm q}}$ from
Eqs.~(\ref{eq:chiem-qm-re})--(\ref{eq:LAM-qm-im}), and assembling them
in Eq.~(\ref{eq:sigma-abc}). This approach can be used for molecules
and other finite systems but not for bulk crystals, because the
quantum-mechanical expressions in Appendix~\ref{app:multipol} become
ill-defined under periodic boundary conditions.
The problem can be traced back to the integrations by parts carried
out around \equ{J1}, where the boundary terms were discarded. Such
procedure is allowed for finite systems, as the boundary can always be
placed outside the sample. It cannot, however, be rigorously justified
for periodic crystals with delocalized electrons. This is a subtle
but by now well-understood problem. For example, the macroscopic
electric polarization and orbital magnetization of crystals cannot be
calculated under periodic boundary conditions as the first moments of
the charge and orbital current distributions in one crystalline cell
because the result depends on the choice of cell.\cite{resta-review07}
The correct band-theory expressions for ${\bf P}$ and orbital ${\bf
M}$ have been derived in Ref.~\onlinecite{King-Smith} and
Refs.~\onlinecite{timo05,xiao05,ceresoli06,shi07}, respectively.
\subsection{Translationally invariant polarizabilities}
\label{sec:transl-inv}
Already for finite systems the description based on
Eqs.~(\ref{eq:inducedP})--(\ref{eq:inducedM}) is highly redundant, as
the individual polarizabilities are
origin dependent.\cite{barron2004,raab2005} The combination of
polarizabilities on the right-hand side of Eqs.~(\ref{eq:sigma-S}) and
(\ref{eq:sigma-A}) is of course translationally invariant (the
conductivity is a physical observable)
but we shall go one step further and redefine the polarizability
tensors
so that they become {\it individually} origin independent, and hence
well defined for periodic crystals.
To begin, we note that the trace of $\alpha$ drops out from
Eq.~(\ref{eq:sigma-S}), leaving eight magnetoelectric
quantities. These fully specify $\sigma^{\mathrm{S}}_{abc}$ in the static limit
while at finite frequencies the quadrupolar tensor
$\gamma_{abc}=\gamma_{bac}$ contributes 18 additional quantities.
This brings the total number to 26, while $\sigma^{\mathrm{S}}_{abc}$ itself,
being symmetric in the first two indices, only contains 18 independent
quantities. The source of this discrepancy lies in the
origin-dependence of the tensors $\alpha$ and $\gamma$, and it can be
removed by suitably redefining them. To that end we note that any
third-rank tensor $\sigma^{\mathrm{S}}_{abc}$ symmetric under $a\leftrightarrow b$
can be uniquely expanded as
\begin{equation}
\label{eq:alpha_gamma_def}
\sigma^{\mathrm{S}}_{abc}=ic
\left(
\epsilon_{bcd}\wt{\alpha}_{ad}
+\epsilon_{acd}\wt{\alpha}_{bd}
\right)
+
\omega\wt{\gamma}_{abc},
\end{equation}
where
\begin{equation}
\label{eq:alpha-tilde}
\begin{split}
\wt{\alpha}_{da}&=\frac{1}{3ic} \sigma^{\mathrm{S}}_{dbc}\epsilon_{bca}\\
&=\alpha_{da}-\frac{1}{3}{\rm Tr}[\alpha]\delta_{ad}
+\frac{\omega}{3ic}\gamma_{dbc}\epsilon_{bca}
\end{split}
\end{equation}
(here $\delta_{ad}$ is the Kronecker delta) and
\begin{equation}
\label{eq:Gamma-tilde}
\begin{split}
\wt{\gamma}_{abc}&=\frac{1}{3\omega}
\left(
\sigma^{\mathrm{S}}_{abc}+\sigma^{\mathrm{S}}_{cab}+\sigma^{\mathrm{S}}_{bca}
\right)\\
&=\frac13
\left(
\gamma_{abc}+\gamma_{cab}+\gamma_{bca}
\right).
\end{split}
\end{equation}
Replacing Eq.~(\ref{eq:sigma-S}) with Eq.~(\ref{eq:alpha_gamma_def}) removes
the above-mentioned discrepancy, because the totally symmetric tensor
$\wt{\gamma}_{abc}$ has only ten independent quantities, compared to 18
in $\gamma_{abc}$. As for the tensor $\wt{\alpha}$, it reduces in the
static limit to the traceless part of the magnetoelectric tensor
$\alpha$. But while $\alpha$ becomes origin dependent at finite
frequencies,\cite{raab2005} $\wt{\alpha}$ remains origin independent
by admixing some quadrupolar character.
It seems appropriate to interpret the renormalized property
tensor $\wt{\alpha}$ as the traceless {\it optical magnetoelectric
tensor}, and $\wt{\gamma}$ as the purely quadrupolar part of
$\sigma^{\mathrm{S}}_{abc}$.
We now turn briefly to $\sigma^{\mathrm{A}}_{abc}$.
A third-rank tensor antisymmetric
in two indices has nine independent components, however, there are 18
quantities on the right-hand side of Eq.~(\ref{eq:sigma-A}). We
therefore replace it with
\begin{equation}
\label{eq:sigmaa-tilde}
\sigma^{\mathrm{A}}_{abc}=ic
\left(
\epsilon_{bcd}\wt{\beta}_{ad}
-\epsilon_{acd}\wt{\beta}_{bd}
\right),
\end{equation}
where
\begin{equation}
\label{eq:beta-tilde}
\begin{split}
\wt{\beta}_{ab}&=\frac{1}{4ic}\epsilon_{bcd}
(2\sigma^{\mathrm{A}}_{acd}-\sigma^{\mathrm{A}}_{cda})\\
&=\beta_{ab}+\frac{\omega}{4ic}\epsilon_{bcd}(2\xi_{acd}-\xi_{cda}).
\end{split}
\end{equation}
Hence natural optical activity, just like gyrotropic birefringence,
is governed by an origin-independent combination of magnetoelectric
($\beta$) and quadrupolar ($\xi$) terms.\cite{buckingham71}
Alternatively, $\wt{\beta}$ can be interpreted as a renormalized
magnetoelectriclike tensor, in the same way as $\wt{\alpha}$.
Equations (\ref{eq:alpha_gamma_def}) and (\ref{eq:sigmaa-tilde}) for
$\sigma^{\mathrm{S}}_{abc}$ and $\sigma^{\mathrm{A}}_{abc}$ correspond to Eqs.~(21) and
(30) of Ref.~\onlinecite{hornreich68} while
Eqs.~(\ref{eq:alpha-tilde}), (\ref{eq:Gamma-tilde}), and
(\ref{eq:beta-tilde}) express the translationally invariant
property tensors $\wt{\alpha}$, $\wt{\beta}$, and $\wt{\gamma}$ as
combinations of origin-dependent multipole polarizabilities.
\section{Evaluation of the conductivity}
\label{sec:bulk}
In this section we derive, working in the independent-particle
approximation, a quantum-mechanical expression for $\sigma_{abc}(\omega)$.
The expression, valid for band insulators, is conveniently written as
a sum of two terms, which we shall denote by the superscripts (m) and
(e). They arise, respectively, from the $q$ dependence of the transition
matrix elements and of the transition energies.\cite{natori75}
At nonabsorbing frequencies $\sigma_{abc}(\omega)$ is an anti-hermitian tensor
in the first two indices.
The imaginary (symmetric) part is given, at $T=0$, by the sum of
\begin{equation}
\label{eq:im-sigma-s-delta}
\begin{split}
\mathrm{Im}\,\sigma^{({\rm m})}_{{\rm S},abc}(\omega)=\frac{2e^2}{\hbar}\int[d{\bf k}]
\sum_{n,l}^{o,e}\,\frac{\omega_{ln}}{\omega_{ln}^2-\omega^2} \\
\times\mathrm{Im}\left(A_{ln,b}B_{nl,ac}+A_{ln,a}B_{nl,bc}\right)
\end{split}
\end{equation}
and
\begin{equation}
\label{eq:im-sigma-s-delta-prime}
\begin{split}
\mathrm{Im}\,\sigma^{({\rm e})}_{{\rm S},abc}(\omega)=\frac{2e^2}{\hbar^2}\int[d{\bf k}]
\sum_{n,l}^{o,e}\,\frac{\omega_{ln}^3}{(\omega_{ln}^2-\omega^2)^2} \\
\times\partial_c(E_l+E_n)\mathrm{Re}\left(A_{nl,a}A_{ln,b}\right),
\end{split}
\end{equation}
and the real (antisymmetric) part is the sum of
\begin{equation}
\label{eq:re-sigma-a-m}
\begin{split}
\mathrm{Re}\,\sigma^{({\rm m})}_{A,abc}(\omega)=\frac{2e^2}{\hbar}\int[d{\bf k}]
\sum_{n,l}^{o,e}\,\frac{\omega}{\omega_{ln}^2-\omega^2} \\
\times\mathrm{Re}\left(A_{ln,b}B_{nl,ac}-A_{ln,a}B_{nl,bc}\right)
\end{split}
\end{equation}
and
\begin{equation}
\label{eq:re-sigma-a-e}
\begin{split}
\mathrm{Re}\,\sigma^{({\rm e})}_{A,abc}(\omega)=-\frac{e^2}{\hbar^2}\int[d{\bf k}]
\sum_{n,l}^{o,e}\,\frac{(3\omega_{ln}^2-\omega^2)\omega}{(\omega_{ln}^2-\omega^2)^2} \\
\times\partial_c(E_l+E_n)\mathrm{Im}\left(A_{nl,a}A_{ln,b}\right).
\end{split}
\end{equation}
In these expressions the indices $n$ and $l$ run over occupied ($o$)
and empty ($e$) bands, respectively, $[d{\bf k}]$ stands for
$d^3k/(2\pi)^3$, $\partial_c=\partial/\partial_{k_c}$, and
$\hbar\omega_{ln}=E_l-E_n$. All quantities in the integrands are
labeled by the index ${\bf k}$, which has been omitted for brevity. The
matrix $A_{nl,a}=A_{ln,a}^*$, known as the Berry connection, is
defined as
\begin{equation}
\label{eq:A}
A_{nl,a}=i\bra{u_n}\partial_a u_l\rangle
\end{equation}
and the matrix $B_{nl,ac}=-B_{ln,ac}^*$ has both orbital and spin
contributions,
\begin{equation}
\label{eq:B-ac}
B_{nl,ac}=B_{nl,ac}^{({\rm orb})}+B_{nl,ac}^{({\rm spin})},
\end{equation}
given by
\begin{equation}
\label{eq:B-ac-orb}
B_{nl,ac}^{({\rm orb})}=\frac{1}{2\hbar}
\left[
\bra{u_n}(\partial_aH)\ket{\partial_c u_l}
-\bra{\partial_c u_n}(\partial_aH)\ket{u_l}
\right]
\end{equation}
and
\begin{equation}
\label{eq:B-ac-spin}
B_{nl,ac}^{({\rm spin})}=-\frac{i}{m_e}\epsilon_{abc}
\bra{u_n}S_b\ket{u_l},
\end{equation}
where $u_{n{\bf k}}$ is a cell-periodic Bloch state, $H_{{\bf k}}$ is
related to the crystal Hamiltonian ${\cal H}$ by $e^{-i{\bf k}\cdot{\bf r}}{\cal
H} e^{i{\bf k}\cdot{\bf r}}$, and $m_e$ is the electron mass.
The energy (e) terms have purely orbital character, while the matrix
element (m) terms have both orbital and spin components. It can
be verified that the spin part of \equ{im-sigma-s-delta} does not
contribute to \equ{Gamma-tilde}, consistent with the fact that
$\wt{\gamma}_{abc}$ is a purely orbital (electric-quadrupolar)
quantity.
\subsection{Derivation}
The derivation of the equations given above proceeds as follows. We
first evaluate the absorptive (Hermitian) part of $\sigma_{abc}$, and then
insert its symmetric and antisymmetric parts into the Kramers-Kr\"onig
relations
\begin{equation}
\label{eq:kk1}
\mathrm{Im}\,\sigma_{abc}(\omega_0)=-\frac{1}{\pi}\mathrm{P}\int_{-\infty}^\infty\,
\frac{\mathrm{Re}\,\sigma_{abc}(\omega)}{\omega-\omega_0}\,d\omega
\end{equation}
and
\begin{equation}
\label{eq:kk2}
\mathrm{Re}\,\sigma_{abc}(\omega_0)=\frac{1}{\pi}\mathrm{P}\int_{-\infty}^\infty\,
\frac{\mathrm{Im}\,\sigma_{abc}(\omega)}{\omega-\omega_0}\,d\omega,
\end{equation}
respectively.
The Kubo-Greenwood formula
for the absorptive part of the conductivity
at finite $\omega$ and ${\bf q}$ reads
\begin{equation}
\label{eq:sigma}
\begin{split}
\sigma^{\mathrm{H}}_{ab}&({\bf q},\omega)=\frac{\pi e^2}{\hbar\omega}
\int [d{\bf k}]\sum_{nl}\,(f_{n,{\bf k}-{\bf q}/2}-f_{l,{\bf k}+{\bf q}/2}) \\
&\times\bra{\psi_{n,{\bf k}-{\bf q}/2}}I^\dagger_a({\bf q})\ket{\psi_{l,{\bf k}+{\bf q}/2}}
\bra{\psi_{l,{\bf k}+{\bf q}/2}}I_b({\bf q})\ket{\psi_{n,{\bf k}-{\bf q}/2}} \\
&\times\delta\left[\omega-\omega_{ln{\bf k}}({\bf q})\right],
\end{split}
\end{equation}
where $f_{n{\bf k}\pm {\bf q}/2}$ is the occupation factor of the Bloch state
$\psi_{n{\bf k}\pm {\bf q}/2}$ with eigenenergy $E_{n{\bf k}\pm {\bf q}/2}$,
\begin{equation}
\hbar\omega_{ln{\bf k}}({\bf q})=E_{l,{\bf k}+{\bf q}/2}-E_{n,{\bf k}-{\bf q}/2},
\end{equation}
and ${\bf I}({\bf q})$ is related to the velocity and spin operators by
\begin{equation}
{\bf I}({\bf q})=\frac{e^{i{\bf q}\cdot{\bf r}}{\bf v}+{\bf v} e^{i{\bf q}\cdot{\bf r}}}{2}+
\frac{i}{m_{\mathrm{e}}}({\bf S}\times{\bf q})e^{i{\bf q}\cdot{\bf r}}.
\end{equation}
Equation~(\ref{eq:sigma})
reduces in the limit ${\bf q}\rightarrow 0$
to the familiar expression for the optical conductivity in the
electric-dipole approximation.\cite{harrison80}
It can be derived starting from the interaction
Hamiltonian
\begin{equation}
H_{\rm I}=\frac{e}{2c}(\mathbf{A}\cdot{\bf v}+{\bf v}\cdot\mathbf{A})+
\frac{e}{m_ec}({\boldsymbol\nabla}\times{\bf A})\cdot{\bf S}.
\end{equation}
Up to terms linear in ${\bf q}$, the optical matrix element
$\bra{\psi_{n,{\bf k}-{\bf q}/2}}I_a^{\dagger}({\bf q})\ket{\psi_{l,{\bf k}+{\bf q}/2}}$ may be replaced by
\begin{equation}
\label{eq:B-q}
\begin{split}
B_{nl{\bf k},a}({\bf q})&\equiv\bra{u_{n,{\bf k}-{\bf q}/2}}
v_a({\bf k})
-\frac{i}{m_e}(S\times q)_a
\ket{u_{l,{\bf k}+{\bf q}/2}}\\
&=B_{nl,a}^{(0)}+B_{nl,ac}q_c+\cdots
\end{split}
\end{equation}
where ${\bf v}({\bf k})=e^{-i{\bf k}\cdot{\bf r}}{\bf v} e^{i{\bf k}\cdot{\bf r}}$.
Using the relation\cite{blount62} $\hbar{\bf v}({\bf k}) =\partial_a H_{{\bf k}}$
together with \equ{A},
the expansion coefficients in the second line are found to be
\begin{equation}
\label{eq:B-a}
B_{nl,a}^{(0)}=\bra{u_n}v_a\ket{u_l}
=i\omega_{nl}A_{nl,a}+\frac1{\hbar}\delta_{ln}\partial_aE_l
\end{equation}
and Eqs.~(\ref{eq:B-ac})--(\ref{eq:B-ac-spin}) for $B_{nl,ac}$.
We are now ready to calculate $\sigma^{\mathrm{H}}_{abc}$ by differentiating
\equ{sigma} with respect to $q_c$. Because we assume an insulator at
$T=0$,\cite{explan-opt-act-metals} the derivative acts only on the
transition matrix elements and on the $\delta$ function selecting the
transition energies, not on the occupation factors.
Using \equ{B-q} for the matrix elements [note that the second,
intraband, term in \equ{B-a} does not contribute in insulators],
together with
\begin{equation}
\begin{split}
&\left.
\frac{\partial}{\partial_{q_c}}\delta\left[\omega-\omega_{ln{\bf k}}({\bf q})\right]
\right|_{{\bf q}=0}= \\
&\;\;\;\;\;\;\;\;
-\frac{1}{2\hbar}\delta'\big(\omega-\omega_{ln{\bf k}}(0)\big)
\partial_c\left(E_{l{\bf k}}+E_{n{\bf k}}\right)
\end{split}
\end{equation}
and inserting the result for the symmetric and antisymmetric parts of
$\sigma^{\mathrm{H}}_{abc}$ into \equ{kk1} and \equ{kk2} respectively, one easily
obtains Eqs.~(\ref{eq:im-sigma-s-delta})--(\ref{eq:re-sigma-a-e}).
\subsection{Static limit}
\label{sec:static}
In the limit $\omega\rightarrow 0$ the ME tensors $\chi^{\rm em}_{ab}$ and
$\chi^{\rm me}_{ba}$ become identical, and as a result $\sigma^{\mathrm{A}}_{abc}$
[Eq.~(\ref{eq:sigma-A})] vanishes. As for $\sigma^{\mathrm{S}}_{abc}$, we noted
in Sec.~\ref{sec:transl-inv} that its dc limit is governed by
$\wt{\alpha}(0)$, the traceless part of the static ME polarizability
tensor $\alpha(0)$. Since our calculation of $\sigma^{\mathrm{S}}_{abc}$ only
included the purely electronic response to the optical fields, we
should recover in that limit the frozen-ion part of
$\wt{\alpha}(0)$.
We will focus here on the orbital contribution to $\sigma^{\mathrm{S}}_{abc}$, and
compare it with the band-theory expression obtained in
Refs.~\onlinecite{malash2010,essin2010} for the frozen-ion orbital ME
tensor. The corresponding proof for the spin contribution is
elementary.
We begin by recasting the orbital part of
Eqs.~(\ref{eq:im-sigma-s-delta}) and (\ref{eq:im-sigma-s-delta-prime}) at
$\omega=0$ in a form where empty states do not appear explicitly. This is
done in Appendix~\ref{app:static}, where we obtain
\begin{equation}
\label{eq:sigmaSfin}
\begin{split}
&\mathrm{Im}\,\sigma^{\rm (orb)}_{{\rm S},abc}(0)
=\frac{e^2}{\hbar}\int[d{\bf k}]\sum_{nm}^o
\mathrm{Re}\Bigr\{\ip{\partial_au_n}{\partial_cu_m}\ip{u_m}{\partial_bu_n}\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
+\ip{\partial_bu_n}{\partial_cu_m}\ip{u_m}{\partial_au_n}\Bigl\}
\\
&+\frac{e}{\hbar}\int[d{\bf k}]\sum_n^o
\Big\{
\left[
\mathrm{Im}\bra{\partial_cu_n}\partial_a(H+E_n)\ket{\wt{\partial}_{\mathcal{E}_b}u_n}
- a\leftrightarrow c
\right]\\
&\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\,\;\;
+b\leftrightarrow a
\Big\},
\end{split}
\end{equation}
where the covariant field derivative
$\ket{\wt{\partial}_{\mathcal{E}_b}u_n}$ is given by \equ{stern2}.
Equation~(\ref{eq:sigmaSfin}) can now be compared with Eq.~(C.2) of
Ref.~\onlinecite{malash2010} for the static ME tensor, which reads
\begin{equation}
\label{eq:al_da}
\begin{split}
\alpha_{da}^{({\rm orb})}&(0)=
\frac{e^2}{2\hbar c}\epsilon_{abc}\int[d{\bf k}]\sum_{nm}^o\mathrm{Re}\Bigl\{\ip{\partial_bu_n}{\partial_cu_m}\ip{u_m}{\partial_du_n}\Bigr\}\\
&-\frac{e}{\hbar c}\epsilon_{abc}\int[d{\bf k}]\sum_n^o
\mathrm{Im}\bra{\partial_bu_n}\partial_c(H+E_n)\ket{\wt{\partial}_{\mathcal{E}_d}u_n}.
\end{split}
\end{equation}
It is easily verified that inserting Eq.~(\ref{eq:al_da}) into
Eq.~(\ref{eq:sigma-S}) at $\omega=0$ yields Eq.~(\ref{eq:sigmaSfin}),
which proves the result.
\section{Numerical results}
\label{sec:results}
\begin{figure}
\centering\includegraphics{fig1.eps}
\caption{(Color online) The $xxy$ component of the gyrotropic
birefringence tensor
$\mathrm{Im}\,\sigma^{\mathrm{S}}_{abc}$, and the $xyz$ component of
the natural optical activity tensor $\mathrm{Re}\,\sigma^{\mathrm{A}}_{abc}$,
calculated for the tight-binding
model described in the text as a function of frequency.
Solid lines: extrapolation from calculations on finite crystallites.
Dashed lines: calculations on periodic crystals using the $k$-space
formulas derived in this work. The vertical dotted line indicates the
frequency corresponding to the direct band gap.}
\label{fig:spectrum}
\end{figure}
In order to check the expressions derived in the previous section, we
have carried out numerical tests comparing calculations done under
periodic boundary conditions against reference calculations on finite
crystallites.
We chose for our tests the tight-binding model of
Ref.~\onlinecite{malash2010}. This is a spinless model on a
$2\times2\times2$ cubic lattice, where $\mathcal{P}$ symmetry is
broken by assigning random on-site energies and $\mathcal{T}$ symmetry
is broken by complex first-neighbor hoppings. The model parameters in
Table~A.1 of Ref.~\onlinecite{malash2010} were used (one of the
complex hopping phases, labeled $\varphi$ therein, shall be used as a
control parameter), and the two lowest bands were treated as occupied.
The tensor components
$\mathrm{Im}\,\sigma^{\mathrm{S}}_{xxy}$ and $\mathrm{Re}\,\sigma^{\mathrm{A}}_{xyz}$
were evaluated at
nonabsorbing frequencies. The calculations on periodic samples were
done on a $30\times 30\times 30$ mesh of $k$ points using
Eqs.~(\ref{eq:im-sigma-s-delta})--(\ref{eq:B-ac-orb}), together with the
sum-over-states formula for ${\boldsymbol\nabla}_{\bf k}
\ket{u_{n{\bf k}}}$.\cite{malash2010} For the calculations on finite
samples we used \eqs{sigma-S}{sigma-A},
\begin{equation}
\label{eq:sigmas_xxy}
\mathrm{Im}\,\sigma^{\mathrm{S}}_{xxy}=2c\mathrm{Re}\,\alpha_{xz}+\omega\mathrm{Im}\,\gamma_{xxy} \doteq
2c\alpha_{xz}-i\omega\gamma_{xxy}
\end{equation}
and
\begin{equation}
\begin{split}
\label{eq:sigma-a-xyz}
\mathrm{Re}\,\sigma_{xyz}^{\mathrm{A}}&=
-c\mathrm{Im}(\beta_{xx}+\beta_{yy})+\omega\mathrm{Re}\,\xi_{xyz} \\
&\doteq ic(\beta_{xx}+\beta_{yy})+\omega\xi_{xyz},
\end{split}
\end{equation}
together with Eqs.~(\ref{eq:alpha})--(\ref{eq:xi}) and
(\ref{eq:chiem-qm-re})--(\ref{eq:LAM-qm-im}) for the magnetoelectric
($\alpha$, $\beta$) and quadrupolar ($\gamma$, $\xi$) tensors. We
chose cubic samples containing $L\times L\times L$ unit cells, with
$L=1,\, 2,\, 3,\, 4$, and then extrapolated the calculated values
to $L\rightarrow
\infty$.\cite{malash2010}
Figure~\ref{fig:spectrum} shows as solid (dashed) lines the frequency
dependence of $\mathrm{Im}\,\sigma^{\mathrm{S}}_{xxy}$ and $\mathrm{Re}\,\sigma^{\mathrm{A}}_{xyz}$ for finite
(periodic) samples, with the parameter $\varphi$ set to $\pi$. The
natural optical activity spectrum
starts off at zero and increases with frequency,
exhibiting a resonant behavior as the minimum direct gap, denoted by
the vertical dashed line, is approached. The ME optical spectrum
displays a similar behavior, except that it
remains finite as $\omega$ goes to zero. The excellent agreement between
solid and dashed lines demonstrates the correctness of the $k$-space
formulas.
\begin{figure}
\centering\includegraphics{fig2.eps}
\caption{(Color online) The $xxy$ component of
$\mathrm{Im}\sigma^{\mathrm{S}}_{abc}(\omega)$, calculated for the tight-binding model
described in the text as a function of the parameter
$\varphi$. Solid lines: extrapolation from calculations on finite
crystallites using \equ{sigmas_xxy}. Dashed lines: calculations on
periodic crystals using
\eqs{im-sigma-s-delta}{im-sigma-s-delta-prime}. Dotted lines: same
as the dashed lines, but ommiting the contribution coming from
\equ{im-sigma-s-delta-prime}.}
\label{fig:sig_xxy}
\end{figure}
Next we discuss a number of additional numerical tests where we investigate
in more detail the behavior of
$\mathrm{Im}\,\sigma^{\mathrm{S}}_{xxy}$. In these tests the frequency was kept fixed, and
the parameter $\varphi$ was scanned over the
range $[0,2\pi]$.
In Figure~\ref{fig:sig_xxy} we plot $\mathrm{Im}\,\sigma^{\mathrm{S}}_{xxy}$ versus
$\varphi$ for two frequencies, $\omega=0$ and $\hbar\omega=1$. As before,
solid and dashed lines represent calculations on finite and periodic
samples respectively. In addition, we show as dotted lines the result
of a periodic-sample calculation using only the matrix element (m)
term, Eq.~(\ref{eq:im-sigma-s-delta}), i.e., omitting the energy (e) term,
\equ{im-sigma-s-delta-prime}. We see that the energy term gives a
small but visible contribution, which must be included in order
to find agreement with the finite-sample calculation.
We now turn to the decomposition of $\mathrm{Im}\,\sigma^{\mathrm{S}}_{xxy}$
according to \equ{sigmas_xxy}, into
magnetoelectric and quadrupolar parts.
They are plotted separately in Fig.~\ref{fig:orig_dep} for $\hbar\omega=1$
and $L=4$. We chose a specific $L$ because $\alpha$ and $\omega\gamma$
are origin-dependent quantities, and it is therefore not meaningful to
extrapolate them separately to $L\rightarrow\infty$. The dashed lines
show how each of them changes when the position of the sample is
shifted. The change in $\alpha_{zz}$ is exactly compensated by the
change in $\omega\gamma_{xxy}$, so that the resulting $\mathrm{Im}\,\sigma^{\mathrm{S}}_{xxy}$
remains the same to machine precision, demonstrating its translational
invariance.
\begin{figure}
\centering\includegraphics{fig3.eps}
\caption{Origin-dependence of the bare magnetoelectric (upper
panel) and quadrupolar (lower panel) polarizabilities appearing on
the right-hand side of \equ{sigmas_xxy}, calculated at $\hbar\omega=1$
for a finite sample ($L=4$) of the model used in
Fig.~\ref{fig:sig_xxy}. Solid lines: the center of the sample is
placed at the origin. Dashed lines: the sample is displaced by
$\mathbf{r}=(1,1,1)$, in units of the lattice constant
of the $2\times 2\times 2$ cubic cell.}
\label{fig:orig_dep}
\end{figure}
An alternative decomposition of $\mathrm{Im}\,\sigma^{\mathrm{S}}_{xxy}$ is given
by \equ{alpha_gamma_def}:
\begin{equation}
\label{eq:sigmas_xxy-inv}
\mathrm{Im}\,\sigma^{\mathrm{S}}_{xxy}\doteq
2c\wt{\alpha}_{xz}-i\omega\wt{\gamma}_{xxy}.
\end{equation}
Unlike the bare property tensors $\alpha$ and $\omega\gamma$ appearing in
\equ{sigmas_xxy}, the renormalized magnetoelectriclike and
purely quadrupolar tensors $\wt{\alpha}$ and $\omega\wt{\gamma}$ are
origin independent and hence separately well defined for periodic
samples. Figure~\ref{fig:orig_indep} shows as dashed (solid) lines
their values calculated for periodic (finite) samples from the first
(second) equality in Eqs.~(\ref{eq:alpha-tilde}) and
(\ref{eq:Gamma-tilde}). Because $\wt{\alpha}$ reduces to the
traceless part of $\alpha$ as $\omega\rightarrow 0$, we can directly
compare the curve for $\wt{\alpha}_{xz}(0)$ with a $k$-space
calculation of $\alpha_{xz}(0)$ using the formula derived in
Refs.~\onlinecite{malash2010,essin2010} (open circles). The precise
agreement confirms numerically the analysis of Sec.~\ref{sec:static}.
\begin{figure}
\centering\includegraphics{fig4.eps}
\caption{(Color online) Translationally invariant decomposition
[Eq.~(\ref{eq:sigmas_xxy-inv})] of the curves in
Fig.~\ref{fig:sig_xxy} into magnetoelectriclike
(upper panel) and purely quadrupolar (lower
panel) contributions. Solid lines: extrapolation from calculations
on finite crystallites. Dashed lines: $k$-space calculations
on periodic crystals. In the static limit the tensor $\wt{\alpha}$
reduces to the traceless part of the magnetoelectric
polarizability $\alpha$, and the open circles show
$\alpha_{xz}(0)$ calculated in $k$ space according to
Refs.~\onlinecite{malash2010,essin2010}.}
\label{fig:orig_indep}
\end{figure}
\section{Summary and outlook}
\label{sec:sum}
In this work we investigated spatial-dispersion optical effects in
insulators. The main result is a band-theory expression for
$\sigma_{abc}(\omega)$, the spatially dispersive optical conductivity.
Special attention was given to the ${\cal T}$-odd part of this tensor,
which is nonzero in magnetoelectric crystals, and comprises
magnetoelectriclike ($\wt{\alpha}_{ab}$) and purely quadrupolar
($\wt{\gamma}_{abc}$) contributions. We showed that each of them
consists of a translationally invariant combination of separately
origin dependent molecular polarizability tensors.
The magnetoelectriclike tensor $\wt{\alpha}_{ab}$
has both spin and
orbital contributions, and the expression for the orbital part
generalizes to finite frequencies the
recently developed band theory of orbital magnetoelectric
response.\cite{malash2010,essin2010} The generalization is, however,
not complete, as the tensor $\wt{\alpha}_{ab}(\omega)$ is traceless, and
therefore does not include the isotropic ME coupling. The reason why
the latter is not recovered from the present formalism is that
our starting point is the current response of an infinite medium to an
electromagnetic wave while the trace of the ME tensor, known as the
{\it axion} contribution, only affects electrodynamics at
boundaries.\cite{hornreich68,essin2010} The calculation of the axion
piece at finite frequencies remains an open problem.
The bulk expression for $\sigma_{abc}(\omega)$ at transparent frequencies
was validated by performing numerical calculations on a tight-binding
model, and comparing against reference calculations done on finite
samples.
The quantities needed to evaluate that expression are the occupied and
empty energy eigenvalues and their $k$-space gradients, the
off-diagonal Berry connection matrix Eq.~(\ref{eq:A}), and the orbital
and spin matrices Eqs.~(\ref{eq:B-ac-orb}) and (\ref{eq:B-ac-spin}). The
evaluation of all these objects in a first-principles context can be
done efficiently by mapping the electronic structure onto localized
Wannier orbitals, and then using the technique of Wannier
interpolation.\cite{wang06} This approach has already been used to
compute the magnetic circular dichroism spectrum of
ferromagnets.\cite{yates07}
First-principles calculations of the optical spectrum of solids beyond
the electric-dipole approximation are still in their infancy. We hope
that the formalism introduced in this work will be useful for carrying
out realistic calculations of spatial-dispersion phenomena in the
optical range, including natural optical activity, gyrotropic
birefringence, and directional dichroism.
\acknowledgments
This work was supported by NSF under Grant No. DMR-0706493. Computational
resources were provided by NERSC.
|
{
"timestamp": "2010-12-30T02:02:07",
"yymm": "1009",
"arxiv_id": "1009.4238",
"language": "en",
"url": "https://arxiv.org/abs/1009.4238"
}
|
\section{Overview}
Realized and potential applications of microstructured dielectric
media motivate a thorough mathematical study of wave-propagation
governed by nonlinear hyperbolic equations, {\it e.g.} Maxwell's
equations with periodic and nonlinear constituitive laws.
This paper explores a class of nonlinear hyperbolic equations with a
spatially periodic flux function:
\begin{subequations}\label{nonlin-periodic}
\begin{align}
\partial_t{\bf v} + \partial_x{\bf f}(x, {\bf v}) &= {\bf 0} \\
{\bf f}(x, {\bf 0}) &= {\bf 0},\\
{\bf f}(x+2\pi,{\bf v}) &= {\bf f}(x,{\bf v}).
\end{align}
\end{subequations}
In particular, we shall assume that periodic variations are weak (a
low contrast structure) and study solutions, whose amplitude
is small and such that the effects of periodicity-induced dispersion and
nonlinearity are in balance.
Indeed, a non-trivial spatially periodic structure is
dispersive. This can be seen by linearizing \eqref{nonlin-periodic}
about the state ${\bold v}={\bold 0}$, giving the linear system:
\begin{align}
&\partial_t{\bf V} + \partial_x \left( D_{\bf v}{\bf f}(x, {\bf 0})
{\bf V} \right) ={\bf 0},
\label{lin-periodic}\end{align}
which retains periodicity. Floquet-Bloch theory
\cite{Eastham,Reed-Simon-IV} implies that associated to the PDE
\eqref{lin-periodic} is a family of {\it band dispersion functions}
$k\mapsto\omega_j(k),\
k\in\left(-\frac{1}{2},\frac{1}{2}\right]$. Wave propagation is
dispersive since the group velocities, $\omega_j'(k)$, are typically
non-zero. Thus, waves of different wavelengths travel with different
speeds.
Dispersive properties, encoded in the functions $\omega_j(\cdot)$ and
the associated Floquet-Bloch states, can be manipulated by {\it tuning
the periodic structure} through, for example, modification of the
periodic lattice, the maximum and
minimum variations of $ D_{\bold v}{\bold f}(x, {\bold 0})$ (material
contrast), {\it etc}.
It is well-known that for general initial conditions, solutions of
hyperbolic systems of conservation laws with spatially homogeneous
nonlinear fluxes:
\begin{equation} \label{nl-hyp-cons-law}
\partial_t{\bf v}\ + \partial_x{\bf f}({\bf v}) ={\bf 0}
\end{equation}
develop singularities (shocks) in finite time, \cite{lax1964dss ,
klainerman1980fsw}. \medskip
\noindent {\bf Question 1:}\ {\it Is {\it spectral band dispersion}, due to
a periodic structure,
sufficient to arrest shock formation?} \footnote{For example,
though typical smooth initial data for the inviscid Burgers equation
$\partial_tu+u\partial_xu=0$ develop shocks in finite time, the
corresponding solutions of the Korteweg - de Vries (KdV) equation,
$\partial_tu+u\partial_xu+\partial_x^3u=0$, a dispersive
perturbation, remain smooth for all time. }
\\
The ability to control or inhibit the formation of singularities in
nonlinear wave propagation could have significant impact in, for
example, electromagnetics and elasticity. Strictly speaking, the
answer to Question 1 is no. Indeed, for a system of the form
\eqref{nonlin-periodic}, let us suppose that the flux function was
periodically piecewise constant. Finite propagation speed
considerations imply that for appropriate initial data, which are
sufficiently localized within a uniform region, a shock will form.
The dispersive character of the periodic structure is manifested only
on sufficiently large spatial and temporal scales. Thus, the problem
of controlling shock formation should be posed relative to some class
of initial conditions
\bigskip
A second motivation is the study and design of media which support the
propagation of stable soliton-like pulses. These have applications to
optical devices which transfer store or, in general, process information
which is encoded as light pulses. Associated
with dispersive wave-propagation at wavenumber $k_\star$ is a
dispersion length $\sim (\omega''(k_\star))^{-1}$. Soliton formation
is possible on length scales where the dispersion length and the
characteristic length on which nonlinear effects act are
comparable. Technological advances have made it possible to fabricate
microstructured media
with specified dispersion lengths at specified wavelengths. For a
given dispersion length, a balance between dispersion and nonlinearity
is achieved by tuning the strength of the nonlinear effects through
adjusting the field intensity (by an amount which is material
dependent). An example of this balance at work is {\it gap soliton}
formation in periodic structures. These are experiments in optical
fiber periodic structures (gratings) involving highly intense
(nonlinear) light with carrier wave-length satisfying the Bragg
(resonance) condition. The length-scale of such solitons is $10^{-2}$
meters \cite{Eggleton-deSterke-Slusher}.
Theory predicts the existence of gap solitons traveling at any speed,
$v$, between zero and the speed of light, $c$\
\cite{christodoulides-joseph:89,aceves1989sit}. Experiments
\cite{Eggleton-deSterke-Slusher} demonstrate speeds as low as $.3c$ to
$.5c$. Potential applications of gap solitons, based on the design of
appropriate localized defects in a periodic structure, are all-optical
storage devices \cite{Goodman-Slusher-Weinstein:02}. The term gap
soliton is used due to frequency of the gap soliton envelope lying in
the spectral gap of the linearized system. \bigskip
Physical predictions of gap solitons are based on explicit solutions
of {\it nonlinear coupled mode equations} (NLCME), given below in
\eqref{eq:nlcme}. NLCME has been formally derived in, for example,
\cite{desterke1994gs} from \eqref{eq:maxwell1}; see also the
discussion in Section \ref{sec:nlcme}. Rigorous derivations of NLCME,
from models with appropriate dispersion have been presented for the
anharmonic Maxwell-Lorentz equations \cite{goodman01npl} and other
nonlinear dispersive equations; see {\it e.g.} \cite{groves2001mps,
schneider2001ncm,schneider2003eas, pelinovsky2007jcm,
pelinovsky2008mgs}.
Within the approximation of a small amplitude wave field as a
wave-packet with slowly varying envelope and {\it single} carrier
frequency, propagating through a low contrast periodic structure near
the Bragg resonance (\ see scaling in Figure \ref{f:wavepack})\ ),
NLCME is argued to govern the principle forward and backward slowly
varying envelopes of carrier waves; see \cite{desterke1994gs} and
references therein.
{\it As discussed in \cite{goodman01npl} and in section
\ref{sec:nlcme}, if the only source of dispersion is the spatial
dispersion of the periodic medium ({\it e.g. negligible chromatic
dispersion}) for weakly nonlinear waves in low contrast media all
nonlinearity-generated harmonics are resonant and therefore all mode
amplitudes are coupled at leading order. The correct mathematical
description would appear to require infinitely many interacting
modes.} Thus, the classical NLCME are {\bf not} a mathematically
consistent approximation. NLCME may however be satisfactory physical
description, for some purposes.t Indeed, the soliton wave form prediction based on NLCME
appears to describe some features of experiment.
%
\footnote{Physicists argue in two ways that the coupling to higher
harmonics is argued to be negligible : (i)\ The material systems
considered are dissipative at higher wave numbers. Higher wave
numbers are damped and therefore these mode amplitudes can be
ignored, and (ii) {\it Chromatic dispersion} (arising due to the
finite time response of the medium to the field) causes nonlinearly
generated harmonics to be off-resonance. Therefore, an initial
condition exciting the principle modes will not appreciable excite
higher harmonics.
These rationales are somewhat ad hoc since the precise damping
mechanisms are not well-understood and chromatic dispersion is a
much weaker effect than photonic band dispersion for weak periodic
structures.}
\begin{figure}
\centering
\includegraphics[width=4in]{figs/wavepack_annotated}
\caption{A wavepacket (real part, $\Re E_0(z)$, of complex field)
with carrier wave length equal twice that of the waveguide
refractive index ($n(z)$). }
\label{f:wavepack}
\end{figure}
\bigskip
\noindent {\bf Question 2:}\ {\it Do nonlinear periodic
hyperbolic systems have stable coherent structures, and can one
develop a mathematical theory? How
are the classical NLCME related to this theory?
See the discussion in section
\ref{sec:discussion}.}
\medskip
In this article we report on progress on Questions 1 and 2 in the
context of the one-dimensional, nonlinear Maxwell equations governing
the electric ($E$) and magnetic ($B$) fields:
\begin{subequations}\label{eq:maxwell1}
\begin{gather}
\partial_{t} D = \partial_{z} B,\\
\partial_{t} B = \partial_{z} E.
\end{gather}
\end{subequations}
with constitutive law
\begin{align}
\label{DE}
D\ &=\ \epsilon(z,E)\ E\ \equiv\ \left(\ n^2(z) + \chi E^2\ \right)E\\
\label{Ndef}
n(z)\ &=\ n_0\ +\ \epsilon N(z)
\end{align}
$n(z)$ is a {\it linear} refractive index, consisting of a nonzero
background average part, $n_0$, and a fluctuating ({\it
e.g. periodic}) part $\epsilon N(z)$. The nonlinear term $\chi E^2$
is the nonlinear refractive index, arising from the Kerr effect; in
regions of high intensity the refractive index is higher. The consituitive law
\eqref{DE}, prescribes $D$ as a
a local function of $E$. Thus chromatic dispersion, which arises due a
time-nonlocal relation between $D$ and $E$ has been
neglected. {\it For simplicity, we assume $n_0 = 1$, which can be
arranged by a simple scaling.}
\subsection{Summary of results}\label{sec:summary}
\begin{enumerate}
\item In section \ref{sec:observations} we present numerical
simulations of the nonlinear periodic Maxwell equations, \eqref{eq:maxwell1},
for initial data obtained from the
explicit NLCME soliton. Under this time-evolution there is robust spatially localized structure
{\it on the scale of the NLCME soliton
envelope}. The persistence of a localized structure and speed
of propagation are consistent with that of the NLCME soliton. There
is, however, a deviation from the NLCME soliton related to {\it
third harmonic} generation; these are the two accessory pulses
around the principle wave in Figure \ref{f:intro_third_harmonic}
(a).
\begin{figure}
\centering
\subfigure[]{\includegraphics[width=2.3in]{figs/_plots_soliton2ncos1_print_20000/matlabfig396}}
\subfigure[]{\includegraphics[width=2.3in]{figs/plots_print_standing_wave_M16_N4096_Zmax32/fig100}}
\caption{On the left is a simulation of the Maxwell equations. On
the right is the simulation of a truncated asymptotic system,
resolving the first and third harmonics. Both simulations were
initiated with the same initial conditions. The two side pulses
about the main wave appear to be the result of third harmonic
generation.}
\label{f:intro_third_harmonic}
\end{figure}
\item {\it On the microscopic scale of the carrier} there is nonlinear steepening
and shock formation. Therefore, the solution does not evolve as a
slowly varying envelope of a single frequency carrier wave. The
long-lived and spatially localized coherent structure
which emerges has the character of a slowly
varying envelope of a train of shocks. We call this an {\it envelope carrier-shock train}.
Figure \ref{f:intro_carrier_shock} illustrates the shock-like small
spatial scale behavior under slowly varying envelope.
\begin{figure}
\centering
\subfigure[]{\includegraphics[width=2.4in]{figs/_plots_soliton2ncos1_shock_print20000/matlabshockfig96}}
\subfigure[]{\includegraphics[width=2.4in]{figs/plots_print_standing_wave_M64_N16384_Zmax32/shock_fig25}}
\caption{On the left is a simulation of the Maxwell equations. On
the right is the simulation of a truncated asymptotic system.
Both simulations were initiated with the same initial conditions.
There is an indication of shock formation in the left. On the
right, we see that once sufficiently many harmonics are included,
the Gibbs effect appears, confirming shock formation.}
\label{f:intro_carrier_shock}
\end{figure}
\item Numerical solution of the nonlinear Maxwell equation
\eqref{eq:maxwell1} is non-trivial due to the cubic nonlinearity.
As a hyperbolic system, it is neither genuinely nonlinear nor
linearly degenerate\cite{smoller1994swa,lax2006hpde,
liu2000hyperbolic}. To solve by finite volume methods, as we do,
an explicit solution of the Riemann problem must be constructed.
Details of this are given in Appendix \ref{sec:methods}.
The appropriate
entropy condition could, in principle be derived from physical
regularization mechanisms, which play the role of viscosity in gas
dynamics. However, these mechanisms are not well understood.
However, such mechanisms and the appropriate notion of
weak solution would respect thermodynamic principles, which are built into our numerical scheme.
\item Using a nonlinear geometric optics expansion
\cite{Hunter:1983vn,Hunter:1986kx,Majda:1984uq,Donnat:1997p97,Joly:1993p4514,Joly:1996p4513,Lannes:1998p4610},
systematically keeping all resonances, we obtain nonlocal equations
governing the interaction of all forward and backward propagating
modes. Our asymptotic nonlocal system captures the slowly varying
envelope of carrier-shock structures described above; see below.
\bigskip
Specifically, we introduce the general wave-form (much more general
than a slowly varying envelope of a nearly monochromatic carrier
plane wave), which includes all harmonics
\begin{equation}
E(z,t) = \epsilon^{1/2} \left(\ E^+(z - t, \epsilon z, \epsilon t ) + E^-(z +
t, \epsilon z, \epsilon t)\ +\ \mathcal{O}(\epsilon)\ \right).
\label{Eansatz}\end{equation}
Let
\begin{equation}
\phi_\pm = z \mp t,\ \ \epsilon t = T,\ \ {\rm and}\ \ \epsilon z = Z\ .
\nonumber\end{equation}
At leading order, the slow evolution of backward and forward
components is governed by the coupled \emph{integro-differential} equations:
\begin{subequations}
\label{eq:integro_diff_intro}
\begin{gather}
\begin{split}
\partial_{T} E^+ + \partial_{Z} E^+&= {\partial_\f} \mean{N( \phi_+ + s) E^-( \phi_+ + 2 s,Z,T)}_s\\
&\quad +\Gamma {\partial_\f}\left[
\frac{1}{3}\paren{E^+}^3+E^+\mean{\paren{E^-}^2} \right],
\end{split}
\\
\begin{split}
\partial_{T} E^- - \partial_{Z} E^-&=-{\partial_\f}\mean{N( \phi_- - s)E^+( \phi_- -2 s, Z,T) }_s\\
&\quad -\Gamma
{\partial_\f}\left[\mean{\paren{E^+}^2}E^-+\frac{1}{3}\paren{E^-}^3\right].
\end{split}
\end{gather}
\end{subequations}
Here $\mean{\cdot}$ is an averaging operation in the $\phi$
argument;
\begin{equation}
\langle\ f\ \rangle\ \equiv\ \lim_{T\to\infty}\ \frac{1}{T}\ \int_0^T\ f(s)\ ds;
\label{avg-def}
\end{equation}
see also section \ref{sec:asympt}. Equations
\eqref{eq:integro_diff_intro} arise as a constraint on
$E^\pm(\phi_\pm,Z,T)$ ensuring that the $\mathcal{O}(\epsilon)$
error term \eqref{Eansatz} in remains small on large time scales:
$T=\mathcal{O}(1)$ or equivalently $t=\mathcal{O}(\epsilon^{-1})$.
Spatial variations in the refractive index, $N(z)$, give rise to a
coupling of backward and forward waves. Indeed, if $N(z)\equiv0$ and
one specifies data for the system \eqref{eq:integro_diff_intro} at
$t=0$ with non-zero forward components ($E^+\ne0$) and, no backward
components ($E^-=0$) then, formally, $E^-$ remains zero for all
time, {\it i.e.} no backward waves are generated. Continuing with
this assumption of $N = 0$ and $E^-_0 = 0$, if we let $V(\phi, T) =
E^+(\phi, Z_0 - T, T)$, with $Z_0$ arbitrary, then $V$ satisfies
\[
\partial_T V = \tfrac{\Gamma}{3} \partial_\phi(V^3).
\]
This generalized Burger's equation will gives rise to a finite time
singularity. We revisit this observation in the discussion, section
\ref{sec:discussion}, when considering how singularities might
appear when the linear coupling between backwards and forwards waves
is restored, $N \neq 0$.
\item The nonlocal equations may also be written as an infinite system
of coupled mode equations. In the case where $E^\pm$ is $2\pi-$
periodic in $\phi_\pm$, the integro-differential equation
\eqref{eq:integro_diff_intro} reduces to an infinite system of
coupled mode equations for the Fourier coefficients
$\left\{E^\pm_p(Z,T) : p\in\mathbb{Z} \right\}$:
\begin{subequations}\label{e:mode_intro}
\begin{gather}
\label{eq:mode_intro_p}
\begin{split}
\partial_{T} E^+_p + \partial_{Z} E^+_p = \mathrm{i}p N_{2p}{E^-_p} +
\mathrm{i}p\frac{\Gamma}{3}&\left[\sumE^+_q
E^+_r E^+_{p-q-r} \right.\\
&\quad\left.+3\paren{\sum \abs{E^-_q}^2} E^+_p \right],
\end{split}
\\
\label{eq:mode_intro_m}
\begin{split}
\partial_{T} E^-_p - \partial_{Z} E^-_p = \mathrm{i}p \bar{N}_{2p}{E^+_p}
+\mathrm{i}p\frac{\Gamma}{3} &\left[\sum E^-_q
E^-_r E^-_{p-q-r} \right.\\
&\quad \left.+3\paren{\sum \abs{E^+_q}^2} E^-_p \right].
\end{split}
\end{gather}
\end{subequations}
We call this system the {\it extended nonlinear coupled mode
equations} (xNLCME).
%
xNLCME reduces to the classical NLCME if we neglect higher
harmonics.
\item Simulations of successively higher dimensional mode truncations
of \eqref{e:mode_intro} show improved resolution of the carrier shocks under a slowly
varying envelope, whose scale is captured by a comparatively low
order truncation. Indeed, Figure \ref{f:intro_third_harmonic} (b)
shows that inclusion of the third harmonic in the asymptotic system
resolves the large scale feature, while inclusion of additional
harmonics in Figure \ref{f:intro_carrier_shock} (b) shows the Gibbs
effect, expected for a finite Fourier representation of a
discontinuous function. This demonstrates that our asymptotic
analysis leads to equations capturing the essential features of
nonlinear Maxwell. However, if we consider how energy, initially
only in the first harmonic, is redistributed in time, we see in
Figure \ref{f:eng_intro} that most of the energy persists in the
first harmonic. This reflects the partial success of NLCME as a
model for periodic nonlinear Maxwell.
\end{enumerate}
\begin{figure}
\centering
{\includegraphics[width=3in]{figs/plots_print_standing_wave_M64_N16384_Zmax32/eng_dist_8}}
\caption{Truncating \eqref{e:mode_intro} to odd harmonics
$\abs{p}\leq 16$, we simulate the initial value problem an NLCME
soliton in the first harmonic, and the others zero. The above
time series of the energy associated with each harmonic, $e_p$,
shows that most of the energy continues to reside in the first
harmonic.}
\label{f:eng_intro}
\end{figure}are
\noindent{\bf Relation to previous work:}\ Some of the earliest
examinations on optical shocks can be found in Rosen,
\cite{Rosen:1965p6493}, and, DeMartini {\it et al.}
\cite{DeMartini:1967p6494}. In these works, the authors applied the
method of characteristics to a unidicretional model.
Kinsler and Kinsler {\it et al.} have continued to examine this
problem, and have developed an algorithm for detecting the onset of
shock formation, \cite{Kinsler:2007p6495,Kinsler:2007p6542}. Carrier shocks were also
examined by Flesch, Moloney, \& Mlejnek, \cite{flesch1996cws}, for spatially homogeneous
Maxwell system with chromatic dispersion, modeled via a time-nonlocal
Lorentzian polarization response. Ranka, Windeler, \& Stentz have found experimental evidence of
optical shocks, \cite{Ranka:2000p6690}. In their work, a
monochromatic pulse with sufficient power steepened and
generated a broadband optical continuum.
Coherent structures in nonlinear and periodic media have also been
studied by LeVeque, LeVeque \& Yong, and Ketcheson
\cite{leveque2002fvm,leveque2003swl,Ketcheson:2009fk} in a model for
heterogeneous nonlinear {\it elastic} media. They considered order
one solutions in high contrast, rapidly varying, periodic
structures. Their simulations yielded localized structures on the
scale of many periods with oscillations on the scale of the
period. For piecewise constant (discontinuous) periodic structures,
they have a discontinuous carrier shock-like character on the scale of
the period, though this is due to discontinuities in the medium, the
fluxes remain continuous. A two-scale (homogenization) expansion
yields a nonlinear dispersive equation, with solitary waves, similar
to the computed solution envelope. In their physical regime, the
variations in the properties of the media and the nonlinearity are
$\mathcal{O}(1)$. In contrast, we consider an asymptotic regime where
the constrast of the periodic structure and nonlinearity are of the
same order, $\mathcal{O}(\epsilon)$. Furthermore, the initial condition
has two scales (envelope and carrier scales), where the carrier wave
length is of the same order, indeed in resonance with, the periodic
structure. These different scalings lead to different asymptotic
descriptions. An early example of the interactions between nonlinearity and
a periodic structure was in atmospheric science, studied by Majda {\it
et al.}, \cite{Majda:1999p2}. In this work, a model of the
interaction of equatorial waves with topography gives rise to
nonsmooth profiles (in this case, solitary waves with corner
singularities).
Finally, systems of coupled modes have also been examined in prior
works, though the work is typically limited two just two harmonics,
such as a first and second harmonic system or a first and third
harmonic system. Such a system was studied by Tasgal, Band, \& Malomed
\cite{Tasgal:2005p6335}, who were able to find stable {\it polychromatic}
solitons in a first and third harmonic system.
\bigskip
An outline of this paper is as follows. In Section \ref{sec:nlcme},
we review how NLCME arises as an approximation of nonlinear Maxwell.
Results of Maxwell Simulations, showing the coherent structures and
shocks, are given presented in Section \ref{sec:observations}. We
then present our derivation of xNLCME in Section \ref{sec:asympt},
followed by simulations of this system in Section \ref{s:xnlcme_sims}.
We discuss all of these results in Section \ref{sec:discussion}.
{\bf Acknowledgements:} The authors would like to thank R.R. Rosales
for discussions during the early stages of this work on the use
nonlinear geometrical optics. We also thank M. Pugh, D. Ketcheson, R.J. LeVeque,
and C. Sulem for helpful discussions. GS was supported in part by NSF-IGERT grant
DGE-02-21041, NSF-CMG grant DMS-05-30853, and NSERC. MIW was supported in part by
NSF grants DMS-07-07850 and DMS-10-08855. MIW would also like to
acknowledge the hospitality of the Courant Institute of Mathematical
Sciences, where he was on sabbatical during the preparation of this
article.
\section{Nonlinear Maxwell and NLCME}
\label{sec:nlcme}
In this section we briefly review how NLCME arises from nonlinear
Maxwell with a periodically varying index of refraction. We also
identify the mathematical inconsistency of NLCME as a description of the
wave-envelope.
First, we write the nonlinear Maxwell equation \eqref{eq:maxwell1} as
\begin{equation}
\label{e:maxwell_scalar}
\partial_{t}^2\paren{n(z)^2E + \chi E^3} = \partial_{z} ^2 E
\end{equation}
with index of refraction
\begin{equation}
n(z) = 1 + \epsilon N(z), \quad 0 < \epsilon \ll 1,
\end{equation}
where $N(z)$ is $2 \pi$ periodic with mean zero and Fourier series:
\begin{equation}
N(z) = \sum_{p \in \mathbb{Z}\setminus\{0\}} N_p e^{\mathrm{i} p z}.
\label{N-Fourier}
\end{equation}
We shall seek solutions which
incorporate \begin{inparaenum}[(i)] \item slow variations in time and
space, due to the weak modulation about a constant refractive
index; \item a scaling of the wave-field which seeks solutions in
which the effects of dispersion and nonlinearity are in balance:
\begin{equation}
E^\epsilon(z,t)\ =\ \epsilon^{1\over2}\ \mathcal{E}^\epsilon(z,t;Z,T),\ \ Z=\epsilon z,\ \ T=\epsilon t.
\label{Eassumptions}
\end{equation}
\end{inparaenum}
Rewriting \eqref{e:maxwell_scalar} in terms of new variables dependent
$\mathcal{E}^\epsilon$ and independent $(z,t,Z,T)$ variables, we
obtain:
\begin{equation}
\left(\partial_t^2-\partial_z^2\right) \mathcal{E}^\epsilon\
+\ \epsilon\left(2\partial_t\partial_T \mathcal{E}^\epsilon-2\partial_z\partial_Z \mathcal{E}^\epsilon+2N(z) \mathcal{E}^\epsilon+\chi\left( \mathcal{E}^\epsilon\right)^3\ \right)\ +\ \mathcal{O}(\epsilon^3)\ =\ 0.
\nonumber\end{equation}
Formally expanding $ \mathcal{E}^\epsilon$ as
\begin{equation}
\mathcal{E}^\epsilon(z,t,Z,T)\ =\ \mathcal{E}_0(z,t,Z,T)\ +\ \epsilon\ \mathcal{E}_1(z,t,Z,T)\ +\ \dots
\nonumber\end{equation}
we obtain the following hierarchy for $\mathcal{E}_j(z,t,Z,T)$, $j
\geq 0$:
\begin{equation}
\begin{split}
\mathcal{O}(\epsilon^0) & \left(\partial_t^2-\partial_z^2\right)
\mathcal{E}_0=0 \\
\mathcal{O}(\epsilon^1) & \left(\partial_t^2-\partial_z^2\right) \mathcal{E}_1 = -2\partial_t\partial_T \mathcal{E}_0+2\partial_z\partial_Z \mathcal{E}_0-2N(z) \mathcal{E}_0-\chi\left( \mathcal{E}_0\right)^3 \\
&\vdots\\
\mathcal{O}(\epsilon^j) & \left(\partial_t^2-\partial_z^2\right)
\mathcal{E}_j =\text{expressions in terms of}\ \mathcal{E}_l,\ \ 0\le l\le j-1 \\
& \vdots
\end{split}
\label{hierarchy}
\end{equation}
Solving the $\mathcal{O}(\epsilon^0)$ equation yields:
\begin{equation}
\mathcal{E}_0(z,t,Z,T) =
\mathcal{E}^+(Z,T)e^{i(z- t)} + \mathcal{E}^-(Z, T)e^{-i(z+ t)} + \mathrm{c.c.}
\label{svea}\end{equation}
Thus, the leading order consists of backward and forward propagating waves,
modulated by the slow envelope amplitude functions
$\mathcal{E}^\pm(Z,T)$, which are to be determined.
Substitution of \eqref{svea} into the $\mathcal{O}(\epsilon^1)$
equation for $\mathcal{E}_1$ yields the equation: {\begin{equation}
\begin{split}
& \left(\partial_t^2-\partial_z^2\right) \mathcal{E}_1\\
&=\bracket{2i \partial_T \mathcal{E}^+ - 2i \partial_Z
\mathcal{E}^+ - 2 N_2 \mathcal{E}^- - 3 \chi\paren{
\abs{\mathcal{E}^+}^2 + 2 \abs{\mathcal{E}^-}^2} \mathcal{E}^+ } e^{i(z- t)}\\
&+\bracket{2i \partial_T \mathcal{E}^- - 2i \partial_Z
\mathcal{E}^+ - 2 \bar{N}_2 \mathcal{E}^+ - 3 \chi\paren{
\abs{\mathcal{E}^-}^2 + 2 \abs{\mathcal{E}^+}^2 } \mathcal{E}^-}e^{-i(z+ t)} \\
&+\paren{\mathcal{E}^+}^3 e^{3i(z- t)} +\paren{\mathcal{E}^-}^3
e^{-3i(z+ t)} + \mathrm{c.c.} + \text{non-resonant terms}
\end{split}
\label{E1eqn}
\end{equation}}
We have used that $N_0 = 0$ and
\begin{equation}
\begin{split}
&N(z) (\mathcal{E}^+ e^{i(z- t)} + \mathcal{E}^- e^{-i(z+t)})\\
&\quad = N_{-2} \mathcal{E}^+ e^{-i(z+ t)} + N_2 \mathcal{E}^-
e^{i(z- t)} +\mathrm{c.c.}+ \text{non-resonant terms}.
\end{split}
\end{equation}
Each term, explicitly written on the right hand side of \eqref{E1eqn}, is resonant with the
kernel of $\left(\partial_t^2-\partial_z^2\right)$ . It follows that
the coefficients of {\bf all} harmonic plane waves: $e^{\pm i q (z-
t)}$ and $e^{\pm i q (z+ t)},\ q\in\mathbb{Z}$ must vanish for
$\mathcal{E}_1$ to be bounded in $t$.
The vanishing of the coefficients of $e^{i(z- t)}$ and $e^{-i(z+ t)}$
yields the nonlinear coupled mode equations (NLCME):
\begin{subequations}
\label{eq:nlcme}
\begin{align}
\partial_{T} \mathcal{E}^+ + \partial_{Z} \mathcal{E}^+ = i N_2 \mathcal{E}^- + i\Gamma\paren{\abs{\mathcal{E}^+}^2 + 2 \abs{\mathcal{E}^-}^2 } \mathcal{E}^+,\\
\partial_{T} \mathcal{E}^- - \partial_{Z} \mathcal{E}^- = i \bar{N}_{2}
\mathcal{E}^+ + i \Gamma \paren{\abs{\mathcal{E}^-}^2 + 2
\abs{\mathcal{E}^+}^2 } \mathcal{E}^-,
\end{align}
\end{subequations}
where $ \Gamma \equiv \frac{3}{2}\chi$ and $\bar{N}_2=N_{-2}$. The
initial value problem for \eqref{eq:nlcme} is well-posed
\cite{goodman01npl}. NLCME also has explicit family of {\it
gap-soliton} solutions; see Appendix \ref{s:nlcme_soliton}.
However, requiring $\mathcal{E}^\pm$ to satisfy \eqref{eq:nlcme} removes only
the lowest harmonic resonances. This is the approximation invoked in
the physics literature; see the survey \cite{desterke1994gs} and
references cited therein.
Note however that the remaining explicitly displayed terms on the right hand side of
\eqref{E1eqn} are resonant as well and induce linear in time
growth. If we choose to remove the resonant terms proportional to
$e^{3i(z- t)}$ and $e^{-3i(z+ t)}$ by including slow modulations of
these plane waves at $\mathcal{O}(\epsilon^0)$, nonlinearity and
parametric forcing through $N(z)$ will generate yet other resonant
harmonics.
{\it A leading order solution which does not generate resonant terms
at higher order must contain \underline{all} harmonics. Thus, NLCME
is mathematically inconsistent. In section \ref{sec:asympt} we derive
an integro-differential equation, which consistently incorporates all
resonances. As seen from our numerical
and asymptotic studies, this nonlocal nonlinear
geometrical optics equation more accurately capture features on
both small and large spatial scales, {\it e.g.} changes in the envelope
due to higher harmonic generation, as well as carrier shock formation. }
\section{Simulations of nonlinear periodic Maxwell}
\label{sec:observations}
In this section we discuss the results of numerical simulations, based
on the algorithms of Appendix \ref{sec:methods}, of the nonlinear and
periodic Maxwell equations \eqref{e:maxwell_scalar}.
\begin{itemize}
\item In section \ref{s:maxwell_solitons} we show that for Cauchy
initial data derived from the classical NLCME soliton, there evolve
spatially localized soliton-like states which persist on long time
scales. We discuss aspects of the large scale (envelope) structure
of such states, which are consistent with the NLCME soliton, as well
as significant deviations.
%
\item In section \ref{s:maxwell_shocks} we show that smoothness breaks
down in finite time. In particular, we observe shock formation on
the fast spatial scale of the carrier wave, while a slowly varying
envelope evolves smoothly.
\end{itemize}
We begin by expressing \eqref{e:maxwell_scalar} as a first order
system:
\begin{equation}
\partial_{t} \begin{pmatrix} n(z)^2 E + \chi E^3 \\ B\end{pmatrix} +
\partial_{z} \begin{pmatrix} -B\\-E\end{pmatrix}=0.
\end{equation}
We introduce the scaling $(E,B,D)^T = \epsilon^{1/2}
(\tilde{E},\tilde{B},\tilde{D})$, and expressing the equations in
terms of the variables: $(\tilde{D}, \tilde{B})$ coordinates.
Dropping tildes, this is
\begin{equation}
\label{e:rescaled_maxwell}
\partial_{t} \begin{pmatrix}D \\ B\end{pmatrix} +
\partial_{z} \begin{pmatrix} -B\\-E(D,z)\end{pmatrix}=0.
\end{equation}
where $E(D,z)$ is the unique real solution of
\begin{equation}
\label{e:rescaled_closure}
D = n(z)^2E + \epsilon \chi E^3
\end{equation}
\subsection{Soliton-like coherent structures}
\label{s:maxwell_solitons}
As is well known \cite{christodoulides-joseph:89,aceves1989sit} NLCME
has spatially localized gap soliton solutions. We use the analytical
expression for the gap soliton to generate Cauchy initial data,
$E(z,0),\ \partial_tE(z,0)$ for \eqref{e:maxwell_scalar} and
numerically simulate the evolution.
Using \eqref{svea} and the leading order approximation for the
magnetic field $B_1^\pm = \mp E_1^\pm$, NLCME soliton data (see
\eqref{eq:NLCME_soliton} in Appendix \ref{s:nlcme_soliton}) can be
seeded into Maxwell using
\begin{subequations}
\begin{align}
E &= \mathcal{E}^+(\epsilon z, \epsilon t)e^{i(z-t)} + \mathcal{E}^-(
\epsilon z, \epsilon t) e^{-i(z+t)} + \mathrm{c.c.},\\
B &= - \mathcal{E}^+(\epsilon z, \epsilon t)e^{i(z-t)} +
\mathcal{E}^+(\epsilon z, \epsilon t)e^{i(z-t)} + \mathrm{c.c.}
\end{align}
\end{subequations}
We obtain $D$ via \eqref{e:rescaled_closure} and evaluate at $t=0$ to
get the initial condition.
For a spatially varying index of refraction, we take
\begin{equation}
\label{e:index}
N(z) = \tfrac{4}{\pi} \cos( 2 z), \quad {\it i.e.}\ N_2 = N_{-2} =\tfrac{2}{\pi},\ \ N_p=0,\ \ |p|\ne2,
\end{equation}
$\epsilon = 0.0625$, and $\chi=1$ ($\Gamma = \tfrac{3}{2}$). The results
of our simulations appear in a - d of Figures
\ref{f:traveling_soliton} and \ref{f:standing_soliton}. While there
is attenuation in amplitude and some dispersive spreading of energy,
the solution remains spatially localized over long time intervals. Not
only is there a persistence of the localization (with the periodic
medium), but also there is good pointwise agreement with the NLCME
approximation; see Figure \ref{f:traveling_soliton_zoom}.
Frames e - h of Figures \ref{f:traveling_soliton} and
\ref{f:standing_soliton} display the corresponding results in the
absence of a periodic structure, {\it i.e.} $N(z) \equiv 0$. The
delocalization, dispersive spreading and attenuation of the wave
amplitude is greatly enhanced. To understand this heuristically, note
that a gap soliton is a localized state whose frequency lies in the
spectral gap of the linearized PDE at the zero solution. A focusing
nonlinearity adds a (self-consistent) potential well, creating a
(nonlinear) defect mode with frequency lying in this spectral gap. If
$N(z)\equiv0$ then the linearization at the zero state has {\it no}
spectral gap. Thus, a oscillating with the gap soliton frequency would
couple to radiation modes and dispersively spread and attenuate.
This mechanism is discussed, for example, in \cite{Soffer-Weinstein:98}.
\begin{figure}
\centering
Simulations with varying refractive index, \eqref{e:index}:
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton1ncos1_print20000/matlabfig0}}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton1ncos1_print20000/matlabfig150}}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton1ncos1_print20000/matlabfig300}}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton1ncos1_print20000/matlabfig400}}
{Simulations with constant refractive index, $N(z) = 0$:}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton1nconst1_print_20000/matlabfig0}}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton1nconst1_print_20000/matlabfig150}}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton1nconst1_print_20000/matlabfig300}}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton1nconst1_print_20000/matlabfig400}}
\caption{Rescaled Maxwell equation, \eqref{e:rescaled_maxwell},
time-evolution for data generated by the NLCME soliton with parameters
$v=.9$ and $\delta = .9$; see \eqref{eq:NLCME_soliton}. The
solutions are computed with 20000 grid points on the domain
$[-500,500]$.
}
\label{f:traveling_soliton}
\end{figure}
\begin{figure}
\centering
{Simulations with varying refractive index, \eqref{e:index}:}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton2ncos1_print_20000/matlabfig4}}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton2ncos1_print_20000/matlabfig196}}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton2ncos1_print_20000/matlabfig296}}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton2ncos1_print_20000/matlabfig396}}
{Simulations with constant refractive index, $N(z) = 0$:}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton2nconst1_print_20000/matlabfig4}}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton2nconst1_print_20000/matlabfig196}}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton2nconst1_print_20000/matlabfig296}}
\subfigure[]{\includegraphics[width=2.2in]{figs/_plots_soliton2nconst1_print_20000/matlabfig396}}
\caption{Solution of rescaled nonlinear periodic Maxwell equation,
\eqref{e:rescaled_maxwell}, for initial data generated by the
NLCME soliton with parameters $v=0$ and $\delta =\pi/2$; see
\eqref{eq:NLCME_soliton}.The solutions are computed with
20000 grid points on the domain $[-500,500]$.
}
\label{f:standing_soliton}
\end{figure}
\begin{figure}
\centering
\subfigure[]{\includegraphics[width=2.35in]{figs/_plots_soliton1ncos1_nlcme_print20000/frame0000fig1}}
\subfigure[]{\includegraphics[width=2.35in]{figs/_plots_soliton1ncos1_nlcme_print20000/frame0150fig1}}
\subfigure[]{\includegraphics[width=2.35in]{figs/_plots_soliton1ncos1_nlcme_print20000/frame0300fig1}}
\subfigure[]{\includegraphics[width=2.35in]{figs/_plots_soliton1ncos1_nlcme_print20000/frame0400fig1}}
\caption{Comparison of the solution appearing in Figure
\ref{f:traveling_soliton} a - d, with the exact NLCME soliton.}
\label{f:traveling_soliton_zoom}
\end{figure}
We note that it is also essential that the data be properly prepared to see a
persistence of localization. For the initial condition
\begin{subequations}\label{e:sech_data}
\begin{align}
D &= 0.5 \cos(z) \mathrm{sech}(\epsilon z),\\
B & = - D,
\end{align}
\end{subequations}
we see in Figure \ref{fig:primitivesech} substantial spreading. This
data mimics the gap soliton's amplitude, slowly varying envelope, and
carrier wave, but is apparently too far outside the basin of
attraction to converge to a localized state. Similar results were
observed with Gaussian wave packet initial conditions.
\begin{figure}
\centering
\subfigure[]{\includegraphics[width=2.35in]{figs/_plots_sech1ncos1_print_20000/frame0000fig1}}
\subfigure[]{\includegraphics[width=2.35in]{figs/_plots_sech1ncos1_print_20000/frame0200fig1}}
\subfigure[]{\includegraphics[width=2.35in]{figs/_plots_sech1ncos1_print_20000/frame0300fig1}}
\subfigure[]{\includegraphics[width=2.35in]{figs/_plots_sech1ncos1_print_20000/frame0400fig1}}
\caption{Solution of rescaled nonlinear periodic Maxwell equation,
\eqref{e:rescaled_maxwell} with periodic refractive index
\eqref{e:index}, for initial data \eqref{e:sech_data} . In
contrast to the NLCME soliton data, the shape of the solution does
not persist. The solution is computed with 20000 grid points on
the domain $[-500,500]$.}
\label{fig:primitivesech}
\end{figure}
\subsection{ Envelope carrier-shock trains}
\label{s:maxwell_shocks}
Although ther the slowly varying NLCME envelope
{\it shape} is robust, for the nonlinear Maxwell time-evolution,
there is evidence of nonlinear steepening and shock formation
on the short (carrier) microstructure spatial scale. Thus, the nearly monochromatic
slowly varying envelope approximation of NLCME is violated.
Figure \ref{f:shock_comparison} displays the time-evolution for
(a) moving and (b) stationary NLCME - gap soliton data.
For each initial condition, the nonlinear Maxwell evolution is simulated
for different grid spacings. As we increase the number of
grid points, sharp features are better resolved by the shock capturing
algorithm. One can also examine the Fourier transform of the output
to see that we obtain an algebraically decaying solution in wave
number, with peaks at the integer wave number values.
In summary, our observations support the emergence of an
{\it envelope carrier-shock train};
persistence of
coherent, slowly varying, wave envelope
and shock formation on the carrier scale.
\begin{figure}
\centering
\subfigure[$v=.9$, $\delta =
.9$]{\includegraphics[width=2.4in]{figs/_plots_soliton1ncos1_shock_compare/frame0040fig1}}
\subfigure[$v=0$, $\delta =
\pi/2$]{\includegraphics[width=2.4in]{figs/_plots_soliton2ncos1_shock_compare/frame0096fig1}}
\caption{Increasing the number of grids points better resolves the
shocks in the carrier wave. For NLCME soliton data with the
indicated $v$ and $\delta$ (see \eqref{eq:NLCME_soliton}), with
index of refraction $N(z)$ given by \eqref{e:index}).}
\label{f:shock_comparison}
\end{figure}
\section{Resonant nonlinear geometrical optics and nonlinear
spatially inhomogeneous Maxwell equations}
\label{sec:asympt}
In this section we
derive a system of equations, which incorporates
all wave-resonances and which our numerical simulations show,
captures the key features of the nonlinear Maxwell time-evolution,
in particular, the presence of robust envelope carrier-shock train solutions.
We derive this system, for general non-homogeneous media, using a nonlinear geometrical optics expansion;
see, for example,
\cite{Hunter:1983vn,Hunter:1986kx,Majda:1984uq}. The equations
obtained are the general integro-differential equations
\eqref{eq:integro_diff_intro}. In the case of a periodic medium,
they reduce to an infinite set of local equations, which we
call the {\it extended nonlinear coupled mode equations}
(xNLCME). If, in xNLCME, we neglect all but principle resonances,
xNLCME
reduces to NLCME.
As we shall see, in our numerical simulations of increasing high
dimensional truncations of xNLCME (section \ref{s:xnlcme_sims}),
this theory appears to accommodate the observed carrier shocks and
large scale coherent structures.
\subsection{Nonlinear geometric optics expansion}
In contrast to the ansatz of Section \ref{sec:nlcme}, we assume the
more general form
\begin{equation}
\label{eq:expansion_ansatz}
\mathbf{u}(z,t) = \mathbf{u}^{(0)}(z,t,Z,T) + \epsilon \mathbf{u}^{(1)}(z,t,Z,T)+\epsilon^2 \mathbf{u}^{(2)}(z,t,Z,T)+\ldots.
\end{equation}
where ${\bf u} = ( E, B)^T$ and $Z = \epsilon z$, $T = \epsilon t$. Inserting
\eqref{eq:expansion_ansatz} into \eqref{e:rescaled_maxwell},
\eqref{e:rescaled_closure}, the first order system
\begin{equation*}
\partial_{t} \begin{pmatrix} n(z)^2E + \epsilon \chi E^3 \\ B\end{pmatrix} +
\partial_{z} \begin{pmatrix} -B\\-E\end{pmatrix}=0.
\end{equation*}
we expand to get,
\begin{equation*}
\begin{split}
\paren{ \partial_{t} +B^{(0)}\partial_{z} }\bold{u}^{(0)}+
\epsilon&\left[\paren{ \partial_{t}+B^{(0)}\partial_{z} }\bold{u}^{(1)} +\paren{ \partial_{T}+B^{(0)}\partial_{Z} }\bold{u}^{(0)}\right.\\
&\left.\quad
+A^{(1)}(z,\mathbf{u})\partial_{t}\bold{u}^{(0)}\right]=\mathcal{O}(\epsilon^2)
\end{split}
\end{equation*}
with matrices
\begin{equation}
B^{(0)} = \begin{pmatrix} 0 & -1 \\ -1 & 0 \end{pmatrix}, \quad
A^{(1)} = \begin{pmatrix} 2 N(z) + 3 \chi E^2 & 0 \\ 0 &
0 \end{pmatrix}
\end{equation}
At $\mathcal{O}(\epsilon^0)$,
\begin{equation}
\paren{\partial_{t} +B^{(0)}\partial_{z} }\bold{u}^{(0)} =0.
\end{equation}
Solving this as the generalized Eigenvalue problem,
\begin{equation*}
\paren{B^{(0)} -\lambda I} \bold{r}=0
\end{equation*}
the solutions are:
\begin{equation}
\lambda_{\pm} = \pm 1,\quad \bold{r}_{\pm} = \begin{pmatrix} 1 \\ \mp 1\end{pmatrix}.
\end{equation}
The corresponding left eigenvectors are
\begin{equation}
\mathbf{l}_{\pm} = \tfrac{1}{2}\begin{pmatrix} 1 & \mp 1 \end{pmatrix}
\end{equation}
With this normalization, $\mathbf{l}_i A^{(0)} \mathbf{r}_j =
\delta_{i,j}$. The leading order fields are then
\begin{subequations}\label{eq:fields0}
\begin{align}
\mathbf{u}^{(0)} &= E^+(\phi_+,Z,T) \mathbf{r}_+ + E^-(\phi_-,Z,T) \mathbf{r}_-,\\
E^{(0)} & = E^+(\phi_+,Z,T) + E^-(\phi_-,Z,T),\\
\phi_\pm& = z\mp t.
\end{align}
\end{subequations}
This expression is much more general than \eqref{svea} used in the
derivation of NLCME.
At $\mathcal{O}(\epsilon)$, the equation is
\begin{equation}
\label{eq:order1}
\begin{split}
\paren{\partial_{t} +B^{(0)}\partial_{z} } \mathbf{u}^{(1)} = - \paren{\partial_{T}
+B^{(0)}\partial_{Z} } \mathbf{u}^{(0)} - A^{(1)}(z,\mathbf{u}^{(0)})\partial_{t}
\mathbf{u}^{(0)}
\end{split}
\end{equation}
If we assume
\begin{equation}
\mathbf{u}^{(1)}(z,t) =m^+(z,t) \mathbf{r}_+ +m^-(z,t) \mathbf{r}_-,
\end{equation}
and substitute into \eqref{eq:order1}, then left multiply by
$\mathbf{l}_+$ and then by $\mathbf{l}_-$, we get the two equations
\begin{align}
-\paren{\partial_{t} m^+ + \partial_{z} m^+}&= \partial_{T} E^+ + \partial_{Z} E^++ \mathbf{l}_+
A^{(1)}(\mathbf{u}^{(0)}) \times\nonumber \\
&\quad \paren{-\partial_{\phi_+ }E^+\bold{r}_+
+\partial_{\phi_-}E^-\bold{r}_- },\label{mplus}\\
&\nonumber\\
-\paren{\partial_{t} m^- - \partial_{z} m^-}&= \partial_{T} E^- - \partial_{Z} E^-+ \mathbf{l}_-
A^{(1)}(\mathbf{u}^{(0)}) \times \nonumber\\
&\quad \paren{-\partial_{\phi_+ }E^+\bold{r}_+
+\partial_{\phi_-}E^-\bold{r}_- }.
\label{mminus}
\end{align}
The last term is the same in both equations,
\begin{equation}
\begin{split}
\bold{l}_\pm &A^{(1)}(\bold{u}^{(0)})\paren{-\partial_{\phi_+
}E^+\bold{r}_+ +\partial_{\phi_-}E^-\bold{r}_- }\\
\quad &= \tfrac{1}{2}\paren{2 N(z) +
3\chi{E^{(0)}}^2 } \paren{- \partial_{\phi_+ }E^+ +\partial_{\phi_-}E^- }\\
\quad &={N(z)}\paren{-\partial_{\phi_+ } E^+ + \partial_{\phi_-} E^-} \\
&\quad\quad + \tfrac{3}{2} \chi\paren{E^+ + E^-}^2 \paren
{-\partial_{\phi_+ } E^+ + \partial_{\phi_-} E^-}.
\end{split}
\end{equation}
Integration of \eqref{mplus} along the characteristic $\partial_t z_+
=1$ from $t=0$ to $t=L$, yields
\begin{equation}
\begin{split}
-&\paren{m^+(z_+(L),L)-m^+(z_+(0),0) } =\\
&\int_{0}^L \partial_{T} E^+(Z,T, z_+(0)) + \partial_{Z} E^+(Z,T,z_+(0))ds\\
&- \int_{0}^L {N(z_+(s))} \partial_{\phi_+ } E^+(Z,T,z_+(0))ds\\
& + \int_{0}^L {N(z_+(s))}\partial_{\phi_- } E^-(Z,T,z_+(s)+s)ds\\
&-\int_0^L \left[\tfrac{3}{2} \chi \paren{
E^+(Z,T,z_+(0))-E^-(Z,T,z_+(s)+s)}^2\right. \\
&\qquad \left. \times\partial_{\phi_+ } E^+(Z,T,z_+(0))\right]ds\\
&+\int_0^L\left[ \tfrac{3}{2} \chi \paren{
E^+(Z,T,z_+(0))-E^-(Z,T,z_+(s)+s)}^2 \right.\\
&\qquad \left. \times\partial_{\phi_-} E^-(Z,T,z_+(s)+s)\right]ds.
\end{split}
\end{equation}
Similarly, integration of \eqref{mminus} along the characteristic
$\partial_t z_- = -1$, yields
\begin{equation}
\begin{split}
-&\paren{m^-(z_-(L),L)-m^-(z_-(0),0) } = \\
&\int_{0}^L \partial_{T} E^-(Z,T, z_+(0)) -\partial_{Z} E^-(Z,T,z_+(0))ds\\
& -\int_{0}^L {N(z_-(s))} \partial_{\phi_+ } E^+(Z,T,z_-(s)-s)ds\\
& + \int_{0}^L {N(z_-(s))}\partial_{\phi_- } E^-(Z,T,z_-(0))ds\\
& -\int_0^L \left[\tfrac{3}{2} \chi \paren{ E^+(Z,T,z_-(s)-s)-E^-(Z,T,z_-(0))}^2\right.\\
&\qquad \left.\times\partial_{\phi_+ } E^+(Z,T,z_-(s)- s)\right]ds\\
& +\int_0^L \left[\tfrac{3}{2} \chi \paren{E^+(Z,T,z_-(s)- s)-E^-(Z,T,z_-(0))}^2 \right.\\
&\qquad \left.\times\partial_{\phi_-} E^-(Z,T,z_-(0))\right]ds.
\end{split}
\end{equation}
Necessary conditions for $m_\pm$ to grow sublinearly in $t$ as $t\to
\infty$ are the solvability conditions:
\begin{subequations}
\begin{gather}
\begin{split}
&\partial_{T} E^+(Z,T, z_+(0)) + \partial_{Z} E^+(Z,T,z_+(0))=\\
&-\lim_{L\to\infty}\frac{1}{L}\int_{0}^L {N(z_+(s))}\partial_{z_+(0) } E^-(Z,T,z_+(s)+s)ds\\
&+ \lim_{L\to\infty}\frac{1}{L}\int_0^L \left[\tfrac{3}{2}
\chi \paren{ E^+(Z,T,z_+(0))
+E^-(Z,T,z_+(s)+s)}^2\right.\\
&\qquad \left. \times \partial_{z_+(0) }
E^+(Z,T,z_+(0))\right]ds,
\end{split}
\\ \\
\begin{split}
&\partial_{T} E^-(Z,T, z_-(0)) -\partial_{Z} E^-(Z,T,z_-(0))=\\
&\lim_{L\to\infty}\frac{1}{L}\int_{0}^t {N(z_-(s))} \partial_{z_-(0) } E^+(Z,T,z_-(s)-s)ds\\
&- \lim_{L\to\infty}\frac{1}{L}\int_0^L \left[\tfrac{3}{2}
\chi \paren{
E^+(Z,T,z_-(s)-s)-E^-(Z,T,z_-(0))}^2\right.\\
&\qquad\left.\partial_{z_-(0) } E^-(Z,T,z_-(0))\right]ds.
\end{split}
\end{gather}
\end{subequations}
Given $(z,t)$, $z_+(0) = z- t = \phi_+$ and $z_-(0) = z+ t =\phi_-$.
Defining
\begin{equation}
\label{eq:mean_infty}
\mean{f} = \lim_{L\to \infty} \frac{1}{L}\int_{0}^L f(s)ds
\end{equation}
the equations may be compactly expressed as:
\begin{subequations}
\label{e:E_full}
\begin{gather}
\label{eq:Ep_full}
\begin{split}
&\partial_{T} E^+ + \partial_{Z} E^+=-\mean{N( \phi_+ + s) \partial_\phi E^-(\phi_+ + 2 s}_s\\
&\quad +\tfrac{3}{2} \chi\paren{\paren{E^+}^2 + 2 E^+ \mean{E^-}
+ \mean{\paren{E^-}^2}}\partial_\phi E^+,
\end{split}
\\
\label{eq:Em_full}
\begin{split}
&\partial_{T} E^- - \partial_{Z} E^-=\mean{N(\phi_- - s)\partial_\phi E^+(\phi_- - 2 s)}_s\\
&\quad - \tfrac{3}{2}\chi\paren{\paren{E^-}^2 + 2 E^- \mean{E^+}
+ \mean{\paren{E^+}^2}} \partial_\phi E^-.
\end{split}
\end{gather}
\end{subequations}
It is important to recognize that the arguments of the fields in
\eqref{eq:Ep_full} are $\phi_+ = z - t$, $Z$, and $T$, while in
\eqref{eq:Em_full}, they are $\phi_- = z + t$, $Z$, and $T$. As in
our derivation of NLCME in Section \ref{sec:nlcme}, $\Gamma \equiv
\tfrac{3}{2}\chi$. With this notation, \eqref{e:E_full} can be
rewritten, after an integration by parts, in conservation law form,
\begin{subequations}\label{e:E_con}
\begin{gather}
\label{eq:Ep_con}
\begin{split}
&\partial_{T} E^+ + \partial_{Z} E^+= {\partial_\f} \mean{n_1( \phi_+ + s) E^-(\phi_+ + 2
s}_s\\
&\quad +\Gamma {\partial_\f}\bracket{\frac{1}{3}\paren{E^+}^3
+ \paren{E^+}^2 \mean{E^-} + E^+\mean{\paren{E^-}^2}},
\end{split}
\\
\label{eq:Em_con}
\begin{split}
&\partial_{T} E^- -\partial_{Z} E^-=-{\partial_\f}\mean{n_1( \phi_- - s)E^+(
\phi_- - 2 s) }_s\\
&\quad -\Gamma {\partial_\f}\bracket{\mean{\paren{E^+}^2
}E^-+\mean{E^+}\paren{E^-}^2+\frac{1}{3}\paren{E^-}^3}.
\end{split}
\end{gather}
\end{subequations}
Equations \eqref{e:E_con} corresponds to the integro-differential
equations of the introduction, if we omit the $\mean{E^\pm}$ terms.
Since $\mean{E^\pm}$ is time-invariant (see section \ref{s:hamiltonian_structure})
by choosing initial conditions for which $\mean{E^\pm}(T=0)=0$,
these terms can be dropped from \eqref{e:E_con}.
Finally, note that \eqref{e:E_full} are applicable to a {\it
general heterogeneous dielectric material} with the appropriate
scalings.
\subsection{Periodic Media and xNLCME}
We now specialize to the periodic case. Assume now that $N(z+2\pi)=N(z)$. Then
\eqref{e:E_con} is invariant under the discrete translation: $\phi\mapsto\phi+2\pi$, {\it i.e.}
\begin{subequations}
\begin{align}
E^+(\phi, Z,T) &\mapsto E^+(\phi + 2\pi, Z,T)\\
E^-(\phi, Z,T) &\mapsto E^-(\phi + 2\pi, Z,T)\ .
\end{align}
\end{subequations}
Thus, under the assumption of existence and uniqueness of solutions to
\eqref{e:E_con}, if the initial data are $2\pi$ in the $\phi$
argument, then the solutions remain $2 \pi $ periodic in $\phi$. In
the periodic setting, the averaging operator, \eqref{eq:mean_infty},
simplifies to
\[
\mean{f} = \frac{1}{2\pi}\int_{0}^{2\pi} f(s)ds.
\]
We now expand $N(z)$ and $E^\pm$ in Fourier series,
\begin{align}
N(z) &= \sum_{p \in \mathbb{Z}} N_p e^{\mathrm{i} p
z},\\\label{e:efield_fourier} E^\pm(\phi ,Z,T) &= \sum_{p}
E^{\pm}_p(Z,T) e^{\pm\mathrm{i} p \phi},
\end{align}
where $\bar{N}_p= N_{-p}$ and $\bar{E}^\pm_p= E^\pm_{-p}$ since $N$
and $E^\pm$ are real valued. In this case, the system of Fourier
coefficients $\{E^\pm_p(Z,T): p\in \mathbb{Z}\}$ satisfy the
infinite system of {\it extended nonlinear coupled mode equations}
(xNLCME):
\begin{subequations}\label{e:E_mode}
\begin{gather}
\label{eq:Ep_mode}
\begin{split}
\partial_{T} E^+_p + \partial_{Z} E^+_p &= \mathrm{i}p N_{2p}{E^-_p} +
\mathrm{i}p\frac{\Gamma}{3}\left[\sum_{q,r} E^+_q
E^+_r E^+_{p-q-r}\right.\\
&\quad \left. + 3 E^-_0 \sum_q E^+_q E^+_{p-q} +3\paren{\sum_q
\abs{E^-_q}^2} E^+_p\right],
\end{split}
\\
\label{eq:Em_mode}
\begin{split}
\partial_{T} E^-_p - \partial_{Z} E^-_p &= \mathrm{i}p
\bar{N}_{2p}{E^+_p}+\mathrm{i}p\frac{\Gamma}{3} \left[\sum_{q,r} E^-_q
E^-_r E^-_{p-q-r}\right.\\
&\quad \left. + 3 E^+_0 \sum_q E^-_q E^-_{p-q}+3\paren{\sum_q
\abs{E^+_q}^2} E^-_p\right].
\end{split}
\end{gather}
\end{subequations}
\subsection{Conservation Laws and Hamiltonian Structure}
\label{s:hamiltonian_structure}
Equation \eqref{e:E_con}, and alternatively \eqref{e:E_mode}, have two
conservation laws:
\begin{prop}
Assume that $E^\pm$ is a sufficiently smooth and sufficiently $Z-$ decaying solution of \eqref{e:E_con}
and that $\{E_p(Z,T)\}_{p\in\mathbb{Z}}$ is the corresponding solution of xNLCME. Then,
\begin{subequations}
\begin{gather}
\frac{d}{dT}\ \int \mean{E^+(\cdot,T)}\ dZ = \ \frac{d}{dT}\ \int E_0^+(\cdot,T)\ dZ\ =\ 0\\
\frac{d}{dT}\ \int \mean{E^+(\cdot,T)}\ dZ = \frac{d}{dT}\ \int E_0^-(\cdot,T)\ dZ\ =\ 0\\
\frac{d}{dT}\ \int \mean{(E^+)^2(\cdot,T)}\ +\ \mean{(E^-)^2(\cdot,T)}\ dZ\\
=\
\frac{d}{dT}\ \sum_p \int \abs{E^+_p(\cdot,T)}^2 +
\abs{E^-_p(\cdot,T)}^2\ dZ\ =\ 0\ .
\end{gather}
\end{subequations}
\end{prop}
\begin{proof}
Setting $p=0$ in \eqref{e:E_mode},
\begin{align*}
\partial_{T} E^+_0 + \partial_{Z} E^+_0&=0,\\
\partial_{T} E^-_0 - \partial_{Z} E^-_0&=0.
\end{align*}
Integrating in $Z$ establishes the first two conservation laws in
terms of the Fourier modes. Integrating \eqref{e:efield_fourier} in
$\phi$ over $[0, 2\pi)$ relates $\mean{E^\pm}$ to $E_0^\pm$.
\bigskip
Multiplying \eqref{eq:Ep_mode} by $\bar{E}^+_p$, summing over $p$,
and adding its complex conjugate,
\begin{equation*}
\begin{split}
\sum \partial_T \abs{E_p^+}^2 + \partial_Z \abs{E_p^+}^2 &= \sum_p
ip N_{2p} E_p^- \bar{E}_p^+ +\ \frac{\Gamma}{3}\ \sum_p\ ip \left[\sum_{q,r}
E^+_q
E^+_r E^+_{-p}E^+_{p-q-r}\right.\\
&\quad + 3 E^-_0 \sum_q E^+_q E^+_{p-q}E^+_{-p} \\
&\quad \left.+3\paren{\sum_q
\abs{E^-_q}^2} \ \abs{E^+_p}^2\ \right] + \mathrm{c.c.}
\end{split}
\end{equation*}
The quartic terms will all vanish. Consider the first quartic term,
and note that
\begin{equation}
\begin{split}
\sum_{p,q,r} p E^+_qE^+_r E^+_{-p}E^+_{p-q-r} &= \sum_{k_1 + k_2 + k_3
+ k_4 = 0} k_1 E^+_{k_1}E^+_{k_2} E^+_{k_3}E^+_{k_4}\\
&=\sum_{k_1 + k_2 + k_3
+ k_4 = 0} k_2 E^+_{k_1}E^+_{k_2} E^+_{k_3}E^+_{k_4}\\
&=\sum_{k_1 + k_2 + k_3
+ k_4 = 0} k_3 E^+_{k_1}E^+_{k_2} E^+_{k_3}E^+_{k_4}\\
&=\sum_{k_1 + k_2 + k_3
+ k_4 = 0} k_4 E^+_{k_1}E^+_{k_2} E^+_{k_3}E^+_{k_4}
\end{split}
\end{equation}
Hence,
\begin{equation}
\begin{split}
&\sum_{p,q,r} p E^+_qE^+_r E^+_{-p}E^+_{p-q-r} \\
&\quad= \frac{1}{4}\sum_{k_1 + k_2 + k_3
+ k_4 = 0} ({k_1 + k_2 + k_3
+ k_4 }) E^+_{k_1}E^+_{k_2} E^+_{k_3}E^+_{k_4}=0
\end{split}
\end{equation}
The second quartic term vanishes using a similar analysis. The last
quartic term,
\begin{equation}
\sum_p p \paren{\sum_q \abs{E^-_q}^2}\abs{E^+_p}^2
\end{equation}
will vanish because the $p$ and $-p$ terms will cancel one another.
Similar analysis holds for \eqref{eq:Em_mode}, leaving
us with the two equations
\begin{align}
\sum \partial_T \abs{E_p^+}^2 + \partial_Z \abs{E_p^+}^2 &= \sum
ip N_{2p}
E_p^- \bar{E}_p^+ - ip \bar{N}_{2p} \bar{E}_p^- E_p^+,\\
\sum \partial_T \abs{E_p^-}^2 - \partial_Z \abs{E_p^-}^2 &= \sum
ip \bar{N}_{2p} \bar{E}_p^- {E}_p^+ - ip {N}_{2p} {E}_p^-
\bar{E}_p^+.
\end{align}
Summing these two, and integrating in $Z$ gives the $L^2$
conservation law.
\end{proof} {\bf To simplify our analysis we assume $E^\pm_0$ are
initially zero from here on}. The equations reduce to
\begin{subequations}\label{e:E_con_zeromean}
\begin{gather}
\label{eq:Ep_con1}
\begin{split}
\partial_{T} E^+ + \partial_{Z} E^+&= {\partial_\f} \mean{N( \phi_+ + s) E^-(\phi_+ + 2s}_s\\
&\quad +\Gamma {\partial_\f}\bracket{\frac{1}{3}\paren{E^+}^3 +
E^+\mean{\paren{E^-}^2}},
\end{split}
\\
\label{eq:Em_con1}
\begin{split}
\partial_{T} E^- - \partial_{Z} E^-&=-{\partial_\f}\mean{N( \phi_- - s)E^+( \phi_- - 2 s) }_s\\
&\quad -\Gamma {\partial_\f}\bracket{\mean{\paren{E^+}^2
}E^-+\frac{1}{3}\paren{E^-}^3}.
\end{split}
\end{gather}
\end{subequations}
and
\begin{subequations}\label{e:E_mode_zeromean}
\begin{gather}
\label{eq:Ep_mode1}
\begin{split}
\partial_{T} E^+_p + \partial_{Z} E^+_p = \mathrm{i}p N_{2p}{E^-_p} +
\mathrm{i}p\frac{\Gamma}{3}&\left[\sumE^+_q
E^+_r E^+_{p-q-r} \right.\\
&\quad\left.+3\paren{\sum \abs{E^-_q}^2} E^+_p \right],
\end{split}
\\
\label{eq:Em_mode1}
\begin{split}
\partial_{T} E^-_p - \partial_{Z} E^-_p = \mathrm{i}p \bar{N}_{2p}{E^+_p}
+\mathrm{i}p\frac{\Gamma}{3} &\left[\sum E^-_q
E^-_r E^-_{p-q-r} \right.\\
&\quad \left.+3\paren{\sum \abs{E^+_q}^2} E^-_p \right].
\end{split}
\end{gather}
\end{subequations}
These are equations \eqref{eq:integro_diff_intro} and
\eqref{e:mode_intro} from the introduction. Truncating
\eqref{e:E_mode_zeromean} to just mode $E^\pm_{\pm 1}$, recovers the
NLCME, subject to the identification of $\mathcal{E}^\pm $ with
$E^\pm_{1}$.
Another time-invariant functional is a consequence of the
Hamiltonian structure given in the following result, which is straightforward to verify:
\begin{prop}
The system \eqref{e:E_mode_zeromean} is a Hamiltonian system:
\begin{equation}
\partial_{T} E^+_p = - \mathrm{i} p \frac{\delta H}{\delta \overline{E}^+_p},
\quad \partial_{T} E^-_p = - \mathrm{i} p \frac{\delta H}{\delta \overline{E}^-_p},
\end{equation}
where with time-invariant Hamiltonian
\begin{equation}
H[E^\pm,\overline{E^\pm}] = \int\ \mathcal{H}(\cdot,T)
\nonumber\end{equation}
and Hamiltonian density
\begin{equation}
\begin{split}
\mathcal{H}(Z,T) &= \frac{\mathrm{i}}{2} \sum_{p_1 =1}^\infty
\frac{1}{p_1}\paren{E^+_{p_1} \partial_{Z} {\bar{E}^+}_{p_1} -
E^-_{p_1} \partial_{Z} {\bar{E}^-}_{p_1}}- \sum_{p_1=1}^\infty N_{2 p_1} \bar{E}^+_{p_1}{E^-_{p_1}}\\
&\quad- \frac{\Gamma}{3}
\frac{1}{2}\frac{1}{4}\sum_{p_1+p_2+p_3+p_4=0}
E^+_{p_1}E^+_{p_2}E^+_{p_3}E^+_{p_4}+E^-_{p_1}E^-_{p_2}E^-_{p_3}E^-_{p_4}\\
&\quad - \Gamma
\frac{1}{2}\frac{1}{2}\paren{\sum_{p_1}\abs{E^+_{p_1}}^2}\paren{\sum_{p_1}\abs{E^-_{p_1}}^2}+
\mathrm{c.c.}\ \ \ .
\end{split}
\end{equation}
\end{prop}
\section{Simulations of the Truncated xNLCME}
\label{s:xnlcme_sims}
In this section we simulate truncations of the infinite dimensional
xNLCME system, performed pseudo-spectrally with fourth order
Runge-Kutta time stepping. These simulations suggest that
\begin{itemize}
\item xNLCME has its own localized soliton-like structures which
better capture the dynamics of the nonlinear periodic Maxwell
equation for our class of initial conditions than NLCME and
\item xNLCME has singular solutions, $\{E^\pm_p(Z,T)\}$ with a cascade of energy to higher wave numbers, $p$.
The physical electric field
\begin{align}
E(z,t)\ &\ \approx \epsilon^{1\over2}\left( E^+(z-t,,\epsilon z,\epsilon t) + E^-(z+t,,\epsilon z,\epsilon t)\ \right)\nonumber\\
&\ =\ \epsilon^{1\over2}\ \sum_{p\in\mathbb{Z}\setminus{0}}\ \left(\ E^+_p(Z,T)e^{ip(Z- T)/\epsilon} +
E^-_p(Z,T)e^{-ip(Z+ T)/\epsilon}\ \right)\nonumber\\
&\ \ \ \ \ \ \ \ \ + \mathrm{c.c.}\ \nonumber
\end{align}
develops a carrier-shock train structure.
\end{itemize}
As we saw in Section \ref{s:maxwell_solitons}, particularly Figure
\ref{f:standing_soliton}, though the NLCME soliton data appeared
robust, there was some escape of energy. This can be accounted for in
the xNLCME through the inclusion of additional modes.
Starting with the same initial conditions, we simulate the NLCME
soliton of $E_{\pm 1 }^\pm$ with soliton parameters $v=0$ and $\delta
= \tfrac{\pi}{2}$, and material parameters
\[
\Gamma = 1, \quad N_{\pm 2}=\tfrac{2}{\pi}, \quad N_{j \neq \pm 2} = 0
\]
in \eqref{e:E_mode_zeromean} resolving only a finite number of
harmonics. The primitive electric field is reconstructed from these
simulations as
\begin{equation}
E = \sum_{p=-p_{\max}}^{p_{\max}} E^+_p(Z,T)e^{ip(Z- T)/\epsilon} +
E^-_p(Z,T)e^{-ip(Z+ T)/\epsilon} + \mathrm{c.c.}
\end{equation}
$E$ is plotted in Figures \ref{f:standing_soliton3}, and
\ref{f:standing_soliton15}, which resolve odd modes up to 3 and 15,
respectively. Comparing with Figure \ref{f:standing_soliton}, we
infer that the two smaller pulses symmetrically expelled from the main
wave were transferred into $E^\pm_{\pm 3}$, since these clearly appear
in Figure \ref{f:standing_soliton3}. This addresses the {\it
macroscopic} discrepancy between NLCME and Maxwell.
\begin{figure}
\centering
\subfigure{\includegraphics[width=2.3in]{figs/plots_print_standing_wave_M16_N4096_Zmax32/fig2}}
\subfigure{\includegraphics[width=2.3in]{figs/plots_print_standing_wave_M16_N4096_Zmax32/fig50}}
\subfigure{\includegraphics[width=2.3in]{figs/plots_print_standing_wave_M16_N4096_Zmax32/fig75}}
\subfigure{\includegraphics[width=2.3in]{figs/plots_print_standing_wave_M16_N4096_Zmax32/fig100}}
\caption{Evolution of an NLCME soliton in the xNLCME, resolving
odd modes $\abs{p}\leq 4$. Computed with $4096$ grid points in
the $Z$ coordinate. Compare with Figure
\ref{f:standing_soliton}.}
\label{f:standing_soliton3}
\end{figure}
\begin{figure}
\centering
\subfigure{\includegraphics[width=2.3in]{figs/plots_print_standing_wave_M64_N16384_Zmax32/fig2}}
\subfigure{\includegraphics[width=2.3in]{figs/plots_print_standing_wave_M64_N16384_Zmax32/fig50}}
\subfigure{\includegraphics[width=2.3in]{figs/plots_print_standing_wave_M64_N16384_Zmax32/fig75}}
\subfigure{\includegraphics[width=2.3in]{figs/plots_print_standing_wave_M64_N16384_Zmax32/fig100}}
\caption{Evolution of an NLCME solition in the xNLCME, resolving
odd modes $\abs{p}\leq 16$. Computed with $16384$ grid points
in the $Z$ coordinate. Compare with Figure
\ref{f:standing_soliton}.}
\label{f:standing_soliton15}
\end{figure}
Including the additional modes also suggests shock formation by
re-examining Figure \ref{f:shock_comparison}. The sharper, shock
like features, can only be resolved by the inclusion of the the
higher harmonics. The contrast between different truncations is
shown in Figure \ref{f:shock_harmonics}. Indeed, we see the Gibbs
phenomenon that would be expected from taking a truncated Fourier
representation of a discontinuous function.
\begin{figure}
\centering \subfigure[Resolves odd harmonics $\abs{p}\leq
2$]{\includegraphics[width=2.35in]{figs/plots_print_standing_wave_M8_N4096_Zmax32/shock_fig25}}
\subfigure[Resolves odd harmonics $\abs{p}\leq
4$]{\includegraphics[width=2.35in]{figs/plots_print_standing_wave_M16_N4096_Zmax32/shock_fig25}}
\subfigure[Resolves odd harmonics $\abs{p}\leq
8$]{\includegraphics[width=2.35in]{figs/plots_print_standing_wave_M32_N8192_Zmax32/shock_fig25}}
\subfigure[Resolves odd harmonics $\abs{p}\leq
16$]{\includegraphics[width=2.35in]{figs/plots_print_standing_wave_M64_N16384_Zmax32/shock_fig25}}
\caption{Comparison of the features that develop on the scale of the
medium in different truncations of the equations. Including
additional harmonics better captures the shocks seen in Figure
\ref{f:shock_comparison}. }
\label{f:shock_harmonics}
\end{figure}
Despite this, NLCME still gets certain leading order effects correct,
such as the main structure in the Maxwell simulations. The robustness
of NLCME can also be seen by exploring how energy is partitioned
amongst the harmonics. Let
\begin{equation}
\label{e:energy_density}
e_p \equiv \int \paren{\abs{E^+_p}^2 + \abs{E^+_{-p}}^2+\abs{E^-_p}^2
+ \abs{E^-_{-p}}^2}dZ, \quad p = 1,3,\ldots p_{\max}.
\end{equation}
This is the energy associated with mode $p$. Their sum is
conserved. Plotting this for the above simulations in Figure
\ref{f:eng_dist}, we see that most of the energy remains in mode one,
some migrates into mode three, and less in the subsequent modes.
\begin{figure}
\centering \subfigure[Resolves odd harmonics $\abs{p}\leq
4$]{\includegraphics[width=2.35in]{figs/plots_print_standing_wave_M16_N4096_Zmax32/eng_dist_2}}
\subfigure[Resolves odd harmonics $\abs{p}\leq
8$]{\includegraphics[width=2.35in]{figs/plots_print_standing_wave_M32_N8192_Zmax32/eng_dist_4}}
\subfigure[Resolves odd harmonics $\abs{p}\leq
16$]{\includegraphics[width=2.35in]{figs/plots_print_standing_wave_M64_N16384_Zmax32/eng_dist_8}}
\caption{Energy distribution, \eqref{e:energy_density}, for
truncated xNLCME simulations with different numbers of harmonics.
In all cases, the energy initially residing in mode one tends to
stay there.}
\label{f:eng_dist}
\end{figure}
\section{Summary and discussion}
\label{sec:discussion}
We first numerically simulated the one-dimensional nonlinear Maxwell
equations in the regime of weak nonlinearity, low contrast periodic
structure (weak dispersion) with wave-packet data satisfying a {\it
Bragg resonance condition}, {\it i.e.} carrier wavelength equal to
twice the medium periodicity. We observe strong evidence of the
emergence of a coherent structure evolving as slowly varying envelope
with a {\it carrier-shock train}. This violates the nearly-monochromatic
assumption underlying the classical nonlinear coupled mode equations.
We propose our nonlocal integro-differential equations governing
coupled forward and backward waves, derived via a nonlinear
geometrical optics expansion, as the physically correct,
mathematically consistent description of waves governed by nonlinear
Maxwell in a periodic structure with negligible chromatic (nonlocal in
time) dispersion. These equations are equivalent to an infinite
dimensional system of couple first order PDEs, the {\it extended
coupled mode system} (xNLCME). The electric field, $E$, obtained
from numerical solution of successively higher truncations of xNLCME
converges toward the envelope carrier-shock trains observed in direct
simulations of the nonlinear Maxwell equations.
Finally we mention that our methods could be applied to study the long time evolution
of wave-packet type initial conditions for the problem of quadratically
nonlinear elastic media,
consider in \cite{leveque2002fvm,leveque2003swl,Ketcheson:2009fk}
We obtain nonlocal equations of resonant
nonlinear geometrical optics (or equivalently an infinite family of nonlinear coupled mode equations),
governing interacting forward and backward propagating waves \cite{Simpson:2010uq}.
A difference between the quadratic and cubic case is that the smallest
truncated system that retains nonlinear interactions contains four
modes, $p=\pm 1, \pm 2$. Nonlinear effects occur through second harmonic generation,
a process well-known in nonlinear optics.
\noindent {\bf Open problems and conjectures:} As our simulations
show, there is agreement between finite mode truncations of the
integro-differential equations and the primitive Maxwell system.
Assessing, and proving the time of validity of this approximation is
one upon problem.
Following up on assessing the time of validity, there is also the
question of the time of existence and the well-posedness of the
equations. We expect that solutions of xNLCME for initial data having
a finite number of nonzero mode amplitudes, {\it e.g.} NLCME gap
soliton data, will give rise to solutions of xNLCME that develop
singularities in finite time. The nature of this blowup is expected
to occur via a cascade to high mode amplitudes (higher index, $p$),
corresponding to modes necessary to resolve the carrier shock
structure in the small scale. As we mentioned in the discussion,
there is clearly singularity formation when the heterogeneity is
turned off ($N=0$), and either $E^+$ or $E^-$ is initially zero. It
is an open problem as to whether this particular mechanism for
singularity formation will persist when coupling is restored.
As pointed out in the introduction, the success in modeling
experiments with NLCME suggests that, although there is such a (weakly
turbulent) cascade, it is only a small part of the optical power that
is transferred to high wavenumbers and that this energy contributes
mainly to resolving the small-scale shocks. To explore this, one needs
to simulate the xNLCME equations with many more harmonics. Plotting
the Fourier transform (in the $Z$ coordinate) of the simulations in
Section \ref{s:xnlcme_sims} in figure \ref{f:fourier_dist}, we see
that the spectral support grows as we increase the number of resolved
envelopes (the $E_p^\pm$'s). A related question is whether or not the
primitive Maxwell system, the xNLCME system, or one of its
truncations possess genuine coherent structures. In
\cite{Tasgal:2005p6335}, the authors found such solutions for a first
and third harmonic system. This shall be further
explored in the forthcoming publication, \cite{Simpson:fk}.
\begin{figure}
\centering \subfigure[Resolves odd harmonics $\abs{p}\leq
2$]{\includegraphics[width=2.35in]{figs/plots_print_standing_wave_M8_N4096_Zmax32/fourier_fig101}}
\subfigure[Resolves odd harmonics $\abs{p}\leq
4$]{\includegraphics[width=2.35in]{figs/plots_print_standing_wave_M16_N4096_Zmax32/fourier_fig101}}
\subfigure[Resolves odd harmonics $\abs{p}\leq
8$]{\includegraphics[width=2.35in]{figs/plots_print_standing_wave_M32_N8192_Zmax32/fourier_fig101}}
\subfigure[Resolves odd harmonics $\abs{p}\leq
16$]{\includegraphics[width=2.35in]{figs/plots_print_standing_wave_M64_N16384_Zmax32/fourier_fig101}}
\caption{Fourier transforms of the solutions to truncations of the
xNLCME equations. Increasing the number of envelopes expands the
support in Fourier space.}
\label{f:fourier_dist}
\end{figure}
Finally, our computations in Section \ref{sec:observations} invoked of
a gas-dynamics entropy condition. Such a condition is necessary to
use finite volume methods. Although thermodynamically consistent,
we do not know whether this is the correct regularizing mechanism of electrodynamics.
\bigskip\bigskip
\bibliographystyle{abbrv}
|
{
"timestamp": "2010-11-19T02:00:51",
"yymm": "1009",
"arxiv_id": "1009.3675",
"language": "en",
"url": "https://arxiv.org/abs/1009.3675"
}
|
\section{Introduction}
Magnetohydrodnamic turbulence in natural objects is often subject to global rotation or applied magnetic field, or both.
In the Earth's core the turbulence occurs under the fast rotation of the planet and is embedded in the dipolar magnetic field
produced by dynamo action. Such double effect is currently studied in an experiment with liquid sodium \cite{Schmitt08}.
Waves of different types have been measured that might be attributed to either Alfv\'en or Rossby waves or a combination of both.
The frequency spectra show a series of bumps, attributed to wave frequencies, in addition to piecewise slopes.
A proper understanding of such rotating MHD-turbulence would require a non-isotropic formalism.
Several ones
have been developed for fast rotation \cite{Galtier03,Schaeffer06,Bellet06}. Phenomenological approaches relying on three-wave \cite{Galtier05} or four-wave \cite{Goldreich97} resonant interactions have been developed for an applied field and documented numerically \cite{Bigot08}.
In the present paper we come back to the Iroshnikov \cite{Iroshnikov63} and Kraichnan \cite{Kraichnan65} phenomenology for isotropic MHD turbulence. They argue that the destruction of phase coherence by Alfv\'en waves traveling in opposite directions introduces a new time-scale $\tau_A$. It might control the energy transfer, provided it is shorter than the eddy turn-over time-scale $\tau_K$.
Applying the same idea, Zhou \cite{Zhou95} suggests that due to global rotation the kinetic energy spectrum is affected through phase scrambling, leading to a third time-scale $\tau_{\Omega}$ associated to the rotation frequency. The generalization to both global rotation and applied magnetic field is therefore straightforward (see section \ref{Phenomenology}), the energy transfers being controlled by the shortest time-scale between $\tau_K$, $\tau_A$ and $\tau_{\Omega}$.
An advantage of assuming isotropy is that it can be tested against simulations with shell models.
Shell models are toy-models that mimic the original Navier-Stokes and induction equations projected in Fourier space, within shells which are logarithmically spaced.
There are only two complex variables per shell, one corresponding to the velocity, the other to the magnetic field
\cite{Frick98,Stepanov06}.
Depending on the model, the energy transfers can be considered as local or not \cite{Plunian07}. Such models
allow for simulations at realistically low viscosity $\nu$ and magnetic Prandtl number $\Pm=\nu/\eta$ \cite{Stepanov08}, where $\eta$ is the magnetic diffusivity.
The time dependency of the solutions is strongly chaotic, eventually leading to intermittency. Therefore, though all geometrical details of velocity and magnetic fields are lost, shell models give relevant informations on spectral quantities like energies, helicities, energy transfers, etc. In section \ref{shell model} we introduce such a shell model of rotating MHD turbulence, taking care to keep the terms corresponding to rotation and applied magnetic field as simple as possible. For $\Pm\le 1$ we calculate the spectra for different values of rotation $\Omega$ and applied field $V_A$.
We also calculate the ratio of the joule dissipation over the viscous dissipation, which cannot be estimated from scaling laws.
\section{Phenomenology}
\label{Phenomenology}
\subsection{Time scales}
Following \cite{Kraichnan65} (see also \cite{Zhou95} and \cite{Matthaeus89}), we assume that for
homogeneous isotropic statistically steady turbulence the decay of triple correlations,
occurring in a time scale $\tau_3(k)$, is responsible for the turbulent spectral transfer
$\varepsilon$ from wavenumbers lower than $k$ to higher wavenumbers.
This implies $\tau_3(k) \sim \varepsilon$. Assuming in addition that $\varepsilon$
depends only on the wave number $k$ and the kinetic energy spectral density $E(k)$,
a simple dimensional analysis
leads to
\begin{equation}
\varepsilon \sim \tau_3(k) E^2(k) k^4.
\end{equation}
The kinetic energy spectral density is defined as $E(k)=k^{-1}u^2(k)$ where $u(k)$ is the
characteristic velocity of eddies at scale $k$.\\
In absence of applied magnetic field and rotation, the time scale for $\tau_3(k)$ is the
eddy turn-over time
\begin{equation}
\tau_{K}(k)=\left[ku(k)\right]^{-1},
\end{equation}
leading to the Kolmogorov turbulence energy spectrum $E(k) \sim \varepsilon^{2/3}
k^{-5/3}$.\\
For fully developed MHD turbulence at $\Pm=1$ the same Kolmogorov spectrum
is assumed for both kinetic and magnetic energy provided that the system is much above the onset for dynamo action \cite{Stepanov06}. In that case
$E(k)$ denotes either the kinetic or magnetic energy spectral density.
In presence of an applied magnetic field $\bB_0$ an other
possible time scale for $\tau_3(k)$ is the Alfv\'en time scale
\begin{equation}
\tau_A (k)=\left[kV_A\right]^{-1},
\end{equation}
leading to the Alfv\'en turbulence energy spectrum
$E(k) \sim V_A^{1/2} \varepsilon^{1/2} k^{-3/2}$.\\
Finally for rotating turbulence caused by uniform rotation $\Omega$ a third possible
time scale for $\tau_3(k)$ is the rotating frequency
\begin{equation}
\tau_{\Omega}=\Omega^{-1},
\end{equation}
leading to the rotating turbulence energy spectrum
$ E(k) \sim \Omega^{1/2} \varepsilon^{1/2} k^{-2}$.
The value of $\tau_3(k)$ is naturally defined by
\begin{equation}
\tau_3(k)=\min\left\{\tau_{K}(k),\tau_A (k), \tau_{\Omega} \right\}.
\end{equation}
It corresponds to the fastest way to transfer energy to smaller scales, between non-linear eddy cascade,
Alf\'en waves interactions and phase scrambling due to rotation.
In addition we define the magnetic dissipation time scale by
\begin{equation}
\tau_{\eta}(k)=(k^2 \eta)^{-1}.
\end{equation}
The dissipation range corresponds to $k\ge k_{\eta}$
with $k_{\eta}$ defined
by $\tau_3(k_{\eta})=\tau_{\eta}(k_{\eta})$.
Therefore at each scale $k^{-1}$, we
have to compare the four time scales $\tau_{K}(k),\tau_A (k), \tau_{\Omega}$ and
$\tau_{\eta}(k)$ to figure out what kind of turbulence occurs.
\subsection{Spectra for $\Pm=1$}
At $k\ll1$, $\tau_{\Omega} < \min\left\{\tau_{K}(k), \tau_A (k), \tau_{\eta} (k) \right\}$
implying that $\tau_3(k)=\tau_{\Omega}$, unless $\Omega=0$.
This corresponds to a rotating turbulence with $E(k)=\Omega^{1/2}\varepsilon^{1/2}
k^{-2}$.
For larger $k$, $\tau_K(k), \tau_A(k)$ and $\tau_{\eta}(k)$ decrease while $\tau_{\Omega}$
stays constant. Therefore, provided that the dissipation is not too strong, a first
transition occurs at a scale for which
$\tau_{\Omega}=\min\left\{\tau_K(k),\tau_A(k)\right\}$. This scale is, either
(i) $k_1=(\Omega^3/\varepsilon)^{1/2}$ if $\varepsilon \ge \Omega V_A^2$, or
(ii) $k_1=\Omega/V_A$ if $\varepsilon \le \Omega V_A^2$.
This transition leads to, either (i) a Kolmogorov $E(k)=\varepsilon^{2/3}k^{-5/3}$, or
(ii) an Alfv\'en $E(k)=V_A^{1/2}\varepsilon^{1/2}k^{-3/2}$ turbulence. This transition
does not occur if the dissipation overcomes the Kolmogorov and Alfv\'en turbulence, namely
if (i) $\eta \ge \varepsilon / \Omega^2$ and (ii) $\eta \ge V_A^2 / \Omega$. In
that case the dissipation scale is given by $k_{\eta}=(\Omega/\eta)^{1/2}$.
In case (i) provided again that dissipation is not too strong, a second transition occurs
at $k_2=\varepsilon / V_A^3$ . This transition leads to an Alfv\'en turbulence
$E(k)=V_A^{1/2}\varepsilon^{1/2}k^{-3/2}$ until the dissipation becomes dominant for
$k\ge k_{\eta}$ with $k_{\eta}=V_A/\eta$.
If $V_A^4\le \eta \varepsilon$ the dissipation overcomes the Alfv\'en turbulence and
the
dissipation scale is given by $k_{\eta}=\varepsilon^{1/4}\eta^{-3/4}$.
In case (ii) a second transition toward a Kolmogorov turbulence is not possible. Indeed,
it would occur at $k=\varepsilon / V_A^3$ which can not be larger than $k_1$ from the
condition $\varepsilon \le \Omega V_A^2$. In that case the Alfv\'en turbulence simply
extends to the dissipation scale given by $k_{\eta}=V_A/\eta$.
\begin{figure*}
\begin{tabular}{@{\hspace{0cm}}c@{\hspace{-7.3cm}}c@{\hspace{1cm}}c@{\hspace{5.7cm}}c@{}}
\includegraphics[width=0.6\textwidth]{fig1a.eps}&&\raisebox{0.8cm}{\hspace{-0.5cm}$k$}&\raisebox{0.8cm}{\quad\quad$k$}
\\*[0cm]
\includegraphics[width=0.6\textwidth]{fig1b.eps}&\raisebox{0.8cm}{\hspace{-1.cm}$k$}&&\raisebox{0.8cm}{\quad\quad$k$}\\*[0cm]
\end{tabular}
\caption{Possible inertial regimes of energy spectral density in rotating MHD turbulence for $\Pm=1$. The capital letters $R, A$ and $K$ denote a
rotating, Alfv\'en or Kolmogorov turbulence.}
\label{Spectra}
\end{figure*}
The four possible types of inertial regimes are sketched in Fig.~\ref{Spectra} in which the spectral energy density is
plotted versus $k$ for $\Pm=1$. The slopes and characteristic wave numbers are indicated.
The
conditions to get one of these four possible inertial regimes are summarized in the plane
$(V_A, \Omega)$ in Fig.~\ref{map}.
The case without rotation corresponds to the abscissa axis. Then two regimes KA
or K are possible depending whether $\eta / V_A^2 \le V_A^2 / \varepsilon$ or not. The
case without applied magnetic field corresponds to the vertical axis.
Then the two regimes R or RK are possible depending whether $\eta \Omega/\varepsilon
\ge \Omega^{-1}$ or not. Without both rotation and applied magnetic field
a K type of turbulence is found.
From our analysis we note that inertial regimes of type AK or RAK are never possible. On the other hand
inertial regimes of type KA, A, or K are possible provided the forcing scale is sufficiently small.
In Fig.~\ref{Spectra} it corresponds to begin the spectra at a larger wave number.
For $\Pm<1$ the inertial range of the kinetic energy spectrum prolongates at scales smaller than $k_{\eta}$
with either an R, K or RK spectrum.
\begin{figure}
\includegraphics[width=0.45\textwidth]{fig2.eps}
\caption{The four possible turbulent inertial regimes given in the map $(V_A,\Omega)$.}
\label{map}
\end{figure}
\section{Shell model}
\label{shell model}
\subsection{The model}
The equations of MHD turbulence for an incompressible fluid embedded in an external uniform
magnetic field $\bB_0$ and subject to rotation $\bOmega$ write
\begin{eqnarray}
\partial \bu/\partial t &+& (\bu \cdot \nabla)\bu - ((\bB+\bv_A) \cdot \nabla)\bB + 2\bOmega
\times \bu \nonumber
\\&=& \nu \nabla^2 \bu + \bF - \nabla P_t \label{U}\\
\partial \bB/\partial t &+& (\bu \cdot \nabla)\bB - ((\bB+\bv_A) \cdot \nabla)\bu = \eta
\nabla^2 \bB \label{B}\\
\nabla \cdot \bu &=& \nabla \cdot \bB = 0 \label{incomp}
\end{eqnarray}
in which $\textbf{v}_A=\bB_0 / \sqrt{\mu \rho}$ is the Alfv\'en velocity (where
$\mu$ and $\rho$ are respectively the fluid magnetic permeability and density) and $\bB$ is given in unit of $V_A=|\bv_A|$. The total
pressure $P_t = P + b^2 /2$ is a functional of $\bu$ and $\bB$ owing to the
incompressibility condition (\ref{incomp}). The forcing $\bF$ insures the fluid motion.
From (\ref{U}) (\ref{B}) (\ref{incomp}) we derive the following shell model
\begin{eqnarray}
\dot{U}_n &=& i k_n \left[Q_n(U,U)-Q_n(B,B)\right] \nonumber \\
&+& i k_n V_A(t) B_n + i \Omega(t)
U_n
- \nu k_n^2 U_n + F_n(t), \label{eq_u} \\
\dot{B}_n &=& i k_n \left[Q_n(U,B)-Q_n(B,U)\right] \nonumber \\&+& i k_n V_A(t) U_n
- \eta k_n^2 B_n, \label{eq_b}
\end{eqnarray}
where
\begin{eqnarray}
Q_n(X,Y)= \lambda^2 (X_{n+1}Y_{n+1}+X_{n+1}^*Y_{n+1}^*)
-X_{n-1}^r Y_n \nonumber \\ -X_n Y_{n-1}^r+{\rm i} \lambda(2
X_n^*Y_{n-1}^i+X_{n+1}^r Y_{n+1}^i-X_{n+1}^i Y_{n+1}^r) \nonumber
\\
\label{Qn}
+X_{n-1}Y_{n-1}+X_{n-1}^*Y_{n-1}^* -\lambda^2(X_{n+1}^r Y_n
+X_n Y_{n+1}^r) \nonumber \\+{\rm i} \lambda(2
X_n^*Y_{n+1}^i+X_{n-1}^r Y_{n-1}^i-X_{n-1}^i Y_{n-1}^r), \label{shellnew}
\end{eqnarray}
represents the non linear transfer rates and $F_n$ the turbulence forcing. This model
is based on wavelet decompostion \cite{Zimin95}. Compared to other shell models
\cite{Gledzer73,Ohkitani89,Lvov98} it has the advantage that helicities are much better defined,
like those based on helical wave decomposition \cite{Benzi96a,Lessinnes09a,Lessinnes09b}.
It
has been introduced in
its hydrodynamic form to study spectral properties of helical turbulence \cite{Stepanov09},
and in its MHD form to study cross-helicity effect on cascades \cite{Mizeva09}.
The parameter $\lambda$ is the geometrical factor from which the wave number is defined $k_n=k_0 \lambda^n$.
As explained in \cite{Plunian07} an optimum shell spacing is the golden number
$\lambda = (1 + \sqrt{5}) / 2$.
The terms involving $\Omega$ and $V_A$ were already introduced in several previous papers dealing with either rotation \cite{Hattori04,Chakraborty10} or applied magnetic field \cite{Biskamp94,Hattori01}.
\subsection{Conservative quantities}
Expressions for the kinetic energy and helicity, $E_U$ and $H_U$, magnetic energy and
helicity, $E_B$ and $H_B$, and cross helicity $H_C$, are given by
\begin{eqnarray}
E_U &=& \sum_{n}E_U(n), \; E_U(n)=\frac{1}{2} |U_n|^2, \\
H_U &=& \sum_{n}H_U(n), \; H_U(n)=\frac{i}{2} k_n ((U_n^*)^2-U_n^2), \\
E_B &=& \sum_{n}E_B(n), \; E_B(n)=\frac{1}{2} |B_n|^2, \label{kinetic}\\
H_B &=& \sum_{n}H_B(n), \; H_B(n)=\frac{i}{2} k_n^{-1}((B_n^*)^2-B_n^2),
\label{magnetic} \\
H_C &=& \sum_{n}H_C(n), \; H_C(n)=\frac{1}{2} (U_n B_n^* + B_n U_n^*).\label{crosshelicity}
\end{eqnarray}
In the inviscid and non-resistive limit ($\nu=\eta=0$), the total energy
$E=E_U+E_B$, magnetic helicity and cross helicity must be conserved
($\dot{E}=\dot{H}_B=\dot{H}_C=0$).
Here with the additional Coriolis and Alfv\'enic terms the properties of conservation are not necessarily satisfied.
A summary of theses properties is given in table \ref{conservation} for 3D MHD turbulence.
In the case of pure hydrodynamic turbulence (without magnetic field) the kinetic energy
and helicity must be conserved ($\dot{E_U}=\dot{H}_U=0$) even with Coriolis forces.
\begin{table}
\begin{center}
\begin{tabular}{|@{\hspace{0.cm}}c|@{\hspace{0.2cm}}l|@{\hspace{0.2cm}}l|@{\hspace{0.2cm}}l|@{\hspace{0.2cm}}l|@{}}
$\Omega$ & =0 & $\neq0$ & $=0$ & $\neq0$ \\*[0cm]
$V_A$ & =0 & $=0$ & $\neq 0$ & $\neq0 $ \\*[0cm]
\hline
$E$ & Y & Y & Y & Y \\*[0cm]
$H_C$ & Y & N & Y & N \\*[0cm]
$H_M$ & Y & Y & N & N \\*[0cm]
\end{tabular}
\caption{In 3D MHD turbulence, conservation properties of total energy $E$, cross-helicity $H_C$ and magnetic helicity $H_M$
depending on global rotation $\Omega$ and applied field $V_A$.}
\label{conservation}
\end{center}
\end{table}
\subsection{Time-scales}
In (\ref{eq_u}) and (\ref{eq_b}) the forcing $F_{n_F}(t)$ (applied at some scale $k_{n_F}^{-1}$), the global rotation $\Omega(t)$ and the applied field
$V_A(t)$ have constant intensities $|F_{n_F}|, \Omega$ and $V_A$. Only their sign may change after a period of time
$t_F$, $t_{\Omega}$ and $t_{V_A}$, the probability of changing from one period to the next being random.
Such a trick allows to control the two characteristic times $\tau_{\Omega} \approx t_{\Omega}$ and
$\tau_{V_A} \approx t_{V_A}$. In the simulations we take $t_{\Omega}=1/\Omega$ and $t_{V_A}=1/(k_{n_F}V_A)$.
It is in same spirit than the one used in \cite{Hattori01} and \cite{Hattori04} though much simpler.
Incidentally the random change of sign of $\Omega(t)$ insures that there is no injection of kinetic helicity on average.
Taking a random sign in $F_{n_F}(t)$ we insure that the forcing intensity satisfies $|F_{n_F}|\approx \sqrt{2 \varepsilon / t_F}$.
It is also important that $t_F$ is the shortest among all other characteristic times of the problem $\tau_K, \tau_{\Omega}$ and $\tau_{V_A}$ (and of course $\tau_{\eta}$). We choose $t_F\le \frac{1}{10}\min\left\{\tau_K, \tau_{\Omega}, \tau_{V_A}\right\}$.
\subsection{No injection of cross-helicity}
In addition it is important to control the injection of cross-helicity as was shown in \cite{Mizeva09}. Indeed any spurious injection
of cross-helicity may lead to a supercorrelation state where $U_n \approx B_n$ implying equality not only in intensity (as in equipartition) but also in phase. In that case the flux of kinetic energy is depleted, implying an accumulation of energy at large scale and steeper spectral slopes. In order to compare the results to the phenomenological approach we impose the injection of cross-helicity to be zero.
For that we could use the forcing
\begin{equation}
\frac{F_{n_F}}{|F_{n_F}|}=\pm i \frac{B_{n_F}}{|B_{n_F}|}
\label{forcing1}
\end{equation}
where again the
sign is randomly changed after each period of time $t_F$.
This forcing is however ill-defined as soon as $|B_{n_F}|\ll |U_{n_F}|\approx 1$. To fix this problem we use the following forcing
\begin{equation}
\frac{F_{n_F}}{|F_{n_F}|}=\frac{a e^{i \varphi} \pm i \zeta \frac{B_{n_F}}{|B_{n_F}|}}{a + \zeta}
\end{equation}
with $\zeta=|B^2_{n_F}|/|U^2_{n_F}|$,
in which $\varphi$ is a phase randomly changed after each period of time $t_F$, and $a$ an additional parameter.
In the case $\zeta \gg a$, (\ref{forcing1}) is recovered, and the phase of $F_{n_F}$ is mainly determined by the phase of $B_{n_F}$ so that it corresponds to zero injection of cross-helicity.
In the case $\zeta \ll a$ the phase of $F_{n_F}$ is controlled by the random phase $\varphi$. Since $B_{n_F}$ is small there is no cross-helicity injection too.
The value $a=10^{-6}$ provides a robust forcing with always a low level of cross-helicity.
\subsection{Dissipations}
We define the dissipation of $U$ and $B$ at scale $k_n$ by $D_U(k_n)=\nu k_n^2 |U_n|^2$ and
$D_B(k_n)=\eta k_n^2 |B_n|^2$.
From the phenomenological formalism above we expect the total dissipation to be equal to the injection rate of energy at the forcing scale
$\varepsilon_{\nu}+\varepsilon_{\eta}=\varepsilon$, with $\varepsilon_{\nu}=\sum_n D_U(k_n)$ and $\varepsilon_{\eta}=\sum_n D_B(k_n)$.
Equivalently in pure HD we would have $\varepsilon_{\nu}=\varepsilon$.
On the other hand the ratio of both dissipations $\rho=\varepsilon_{\eta}/\varepsilon_{\nu}$ cannot be predicted. It can only be calculated numerically.
\section{Results}
\subsection{Spectra for $\Pm=1$}
\begin{figure}
\begin{center}
\begin{tabular}{@{}c@{\hspace{0em}}c@{\hspace{0em}}c@{\hspace{0em}}c@{}}
\rotatebox{90}{$\quad \quad \quad \quad \Omega^{-1/2}\varepsilon^{-1/2}k^2E(k)$} & \includegraphics[width=0.45\textwidth]{fig3a.eps}\\
& $\Omega^{-3/2}\varepsilon^{1/2}k$ \\
\rotatebox{90}{\quad \quad \quad \quad $\varepsilon^{-2/3}k^{5/3}E(k)$} & \includegraphics[width=0.45\textwidth]{fig3b.eps}
\\*[0.0cm]
& $V_A^{3}\varepsilon^{-1}k$\\
\rotatebox{90}{\quad \quad \quad \quad $k^{3/2}\varepsilon^{-1/2} V_A^{-1/2}E(k)$} & \includegraphics[width=0.45\textwidth]{fig3c.eps}&&\\*[0.0cm]
& $V_A\Omega^{-1}k$ &&
\end{tabular}
\end{center}
\caption{(Color online) Normalized spectra for $\nu=10^{-7}$ and $\Pm=1$. Curves (a) are shown for $V_A=0$ and $\Omega=12.5,25,50,10,200,400,800,1600$ (from right to left, from darker to lighter).
Curves (b) are shown for $\Omega = 0$ and $V_A=0.16, 0.32, 0.64,1.28,2.56,5.12,10.24,20.48$ (from left to right, from darker to lighter).
Curves (c) are shown for
$(V_A,\Omega)=(0.16,800); (0.32, 400); (0.64,200); (1.28,100); (2.56,50);$ $(5.12,25); (10.24,12.5); (20.48,6.25)$ (from left to right, from darker to lighter).}
\label{Pm=1}
\end{figure}
In Fig.~\ref{Pm=1} the spectra are plotted for $\nu=10^{-7}$ and $\Pm=1$ in the three cases $V_A=0$,
$\Omega = 0$, and $V_A \Omega \ne 0$. For $V_A=0$, the horizontal and $k^{1/3}$ dashed lines
disclose a RK regime. For $\Omega = 0$, the dashed line $k^{1/6}$ disclose a KA regime.
For $V_A \Omega \ne 0$, the $k^{-1/2}$ and horizontal dashed lines disclose a RA regime.
In each case the transition between two power laws is rather smooth and occurs over a scales range
of about two orders of magnitude.
For $\varepsilon \approx 1$ and taking the numerical values for $\Omega, V_A$ and $\eta$
given in Fig.~\ref{map}
we find that
the three sets of spectra found with the shell model belong indeed to the three parts
RK, (R)KA and RA of Fig.~\ref{map}.
We tried to track the transition from one part to the other, varying $\Omega$ and $V_A$. It is however not possible to handle it numerically as the spectral slopes
are not so well defined at the neighborhood of the frontiers delimiting the four parts of Fig.~\ref{map}.
\begin{figure*}
\begin{center}
\begin{tabular}{@{}c@{\hspace{0em}}c@{\hspace{0em}}c@{\hspace{0em}}c@{}}
\rotatebox{90}{\quad \quad \quad \quad \quad \quad $k^2E(k)\varepsilon^{-1/2}$} &
\includegraphics[width=0.45\textwidth]{fig4a.eps}&
\rotatebox{90}{\quad \quad \quad \quad \quad \quad $k^2E(k)\varepsilon^{-1/2}$}&
\includegraphics[width=0.45\textwidth]{fig4b.eps}\\*[0.0cm]
& $k\varepsilon^{1/2}$ && $k\varepsilon^{1/2}$\\*[0.0cm]
\rotatebox{90}{\quad \quad \quad\quad \quad \quad $k^{5/3}E(k)\varepsilon^{-2/3}$} &
\includegraphics[width=0.45\textwidth]{fig4c.eps}&
\rotatebox{90}{\quad \quad \quad\quad \quad \quad $k^{3/2}E(k)\varepsilon^{-1/2}$}&
\includegraphics[width=0.45\textwidth]{fig4d.eps}\\*[0.0cm]
& $k\varepsilon^{-1}$ && $k\varepsilon^{-1}$\\*[0.0cm]
\rotatebox{90}{\quad \quad \quad\quad \quad \quad $k^{2}E(k)\varepsilon^{-1/2}$} &
\includegraphics[width=0.45\textwidth]{fig4e.eps}&
\rotatebox{90}{\quad \quad \quad\quad \quad \quad $k^{3/2}E(k)\varepsilon^{-1/2}$} &
\includegraphics[width=0.45\textwidth]{fig4f.eps}\\*[0.0cm]
& $k\varepsilon^{-1}$ && $k\varepsilon^{-1}$
\end{tabular}
\end{center}
\caption{(Color online) Normalized kinetic (full) and magnetic (dashed) spectra for $\nu=10^{-7}$:
(a) $(V_A,\Omega)=(0,100)$, (b) $(V_A,\Omega)=(0,1600)$.
(c) $(V_A,\Omega)=(0.08,0)$, (d) $(V_A,\Omega)=(1.28,0)$.
(e) $(V_A,\Omega)=(0.32,400)$, (f) $(V_A,\Omega)=(20.48,6.25)$.
For each set of curves for
$\Pm=10^{-5},10^{-4},10^{-3},10^{-2},10^{-1}, 1$ (from lighter to darker). Note, that the kinetic and magnetic spectra are superposed in the case (f).
}
\label{Pm<1}
\end{figure*}
\subsection{Spectra for $\Pm<1$}
In Fig.~\ref{Pm<1} the kinetic and magnetic spectra are plotted for $\nu=10^{-7}$ and several values of $\Pm$,
for the three previous cases.
For $V_A=0$ (a,b) increasing $\Pm$ decreases the magnetic dissipation scale while the viscous scale
is not significantly changed. This is in agreement with a simple Kolmogorov phenomenology
\cite{Stepanov08}, the ratio of dissipation scales being given by $k_{\nu}/k_{\eta}\propto \Pm^{-3/4}$.
For $\Pm\ge 10^{-2}$ the effect of rotation is visible in the spectra flatness.
At smaller values of $\Pm$ it is however difficult to determine any slope at all.
For $\Omega=0$ and $V_A=1.28$ (c,d)
both kinetic and magnetic spectra are almost the same whatever the value of $\Pm$.
The effect of an applied magnetic field is to correlate both fields as expected in Alfv\'en waves.
In particular the dissipation scale is governed by the magnetic diffusivity, with $k_{\nu} \approx k_{\eta}$ .
The same conclusions are found for $(V_A,\Omega)=(0.32,400)$ (e) and $(V_A,\Omega)=(20.48,6.25)$ (f).
In these two cases the horizontal slopes are due to rotation (e) and applied magnetic field (f).
We note that for $\Omega=0$ and $V_A=1.28$ (d) the normalized curves are not horizontal. They correspond
to spectral energy density slopes between $k^{-5/3}$ and $k^{-3/2}$. The latter is obtained for values of
$V_A$ about ten times larger.
\begin{figure*}
\begin{center}
\begin{tabular}{@{}c@{\hspace{0em}}c@{}}
\includegraphics[width=0.45\textwidth]{fig5a.eps}&
\includegraphics[width=0.45\textwidth]{fig5b.eps}\\
\includegraphics[width=0.45\textwidth]{fig5c.eps}&
\includegraphics[width=0.45\textwidth]{fig5d.eps}
\end{tabular}
\end{center}
\caption{Dissipation ratio versus $\Pm$.
On panel (a) $V_A=\Omega=0$ and the full curves from right to left correspond to $\nu=10^{-5}$, $10^{-6}$, $10^{-7}$, $10^{-8}$. The dashed curves from bottom to top correspond to $\eta=1/4$, $1/8$, $1/16$, $1/32$, $1/64$, $1/128$, $1/256$.
On panel (b) $V_A=0$, $\nu=10^{-7}$ and the curves correspond to $\Omega=0$ (full curve), $100$ (dashed), $400$ (dotted), 1600 (dot-dashed).
On panel (c) $\Omega = 0$, $\nu=10^{-7}$ and the curves correspond to $V_A=0$ (full curve), $0.08$ (dashed), $0.32$ (dotted), 1.28 (dot-dashed).
On panel (d) $\nu=10^{-7}$ and the curves from bottom to top correspond to
$(V_A,\Omega)=(0,0)$, (0.32, 400), (2.56,50) and (20.48,6.25).
}
\label{dissipation_ratio}
\end{figure*}
\subsection{Dissipation ratio}
In Fig.~\ref{dissipation_ratio} the ratio $\rho=\varepsilon_{\eta}/\varepsilon_{\nu}$ is plotted versus $\Pm$
for $V_A=\Omega=0$ (a), $V_A=0$ (b), $\Omega=0$ (c) and $V_A \Omega \ne 0$ (d).
In the limit $\Pm\rightarrow 0$ the dynamo action does not occur, implying $\rho\rightarrow 0$.
For $\Pm = 1$ both kinetic and magnetic spectra are identical, implying $\varepsilon_{\nu}=\varepsilon_{\eta}=\varepsilon/2$, and then $\rho=1$.
We always find an intermediate value of $\Pm$ for which $\rho$ reaches a maximum.
This is related to a super-equipartition state in which the magnetic energy is higher than the kinetic energy at large scales. Varying $V_A$ and $\Omega$ we find that this maximum value can increase by several orders of magnitude
and that it does not occur at the same $\Pm$.
For the two last cases an asymptotic curve $\rho=O(\Pm^{-1})$ is obtained for large values of $V_A$.
This is a direct consequence of the equipartition regime $|U_n| \approx |B_n|$ obtained at any scale (see Fig.~\ref{Pm<1}).
In that cases the definition of $\rho$ directly implies the scaling $O(\Pm^{-1})$.
\section{Discussion}
For $\Pm=1$ both approaches, phenomenological and shell model, give consistent results in terms of inertia regimes. They are controlled by the shortest time-scale corresponding either to rotation, applied magnetic field, inertia, or a combination of them. For $\Pm <1$ the magnetic dissipation occurs at a scale larger than the viscous scale implying that the different regimes are not so easy to discriminate. However for a sufficiently strong applied magnetic field both kinetic and magnetic energy spectra are merged, implying a strong increase of the viscous dissipation scale. Whether this is due to our isotropic assumption is not clear and cannot be answered with our models. A consequence is that, for a strong applied field, the ratio of magnetic to kinetic dissipation scales like $O(\Pm^{-1})$ and can reach very high values for $\Pm \ll 1$. Without applied field, this ratio is also maximum for some value of $\Pm \ll 1$, depending on the fluid viscosity and global rotation.
\begin{acknowledgments}
This work benefited from the support of a RFBR/CNRS 07-01-92160 PICS grant and of a Russian Academy of Science project 09-P-1-1002.
It was also completed during the Summer Program on MHD Turbulence at the Universit\'e
Libre de Bruxelles in July 2009.
We warmly thank G. Sarson for enlightening discussions and anonymous referees for helping us in improvement of the paper.
\end{acknowledgments}
\bibliographystyle{apsrev}
|
{
"timestamp": "2010-10-18T02:02:06",
"yymm": "1009",
"arxiv_id": "1009.3549",
"language": "en",
"url": "https://arxiv.org/abs/1009.3549"
}
|
\section{Introduction}
VUV and X-ray small-scale brightenings are often detected in
images or spectra of the solar atmosphere
\citep[e.g.][]{berghmans98, aletti00,
aschwanden02, christe08}. These observations and theoretical
considerations indicate that these brightenings may be one
manifestation of small-scale impulsive heating acting on the
unresolved fine scale of the corona. Modelling unresolved small-scale brightenings can help, for example, to explain the long
lifetimes of active region loops. Understanding the properties of this heating and its contribution to global coronal heating are among the most challenging questions in solar physics. important starting point is identifying key
observables that can diagnose proposed heating functions.
The study of the frequency distribution of the intensity of coronal VUV
emission lines is one method for investigating this
problem. The distributions derived from observations with a variety of
instruments and data sets show power laws
with indices mainly between $-1.5$ and $-2$
\citep[e.g][]{aschwanden05}. Converting the frequency distribution into
a distribution of thermal energy for the heating events yields a similar power-law, and this is assumed to represent the power-law index of the
(unknown) coronal heating function.
However, behind this method there is the assumption that the
energy conversion mechanisms do not modify the original heating energy distribution, so that the measured events distribution is equivalent to the heating distribution.
\cite{parenti06} investigated this aspect using a forward
modelling approach: a statistical model of
coronal heating was posited and properties of the predicted emission
line intensities and thermal energies were deduced.
Their main result indicated that only high
temperature lines, formed when the main cooling mechanism in the loop was conduction, are reliable.
These authors showed that the statistical properties of the heating function are conserved by the line intensity statistical distribution,
only during such a period. The lines analyzed by these authors did not
belong to the lithium isoelectronic sequence.
As with many other
ions formed in the solar atmosphere, the Li isoelectronic ions are formed principally
over a narrow temperature interval ($\approx 0.3$~dex in
$\log\,T$), although, unlike most other ions, they also have a tail in
their temperature distributions that extends to high temperatures
(e.g. Fig.~\ref{GT}), which leads to the ions sampling a wider
temperature range than most other ions.
In the present work, the results of Parenti et al. (2006) will be extended to investigate the following questions:
\begin{itemize}
\item How are the statistical distributions affected if the emission lines belong to the lithium-like isoelectronic sequence?
\item How will a statistical distribution measured from a wideband imaging instrument compare with a distribution from a single spectral line observed by a spectrometer?
\end{itemize}
The work presented here is particularly useful in the light of upcoming
multi-channel, high resolution solar VUV instruments. The
Atmospheric Imager Assembly \citep[AIA,][]{golub06} is scheduled to be flown on the
Solar Dynamics Observatory (SDO) in 2009 and has eight distinct
channels, seven of which are narrow bandpass channels centred on strong
emission lines. Solar Orbiter is planned to be launched in 2015 and
VUV imaging and spectroscopic instruments -- the Extreme Ultraviolet
Imager (EUI) and Extreme Ultraviolet Spectometer (EUS), respectively
-- are part of the strawman
payload \citep{hochedez07, young07a}.
Due to the multiple hot channels present on SDO/AIA, it appears to be a
promising instrument, in terms of statistics, for observing small-scale heating events.
In a more general context,
the plasma conditions in the solar atmosphere can be diagnosed if a set of lines formed at each layer of the solar atmosphere, can be detected.
The coronal Li-like ions, as well as some of the \ion{Fe}{} ions studied here, have lines at long UV wavelengths that can be simultaneously observed with chromospheric and transition region lines (e.g. \ion{Fe}{xviii} and \ion{Fe}{xix} in the SUMER waveband). This is an important consideration for future spectrometers where we want access to all layers of the solar atmosphere with just two or three spectral bands \citep{young07a}.
Due to the large number of current and planned VUV instruments, the present work concentrates on VUV lines and we do not investigate
the consequences of the small-scale brightening model on X-ray
emission. However, to our knowledge, soft X-ray microflare
studies of this kind have not been completed previously, although results from the Hinode Soft X-ray Telescope (SXT) may be forthcoming.
The paper is organised as follows.
Section 2 introduces the hydrodynamical model used in the simulations.
Section 3 describes how the synthetic spectra are built.
Section 4 provides the results, and conclusions are drawn in Sect. 5.
\section{The model}
The hydrodynamical model used was described in detail by
\cite{parenti06}, and we provide only a brief description.
A coronal loop was modelled by a bundle of unresolved, identical
threads following the approach of \cite{cargill94}. Two thousand threads were
used each with a half length, $L$, of $10^9 ~\rm cm$, and a
cross-sectional area, $A$, of $2\times 10^{14}~ \rm cm^2$. At each
time $t$, a thread was described by a single temperature, $T$, and single density, $N$. The model simulates the heating-cooling cycle of each thread independently. After the thread was heated impulsively, the cooling proceeded first by thermal conduction (at high temperatures) and then by radiation (at low temperatures). During the period dominated by conduction, the strand was filled with plasma by evaporation from the chromosphere, while it drains during the radiation phase.
The time evolution of the density and temperature in each thread was used to calculate the time-dependent synthetic spectra in the entire loop. The assumed energy loss was $2.5 \times 10^{-4}~ \mathrm{erg~ cm^{-3}~ s^{-1}}$.
The heating function employed by \cite{parenti06} was derived from a
model of coronal turbulence developed by \cite{buchlin03}. For the present work,
a synthetic heating function was used,
which simulated a sequence of small energy-impulsive events ($\approx 10^{21} - 10^{24}~ \mathrm{erg}$), each involving only one thread at a time.
The heating function had a log-normal distribution with index $\alpha
= -1.7$, over the energy range $\approx 10^{22} - 10^{24}~
\mathrm{erg}$ (Fig. \ref{heating_f}).
Our use of this synthetic heating function is justifiable because power-law
distributions of observed energy events have been reported, and
theoretical models have also been shown to generate a power-law
distribution of energies \citep{parenti06}. A synthetic function can
also be easily modified, to allow a full investigation of model parameters.
\begin{figure}
\centering
\includegraphics[scale=0.5]{9928fig1.eps}
\caption{Probability Distribution Function of the heating function used for our simulations.}
\label{heating_f
\end{figure}
\section{The synthetic spectra}
To study the behaviour of the optically thin FUV-UV line intensities, it is helpful to consider the column Differential Emission Measure (DEM) distribution, which represents the amount of material at a given temperature along the line of sight ${h}$:
\begin{equation}
\mathrm{DEM}(T) = N^2 \left({d T}\over{d h}\right )^{-1}
\end{equation}
\noindent where $N$ is electron density along $h$ at a given temperature ($T$).
This quantity is linked to a line intensity ($I$) for an optically thin line by:
\begin{equation}\label{eq_dem}
I = \int_0^\infty \mathrm{ A(X)}~ G(T) ~ \mathrm{DEM}(T) ~ dT ~~~ \mathrm{[erg ~ cm^{-2}~ s^{-1}~ sr^{-1}]}
\end{equation}
\noindent where
$G(T)$ is the contribution function that contains the atomic physics
information for the emission line and is predominantly a function of temperature for most allowed transitions, and $\mathrm{A(X)}$ is the abundance of the element with respect to hydrogen.
In our model, we assumed that the line of sight was perpendicular to the loop axes, so that the line of sight was given by the diameter of the strand multiplied by the number of strands emitting at the given temperature.
Under these conditions, the shape of the DEM can be represented by the distribution of $N^2$ versus $T$. (For details on the dependence of $\mathrm{DEM}(T)$ on the model parameters see \cite{parenti06} and references therein.)
Figure \ref{figem} shows the logarithm of ${N^2}$ as a function of the logarithm of temperature, integrated over the entire loop system (2000 strands) and simulation time ($10^5~ \mathrm{s}$).
This plot helps to interpret our results.
\begin{figure}[h]
\includegraphics[scale=0.5]{9928fig2.eps}
\caption{ Variation in ${\log~ N^2}$ with ${\log~T}$ inside the loop.}\label{figem}
\end{figure}
As mentioned earlier, in this model the cooling of a
strand proceeded in two separate phases: the conduction phase, where the strand is still at a high temperature; and the
radiation phase, for lower temperatures. For this reason, we are able to follow the cooling process in Fig. \ref{figem}.
The plasma evaporation from the chromosphere acting during the period dominated by thermal conduction fills the high temperature part (on the right side from the peak) of the ${N^2}$--${T}$ distribution. The lower temperature part of the distribution is filled during the radiation phase. The peak corresponds to the temperature location at which part of the strands are cooling by conduction and part by radiation.
One of the main results of
\cite{parenti06} was that the intensity distribution of a spectral line
is representative of the heating function distribution, if the
formation temperature of the line is within the high temperature region
of the DEM distribution. This is due to the proximity with the
moment of heat injection, which produces radiative
emission that inherits the properties of the heating function.
For the case presented here, the DEM peaks at about $\log~T =
6.6$. This implies that lines emitted at lower temperatures
are formed while the cooling in the strand is dominated by
radiation. For $\log~T > 6.9$, the cooling is dominated by
conduction. An intermediate condition is found between these two
temperatures, in which the line emission loses all information about the
heating function.
Table \ref{tab1} lists the lines modelled in the present work, and their formation temperatures. These lines are observable with the instruments SOHO/SUMER, EIT and Hinode/EIS. Where possible, we chose two lines with similar formation temperatures but belonging to ions of different isoelectronic sequences. Lines belonging to the
Li isoelectronic sequence are highlighted. The SUMER
instrument
on board SOHO is an
ultraviolet spectrometer observing in the wavelength range
500--1600~\AA\, which was described by \cite{wilhelm95}. This is one of
the most useful bands in the VUV because it contains lines emitted from the chromosphere to the corona.
Another SOHO instrument is EIT, which acquires solar
images in four wavelength bands by using multilayer optical coatings
\citep{delaboudiniere95}.
EIS is an
ultraviolet spectrometer on board the Hinode satellite operating in
the wavelength ranges 170--211~\AA\ and 246--292~\AA\ and was described
in \cite{culhane07}.
\begin{figure}[th]
\includegraphics[scale=0.5]{9928fig3a.eps}\\
\includegraphics[scale=0.5]{9928fig3b.eps}
\caption{Contribution functions for the spectroscopic lines (top) and the temperature response function for the EIT 171 and 195 filters (bottom).
}\label{GT}
\end{figure}
\begin{table}[h]
\caption[]{List of lines synthesised in this work, and with the logarithm of the temperature of their maximum emission. These lines can be observed by SOHO/SUMER (\ion{Ne}{viii}, \ion{Mg}{x}, \ion{Fe}{xviii}, \ion{Fe}{xix}), SOHO/EIT (\ion{Fe}{ix}, {\bf \ion{Fe}{xii}}) and Hinode/EIS (\ion{Fe}{ix}, \ion{Fe}{xii}, \ion{Fe}{xxiv})}
\label{}
\centering
\begin{tabular}{c l l }
\hline\hline
$\log~(T_{\rm max})$ & Li-like & Others \\
\hline
5.8 & \ion{Ne}{viii} 770.41 \AA& \ion{Fe}{ix} 171.07 \AA\\
6.1 & \ion{Mg}{x} 624.94 \AA & \ion{Fe}{xii} 195.12 \AA\\
6.8 & & \ion{Fe}{xviii} 974.86 \AA \\
6.9 & & \ion{Fe}{xix} 1118.06 \AA \\
7.24 & \ion{Fe}{xxiv} 192.03 \AA & \\
\hline
\end{tabular}\label{tab1}
\end{table}
\begin{figure*}[fth]
\centering
\includegraphics[scale=.5]{9928fig4a.eps}
\includegraphics[scale=.5]{9928fig4b.eps}\\
\includegraphics[scale=.5 ]{9928fig4c.eps}
\includegraphics[scale=.5 ]{9928fig4d.eps}
\caption{Top: PDFs for the \ion{Fe}{ix} (on the right) and \ion{Ne}{viii} (on the left) lines. Bottom: PDFs for the \ion{Fe}{xii} (on the right) and \ion{Mg}{x} (on the left) lines.}
\label{Fig3
\end{figure*}
For the theoretical calculation of the line intensities, we used the
CHIANTI \citep[v. 5.2,][]{dere97, landi06} atomic database and software,
adopting the \cite{mazzotta98} ion fractions and photospheric element
abundances \citep{grevesse98}. We simulated the measured data numbers (DN) from the
spectrometers by assuming observations with a 1 arcsec slit and 1s
cadence and using the standard calibration software available in the
SolarSoft IDL package. In the case of EIT, we simulated the total emission
in the channel by integrating the emission from all lines in the
waveband, as predicted by the CHIANTI atomic database, taking into
account the effective area of the instrument.
The \ion{Fe}{ix} 171 channel of EIT in particular, is affected by significant
contributions from the slightly hotter \ion{Fe}{x} 174.53~\AA~ line, which therefore alters its response to the heating function. The 195 channel has a hot component due to \ion{Fe}{xxiv} 192.03~\AA.
\noindent Figure \ref{GT} shows the contribution functions
$G(T)$ for all lines (top) and the EIT response function for the
channels (bottom) used in this work. The $G(T)$ functions of the lithium isoelectronic lines in the top plot can be identified by their asymmetry towards high temperatures.
We anticipate in our results that this a high temperature tail will increase the number of
events with low intensity.
\section{Line intensity distributions}
\subsection{``Warm'' line properties}
We investigate the intensity distribution of
``warm'' coronal lines, i.e., those lines that form at the
average temperature of the corona (1--2~MK).
Figure \ref{Fig3} shows the PDFs for \ion{Fe}{ix} and
\ion{Ne}{viii}
($\log~T_{\rm max} = 5.8$; top panels), \ion{Fe}{xii}, and
\ion{Mg}{x} ($\log~T_{\rm max} = 6.1$; bottom panels).
The left panels in the Fig. show the PDFs from the Li-like lines.
The power-law behaviour of the heating function was transmitted to the
predicted PDFs of each wavelength, although, in each case the index
$\alpha$ was far smaller than the $-1.7$ value of the heating function.
This behaviour is consistent with the findings of \cite{parenti06}.
The minimum value of $\alpha$ was reached for the two Li-like lines,
because of the increase in the number of strands emitting a weak intensity; this was caused by the extension of their contribution function to higher temperatures.
To confirm that high temperature tails produced the lower
$\alpha$ values for Li-like ions, the
contribution functions
were truncated at high temperature and the calculations repeated. The
PDFs were found to then have $\alpha$ values that were consistent with
the \ion{Fe}{ix} and \ion{Fe}{xii} lines, confirming our hypothesis.
This comparison demonstrated that the Li-like ions had a
different response to the heating function than other ions formed at
the same temperature, due to the high temperature tails of
the contribution functions.
\subsection{Imager versus spectrometer}
We then investigated weather the difference in the bandwidth of the instrument could have an effect on the shape of the PDF. We simulated the EIT intensities in
the 171 and 195 channels, which were compared with the results presented in
the previous section (Figure \ref{fig4}) for the 171 and 195 emission
lines observed by Hinode/EIS.
Comparing Figs.~\ref{Fig3} and \ref{fig4}, the PDFs for the EIS lines
have indices higher than those derived for the two EIT channels. We
have therefore demonstrated that observing emission lines with a wide
band imager can affect the PDF of the emission line.
We note that the EIS \ion{Fe}{ix} line is situated at the border of the
instrument waveband for which the response is low.
The predicted DN values are therefore very small and the line is not
useful in practice for the study of small-scale brightenings.
\begin{figure}[h]
\centering
\includegraphics[scale=.5 ]{9928fig5a.eps}\\
\includegraphics[scale=.5]{9928fig5b.eps}
\caption{PDF for the counts in the EIT 171 (top) and 195 (bottom) channels.}\label{fig4}
\end{figure}
To compare with the spectrometer lines,
the intensities from the EIT channels were calculated by assuming a 1s
exposure time, which is generally far lower than true EIT exposure times. For a more realistic case, we also calculated the
PDFs assuming $60~$s exposure times (similar exposure times are
reached during the high cadence EIT ``Shutterless''
program\footnote{See http://sidc.oma.be/EIT/High-cadence.}), and
verified that the PDFs retained the same power-law indices as for the
1s case.
\subsection{``Hot'' line properties}
By hot lines, we indicate those lines that form at temperatures higher
than the average coronal temperature.
\begin{figure}[h]
\includegraphics[scale=0.5]{9928fig6a.eps}\\
\includegraphics[scale=0.5]{9928fig6b.eps}
\caption{PDFs of \ion{Fe}{xviii} (top) and \ion{Fe}{xix} (bottom).}\label{feca}
\end{figure}
The forbidden lines \ion{Fe}{xviii} 974.86 \AA~ and \ion{Fe}{xix}
1118.06 \AA~ are important because they appear at long UV wavelengths
close to cooler lines from the transition region and chromosphere.
A spectrometer observing these lines will therefore observe simultaneously lines
from distinct regions of the solar atmosphere, a feature valuable for
understanding the multithermal nature of coronal structures.
At $\log~T =6.8$ \ion{Fe}{xviii} forms, close to the DEM peak (see
Figs. \ref{figem} and \ref{GT}), where the loop threads can cool
by both conduction and radiation. \ion{Fe}{xix} forms during the cooling phase dominated by thermal conduction only ($\log~T =6.9$).
Figure \ref{feca} shows the PDFs for these lines (\ion{Fe}{xviii} at the top and \ion{Fe}{xix} at the bottom). The PDF for \ion{Fe}{xviii} does not have a power-law distribution, while \ion{Fe}{xix} reproduces well the statistical behaviour of the heating function.
The \ion{Fe}{xix} line was also modelled by \cite{parenti06}, who
found similar results. This finding is then a positive test for our
heating function, which, in contrast to \cite{parenti06}, has been completely synthesised.
Unfortunately there are no strong observed EUV Li-like lines that form
at the same temperatures as \ion{Fe}{xviii} and \ion{Fe}{xix}. For this reason, we cannot directly compare with the results shown in Fig. \ref{feca}.
As a final test, we compiled the intensity distribution for the most prominent line of the Li-like
\ion{Fe}{xxiv} ion, which exibits a key flare doublet in the EIS
wavelength bands at 192.03~\AA~ and 255.11~\AA\ and whose emission peaks at $\log~T=7.24$ \citep{young07b}. If nanoflares are
responsible for the heating in ARs, we would expect faint, non flaring,
emission at these temperatures \citep[e.g.][]{patsourakos06}. In the
simulation used here,
the loop system has low DEM values at these high temperatures (Fig.
\ref{figem}), but emission is still found at the \ion{Fe}{xxiv}
line, which will be
completely formed during the conduction phase. For the final reason, we would expect that the high temperature wing of the line would not affect the slope of the PDF.
Figure \ref{fe24} shows the resulting PDF. As expected from the previous results, the distribution follows a power-law and the index $\alpha$ is close to that of the heating function
but in fact a little higher, in contrast to results for the other ions that have
been studied.
We believe that this is because the line forms at the very end of
the heating function energy range, for which the statistics are probably too small
to be representative (see also the drop of the $\mathrm{DEM}(T)$ at high temperatures in Fig. \ref{figem}). This may be a limit to our diagnostic method.
\begin{figure}[h]
\includegraphics[scale=0.5]{9928fig7.eps}
\caption{The PDF of \ion{Fe}{xxiv}.} \label{fe24}
\end{figure}
\section{Conclusions}
In this work, we have modelled the statistical behaviour of emission line
intensities from a coronal loop, whose heating function is distributed
as a log-normal distribution in energy.
We investigated the plasma response for a wide range of
temperatures, by simulating measurements from SOHO/SUMER, Hinode/EIS,
and SOHO/EIT lines, and comparing with the earlier work of
\cite{parenti06}.
Our results may be summarised as follows:
\begin{itemize}
\item The synthesised heating function used in this work reproduces the statistical behaviour of a similar function derived by a model used in a previous work \citep{parenti06}. This would support the use of a modified version of this function in further tests with heating distributions of different power-law indices.
\item We confirm the previous results of \cite{parenti06} that the power-law index of the heating function is preserved only by the intensity
distributions of hot lines ($\log\,T\ge6.9$).
\item The shape of the PDF of the line intensities depends not only
on the temperature of the line formation but also on the
iso-electronic sequence of the emitting ion. In particular, we have
shown that the ``warm'' Li-like lines are inappropriate for this
type of diagnostic, due to the high temperature tail of their
contribution functions.
\item The behaviour of Li-like \ion{Fe}{xxiv} is not, however,
compromised by the high temperature tail of the contribution
function and the index of the PDF is close
to that of the heating function.
\item We identify a weakness in the use of imaging instruments
for statistical studies of coronal heating:
their wavelength bands generally contain more than one strong emission line that can affect the PDF.
\end{itemize}
{
In light of the present results, the high temperature channels on AIA (\ion{Fe} {xxiii} 133 \AA) and Solar Orbiter (EUI) or the flaring spectroscopic lines on Solar Orbiter/EUS could be an important source of information for the coronal heating problem. At the same time, sensible contributions from other lines in the large band instruments need to be carefully investigated.
The results presented here were obtained using a simple, hydrodynamical
model, halthough it has been shown that the model could reproduce the general properties of the plasma well.
In the future, we will search for further confirmation of our findings by
using an evolved version of our model.
\begin{acknowledgements}
SP would like to thank David Berghmans for the fruitful discussion.
SP acknowledge the support from the Belgian Federal Science Policy Office through the ESA-PRODEX programme.
This work was partially supported by the International Space Science Institute in the framework of an international working team (n. 108).
CHIANTI is a collaborative project involving the NRL (USA), RAL (UK), MSSL (UK), the Universities of Florence (Italy) and Cambridge (UK), and George Mason University (USA).
\end{acknowledgements}
\bibliographystyle{aa}
|
{
"timestamp": "2010-09-22T02:02:24",
"yymm": "1009",
"arxiv_id": "1009.4112",
"language": "en",
"url": "https://arxiv.org/abs/1009.4112"
}
|
\section{Introduction}
A new morphological class of supernova remnants (SNRs) known as the
mixed-morphology (MM) SNRs
has been firmly established in the recent literature \citep{Rho98, Shelton99}.
The defining characteristics of SNRs of this class include
a shell-like radio morphology combined with a centrally-filled X-ray
morphology. X-ray observations of these SNRs made with the
$\it{Roentgensatellit}$ ({\it ROSAT}), the Advanced Satellite for
Cosmology and Astrophysics ({\it ASCA}), {\it Chandra} and {\it XMM-Newton}
have found that the central
X-ray emission from these SNRs is not non-thermal emission
(as would be expected from a central plerion) but instead thermal from
shock-heated swept-up interstellar material. Examples of well-known
MM SNRs include W28 \citep{Rho02}, G290.1$-$0.8 (MSH 11$-$6$\it{1}$A)
\citep{Slane02} and IC 443 \citep{Kawasaki02}. \citet{Rho98} suggested
that as many as 25\% of the entire population of Galactic SNRs may belong
to this morphological class. Based on CO and infrared observations
\citep{Koo01, Reach05},
it appears that many MM SNRs are interacting with nearby molecular and HI
clouds. This result suggests a connection between the contrasting X-ray and
radio morphologies of these SNRs with the interaction between these sources
and adjacent clouds, but such a connection
lacks a detailed theoretical basis at this time. Two leading scenarios
have been advanced to explain the origin of the contrasting radio and
X-ray morphologies of MM SNRs: in the first scenario -- known as the
evaporating clouds scenario -- molecular clouds overrun by the expanding
shock front of the SNR survive passage through the shock and eventually
evaporate, providing a source of material that increases the density of the
interior X-ray-emitting plasma of the SNR \citep{Cowie77, White91}. In
the second scenario -- known as the radiative shell model -- the SNR has
evolved to an advanced stage where the shock temperature is low and
very soft X-ray emission from the shell is absorbed by the interstellar
medium (ISM): therefore,
the only detectable X-ray emission is from the interior of the SNR
\citep{Cox99, Shelton99, Shelton04}. In the current paper we analyze and
discuss X-ray emission from two Galactic SNRs -- HB21 and CTB 1 -- which
have both been previously classified as MM SNRs by \citet{Rho98}.
\par
HB21 (G89.0$+$4.7) was discovered in a radio survey by \citet{Brown53}.
The radio angular extent of this SNR is large --
120$\times$90 arcminutes \citep{Green09a} -- and the radio morphology is a
closed shell. The shell appears to be flattened
along the eastern boundary and features bright regions along the northern and
southern boundaries with a prominent indentation seen along the northern
boundary. The radio emission from HB21 is strongly polarized (3.7$\%$$\pm$0.4$\%$) with a
projected magnetic field tangential to the shell \citep{Kundu71,Kundu73,Kothes06},
suggesting that the shell was compressed during the radiative evolutionary stage of the
SNR. The measured radio spectral index for this SNR is $\alpha$ $\sim$ 0.4
($S_\nu \propto \nu^{-\alpha}$) \citep{Leahy06, Green09b} but significant variations
in the values (from 0.0-0.8 with a standard deviation of 0.16) of the spectral index across
the face of the SNR were observed by \citet{Leahy06}. Based on {\it IRAS} observations,
\citet{Saken92} detected clumpy infrared filaments associated with this SNR.
Filamentary optical emission from this SNR with an angular extent comparable to the
radio shell was detected by \citet{Mavromatakis07}. Extensive evidence exists that
indicates HB21 is interacting with adjacent molecular clouds: this evidence includes
CO observations \citep{Tatematsu90, Koo01, Byun06} as well as near- and mid-infrared
observations of the sites of shock-molecular cloud interactions along the northern and
southern parts of the SNR \citep{Shinn09,Shinn10}. HI
observations toward HB21 \citep{Tatematsu90, Koo91} have revealed a high velocity
expanding shell associated with this SNR. Finally, HB21 has
been the subject of prior pointed X-ray observations made by {\it Einstein}
\citep{Leahy87} and {\it ROSAT} \citep{Rho95, Leahy96}: we include in this
paper
the {\it ROSAT} images presented previously by \citet{Rho95} in her
PhD thesis work. No pulsars or $\gamma$-ray sources are believed to be
associated with HB21: radio searches for a pulsar associated with this SNR were
conducted by \citet{Biggs96} and \citet{Lorimer98} but no candidate sources were found.
The distance to HB21 is not well known: \citet{Tatematsu90} argued for a distance of
only 0.8 kpc based on an association between the SNR and molecular material that
belongs to the Cygnus OB7 association \citep{Humphreys78}.
However, \citet{Yoshita01} suggested a distance
of $\geq$1.6 kpc based on a correlation that those authors found between
X-ray absorbing column density and extinction, and \citet{Byun06}
suggested a distance of 1.7 kpc based on CO observations.
In this paper, we have adopted a distance of 1.7 kpc to HB21.
\par
CTB 1 (G116.9$+$0.2) was discovered in a survey of Galactic radio emission
at 960 MHz by \citet{Wilson60}. Subsequent radio observations of this SNR \citep{Velusamy74,
Angerhofer77,Landecker82, YarUyaniker04,Tian06,Kothes06} reveal a radio morphology
that may be described as a nearly complete circular shell with a diameter of approximately
34 arcminutes \citep{Green09b}. The
radio emission is brightest along the western rim and a prominent gap is seen along the
northern and northeastern sector of the circular emission. Like HB21, the magnetic field is
aligned in the tangential direction, also suggesting that the shell was compressed during the
radiative stage of the evolution of the SNR, but compared to HB21 the degree of polarization
is much lower (0.4\%$\pm$0.1\% -- see \citet{Kothes06}). The measured spectral index
of the observed radio emission is $\alpha$ $\sim$ 0.6 \citep{Landecker82, Kothes06,
Tian06, Green09b}. CTB 1 has also been detected at
optical wavelengths, in emission lines such
as [\,OIII\,] $\lambda$5007 and [\,SII\,] $\lambda\lambda$ 6716, 6731; the
observed optical shell-like morphology
closely matches the radio shell. The optical images of CTB 1 presented by
\citet{Fesen97} depict a remarkable contrast in the emission-line
properties of this SNR: while [\,SII\,] emission is seen from roughly the
entire optical shell (with the greatest amount of emission in the
south), the [\,OIII\,] emission appears to be almost entirely localized to
the western rim of the shell. \citet{Saken92}
detected infrared emission from CTB 1 in the 60 $\mu$m and 100 $\mu$m
{\it IRAS} bands: an arc of infrared emission was seen in
the 60 $\mu$m/100 $\mu$m ratio image that appears to be coincident with
the radio shell. CTB 1 was observed in X-rays with {\it ROSAT}
\citep{Hailey95,Rho95,Craig97}: like other MM SNRs, the X-ray emission from this SNR
(which has a thermal origin) lies interior to the radio and optical shells. Remarkably, the
X-ray emission is also seen to extend through the known northern gap of the SNR.
Like HB21, no pulsars or $\gamma$-ray sources are believed to be
associated with CTB 1: a radio search conducted by \citet{Lorimer98} for a pulsar
revealed no candidate sources. Published distance estimates
for this SNR have ranged from 1.6 to 3.5 kpc; in this paper we have adopted a
distance to CTB 1 of 3.1$\pm$0.4 kpc as measured by \citet{Hailey94}.
\par
The organization of this paper is as follows: in
Section \ref{ObsandReduction} we describe the {\it ASCA} X-ray observations of HB21 and
CTB 1 and the {\it Chandra} observations of CTB 1, including details of
data reduction. {\it ROSAT} and radio observations of these SNRs
are also described in this section.
In Section 3 we present the results of our spectral analyses for
both HB21 and CTB 1 (in Section 3.1 and Section 3.2, respectively).
In Section 4 we discuss the nature of the hard discrete X-ray source 1WGA
J0001.4$+$6229: we have discovered weak evidence for pulsed X-ray emission
from this source
(which is seen in projection against CTB 1) and consider the possibility
that it is a neutron star associated with CTB 1. We also present a search for radio
pulsations from 1WGA J0001.4$+$6229. Interpretations of our X-ray results for HB21 and CTB 1
are presented in Section 5 and Section 6, respectively.
We also detected hard X-ray emission from CTB 1: we discuss the nature of this emission
in Section \ref{CTB1HardSection}. Our preliminary results of this paper have been presented in
\citet{Pannuti04} after which we note that similiar data sets were analyzed and presented by
\citet{Lazendic06}. Our primary results of HB21 are in agreement with
and strengthen those of \citet{Lazendic06}; for CTB 1,
our paper presents extensive and thorough analysis of the {\it Chandra} and
{\it ASCA} data in
smaller-scale regions. We also report important new results for this SNR, including
the probable detection of oxygen-rich ejecta from CTB 1 as well as spectral variations across
the object. In addition, an X-ray hard point-like source is identified and an analysis
of its X-ray and radio properties is presented. Finally, the
conclusions of this work are summarized in Section 8.
\section{Observations and Data Reduction\label{ObsandReduction}}
\subsection{{\it ASCA} Observations of HB21 and CTB 1}
Because the X-ray emission from both HB21 and CTB 1 cover a large angular
extent on the sky, two pointed observations were made
of each SNR with {\it ASCA} \citep{Tanaka94}, namely the southeast and northwest regions of
HB21 and the southwest and northeast regions of CTB 1 (see Table \ref{ASCAObsTable}
for details of these observations). These observations provided almost
a complete spatial coverage of the X-ray emitting gas in both SNRs.
The data reduction was conducted
using the ``XSELECT" program (Version 2.2), which is available from the
High Energy Astrophysics Science Archive Research Center
(HEASARC\footnote{see http://heasarc.gsfc.nasa.gov.}). There were two
types of instruments onboard {\it ASCA} -- the Gas Imaging Spectrometer
(GIS) and the Solid-State Imaging Spectrometer (SIS) -- and both
of these instruments were composed of two units denoted as GIS2, GIS3,
SIS0 and SIS1, respectively. A single GIS unit sampled a field of view
$\sim 50'$ in diameter and a background
count rate of the GIS is 5 $\times$ 10$^{-4}$ counts cm$^{-2}$ sec$^{-1}$
keV$^{-1}$; in comparison, a single SIS unit sampled a field of view
approximately $44' \times 44'$ in size. The
nominal FWHM angular resolution of
both the GIS and SIS units were approximately 1 arcminute.
The standard REV2 screening criteria were applied when reducing both
the raw GIS and SIS datasets. We used ``FMOSAIC"
from the FTOOLS software package to generate an X-ray
map by combining the GIS2 and GIS3 maps.
We used the FTOOL ``MKGISBGD" to prepare
blank-sky background spectra and images for each extracted GIS source
spectra: for these background datasets, point sources which are brighter than
approximately 10$^{-13}$ ergs cm$^{-2}$ sec$^{-1}$ have been removed.
Similarly, background spectra were generated using standard SIS blank-sky
datasets for analyzing the extracted SIS source spectra.
The standard GIS2 and GIS3 response matrix files (RMFs) were used for
analyzing extracted the GIS source spectra while the FTOOL ``sisrmg" was
used to prepare RMFs for the extracted SIS source spectra. Finally, the
FTOOL ``ASCAARF" was used to prepare ancillary response files (ARFs) for
each extracted GIS and SIS source spectra.
\subsection{{\it Chandra} Observation of CTB 1}
We have also analyzed an archival dataset from an additional X-ray
observation of CTB 1 made with {\it Chandra} \citep{Weisskopf02}.
The corresponding ObsID of this observation is 2810 (PI: S. Kulkarni)
and it was conducted as part of a search for
central X-ray sources associated with Galactic SNRs. This observation
was conducted in FAINT Mode on 14 September 2002 with the Advanced
CCD Imaging Spectrometer (ACIS) at a focal plane temperature of
$-$120$^{\circ}$C such that the ACIS-I array of chips sampled a significant
portion of the X-ray emitting plasma located
interior to the radio shell of the SNR. The ACIS-I array is composed
of four front-illuminated CCD chips: each chip is 8$\farcm$3 $\times$
8$\farcm$3 and the field of view of the entire array is
approximately 17$'$ $\times$ 17$'$.
These chips are nominally sensitive to photons in
the 0.2 through 10 keV energy range: the maximum effective collecting
area for each chip is approximately 525 cm$^{2}$ at 1.5 keV. The full-width at half-maximum
(FWHM) angular resolution of each chip at 1 keV is 1$"$ and finally the spectral resolution
at 1 keV of each chip is 56.
These data were reduced using the {\it Chandra} Interactive Analysis of
Observations (CIAO\footnote{http://cxc.harvard.edu/ciao/}) package
(Version 3.1) with the calibration database (CALDB) version 2.29.
Standard processing was applied to this dataset: in particular, the task
``acis$\_$process$\_$events" was used to generate a new event file where
corrections for charge transfer inefficiency and time-dependent gain
were applied. The data were also filtered for bad pixels, background
flare activity and events which had a GRADE value of 1, 5 or 7.
Finally, we applied the good time interval (GTI) file supplied by
the pipeline (as well as the GTI file prepared when filtering for
background flares) and the resulting total effective exposure time
of the observation was 48.9 kiloseconds.
\par
Discrete sources were identified with the {\it CIAO}
wavelet detection routine ``wavdetect" \citep{Freeman02}: in making a
final image, these sources were excluded and the image was exposure-corrected
and smoothed with the {\it CIAO} task ``csmooth." We extracted spectra from several regions
of the diffuse X-ray emission seen in the {\it Chandra} images using the {\it CIAO} task
``dmextract": results of the spectral analysis are presented in Section
\ref{CTB1SubSection}. When extracting spectra, we
excluded point sources identified by ``wavdetect" to help reduce
confusion with emission from background sources. We prepared ARFs and RMFs using
the CIAO tools ``mkwarf" and ``mkrmf," respectively; background spectra
were generated using a reprojected blank sky observation made with the
ACIS-I array and available from the {\it Chandra} X-ray Center via the
World Wide Web.\footnote{See http://cxc.harvard.edu/contrib/maxim/acisbg/.}
\subsection{Additional Observations}
We also included {\it ROSAT} Position Sensitive Proportional Counter (PSPC)
observations of HB21 in our analysis: these observations were discussed
already in some detail in the Ph.D thesis of \citet{Rho95}. They extend
beyond the two fields observed by {\it ASCA} and provide the complete
spatial coverage of the SNR. The PSPC images are exposure and
particle background corrected and merged together using the analysis
techniques for extended objects developed by \citet{Snowden94}. The
smoothing technique includes neighboring pixels within a
circle of increasing radius until a selected number of counts is
reached to optimize the signal-to-noise ratio.
Lastly, we have augmented the X-ray datasets analyzed in this work
with radio data provided by the Canadian Galactic Plane Survey
(CGPS) \citep{Taylor03}. From this survey
we have obtained radio images of HB21 and CTB 1 at the frequencies of
408 MHz and 1420 MHz. The angular resolution and sensitivity of
the 408 MHz data are 3.$'$4 $\times$ 3.$'$4 csc $\delta$ and 0.75 sin
$\delta$ K (3.0 mJy beam$^{-1}$), respectively, while the angular resolution
and sensitivity of the 1420 MHz data are 1' $\times$ 1' csc $\delta$ and 71
sin $\delta$ K (0.3 mJy beam$^{-1}$), respectively. The reader is referred to
\citet{Taylor03} for more description about the CGPS radio observations
and the accompanying data reduction process.
\section{Results}
\subsection{HB21\label{HB21Section}}
In Figure \ref{hb21asca} we present our broadband (0.7-10.0 keV)
exposure-corrected mosaicked {\it ASCA} GIS image of HB21: we have
overlaid radio emission contours using CGPS observations at a
frequency of 408 MHz to show the extent of the SNR radio shell.
In Figure \ref{hb21rosat} we present
a mosaicked {\it ROSAT} PSPC image of HB21 (with the same radio
contours overlaid) which depicts the entire extent of X-ray emission
from the SNR (which extends beyond the two fields observed by {\it ASCA}).
It is clear from inspection of these images that the X-ray
emission is located in the interior of the well-defined SNR radio shell:
this combination of X-ray and radio morphologies exemplifies the
defining characteristics of mixed-morphology SNRs. The bulk of the
interior X-ray emitting plasma is located just south of a prominent
bend in the northern edge of the radio shell. We note that
\citet{Koo01} detected broad CO emission lines from the location of
this bend and \citet{Shinn09} presented near- and mid-infrared images of this
same region which showed shock-cloud interaction features. Both of these
studies indicated that this is a site of an interaction between
HB21 and a neighboring molecular cloud complex. In Figure
\ref{hb21softhard} we present additional mosaicked {\it ASCA} GIS
images which depict soft and hard emission (corresponding to the energy
ranges of E$<$1 keV and E$>$1 keV, respectively) from this SNR.
The GIS2 count rates for E$<$1 keV and E$>$1 keV are
1.91($\pm$0.08)$\times$10$^{-2}$ cts s$^{-1}$ and
3.21($\pm$0.15)$\times$10$^{-2}$ cts s$^{-1}$, respectively, while the
respective count rates for E$<$2 keV and E$>$2 keV are
4.56($\pm$0.13)$\times$10$^{-2}$ cts s$^{-1}$ and
0.56($\pm$0.08)$\times$10$^{-2}$ cts s$^{-1}$. Significantly more X-ray
emission is detected from HB21 at E$<$2 keV X-ray energies than at E$>$2 keV
energies, illustrating the soft spectral nature typical of SNRs.
\par
The GIS2/GIS3 spectra were extracted from elliptical regions approximately
23$\arcmin$ in size carefully selected to include most of the emission and
avoid the edges of the field of view. Likewise, the SIS0/SIS1
spectra were extracted from square-shaped regions approximately 10
arcminutes on a side and again the edges of the fields of view were
avoided. The spectral extraction regions are marked in Figure
\ref{hb21asca}. We analyzed the extracted spectra using the software
package {\it XSPEC}\footnote{http://heasarc.gsfc.nasa.gov.docs/xanadu/xspec/.}
Version 11.3.1 \citep{Arnaud96}.
For our spectral fitting we used two thermal models: the
thermal model VAPEC, which describes an emission spectrum from a
collisionally-ionized diffuse gas with variable elemental abundances
\citep{Smith00, Smith01a, Smith01b}\footnote{Also see
http://hea-www.harvard.edu/APEC.} and VNEI, which is
a non-equilibrium collisional plasma model which assumes a constant
temperature and single ionization parameter \citep{Hamilton83,
Liedahl95, Borkowski01}. Photoelectric absorption along the line of sight was
accounted for with the PHABS model; finally, we allowed the abundances
of silicon and sulfur to vary during the fitting process
(because lines associated with these particular elements are
noticeable in the spectra) while
leaving the abundances of the other elements frozen to solar values.
\par
In Table \ref{HB21SpectralTable} we present results of our simultaneous
fits to the GIS2/GIS3 and SIS0/SIS1 spectra for both the northwestern
and southeastern regions of HB21. We have obtained
statistically acceptable fits (with $\chi$$^2_{\nu}$ values of $\sim$ 1.04-1.06)
to the extracted spectra using
a single thermal component (that is, either the VAPEC model or the
VNEI model) for both the northwestern and southeastern
regions. The column density and temperature are
similar for both thermal models, namely $N$$_H$
$\sim$ 2-3$\times$10$^{21}$ cm$^{-2}$ and $\it{kT}$ $\sim$ 0.66-0.68 keV.
In Figure \ref{hb21spectra}, we present the extracted GIS2, GIS3, SIS0 and SIS1
spectra for the northwest region of HB21: in each case, the fits obtained
using ionization equilibrium (CIE) and nonequilibrium ionization models
are comparable in quality. In addition, the abundances of silicon
and sulfur in our spectral fits exceed solar abundances: these {\it ASCA} observations
are the first to reveal enhanced abundances of heavy elements in the X-ray spectra of HB21
(as also noticed by \citet{Lazendic06}). Our results are consistent with previous analyses of
X-ray emission from HB21 \citep{Leahy87, Rho95} where our present analysis of the broadband
{\it ASCA} spectra has yielded similar temperatures to those derived from {\it Einstein}
and {\it ROSAT} observations. However, with the data from the {\it ASCA} observations we
may establish more stringent constraints on the
ionization timescale and the abundances of sulfur and silicon.
The ionization timescales derived with the VNEI model for the two regions
are long ($\tau$ = 5.9 ($>$3.2)$\times$10$^{11}$ cm$^{-3}$ s
and $\tau$ = 4.1$^{+5.9}_{-1.1}$$\times$10$^{11}$ cm$^{-3}$ s for the northwestern and
southeastern regions of HB21, respectively) and within the error bounds for this
parameter, CIE is included ($\tau$ $\geq$ 10$^{12}$ cm$^{-3}$ s -- see \citet{Smith10}).
For this reason and because both the VAPEC and the
VNEI models return equally acceptable fits, we argue that the X-ray emitting
plasma located within the interior of HB21 is close to ionization equilibrium.
Similar to the results presented by \citet{Lazendic06},
we also observed slightly enhanced silicon and sulfur abundances in HB21: based on our
PHABS$\times$VAPEC (PHABS$\times$VNEI) fits, our measured abundances are
Si=1.3$^{+0.3}_{-0.2}$ (1.8$\pm$0.05) and S=2.4$\pm$1.0 (3.6$^{+1.7}_{-1.9}$)
for the northwestern region and Si=1.4$\pm$0.3 (2.0$\pm$0.4) and S=1.7$^{+0.9}_{-0.8}$
(3.0$\pm$1.4) for the southeastern region (see Table \ref{HB21SpectralTable}).
The slightly different abundances between the VAPEC and VNEI fits may be due
to either differences in the predicted line strengths between the non-equilibrium and equilibrium
conditions or different sets of atomic
data incorporated in these models. The respective lower limits for the abundances of Si (S) for
the northwestern region are 1.1 (1.3) and 1.4 (1.7), while the lower limits for the
southeastern region are 1.1
(1.6) and 0.9 (1.6). The northwestern region shows modestly more consistent evidence of
slightly enhanced silicon and sulfur abundances than the southeastern region.
This suggests a contribution to the observed X-ray spectra from ejecta material.
\par
In Figure \ref{HB21ConfContours} we present a plot of confidence contours for the
silicon and sulfur abundances based on the PHABS$\times$VAPEC fit to the spectrum of the
northwestern region. No significant spectral differences are seen between the
northwestern and southeastern regions of this SNR. In their analysis
of spectra extracted from the {\it ASCA} observations of HB21, \citet{Lazendic06}
found that fits obtained using the VNEI model to the spectra extracted for both regions
gave better fits at a statistically-significant level (4$\sigma$) than fits obtained by a thermal
plasma. In contrast, we find that fits obtained with VNEI and fits obtained with a standard
thermal plasma are comparable in quality. Also, \citet{Lazendic06} derived comparable
(though slightly larger) values for $N$$_H$ for both regions: those authors also found a
similar trend where $N$$_H$ is modestly elevated for the southeastern region compared
to the northwestern region. We suspect that the minor differences between our
results and those obtained by \citet{Lazendic06} may be attributed to small differences in
data reduction and spectral
analysis techniques (such as background subtraction). We also point out that
\citet{Lazendic06} commented that the derived column density
values seemed to be rather high if HB21 is indeed only 0.8 kpc distant (which was their
assumed distance to the SNR). They noted that an
elevated column density (at least for the eastern portion of the SNR) is comparable
to the column density of the complex of molecular clouds seen toward HB21 as described
by \citet{Tatematsu90}. We point out that if the larger distance to HB21 that we have assumed
(1.7 kpc) is adopted, the measured values for $N$$_H$ seem more reasonable.
\par
Lastly, to help determine more stringent constraints on the properties of
the X-ray emitting plasma (specifically the abundances of silicon and sulfur), we
simultaneously fit the GIS2 spectra extracted for the northwestern and southeastern portions
of the SNR with first the PHABS$\times$VAPEC model and then with the
PHABS$\times$VNEI model. For both models we obtained statistically acceptable fits
(with $\chi$$^2$$_{\nu}$ values of $\sim$ 1.1) with values for $kT$ and $N$$_H$ that
were consistent with those obtained for fits of the individual regions (that is, $kT$ $\sim$
0.6 keV and $N$$_H$ $\sim$ 0.2 $\times$ 10$^{22}$ cm$^{-2}$). The abundances for
both silicon and sulfur were indeed enhanced relative to solar: for the PHABS$\times$VAPEC
(PHABS$\times$VNEI) model, the abundances were 1.7$\pm$0.4 (1.9$^{+0.5}_{-0.4}$) for
silicon and 3.5$^{+1.9}_{-1.0}$ (4.3$^{+2.2}_{-1.8}$) for sulfur. The ionization timescale
derived for the PHABS$\times$VNEI model -- $\tau$ $\sim$ 4($>$0.03)$\times$10$^{13}$
cm$^{-3}$ s -- is also consistent with ionization equilibrium.We summarize the results
for these fits in Table \ref{HB21SpectralTable}.
\subsection{CTB 1\label{CTB1SubSection}}
A broadband (0.7--10.0 keV) exposure-corrected and mosaicked {\it ASCA} GIS
image of CTB 1 is presented in both Figures \ref{ctb1asca} and \ref{ctb1asca2}
with radio contours from the CGPS overlaid. Both HB21 and CTB 1 show the
typical center-filled X-ray morphology combined with a
shell-like radio morphology that characterizes MM SNRs: in the case of CTB 1,
however, as noted previously the X-ray emission is seen to extend through a gap along the
northeastern portion of the radio shell.
Figure \ref{ctb1softhard} shows mosaicked images of the
soft and hard X-ray emission (again corresponding to the energy ranges
E$<$1 keV and E$>$1 keV, respectively); there is a noticeable
difference in the spatial structure between the soft and the hard emission.
In Figure \ref{ctb1optical} we present an optical H$\alpha$ image of CTB 1
(courtesy of Robert Fesen) with the contours of the X-ray emission overlaid:
the optical morphology of CTB 1
is quite similar to the radio morphology with the same shell-like structure
and a prominent gap in the northeast \citep{Fesen97}. Little
H$\alpha$ emission is seen in the interior of CTB 1, nor is any optical
emission seen where the X-ray emission extends through the gap in the
optical and radio shell in the northeast. The observed slight extension of X-ray emission
in the west through the optical and radio shell is likely either
residuals due to the broad {\it ASCA} point-spread-function or point sources
rather than true emission from the SNR.
The {\it ASCA} hard image reveals a point-like source seen in projection against
the diffuse emission of CTB 1: it is located at RA (J2000.0) 00$^h$
01$^m$ 25.$^s$5, Dec (J2000.0) $+$62$^{\circ}$ 29$\arcmin$ 40$\arcsec$ and it lies
very close to the eastern edge of the optical and radio shell.
This source (denoted as 1WGA J0001.4$+$6229) may be
a neutron star possibly associated with CTB 1; we discuss it in
detail in the next section. In contrast to HB21, a significant
amount of emission from CTB 1 is seen at energies above 1 keV.
\par
Similar to our spectral analysis performed with HB21, we extracted GIS2
and GIS3 spectra from elliptical regions approximately 23$\arcmin$
in size from both the southwestern and northeastern portions of the
X-ray emitting plasma. When extracting GIS2 and GIS3 spectra from the
northeastern region, we excluded a region 4 arcminutes in diameter
centered on the position of the discrete X-ray source 1WGA
J00001.4$+$6229 to avoid spectral contamination by this source.
We also extracted SIS0 and SIS1 spectra from both the
southwestern and northeastern portions of the SNR, again using
square-shaped regions approximately 10 arcminutes on a side.
Unfortunately, the signal-to-noise ratio of the extracted SIS0 and SIS1
spectra for the northeastern region was not sufficient for spectral
analysis and we therefore omitted these spectra from our analysis.
We attempted to fit the extracted {\it ASCA} spectra using
the thermal models VAPEC and VNEI along with the model PHABS for the
photoelectric absorption. In contrast to HB21, our fits with either VAPEC
or VNEI {\it did not} produce a $\chi$$^2_{\nu}$ lower than 1.3 for either
region and failed to account for the
hard X-ray emission seen above $\sim$3 keV from both regions.
A two-component model with a soft thermal component with a temperature of
0.3-0.4 keV (either VAPEC or VNEI) and a second thermal component with a
higher temperature or a power law component was required
for an acceptable fit to the extracted spectra from both the
southwest and northeast portions of the SNR (see Table \ref{ctb1ascafit}).
Thus, a hard excess is present in the
{\it ASCA} spectra of CTB 1: \citet{Lazendic06} also identified a second
thermal component of X-ray emission from this SNR based on analysis of extracted
{\it ASCA} spectra but only for the southwestern region.
This hard X-ray
emission was not detected previously because the X-ray observatories
employed in prior observations of CTB 1 (such as {\it ROSAT}) lacked
the required sensitivity at high energies.
\par
We examine first the high-spatial resolution {\it Chandra} data before
discussing the hard emission in more detail.
In Figure \ref{ctb1chandra} we present a three-color {\it Chandra} image
of the interior X-ray emission surrounded by the well-defined optical and
radio shell of CTB 1. To illustrate the spectral properties of this
emission, in this Figure we have depicted soft (0.5-1.0 keV), medium
(1.0-2.0 keV) and hard (2.0-8.0 keV) emission in red, green and blue,
respectively. Many features with different spectral properties are
visible: most importantly, features with primarily medium and hard
spectra are clearly mixed together within the X-ray plasma.
At the angular resolution of {\it Chandra}, the hard
emission (such as the structure $\sim$ 1 arcminute in
size located at approximately RA (J2000.0) 23$^h$ 59$^m$ 01.0$^s$, Dec
(J2000.0) $+$62$^{\circ}$ 30$\arcmin$ 57$\arcsec$) is clearly diffuse and
not point-like.
\par
To investigate variations in the spectral properties of the X-ray-emitting
plasma of CTB 1 as revealed by {\it Chandra}, we extracted spectra
from three different {\it Chandra} chips. These three extraction regions
are as follows: the first region (which we refer to as the ``diffuse"
region) is on the ACIS-I2 chip and
covers most of the area of this chip. A second region (which we
refer to as the ``soft" region) is located on the ACIS-I3 chip and covers most
of the area of that chip as well. Lastly we extracted spectra from a third
region (which we will call the ``hard" region) which corresponds to a region
of hard emission mentioned above and is located on the ACIS-I1 chip:
the positions of all three regions are indicated in Figure \ref{ctb1chandra}.
A region of excess medium energy seen toward the middle of the field
in Figure \ref{ctb1chandra} largely fell into gaps between the {\it Chandra} detector chips:
for this reason, a detailed analysis of its X-ray spectrum could not be conducted.
\par
We present the extracted spectra of all three
regions in Figure \ref{ctb1chandraspec}: spectral variations are present across
the X-ray emitting plasma of CTB 1. For example, the spectrum from the
``diffuse" region shows prominent O Ly
$\alpha$ (0.65 keV), Ne IX (0.9 keV), and Mg XIII (1.35 keV) lines, together
with Fe L-shell line emission: in this spectrum the O and Ne lines are stronger
than the Fe L-shell line emission.
In contrast, the lines are hardly noticeable in the ``hard" region spectrum.
We derived an acceptable fit to the spectrum of the diffuse region
using a thermal model (VAPEC) with a temperature ${\it kT}$ =
0.28$\pm$0.03 keV and $N$$_H$ = 0.64$\pm$0.08$\times$10$^{22}$ cm$^{-2}$; this
fit reveals enhanced oxygen and neon abundances and a low iron abundance (see
Table \ref{ctb1chandrafit}). The column density
$N$$_H$=0.64$\pm$0.08 $\times$ 10$^{22}$ cm$^{-2}$ is consistent with the optical
extinction of E(B-V)=0.7-1 derived by \citet{Fesen97}, which is equivalent
to $N$$_H$$\sim$0.5-0.7 $\times$ 10$^{22}$ cm$^{-2}$.
In Figures \ref{ctb1OFeconf}a and \ref{ctb1OFeconf}b, we present
confidence contour plots for the abundances of oxygen compared to iron and neon
compared to iron (respectively) based on the fit derived from the PHABS$\times$VAPEC
model to the spectrum of the ``diffuse" region. These figures show that the oxygen and
neon abundances are both above solar while the iron abundance is below solar:
such relative abundances are a typical characteristic of oxygen-rich SNRs
\citep{Woosley95}.
\par
Our detection and analysis presented here of probable oxygen-rich ejecta in CTB 1
is the first detailed study presented
of oxygen-rich ejecta associated with an MM SNR. Based on X-ray spectral analysis,
\citet{Lazendic06} suggested that another MM SNR -- HB 3 -- may also feature oxygen-rich
ejecta, though those authors could not determine if the plasma associated with that SNR
has significantly enhanced abundances of oxygen, neon and magnesium or marginally
enhanced abundances of magnesium and underabundant iron.
The {\it Chandra} spectra are not well fit by a
single temperature thermal model (see the poor match to the
Mg XII Ly$\alpha$ line as seen in Figure \ref{ctb1chandrafit}),
so it is not possible to have oxygen,
neon and magnesium all residing in the same constant temperature plasma in
ionization equilibrium. Magnesium appears
to be more ionized than neon. This may be caused by higher
temperatures in magnesium-rich ejecta than in the oxygen- and neon-rich
ejecta.
It is also possible that the high temperature plasma could be
overionized; an overionized plasma has been reported in
the MM SNR IC 443 \citep{Kawasaki02}.
When we fit the spectrum of the
``soft" region using a VAPEC model combined with
the PHABS model for the interstellar photoelectric absorption,
the iron abundance is higher and the neon abundance is lower when compared with
the ``diffuse" region spectra (see Table
\ref{ctb1chandrafit}), suggesting spatial variations in the chemical composition
within the X-ray emitting gas of CTB 1.
\par
Lastly, we describe the
spectral analysis of the third (``hard") region which features a harder
spectrum when compared to the other two spectra (see Figure
\ref{ctb1chandraspec}).
Such localized hard emission has been reported in several
other MM SNRs, either as non-thermal diffuse knots seen in
{\it XMM-Newton} observations of IC 443 \citep{Bocchino03} or as a
pulsar wind nebula (also in IC 443 -- \citet{Olbert01}).
Because of the lack of lines
in the spectrum, we first fit the spectra with a power law, which resulted in
an acceptable fit, but with an unrealistically high photon index of
$\Gamma$$\sim$6.4$^{+5.6}_{-2.9}$ (see Table \ref{ctb1chandrafit}).
In comparison, a VAPEC thermal model yielded a comparable-quality
fit with a temperature of $\it{kT}$ =
0.66$^{+0.27}_{-0.41}$ keV: for this fit, we froze the abundance of oxygen
to 1.7 (to be consistent with the fits derived to the spectra of the
``diffuse" and ``soft" regions) or 1 while fitting for the abundances of neon
and iron.
We also fit the spectrum of this region with a
combination of an APEC thermal component and a non-thermal power-law
component; to reduce the number of free parameters we
fixed the temperature of the thermal component to ${\it kT}$ = 0.28 keV,
equal to the temperature derived from fits to the other two regions. The
results are
given in Table \ref{ctb1chandrafit}: unfortunately, we do not have enough
counts in the spectrum of the ``hard" region
to distinguish between non-thermal and thermal interpretations
of the spectrum of this source. However, the temperature derived from
the thermal fit ($kT$ = 0.66 keV) is significantly
higher than the temperature of the ``soft" and ``diffuse" regions. Also,
thermal models systematically underpredict {\it Chandra} spectra at high
energies in all the regions: we interpret this result as evidence
that the hard excess observed toward CTB 1 has a diffuse origin.
\par
In Figure \ref{harddiffuseconf}, we present a plot of confidence contours for
$N$$_H$ and $kT$ that correspond to the fit to the spectrum of the ``hard" region as
fit with the PHABS$\times$VAPEC model with fixed solar abundances. The spectrum
of this region demonstrates a bimodality with temperatures of $kT$ $\sim$ 0.28 and
$kT$ $\sim$ 0.66 keV, suggesting the presence of an additional component in addition
to the thermal component associated with the ``diffuse" region.
We will discuss the nature of this hard component further below and in
Section \ref{CTB1HardSection}. We also note differences between our results from analyzing
{\it Chandra} ACIS-I spectra and the results presented by \citet{Lazendic06}: those
authors jointly fit four spectra taken from each ACIS-I chip (they did not attempt any spectral
analysis of small regions like the ``hard" region") and presented an acceptable fit
obtained using two VNEI components: one with a temperature $kT$ = 0.20$^{+0.04}_{-0.01}$
keV and solar abundances and the other with a temperature $kT$=0.86$^{+0.03}_{-0.06}$
and an elevated magnesium abundance (Mg=3.1$^{+1.0}_{-0.4}$). Those authors
did comment on the presence of lines associated with oxygen as well as the neon and
iron line blend in the extracted ACIS spectra but they did not present an analysis of the
abundance of those elements.
\par
After establishing the spectral properties of the soft X-ray emitting plasma
from {\it Chandra} spectra, we fit the {\it ASCA} GIS2/3 and SIS0/1 spectra
of the southwestern portion of CTB 1. We froze the oxygen abundance to
1.7 (or 1) for all spectral fits while allowing the neon and iron abundances
to vary. Because models with a single thermal component were not sufficient to
describe the X-ray spectra, we used a combination of the thermal models
VAPEC and VNEI along with a power law model to jointly fit the {\it ASCA}
spectra: the results of these spectral analyses are given in Table
\ref{ctb1ascafit}. From these
fits, we estimate a column density $N$$_H$ $\sim$ 0.5-0.6 $\times$
10$^{22}$ cm$^{-2}$ and a temperature $\it{kT_{\rm{soft}}}$
$\sim$ 0.2-0.3 keV for the soft emission. The soft
thermal component has most likely attained
thermal equilibrium because fits to this soft emission using the VNEI
model resulted in a long ionization timescale ($\tau$ $\sim$
1 ($>$0.2) $\times$ 10$^{11}$ cm$^{-3}$ s). The hydrogen column density $N$$_H$
and the temperature of the soft component $\it{kT_{\rm{soft}}}$ are in
agreement with the analysis of {\it ROSAT} spectra \citep{Rho95, Craig97}.
The inclusion of a second component was necessary for obtaining
statistically-acceptable fits to the {\it ASCA} spectra of the southwestern
region (with values for the $\chi$$^2_{\nu}$ $\sim$1.1): the addition
of a power law with a photon index $\Gamma$ $\sim$ 2-3 or a second
thermal component with a temperature ${\it kT}$ $\sim$
3 keV yields fits with a comparable quality. Unfortunately due
to a low number of counts in the spectra at the higher X-ray energies,
we cannot distinguish between different models for the high-energy emission.
In Figure \ref{ctb1swspectra}
we present the GIS and SIS X-ray spectra of the southwest
region, fit with the combination of a thermal (VAPEC) and non-thermal
(power law) model. Our results differ from \citet{Lazendic06} who fit the
extracted ASCA spectra from this region with two thermal components in CIE: the softer
temperature component featured a temperature $kT$ = 0.19$^{+0.09}_{-0.03}$ keV and
a magnesium abundance fixed at solar while the harder temperature component featured
a temperature $kT$=0.82$^{+0.09}_{-0.06}$ keV and an elevated magnesium abundance
(Mg=2.7$^{+0.9}_{-0.5}$).
\par
Finally, we examine the {\it ASCA} spectra of the northeast region of
CTB 1 which corresponds to the known ``break-out" site seen in optical and radio images
of this SNR.. Like the southwest region, a second component
is needed (in addition to a soft thermal component) to derive a statistically
acceptable fit. A fit to
these spectra using a power-law component for the hard emission is
presented in Table \ref{ctb1ascafit} and the GIS spectra are presented in
Figure \ref{ascactb1gisspectra}. For the thermal component we have first
assumed the abundances of oxygen, neon and iron to be 1.7, 1.6 and 0.4, respectively,
equal to the abundances in the ``diffuse" region.
Secondly, we assumed solar abundances for the hard component. Although the
GIS spectra of the northeastern region feature a stronger Fe L-shell line
complex when compared with the spectra of the southwestern region,
our spectral fits could not confirm
non-solar abundances and the abundances are consistent with solar values.
\par
The photon index derived from fits to the northeast is flatter than the
photon index derived from fits for the southwest region: $\Gamma =
1.4$ compared to $\Gamma = 2-3$, respectively. However, the photon statistics
at higher energies is poor, making it difficult to determine
the true nature of this hard emission. Although the northeast region does
correspond to the prominent breakout
feature, we do not find evidence for any major differences in the X-ray
properties between the northeast and southwestern regions of CTB 1.
The fact that no significant variations are seen on large scales in the X-ray
properties of CTB 1, coupled with the variations seen on small scales as
revealed by the {\it Chandra} observation, indicate that the X-ray emission
from this SNR is complex. A parallel may be drawn with the results from
{\it Chandra} observations of 3C 391, another MM SNR: in the case of that
source, local spectral differences appeared to be stronger than global ones
\citep{Chen04}. Here again we find that our results differ from
those presented by \citet{Lazendic06}: for this region, the authors derived an acceptable fit
using a single thermal component ($kT$ = 0.18$^{+0.00}_{-0.01}$ keV) in CIE with solar
abundances.
\section{1WGA J0001.4+6229 -- An X-ray Pulsar Associated with
CTB 1?\label{WGASection}}
The {\it ASCA} hard energy image ($E$ $>$ 1 keV) revealed a hard source in the
northeastern region of CTB 1 which is located just inside the eastern
shell of the SNR: the position of this source is RA (J2000.0) 00$^h$
01$^m$ 25.$^s$5, Dec (J2000.0) $+$62$^{\circ}$ 29$\arcmin$ 40$\arcsec$
with a positional uncertainty of 13$\arcsec$. This is the discrete X-ray
source 1 WGA J0001.4$+$6229
in the Catalog of {\it ROSAT} PSPC WGA Sources \citep{White94, White97,
Angelini00}\footnote{Also see http://wgacat.gsfc.nasa.gov.}. This X-ray
source may possibly be a neutron star associated with CTB 1. We
therefore conducted spectral and timing analysis of the X-ray emission
from this source using the {\it ASCA} GIS2 and GIS3 datasets.
\par
The procedure for extracting GIS2 and GIS3 spectra of 1WGA J0001.4$+$6229
and performing a spectral analysis was the same as for
extracting GIS2 and GIS3 spectra of the diffuse emission from HB21 and
CTB 1: a circular region
four arcminutes in diameter centered on the source position was used to
extract spectra. The total
number of counts and the corresponding count rate (over the energy
range from 0.6 keV to 10 keV) for our GIS2 and GIS3 observations were
115 and 103 counts, and 5.71$\pm$0.53$\times$10$^{-3}$ and
5.11$\pm$0.51$\times$10$^{-3}$ counts per second, respectively.
We derived a
statistically-acceptable joint fit to the spectra using a simple power
law model combined with the same PHABS model mentioned previously for
the photoelectric absorption along the line of sight. The parameters of
this fit were a column density of $N$$_H$=0.3 ($<$0.65$)\times
10^{22}$ cm$^{-2}$ and a photon index $\Gamma$=2.2$^{+0.5}_{-1.2}$:
in Figure \ref{ctb1conf} we present the extracted GIS2/GIS3 spectra
together with the best-fit model, and a confidence contour plot for
$N$$_H$ and $\Gamma$.
The derived photon index is typical for
rotation-powered pulsars. The column density is consistent
with the range of column densities derived in our fits (see Tables
\ref{ctb1chandrafit} and \ref{ctb1ascafit}) to the
CTB 1 spectra, hinting at a possible association.
If we fit the extracted spectra for this source with a blackbody model,
a temperature of kT$\sim$1.1 keV is derived (although this fit with
$\chi$$^2_{\nu}$=0.92 for 48 degrees of freedom (DOF) is
inferior to the power-law fit with $\chi$$^2_{\nu}$=0.79). Because of the low derived
column density, our estimated absorbed and unabsorbed fluxes for this
source are virtually identical: for the GIS2 (GIS3) spectrum, the flux
is 5.4$\times$10$^{-13}$ (6.4$\times$10$^{-13}$) ergs cm$^{-2}$
sec$^{-1}$; at the assumed distance to CTB 1, these
fluxes correspond to luminosities of 6.2$\times$10$^{32}$
(7.4$\times$10$^{32}$) ergs s$^{-1}$, respectively. We also performed a
timing analysis using the GIS2/GIS3 datasets to search for pulsed X-ray
emission from this source and detected
a period of 47.6154 milliseconds using the Rayleigh test (a maximum signal
of $Z$$^2$ = 31.4), but the detection is not statistically significant.
\par
We further searched for pulsations from 1WGA J0001.4$+$6229
with the 100-meter Green Bank Telescope (GBT) of the National Radio
Astronomy Observatory (NRAO\footnote{The National Radio Astronomy
Observatory is a facility of the National Science Foundation, operated
under cooperative agreement by Associated Universities, Inc.}) on 2004
December 7. The
target position was observed for 7.6 hours at a center frequency of 825
MHz. The frontend was the GBT Prime Focus 1 receiver to feed the Pulsar
Spigot \citep{Kaplan05} and Berkeley-Caltech Pulsar Machine (BCPM)
backends. The receiver provided 50 MHz of bandwidth in two orthogonal
polarizations that were summed and synthesized into 1024 frequency
channels every 81.92 $\mu$s in the Spigot and 96 channels of 0.25 MHz
width every 144 $\mu$s in the BCPM. The interstellar dispersion toward
CTB1 is unknown, but we can estimate it with the latest model of
Galactic electron density \citep{Cordes02}, which predicts a dispersion
measure (DM) of 105 pc cm$^{-3}$ for a distance of 3.1 kpc or DM = 33
pc cm$^{-3}$ at a distance of 1.6 kpc (corresponding to the two distances
to CTB 1 that have been published in the literature). We therefore take
a conservative upper limit
of DM = 1000 to CTB 1 (given the narrow channels and relative long
pulsation period the search is not highly DM dependent). The data set
was dedispersed with DMs from 0 to 1000 and searched for periodicities
using standard folding and fast Fourier-Transform (FFT)-based
techniques. Based on these analyses, we find no significant evidence
for pulsations with any period from 1WGA J0001.4$+$6229.
\par
Assuming that 1WGA J0001.4$+$6229 is in fact a neutron star associated
with CTB 1, a transverse velocity can be estimated. The angular displacement
of 1WGA J0001.4$+$6229 from the center of CTB 1 is
14$'$ while the radius of the SNR itself is 17$'$. Therefore, we calculate
a transverse velocity $v$ = 850 d$_{3.1}$
t$_{1.6}^{-1}$ km s$^{-1}$, where d$_{3.1}$ is the distance to CTB 1 in units
of 3.1 kpc and t$_{1.6}$ is the age of the SNR \citep{Fesen97} in units of 1.6
$\times$ 10$^4$ yr.
This estimated transverse velocity is high but this may be
an overestimate because of the considerable uncertainties associated with
estimates of the distance and age of CTB 1: if we assume a distance
to the SNR of 1.6 kpc, the tranverse velocity is only
420 km s$^{-1}$.
In particular, the published age estimates are based on simple one-dimensional
SNR models: the obvious breakout morphology of CTB 1 clearly indicates
that such models are not applicable in this case. For comparison, the
transverse velocities for neutron stars located off center
in their associated SNRs are 375 km s$^{-1}$ in the case of the SNR
W44 \citep{Frail96} and 250$\pm$50 km s$^{-1}$ in the case of the SNR IC 443
\citep{Olbert01}.
It is plausible that 1WGA J0001.4$+$6229 is associated with CTB 1
but deeper high-spatial resolution X-ray observations are needed to
examine its spectrum in more detail and search for possible pulsations.
\section{Plasma Conditions in HB21\label{HB21SubSection}}
We first estimate the density and mass of the X-ray emitting
plasma associated with HB21 based on the emission measures derived from
our spectral fitting. Our GIS spectral extraction regions extend over
approximately 11$\farcm$5 $\times$ 11$\farcm$5 or 5.7 pc $\times$
5.7 pc (2.7 pc $\times$ 2.7 pc) at the assumed 1.7 (0.8) kpc distance to HB21. Assuming a
cylindrical geometry with the long axis equal to the observed
extent of the X-ray plasma (35$\farcm$8, corresponding to 17.7 (8.3) pc),
the volume of each region is approximately 5.4$\times$10$^{58}$ (5.6$\times$10$^{57}$)
cm$^{3}$. From the mean values of our derived emission measures (which
are approximately the same for all regions and all
models), we calculate an electron density $n$$_e$ $\approx$ 0.06 (0.08) cm$^{-3}$
(where we have assumed $n$$_e$ $\approx$ 1.2$n$$_H$) and a volume filling
fraction of unity based on the smooth appearance and isothermal nature
of the X-ray emitting gas. Based on this value, we estimate the total mass of the X-ray
emitting plasma within the field of view of the {\it ASCA} observations to be only
$\approx$$2.6 (1.5) M_{\odot}$. When we account for the incomplete spatial coverage of
HB21 by the {\it ASCA} observations, the total
X-ray mass could be higher by a factor of 9, which amounts to a total of
23.4 (14) $M_\odot$. The corresponding Si and S masses are
$1.7 \times 10^{-3} M_{\odot}$ and
$1.3 \times 10^{-3} M_{\odot}$, respectively.
The presence of a bright radio shell without associated X-ray emission
combined with the detections of an expanding HI shell and infrared emission from shock-cloud
interaction regions
\citep[][and the references therein]{Koo91,Shinn09,Shinn10} imply that HB21 is in a
radiative cooling stage.
The SNR age inferred from the presence of an expanding
($v_{exp}=124$ km s$^{-1}$) HI shell is $t$$_d$ = 4.5$\times$10$^4$ yr
\citep{Koo91}. We can also infer the pressure within HB21 from properties
of the X-ray emitting gas. The total number of particles is
$n$$_{total}$ = $n$$_e$+$n$$_H$+$n$$_{He}$$\approx$2$n$$_e$ for a plasma
with cosmic abundances: from our estimated values for electron density
and temperature, the corresponding pressure is P/k = 2$n$$_e$$T$ =
0.9 (1.2)$\times$10$^6$ K cm$^{-3}$, which as about two orders of
magnitude higher than the typical ISM pressure. The physical properties of HB21 are
summarized in Table \ref{physical}.
\par
The X-ray properties of HB21 are similar to those measured for many other
MM SNRs. First, the presence of an isothermal plasma with a temperature of $kT$ $\sim$
0.2-0.7 keV is consistent with other MM SNRs such as 3C391 \citep{Rho96, Chen04}, W44
\citep{Rho94, Shelton04}, 3C400.2 \citep{Yoshita01}, W51C
\citep{Koo02}, W63 \citep{Mavromatakis04}, and Kes 79 \citep{Sun04}.
This result supports the interpretation that
these SNRs are evolved and in the radiative phase as suggested by
the presence of infrared, optical and HI shells for many of
these SNRs. There are no temperature variations and no pronounced enhancements
of chemical abundances in HB21 either, just as in many other MM SNRs like
3C 391.
These properties of HB21 exemplify the typical X-ray properties characteristic
of MM SNRs as defined by \citet{Rho98}. At a sufficiently old age ($\sim 10^6$
yr) age, a SNR should exhibit a centrally-filled X-ray morphology and
eventually merge with the hot ISM gas \citep{Cui92}: however, MM SNRs
attain this state at a much earlier age ($\sim 10^4$ yr).
When a distance of 1.7 kpc to HB21 is assumed, the calculated X-ray
emitting mass of this SNR is comparable to those of other MM SNRs.
For standard radiative SNR models, we expect
$\sim 100$ M$_\odot$ of X-ray emitting gas at the HB21 age of
$4 \times 10^4$ yr based on equations given by \citet{Cui92} and models
presented by \citet{Hellsten95}. Even more X-ray emitting gas is
expected in conduction models of \citet{Cox99}. This discrepancy between
the observed and predicted X-ray emitting mass is also present in W28
\citep{Rho02}, but unlike in W28 (where large temperature gradients
have been detected), the presence of an isothermal plasma at the center of HB21
suggests that the electron thermal conduction is important \citep{Chevalier99}.
The conduction model of \citet{Cox99} overpredicts the mass of
X-ray emitting gas: however, this model assumes a uniform ambient ISM while HB21 is
known to be interacting with with clumpy molecular clouds \citep{Koo01,Shinn09,Shinn10}.
More elaborate X-ray emission models
of SNRs in molecular clouds are needed to account for the observed X-ray
properties of HB21 and similar MM SNRs.
\section{Supernova Ejecta in CTB 1\label{CTB1EjectaSection}}
The enhanced abundances of oxygen and neon and low iron abundances in
the ``diffuse" region (see Table \ref{ctb1chandrafit}) indicate that CTB 1 is likely an
oxygen-rich SNR. This SNR was likely produced by a core-collapse SN explosion,
because such explosions produce O- and Ne-rich, and Fe-poor ejecta
\citep{Nomoto97,Woosley95}. This finding is consistent with the presence of
a massive star forming environment near CTB 1 \citep{Landecker82}: in addition, the
scenario for the creation of this SNR by a core-collapse SN explosion would be further
supported if the discrete X-ray source discussed earlier -- 1WGA J0001.4$+$6229 --
is shown to be a neutron star associated with this SNR.
Examples of oxygen-rich SNRs include young sources like Cas A, N132D,
and E0102.2$-$72.3; recently two older SNRs located in the Small Magellanic Cloud
(SMC) -- SNR B0049-73.6
\citep{Hendrick05} and B0103-72.6 \citep{Park03} -- were also classified as
oxygen-rich SNRs. Both of these SMC SNRs show the ejecta material
in their interiors surrounded by shells of swept-up ambient material at
relatively low X-ray emitting temperatures. An X-ray emitting shell might
be present in CTB 1, but its detection may be prevented by substantial interstellar
absorption in this direction combined with
an expected low temperature of the shocked ambient gas.
\par
We estimated the X-ray mass and density of CTB 1 from our fits to the
{\it ASCA} spectra, assuming metal abundances
derived from the {\it Chandra} spectra of the ``diffuse" and ``soft"
regions. The spectral fits to the GIS spectra in the
11$\farcm$5 $\times$ 11$\farcm$5 region
imply an electron density of $0.16 f^{-1/2}_{soft}$ cm$^{-3}$ for the soft
thermal component (where $f_{soft}$ is the volume filling factor for this
component). The corresponding hydrogen density $n_H$ is equal to
$n_e/1.2 = 0.13 f^{-1/2}_{soft} {\rm cm}^{-3}$; based on this value we
estimate the total mass of the X-ray emitting plasma to be $\approx
40 f^{1/2}_{soft} M_{\odot}$.
From our derived abundances of oxygen, neon and iron based on {\it
Chandra} data, we estimate the oxygen, neon and iron masses to be
$0.66 f^{1/2}_{soft}$, $0.11 f^{1/2}_{soft}$, and $0.03 f^{1/2}_{soft}$
M$_{\odot}$,
respectively. The ratio of [O/Fe] is 4.3$^{+10.2}_{-2.5}$ and [Ne/Fe] is
4.0$^{+8.0}_{-2.2}$ for CTB 1.
The expected ratio of [O/Fe] is 0.75 for a Type Ia explosion and greater
than 4 for a core-collapse explosion.
These abundances imply that CTB 1 is a remnant of a core-collapse explosion
and are consistent with the predictions for a stellar progenitor with a mass
of 13 - 15 M$_{\odot}$ \citep{Woosley95, Nomoto97}, but higher mass stellar
progenitors are not excluded.
\par
Finally, we estimate the pressures of the
soft and hard components of the X-ray emitting gas using the parameters of
the PHABS$\times$(VAPEC+VAPEC) model: for the soft component we calculate a
corresponding pressure P/k = 1.1$\times$10$^6$ $f$$_{soft}^{-1/2}$ K
cm$^{-3}$. For the hard component (assuming a thermal origin), we first
need to calculate the corresponding electron density which we can determine
from the electron density of the soft component and the emission
measures (EMs) of the soft and hard components (i.e., $n_{e}$(hard) =
$n_e$(soft) [EM$_{hard}$/EM$_{soft}$]$^{1/2}$). From this relation, we
obtain $n$$_e$(hard) = 0.029 $f$$_{hard}^{-1/2}$ cm$^{-3}$
(here $f$$_{hard}$ is the volume filling factor for the hard component)
and therefore a corresponding pressure P/k = 2.0$\times$10$^6$
$f$$_{hard}^{-1/2}$ K cm$^{-3}$. This result implies a factor of
four larger filling factor for the hotter gas than the cooler gas if
these two components are in pressure equilibrium: a higher filling
factor for the hot gas is typical. Assuming $f_{soft}+f_{hard}=1$, pressure
within CTB 1 is $2.8 \times 10^6$ K cm$^{-3}$.
We summarize these inferred physical properties for CTB 1 in
Table \ref{physical}.
\par
CTB 1 therefore belongs to a growing number of known evolved SNRs which feature
an enhanced metal abundance in their interiors. An example of another MM SNR
which features such enhanced abundances is W44 \citep{Shelton04}: other
similar sources are identified by \citet{Lazendic06}
(including HB21, which was analyzed both in their study and in the
study presented here.)
In addition, two other Galactic SNRs -- the Cygnus Loop \citep{Miyata98}
and G347.7$+$0.2 \citep{Lazendic05} -- feature enhanced abundances of metals as well.
W49B \citep{Hwang00} shows highly enhanced abundances but its age is estimated to be
2000 years \citep{Hwang00} and \cite{Rho98} describe the source as an atypical MM SNR.
Because MM SNRs like CTB 1 are commonly believed to be evolved sources --
age estimates of CTB 1 range from 9000 yr \citep{Craig97} to 4.4$\times$10$^4$ yr
\citep{Koo91} -- their X-ray spectra are dominated by swept-up material \citep{Rho98}. Therefore,
the detection of X-ray-emitting material associated with these sources with enhanced metal
abundances is unexpected. The detection of O-rich ejecta associated with CTB 1 is
particularly noteworthy: CTB 1 may belong to a previously unrecognized class of MM SNRs
whose X-ray emission is dominated by O-rich ejecta located within
their interiors. As noted previously, another possible member of this particular class of
MM SNRs with O-rich ejecta may be
HB3 \citep{Lazendic06}. We note that O-rich ejecta has been previously detected in the evolved
(1.4 $\times$10$^4$ yr old) SMC SNR B0049-73.6 by \citet{Hendrick05} and it is likely
that centrally-located ejecta will be found in a number of relatively old
Galactic SNRs.
\par
The X-ray emitting plasma associated with CTB 1 clearly extends through
the gap in the crescent-shaped radio
shell: \citet{Hailey94} and \citet{Rho95} first noticed this remarkable
extension of X-ray emission based on {\it ROSAT} PSPC observations.
Two scenarios have been proposed to explain the morphology of the
observed X-ray emission: \citet{Craig97} has suggested that the
ambient ISM toward the northeastern portion of the
SNR was cleared by a supernova event which took place prior
to the birth of CTB 1, and thus
a breakout occurred as the SNR expanded into this region of
a dramatically lower density.
A competing theory for the morphology has been proposed by
\citet{YarUyaniker04}, who suggested that the X-ray emission from CTB 1
lies in the interior of a bubble seen in the 21 cm H line, presumably blown by
winds of the CTB 1 stellar
progenitor. Our X-ray images show that the diffuse X-ray
emission in the northeast clearly extends through the relatively narrow
break in the optical and radio shell, with the breakout directed into the
interior of the bubble seen in the neutral hydrogen line. Such a morphology
favors a scenario where a supernova explosion occurred within the HI shell,
followed by subsequent breakout into
the bubble and not an explosion within the bubble itself.
Our {\it ASCA} spectra show little difference
between the X-ray properties of the southwest and northeast regions:
hints of variations in temperature and abundances exist but
better X-ray data are needed to determine whether they are real and not just
statistical fluctuations.
\section{The Nature of the Hard X-ray Emission from
CTB 1\label{CTB1HardSection}}
The {\it ASCA} spectra of the southwestern portion of CTB 1 revealed
the presence of a hard component in addition to the soft ($kT$ $\sim$ 0.28 keV)
component. This hard
component may be modeled as a second thermal component with a temperature of
$\it{kT}$ $\sim$ 3 keV or as a power-law continuum with a photon index
$\Gamma$ $\sim$ 2-3.
Hard X-ray emission was also detected by {\it ASCA} in the northeast
region of CTB 1: a power-law component with a photon index $\Gamma$
$\sim$ 1.4 (a somewhat lower value compared to the southwest region) combined
with a soft thermal component (again with a temperature $\it{kT}$ $\sim$ 0.28 keV)
yields a statisically acceptable fit. The {\it Chandra} observation of the
southwest region of CTB 1
revealed regions of harder emission patches on the scale of an
arcminute in size: the spectrum of one of these regions can be modeled by
either a single thermal component with an elevated temperature ($\it{kT}$
$\sim$ 0.66 keV) or as the combination of a soft thermal component
($\it{kT}$ $\sim$ 0.28 keV) and a power law component with a photon index
$\Gamma$ $\sim$ 2.0 (see Tables \ref{ctb1chandrafit} and
\ref{ctb1ascafit}). The ``hard" region observed by {\it Chandra}
is only a few arcminutes from the center of CTB 1 and well inside the
radio-emitting shell of the SNR: it is diffuse in nature although the number
of counts detected from the source is limited. It is possible that the
hard X-ray emission detected by {\it ASCA} from CTB 1 may be composed of
localized hard regions such as this one: unfortunately we do not have
enough counts in this ``hard" region to distinguish
between thermal and non-thermal origins.
We note that two other MM SNRs, W28 and IC 443, contain high-temperature
thermal plasmas in their interiors \citep{Rho02, Kawasaki02}.
\par
Several possible explanations may
be considered for the origin of hard X-ray emission
from MM SNRs: first, the hard emission may be caused by
temperature variations within the SNR. In the case of CTB 1, this scenario
is supported by a good fit to the spectrum of the ``hard" region
with a thermal component with a much higher than average temperature.
Supernova ejecta may be inhomogeneous, in which case a multi-temperature
plasma with spatially-varying abundances is expected.
Alternatively, the ``hard" regions may be caused by localized nonthermal
emission: such emission has already been detected in IC 443 \citep{Bocchino03}
and $\gamma$ Cygni \citep{Uchiyama02}.
Additional observations are needed to understand the true nature of the hard
X-ray emission from CTB 1.
\section{Summary}
1. We presented {\it ASCA} observations of the MM SNR HB21. Our
{\it ASCA} images of this SNR are similar to {\it ROSAT} images and
reveal a diffuse centrally filled X-ray emission located
within a radio shell. From X-ray spectra, we measure a column density
toward this source and a temperature for the X-ray emitting plasma of
$N$$_H$ $\sim$ 0.3$\times$10$^{22}$ cm$^{-2}$ and
$\it{kT}$ $\sim$ 0.7 keV, respectively: no significant spatial
differences in temperature are found.
Silicon and sulfur abundances are
slightly enhanced relative to solar, particularly for the northwestern region, and
no hard component to the X-ray emission was detected.
The properties of HB21 are similar to those seen in several other MM SNRs,
such as the presence of isothermal plasma. This result supports the
interpretation that MM SNRs are evolved sources currently in the radiative
phase of evolution: the X-ray properties of HB21 exemplify the primary
characteristics of MM SNRs as defined by \citet{Rho98}.
\par
2. We presented {\it ASCA} and {\it Chandra} observations of the MM SNR CTB 1.
{\it ASCA} observations reveal center-filled X-ray emission located
within the radio shell: the X-ray emission extends outside
the circular shell through the breakout
gap in the northeast. While the global X-ray and radio morphology is
similar to HB21, the X-ray spectra of CTB 1 and HB21 are very different. The
X-ray spectrum of CTB 1 shows several prominent lines such as O Ly$\alpha$
(0.65 keV) and Ne IX (0.9 keV). We find that CTB 1 is likely an oxygen-rich SNR
with enhanced abundances of oxygen and neon: this is surprising for
an evolved SNR such as CTB 1.
The derived abundances are consistent with an explosion of a stellar
progenitor with a mass of 13 - 15 M$_{\odot}$ and possibly even higher.
\par
3. The {\it ASCA} spectra of the southwest region of CTB 1 cannot be fit
with a single thermal
component and instead require the presence of an additional component
to account for an excess emission seen at higher energies. Based on
{\it ASCA} and {\it Chandra} spectra of CTB 1, we derive a column
density $N$$_H$ $\sim$ 0.6$\times$ 10$^{22}$ cm$^{-2}$ and the soft
component temperature $\it{kT}$$_{soft}$$\sim$ 0.28 keV; the hard
emission may be modeled either by a thermal component with a
temperature $\it{kT}$$_{hard}$ $\sim$ 3 keV or by a power law component
with a photon index of $\Gamma$ $\sim$ 2-3. Likewise, the {\it ASCA}
spectra of the northeast region of CTB 1 also show an excess at
higher energies: these spectra are fit best by a power law with a
photon index $\Gamma$ = 1.4 plus the soft thermal component. The {\it Chandra}
observation of the southwestern region reveals localized regions of
hard emission: one such region is $\sim 1'$ in size. The X-ray
spectrum of this region may be fit with either a higher temperature
thermal component ($\it{kT}$ = 0.66 keV) or with the combination of a softer
thermal component ($\it{kT}$ = 0.28 keV) and a power law component
($\Gamma$ $\sim$ 2.0). Because of the poor photon statistics,
its true nature is unclear. Possible scenarios for its origin include
temperature variations within the X-ray emitting plasma of CTB 1, including
the ejecta, or localized non-thermal X-ray emission.
4. The {\it ASCA} hard ($E > 1$ keV) image of CTB 1 reveals a point-like
source seen in projection against
the diffuse emission of CTB 1. This source -- denoted as 1WGA J0001.4$+$6229
and located at RA (J2000.0) 00$^h$ 01$^m$ 25.$^s$5, Dec (J2000.0)
$+$62$^{\circ}$ 29$\arcmin$ 40$\arcsec$ -- may be
a neutron star associated with CTB 1. The
GIS2/GIS3 spectra of this source are well-fit by a power-law continuum
with a photon index $\Gamma$=2.2$^{+0.5}_{-1.2}$ (typical for rotation-powered
pulsars) and the measured column density is comparable to the column density measured
for CTB 1. There is marginal evidence for pulsations in X-ray data at
47.6 msec, but no pulsations have been detected at radio wavelengths.
\acknowledgments
We thank the referee for many useful comments which helped improve the
overall quality of the manuscript.
We acknowledge useful discussions with Steven Reynolds regarding the
nature of the hard X-ray emission seen toward CTB 1. T.G.P. thanks
Ken Ebisawa and Koji Mukai for their assistance with analyzing the
{\it ASCA} data, Keith Arnaud for suggestions during the spectral
fitting process and Ilana Harrus for her assistance with making the
mosaicked X-ray images of HB21 and CTB 1. T.G.P. also thanks Daniel Harris,
Samantha Stevenson and Nicholas Lee for helpful suggestions regarding the
reduction of the {\it Chandra}
observations of CTB 1. We also thank Robert Fesen for kindly
sharing his optical images of CTB 1 with us, Bryan Jacoby for
assistance with the GBT observations and Eric Gotthelf for
his contributions in searching for pulsed X-ray emission from
1WGA J00001.4$+$6229.
This research has made use of NASA's Astrophysics
Data System and data obtained through the High Energy
Astrophysics Science Archive Research Center Online Service, provided by
the NASA/Goddard Space Flight Center. The research presented in this
paper has used data from the Canadian Galactic Plane Survey, a
Canadian project with international partners, supported by the
Natural Sciences and Engineering Resources Council.
|
{
"timestamp": "2010-09-22T02:00:56",
"yymm": "1009",
"arxiv_id": "1009.3987",
"language": "en",
"url": "https://arxiv.org/abs/1009.3987"
}
|
\section{Introduction}
The study of stability of spatially periodic traveling wave
solutions to various classes of partial differential equations
motivates the study of $L^2(\RM;\CC^n)$ (essential) spectra of
periodic-coefficient differential operators
\be\label{e:L1}
L=(\partial_x)^m a_m(x) + \dots + \partial_x a_1(x) + a_0(x)
\ee
on the line, where coefficients $a_j\in \CC^{n\times n}$ are periodic
with period $X$.
By Floquet theory,
it is equivalent to study the $L^2([0,X]; \CC^n)$ point spectra of
the family of
Bloch operators
$$
L_\sigma=(\partial_x+i\sigma)^ma_m(x) + \dots
+(\partial_x+i\sigma)a_1(x) + a_0(x),
$$
where $X$ is the common period of the coefficients and $\sigma\in [0,2\pi)$ acts as a parameter.
Indeed, using this decomposition we have\footnote{Unless otherwise stated, throughout this paper all
functions are assumed to be complex valued and we adopt the notation $L^2(\RM)=L^2(\RM;\CM)$
and similarly for $L^2_{\rm per}([0,X])$.}
\[
{\rm spec}_{L^2(\RM)}\left(L\right)=\bigcup_{\sigma\in[0,2\pi)}{\rm spec}_{L^2_{\rm per}([0,X])}(L_\sigma);
\]
see, for example, \cite{G} for more details.
Due to the mathematical difficulties
involved
in
analytically computing the $L^2(\RM)$ spectrum of such an,
in general, variable-coefficient and vector-valued, operator,
or, equivalently,
computing the periodic spectra of the full family of associated Bloch operators,
the determination of spectrum of periodic-coefficient operators
is typically carried out numerically.
This may be accomplished in a number of ways: for example, shooting,
discretization, or various spectral and Galerkin methods.
See Appendix B, \cite{JZN}, for
further discussion.
A particularly natural
and direct
approach is Hill's method
\cite{DK},\footnote{
A convenient implementation may be found in the
numerical package SpectrUW \cite{CDKK}.}
a spectral Galerkin method carried out in a periodic Fourier basis, which
is exact in the constant-coefficient case.
In this method, to
approximate the spectra of $L_\sigma$ for a fixed $\sigma\in[0,2\pi)$,
one considers the eigenvalue problem
\begin{equation}\label{e:gspec}
L_\sigma v=\lambda v,
\end{equation}
by expressing the coefficients $a_j$ of $L_\sigma$ and the function $v$ as Fourier series in $L^2_{\rm per}([0,X])$,
as an infinite-dimensional matrix equation in $\ell^2$.
Truncating the Fourier modes to frequencies $|k|\leq J$ for each $J\in\NM$,
one then obtains a sequence of finite-dimensional
matrix eigenvalue problem whose eigenvalues approximate true eigenvalues of the operator $L_\sigma$ on $L^2_{\rm per}([0,X])$.
See Section \ref{s:hill} for further details.
This method is fast and easy to use, and in practice
appears to give excellent results
under quite general circumstances \cite{DK,BJNRZ1}.
However, up to now, an accompanying rigorous convergence theory has
been established only in certain commonly occurring
but restricted cases \cite{CuD}.
By convergence, we mean roughly that not
only is Hill's method accurate, meaning that the numerically computed eigenvalues are always close to the
actual eigenvalues of the associated Bloch-operator
(the ``no-spurious modes condition" of \cite{CuD}),
but also that the method is complete in the sense
that it faithfully produces all of $\sigma(L_\sigma)$ for a fixed $\sigma$: see \cite{CuD} for
a more precise discussion of convergence from this point of view.
Here, we make the simpler,
operational definition that on any bounded domain
$B=\{\lambda:\, |\lambda|\le R\}$
whose boundary contains no eigenvalue of $L_\sigma$,
the set of approximate eigenvalues lying in $B$ converges to the
set of exact eigenvalues of $L$ in both location and number;
see Cor. \ref{Lconvthm}.\footnote{This
includes and slightly strengthens the definition of \cite{CuD}.}
Despite its obvious practical interest,
up to now the convergence of Hill's method
has been established to our knowledge
only for self-adjoint operators with principal coefficient $a_m=I$
\cite{CuD}. In particular, though accuracy of Hill's method was
shown in \cite{CuD} under quite general assumptions,
completeness of the method in the non-selfadjoint
case, which arises naturally, for example, in the applications in \cite{BJNRZ1,BJNRZ2},
does not seem to have been fully addressed.
In this short paper,
we give a brief and simple proof of the convergence
of Hill's method applying
to the general class of operators \eqref{e:L1}
such that $a_m$ is symmetric positive definite.
In the scalar case, this condition
on the principal coefficient $a_m$ amounts to the mild requirement
that the operator be nondegenerate type.
In the system case, it is a
genuine restriction,
and it is an
interesting and apparently nontrivial
question,
related to certain properties of Toeplitz matrices,
to what extent the condition can be relaxed.
Notably, our
analysis
applies to the important case where the operator $L_\sigma$ is non-selfadjoint.
The main ingredient of our
our proof is the introduction of a generalized periodic Evans
function, of interest in its own right, consisting of a
$2$-modified Fredholm determinant $D_\sigma$ of an associated
Birman--Schwinger type operator, whose roots we show to agree in
location and multiplicity with the eigenvalues of $L_\sigma$.
For related analysis in the solitary wave case, see \cite{GLZ}.
Once these properties are established, the desired convergence
follows
immediately by the observation that the corresponding 2-modified characteristic polynomial of the
$J^{\textrm{th}}$ Galerkin-truncation of $(L_\sigma-\lambda)v=0$ are
a subclass of the
approximants used to define the aforementioned $2$-modified Fredholm determinant
in the limit as $J\to \infty$, and furthermore that these approximates are a sequence of
analytic functions converging locally uniformly to the generalized periodic Evans function.
A novel feature of the present analysis is that our argument yields
convergence of the spectrum in both location and multiplicity,
whereas the results of \cite{CuD} concerned only location.
On the other hand, there was established in \cite{CuD} a fast rate of
convergence to the smallest (in modulus) eigenvalue in the self-adjoint
case, whereas our methods do not
readily
appear to yield a rate.
A second novelty of our work is to make the connection to the Evans function,
putting the work in a broader context.
\section[Hilbert-Schmidt operators]{Hilbert--Schmidt operators and $2$-modified Fredholm determinants}
We begin
by recalling the basic properties
of $2$-modified Fredholm determinants, defined for
Hilbert--Schmidt perturbations of the identity;
see \cite{GGK1,GGK2}, \cite[Ch.\ XIII]{GGK3},
\cite[Sect.\ IV.2]{GK}, \cite{Si1}, \cite[Ch.\ 3]{Si2}
\cite[Sect.\ 2]{GLZ} for more details.
For a given Hilbert space $\cH$,\footnote{Throughout this paper, we will always assume that our Hilbert spaces are separable.}
the Hilbert--Schmidt class
$\cB_2(\cH)$ is defined as the set of all bounded linear operators $A$ on $\cH$
for which the norm
\[
\|A\|_{\cB_2(\cH)} :=\sum_{j,k} |\langle Ae_j,e_k\rangle|^2=\tr_\cH ( A^* A
\]
is finite, where $\{e_j\}$ is any orthonormal basis.
Evidently, $\|\cdot\|_{\cB_2(\cH)}$ is independent of the basis chosen.
Moreover, every operator in $\cB_2(\cH)$ is compact (Fredholm).
On a finite-dimensional
space $\cH$, we define the $2$-modified Fredholm determinant as
\ba \lb{2.34}
{\det}_{2,\cH} (I_{\cH}-A):= {\det}_{\cH}((I_{\cH}-A)e^{A})
={\det}_{\cH}(I_{\cH}-A) \, e^{\tr_{\cH}(A)},
\ea
where $\det_{\cH}$
and $\tr_\cH$
denotes the usual determinant
and trace, respectively.
From this definition,
we have the useful estimates
\be \lb{2.34b}
|{\det}_{2,\cH}(I_{\cH}-A)| \leq e^{C\|A\|_{\cB_2(\cH)}^2}
\ee
and
\be\label{compare}
|{\det}_{2,\cH}(I_{\cH}-A) - {\det}_{2,\cH}(I_{\cH}-B)| \leq \|A-B\|_{\cB_2(\cH)}
e^{C[\|A\|_{\cB_2(\cH)}+\|B\|_{\cB_2(\cH)}+1]^2},
\ee
where $C>0$ is a constant independent of the dimension of $\cH$.
To extend this notion of a determinant to an infinite dimensional Hilbert space $\cH$, we note that
for any $A\in\cB_2(\cH)$ the estimate \eqref{compare} allows us to
define the
$2$-modified Fredholm determinant unambiguously
as the limit
\be\label{limdef}
{\det}_{2,\cH}(I_{\cH}-A):= \lim_{J\to \infty} {\det}_{2,\cH_J}(I_{\cH_J}-A_J),
\ee
where $\cH_J$ is any increasing sequence of finite-dimensional subspaces
filling up $\cH$, and $A_J$ denotes the Galerkin approximation
$P_{\cH_J}A|_{\cH_J}$, where $P_J:\cH\to\cH_J$ is the orthogonal projection
onto $\cH_J$.
That is, thinking of the infinite-dimensional matrix representation
of $A$, the 2-modified Fredholm determinant is defined as the limit of such determinants
on finite, $J$-dimensional, minors as $J\to \infty $.
Alternatively, denoting the (countably many, since $A$ is Fredholm) eigenvalues
of $A$ as $\{\alpha_j\}_{j=1}^\infty$, and taking $\cH_J$ to be the
(total) eigenspace
associated with the eigenvalues $\{\alpha_j\}_{j=1}^J$
we find that
\be\label{productformula}
{\det}_{2,\cH}(I_{\cH}-A)=\lim_{J\to\infty}\prod_{k=1}^J(1-\alpha_k)e^{\alpha_k},
\ee
which,
by
$\Pi_k(1-\alpha_k)e^{\alpha_k}\lesssim
\Pi_k (1+\alpha_k^2)
\sim e^{\sum_k \alpha_k^2}\le
e^{\|A\|_{\cB_2(\cH)}} $,
is readily seen to converge for all $A\in\cB_2(\cH)$
by Weyl's inequality
$\sum |\alpha_j|^r\le \sum|s_j|^r$ for $r\ge 0$, where
$s_j$ denote the eigenvalues of $|A|:=(A^*A)^{1/2}$ \cite{Si1,W}.
This shows how the renormalization of the
standard determinant
$\det(I_{\cH}-A):=\Pi_j(1- \alpha_j)$
by factor $e^{\tr_\cH(A)}$ cancels the possibly divergent first-order terms
in $ \Pi_k (1-\alpha_k) \sim e^{\sum_k \alpha_k}$,
allowing the treatment of operators $A$
that are not in trace class $\cB_1:=\{A:\,
\||A|^{1/2}\|_{\cB_2(\cH)} <+\infty\}$.\footnote{
For $A\in \cB_1$, $ \tr_\cH(A)=\sum_j \alpha_j$ is absolutely convergent,
by Weyl's inequality with $r=1$, and so
the standard determinant
$\det_\cH (I_{\cH}-A)= \Pi_j (1-\alpha_j)$ converges.
For $A$ self-adjoint,
$\|A\|_{\cB_1}:=\||A|^{1/2}\|_{\cB_2(\cH)}=\sum_{j}|\alpha_j|$
and $\|A\|_{\cB_2(\cH)}=\sum_j|\alpha_j|^2$.
}
\bpr\label{property}
For $A\in \cB_2(\cH)$, the operator $(I_\cH-A)$ is invertible if and only if
${\det}_{2,\cH}(I_\cH-A)$ is non-zero.
\epr
\begin{proof}
By standard Fredholm theory, this is equivalent to the statement that
$0$ is an eigenvalue of $(I_\cH-A)$ if and only if $\det_{2,\cH}(I_\cH-A)= 0$.
Note that, since $A$ is Fredholm, it possesses a countable number of
isolated eigenvalues $\{\alpha_j\}$ of finite multiplicity, except
possibly at zero.
Choosing $J\in\mathbb{N}$ sufficiently large, then,
we may factor the product formula \eqref{productformula} as
\[
{\det}_{2,\cH}(I_\cH-K)=\left(\prod_{j=1}^J (1-\alpha_j)e^{\alpha_j}\right)
\left(\prod_{j=J+1}^\infty (1-\alpha_j)e^{\alpha_j}\right),
\]
where
\[
\prod_{j=J+1}^\infty(1-\alpha_j)e^{\alpha_j}\approx e^{\sum_{j=J+1}^\infty\alpha_j^2}\neq 0.
\]
It follows then that
${\det}_{2,\cH}(I_\cH-A)$ vanishes if and only if $1-\alpha_j=0$
for some $1\le j\le J$, hence, since $J\in\mathbb{N}$ was arbitrary, if and only if $0$ is an eigenvalue of $(I_\cH-A)$.
\end{proof}
\section{Analysis of a simple case}\label{s:simple}
With the above preliminaries in
hand,
we now turn to our proof of convergence.
As a first step in this analysis, we
present a complete proof
in the
case of a second-order
operator with identity principal
part. In later sections, we will then describe the extension of this proof to more
general cases, noting that most of the ideas can be found in this simpler context.
Consider a periodic-coefficient differential operator
\[
L_\sigma=(\partial_x+i\sigma)^2 + (\partial_x+i\sigma)a_1(x)
+ a_0(x)
\]
acting on vector-valued functions in
$L^2_{\rm per}([0,X])$,
$\sigma \in [0,2\pi)$ the Floquet parameter and
$a_j\in L^2([0,X])$ matrix-valued and periodic on $x\in [0,X]$.
We can rewrite this more generally as a family of operators in the simpler form
\be\label{e:L}
L_\sigma= \partial_x^2 + \partial_x A_1(\sigma,x) + A_0(\sigma,x),
\ee
where
$$
A_1= a_1 + 2i\sigma,\quad A_0= a_0-\sigma^2 + i\sigma a_1.
$$
In order to analyze the (necessarily discrete) spectrum of the operator $L_\sigma$, we
introduce a generalization of the periodic Evans function, a complex analytic function whose
roots coincide in location and multiplicity with the eigenvalues of $L_\sigma$ \cite{G},
expressed in terms of a 2-modified Fredholm determinant. To
this end, notice that associated with the eigenvalue problem
\be\label{eig}
(L_\sigma-\lambda)U=0
\ee
is the equivalent problem
\be\label{equiv}
(I+K(\sigma, \lambda))U=0,
\ee
where here $I$ is the identity operator on
$L^2_{\rm per}([0,X])$
and $K=K_1+K_0$, with
\[
K_1=\partial_x (\partial_x^2-1)^{-1} A_1, \quad
K_0= (\partial_x^2-1)^{-1} (A_0+1 -\lambda).
\]
In particular, notice that $\lambda$ is an eigenvalue of $L_\sigma$ if and only if $0$ is an eigenvalue of the operator $(I+K(\sigma,\lambda))$.
Before we can define the appropriate generalization of the Evans function, we need the following fundamental lemma.
\bl\label{l:hs}
For
$A_j\in L^2_{\rm per}([0,X])$,
the operator $K$ is Hilbert-Schmidt.
\el
\begin{proof}
Expressing $K_m$ in matrix form $\cK_m$ with respect to
the infinite-dimensional Fourier basis, we find that the corresponding matrix elements
can be expressed as
$$
[\cK_{1}]_{j,k}= \frac{ij}{1+j^2} \hat A_{1}(j-k),
$$
where $\hat A_1(m)$ denotes the $m^{th}$ Fourier coefficient of $A_1$, and $i:=\sqrt{-1}$.
Computing explicitly, we find by Parseval's Theorem that\footnote{Henceforth, Hilbert-Schmidt spaces $\cB_2$
will always be considered on the Hilbert space $L^2_{\rm per}([0,X])$. That is, we adopt the notation
$\cB_2:=\cB_2(L^2_{\rm per}([0,X]))$.}
$$
\begin{aligned}
\|K_1\|_{\cB_2}&=\|\cK_1\|_{\cB_2}=
\sum_j
\frac{j^2}{(1+j^2)^2} \sum_k |\hat A_1(j-k)|^2\\
& =
\sum_j\frac{j^2}{(1+j^2)^2} \|A_1\|_{L^2_{\rm per}([0,X])}
<+\infty,
\end{aligned}
$$
hence $K_1$ is a Hilbert-Schmidt operator. Similarly, we find that $K_0$ is Hilbert--Schmidt,
with norm
\[
\|K_0\|_{\cB_2}=\sum_{j}\frac{1}{(1+j^2)^2}\sum_k\left|\hat A_0(j-k)+(1-\lambda)\delta_j^k\right|^2,
\]
which implies that
$K=K_1+K_0\in \cB_2$
as claimed.
\end{proof}
\br
{
On the other hand, $K_1$ is not trace class if $\hat A_{1}(0):=\int_0^X A_1(x)dx\ne0$,
since then
$\sum_j |\cK_{1,jj}|= |\hat A_{1}(0)| \sum_j \frac{|j|}{1+ |j|^{2}} =+\infty$.
This illustrates the necessity of our extension of the usual notion of a determinant
to operators in $\cB_2$.
}
\er
\subsection{Generalized Periodic Evans Function}
By Lemma \ref{l:hs} in conjunction with Proposition \ref{property}, it follows that the zero eigenvalues
of $(I_{L^2_{\rm per}([0,X])}-K(\sigma,\lambda))$ can be identified through the use of a 2-modified Fredholm determinant. This
leads us to the following definition.
\begin{definition}
For a fixed $\sigma\in[0,2\pi)$, we define the generalized periodic Evans function $D_\sigma:\CM\to\CM$ by
\be\label{evans2}
D_\sigma(\lambda):={\det}_{2,L^2_{\rm per}([0,X])}(I_{L^2_{\rm per}([0,X])}-K( \sigma, \lambda)).
\ee
\end{definition}
For ease of notation, throughout the rest of our analysis we will drop the dependence on the Hilbert space $L^2_{\rm per}([0,X])$
on the identity operator and all $2$-modified Fredholm determinants. In particular, we will write
$D_\sigma(\lambda)=\det_2(I-K(\sigma,\lambda))$ for the above generalized Evans function.
\bt\label{evansthm}
For $A_j\in L^2_{\rm per}([0,X])$, the function
$D_\sigma$ is complex-analytic in $\lambda$ and continuous in the parameter $\sigma$.
Furthermore, the
roots of
$D_\sigma$ for a fixed $\sigma\in[0,2\pi)$ correspond in location and multiplicity
with the eigenvalues of $L_\sigma$.
\et
\begin{proof}
Following the notation in Lemma \ref{l:hs}, for each $J\in\mathbb{N}$ we let $\cK_J:=([\cK]_{j,k})_{|j|,|k|\leq J}$
be the finite dimensional Galerkin matrix approximation
of the bi-infinite dimensional matrix representation of the operator $K$ defined above.
Clearly, then, for each fixed
$J\in\mathbb{N}$ the finite-dimensional
approximation
$\Delta_J(\sigma,\lambda):=\det_{2}(I-\cK_J(\sigma,\lambda))$
is complex-analytic
in $\lambda$ and continuous in $\sigma\in[0,2\pi)$. Furthermore, as in the proof of Lemma \ref{l:hs}
we have
\begin{equation}\label{e:kconverge}
\|\cK_{1,J}(\sigma,\lambda)-\cK_1(\sigma,\lambda)\|_{\cB_2}\leq\|A_1\|_{L^2([0,X])}\sum_{|j|\geq J+1}\frac{j^2}{(1+j^2)^2},
\end{equation}
where $\cK_{1,J}$ denotes the truncation of $\cK_1$,
and hence we find that $\cK_{1,J}\to\cK_1$ in
$\cB_2$
uniformly in both $\sigma$ and $\lambda$.
Similarly, we find that $\cK_{0,J}(\sigma,\lambda)\to\cK_0(\sigma,\lambda)$ in
$\cB_2$
uniformly in $\sigma$ and locally uniformly in $\lambda$,
and hence
the estimate \eqref{compare} implies\footnote{To use the estimate \eqref{compare} directly, one should
consider the operator $\cK_J$, which is technically defined on the finite-dimensional subspace $H_J$,
as being defined on the larger space $L^2_{\rm per}([0,X])$. Throughout the remainder of our analysis
we will consider this extension without reserve.}
that $\Delta_J\to D_\sigma$
locally uniformly in $\lambda\in \CM$ and uniformly in $\sigma\in[0,2\pi)$.
It follows that the function $(\sigma,\lambda)\mapsto D_\sigma(\lambda)$
inherits the same regularity properties in $\lambda$ and $\sigma$ as the limiting
sequence $\Delta_J$, thus verifying the first claim of the Theorem.
Next, by equivalence of the problems \eqref{eig} and \eqref{equiv} together
with Proposition \ref{property}, we obtain immediately correspondence
in location of the roots of $D_\sigma$ and the eigenvalues of the operator $L_\sigma$.
To obtain agreement in multiplicity, consider an eigenvalue $\lambda_*$
of $L_\sigma$, with corresponding eigenspace $H_*$. Recalling that, by standard Fredholm theory, the eigenvalues
of $L_\sigma$ are countable, isolated, and have finite-multiplicity\footnote{
Note that in this standard theory, one inverts $L_\sigma -\mu I$ rather
than $\cD^2-1$.},
we find that there exists a closed ball $B(\lambda_*, \eps)$ of radius $\eps$, centered at $\lambda_*$, containing
no other eigenvalues of $L_\sigma$.
Consider now an increasing sequence of eigenspaces $\{H_J\}_{j\in\mathbb{N}}$ of $L^2_{\rm per}([0,X])$
such that $\lim_JH_J=L^2_{\rm per}([0,X])$ and $H_*\subset H_J$ for all $J\in\mathbb{N}$.
For each $J$, let $\{r_k\}_{k=1}^J$ be an orthonormal basis of $H_J$ and let $R_J=(r_1,\ldots,r_J)$.
Then we can define the
finite-dimensional approximants
\be\label{factn}
\delta_J(\sigma,\lambda):= {\det}_{2}\left( R^*_J (\partial_x^2-1)^{-1}(L_\sigma-\lambda I)R_J\right)
\ee
Since $D_\sigma $ does not vanish on $\partial B(\lambda_*,\eps)$,
by the correspondence in location of roots and eigenvalues established above, and since
$\delta_J$ converges locally uniformly in $\lambda$ to $D_\sigma$ by \eqref{compare},
Rouch\'e's Theorem implies that there exists a $J^*\in\NM$ sufficiently large such that for $J>J^*$
the winding number of $D_\sigma $ around $\partial B(\lambda_*,\eps)$
is equal to the winding number of $\delta_J$ around the same ball.
Finally, fixing $J_0>J^*$ and noticing that $L_\sigma R_{J_0}=R_{J_0} M_{\sigma,J_0}$, where $M_{\sigma,J_0}$
is an $J_0\times J_0$ matrix representation of $L_\sigma$ on the finite-dimensional
invariant subspace $H_{J_0}$, we find from \eqref{factn} that there exists a constant $C\neq 0$ such that
\[
\delta_{J_0}(\sigma,\lambda)={\det}_{2}\left(R_{J_0}^*(\partial_x^2-1)^{-1}R_{J_0} (M_{\sigma,J_0}-\lambda I)\right)=
C{\det}_{2}\left(M_{J_0}-\lambda I\right),
\]
and hence we see that $\delta_{J_0}$
is a nonvanishing multiple of the characteristic polynomial
of $M_{\sigma,J_0}$. Here, we are using the fact that $R_{J_0}^*(\partial_x^2-1)^{-1}R_{J_0}$
is positive definite, by positive symmetric definiteness of
$(\partial_x^2-1)^{-1}$.
It follows that $\delta_{J_0}$
has a zero at $\lambda_*$ of precisely the algebraic
multiplicity of $\lambda_*$ as an eigenvalue of $L_\sigma$.
Thus, we conclude that the multiplicity of $\lambda_*$ as a root of $D_\sigma$ is
equal to the winding number of $\delta_{J_0}(\cdot,\sigma)$ about the ball $\partial B(\lambda_*,\eps)$,
which in turn is equal to the algebraic multiplicity of $\lambda_*$
as an eigenvalue of $L_\sigma$, completing the proof.
\end{proof}
\br\label{ges}
The truncated winding-number argument for agreement of multiplicity
to our knowledge is new, and seems of general use in similar situations.
It would be interesting to prove this also in a different way by establishing
a direct correspondence between the Fredholm determinant and the standard
periodic Evans function construction of Gardner \cite{G}, as done in the
solitary-wave case in \cite{GLM1, GLMZ2, GM} and in the periodic
Schr\"odinger case in \cite[Sect.\ 4]{GM}.
This would give at the same time an alternative proof of Gardner's fundamental
result of agreement in location and multiplicity of roots of
the standard periodic Evans function with eigenvalues of $L_\sigma$,
through the result of Theorem \ref{evansthm}.
\er
\subsection{Convergence of Hill's method}\label{s:hill}
Next, we
use the machinery developed in the previous section to give a
proof of the convergence of Hill's method.
In order to precisely describe Hill's method,
notice that by taking the Fourier transform,
we may express \eqref{eig} equivalently as
the infinite-dimensional matrix system
\[
(\D^2 +\D\A_1 + \A_0- \lambda I)\U=0,
\]
where for each $m=0,1$ and $j,k\in\ZM$,
\be\label{mats}
\D_{jk}=\delta_j^k ij,
\quad
[\A_m]_{jk}= \widehat{A_m}(j-k),
\quad\textrm{ and }
\U_j= \widehat U(j),
\quad
\ee
where $\hat f(k)$ denotes the discrete Fourier tranform of $f$ evaluated at Fourier frequency $k$ and,
as elsewhere, $i=\sqrt{-1}$.
Hill's method then consists of fixing $J\in\NM$ and truncating the above
infinite-dimensional matrix system
at wave number $J$, that is,
considering the $(2J+1)$-dimensional minor $|(j,k)|\le J$,
and computing the eigenvalues of the finite-dimensional matrix
\begin{equation}\label{approxL}
L_{\sigma,J}:=\D_J^2+\D_J\A_{1,J}+\A_{0,J},
\end{equation}
where $\D_J$ and $\A_{m,J}$ denote the $(2J+1)$-dimensional matrices resulting from truncating the matrices $\D$ and $\A_m$
to frequencies $|(j,k)|\leq J$,
to obtain approximate eigenvalues for $L_\sigma$.
Notice this can be done
quite efficiently by applying modern numerical linear
algebra techniques.
\br\label{divrmk}
In
applications, one may of course
encounter operators $L$ that are not in divergence
form \eqref{e:L}.
In this case, we point out that there is no effect in changing from
nondivergence to divergence form except that we increase the regularity
requirement on $A_1$ from $L^2$ to $H^1$.
Indeed,, we may change
from one form to the other using the Leibnitz rule
$\A_1\D-\D\A_1= (\A_1)'$, where
$$
(\A_1)'_{jk}= i(j-k)\A_1(j-k)=
(\widehat {A_{1,x}})(j-k),
$$
and noting that,
since $\D$ is diagonal, this operation is respected by truncation.
Thus, there is indeed no loss of generality in our representation of
operators in divergence form, as it does not affect the result of
Hill's method.
\er
Following the construction of the generalized periodic Evans function \eqref{evans2},
we may rewrite the truncated eigenvalue equation
\be\label{trunceig}
\left(L_{\sigma,J}-\lambda I\right)\cU=0
\ee
as
\be\label{truncfred}
(I+ \cK_J)\cU=0,
\ee
where $\cK_J=\cK_{1,J}+\cK_{2,J}$ is the truncation of the
Fourier representation $\cK=\cK_1+\cK_2$ of operator $K$ to frequencies $|(j,k)|\leq J$, that is,
\be\label{trun}
\cK_{1,J}=\D_J(\D_J^2-I)^{-1}\A_{1,J}
\quad\textrm{ and }\quad
\cK_{2,J}=(\D_J^2-I)^{-1}(\A_{0,J}+1-\lambda).
\ee
Continuing to follow the above construction of $D_\sigma$, we now define the truncated
periodic Evans function as
\be\label{truncevans}
D_{\sigma,J}(\lambda):={\det}_{2}(I-\cK_J)
\ee
and notice that we have the following preliminary result.
\bl\label{trunccorr}
The zeros of $D_{\sigma,J}$ correspond
in location and multiplicity with those of $L_{\sigma,J}$.
\el
\begin{proof}
This is immediate by the
nonsingularity of $(\cD_J^2-I)^{-1}$ and properties
of the (usual, finite-dimensional) characteristic polynomial,
together with the observation that
\[
{\det}_2(I-\cK_J)=
{\det}_2(\cD_J^2-I)^{-1} {\det}_2(\D_J^2 +\D_J\A_{1,J} + \A_{0,J}-\lambda I).
\]
\end{proof}
With this construction in hand, we now state the main result of this section.
\bt\label{convthm}
For $A_j\in L^2_{\rm per}([0,X])$, the sequence of determinants $D_{\sigma,J}$
converges to $D_\sigma$ as $J\to \infty$ uniformly in $\sigma$ and locally uniformly in $\lambda$
\et
\begin{proof}
This convergence result follows from the proof of Theorem \ref{evansthm}.
Indeed,
noting
that $D_{\sigma,J}$ is exactly such a
sequence of approximate determinants, corresponding here
to the ascending sequence of sinusoidal functions of integer wave number,
by which the generalized periodic Evans function $D_\sigma$ was
defined in \eqref{evans2},
we find by our definition of the 2-modified Fredholm determinant that
$D_{\sigma,J}\to D_\sigma$ pointwise in $\lambda$
as $J\to \infty$ for each fixed $\sigma\in[0,2\pi)$.
Moreover, recalling that the rate of convergence is determined by
the difference between truncated operator $\cK_J$ and $\cK$ in
$\cB_2$
norm, and noting that we have uniformly bounded
$\cB_2$
estimates on each entry of $\cK_J$,
we find that this convergence is uniform in $\sigma$ and locally uniform in $\lambda$.
\end{proof}
From Theorem \ref{convthm} we immediately have convergence of Hill's method, as described in the introduction.
For completeness, we state this result in the following corollary.
\bc\label{Lconvthm}
For $A_j\in L^2_{\rm per}([0,X])$,
the eigenvalues of $L_{\sigma,J}$ defined in \eqref{approxL}
approach the eigenvalues of $L_\sigma$ in location and multiplicity
as $J\to \infty$, uniformly on $|\lambda|\le R$, $\sigma\in [0,2\pi]$,
for any $R$ such that $\partial B(0,R)$ contains no eigenvalues of
$L_\sigma$.
\ec
\begin{proof}
This is immediate from Theorem \ref{evansthm}, Lemma \ref{trunccorr}, and Theorem \ref{convthm},
along with basic
properties of uniformly convergent analytic functions.
\end{proof}
\subsection{Rates of Convergence}
Next, we address the issue of the rates of convergence of $D_{\sigma,J}$ to $D_\sigma$ and of the
approximate spectra to the exact spectra. Assuming slightly more regularity on the function $A_1$
in \eqref{e:L}, we have the following easy convergence result.
\bt\label{Lconvrate}
For $A_j\in H^1_{\rm per}([0,X])$ and each fixed $R>0$, there exists a constant $C=C(R)>0$ such that
for each fixed $|\lambda|\leq R$
\[
|D_{\sigma,J}(\lambda)- D_\sigma(\lambda)|\le CJ^{-1/2}.
\]
In particular, this estimate is locally uniform in $\lambda$ and uniform in
$\sigma$.
\et
\begin{proof}
The rate of convergence is bounded by
$\|\cK_J-\cK\|_{\cB_2}$
from which we readily obtain the result using the Cauchy-Schwarz
estimate
$$
\sum_{|j|\ge J}|\widehat{A^m}(j)|^2\le \sum_{|j|\ge J}|j|^{-2}
\sum_{|j|\ge J}|j|^{2}|\widehat{A^m}(j)|^2\le (C/J)\|A^m\|_{H^1([0,X])}
$$
for each $m\in\{0,1\}$.
For details, see the very similar estimates in the proof of
Theorem 4.9, \cite{GLZ}.
\end{proof}
Notice that Theorem \ref{Lconvrate} does not imply
a rate of convergence of the roots of $D_{\sigma,J}$ to the
roots of $D_\sigma$, or, equivalently, the eigenvalues of $L_{\sigma,J}$ to
the eigenvalues of $L_\sigma$.
Indeed, the above convergence result
is, with or without rate information, essentially an abstract one.
For, though we find convergence the of analytic functions
$D_{\sigma,J}$ to $D_\sigma$,
we don't obtain rates of convergence of their zeros without more structural
information about $D_\sigma$ itself. In particular, we can not conclude
convergence rates of the approximate spectra to the true eigenvalues of $L_\sigma$
using only the knowledge of the
eigenvalues of $L_{\sigma,J}$ computed in the course of Hill's method.
This suggests the idea of computing the approximate Evans function
$D_{\sigma,J}$ directly, instead of using it as a purely analytical tool,
an idea that would be interesting for future investigation. Though
in principle slower due to the need for multiple evaluations of eigenvalues,
this computation is better conditioned, so there might perhaps be some
counterbalancing advantages to this approach, besides the possibility
already mentioned to obtain a posteriori estimates on the error bounds
for eigenvalue approximations.
We leave this as an interesting topic for further investigation,
related to the larger question of relative advantages
of standard periodic Evans function (as in \cite{G})
vs. Hill's computations.
\section[Generalizations]{Generalizations}
\label{s:comp}
Here, we briefly discuss various generalizations of the theory developed in
Section \ref{s:simple}.
\subsection{Operators with nontrivial principal coefficient}\label{s:princ}
Consider now a system of the more general form
\be\label{e:Ln}
L_\sigma= \partial_x^2 A_2 + \partial_x A_1(\sigma,x) + A_0(\sigma,x),
\ee
where $A_2$ is symmetric positive definite, satisfying $ A_2(x)\ge C$
for some $C>0$, uniformly on $x\in [0,X]$.
Define as usual $\cA_2$ to be the infinite-dimensional
matrix representation of $A_2$ under Fourier transform; that is,
$\cA_{2,jk}=\widehat{A_2}(j-k)$.
Then clearly $\cA_2$ is symmetric and, by Parseval's identity,
satisfies $ \cA_2 \ge C$ when considered as a quadratic
form on $\ell^2(\NM)$.
As a consequence, the $J^{\rm th}$ truncation $\cA_{2,J}$, as a principal
minor of a positive definite symmetric matrix, must also be positive definite
and satisfy the same bound $ \cA_{2,J}\ge C$.
In particular, $\cA_2$ is invertible with
$$
\cA_2^{-1} \ge 1/C,
\quad
\cA_{2,J}^{-1} \ge 1/C.
$$
\bl\label{multlem}
$\|AB\|_{\cB_2}\le |A|_{L^2}\|B\|_{\cB_2}$,
where $|\cdot|_{L^2}$ denotes $L^2([0,X])$ operator norm.
\el
\begin{proof}
Straightforward from the definition of $\|\cdot\|_{\cB_2}$.
\end{proof}
\bc\label{c:compose}
For $A_j\in L^2_{\rm per}([0,X])$ and $A_2$ symmetric positive definite with $A_2(x)\ge C$,
the operator $\cM:=\cA_2^{-1}\cK$ is Hilbert-Schmidt where $\cK=\cK_1+\cK_2$ is defined
as in \eqref{trun}.
\ec
In this case, following the notation of Corollary \ref{c:compose}, we
define the generalized Evans function as $D_\sigma(\lambda):=\det_2 (I-\cM)$,
noting that the eigenvalue problem may be written equivalently
as $(I-\cM)\cU=0$.
The associated series of Fredholm approximants is
$ D_{\sigma,J}(\lambda):=\det_2(I-\cM_J)$, with $D_{\sigma,J}(\lambda)
\to D_\sigma(\lambda)$ uniformly as $J\to \infty$, just as before,
and zeros of $D_{\sigma}$ corresponding
in location and multiplicity with eigenvalues of $L_\sigma$.
However, the corresponding object obtained by Hill's method is not
the truncated Fredholm determinant $D_{\sigma,J}$ defined above, but rather the modified version
\be\label{hillapprox}
\check{D}_{\sigma,J}(\lambda):= {\det}_2( I- \cA_{2,J}^{-1} \cK_J),
\ee
and it is this function whose zeros correspond with the eigenvalues
of the Hill approximant operator $L_{\sigma,J}$.
To verify convergence of Hill's method in this case then, it is sufficient to show that
\be \label{key}
\| \cM_J- \cA_{2,J}^{-1} \cK_J\|_{\cB_2}
=\| (\cA_2^{-1}\cK)_J- \cA_{2,J}^{-1} \cK_J\|_{\cB_2}
\to 0
\ee
as $J\to \infty$. Indeed, with this convergence result in hand
we may conclude
by \eqref{compare} that $\lim_{J\to\infty}|\check{D}_{\sigma,J}-D_{\sigma,J}|= 0$,
and
thus $\check D_{\sigma,J} \to D_\sigma$ as $J\to \infty$, yielding
the convergence result as before.
\bt
For operators of the form \eqref{e:Ln}, Hill's method converges in location and multiplicity
provided that $A_j\in L^2_{\rm per}([0,X])$.
\et
\begin{proof}
We sketch the proof of \eqref{key}.
By boundedness of $\|A_2\|_{L^2([0,X])}$, we may truncate $\widehat{A_2}$
at wave number $M$ to obtain an $M$-banded infinite-dimensional diagonal matrix
centered around zero-frequency approximating $\cA_2$ to arbitrarily small order in the $\ell^2(\NM)$
operator norm. Hence, for purposes of this argument, we may assume
without loss of generality that $\cA_2$ is $M$-banded diagonal operator centered about zero-frequency.
Furthermore, noting that since $\widehat{\cA_2^{-1}}$ is bounded in $L^2(\RM)$, for $J\in\NM$ sufficiently large
the columns of $\cA_2^{-1}$ corresponding to frequencies $|j|\leq J-M$ are small off the principal $2J+1-M$ minor
and hence
a brief calculation reveal
that
$$
(\cA_2^{-1})_J \cA_{2,J}=
\bp
E_{M}& 0 & 0\\
0 & I_{2J-2M} & 0\\
0 & 0 &F_{M}\\
\ep,
$$
where $E_M$ and $F_M$ are $M\times M$ matrices that are invertible
by invertibility of
$(\cA_2^{-1})_J \cA_J$, a property of principal minors of positive-definite
symmetric matrices.
By a further left-multiplication by the block-diagonal matrix
$$
\bp
E_{M}^{-1}& 0 & 0\\
0 & I_{2J-2M} & 0\\
0 & 0 &F_{M}^{-1}\\
\ep
$$
we obtain
$I_{2J+1}$,
demonstrating that
$(\cA_{2,J})^{-1}$ agrees with $(\cA^{-1})_{2,J}$ on the central
$2J-2M+1$ dimensional minor.
Recalling that $\|\cK-\cK_J\|_{\cB_2}\to 0$ as $J\to 0$ by \eqref{e:kconverge},
we thus obtain by a straightforward calculation
$$
\| (\cA_2^{-1}\cK)_J- \cA_{2,J}^{-1} \cK_J\|_{\cB_2}
\sim \| (\cA_2^{-1}\cK_J)_J- \cA_{2,J}^{-1} \cK_J\|_{\cB_2}
\to 0,
$$
completing the proof by \eqref{compare}
\end{proof}
\subsection{Composite and Higher-order operators}
The reader may easily verify that all of the arguments of Sections
\ref{s:simple} and \ref{s:princ} carry over to the case when the operator \eqref{e:L1}
is replaced by a general periodic-coefficient operator
$$
L=\partial_x^m a_m(x) + \partial_x^{m-1}a_{m-1}(x)+\dots +a_0(x)
$$
where $a_j\in L^2_{\rm per}([0,X])$ and where the principal coefficient $a_m$ symmetric positive definite.
Indeed, the analysis parallels that of previous sections except that one must
substitute for $(\partial_x^2-1)$
everywhere the positive definite symmetric
Fourier multiplier
\[
|\partial_x^2-1|^{m/2}= \cF^{-1}(|j|^2+1)^{m/2}\cF,
\]
where $j$ denotes the Fourier wave number
and $\cF$ denotes Fourier transform. With these substitutions, our previous arguments
immediately yield convergence of Hill's method in this case as well.
Furthermore, it is straightforward to verify that all of the analysis
of Sections \ref{s:simple} and \ref{s:princ} extends readily to the case of operators
of ``composite'' type
\[
L=\bp
\partial_x^{m_1}a^1_{m_1} +\dots\\
\vdots\\
\partial_x^{m_n}a^n_{m_n} +\dots\\
\ep,
\]
with $a^j_{k}\in L^2_{\rm per}([0,X])$ and $a^j_{m_j}$ symmetric positive definite
for each suitable choice of indices:
that is,
still assuming $L$ is a nondegenerate ordinary differential operator in some sense.
\br\label{compapp}
It is the above observation that applies to the numerics in \cite{BJNRZ1,BJNRZ2}, where
the authors use Hill's method to numerically analyze the spectrum of the linearized
St. Venant equations
\begin{align*
\lambda \tau-c\tau' - u'&= 0,\\
\lambda u-cu'
-(\bar \tau^{_-3}(F^{-1}- 2\nu \bar u_x)\tau)' &=
-(s+1)\bar \tau^s\bar u^r \tau
- r\bar \tau^{s+1}\bar u^{r-1}
+\nu (\bar \tau^{-2}u')'
\end{align*}
about a given periodic or homoclinic orbit $(\bar u,\bar \tau)$,
where $r$, $s$, $F$, and $\nu$ are physical parameters in the problem and $\lambda$
is the corresponding spectral parameter.
\er
\subsection{Operators with general coefficients}
Our results are completely general in the scalar case,
applying to all nondegenerate operators.
However, they are restricted in the system case by the
condition that the principal coefficient(s) be symmetric positive
definite. Whether this condition may be relaxed is an interesting operator-theoretic
question regarding properties of Toeplitz matrices.
Specifically, the property that we need to carry out Hill's method
(and indeed, to complete our entire convergence analysis) is
that the minor $\cA_{2,J}$ of a Toeplitz matrix $[\cA_{2}]_{mn}=\widehat{A_2}(k-n)$
be invertible for $J$ sufficiently large.
The question is what properties of $A_2(x)$ are sufficient
to guarantee this: in particular, is uniform invertibility enough?
Alternatively, what are sufficient conditions on $\widehat{A_2}?$
This seems an interesting problem for further investigation.
\medskip
{\bf Acknowledgement.} Thanks to Bernard Deconink for pointing
out the references \cite{CuD,CDKK,DK}.
|
{
"timestamp": "2010-09-21T02:04:08",
"yymm": "1009",
"arxiv_id": "1009.3908",
"language": "en",
"url": "https://arxiv.org/abs/1009.3908"
}
|
\section{Introduction}
Over the last decade, a lot of progress has been made on the solution of toy models belonging to the Gross--Neveu family \cite{1}. They describe $N$ species of massive or massless fermions in $1+1$ dimensions interacting via a scalar or pseudoscalar four--fermion interaction:
\begin{equation}\label{c1}
{\cal L} = \bar \psi \left( {\rm i} \gamma^\mu \partial_\mu - m\right) \psi + \frac{g^2}{2} (\bar \psi \psi)^2 + \frac{G^2}{2} (\bar \psi {\rm i} \gamma^5 \psi)^2 .
\end{equation}
Summation over flavor indices is implied and we abbreviate $\bar \psi \psi \equiv \sum_{i = 1}^N \bar \psi^{(i)} \psi^{(i)}$. For $G^2 \equiv 0$ and $m \equiv 0$ we recover the original Gross--Neveu model (GN) with discrete chiral symmetry $\psi \to \gamma^5 \psi$ \cite{1}. For $g^2 \equiv G^2$ the theory possesses a continuous chiral symmetry $\psi \to e^{{\rm i} \gamma^5 \theta} \psi$ and corresponds to a $1+1$ dimensional version of the Nambu--Jona-Lasinio model ($\chi$GN) \cite{33}. Over the past years, previous work on the phase diagrams of those models \cite{18,32} has been extended significantly and revised phase diagrams were proposed that respect the particle content of the theories. It was found that all of the above models exhibit crystalline phases, for many of which analytical solutions could be obtained \cite{11,58,59,55,60,61}. The GN model features a kink--antikink crystal at high density whereas the thermodynamically preferred ground state of the $\chi$GN model below a transition temperature is a ''chiral spiral'' --- a helical chiral condensate whose amplitude is determined by the temperature and spatial period by the chemical potential \cite{11}. The phase transition is reminiscent of the Peierls transition in 1d metals with the first band gap opening up at the Fermi surface \cite{35}. While those models provide a rich playground for the study of relativistic field theories, they were not thought to have any application to reality due to their lower dimensionality.
In recent years, however, the expertise on those toy models has been used to study problems related to the ground state of QCD at finite density. Thereby, $1+1$ dimensional models arise in the form of effective field theories when phases with lower-dimensional modulations of the chiral condensate are considered. Such modulations can occur due to a strong external field \cite{30} or can be induced by the Fermi surface \cite{34,20}. Examples include work by Shuster and Son \cite{34} who --- based on a dimensional reduction onto a variant of the chirally invariant Thirring model --- refuted the possibility of the large $N_c$ Deryagin--Grigoriev--Rubakov (DGR) chiral wave ground state \cite{27} for $N_c = 3$. More recently, Peierls-like instabilities were discussed in the context of the Quarkyonic Phase of QCD \cite{21} or in the presence of a very strong magnetic field, where they were named ''Quarkyonic Chiral Spirals'' \cite{20, 20a} and ''Chiral Magnetic Spirals'' \cite{30}, respectively. The exact solution of the massive GN model was used by Nickel in order to investigate the phase diagram of the massive Nambu--Jona-Lasinio model when restricting to one-dimensional condensates by mapping the energy spectrum of the model onto the spectrum of the GN model \cite{1b}. Further examples include work by Bietenholz \emph{et al.} \cite{28}.
Since the models in question are very involved, it is desirable to study examples of dimensional reduction isolated from the context of QCD. For this purpose we investigate dimensional reduction in nonrelativistic quantum field theories in this paper. We establish the equivalence of BCS theory of superconductivity to the massless Gross--Neveu model with discrete chiral symmetry on a mean-field level, provided that the chiral condensate does not break translational invariance. Furthermore, we will show that the mean-field description of a well-known model from condensed matter physics which is used in the description of spin-Peierls systems --- the quasi one-dimensional extended Hubbard model at half-filling --- is equivalent to the massless generalized Gross-Neveu model with two coupling constants. In particular, we do not have to impose any restrictions on the symmetry of the condensate and the phase diagrams of both models can be identified.
On the one hand, we aim to supplement the current literature by elucidating the mechanism of dimensional reduction in the context of selected nonrelativistic field theories. On the other hand, we want to explore the fascinating applications of the GN model to condensed matter physics. Furthermore, since Hubbard models are extensively studied both analytically as well as numerically, our results might point to applications of those theories to the GN model.
This paper is structured as follows: In Sec. II we give a brief overview of the basics of the massless GN model with two coupling constants. Section III presents the dimensional reduction of BCS theory onto the GN model with discrete chiral symmetry. The reduction holds only for homogeneous condensates. Limitations for spatially varying condensates as well as the cutoff dependence of the phase diagram are discussed. Section IV contains the dimensional reduction of the quasi one-dimensional Hubbard model onto the generalized GN model. We give an introduction to the model and discuss its particle hole symmetry. Exploiting this symmetry, the dimensional reduction is formulated. A discussion of the cutoff dependence and comparison with selected experimental results follows. The paper is concluded by a summary and conclusions in section V.
\section{Massless Gross--Neveu model with two coupling constants}
In this paper, we consider the massless GN model with two coupling constants (genGN), i.e. as defined by Eq.~(\ref{c1}) with $m \equiv 0$. While early work on this system was carried out by Klimenko \cite{1a}, the full revised phase diagram and particle content were worked out only recently by Boehmer and Thies \cite{2}.
We are interested in the phase diagram of this model. According to Coleman's theorem, any long-range order in 1+1 dimensions is destroyed by fluctuations \cite{25}. As shown by Witten \cite{31}, those fluctuations are suppressed in the limit $N \rightarrow \infty$. The counting of various orders, as carried out in different contexts by Dolan and Jackiw \cite{29} as well as 't~Hooft \cite{3}, reveals that for $N \rightarrow \infty$ a finite leading order can only be obtained if $g^2 N$ and $G^2 N$ are kept fixed. Subleading orders are suppressed by powers of $N$.
The suppression of fluctuations in the large-$N$ limit admits a semiclassical treatment \cite{1,4}. The Hamiltonian of the genGN model in a mean-field approximation becomes
\begin{equation}
\frac{H}{N} = \int {\rm d}x \bigg\{ \psi^\dagger \left[ -{\rm i} \gamma^5 \partial_x + \gamma^0 S + \gamma^1 P \right] \psi + \frac{S^2}{2 g^2 N} + \frac{P^2}{2 G^2 N} \bigg\} ,
\end{equation}
where the scalar and pseudoscalar fields $S$ and $P$ satisfy the self-consistency relations
\begin{equation}\label{b4}
S = - g^2 \langle \bar \psi \psi \rangle \quad {\rm and} \quad P = - G^2 \langle \bar \psi {\rm i} \gamma^5 \psi \rangle
\end{equation}
and $\langle \ldots \rangle$ denotes the thermal expectation value.
The canonical transformation $\psi \to e^{{\rm i} \gamma^5 \pi/4} \psi$ maps the scalar and pseudoscalar interaction terms onto one another. Hence, we can assume without loss of generality that $0 < G^2 \leq g^2$. Renormalization can be performed using the conditions \cite{2}
\begin{equation}
\frac{\pi}{g^2 N} = \ln \left(\sqrt{\left(\Lambda/2\right)^2 + 1} + \Lambda/2\right) \approx \ln \Lambda \quad {\rm and} \label{a2}
\end{equation}
\begin{equation}
\frac{\pi}{G^2 N} = \ln \Lambda + \xi , \label{a1}
\end{equation}
where we set the scalar condensate equal to 1. $\xi \geq 0$ is the renormalized parameter that describes the imbalance in the scalar and pseudoscalar coupling. The case $\xi \equiv 0$ corresponds to the $\chi$GN model whereas for $\xi \to \infty$ we recover the discrete GN model.
\begin{figure}[b!]
\begin{center}
\scalebox{1.15}{\epsfig{file=figure1.eps}
\caption{Phase diagram of the genGN model as derived in \cite{2}.}
\label{fig7}
\end{center}
\end{figure}
The genGN model contains meson and baryon bound states. For a detailed analysis see \cite{4a} and \cite{2}. The phase diagram of the model is shown in Fig.~\ref{fig7}. At zero density (small $\mu$), there is a homogeneous phase with vanishing pseudoscalar condensate. A sheet of second order critical points separates this phase from a massless homogeneous one. Since in both phases the pseudoscalar condensate vanishes, the critical temperature depends on $\mu$ but not on $\xi$. A sheet of first order transition lines divides the massive homogeneous phase from a phase where the condensate takes the form of a soliton crystal which interpolates between the kink-antikink condensate of the GN model and the helical condensate of the $\chi$GN model. All transition sheets converge in a line of tricritical points. The onset of the inhomogeneous phase at zero temperature corresponds to twice the baryon mass. Thereby, at low density and large $\xi$, the condensate takes the form of separated kinks and antikinks and goes over into a sinusoidal shape for large density. As $\xi \to 0$, the pseudoscalar condensate is less suppressed and the ground state oscillates between the scalar and the pseudoscalar condensate. It is interesting to note that chiral symmetry is never restored at zero temperature. In all cases, the spatial period $a$ of the inhomogeneous condensate is determined by the first gap which opens up at the Fermi surface: $a = \pi/p_F$, where $p_F$ is the Fermi momentum. The fact that a system can lower the energy of its ground state by opening up a band gap at the Fermi surface is a well-known phenomenon in condensed matter physics where it was proposed by Peierls for 1d metals \cite{35}.
\section{BCS theory}
We start with the following observation: The phase diagram of the massless GN model was derived for the first time by Wolff \cite{18} assuming only translationally invariant phases. The diagram is shown in Fig.~\ref{fig1}. However, an equivalent phase diagram had been obtained by Sarma \cite{19} even a decade before the proposal of the GN model when investigating the phase diagram of BCS theory in an external magnetic field \cite{26}. The main difference is that the phase diagram of the GN model shows $T$ vs $\mu$ whereas Sarma derives the critical temperature as a function of an external magnetic field. The massive phase of the GN model corresponds to the superconducting phase of BCS theory and the massless phase with restored chiral symmetry to the gapless normal phase. This indicates strongly that both theories are equivalent on that level. This section presents the proof of this equivalence.
\begin{figure}[b!]
\begin{center}
\scalebox{0.75}{\epsfig{file=figure2.eps}
\caption{Phase diagram of the massless GN model assuming only translationally invariant phases. The continuous line denotes a second order and the dashed line a first order transition. The thin continuous lines mark boundaries of metastability.}
\label{fig1}
\end{center}
\end{figure}
The electrons in a conductor interact via the Coulomb interaction with the nuclei and each other. Since the scale of the excitation in a conductor is much smaller than the binding energies due to Coulomb interaction, the conductor can be treated as consisting of a gas of weakly interacting quasiparticles. We start with the effective grand canonical Hamiltonian describing the low-energy excitations in a conductor \cite{24}:
\begin{equation}\label{b1}
H = \int {\rm d}^3{\bf x} \, \psi^\dagger \left( - \frac{\nabla^2}{2 m} - \mu - \mu_A \sigma^3 \right) \psi - \frac{g}{2} (\psi^\dagger \psi)^2 .
\end{equation}
$\psi({\bf x})$ denotes a fermion spinor with components $\psi_\uparrow({\bf x})$ and $\psi_\downarrow({\bf x})$. $m$ is the effective mass of the quasiparticles and $\mu$ the chemical potential. We introduce an ''axial'' chemical potential $\mu_A$ which imbalances the two spin species: The chemical potential of spin-$\uparrow$ fermions becomes $\mu + \mu_A$ and for spin-$\downarrow$ fermions $\mu - \mu_A$. Physically, $\mu_A$ describes an external magnetic field or the effect of impurities in the material. The coupling $g > 0$ carries dimension $E \Lambda^{-3}$, with $E$ carrying dimension of energy and $\Lambda$ of momentum. We perform a Fierz transformation to rewrite the interaction term:
\begin{equation}
(\psi^\dagger \psi)^2 = - \frac{1}{2} \left[\psi^\dagger C \psi^*\right] \left[\psi^T C \psi\right],
\end{equation}
where $C = {\rm i} \sigma_2$. The low-energy excitations are described by states whose momenta ${\bf p}$ are close to the Fermi surface. This defines a cutoff
\begin{equation}\label{b3}
\left| |{\bf p}| - p_F \right| < \Lambda/2 \ll p_F
\end{equation}
with $p_F = \sqrt{2 m \mu}$. In this region, we can linearize the dispersion relation
\begin{equation}
\frac{{\bf p}^2}{2 m} - \mu \approx v_F (|{\bf p}| - p_F).
\end{equation}
The Fermi velocity is defined by $v_F = p_F/m$. All information about the underlying microscopic theory is absorbed in $v_F$ and $\Lambda$. For this linearization to hold for nonzero $\mu_A$ we assume the hierarchy of scales
\begin{equation}
\mu_A \ll v_F \Lambda/2 \ll \mu .
\end{equation}
Renormalization can be performed using a similar condition as Eq.~(\ref{a2}) \cite{61a}:
\begin{equation}\label{b2}
\frac{2}{g \rho} = \ln v_F \Lambda,
\end{equation}
where we set the energy scale equal to $1$ and define the density of states \cite{62}
\begin{equation}
\rho = \frac{p_F^2}{\pi^2 v_F}.
\end{equation}
In deriving Eq.~(\ref{b2}) we apply a large $N$ argument: Since we only consider states close to the Fermi surface, the fermion propagator $G({\bf p})$ has nonvanishing support in a shell around the Fermi surface defined by Eq.~(\ref{b3}). This means that for $|{\bf q}| \sim {\cal O}(p_F)$ the product of two propagators $\int {\rm d}{\bf p} \, G({\bf p}) G({\bf q + p})$ is phase-space suppressed by a factor of ${\cal O}(\Lambda/p_F)$. To leading order, this selects all ''cactus'' or ''daisy'' diagrams. For further reference see \cite{22,29}.
By virtue of the above argument we analyze (\ref{b1}) in a mean-field approximation. Eq. (\ref{b1}) then becomes
\begin{align}\label{b4}
H &= \int {\rm d}^3{\bf x} \ \huge \bigg\{ \psi^\dagger \left(v_F (|- {\rm i} \nabla| - p_F) - \mu_A \sigma_3\right) \psi
\nonumber \\
&\qquad + \frac{1}{2} \Delta \psi^\dagger C \psi^* - \frac{1}{2} \Delta^* \psi^T C \psi + \frac{\Delta^2}{g}\bigg\},
\end{align}
where the BCS-condensate $\Delta$ satisfies the self-consistency condition
\begin{equation}
\Delta = \frac{g}{2} \langle \psi^T C \psi \rangle
\end{equation}
and $\langle \ldots \rangle$ denotes the thermal average.
\subsection{Dimensional Reduction}
To keep track of the dimensionality of the various quantities we place the system in a box of length $L$. We define the Fourier transform of the spinors by
\begin{equation}
\psi({\bf x}) = \frac{1}{L^{3/2}} \sum_{\bf p} \psi({\bf p}) e^{{\rm i} {\bf p} \cdot {\bf x}} .
\end{equation}
Since the Hamiltonian is rotationally invariant, we can switch to spherical coordinates and formally replace the angular integration by the sum over a number of $N_{\rm pat}$ patches that cover the Fermi surface:
\begin{equation}\label{b5}
\int \frac{{\rm d}^3{\bf p}}{(2 \pi)^3} \to N_{\rm F} \left[\frac{1}{N_{\rm pat}} \sum_i\right] \frac{1}{L^3} \sum_{\rm p} ,
\end{equation}
where we set $N_{\rm F} = p_F^2 L^2/\pi = 4 \pi p_F^2/(2 \pi/L)^2$. Note that this is the number of patches that cover the Fermi surface if the system is enclosed in a finite box of length $L$, hence $N_{\rm pat} \equiv N_{\rm F}$. We introduce the notation $\psi^{(i)}(p)$ for a spinor with momentum $p_F + p$ in the direction of the $i$-th patch. We label the patch opposite to a patch $i$ by $-i$. Eq.~(\ref{b4}) becomes (suppressing the patch labels)
\begin{align}
\frac{H}{N_{\rm pat}} &= \sum_{\rm p} \ \huge \left\{ \psi^\dagger \left(v_F p - \mu_A \sigma_3\right) \psi + \frac{1}{2} \Delta \psi^\dagger C \psi^* \right. \nonumber \\
&\qquad \left. - \frac{1}{2} \Delta \psi^T C \psi \right\} + \frac{L \Delta^2/v_F}{\pi \rho g} .
\end{align}
The factor $g \rho$ is dimensionless. If we perform a partial particle-hole conjugation (not changing the spin quantum number) for spin-$\downarrow$ particles,
\begin{equation}
\left(\begin{matrix}\psi^{(i)}_\uparrow(p) \\ \psi^{(i)}_\downarrow(p)\end{matrix}\right) \rightarrow \left(\begin{matrix}\psi^{(i)}_\uparrow(p) \\ \psi^{(- i) \dagger}_\downarrow(p)\end{matrix}\right) ,
\end{equation}
we obtain after normal ordering the Hamiltonian of the massless Gross--Neveu model in a chiral basis:
\begin{align}
\frac{H}{N_{\rm pat}} &= \sum_{\rm p} \ \psi^\dagger \left(v_F p \, \sigma_3 - \mu_A + \Delta \, \sigma_1\right) \psi + \frac{L \Delta^2/v_F}{\pi g \rho} \nonumber \\
&= \sum_{\rm p} \ \bar{\psi} \left(v_F p \, \gamma^1 - \mu_A \, \gamma^0 + \Delta\right) \psi + \frac{L \Delta^2/v_F}{\pi g \rho}, \label{d1}
\end{align}
where we choose $\gamma^0 = \sigma_1, \gamma^2 = -{\rm i}\sigma_2$ and $\gamma^5 = \sigma_3$. This establishes the equivalence between both models on a mean-field level. Each flavor in the GN model corresponds to a patch on the Fermi sphere. Spin-$\uparrow$ and -$\downarrow$ are mapped onto the right- and left-handed components of the relativistic spinor. The coupling $g \rho$ corresponds to $(2/\pi) \, g^2 N$ in the Gross--Neveu model.
Unlike in a relativistic field theory, the cutoff is a physical quantity in BCS theory. Typical cutoffs are of order ${\cal O}(10^2)$--${\cal O}(10^3)$ in units of the mass scale \cite{48a}. In order to establish the full equivalence we must assure that for typical values of the cutoff the phase diagram of the theory is not significantly distorted and we can take $\Lambda \to \infty$ without loss of generality. The Hamiltonian (\ref{d1}) is just the Hamiltonian of a massive free relativistic Fermi gas with single particle energies $\varepsilon(p) = \pm \sqrt{p^2 + \Delta^2}$ and chemical potential $\mu_A$ plus a \emph{c}--number term. The grand canonical potential density is given by (setting $v_F \equiv 1$)
\begin{align}
\frac{\Omega}{N L} &= \int_{- \Lambda/2}^{\Lambda/2} \frac{{\rm d}p}{2 \pi} \ln \left\{ \left(1 + e^{\beta (\varepsilon(p) + \mu_A)}\right) \left(1 + e^{- \beta (\varepsilon(p) - \mu_A)}\right) \right\} \nonumber \\
&\qquad + \frac{\Delta^2}{2 g^2 N} .
\end{align}
Removing the logarithmic divergences by using the renormalization condition Eq.~(\ref{a1}) (or Eq.~(\ref{b2})) without taking the limit $\Lambda \to \infty$ we obtain
\begin{align}
\frac{\Omega}{N L} &= - \frac{2}{\beta} \int_0^{\Lambda/2} \frac{{\rm d}p}{2 \pi} \ln \left\{ \left(1 + e^{- \beta \sqrt{p^2 + \Delta^2} - \mu_A}\right) \right. \nonumber \\
&\quad \times \left. \left(1 + e^{- \beta \sqrt{p^2 + \Delta^2} + \mu_A}\right) \right\} + \frac{\Delta^2}{2 \pi} \left(\ln \Delta - \frac{1}{2}\right) \nonumber \nonumber \\
&\quad + \frac{\Delta^2}{2 \pi} \ln\left(\frac{\Lambda}{2} + \sqrt{\left(\frac{\Lambda}{2}\right)^2 + 1}\right) \nonumber \\
&\quad - \frac{\Delta^2}{2 \pi} \ln\left(\frac{\Lambda}{2} + \sqrt{\left(\frac{\Lambda}{2}\right)^2 + \Delta^2}\right) \nonumber \\
&\quad + \frac{1}{16 \pi} \frac{\Delta^4}{(\Lambda/2)^2} + \mathcal{O}\left(\Delta^2 \left(\frac{\Delta}{\Lambda/2}\right)^4\right) , \label{f1}
\end{align}
where we subtracted two trivial ''would-be'' divergences
\begin{equation}
- \frac{1}{8 \pi} \Lambda^2 - \frac{\mu_A}{2 \pi} \Lambda ,
\end{equation}
the second one stemming from the infinite fermion density of the Dirac sea. The second line of Eq.~(\ref{f1}) vanishes in the limit $\Lambda \to \infty$. Fig.~\ref{fig2} shows several phase diagrams for cutoff values $\Lambda = 2.5,4$ and $\infty$. As we can see, we can take $\Lambda \to \infty$ for typical values of the cutoff.
\begin{figure}[t!]
\begin{center}
\scalebox{1.09}{\epsfig{file=figure3.eps}
\caption{Phase diagram of the massless GN model assuming only translationally invariant phases for different cutoff values (inner to outer) $\Lambda = 2.5, 4$ and $\infty$. The thin continuous lines mark boundaries of metastability.}
\label{fig2}
\end{center}
\end{figure}
It is not possible, however, to include inhomogeneous phases of superconductors \cite{49,50} in the dimensional reduction in the same vein as sketched above. Consider the Hamiltonian Eq.~(\ref{b4}) for a spherically symmetric condensate,
\begin{align}
&\frac{1}{2} \sum_{\bf p} \sum_{\bf q} \Delta(|{\bf q}|) \psi^\dagger(|{\bf p + q}|) C \psi^*(|{\bf p}|) \nonumber \\
&\qquad - \frac{1}{2} \Delta^*(|{\bf q}|) \psi^T(|{\bf p - q}|) C \psi(|{\bf p}|) ,
\end{align}
where $|{\bf q}| \ll |{\bf p}|$. The requirement for dimensional reduction is
\begin{equation}
|{\bf p \pm q}| = |{\bf p}| \pm \frac{{\bf p} \cdot {\bf q}}{|{\bf p}|} = |{\bf p}| \pm |{\bf q}|
\end{equation}
for all ${\bf p}$. This is only possible for ${\bf q} = 0$, i.e. a homogeneous condensate.
\section{Hubbard model}
In this section we will consider a system from condensed matter physics that displays a one-dimensional instability. For a wide class of organic and some inorganic materials conduction is essentially restricted to one dimension due to their anisotropic structure. They can be described as a family of weakly coupled chains. This weak transverse coupling allows these materials to circumvent the Coleman--Mermin--Wagner theorem and exhibit long-range order. Such phases are characterized by a one-dimensional inhomogeneous charge or spin distribution and are, therefore, called charge and spin density waves (CDW, SDW). These materials are known as spin-Peierls systems. For further reference see \cite{44,63}.
As for BCS theory, the phase diagram of the Hubbard model does not depend on the chemical potential $\mu$ but on an external magnetic field $h$. At low temperature and small magnetic field the system possesses a CDW${}_0$ ground state. The CDW${}_0$ condensate takes the form of a plane wave $\Delta e^{{\rm i} {\bf Q}_n \cdot {\bf x}}$. The wavevector ${\bf Q}_n$ is tilted by an angle depending on the lattice spacing with respect to the preferential direction. At high density, this condensate is modulated in $x$ direction. This is called a CDW${}_x$ phase. The modulation is of order $\cal{O}\left( {\bf Q_n} \right)$. A CDW${}_y$ condensate which is modulated in the perpendicular directions is possible as well, but is excluded for the parameter values that we consider in this paper \cite{39}. At high temperature a transition to a homogeneous normal phase occurs where the gap vanishes. In this section we will show that the phase diagram of the Hubbard model can be identified with the phase diagram of the genGN model, Fig.~\ref{fig1}. The massive homogeneous phase in the genGN model corresponds to a CDW${}_0$ phase, the inhomogeneous phase to a CDW${}_x$ phase and the massless chirally symmetric phase to the normal phase.
Early work on the relationship between the original Hubbard model \cite{38} with repulsive interaction and the chirally invariant Thirring model was carried out by Filev \cite{23} and extended by Melzer \cite{36} as well as Woynarovich and Forg\'acs \cite{37}. Since we are investigating a more general model which contains two competing interaction terms and admits a nontrivial phase diagram \cite{2}, there is almost no overlap between our work and theirs.
\subsection{Definition, Symmetries and Anisotropic Hopping}
We start with a system of fermions on a hypercubic lattice with $N_{\rm lat}$ sites that are allowed to tunnel (or ''hop'') to nearest lattice sites and to interact with their nearest neighbors via a spin-dependent repulsive quartic interaction. The lattice is bipartite, i.e. it can be divided into two sublattices A and B such that the interaction takes place between fermions on different sublattices:
\begin{align}\label{v1
H &= - \sum_\sigma \sum_{<{\bf i j}>} t_{ij} \psi_\sigma^\dagger({\bf j}) \psi_\sigma({\bf i}) - h \sum_\sigma \sum_{\bf j} \sigma \, \psi_\sigma^\dagger({\bf j}) \psi_\sigma({\bf j}) \nonumber \\
&\quad - \frac{1}{2} \sum_{\sigma, \sigma'} \frac{1}{N_\mu} \sum_{<{\bf i j}>} V_{\sigma \sigma'} \, \Big(\psi_\sigma^\dagger({\bf i}) \psi_\sigma({\bf i}) - n_{\sigma}({\bf i})\Big) \nonumber \\
&\quad \times \Big(\psi_{\sigma'}^\dagger({\bf j}) \psi_{\sigma'}({\bf j}) - n_{\sigma'}({\bf j})\Big) ,
\end{align}
The sum runs over the spin indices $\sigma$ and lattice sites ${\bf i}$. $<{\bf i j}>$ denotes the sum over nearest neighbors and $N_\mu$ the number of nearest neighbors ($N_\mu = 2 d$ for a hypercubic lattice). $\psi_\sigma({\bf i})$ is a fermion spinor at site ${\bf i}$ with spin $\sigma$ and $n_\sigma({\bf j}) = \langle \psi_\sigma^\dagger({\bf j}) \psi_\sigma({\bf j}) \rangle$ is the mean occupation number of a site. $t_{ij}$ is the hopping amplitude between adjacent lattice sites which we suppose to depend only on the lattice direction. As for the BCS theory, $h$ describes the effect of impurities or an external magnetic field. The most general symmetric form of the coupling is
\begin{equation}
V_{\sigma \sigma'} = U_c - U_s \sigma \sigma'
\end{equation}
with $U_c < 0$ and $0 \leq U_s \leq |U_c|$. A coupling of the form $- U_s (1 + \sigma \sigma')$ is repulsive for spins of the same type and favors the formation of an alternating pattern of spin-$\uparrow$ and -$\downarrow$ fermions --- a spin density wave (SDW). A coupling of the form $U_c < 0$ enhances an inhomogeneous charge distribution --- a charge density wave (CDW).
The Hamiltonian (\ref{v1}) is symmetric under a particle hole conjugation defined by
\begin{align}
U^\dagger \psi_\sigma({\bf j}) U &= \left\{\begin{matrix} \psi_{-\sigma}^\dagger({\bf j}) & {\bf j} \in A \\ - \psi_{-\sigma}^\dagger({\bf j}) & {\bf j} \in B\end{matrix}\right. = e^{{\rm i} {\bf Q}_n \cdot {\bf x}_j} \, \psi_{-\sigma}^\dagger({\bf j}) ,
\end{align}
where ${\bf Q}_n = (\pi/a_x, \pi/a_y, \pi/a_z)$ and ${\bf x}_j = (j_1 a_x, j_2 a_y, j_3 a_z)$ and $a_k$ is the lattice spacing in $k$ direction. The minus sign ensures that the kinetic term is invariant under this transformation for a bipartite lattice and the factor $\sigma$ in the magnetic term provides the invariance of this term. The invariance of the interaction term follows from $U^\dagger \psi_\sigma^\dagger({\bf i}) \psi_\sigma({\bf i}) U = 1 - \psi_{-\sigma}^\dagger(\bf{i}) \psi_{-\sigma}(\bf{i})$ and the symmetry in the summation over $\sigma$ and $\sigma'$ as well as $<\bf{i j}>$. The momentum decomposition of the transformed fields is
\begin{align}\label{v4}
U^\dagger \psi_\sigma({\bf p}) U &= \frac{1}{N_{\rm lat}^{1/2}} \sum_{\bf j} U^\dagger \psi_\sigma({\bf j}) U \, e^{-{\rm i} {\bf p} \cdot {\bf x}_j} \nonumber \\
&= \psi_{-\sigma}^\dagger({\bf Q}_n - {\bf p}) .
\end{align}
Note that the transformation reflects the momentum on the Fermi surface. For further reference, we note an identity that holds for the expectation value of operator bilinears (when ${\bf q} \neq 0$) \cite{37a}:
\begin{align}
\langle \psi_\sigma^\dagger({\bf p} - {\bf q}) \, \psi_\sigma({\bf p}) \rangle &= \frac{{\rm tr}\left\{\psi_\sigma^\dagger({\bf p} - {\bf q}) \psi_\sigma({\bf p}) \exp\left[- \beta H\right] \right\}}{{\rm tr} \left\{\exp\left[- \beta H\right]\right\}} \nonumber \\
&= - \langle \psi_{-\sigma}^\dagger({\bf Q}_n - {\bf p}) \, \psi_{-\sigma}({\bf Q}_n - {\bf p}+ {\bf q}) \rangle.
\end{align}
For ${\bf q} = 0$ the same calculation yields the expression $\langle \psi_\sigma^\dagger({\bf j}) \psi_\sigma({\bf j}) \rangle + \langle \psi_{-\sigma}^\dagger({\bf j}) \psi_{-\sigma}({\bf j}) \rangle = 1$. This symmetry implies half-filling of the system, i.e. half of the number of states are occupied:
\begin{equation}
\sum_\sigma \sum_{\bf j} \langle \psi_\sigma^\dagger({\bf j}) \psi_\sigma({\bf j}) \rangle = N_{\rm lat} .
\end{equation}
The low-energy behavior is determined by the modes close to the Fermi surface. For ${\bf Q} \approx 0$ only a fluctuation term survives in Eq.~(\ref{v1}) which we will ignore. It is customary to neglect scattering into higher states and assume ${\bf Q} \approx \pm {\bf Q}_n$. Hence, we obtain the Hamiltonian
\begin{align}\label{v5}
H &= \sum_\sigma \sum_{\bf p} \varepsilon_\sigma({\bf p}) \psi_\sigma^\dagger({\bf p}) \psi_\sigma({\bf p}) + \frac{1}{2} \sum_{\sigma, \sigma'} \sum_{{\bf p}, {\bf p}', {\bf Q}} V_{\sigma \sigma'} \psi_\sigma^\dagger({\bf p} + {\bf Q}) \nonumber \\
&\quad \times \psi_\sigma({\bf p}) \psi_{\sigma'}^\dagger({\bf p}' - {\bf Q}) \psi_{\sigma'}({\bf p}') ,
\end{align}
where the single particle energy-spectrum $\varepsilon_\sigma({\bf p})$ is given by
\begin{equation}
\varepsilon_\sigma({\bf p}) = \varepsilon({\bf p}) - \sigma h = - \sum_i 2 t_i \cos p_i a_i - \sigma h .
\end{equation}
We note that this Hamiltonian is still symmetric under a particle hole transformation. The spin-independent part of the energy is antisymmetric under a shift by ${\bf Q}_n$:
\begin{equation}\label{v3}
\varepsilon({\bf p} + {\bf Q}_n) = - \varepsilon({\bf p}) .
\end{equation}
In particular, every point on the Fermi surface of the free field theory at half-filling and $h = 0$ is mapped onto another point on the Fermi surface under a shift by ${\bf Q}_n$. This is called nesting and ${\bf Q}_n$ is known as the nesting vector.
\subsection{Dimensional Reduction}
In a quasi one-dimensional system the hopping amplitude in one direction is much larger compared to the others: $t_x \gg t_y, t_z$. If we assume that the hopping amplitude is proportional to the overlap of atomic orbitals this reflects directly the orbital structure and the configuration of the lattice. A slice through the Fermi surface is shown in Fig.~\ref{fig3}. We linearize the energy around $p_F$:
\begin{figure}[b!]
\begin{center}
\scalebox{0.75}{\epsfig{file=figure4.eps}
\caption{Slice through the Fermi surface of the quasi one-dimensional Hubbard model.}
\label{fig3}
\end{center}
\end{figure}
\begin{equation}
\varepsilon_\sigma({\bf p}) \approx v_F (p - p_F) - 2 t_y \cos p_y a_y - 2 t_z \cos p_z a_z - \sigma h,
\end{equation}
where the Fermi velocity is defined by
\begin{equation}
v_F = \left. \frac{\partial \varepsilon({\bf p})}{\partial p_x} \right|_{p_x = \pi/2 a_x,p_y = p_z = 0} = 2 t_x a_x .
\end{equation}
Again we assume the hierarchy $h,t_y,t_z \ll v_F p_F$.
We analyze the Hamiltonian in a mean-field approximation
\begin{align}\label{v5a}
H &= \sum_{{\bf p}, \sigma} \varepsilon_\sigma({\bf p}) \psi_\sigma^\dagger({\bf p}) \psi_\sigma({\bf p}) + \sum_{{\bf Q}, {\bf p}, \sigma} \psi_\sigma^\dagger({\bf p} + {\bf Q}) \psi_\sigma({\bf p}) \Delta_{{\bf Q} \sigma} \nonumber \\
&\quad - \frac{1}{2} \sum_{{\bf Q}, \sigma} D_{- {\bf Q} \sigma} \Delta_{{\bf Q} \sigma} ,
\end{align}
where we define
\begin{align}
D_{{\bf Q} \sigma} &= \sum_{\bf p} \langle \psi_{\sigma}^\dagger({\bf p} - {\bf Q}) \psi_{\sigma}({\bf p}) \rangle \quad {\rm and} \nonumber \\
\Delta_{{\bf Q} \sigma'} &= \sum_{\sigma'} V_{\sigma \sigma'} D_{{\bf Q} \sigma'} .
\end{align}
Because of the nesting symmetry (\ref{v3}) the Fermi surfaces for $p_x > 0$ and $p_x < 0$ can be mapped onto one another for small perturbations. In this case, each $p_x$ value corresponds to two patches on the Fermi surface with $p_x > 0$ and $p_x < 0$, respectively. For the right (upper) Fermi surface (i.e. $p_x > 0$) we introduce the notation $\psi_{R\sigma}^{(i)}(p) = \psi_\sigma({\bf l}^{(i)} + (p,{\bf 0}_\perp))$, where ${\bf l}^{(i)}$ points to the i-th patch on the right Fermi surface. If we define left-moving spinors by $\psi_{L\sigma}^{(i)}(p) = \psi_\sigma({\bf l}^{(i)} - {\bf Q}_n + (p,{\bf 0}_\perp))$ the mean-field Hamiltonian separates into a sum over all patches. The construction is illustrated in Fig. \ref{fig3}. In this notation the action of the particle-hole transformation (\ref{v4}) on the spinors reads
\begin{equation}
U \psi_{L/R \sigma}(p) U^\dagger = \psi_{L/R -\sigma}^\dagger(-p) .
\end{equation}
Note that by definition $D_{{\bf Q} \sigma}^\dagger = D_{- {\bf Q} \sigma}$ and by the particle-hole symmetry (\ref{v4})
\begin{equation}\label{v6}
D_{{\bf Q}_n + q \sigma} = D_{- {\bf Q}_n + q -\sigma} .
\end{equation}
For our choice of couplings $U_c < 0$ and $|U_c| \geq U_s > 0$ it can be shown that the condensate $\Delta$ is only modulated in $x$ direction \cite{39}. Such modulations are called CDW${}_x$ phases (as opposed to CDW${}_y$ phases, where the modulation is perpendicular to the conducting direction). We can now split the summation over ${\bf p}$ into a sum over patches on the Fermi sphere and a summation over the $p_x$-component. Equation~(\ref{v5a}) becomes
\begin{widetext}
\begin{align}\label{v7}
\frac{H}{N} &= \sum_\sigma \sum_{p, q} \left(\psi_{R\sigma}^\dagger(p + q) \ \psi_{L\sigma}^\dagger(p)\right) \left(\begin{matrix} \left[v_F (p + q) - \sigma h\right] \delta_{q,0} & \Delta_{{\bf Q}_n + q \, \sigma} \\ \Delta_{{\bf Q}_n + q \, \sigma}^* & \left[- v_F p - \sigma h\right] \delta_{q,0} \end{matrix}\right) \left(\begin{matrix}\psi_{R\sigma}(p + q) \\ \psi_{L\sigma}(p)\end{matrix}\right) \nonumber \\
&\qquad \qquad \qquad - \frac{1}{2 N} \sum_{\sigma, \sigma'} \sum_q \, V_{\sigma \sigma'} \left(D_{{\bf Q}_n + q \, \sigma}^* D_{{\bf Q}_n + q \, \sigma'} + D_{{\bf Q}_n + q \, \sigma} D_{{\bf Q}_n + q \, \sigma'}^*\right) .
\end{align}
\end{widetext}
We transform the spinors according to
\begin{equation}
\left(\begin{matrix}\psi_{R\uparrow}(p) \\ \psi_{L\uparrow}(p) \\ \psi_{R\downarrow}(p) \\ \psi_{L\downarrow}(p)\end{matrix}\right) \rightarrow \left(\begin{matrix}\psi_{R\uparrow}(p) \\ \psi_{L\uparrow}(p) \\ \psi_{L\downarrow}^\dagger(- p) \\ \psi_{R\downarrow}^\dagger(- p)\end{matrix}\right) .
\end{equation}
This transformation maps $D_{{\bf Q}_n + q \, \sigma} \to \sigma D_{{\bf Q}_n + q \, \sigma}$ and the \emph{c}-number term of Eq.~(\ref{v7}) becomes
\begin{equation}\label{va8}
- \frac{1}{N} \sum_{\sigma, \sigma'} \sum_q \, \left(U_c \, \sigma \sigma' - U_s\right) D_{{\bf Q}_n + q \, \sigma}^* D_{{\bf Q}_n + q \, \sigma'} .
\end{equation}
The condensate becomes
\begin{align}
\Delta_{{\bf Q}_n + q \, \uparrow} &\to S(q) - {\rm i} P(q) \quad {\rm and} \nonumber \\
\Delta_{{\bf Q}_n + q \, \downarrow} &\to S(q) + {\rm i} P(q) \label{Deltadown} ,
\end{align}
where we define
\begin{align}
S(q) &= U_c \left(D_{{\bf Q}_n + q \, \uparrow} - D_{{\bf Q}_n + q \, \downarrow}\right) \nonumber \\
&= \frac{U_c}{2} \sum_\sigma \sigma \left(D_{{\bf Q}_n + q \, \sigma} - D_{- {\bf Q}_n + q \, \sigma}\right) \quad {\rm and} \\
{\rm i} P(q) &= U_s \left(D_{{\bf Q}_n + q \, \uparrow} + D_{{\bf Q}_n + q \, \downarrow}\right) \nonumber \\
&= \frac{U_s}{2} \sum_\sigma \left(D_{{\bf Q}_n + q \, \sigma} + D_{- {\bf Q}_n + q \, \sigma}\right) .
\end{align}
The particle hole symmetry (\ref{v6}) implies the relations
\begin{equation}
S(q)^* = S(-q) \qquad ({\rm i} P(q))^* = - {\rm i} P(-q)
\end{equation}
and we can rewrite Eqs.~(\ref{v7}) and (\ref{va8}) enclosing the system in a box with length $L$ in $x$ direction:
\begin{align}
\frac{H}{N} &= \sum_\sigma \sum_{p, q} \left(\psi_{R\sigma}^{\dagger}(p + q) \ \psi_{L\sigma}^{\dagger}(p)\right) \nonumber \\
&\quad \times \left(\begin{matrix} \left[v_F (p + q) - h\right] \delta_{q,0} & \sigma S(q) - {\rm i} P(q) \\ \sigma S(-q) + {\rm i} P(-q) & \left[- v_F p - h\right] \delta_{q,0} \end{matrix}\right) \nonumber \\
&\quad \times \left(\begin{matrix}\psi_{R\sigma}^{}(p + q) \\ \psi_{L\sigma}^{}(p)\end{matrix}\right) - \sum_q \frac{L |S(q)|^2}{U_c L N} + \sum_q \frac{L |P(q)|^2}{U_s L N} .
\end{align}
Finally, if we transform the spin-$\downarrow$ spinors according to
\begin{equation}
\left(\begin{matrix}\psi_{R\downarrow}(p) \\ \psi_{L\downarrow}(p)\end{matrix}\right) \rightarrow \left(\begin{matrix}- \psi_{L\downarrow}(p) \\ \psi_{R\downarrow}^\dagger(p)\end{matrix}\right) ,
\end{equation}
we can recast the theory in the form of the genGN model in a chiral basis with coupling constants $g^2 N = - U_c L N/2 v_F$ and $G^2 N = U_s L N/2 v_F$:
\begin{align}
\frac{H}{N} &= \sum_\sigma \int {\rm d}x \, \psi^\dagger(x) \left[- {\rm i} \gamma^5 \partial_x - h + S(x) \gamma^0 \right. \nonumber \\
&\quad + \left. {\rm i} P(x) \gamma^1\right] \psi(x) + \frac{S^2(x)/v_F}{2 (- U_c L N/2 v_F)} \nonumber \\
&\quad + \frac{P^2(x)/v_F}{2 (U_s L N/2 v_F)} ,
\end{align}
where we redefine $S/P(q) \to L^{1/2} S/P(q)$ in order to give proper engineering dimension. The additional spin degree of freedom results in a doubling of flavors in the genGN model. The transformed condensates are
\begin{align}
S(q) &= \frac{U_c}{2} \sum_\sigma \left(D_{{\bf Q}_n + q \, \sigma} + D_{- {\bf Q}_n + q \, \sigma}\right) \nonumber \\
&= \frac{U_c}{2} \sum_\sigma \sum_p \langle \overline{\psi}(p - q) \, \psi(p) \rangle \quad {\rm and} \\
{\rm i} P(q) &= \frac{U_s}{2} \sum_\sigma \left(D_{{\bf Q}_n + q \, \sigma} - D_{- {\bf Q}_n + q \, \sigma}\right) \nonumber \\
&= - \frac{U_s}{2} \sum_\sigma \sum_p \langle \overline{\psi}(p - q) {\rm i} \gamma^5 \, \psi(p) \rangle ,
\end{align}
as one would expect from varying the grand canonical potential density $\Omega = - \ln {\rm tr} \, e^{- \beta H} /\beta L$ with respect to $S$ and $P$.
As for BCS theory, the cutoff is a physical quantity. The cutoff dependence of the second order transition line between massive and massless homogeneous phases has already been determined in Sec.~III. As pointed out in \cite{2}, the genGN condensate can be approximated by the variational ansatz
\begin{equation}
S(x) = 2 S_1 \cos(2 p_F x) \ {\rm and} \ P(x) = 2 P_1 \cos(2 p_F x)
\end{equation}
in the vicinity of the second order transition between inhomogeneous and homogeneous massless phase. The single particle energies can be calculated perturbatively using almost degenerate perturbation theory which allows us to determine the correction to the grand canonical potential of the free Fermi gas:
\begin{align}
\Psi &= \Psi_{\rm normal} + \delta \Psi \nonumber \\
&= \Psi_{\rm normal} + {\cal M}_{11} S_1^2 + 2 {\cal M}_{12} S_1 P_1 + {\cal M}_{22} P_1^2 .
\end{align}
Varying this expression with respect to the parameters $S_1, P_1$ and $p_F$ yields two conditions which determine the critical values at the phase transition,
\begin{align}
\det {\cal M} &= {\cal M}_{11} {\cal M}_{22} - {\cal M}_{12}^2 = 0 \quad {\rm and} \quad \frac{\partial \det {\cal M}}{\partial p_F} = 0 ,
\end{align}
where the coefficients ${\cal M}_{ij}$ are:
\begin{figure}[b!]
\begin{center}
\scalebox{0.68}{\epsfig{file=figure5.eps}
\caption{Second order phase transition sheet of the genGN model for the cutoff values $\Lambda = 4,5$ and $\infty$.}
\label{fig5}
\end{center}
\end{figure}
\begin{align
{\cal M}_{11} &= {\rm PV} \int_0^{\Lambda/2} \frac{{\rm d}p}{2 \pi} \frac{4 p_F}{p^2 - p_F^2} \left(\frac{1}{1 + e^{\beta (p - \mu)}} + \frac{1}{1 + e^{\beta (p + \mu)}}\right) \nonumber \label{v8} \\
&\quad + \frac{1}{\pi} \ln \left[\frac{\Lambda}{2} + \left(\left(\frac{\Lambda}{2}\right)^2 + 1\right)^{1/2}\right] \nonumber \\
&\quad - \frac{1}{2 \pi} \ln\left[ \left(\frac{\Lambda}{2 p_F}\right)^2 - 1 \right] \\
{\cal M}_{22} &= {\cal M}_{11} + \frac{\xi}{\pi} \\
{\cal M}_{12} &= {\rm PV} \int_0^{\Lambda/2} \frac{{\rm d}p}{2 \pi} \frac{2 p_F}{p^2 - p_F^2} \left(\frac{1}{1 + e^{\beta (p - \mu)}} - \frac{1}{1 + e^{\beta (p + \mu)}}\right) \nonumber \\
&\quad + \frac{1}{2 \pi} \left(\ln\left[\frac{\Lambda}{2 p_F} - 1\right] - \ln\left[\frac{\Lambda}{2 p_F} + 1\right]\right) \label{v9} .
\end{align}
\noindent ${\rm PV}$ denotes a principal value integration. As $\Lambda \to \infty$ the second lines of Eq.~(\ref{v8}) equals $- 1/\pi \ln 2 p_F$. The second line of Eq.~(\ref{v9}) vanishes in this limit and Eqs.~(\ref{v8})-(\ref{v9}) are, of course, equivalent to Eq.~(129) of \cite{2}. Examples of second order transition sheets are shown in Fig.~\ref{fig5}. As expected, even for moderate values of $\Lambda$ the distortion of the phase diagram is negligible. We refrain from determining the cutoff dependence of the first oder transition sheet which would require an extensive numerical Hartree-Fock calculation \cite{47,48} and would not yield much physical insight.
The phase diagram of various spin-Peierls compounds has been measured with high accuracy \cite{43,44,45,46} whereas theoretical investigations \cite{39,41,42} have not revealed the full phase diagram to the best of our knowledge. In particular, the first order transition line between massive homogeneous (CDW${}_0$) and inhomogeneous (CDW${}_x$) phase has never been determined. We are now able to exploit our mapping and confront experimental data with a full theoretical phase diagram for the first time. Figure~(\ref{fig4}) shows a fit to data obtained by Hase \emph{et al}. \cite{46} for the inorganic spin-Peierls system CuGeO${}_3$~. The theoretical phase diagram is fit to the scale of experimental data. This corresponds to fitting the scale parameters $\Delta_0$ and $v_F$ which are set equal to $1$ in the analysis of the genGN model. The phase diagram of the genGN model was determined in the $\mu$-$T$ plane for fixed values of $\xi = 0, 0.1, 0.2, 0.4, 0.8, 1.2, 2, 3, 5$ and $10$ and the fit $\xi = 2$ was chosen from this ensemble.
\begin{figure}[t!]
\begin{center}
\scalebox{1.05}{\epsfig{file=figure6.eps}
\caption{Phase diagram of the inorganic spin-Peierls cuprate CuGeO${}_3$. The fit is done with $\xi \approx 2$. The theoretical curve is fit to the scale of the data. The data is taken from \cite{46}.}
\label{fig4}
\end{center}
\end{figure}
\section{Summary and conclusions}
In this work we explored applications of the Gross--Neveu model to nonrelativistic field theories. Starting from the striking observation that the phase diagram of BCS theory and Gross--Neveu model coincide when restricting to phases with translational invariance, we were able to map BCS theory onto the massless Gross--Neveu model with discrete chiral symmetry on a mean-field level. We were able to show that the mean-field Hamiltonian of the quasi one-dimensional extended Hubbard model, which is widely used in condensed matter physics in the description of spin-Peierls systems, is equivalent to a generalized Gross--Neveu model with two coupling constants. In particular, the phase diagrams of both models are equivalent including inhomogeneous phases. This model was worked out in detail only recently by Boehmer and Thies \cite{2}. Relying on their results we were able to complete the phase diagram of the Hubbard model and confront experimental data with the full phase diagram for the first time
It is interesting to note that although all models have been subject to intensive research over the last three decades the correspondence of the phase diagrams has not been noticed. In particular inhomogeneous phases of the Hubbard model had been discussed long before the first phase diagram of the Gross--Neveu model was even proposed \cite{41,18}.
This work supplements current efforts to use the expertise on quantum field theoretical toy models in the study of phases of QCD at high density by providing systems that can be dimensionally reduced most clearly. The equivalence of the Gross--Neveu and Hubbard model might lay the ground for further work. In particular, analytical solutions for phases of the Gross--Neveu model with continuous chiral symmetry that do not affect the phase diagram might find a physical counterpart \cite{54,55}. Recently, there has been a lot of interest in baryon scattering and other dynamical phenomena in the Gross--Neveu model \cite{52,53}. The extensive analytical work over the past decades that was devoted to the study of dynamical phenomena in spin-Peierls systems (''sliding'' of CDWs) \cite{51} might be useful.
\section*{Acknowledgements}
I would like to thank Michael Thies for suggesting to work out a correspondence between the massless Gross--Neveu model and BCS theory. Furthermore, I am indebted to him for many useful discussions and a critical reading of this manuscript. I would like to thank Christian Boehmer and Dominik Nickel for helpful comments concerning their work and Gerald V. Dunne for pointing out Ref.~\cite{44} and a helpful comment.
|
{
"timestamp": "2010-12-17T02:03:00",
"yymm": "1009",
"arxiv_id": "1009.4071",
"language": "en",
"url": "https://arxiv.org/abs/1009.4071"
}
|
\section{Introduction}
The radiation emitted by accelerated charges produces reaction forces acting back on them. For rotating charged particles (e.g., electric\cite{B1988} and magnetic\cite{P1955} dipoles), this gives rise to reaction torques.\cite{C1967} Likewise, accelerated neutral bodies are known to experience friction because they emit light due to the absolute change in the boundary conditions of the electromagnetic field. This is the so-called Casimir radiation.\cite{KG99,KN02}
A spinning sphere presents a more challenging situation: its surface appears to be unchanged, although it experiences a centripetal acceleration. So, the question arises, does a homogeneous, neutral sphere emit light simply by rotating? Is such a particle slowing down when spinning in vacuum? We know the inverse process to be true: the angular momentum carried by light can be transformed into mechanical rotation of neutral particles.\cite{TYL05} However, this type of problem requires a delicate analysis, somehow related to the non-contact friction predicted to occur between planar homogeneous surfaces set in relative uniform motion,\cite{P97} which is currently generating a heated debate.\cite{controversy}
In this paper, we investigate the friction produced on rotating neutral particles by interaction with the vacuum electromagnetic fields. Friction is negligible in dielectric particles possessing large optical gap compared to the rotation and thermal-radiation frequencies. For other materials (e.g., metals), in contrast to previous predictions,\cite{P06} we find nonzero stopping even at zero temperature. The dissipated energy is transformed into radiation emission and thermal heating of the particle, although cooling relative to the surrounding vacuum is shown to take place under very common conditions. We formulate a theory that describes these phenomena and allows us to predict experimentally measurable effects.
\section{Theoretical description}
We consider an isotropic particle at temperature $T_1$ spinning with frequency $\Omega$ and embedded in a vacuum at temperature $T_0$ (see Fig.\ \ref{Fig1}). The particle experiences a torque $M$ by interaction with the surrounding radiation field and it is also capable of exchanging photons, with net emission power $P^{\rm rad}$. For simplicity, we assume the particle radius $a$ to be small compared to the wavelength of the involved photons, so that we can describe it through its frequency-dependent polarizability $\alpha(\omega)$. Since the maximum frequency of exchanged photons is controlled by the rotation frequency and the thermal baths at temperatures $T_0$ and $T_1$, this approximation implies that both $\Omega a/c$ and $k_BT_ja/c\hbar$ are taken to be small compared to unity. These conditions are fulfilled in very common situations (for instance, for $a=50\,$nm, one has $\Omega\ll6\times10^3\,$THz and $T_j\ll4.6\times10^4\,$K).
\begin{figure}
\centerline{\includegraphics*[width=7cm]{fig1.pdf}}
\caption{Sketch of a spherical rotating particle and parameters considered in this work. The particle is at temperature $T_1$ and rotates with frequency $\Omega$. The interaction with vacuum at temperature $T_0$ produces a frictional torque $M$ and a radiated power $P^{\rm rad}$.} \label{Fig1}
\end{figure}
Friction originates in fluctuations of both (i) the vacuum electromagnetic field $\Eb^{\rm fl}$ and (ii) the particle polarization $\pb^{\rm fl}$. We calculate the emitted power from the work exerted by the particle dipole,
\begin{eqnarray}
P^{\rm rad}=-\left\langle\Eb^{\rm ind}\cdot\partial\pb^{\rm fl}/\partial t+\Eb^{\rm fl}\cdot\partial\pb^{\rm ind}/\partial t\right\rangle,\label{P00}
\end{eqnarray}
where $\Eb^{\rm ind}$ is the field induced by $\pb^{\rm fl}$, and $\pb^{\rm ind}$ is the dipole induced by $\Eb^{\rm fl}$. Likewise, the torque is obtained from the action of the field on the dipole,
\begin{eqnarray}
{\bf M}=\left\langle\pb^{\rm fl}\times\Eb^{\rm ind}+\pb^{\rm ind}\times\Eb^{\rm fl}\right\rangle.\label{M00}
\end{eqnarray}
The result is quadratic in $\Eb^{\rm fl}$ for contribution (i) and in $\pb^{\rm fl}$ for contribution (ii). The brackets $\left\langle\right\rangle$ represent the average over these quadratic fluctuation terms, which we perform using the fluctuation-dissipation theorem (FDT) (see Appendix).
Rotational motion enters here through the transformation of the field and the polarization back and forth between rotating and lab frames. This is needed because the particle polarizability can only be applied in the rotating frame, in which the electronic and vibrational excitations participating in $\alpha$ are well defined and $\Omega$-independent. In contrast, the effective polarizability in the lab frame has a dependence on $\Omega$. Further details of this formalism are given in the Appendix. The resulting radiated power reads (see Appendix for a detailed derivation)
\begin{eqnarray}
P^{\rm rad}=\int_{-\infty}^\infty\hbar\omega\;d\omega\;\Gamma(\omega)\label{P},
\end{eqnarray}
where
\begin{eqnarray}
\Gamma(\omega)&=&(2\pi\omega\rho^0/3)\,\Big\{2g_\perp(\omega-\Omega)\,\big[n_1(\omega-\Omega)-n_0(\omega)\big]\nonumber\\
&&+g_\parallel(\omega)\,\big[n_1(\omega)-n_0(\omega)\big]\Big\}\label{Pw}
\end{eqnarray}
is the spectral distribution of the rate of emission (when $\omega\Gamma>0$) or absorption ($\omega\Gamma<0$),
$n_j(\omega)=[{\exp(\hbar\omega/k_BT_j)-1}]^{-1}$ is the Bose-Einstein distribution function at temperature $T_j$,
\begin{eqnarray}
g_l(\omega)={\rm Im}\{\alpha_l(\omega)\}-\frac{2\omega^3}{3c^3}|\alpha_l(\omega)|^2\nonumber
\end{eqnarray}
are odd functions of $\omega$ describing particle absorption for polarization either parallel ($l=\parallel$) or perpendicular ($l=\perp$) with respect to the rotation axis, and $\rho^0=\omega^2/\pi^2c^3$ is the free-space local density of photonic states. These results apply to particles with orthogonal principal axes of polarization, rotating around one of them, and with $\alpha_\perp$ given by the average of the polarizability over the remaining two orthogonal axes. The torque $M$ takes a similar form,
\begin{eqnarray}
M=-\int_{-\infty}^\infty d\omega\;\hbar\Gamma(\omega).\label{M}
\end{eqnarray}
Incidentally, the $g_\parallel$ term of Eq.\ (\ref{Pw}) vanishes under the integral of Eq.\ (\ref{M}), and furthermore, $M=0$ for $\Omega=0$. In the $T_0=T_1=0$ limit, one has $n_j(\omega)=-\theta(\omega)$, from which we find the integrals to be restricted to the $(0,\Omega)$ range: only photons of frequency below $\Omega$ can be generated.
Unfortunately, Eqs.\ (\ref{P00}) and (\ref{M00}) do not account for radiative corrections coming from the elaborate motion of induced charges in the rotating particle. Although such corrections are insignificant for small particles, we incorporate them here for spheres in a phenomenological way through the term proportional to $|\alpha|^2$ in Eq.\ (\ref{Pw}), preceded by a coefficient chosen to yield $g_l=0$ (and consequently, $M=0$) in non-absorbing particles:\cite{V1981} internal excitations (i.e., absorption) are necessary to mediate the coupling between the rotational state and radiation.\cite{friction2} Furthermore, we neglect magnetic polarization, which can be important for large, highly conductive particles.\cite{frictionn}
\section{Metallic particles}
This case is representative for absorbing particles. At low photon frequencies $\omega$ below the interband transitions region, metals can be well described by the Drude model, characterized by a DC electric conductivity $\sigma_0$ and a dielectric function $\epsilon=1+{\rm i}} \def\ee{{\rm e}} \def\mb{{\bf m}} \def\vb{{\bf v}\,4\pi\sigma_0/\omega$.\cite{AM1976} For a spherical particle of radius $a$, we have $\alpha\approx a^3(\epsilon-1)/(\epsilon+2)$, and consequently
\begin{eqnarray}
{\rm Im}\{\alpha(\omega)\}\approx3\omega a^3/4\pi\sigma_0.\label{drude}
\end{eqnarray}
For sufficiently small particles, absorption dominates over radiative corrections, so that we can overlook terms proportional to $|\alpha|^2$ in Eqs.\ (\ref{P})-(\ref{M}). Then, we find the closed-form expressions
\begin{eqnarray}
P^{\rm rad}_{\rm D}=\frac{\hbar a^3}{60\pi^2c^3\sigma_0}\Big[2\Omega^6+5\Omega^4\theta_1^2+3\Omega^2\theta_1^4
+\frac{5}{14}(\theta_1^6-\theta_0^6)\Big]
\label{Pdrude}
\end{eqnarray}
and
\begin{eqnarray}
M_{\rm D}=\frac{-\hbar a^3\Omega}{120\pi^2c^3\sigma_0}\Big[6\Omega^4+10\Omega^2\theta_1^2+\theta_0^4+3\theta_1^4\Big],
\label{Mdrude}
\end{eqnarray}
where the subscript D refers to the Drude model and
\begin{eqnarray}
\theta_j=2\pi k_BT_j/\hbar.\nonumber
\end{eqnarray}
Equations\ (\ref{Pdrude}) and (\ref{Mdrude}) show that vacuum friction is always producing stopping ($M\Omega<0$), whereas the balance of radiation exchange between particle and free space can change sign depending on their relative temperatures. The general trend of these expressions is shown in Fig.\ \ref{Fig2}(b). At low $\Omega$, the torque scales as $\Omega$, whereas a steeper $\Omega^5$ dependence is observed at faster velocities. Interestingly, a nonzero torque $M\propto\Omega^5$ is predicted at $T_0=T_1=0$, despite the axial symmetry of the particle.
\section{Equilibrium temperature}
The power absorbed by the particle in the form of thermal heating $P^{\rm abs}$ can be obtained from energy conservation, expressed by the identity $-M\Omega=P^{\rm rad}+P^{\rm abs}$, where the left-hand side represents mechanical energy dissipation (stopping power). Using Eqs.\ (\ref{Pdrude}) and (\ref{Mdrude}), we find
\begin{eqnarray}
P^{\rm abs}_{\rm D}=\frac{\hbar a^3}{120\pi^2c^3\sigma_0}\Big[2\Omega^6+\Omega^2(\theta_0^4-3\theta_1^4)
+\frac{5}{7}(\theta_0^6-\theta_1^6)\Big].
\label{PaDrude}
\end{eqnarray}
The particle equilibrium temperature is determined by the condition $P^{\rm abs}=0$, and it is stable because $\partial P^{\rm abs}/\partial T_1<0$ [this inequality is obvious from Eq.\ (\ref{PaDrude}), but it can be easily derived in the general case from Eqs.\ (\ref{P})-(\ref{M})]. Unlike conventional friction of a spinning object immersed in a fluid, vacuum friction is not always leading to particle heating, as shown in Fig.\ \ref{Fig2}(a) from the solution of $P^{\rm abs}_{\rm D}=0$. Actually, $T_1<T_0$ for finite temperatures and rotation velocities below $\Omega=\theta_0$, whereas particle heating occurs at higher $\Omega$. The crossing point between these two types of behavior is independent of particle size $a$ and conductivity $\sigma_0$.
At $T_0=0$, we find $\theta_1\approx0.867\,\Omega$, so that the $\Omega^5$ dependence of $M_{\rm D}$ is maintained with the particle at equilibrium temperature. The loss of mechanical energy is then fully converted into a radiated power $P^{\rm rad}_{\rm D}\approx0.013\,\hbar a^3\Omega^6/c^3\sigma_0$.
It should be noted that having the particle at equilibrium temperature or at the same temperature as the vacuum results in significant differences in the stopping power [Fig.\ \ref{Fig2}(b), calculated from Eqs.\ (\ref{drude})-(\ref{PaDrude})].
\begin{figure}
\centerline{\includegraphics*[width=8cm]{fig2.pdf}}
\caption{Equilibrium temperature and stopping of a metallic sphere. {\bf (a)} Normalized particle temperature at equilibrium ($T_1/T_0$) as a function of $\Omega/\theta_0$, where $\theta_0=2\pi k_BT_0/\hbar$ (see Fig.\ \ref{Fig1}). {\bf (b)} Universal normalized stopping power both at equilibrium temperatures (solid curve) and at equal temperatures ($T_0=T_1$, broken curve).} \label{Fig2}
\end{figure}
\section{Emission spectra}
The probability of emitting photons at frequency $\omega$ is given by $\Gamma(\omega)-\Gamma(-\omega)$ [see Eq.\ (\ref{Pw})], which is normalized per unit of emission-frequency range. The emission profile at low rotation velocities ($\Omega=0.05\,\theta_0$ curve in Fig.\ \ref{Fig3}) mimics the absorption spectrum from a static cold particle (dashed curve), also peaked around $\hbar\omega\approx5k_BT_1$ for Drude spheres. However, the maximum of emission is driven by $\Omega$ for faster rotations (see inset and $\Omega=5\,\theta_0$ curve in Fig.\ \ref{Fig3}), thus signalling a significant departure from standard black-body theory.
\begin{figure}
\centerline{\includegraphics*[width=8cm]{fig3.pdf}}
\caption{Power spectrum $dP^{\rm rad}/d\omega=\hbar\omega[\Gamma(\omega)-\Gamma(-\omega)]$ [see Eq.\ (\ref{Pw})] radiated by a metallic spinning particle for various rotation frequencies. Solid curves: emission at equilibrium temperatures. Dashed curve: absorption by a particle at rest and $T_1=0$. The emitted-photon frequency $\omega$ is normalized to $\theta_0=2\pi k_BT_0/\hbar$. The inset shows the frequency of maximum emission at equilibrium as a function of $\Omega/\theta_0$.} \label{Fig3}
\end{figure}
\section{Stopping time}
At low rotation velocity and finite temperature, the frictional torque acting on a metallic particle is proportional to $\Omega$ [see Eq.\ (\ref{Mdrude})]. The correction to the particle equilibrium temperature [$\theta_1\approx\theta_0-(7/15)\Omega^2/\theta_0$] can be then neglected to first order in $\Omega$, so the torque becomes $M\approx-\beta\Omega$, where $\beta=\hbar a^3\theta_0^4/30\pi^2c^3\sigma_0$. From Newton's second law, we find an $\Omega(t)=\Omega(0)\exp(-t/\tau)$ time dependence of the rotation velocity, where $\tau=I/\beta$ is the characteristic stopping time and $I$ is the moment of inertia. For a spherical Drude particle, we find
\begin{eqnarray}
\tau=\frac{(\hbar c)^3}{\pi}\frac{\rho a^2\sigma_0}{(k_BT_0)^4},\label{tau}
\end{eqnarray}
where $\rho$ is the particle density.
\begin{figure}
\centerline{\includegraphics*[width=8cm]{fig4.pdf}}
\caption{Characteristic stopping time of spinning graphite particles as a function of vacuum temperature. Solid curves: full calculation using measured dielectric functions for the graphite particles.\cite{D03} Broken curves: analytical Drude approximation [Eq. (\ref{tau})]. Various particle sizes and shapes are considered: spheres of radius 10\,nm and 100\,nm, and an oblate ellipsoid of radius 10\,nm and aspect ratio $\eta=0.2$. Low rotation velocities $\Omega\ll k_BT_0/\hbar$ are assumed (e.g., $\Omega\ll21\,$GHz at $T_0=1\,$K).} \label{Fig4}
\end{figure}
Graphite particles are abundant in interstellar dust,\cite{HW1962} so we focus on them as an important case to study the rotation stopping time. The frequency-dependent dielectric function of graphite is taken from optical data,\cite{D03} tabulated for different particle sizes, which differ due to nonlocal corrections. The low-$\omega$ behavior is well approximated by the Drude model with $\sigma_0=2.3\times10^4\;(2.0\times10^5)\,\Omega^{-1}$m$^{-1}$ for spherical particles of radius $a=10\,(100)\,$nm, where the response has been averaged over different crystal orientations. Plugging this into Eq.\ (\ref{tau}), we obtain the results shown in Fig.\ \ref{Fig4} by broken lines. Interband transitions become important in the response of graphite at frequencies above $\hbar\omega\sim10^{-2}\,$eV, so we expect a deviation from Drude behavior at temperatures above $\sim100\,$K in this material. This is indeed confirmed by numerically integrating Eq.\ (\ref{M}) with the full tabulated response of graphite to obtain $\tau$ (Fig.\ \ref{Fig4}, solid curves). For the particle sizes under consideration, stopping times are small on cosmic scales within the plotted range of temperatures, which are often encountered in hot dust regions.\cite{HW1962} In cooler areas ($T_0=2.7\,$K), 100\,nm graphite particles have a stopping time $\tau\sim\,0.6$ billion years.
Dust particles can adopt non-spherical shapes. In particular, for oblate ellipsoids Eq.\ (\ref{drude}) [${\rm Im}\{\alpha(\omega)\}$] must be corrected by a factor $\eta/9L^2$, where $\eta$ is the aspect ratio (see inset in Fig.\ \ref{Fig4}) and $L$ is the depolarization factor for equatorial polarization, approximately linear in $\eta$.\cite{friction5} Also, $I$ is linear in $\eta$, thus leading to a $\tau\propto\eta^2$ dependence for fixed radius. We show in Fig.\ \ref{Fig4} the case $a=10\,$nm and $\eta=0.2$, which exhibits a significant reduction in $\tau$ compared to spherical particles of the same radius. In a related context, translational motion leads to thermal drag,\cite{drag} only at nonzero temperature and with similar stopping times.
\section{Concluding remarks}
The present results can be relevant to study the distribution of rotation velocities of cosmic nanoparticles, which could be eventually examined through measurements of rotational frequency shifts.\cite{BB97MHS05} Besides, relatively small stopping times are predicted for graphite nanoparticles, which ask for experimental corroboration (for example, using in-vacuo optical trapping setups). By analogy to the Purcell effect,\cite{P1946} the frictional torque can be altered due to the presence of physical boundaries that modify the density of states appearing in Eq.\ (\ref{Pw}), thus opening new possibilities for controlling the degree of friction (e.g., the torque can be strongly reduced at low temperature and small rotation frequency by placing the particle inside a metallic cavity, which produces a threshold of $\rho^0$ in $\omega$).
\section*{ACKNOWLEDGMENT}
This work has been supported by the Spanish MICINN (MAT2007-66050 and Consolider NanoLight.es). A.M. acknowledges an FPU scholarship from ME.
|
{
"timestamp": "2010-09-22T02:02:20",
"yymm": "1009",
"arxiv_id": "1009.4107",
"language": "en",
"url": "https://arxiv.org/abs/1009.4107"
}
|
\section{Introduction}
The young X-ray pulsar \hbox{PSR~J0821$-$4300}, associated with the \hbox{Puppis~A}\ supernova
remnant (SNR), is one of three pulsars in SNRs
that are spinning down nearly imperceptibly \citep[][Paper I]{got09}.
The age of \hbox{PSR~J0821$-$4300}\ is $3.7$~kyr based on the
proper motion of oxygen knots in \hbox{Puppis~A}\ \citep{win88}, and
its distance is $2.2$~kpc from \ion{H}{1} absorption features
to the SNR \citep{rey95}.
In the context of the magnetic dipole model, these pulsars
are a new physical manifestation of neutron stars (NSs),
``anti-magnetars'' born with weak
magnetic fields possibly related to slow natal spin \citep{got08}.
They were drawn from the previously defined
class of central compact objects (CCOs) in SNRs,
which are characterized by their steady, predominantly thermal X-ray emission,
lack of optical or radio counterparts, and absence of pulsar wind
nebulae (see reviews by
\citealt{pav04} and \citealt{del08}). Currently, 7--10
objects are known or proposed to be CCOs
\citep[for a list, see][]{hal10},
and are therefore candidates for anti-magnetars.
The \xmm\ discovery observations of \hbox{PSR~J0821$-$4300}\ revealed an
abrupt $180^{\circ}$ phase reversal of its quasi-sinusoidal
pulse profile at an energy of around $1.2$~keV (Paper~I).
The X-ray spectrum of \hbox{PSR~J0821$-$4300}\ was
fitted with a two-temperature blackbody model,
both temperatures being seen at all rotation phases,
while the cross-over energy of the spectral components
coincides with the energy where the pulse reverses phase.
These detailed properties afford the opportunity to construct
a highly constrained model of the NS surface emission geometry.
In this Paper, we present a quantitative verification of the
geometrical model for \hbox{PSR~J0821$-$4300}\ proposed in Paper~I. By reproducing the
detailed pulse profile behavior, we are able to specify the surface
emission areas and viewing geometry to within $< 2^{\circ}$. Our
treatment includes general relativistic effects of light deflection
and gravitational redshift, and examines the effects of local
anisotropy (beaming) in the emitted radiation. We describe the
antipodal hot-spot model in Section~2, compare the energy-dependent
modulation to the data for the range of allowed geometries in
Section~3, and explore if it is possible to limit the neutron star
radius in conjunction with the estimated distance. In Section~4 we
discuss some implications of the model results, and in Section~5
compare with pulsars of other types.
\section{The Emission model}
Our method of modeling the emission from spots on the surface of
a NS follows the derivation given by \citet{pec83},
with some generalizations introduced by \citet{per08}.
The radiation comes from a hot spot of blackbody temperature $T_h$ and
angular radius $\beta_h$, and an antipodal warm spot of
lower temperature $T_w$ and angular radius $\beta_w$.
The remainder of the surface is assumed to be at a uniform
temperature $T_{\rm NS} < T_w$.
The geometry is indicated in Figure~\ref{fig:NS}.
We use $\gamma(t)$ to indicate the phase of rotation
instead of the common notation $\phi(t)$.
Phase $\gamma=0$ corresponds to the closest
approach of the hot spot to the observer, while the phase
of rotation is related to the angular rotation
rate of the star $\Omega$ through $\gamma(t)=\Omega t$.
We indicate with $\alpha_h(t)$ the angle that the hot-spot axis
makes with the line-of-sight. $\alpha_h(t)$ is a function of the angle $\xi$
between the hot-spot axis and the rotation axis and the angle $\psi$
between the line-of-sight and the rotation axis, by means of the relation
\beq
\alpha_h(t)=\arccos[\cos\psi\cos\xi+\sin\psi\sin\xi\cos\gamma(t)]\;.
\label{eq:alpha}
\eeq
For each set of angles $\xi$ and $\psi$, the angle $\alpha_w$ that
the axis of the opposing warm spot makes with respect to the line-of-sight
is simply $\alpha_w(t)=\pi-\alpha_h(t)$.
The spherical coordinate system $(\theta,\phi)$ is
defined with respect to the line-of-sight as the $z$-axis.
Due to general relativistic effects,
a photon emitted at a colatitude $\theta$ reaches the observer
only if emitted at an angle $\delta$ with respect to the perpendicular to the NS surface.
The two angles are related by the
ray-tracing function\footnote{To improve the computational efficiency
of this equation we use the approximation presented in
\citet{bel02}.} \citep{pec83,pag95a}
\beq
\theta(\delta)=\int_0^{R_s/2R}x\;du\left/\sqrt{\left(1-\frac{R_s}{R}\right)
\left(\frac{R_s}{2R}\right)^2-(1-2u)u^2 x^2}\right.\;,
\label{eq:teta}
\eeq
having defined $x\equiv\sin\,\delta$. Here, $R/R_s$ is the ratio of the
NS radius
to Schwarzschild radius, $R_s=2GM/c^2$ (we will assume $M=1.4\,M_\odot$).
The hot spot is bounded by
the conditions:
\beq
\theta\le\beta_h\;\;\;\;\;\;\;\;\;\;\;\; \rmmat{if}\;\;\; \alpha_h=0\;
\label{eq:con1}
\eeq
and
\beq
\left\{
\begin{array}{ll}
\alpha_h-\beta_h\le\theta\le\alpha_h+\beta_h \\
2\pi-\phi_p^h\le\phi\le\phi_p^h \;\; \;\;\;\;\rmmat{if}
\;\;\;\alpha_h\ne 0\;\;\;\rmmat{and} \;\;\;\beta_h\le\alpha_h,\\
\end{array}\right.\;
\label{eq:con2}
\eeq
where
\beq
\phi_p^h=\arccos\left[\frac{\cos\beta_h-\cos\alpha_h\cos\theta}{\sin\alpha_h\sin\theta}\right]\;.
\label{eq:phip}
\eeq
On the other hand, it is identified through the condition
\beq
\theta\le\theta^h_*(\alpha_h,\beta_h,\phi)\;\;\;\;\;\
\rmmat{if}\;\;\; \alpha_h\ne 0\;\;\;\rmmat{and}\;\;\;\beta_h > \alpha_h\;,
\label{eq:con3}
\eeq
where the outer boundary $\theta^h_*(\alpha_h,\beta_h,\phi)$ of the spot is computed by numerical
solution of the equation
\beq
\cos\beta_h = \sin\theta_*^h\sin\alpha_h\cos\phi + \cos\theta_*^h\cos\alpha_h\;.
\label{eq:t*}
\eeq
The antipodal warm spot is described on the surface of the star through
the same conditions, but with the substitutions $\beta_h\rightarrow\beta_w$
and $\alpha_h\rightarrow\alpha_w$.
We assume that the emission from the hot and warm spots is blackbody,
of uniform temperatures $T_h$ and $T_w$, respectively. The spectral
function is then given by $n(E,T)=1/[\exp(E/kT)-1]$, where the
temperature $T(\theta,\phi)$ is equal to $T_h$ or $T_w$ if $\theta$
and $\phi$ are inside either of the spots, respectively, and it is
equal to zero outside. Given that the presence of NS atmospheres and
their elemental composition are yet to be firmly established, here we
first model isotropic radiation. Then, we explore how the obtained
constraints depend on the assumption of forward beaming of the
radiation, as found in magnetized, light element atmosphere models
\citep[e.g.,][]{pav94}, by approximating beaming\footnote{This
approximation is based on the assumption that the magnetic field is
normal to the surface in the heated regions; this is plausible since
heat transport along the B field lines is enhanced in the outer
envelope, at least for $B\ga 10^{10}$~G. See
\S4 for further discussion of this issue.} as intensity
$I(\delta) \propto \cos\,\delta^n$. The observed spectrum as a
function of phase angle $\gamma$ is obtained by the standard method of
integrating the local emission over the observable surface of the
star, accounting for the gravitational redshift of the radiation
following \citet{pag95a}:
\begin{eqnarray}
F(E_\infty,\gamma) =\frac{2 \pi}{c\,h^3}\frac{R_\infty^2}{D^2}\;E_\infty^2
e^{-N_{\rm H}\sigma(E_\infty)} \int_0^1 2xdx\nonumber \\
\times \int_0^{2\pi} \frac{d\phi}{2\pi}\;
I_0(\theta,\phi) \;n[E_\infty e^{-\Lambda_s};T(\theta,\phi)]\;
\label{eq:flux}
\end{eqnarray}
in units of photons cm$^{-2}$ s$^{-1}$ keV$^{-1}$. In equation~(\ref{eq:flux}),
the NS radius and photon energy as observed at infinity are given by
$R_\infty= Re^{-\Lambda_s}$ and $E_{\infty}= E e^{\Lambda_s}$, where
$E$ is the energy emitted at $R$,
and ${\Lambda_s}$ is defined as
\beq
e^{\Lambda_s}\equiv\sqrt{1-{\frac{R_s}{R}}}.
\eeq
The phase-averaged flux is computed as $F_{\rm avg}(E_\infty)=
1/2\pi\int_0^{2\pi}d\gamma F(E_\infty,\gamma)$.
\begin{figure}[t]
\plotone{pupa_model_geometry.ps}
\caption{Emission geometry on the surface of the NS for
the model presented herein: a hot spot of temperature $T_h$ and
angular size $\beta_h$ and an antipodal spot of temperature
$T_w$ and angular size $\beta_w$. As the NS
rotates with angular velocity $\Omega$, the angle $\alpha(t)$ is a
function of the phase angle $\gamma(t)=\Omega t$, the angle
$\xi$ between spin axis and hot spot axis, and the angle $\psi$
between spin axis and line-of-sight.}
\label{fig:NS}
\end{figure}
In addition to the basic two-temperature antipodal spot model
described above, the spectrum of \hbox{PSR~J0821$-$4300}\ requires an additional narrow
line-like component around 0.77 keV, possibly an electron cyclotron
feature in emission (Paper~I). Furthermore, as shown in Paper~I, this
spectral feature is associated exclusively with the larger spot, of
temperature $T_w$. In the current study, we include the best fitted
Gaussian line as an additive component to our basic model. With no
other information about its spatial distribution, this emission is
assumed to be spread uniformly over the surface of the warm spot
only. This line emission is shown to account for a notable increase in
the observed modulation below 1~keV, as described in Section~3.
\section{Modeling the Energy-Dependent Modulation}
The surface emission geometry of \hbox{PSR~J0821$-$4300}\ is highly constrained by its
unique energy dependent pulse profile. As shown in Paper~I, the
quasi-sinusoidal signal has a background subtracted pulsed fraction of
$\approx 11\%$ in the energy band $0.5-4.5$~keV, with an abrupt
$180^{\circ}$ change in phase at $1.2$~keV, around which the
modulation evidently cancels out. This behavior is indicative of a
geometry having the symmetry of Figure~\ref{fig:NS}, namely, a pair of
antipodal spots of different temperatures. Our goal is to match the
observed pulse profile (modulation and phase) in three
interesting energy bands, $0.5-1.0$, $1.0-1.5$, and $1.5-4.5$~keV, using
the antipodal model, by exploring the range of all possible viewing
and hot-spot geometry pairs ($\xi,\psi$; see Figure~\ref{fig:NS}),
and fitting for the correct one.
We summarize our proceedure as follows. We started by fitting the
X-ray spectrum of Paper~I using the antipodal model to compute the
temperatures and spot-sizes corresponding to all spot and viewing
angles $\xi$ and $\psi$. We then used these models to compute
predicted pulse profiles in the three bands as a function of
($\xi$,$\psi$), to compare with the observed profiles. Clearly, only
certain geometries will produce a phase shift; for example, there will
be no shift if the spot axis is nearly coaligned with both the viewing
direction and spin axis, as only one spot remains in view as the star
rotates. Finally, we considered the effect of radiative beaming, and
repeated our analysis for a range of NS radii. In the following we
describe our procedure and results in detail.
\begin{figure*}
\hfill
\includegraphics[angle=270,width=0.47\linewidth,clip=]{pupa_cco_spot_size_only_results_r12_b1.ps}
\hfill
\includegraphics[angle=270,width=0.47\linewidth,clip=]{pupa_cco_spot_size_only_results_r12_b2.ps}
\hfill
\caption{Best fitted values for the sizes of the two emitting spots
on the surface of \hbox{PSR~J0821$-$4300}\ as a function of geometry
parameters ($\xi,\psi$), for a NS radius of $R=12$~km.
Left: Map of warm-spot size parameterized by its angular radius $\beta_w$.
Right: Corresponding map for hot-spot angular radius $\beta_h$.
The spot size is computed at intervals of $10^{\circ}$
in $\psi$ and $\xi$, and interpolated to $1^{\circ}$.
Note the very different size ranges of each spot.
Using these results, the modulation as a
function of viewing geometry is then computed and
compared to the data, constraining the allowed geometry
as shown in Figure~\ref{fig:pf}.
This procedure is repeated for a range of NS radii,
each requiring a new pair of $\beta$ maps, with results
given in Table~\ref{tab:spectable}.}
\vspace{0.1in}
\label{fig:size}
\end{figure*}
\begin{figure*}
\hfill
\includegraphics[angle=270,width=0.47\linewidth,clip=]{pupa_cco_spot_size_only_results_r12_modulation_1_1.ps}
\hfill
\includegraphics[angle=270,width=0.47\linewidth,clip=]{pupa_cco_spot_size_only_results_r12_modulation_1_3.ps}
\hfill
\caption{The model modulation derived in two energy bands
as a function of geometry parameters ($\xi,\psi$), for a NS radius of
$R=12$~km. The pulsed fraction ranges from 0--65\% and is scaled
linearly, with yellow denoting the largest modulation. The solid line
in each panel indicates the contour of the measured pulsed fraction in
the soft energy band (left panel) and hard energy band (right panel),
while the dashed lines give the $1\sigma$ error range. The possible
geometry is then strongly constrained by the intersection of the two
contour regions, as shown in Figure~\ref{fig:contour}. Note that {\it
both} antipodal spots contribution to the modulation shown in each of
these plots. }
\label{fig:pf}
\end{figure*}
\begin{figure}
\centerline{
\includegraphics[angle=270,width=0.90\linewidth,clip=]{pupa_cco_spot_size_only_results_r12_chi2map.ps}
}
\caption{
Contours of $\chi_{\nu}^2$ obtained by comparing the pulse profiles of
the antipodal model with the data in three energy bands, for a range
of angles $\xi$ and $\psi$, as described in the text. The $1\sigma$,
$2\sigma$, and $3\sigma$ confidence levels are shown for the best
match for a NS radius of $R=12$~km. The results are degenerate with
respect to an interchange of $\xi$ and $\psi$. The minimum
$\chi^2_\nu$ for the viewing geometry parameters is obtained at
$\psi=86^{\circ}$ and $\xi=6^{\circ}$, evidently providing a strong
constraint. The geometries that manifest a phase shift (or not) are
indicated.}
\vspace{0.1in}
\label{fig:contour}
\end{figure}
\begin{figure}
\includegraphics[angle=270,width=0.97\linewidth,clip=]{model_data_profiles3.ps}
\caption{Pulse profiles generated using the antipodal model
(solid line) that best matches the observed data (histogram)
across the three selected energy bands for \hbox{PSR~J0821$-$4300}. The parameters
of this model assuming a NS radius $R=12$~km are given in
Table~\ref{tab:spectable}. The measured background in each band
has been added to the model to enable a direct comparison.}
\label{fig:profile}
\end{figure}
For a given set of viewing angles the antipodal flux model can be integrated
over phase to provide a direct comparison with the observed spectra. We have
incorporated equation~(\ref{eq:flux}) into an ``additive model'' for use in
the {\tt XSPEC} spectral fitting software \citep{arn96}.
The coded model comprises 13 parameters: the NS radius and distance
($R,D$), three blackbody temperatures ($T_w,T_h,T_{\rm NS}$), two spot
angular sizes ($\beta_w,\beta_h$), two geometrical angles ($\xi,\psi$),
the rotation phase ($\gamma$), and the Gaussian emission-line
center, width, and flux. The column density is fixed at the best
value determined in Paper~I from a fit to the overall spectrum,
$N_{\rm H} = 4.8 \times 10^{21}$~cm$^{-2}$.
In the spectral fits, the normalization is set to unity
so that the flux is determined by $R$ and $D$, and
implicitly takes into account all relativistic effects
noted in Section~2.
\begin{figure}
\includegraphics[angle=270,width=0.97\linewidth,clip=]{model_comp_counts2.ps}
\includegraphics[angle=270,width=0.97\linewidth,clip=]{phase_vs_energy4.ps}
\end{figure}
\begin{figure}
\caption{
The antipodal explanation for the observed energy-dependent
modulation and phase shift between the soft and hard X-ray bands
seen in \hbox{PSR~J0821$-$4300}. Top: Contribution of each phase-averaged
model spectral component to the two-blackbody model, plus a Gaussian
emission line. The dotted line shows the warm blackbody component
without the emission line contribution. Bottom: The model pulse modulation
as a function of energy. The strong energy dependence
results from the relative contributions of (out of phase)
overlapping (in energy) flux from the two spectral components. The maximum modulation is
19.6\% at the highest energy. However, the modulation in Band~3 is
smaller, as it is weighted by lower-energy photons.
The emission-line contribution is evident as the difference
between the solid and dotted lines.
The phase of the peak of the light curve follows the dominant
spectral component at each energy, and is restricted by
the symmetry of the model
to either $0$ or $0.5$ cycles. Where the spectral components cross,
the phase must shift by $180^{\circ}$ as observed (see Figure~\ref{fig:profile} and Paper~I).}
\label{fig:explain}
\end{figure}
The model allows for a uniform temperature $T_{\rm NS}$ in the
inter-spot area, but since none is necessary for an acceptable fit to
the spectrum or pulse profiles, we set $T_{\rm NS}=0$. Nevertheless,
to place a model-independent upper limit on $T_{\rm NS}$, we have
simulated blackbody spectra in {\tt XSPEC} for $R=12$~km and
$D=2.2$~kpc (Reynoso et al.\ 1995), increasing the temperature until
the model exceeds the spectrum of \hbox{PSR~J0821$-$4300}\ at low energies. Since the
model counts depend on the interstellar column density, we assumed
here the largest value measured by \citet{kas10} for filaments in the
\hbox{Puppis~A}\ remnant in {\em Chandra} data, $N_{\rm H}=5.5 \times
10^{21}\;{\rm cm}^{-2}$. This yields a conservative $3\sigma$ upper
limit of $T^{\infty}_{\rm NS}< 0.15$~keV;
\citet{hwa08} report a significantly lower value of
$N_{\rm H}=3 \times 10^{21}\;{\rm cm}^{-2}$ from \suzaku\
measurements, which would allow a smaller limit on $T^{\infty}_{\rm
NS}$, if applicable.
To map the antipodal model for \hbox{PSR~J0821$-$4300}\ as a function of ($\xi,\psi$) we
generated best fit model parameters over the grid spanning
($0^{\circ}<\xi <90^{\circ};\; 0^{\circ}<\psi<90^{\circ}$) in steps of
$10^{\circ}$ by systematically fitting for $T_w$, $T_h$, $\beta_w$,
and $\beta_h$. Since we fitted the phase-averaged spectrum (presented
in Paper~I and described therein), a reasonable approximation for the
equivalent phase-averaged model flux is $F_{\rm avg} =
[F(\gamma=0^{\circ})+ F(\gamma=180^{\circ}) ] /2$. The flux of each
blackbody component depends on the size of its respective emission
spot as parameterized by its $\beta$, which sets the normalization for
that component. As we consider the range of viewing orientations, the
size of the spots must be adjusted to keep the model flux fixed to
that of the observed value. This is because the projected flux from
the phase integrated emission strongly depends on the viewing geometry
($\xi,\psi$), which is not known {\it a priori}. In comparing with
data, for each trial ($\xi,\psi$) pair, we fit for the spot sizes
($\beta_w,\beta_h$) that correctly normalize the blackbody components
to match the observed spectrum.
The grid of best fitted models with values of
($\beta_w,\beta_h,T_w,T_h$) describes the flux from \hbox{PSR~J0821$-$4300}\ as a function
of energy and phase ($F[E_{\infty},\gamma]$)
according to equation~(\ref{eq:flux}) for any realizable
geometry\footnote{Not all values of ($\xi,\psi$)
can fit the spectrum, as the available flux in some cases,
e.g., $\xi,\psi$ both near zero, are
insufficient to match the data even with $\beta_w$ at its maximum
value of $90^{\circ}$. Such cases produce unacceptable $\chi^2$
statistics.},
the latter parameterized by ($\xi,\psi$) and a set of fixed $R,D$,
and emission-line parameters. We find that the spot temperatures
remain almost constant in ($\xi,\psi$) and use the average value given in
Table~1 in our final fits. The resulting fitted values of
$\beta_w$ and $\beta_h$ for the case $R=12$~km
are interpolated to $1^{\circ} \times 1^{\circ}$ pixels on a
$90^{\circ}\times90^{\circ}$ grid and displayed in Figure~\ref{fig:size}.
We are now prepared to compute the set of phase-resolved modulations
in the three interesting energy bands, which manifest the phase shift.
Specifically, the pulse in the soft $0.5-1.0$~keV band peaks at
rotation phase $\gamma=180^{\circ}$, the hard $1.5-4.5$~keV band peaks
at $\gamma=0^{\circ}$, while the modulation in the intermediate
$1.0-1.5$~keV band cancels out to $< 1\%$.
The comparison is best done in counts space rather than flux,
to make use of the counting statistics for errors. The background
counts are measured for each band and added to the model counts.
For each ($\xi,\psi$) pair, the antipodal flux model, including
interstellar absorption, is folded through the \xmm\ EPIC response
function used for the spectral fits (see Paper~I).
For each energy band, we record the magnitude of the model
modulation $f_p$ defined as
\begin{equation}
f_p =\frac{F_{\rm tot}-N \ {F_{\rm min}}}{F_{\rm tot}}\;,
\label{eq:pf}
\end{equation}
where the flux $F_{\rm tot}$ is the total flux in the band,
$N$ is the number of bins in the light curve (ten in this case),
and $N \ {F_{\rm min}}$ is the ``unpulsed'' flux in the band,
determined from the bin of minimum flux. The results are
shown in Figure~\ref{fig:pf} for the soft and hard energy bands.
For a given ($\xi,\psi$) pair, the model profile in ten
phase bins is compared to the observed counts in the three bands,
for a total of 30 bins, using the $\chi^2$ statistic.
Since the total count normalizations in each band were fixed earlier
by the spectral fits via the $\beta$ parameters, the mean counts of
the models match the means of the three observed light curves. This
reduces the number of degrees of freedom by three, from 30 to 27.
The ($\xi,\psi$) region that best matches the measured pulsed fraction
in the soft and hard bands, $f_p=11\% \pm 2\%$ corrected for
background, is indicated by the contours in Figure~\ref{fig:pf}.
While there is a range of allowed angles in each band separately,
they overlap in only two small regions corresponding to the minimum
$\chi^2$. This uniquely identifies the geometry of the system, within
statistical uncertainty.
More specifically, we find that, for
$R=12$~km, the most likely geometry is specified by the combinations
$(\psi,\xi)=(86^{\circ},6^{\circ})$ or $(6^{\circ},86^{\circ})$. Since
the flux depends on the angles $\xi$ and $\psi$ only through the
parameter $\alpha_h$ in equation~(\ref{eq:alpha}), it is
symmetric with respect to an exchange of $\xi$ and $\psi$, yielding
the two possible solutions. This is evident in the $\chi^2$ map that
compares the model and observed profiles in the three bands
(Figure~\ref{fig:contour}). In one solution the spin axis is nearly
perpendicular to the line of sight, while in the other solution the
two lines are nearly parallel. Figure~\ref{fig:profile} shows the fit
to the pulse profiles for these equivalent best solutions.
Finally, the
parameters for the model that resulted in the smallest $\chi^2$ for
each test radius $R$ are given in Table~\ref{tab:spectable}. In all
cases, the reduced $\chi^2_{\nu}$ is near unity, indicating an
excellent match to the observed pulse profiles.
Figure~\ref{fig:explain} provides a graphic explanation of the origin
of the observed energy-dependent modulation and phase-shift seen from
\hbox{PSR~J0821$-$4300}. For the set of model parameters that best fit the data,
we graph separately the phase-averaged fluxes for the two spots to
gauge their contribution to the light curve. At the lower energies,
the large warm spot dominates the spectrum and the light curve peaks
when this component is in view, while at the higher energies, the
small hot spot dominates and it peaks in view 0.5 cycles later. Thus,
the energy at which the dominant spectral component switches is around
1.3~keV, and the peak phase necessarily shifts by $180^{\circ}$ at
this energy due to the antipodal symmetry in longitude. This phase
reversal, at this energy, is thus a direct consequence of the crossing
of the spectral components of different temperatures. The agreement
between the energy of phase reversal and the energy of the spectral
cross-over point provides direct evidence of the correctness of the
model.
Figure~\ref{fig:explain} also shows that the predicted modulation is
19.6\% at the highest energies, where 100\% of the flux comes from the
$T_h$ blackbody component. At lower energies, the modulation is
reduced exactly in proportion to the increased contribution of the
$T_w$ or $T_h$ spot, depending on which one dominates the flux at a
given energy, $f_p(E) = 19.6 \% \times |F(E;T_w) - F(E;T_h)| /
[F(E;T_w) + F(E;T_h)]$. For our broad energy bands, used to compare
the model with data, the observed modulation is evidently weighted by
the total flux over the band. This is especially clear in the highest
energy band (1.5--4.5 keV), in which most of the photons are from the
lower end of the band where the modulation is significantly less than
19.6\%. The modulation in the middle band, which spans the phase shift
with equal counts, is mostly canceled out. That a similar pulsed
fraction is measured in the lower and higher bands is largely
coincidence in this case. Figure~\ref{fig:explain} also shows the
contribution of the spectral line at $0.77$~keV; a small but
significant increase in the modulation results. The photon statistics
of the current data do not allow a more detailed comparison of the
modulation as a function of energy.
Next we explored the dependence of the fitted parameters on the radius
$R$ of the NS. As discussed above, due to general-relativistic effects
$R$ is not simply a normalization of the flux; this requires us to
compute $\beta$ maps for each test value of $R$. Therefore, we
repeated our full analysis for the range of values $9\le R \le 14$~km,
in 1~km increments. These results are presented in
Table~\ref{tab:spectable}. In principle, the relativistic effects can
lead to a preferred $R$, but statistically, no unique radius is
suggested here. The general trend is that the angles $\beta_w,
\beta_h$, and $\xi$ increase with decreasing radius. Two counteracting
effects, both due to flux conservation, influence the spot sizes.
Gravitational redshift decreases the inferred emission area on a more
relativistic (smaller) star to compensate for the larger needed
surface temperature. On the other hand, for a fixed distance $D$, the
spot angular size increases on the smaller star. For the values of the
fitted parameters here, the latter effect tends to dominate over the
former, reducing the modulation from a smaller star for the same
viewing geometry. However, the most important effect influencing the
amplitude of the modulation is the gravitational deflection of light
rays, which acts to suppress the pulsed flux for the smaller star.
Therefore, in order to reproduce the same observed level of
modulation, a smaller $R$ requires a larger modulation of the viewing
angle $\alpha(t)$, which, according to equation~(\ref{eq:alpha}), is
obtained by increasing either $\xi$ or $\psi$. This explains the
trends in Table~\ref{tab:spectable}\footnote{The larger of the two
angles ($\xi,\psi$) is unchanged as a function of radius relative to
its $2^{\circ}$ error, while the smaller one is
clearly decreasing with radius as compared to its $1^{\circ}$ error.}
We then repeated our analysis for a locally anisotropic intensity
pattern, $I(\delta) \propto \cos\,\delta$. For the nominal radius
$R=12$~km, the combined constraints from the pulsed fraction and the
phase shifts moves the best fitted viewing angles to
$(\xi,\psi)=(84^\circ,3^\circ)$, with a larger uncertainty than found
for the unbeamed case. This behavior
is readily accounted for. Forward beaming enhances the emission
in the direction of the spot axis, hence increasing the differences in
observed flux as the axis of the spot moves toward and away from the
observer. As a result, a fit to the observed level of modulation
requires smaller values of the viewing angles. The pulse profile is
found to remain sinusoidal and is statistically indistinguishable from
the isotropic intensity case. For the specific geometry of \hbox{PSR~J0821$-$4300}, such
beaming is a weak effect in narrowing the pulse profile
because of our relatively unmodulated views of the two antipodal spots
at glancing angles.
In the antipodal model, the energy dependent phase reversal is a
direct consequence of switch in dominance between the two blackbody
spectral components of different temperatures. As mentioned previously,
this agreement between the cross-over energy in phase and spectra is
distinct feature of this symmetric model, where the emission spots are
exactly antipodal. However, similar light curves can be obtained if
the spots remain opposite in longitude, but are allowed to move closer
in latitude. Such geometry can still produces a $180^{\circ}$ phase
reversal. Allowing this additional degree of freedom, the angles
$(\xi,\psi$) would not be so strongly constrained as in the antipodal
model. More generally, if the spot locations differ in longitude by
$\Delta\gamma<180^{\circ}$, such an ``offset'' model allows the
possibility of a continuous phase shift as a function of energy. The
effect of asymmetric spot locations is clear in the energy-dependent
model profiles, but any such effect in \hbox{PSR~J0821$-$4300}\ is not apparent, and
would require higher quality data to discern. Description of the
geometry of offset models can be found in Bogdanov et al.\ (2007, 2008),
for example.
\begin{deluxetable}{lccccccc}
\tablecolumns{8}
\tighten
\tablewidth{0.0pt}
\tablecaption{Model Results as a Function of NS Radius}
\tablehead{
\colhead{Parameter} & \colhead{Unc.$^a$} & \colhead{} & \colhead{} & \colhead{$R$ (km)} & \colhead{} & \colhead{} & \colhead{} \\
\colhead{} & \colhead{} & \colhead{9} & \colhead{10} & \colhead{11} & \colhead{12} & \colhead{13} & \colhead{14}
}
\startdata
$kT_w$ (keV) & 3\% & 0.29 & 0.28 & 0.27 & 0.26 & 0.26 & 0.25 \\
$kT_h$ (keV) & 3\% & 0.57 & 0.54 & 0.53 & 0.51 & 0.50 & 0.50 \\
$\beta_w$ & 6\% & $39\!^{\circ}$ & $37\!^{\circ}$ & $35\!^{\circ}$ & $34\!^{\circ}$ & $32\!^{\circ}$ & $31\!^{\circ}$ \\
$\beta_h$ & 11\% & $ 8\!^{\circ}$ & $ 7\!^{\circ}$ & $ 7\!^{\circ}$ & $ 7\!^{\circ}$ & $ 6\!^{\circ}$ & $ 6\!^{\circ}$ \\
$\xi$ or $\psi$ & $2^{\circ}$ & $86^{\circ}$ & $87^{\circ}$ & $87^{\circ}$ & $86^{\circ}$ & $87^{\circ}$ & $87^{\circ}$ \\
$\psi$ or $\xi$ & $1^{\circ}$ & $ 9^{\circ}$ & $ 7^{\circ}$ & $ 7^{\circ}$ & $ 6^{\circ}$ & $ 5^{\circ}$ & $ 5^{\circ}$ \\
$\chi^2_{\nu}$~(27 DoF) &\dots & 0.94 & 1.00 & 0.99 & 1.00 & 0.95 & 0.97
\enddata
\tablecomments{Spectral fits with fixed parameters
$N_{\rm{H}}=4.8\times 10^{21}$~cm$^{-2}$ and $D=2.2$~kpc. Gaussian
line model parameters are fixed at energy $E^{\infty} = 0.77$~keV,
width $\sigma=0.05$~keV, and flux normalization
$1.8\times10^{-4}$~ph~cm$^{-2}$~s$^{-1}$.
\\$^a$ The $1\sigma$ uncertainties in the spectral parameters are estimated by running the
XSPEC {\tt error} command; the uncertainties in $\xi$ and $\psi$ are
determined from the $\chi^2$ map of Figure~\ref{fig:contour}.}
\label{tab:spectable}
\end{deluxetable}
\section{Discussion of Model Results}
Using an antipodal spot model, we have accounted for all of the
details of the \xmm\ observations of \hbox{PSR~J0821$-$4300}\ described in
Paper~I. In particular, we can reproduce the overall spectral shape,
energy-dependent pulsed modulation, and abrupt $180^{\circ}$ phase
reversal at the cross-over energy of the fitted blackbody components.
In so far as no observed phenomena remains unmodeled, and
no unobserved features are predicted, the antipodal model provides a
credible description of the geometry of emission from
the CCO in \hbox{Puppis~A}.
The full data set can be reproduced, with slight differences
in the best fitted parameters, assuming either isotropic or
forward-beamed emission. Differentiating between these assumptions
will require observations with higher statistics.
By matching the observed modulation in three broad energy bands, we
are able to restrict the angles that the hot-spot axis and the line of
sight make with respect to the spin-axis to within $<2^{\circ}$, up to
the degeneracy between these two angles. Either the spin axis lies
nearly parallel ($6^{\circ}$) to the line-of-sight, with the hot-spot
axis at $86^{\circ}$, or the hot-spot axis is nearly co-aligned with the
spin-axis, but perpendicular to the line-of-sight. In the absence of
a strong physical motivation to prefer one of these configurations
over the other, we note that the a priori probability of the spin axis
lying $6^{\circ}\pm 1^{\circ}$ from the line of sight is only $3.6
\times 10^{-3}$, while the probability that it is at $86^{\circ}\pm
2^{\circ}$ is $7.0 \times 10^{-2}$, a factor of 20 larger, although
still small. We note that the specific orientations that fit the
observations of \hbox{PSR~J0821$-$4300}\ are not the only ones that allow phase reversals
in the two-temperature model. Rather, phase reversals are found in
the majority of configurations of Figure~\ref{fig:contour}.
Comparing with the other CCO pulsars, we see that PSR J1852+0040
in the SNR Kesteven 79 \citep{hal10} also has a two-temperature
X-ray spectrum, but its highly modulated pulse ($f_p = 64\%$)
is single-peaked and virtually invariant with energy.
Because of this, its emitting regions are likely
to be concentric, or nearly so. In the case of 1E~1207.4$-$5209
in {\rm PKS~1209$-$51/52}, there are large variations in pulse phase
and amplitude as a function of energy \citep{pav02b,del04},
with the largest pulsed fraction coinciding with the strong
absorption lines in the unique spectrum of this pulsar.
This effect may be a manifestation of
angle-dependent scattering in cyclotron lines, which is
the favored identification of the spectral features considering
the upper limit of $B_s < 3.3 \times 10^{11}$~G on the surface
dipole field from the absence of spin-down \citep{got07}.
The data on 1E~1207.4$-$5209 should be fitted
with detailed atmosphere models that include quantum treatment
of the cyclotron harmonics \citep{sul10}.
Applying this model to surface thermal emission from CCOs with
weak magnetic fields (anti-magnetars) is an especially apt use,
in that additional complicating emission mechanisms that are evident
in other classes of pulsars (see Section~5)
are absent in CCOs. Such contributions
include nonthermal magnetospheric emission in spin-powered pulsars,
polar-cap heating from backflowing particles, and transient and
variable heating from magnetic field decay in magnetars.
The first two extra contributions can be
significant even for recycled millisecond pulsars,
which are now known to be efficient $\gamma$-ray emitters
\citep{abd09a,abd09b}.
Observations and upper limits on spin-down of CCOs indicate
spin-down luminosities that are smaller than their thermal
X-ray luminosities, and dipole magnetic fields of order $10^{10-11}$~G,
remarkably small for young pulsars. These properties imply that
none of the above-mentioned emission and surface heating mechanisms
can be significant, and
constrain the effects that may be responsible for the
multiple temperatures that are a ubiquitous feature of CCO spectra,
even those that have not yet been observed to pulse.
For the assumed distance to \hbox{PSR~J0821$-$4300}\ of 2.2~kpc and a radius of 12~km,
the best match for the modulation fixes the extent of the hot and warm
regions to angles $\beta_h = 6.\!^{\circ}6\pm0.\!^{\circ}5$ and
$\beta_w = 34.\!^{\circ}0\pm2\!^{\circ}$, representing $0.33\%$ and
$8.5\%$ of the surface area, respectively. The existence of a hot
spot is difficult to understand in the context of a weakly magnetized
NS, as it requires a mechanism to confine the heat to such a small
area. Using \xmm, we obtained a new period measurement of \hbox{PSR~J0821$-$4300}\ on
2010 May~2 using the identical observational setup and analysis as
described in Paper~I. We obtained another measurement of the
pulsations on 2010 Aug~16, from a \chandra\ CC-mode observation.
These results will be presented in a future publication. The period
is found to be unchanged from the values observed in 2001. In
combination with the previous measurements listed in Paper~I, this
places a $2\sigma$ limit on $\dot P$ of $<3.5 \times 10^{-16}$
and, under the assumption of dipole spin-down,
$B_s < 2.0 \times 10^{11}$~G, confirming \hbox{PSR~J0821$-$4300}\ as an
anti-magnetar. Given the corresponding upper limit on spin-down
luminosity of $<1 \times 10^{34}$ erg~s$^{-1}$, the hot-spot
luminosity of $\approx 2 \times 10^{33}$ erg~s$^{-1}$ can hardly be
attributed to external heating by backflowing particles. The same
problem was discussed in the context of the highly pulsed emission
from PSR J1852+0040 in Kes~79 \citep{hal10}.
Possible explanations for the properties of CCOs are largely
focused on magnetic field induced anisotropies
in the surface temperature of a NS, as proposed by \citet{gre83},
in which strongly enhanced conductivity in the direction parallel
to the magnetic field is matched by a corresponding reduction in the
perpendicular direction.
The effect of the magnetic field on the heat transport of the
crust and envelope of neutron stars has been investigated by a number
of authors \citep[e.g.,][]{hey98,hey01,pot01,lai01,gep04,gep06,per06a,pon09}.
While heat transport in the core ($\rho\ga
1.6\times 10^{14}$ g~cm$^{-3}$) is expected to be roughly isotropic
due to proton superconductivity, anisotropy of heat
transport becomes pronounced in the outer envelope
($\rho\la 10^{10}$ g~cm$^{-3}$) for field strengths $B\ga 10^{10}$~G,
and it extends deeper into the whole crust for higher fields,
$B\ga 10^{12}-10^{13}$~G. The main question is whether subsurface
fields in CCOs can be strong enough to affect heat transport to
the extent required, while not exceeding the weak external dipole
field as constrained by their spin-down properties.
\citet{gep04} discussed the differing effects of a poloidal magnetic
field in the core of the NS, versus one confined to the crust,
the true configuration being a matter of uncertainty. From a core field,
any surface temperature anisotropy is expected to be small, while
a tangential crustal field insulates the magnetic equator and conducts
heat to the magnetic poles. A tangential crustal field may be indicated
for CCOs, because it can lead to small hot regions where
the field emerges normal to the surface, while contributing
very little to the external dipole field. Of particular
interest here, \citet{gep06} found that, if the crustal field
consists of both a dipolar poloidal and a toroidal component,
then configurations can be realized in which two warm regions of
{\em different sizes} are separated by a cold equatorial
belt. However, their case study included large poloidal
magnetic fields, $B \ga 10^{12}$~G in both core and crust components,
which would tend to violate the observed
spin-down limit of $B_s < 2.0 \times 10^{11}$~G for \hbox{PSR~J0821$-$4300}.
\section{Comparison with Other Pulsars}
The ultimate goal of this field is to infer the equation of state and
measure the radius of the NS.
Some progress on these fronts has been made with high-quality data
from millisecond pulsars (MSPs). Using an unmagnetized hydrogen
atmosphere model fitted to the spectra and pulse profile of the
nearest known MSP J0437$-$4715, \citet{bog07} derived $6.8 < R <
13.8$~km (for $M = 1.4\;M_{\odot}$). \citet{bog08,bog09} also obtained
lower limits on $R$ modeling X-ray observations of MSPs J2124$-$3358
and J0030+0451. Blackbody emission was not able to fit the pulse
profiles, thus requiring a NS atmosphere.
\citet{bog07} assumed an identical pair of
polar caps, but fitted two temperatures to each,
as required by the data, which can be understood as non-uniform
heating by backflowing particles from the magnetosphere
giving a concentric temperature gradient, as originally
modeled by \citet{zav98}. \citet{bog07}
concluded that the magnetic dipole is not centered on the star,
but must be offset by $\sim 1$~km to account for
an asymmetry in the observed pulse profile. The data on
\hbox{PSR~J0821$-$4300}\ in \hbox{Puppis~A}\ are not yet of a quality to search for such effects.
On the other hand, the geometrical angles ($\xi,\psi$)
are not nearly as well constrained in the MSPs.
(In the case of the binary MSP J0437$-$4715.
it could be assumed that $\psi=42^{\circ}$ because that
is the inclination angle of its binary orbit.)
Pulsed light curves of the middle-aged pulsars PSR B0656+14, B1055$-$52, and
Geminga, whose X-ray spectra are dominated by surface thermal emission,
have been modeled by \citet{pag95a}, \citet{pag95b}, \citet{pag96},
and \citet{per01}. Beginning with {\it ROSAT}\ data,
it appeared that PSR B0656+14 \citep{pos96,gre96}, PSR B1055$-$52 \citep{ogl93},
and Geminga \citep{hal93} had two thermal components with pulse-phase shifts
of between 0.1 and 0.3 cycles, the hotter component being attributed to a
heated polar cap. Follow-up observations at higher energy with {\it ASCA\/}
found that the harder components from Geminga \citep{hal97} and
PSR B1055$-$52 \citep{wan98}
are better fitted by non-thermal power laws. As beamed emission
from the magnetosphere, their hard X-ray pulses need not bear a simple phase
relationship to the soft thermal components. Only PSR B0656+14 continued
to have two clear temperatures when observed at higher energy
\citep{pav02a}, with only a weak nonthermal tail.
Detailed study of the energy-dependent pulse profiles of these
primarily thermal pulsars with \xmm\ \citep{del05} confirm that
PSR B0656+14 has two thermal components, with the hotter one
interpreted as a small polar cap, shifted in phase by $\sim 0.2-0.3$
cycles from the softer emission. The spectrum of PSR B1055$-$52
was fitted with two temperatures and a non-thermal power law,
although it is difficult to explain why the hotter blackbody component
has a pulsed amplitude of $\sim 100\%$.
A case for a hot polar cap on Geminga was made by \citet{car04}
and \citet{del05}, but \citet{jac05}, analyzing the same data,
did not find it to be necessary.
Such a component does not make a significant contribution to the spectrum
of Geminga at any energy, and its fitted pulse profile appears to have
the same phase and similar shape as the power-law component,
suggesting that it is a distinction without a difference.
Despite these difficulties, the pulsed amplitudes and fitted areas
of thermal X-ray emission from cooling neutron stars indicate that
most have highly nonuniform surface temperatures that may be
regulated by their crustal magnetic field geometry. For example,
the \xmm\ observation of the middle-aged pulsar B1706$-$44
\citep{mcg03} shows an asymmetric, double-peaked pulse profile whose
$T^{\infty} = 8 \times 10^5$~K spectrum is compatible with the full NS area,
while having a 22\% pulsed fraction.
In contrast, PSR J0538+2817 appears to have only a hot polar cap of
$T^{\infty} = 2.2 \times 10^6$~K \citep{mcg04}. One of the most
unusual results is the apparently thermal ($T^{\infty} = 2.4 \times 10^6$~K)
spectrum of the high $B$-field ($4.1 \times 10^{13}$~G) PSR~J1119$-$6127
\citep{gon05}, which has pulsations of amplitude $74\% \pm 14\%$ that
are only detected below 2~keV.
Another family of thermally emitting pulsars are the nearby, isolated
neutron stars (INSs) \citep{hab07,kap09} with periods of $3.4-11.4$~s,
and pulsed fractions that range from 1.2\% for RX J1856.6$-$3754
\citep{tie07} to 52\% for RX~J1308.6+2127 \citep{sch05}. The latter
authors fitted the double-peaked pulse profile of RX~J1308.6+2127 to a
model of two small spots with temperatures of $kT_1^{\infty} = 92$~eV
and $kT_2^{\infty} = 84$~eV separated by $~\sim 160^{\circ}$ in phase.
Timing measurements have revealed that these INSs have somewhat larger
dipole magnetic fields than most young pulsars, with $B_s \ga
10^{13}$~G \citep{kap09}, and may be significantly heated by
continuing magnetic field decay \citep{pon09}. In this sense, they
may be the $\sim 10^6$~year old descendants of magnetars. Several of
their spectra have very broad absorption features that have been
interpreted as ion cyclotron lines or, in the case of multiple
features, possibly atomic lines (Haberl 2007; Schwope et al.\ 2007).
One of the best studied objects of this class, RX J0720.4$-$3125,
shows a pulse phase shift of $\sim 0.1$ between soft and hard X-rays
\citep{cro01}, which suggests that there could be two spots of
different temperatures. However, this interpretation is complicated
by long-term (years) changes in the shape of the spectrum and pulse
profile \citep{dev04,hoh09}, which lends support to the idea that,
similar to the case of magnetars, localized and variable heating by
magnetic field decay is responsible for relatively short-lived surface
thermal structure. This is evidently the case for the transient
magnetar XTE J1810$-$197, whose declining hot spot temperatures and
areas were modeled by \citet{got05,got07}, \citet{ber09},
\citet{alb10}, and by \citet{per08}, who used a similar treatment as
that presented herein.
\citet{per06b}, \citet{zan06} and \citet{zan07} investigated models for INSs
involving a combination of star-centered dipole and quadrupole magnetic field
components to explain their asymmetric pulse profiles.
The properties of \hbox{PSR~J0821$-$4300}\ in \hbox{Puppis~A}\ may ultimately be ascribed to these same
effects, but in a simpler system that is not variable in time.
\section{Conclusions}
We modeled the \xmm\ spectra and pulsed light curve of
\hbox{PSR~J0821$-$4300}\ in \hbox{Puppis~A}, one of three CCO pulsars whose dipole magnetic
field strengths are measured to be less than those of all spin-powered
pulsars of similar age ($B_s < 2.0 \times 10^{11}$~G in the case of
\hbox{PSR~J0821$-$4300}). The sizes and configurations of the surface hot and warm spots
on \hbox{PSR~J0821$-$4300}\ are particularly well constrained. The two emitting areas
differ by a factor of 2 in temperature and 20 in area, which
conveniently endows them with similar luminosities that fall in the
\xmm\ bandpass. The $180^{\circ}$ phase reversal between the soft and
hard X-ray pulse profiles reveals the antipodal geometry. It is
especially significant that the X-ray spectra and pulse profiles of
CCOs indicate considerably nonuniform surface temperatures. Many of
the mechanisms that are held responsible for such effects in other
classes of NSs are not expected to be operating in these
anti-magnetars, which appear to be simple cooling neutron stars whose
conduction of heat from the interior is highly anisotropic. The
essential problem in understanding CCOs is to explain how this is
accomplished without creating a strong external dipole magnetic
field. Our tentative hypothesis is that even CCOs have strong
tangential fields buried in the crust that channel heat toward the
magnetic poles, or external quadupole fields. But what is the geometry of that
magnetic field? Although the orientation of the hot spots in \hbox{PSR~J0821$-$4300}\ is
determined to within $2^{\circ}$, the degeneracy of the model does not
allow us to decide if the axis of the hot spots is nearly aligned with
the NS spin axis or nearly perpendicular to it. Geometrical
probability, as well as the observation of larger pulsed fractions in
other NSs, would suggest the latter. The actual geometry is probably
fixed during the genesis of magnetic fields in these young NSs, which,
in the case of CCOs, have not spun down since their birth and are
likely to have preserved the natal $B$-field configuration.
\acknowledgements
We thank the referee, Silvia Zane, for helpful comments on the paper.
This work is based on observations obtained with \xmm, an ESA science
mission with instruments and contributions directly funded by ESA
Member States and NASA. This work was supported by NASA XMM grant
NN08AX71G.
|
{
"timestamp": "2010-09-24T02:00:22",
"yymm": "1009",
"arxiv_id": "1009.4473",
"language": "en",
"url": "https://arxiv.org/abs/1009.4473"
}
|
\section{Introduction}
In this article, we study the range, local times,
and periodicity or ``parity'' statistics of nearest-neighbor
symmetric, weakly asymmetric, and asymmetric random walks up to the
time of exit from an interval of $N$ sites.
We derive several associated scaling limits which appear curious, some which
connect with the entropy of an exit distribution, generalized
Ray-Knight constructions, and Bessel and Ornstein-Uhlenbeck square processes, among other
objects.
The study of the range of random walk is of course an old subject.
However, examining the range and related statistics at the time the random walk leaves an interval, although a simple, natural concern, seems unexplored. We refer to Bass-Chen-Rosen \cite{BCR}[Ch. 2] and Den Hollander-Weiss \cite{DenHollander_Weiss} for exhaustive references on the range and related statistics of random walk in various settings.
From another view, indeed our initial motivation for this problem, the study of the range and other structures of random walk when it exits an interval can be thought of as a stochastic version of the ``locker'' problem, popular in university curriculum:
Suppose there is a hallway of lockers labeled from $1$ to $N$, for $N\geq
1$, which are initially closed. Let persons $L$, for $L\geq 1$, walk through the hallway, toggling every $L$th locker, that is opening it if closed and closing it if open.
The question is then to find out those lockers which will be open after the first $N$ people walk through. The lockers whose labels are the squares, $1$, $4$, $9$, etc., are exactly those with an odd number of factors. Consequently, these lockers are
the open ones. Other variations of this problem can be found in
Tanton \cite{tanton} and references therein.
In our random walk setting, we can imagine each site in the interval to be either open or closed, and the random walker toggling a site on each visit (from open to closed and vice versa) before it exits the interval. In comparison to the ``locker'' problem, we address the following questions:-
\begin{itemize}
\item[(1)] What fraction of sites will be visited when the walker exits, e.g. the range?
\item[(2)] How many times will each site be visited before exit, e.g local times across sites?
\item[(3)] And, given a set of sites
that have been visited, what is the joint distribution of their open status at the time of exit, e.g parity of the visits to points in the interval?
\end{itemize}
The specific answers naturally depend on the type of random walk
considered. A goal of the paper is to see how the behaviors under
symmetric and asymmetric walks are interpolated in terms of weakly
asymmetric walks.
For the first question, we derive
the limiting distribution for the range
(Proposition \ref{rangeprop}), and observe as a consequence,
which seems surprising, that the scaled range, when starting at
random, is uniformly distributed on $[0,1]$ no matter the dynamics
(Proposition \ref{uniformprop}). Also, curious values for the expected scaled range
under symmetric walks, and the chance a given point is in the range,
when starting at random are found (Remarks
\ref{entropy} and \ref{point_visitedrmk}).
For the second question, we find the scaling limit of the local times
through a ``Ray-Knight'' construction involving Bessel and
Orstein-Uhlenbeck squared processes (Propositions \ref{Besqprop},
\ref{OUprop} and \ref{asym_limprop}).
For the third question, we show that
the parities of well-separated points, given that they are visited, are independent and
identically distributed Bernoulli variables, and fair in the
symmetric/weakly asymmetric case, and biased in the asymmetric
situation (Proposition \ref{thm1_question3} and \ref{thm2_question3}).
\vskip .1cm
{\bf Set up:-}
Let $\mathcal T_{N} = \{0,1,2,\ldots, N\}.$ Let $X_n$ be the position
of a random walk on $\mathcal T_N$ at times $n\geq 1$. At each
time step, the walk moves to the nearest point to its left
(right) with probability $q_N$ ($p_N$) where $p_N+q_N=1$. The walk stops the moment it
is at either $0$ or $N$. When $p_N=q_N=1/2$, the walk is of course referred to as the symmetric random
walk. When $q_N = 1/2 -c/N$ (and so $p_N = 1/2+c/N$) for some
constant $c>0$ and $N$ large enough so that $0< p_N,q_N<1$, we say the
walk is weakly asymmetric. When $q_{N}= q < p = p_{N} $, the walk
is asymmetric.
Define $T_a = \inf \{n \geq 1: X_{n} = a\}$ as the hitting time of $a\in \mathcal T_N$. Then, $\tau_N = T_0\wedge T_N$ is the ``exit'' time from the strip.
Clearly, starting from $1\leq x\leq N-1$, $\tau_N$ is
finite: $P_x(\tau_N<\infty)=1$ where we denote
$P_x(A)=P(A|X_0=x)$ as the conditional probability of the event $A$
with respect to the walk starting from $X_0=x$.
Then, the number of visits
to $y \in \mathcal T_{N}$ before exiting is
$G(y) = \sum_{k=0}^{\tau_N} 1_{y}(X_k)$.
Hence, the event $y$ is visited at all corresponds to $G(y)\geq 1$.
In this case, we say the parity of $y$ is ``even'' (locker $y$ is closed) if $G(y)\geq 1$ and $G(y)=0 \ {\rm mod}_2$. Correspondingly, the parity of $y$ is ``odd'' (locker $y$ is open) when $G(y)\geq 1$ and
$G(y)=1\ {\rm mod}_2$.\vskip .1cm
The plan of the article is to address questions (1),(2) and (3) in
sections \ref{distributional}, \ref{ray-knight}, and \ref{independent}
respectively.
\section{Question 1: Range of random walk in $\mathcal T_N$}
\label{distributional}
In this section, we obtain
distributional limits of the range up to the exit time when starting
from a point, and at random in
subsections \ref{distributional1}, \ref{random}.
\subsection{The range starting from a point}
\label{distributional1}
Denote $R_N$ as the number of locations visited before exit, the range of the walk on $\mathcal T_N$, that is
$$R_N \ = \ \#\{y\in \mathcal T_N: G(y)\geq 1\}.$$
Observe, when starting from $[ \alpha N]$, necessarily $[ \alpha N\wedge (1-\alpha) N] \leq R_N\leq N$.
\begin{proposition}
\label{rangeprop} Let $X_{0}= [ \alpha N]$ for $0<\alpha<1$. For symmetric and weakly asymmetric walks, $R_N/N$ converges in distribution to absolutely continuous measures on $[0,1]$, respectively $G_{0,\alpha}$ and $G_{c,\alpha}$ defined in (\ref{G_symmetric}) and (\ref{G_weakly}). For asymmetric walks, $R_N - [(1-\alpha)N] \Rightarrow Z$ where $Z$ is Geometric$(q/p)$.
\end{proposition}
\proof First, we observe, for $0<\beta<1$, starting from location $x=[\alpha N]$, since the motion is nearest-neighbor,
\begin{eqnarray*}
\{R_N \geq \beta N\} & = & \{R_N \geq \beta N, \tau_N = T_0\} \cup \{R_N \geq \beta N, \tau_N = T_N\}\\
&=& \{T_{[ \beta N]} < T_0<T_N\} \cup \{T_{N-[ \beta N]}<T_N<T_0\}.
\end{eqnarray*}
We now specialize arguments to the three types of random walks.
\vskip .1cm
{\it Symmetric Walk:-}
When the walk is symmetric $p_N=q_N=1/2$, recall the standard Gambler's ruin identity: For $a,b,z \in \mathcal T_{N},$ such that $a<z<b $,
\begin{equation}\label{gamblers_symmetric}
P_z(T_a<T_b) \ = \ \frac{b-z}{b-a}.\end{equation}
For $\beta> \alpha$, compute
\begin{eqnarray*}
P_{[ \alpha
N]}(R_N \geq [ \beta N], \tau_N = T_0)
&=& \frac{[ \alpha
N]}{[ \beta N]}\frac{N-[ \beta N]}{N} \
\rightarrow\ \frac{\alpha(1-\beta)}{\beta}.\end{eqnarray*}
When, $\beta> 1-\alpha$, we have
\begin{eqnarray*} P_{[ \alpha
N]}(R_N \geq [ \beta N], \tau_N = T_N)
&=& \frac{N-[ \alpha
N]}{[ \beta N]}\frac{N-[ \beta
N]}{N}\ \rightarrow \
\frac{(1-\alpha)(1-\beta)}{\beta}.\end{eqnarray*}
Putting these expressions together, along with simple calculations, we have
$$\lim_{N\uparrow \infty}P_{[ \alpha
N]}(R_N/N\geq \beta) \ = \ \left\{\begin{array}{rl}
1&\ {\rm when \ }0\leq \beta\leq \alpha \wedge (1-\alpha)\\
\frac{\alpha \wedge (1-\alpha)}{\beta} &\ {\rm when \
}\alpha \wedge (1-\alpha) <\beta
<\alpha \vee (1-\alpha)\\
\frac{1-\beta}{\beta}&\ {\rm when \ }
\alpha \vee (1-\alpha) \leq \beta \leq 1\\
0&\ {\rm when \ }\beta>1.\end{array}\right.$$
The right-side defines a distribution $G_{0,\alpha}$, supported on $[\alpha\wedge (1-\alpha), 1]$ whose density
\begin{equation}
\label{G_symmetric}
g_{\alpha}(\beta) \ = \ \left\{\begin{array}{rl}
\frac{\alpha \wedge (1-\alpha)}{\beta^2}& \ {\rm for \
}\alpha \wedge (1-\alpha) <\beta
<\alpha \vee (1-\alpha)\\
\frac{1}{\beta^2}& \ {\rm for \ }\alpha \vee (1-\alpha)\leq \beta \leq 1\\
0&\ {\rm otherwise.}\end{array}\right.\end{equation}
\vskip .1cm
{\it Weakly-asymmetric walk:-} In the weakly asymmetric case, $q_N= 1/2 - c/N$ and $p_N = 1/2 + c/N$ with $c>0$, let
$$s_N \ := \ \frac{q_N}{p_N} \ = \ \frac{1/2 -c/N}{1/2+c/N} \ = \ 1-\frac{4c}{N} + O(N^{-2}).$$
The corresponding gambler's ruin identity becomes, for $a<z<b$,
\begin{equation}\label{gamblers_weakly}
P_z(T_a<T_b) \ = \ \frac{(q_N/p_N)^z - (q_N/p_N)^b}{(q_N/p_N)^a - (q_N/p_N)^b}.\end{equation}
Then, following the symmetric argument, when $\beta> \alpha$,
\begin{eqnarray*}
P_{[ \alpha
N]}(R_N \geq [\beta N], T_0= \tau_N)
&=& \frac{1-s_N^{[ \alpha N]}}{1-s_N^{[ \beta N]}} \frac{s_N^{[ \beta N]} - s_N^N}{1-s_N^N}\\
&\rightarrow& \frac{1-e^{-4\alpha c}}{1-e^{-4\beta c}}\frac{e^{-4\beta c}-e^{-4c}}{1-e^{-4c}}\ := \ A_1(\alpha,\beta,c).
\end{eqnarray*}
When $\beta>1-\alpha$,
\begin{eqnarray*}
P_{[ \alpha
N]}(R_N\geq [\beta N], T_N=\tau_N)
&=& \frac{s_N^{[ \alpha N]}-s_N^N}{s_N^{N-[ \beta N]}-s_N^N}\frac{1-s_N^{N-[ \beta
N]}}{1-s_N^N}\\
&\rightarrow &
\frac{e^{-4\alpha c} - e^{-4c}}{e^{-4(1-\beta)c}-e^{-4c}}\frac{1-e^{-4(1-\beta) c}}{1-e^{-4c}} \ := \ A_2(\alpha,\beta,c).\end{eqnarray*}
Noting
$$
\lim_{N\uparrow\infty}P_{[ \alpha N]}(T_N<T_0) \ =\ \frac{1-e^{-4\alpha c}}{1-e^{-4c}},\ \ {\rm and \ \ }
A_1+A_2 \ = \ \frac{e^{-4\alpha c}(e^{4c(1-\beta)}-1)}{1-e^{-4c\beta}},
$$
we have
\begin{equation}
\label{G_weakly}
\lim_{N\uparrow \infty}P_{[ \alpha
N]}(R_N/N\geq \beta) \ = \ \left\{\begin{array}{rl}
1&\ {\rm when \ }0\leq \beta\leq \alpha \wedge (1-\alpha)\\
\frac{1-e^{-4c\alpha}}{1-e^{-4c\beta}} &\ {\rm when \
}\alpha \leq \beta
\leq 1-\alpha\\
\frac{1-e^{4c(1-\alpha)}}{1-e^{4c\beta}}& \ {\rm when \ } 1-\alpha \leq \beta \leq \alpha\\
\frac{e^{-4\alpha c}(e^{4c(1-\beta)}-1)}{1-e^{-4c\beta}}&\ {\rm when \ }
\alpha \vee (1-\alpha) \leq \beta \leq 1\\
0&\ {\rm when \ }\beta>1.\end{array}\right.\end{equation}
The right-side defines a distribution $G_{c,\alpha}$, supported on $[\alpha\wedge (1-\alpha), 1]$, whose density, although messy, can be easily derived.
\vskip .1cm
{\it Asymmetric walk:-} In the asymmetric case, $q_N=q$, $p_N=p$, and $p>q$, and it is not difficult to see that we cannot go left too many times. The gambler's ruin identity (\ref{gamblers_weakly}) also holds in this case and,
for $x= [\alpha N]$, $P_{x}(T_0<T_N) = \exp\{-CN\}$ for some constant $C>0$.
To complete the proof, for integers $z\geq 0$, compute
\begin{eqnarray*}
P_x(R_N \geq N-x + z) &=& P_x(T_N<T_0, R_N\geq N-x +z) + o(1)\\
&=& P_x(T_{x-z}<T_N<T_0) + o(1)\\
&=& (q/p)^{z} + o(1).
\end{eqnarray*}
\hfill$\nabla$
\begin{remark}
\label{entropy}
\rm
1. As expected, $G_{c,\alpha}$ interpolates between the symmetric and asymmetric cases: Namely, as $c\downarrow 0$, $G_{c,\alpha}\Rightarrow G_{0,\alpha}$, and as $c\uparrow\infty$, $G_{c,\alpha}$ converges to the constant $1-\alpha$.
2. It is curious to observe, for symmetric walks, that starting from $x = [ \alpha N]$, with $\alpha \in (0,1/2]$, the expected range
$$
\int_0^1 \beta g_\alpha(\beta)d\beta
\ = \ \int_{\alpha}^{1-\alpha} \beta\frac{\alpha}{\beta^2}d\beta + \int_{1-\alpha}^1 \beta\frac{1}{\beta^2}d\beta
\ =\ -(1-\alpha)\log(1-\alpha)
-\alpha\log(\alpha)
$$
is the entropy of the exit distribution $\langle 1-\alpha,\alpha\rangle$ where $1-\alpha$ is the probability of leaving by the left endpoint, and $\alpha$ the chance of exiting right! The maximum value $\log 2$ occurs when $\alpha=1/2$.
3. For symmetric and weakly asymmetric walks, the limit distributions may also be derived in terms of Brownian motion and diffusion estimates.
\end{remark}
\subsection{The range when starting at random} \label{random} We derive now the limiting law of $R_N/N$ when starting at random, that is the uniform distribution on $\mathcal T_N$. It seems nonintuitive that the limit law is U$[0,1]$ no matter the type of random walk.
\begin{proposition}
\label{uniformprop}
For symmetric, weakly asymmetric and asymmetric random walk, when starting at random in $\mathcal T_N$, $R_N/N$ converges weakly to the uniform distribution $U[0,1]$.
\end{proposition}
\proof Suppose our starting point was random. In the symmetric and weakly asymmetric cases, the limiting distribution of ${R_{N}}/{N}$, from straightforward considerations, is found by integrating the density $g_\alpha$ and tail of $G_{c,\alpha}$ with respect to $\alpha$ (denoted by $G_{c,\alpha}([\beta,1])$).
In the symmetric case, when $\beta\leq 1/2$,
\begin{eqnarray*}
\int_0^1 g_\alpha(\beta)d\alpha &=& \int_0^\beta \frac{\alpha}{\beta^2} d\alpha + \int_\beta^{1-\beta} 0\; d\alpha + \int_{1-\beta}^1 \frac{1-\alpha}{\beta^2} d\alpha \ = \ 1.
\end{eqnarray*}
But, also, when $\beta>1/2$,
\begin{eqnarray*}
\int_0^1 g_\alpha(\beta)d\alpha &=& \int_0^{1-\beta} \frac{\alpha}{\beta^2}d\alpha + \int_{\beta}^{1-\beta} \frac{1}{\beta^2}d\alpha + \int_{\beta}^1 \frac{1-\alpha}{\beta^2}d\alpha \ = \ 1.
\end{eqnarray*}
On the other hand, in the weakly asymmetric case, we have, when $\beta\leq 1/2$,
\begin{eqnarray*}\int_0^1 G_{c,\alpha}([\beta,1])d\alpha &=&
\int_0^\beta \frac{1-e^{-4c\alpha}}{1-e^{-4c\beta}}d\alpha + \int_\beta^{1-\beta}1\; d\alpha \\
&&+ \int_{1-\beta}^1 \frac{e^{-4c\beta}(e^{4c(1-\alpha)} - 1)}{e^{-4c(1-\beta)}-e^{-4c}}d\alpha\ = \ 1-\beta.
\end{eqnarray*}
Similarly, when $\beta>1/2$, $\int_0^1 G_{c,\alpha}([\beta,1])d\alpha$ equals
\begin{eqnarray*}
&&\frac{1}{1-e^{-4c\beta}}\Big[\int_0^{1-\beta}{1-e^{-4c\alpha}}d\alpha + \int_{1-\beta}^\beta e^{-4c\alpha}(e^{4c(1-\beta)}-1)d\alpha\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ + \int_\beta^1 e^{-4c\beta}(e^{4c(1-\alpha)}-1)d\alpha\Big] \ = \ 1-\beta.
\end{eqnarray*}
Consequently, for symmetric and weakly asymmetric walks, the limiting distribution is $U[0,1]$ when the starting position is uniformly chosen.
However, in the asymmetric case, from Proposition \ref{rangeprop}, $R_N/N \rightarrow 1-\alpha$ in probability starting from $x=[ \alpha N]$. Then, starting at random in $\mathcal T_N$, we have that $R_N/N\rightarrow Y$ in probability where $Y$ is a $U[0,1]$ distributed random variable. \hfill$\nabla$
\begin{remark}
\label{point_visitedrmk}
\rm
One might ask, on the other hand, with what probability a point $y= [ \beta N]$ belongs to the range when starting at random. This is the same as asking when $y$ is visited by the walk. For symmetric walk, it is not difficult to use the gambler's ruin identity (\ref{gamblers_symmetric}) to see,
as $N\uparrow\infty$, that the probability tends to
$$\int_\beta^1 \frac{1-\alpha}{1-\beta}d\alpha +
\int_0^\beta \frac{\alpha}{\beta}d\alpha\ = \ \frac{1}{2}.$$
It seems curious that the limit does not depend on $\beta$.
For asymmetric walk, starting from $[ \alpha N]$, when $\alpha>\beta$, the point $y$ cannot be reached with positive probability in the limit. Then, the chance $y$ belongs to the range, when starting at random, is $\beta$.
For weakly asymmetric walks, using (\ref{gamblers_weakly}), the limit is $\frac{\beta}{1-e^{-4c\beta}} - \frac{1-\beta}{1-e^{4c(1-\beta)}}$ which interpolates between the other cases as $c\downarrow 0$ and $c\uparrow\infty$
\end{remark}
\section{Question 2: Characterization of local times}
\label{ray-knight}
To capture the local times of the random walk before its exit, we use the ``Ray-Knight'' or ``Kesten-Kozlov-Spitzer'' representation, and some martingale characterizations. Our treatment and proofs will be similar to those in Toth \cite{Toth} which considered certain self-interacting random walks.
Let $0 < \alpha < 1$. Suppose the walk starts at $[\alpha N]$, and exits at the right endpoint $N$. Let $\zeta^N_{j}$ be the number of left crossings of the bond $(N-(j-1),N-j)$ before exit. Then, $\zeta^N_0=0$, and $\zeta^N_1$ is distributed as $D_N$, a Geometric$(q_N)$ random variable minus $1$,
$P(D_N=n) = p_Nq_N^{n}$ for $n\geq 0$. In the following, we drop the script $N$.
Let $\{\xi_{j,i}\}_{i,j\geq 0}$ be i.i.d. random variables with distribution $D_{N}$. A moment's thought convinces that $\{\zeta_j\}_{0\leq j\leq N}$ is a Markov chain with representation
\begin{equation}
\label{representation}\zeta_{j+1} \ = \ \left\{\begin{array}{rl}\sum_{i=0}^{\zeta_j} \xi_{j,i}& \ {\rm for \ } 0\leq j<[(1-\alpha)N]\\
\sum_{i=1}^{\zeta_j} \xi_{j,i} & \ {\rm for \ } [(1-\alpha)N]\leq j\leq N-1\end{array}\right.
\end{equation}
such that
\begin{equation}
\label{restriction}\zeta_j=0 {\rm \ \ \ for\ some\ \ \ } [(1-\alpha)N]\leq j<N,\end{equation}
with the convention that empty sums vanish.
Note that for $j< [(1-\alpha)N]$, the sum starts with index $i=0$ since, even if $\zeta_j=0$, given exit at the right, the walk must visit locations $[\alpha N]\leq x\leq N$ and may have left crossings of $(x-1,x)$.
However, for $j\geq [(1-\alpha)N]$, since the walk is not guaranteed to visit sites to the left of $[\alpha N]$, $\zeta_j$ is the size of a Branching process, with initial value $\zeta_{[(1-\alpha)N]}$, which must vanish before time $j=N$.
Then, the local time of the walk is $$G(y) \ =\ \left\{\begin{array}{rl}
\zeta_{N-y}& \ {\rm for \ }0 \leq y < [\alpha N]\\
\zeta_{N-y} +1 & \ {\rm for \ }[\alpha N] \leq y\leq N. \end{array}\right.
$$
In the following, to analyze $\{\zeta_j\}_{0\leq j\leq N}$, it will be helpful to consider the Markov chain $\eta_j$, such that $\eta_0=0$ and $\eta_1\stackrel{d}{=}D_N$, for which representation (\ref{representation}) holds in terms of the variables $\{\xi_{j,i}\}_{i,j\geq 0}$, but {\it without} the restriction (\ref{restriction}).
When the walk exits at the left endpoint $0$, one considers an analogous Markov chain $\tilde\zeta_j$, corresponding to right-crossings of $(j,j+1)$, where the representation and restriction are reversed. Namely, let $\tilde D_N$ be a ${\rm Geometric}(p_N)$ random variable minus $1$, $P(\tilde D_N = n) = q_Np_N^{n}$ for $n\geq 0$. Define $\tilde\zeta_0=0$, $\tilde \zeta_1 \stackrel{d}{=}\tilde D_N$, and
$$
\zeta_{j+1} \ = \ \left\{\begin{array}{rl}\sum_{i=0}^{\zeta_j} \xi_{j,i}& \ {\rm for \ } 0\leq j<[\alpha N]\\
\sum_{i=1}^{\zeta_j} \xi_{j,i} & \ {\rm for \ } [\alpha N]\leq j\leq N-1\end{array}\right.
$$
such that
$\zeta_j=0$ for some $[\alpha N]\leq j<N$. The local time of the walk in this case is $G(y) = \tilde\zeta_{y}$ for $[\alpha N]< y\leq N$ and $G(y) = \tilde\zeta_y +1$ for $0\leq y\leq [\alpha N]$.
Here also it will be of use to define analogously a Markov chain $\tilde\eta_j$ satisfying $\tilde\eta_0=0$, $\tilde\eta_1\stackrel{d}{=}\tilde D_N$, and the reversed representation but without the restriction that the chain must vanish for $[\alpha N]\leq j<N$.
Finally, define $Y_N(t) = \frac{1}{N}\eta_{[Nt]}$ and $\tilde Y_N(t) = \frac{1}{N}\tilde \eta_{[Nt]}$ for $0\leq t\leq 1$,
and suppose that $Y_N(0)=\tilde Y_N(0)=0$.
\subsection{Symmetric walks}
Consider the following processes. Let $Z_0=0$, and define
$$ Z_t \ = \ \left\{\begin{array}{rl}
t + \int_{0}^{t}\sqrt{2Z_s}dB_s & \ {\rm for \ } 0\leq t\leq 1-\alpha\\
&\\
Z_{1-\alpha} + \int_{1-\alpha}^{t}\sqrt{2Z_s}dB_s & \ {\rm for \ } 1-\alpha \leq t\leq 1.
\end{array}\right.
$$
Observe that
$Z_t$ for $0\leq t\leq 1-\alpha$ is the same in law as Besq$^2(t/2)$ process, and a Besq$^0(t/2)$ process for $1-\alpha\leq t\leq 1$ (cf. Revuz-Yor \cite{RY} for more on the processes Besq$^\delta$ $dX_t = \delta dt + 2\sqrt{X_t}dB_t$). Let $\tau^R_0$ be the first time $Z_t$ hits $0$ after time $t=1-\alpha$. Note that $Z_t$ remains at value $0$ after time $\tau^R_0$.
Define also
$\tilde Z_t$ where $\tilde Z_0=0$ and
$$\tilde Z_t \ = \ \left\{\begin{array}{rl}
t + \int_0^t\sqrt{2\tilde Z_s}dB_s & \ {\rm for \ } 0\leq t\leq \alpha\\
\tilde Z_\alpha + \int_\alpha^t\sqrt{2\tilde Z_s}dB_s & \ {\rm for \ } \alpha \leq t\leq 1.
\end{array}\right.
$$
Let also $\tau^L_0$ be the time $\tilde Z_t$ reaches $0$ after time $t=\alpha$. Here, also, $\tilde Z_t \equiv 0$ for $t\geq \tau^L_0$.
It will turn out that $Z_t$ and $\tilde Z_t$ will be identified respectively, as the scaling limits of the local times when the random walk exits at the right and left endpoints of the interval. The important point in this identification is the next result.
\begin{proposition}
\label{mainprop}
For symmetric walk starting from $x=[\alpha N]$, we have $$Y_{N}(t) \ \Rightarrow \ Z(t) \ \ \ {\rm and \ \ \ } \tilde Y_N(t) \ \Rightarrow \ \tilde Z(t)$$ in $D[0,1]$, in the sup topology.
\end{proposition}
Instead of proving Proposition \ref{mainprop}, which follows steps as in Toth \cite{Toth}, we prove Proposition \ref{mainprop_OU} in the next subsection, with respect to weakly asymmetric random walks, dealing with squared Ornstein-Uhlenbeck processes which are less standard.
Now, with Proposition \ref{mainprop} in hand, since $Y_N(t)$ and $\tilde Y_N(t)$ converge respectively to $Z_t$ and $\tilde Z_t$ in the sup topology, it follows that the conditional distributions of $Y_N(t)$ given $\eta_j$ vanishes for $j\geq [(1-\alpha)N]$ and $\tilde Y_N(t)$ given $\tilde \eta_j$ vanishes for $j\geq [\alpha N]$ converge to the conditional distributions of $Z_t$ given that $1-\alpha\leq \tau^R_0<1$ and $\tilde Z_t$ given that $\alpha\leq \tau^L_0<1$.
Hence, from this discussion, the following characterization holds for the local times of the walk up to time of exit. Recall that $1-\alpha$ and $\alpha$ are the exit probabilities of right and left exit respectively.
\begin{proposition}
\label{Besqprop} For symmetric walk starting from $[\alpha N]$, the local times
$$G([Nt])/N \ \Rightarrow \ \alpha \mu^R + (1-\alpha) \mu^L$$
where $\mu^R$ is the law of the process $Z_{1-t}$ conditioned on $1-\alpha\leq \tau^R_0<1$, and $\mu^L$ is the law of the process $\tilde Z_t$ conditioned on $\alpha\leq \tau^L_0<1$.
\end{proposition}
\vskip .1cm
\subsection{Weakly asymmetric walks}
The development of the local time structure is similar to the symmetric case. Corresponding to right exit, $E D_{N} = q_N/p_N =1-\frac{4c}{N+2c}$ and ${\rm Var}(D_{N})=q^2_N/p^2_N + q_N/p_N = 2- \frac{12c}{N+2c} +\frac{16c^{2}}{(N+2c)^{2}}$. Define the process $Z_t^c$ by
$Z^c_0=0$, and
$$
Z^c_t \ = \ \left\{\begin{array}{rl}
\int_{0}^{t}(1-4cZ^c_s)ds + \int_{0}^{t} \sqrt{2Z^c_s}dB_s & \ {\rm for \ } 0\leq t\leq 1-\alpha\\
&\\
Z^c_{1-\alpha}- \int_{1-\alpha}^{t}4cZ^c_s ds+ \int_{1-\alpha}^{t}\sqrt{2Z^c_s}dB_s & \ {\rm for \ } 1-\alpha \leq t\leq 1
\end{array}\right.
$$
Note $2(Z^c_t+1)$ and $2(Z^c_t -t)$ are the squares of the Ornstein-Uhlenbeck process $dX_t = -4cX_tdt +\sqrt{2}dB_t$ for $0\leq t\leq 1-\alpha$ and $1-\alpha\leq t\leq 1$ respectively.
Also, with respect to left exit, $E \tilde D_{N} = p_N/q_N = 1+4c/N + O(N^{-2})$ and ${\rm Var}(\tilde D_{N}) = 2+O(N^{-1})$. Define
$\tilde Z^c_t$ by $\tilde Z^c_0=0$ and
$$\tilde Z^{c}_t \ = \ \left\{\begin{array}{rl}
\int_{0}^{t}(1+4c\tilde Z^c_s)ds + \int_{0}^{t}\sqrt{2\tilde Z^{c}_s}dB_s & \ {\rm for \ } 0\leq t\leq \alpha\\
\tilde Z^{c}_{\alpha} + \int_{\alpha}^{t}\sqrt{2\tilde Z^{c}_s}dB_s & \ {\rm for \ } \alpha \leq t\leq 1.
\end{array}\right.
$$
As before, let $\hat\tau^R_0$ be the first time after $t=1-\alpha$ that $Z^c_t$ reaches $0$, and $\hat\tau^L_0$ be the first time after $t=\alpha$ that $\tilde Z^c_t$ hits $0$.
Analogous to the symmetric random walk case, we show that $Z^c_t$ and $\tilde Z^c_t$ are the scaling limits of the local times when the weakly asymmetric random walk exits at the right and left endpoints respectively.
\begin{proposition}
\label{mainprop_OU}
For the weakly asymmetric random walk starting from $x=[\alpha N]$, we have $$Y_{N}(t) \ \Rightarrow \ Z^c(t) \ \ \ {\rm and \ \ \ } \tilde Y_N(t) \ \Rightarrow \ \tilde Z^c(t)$$ in $D[0,1]$, in the sup topology.
\end{proposition}
The same argument as in the symmetric case allows to deduce the the following characterization.
\begin{proposition}
\label{OUprop}
For the weakly asymmetric walk, starting from $x=[\alpha N]$, the local times satisfy
$$G([Nt])/N \ \Rightarrow \ R(\alpha) \mu_c^R + (1-R(\alpha)) \mu_c^L$$
where $\mu_c^R$ is the law of the process $Z^c_{1-t}$ conditioned on $1-\alpha\leq \hat\tau^R_0<1$, and $\mu_c^L$ is the law of the process $\tilde Z^c_t$ conditioned on $\alpha\leq \hat\tau^L_0<1$. Here, $R(\alpha) = (1-e^{-4c\alpha})/(1-e^{-4c})$ is the exit probability to the right.
\end{proposition}
{\it Proof of Proposition \ref{mainprop_OU}.} Here, we argue that $Y_N(t) \Rightarrow Z^c(t)$ which corresponds to ``left crossings.'' The argument for $\tilde Y_N(t)\Rightarrow \tilde Z^c(t)$ is similar.
The proof naturally separates into two parts corresponding to when $j\leq [(1-\alpha)N]$ and $j\geq [(1-\alpha)N]$.
The strategy will be to use martingale decompositions of the Markov chain $\{\eta_j\}_{j\geq 0}$.
Define, for $[Nt]\leq [(1-\alpha) N]$, the martingale and its quadratic variation,
\begin{eqnarray*}
M_N(t) & = & \eta_{[Nt]}-\eta_0 - \sum_{j=0}^{[Nt]-1} \left( E[\eta_{j+1}|\eta_j]-\eta_j\right)\\
\langle M_N(t) \rangle &=& \sum_{j=0}^{[Nt]-1} E\big[ \big(\eta_{j+1} - E[\eta_{j+1}|\eta_j]\big)^2\big].
\end{eqnarray*}
Then,
\begin{eqnarray*}
\frac{1}{N}M_N(t) &=& Y_N(t) - Y_N(0) - \frac{1}{N}\sum_{j=0}^{[Nt]-1} (E(D_{N}) (\eta_{j}+1)-\eta_j)\\\
&=& Y_N(t) - Y_N(0) - \frac{1}{N}\sum_{j=0}^{[Nt]-1} ((1-\frac{4c}{N +2c} ) (\eta_{j}+1)-\eta_j) \\
&=& Y_N(t) - Y_N(0) - \frac{1}{N}[Nt] + \frac{4c}{N+2c}\sum_{j=0}^{[Nt]-1} Y_N\left(\frac{j}{N}\right) + \frac{4c[Nt]}{N(N+2c)}
\end{eqnarray*}
and
\begin{eqnarray}
\langle N^{-1}M_N(t)\rangle &=& \frac{1}{N^2} \sum_{j=0}^{[Nt]-1} E\big[(\eta_{j+1}-E(D_{N})(\eta_j+1))^2|\eta_j\big]\nonumber\\
&=& \frac{1}{N^2} \sum_{j=0}^{[Nt]-1} E\left[\left(\sum_{i=0}^{\eta_j} (\xi_{j,i}-E(D_{N}))\right)^2|\eta_j\right]\nonumber\\
&=& \frac{1}{N^2} \sum_{j=0}^{[Nt]-1} (\eta_j +1) \mbox{Var}(D_{N})\nonumber\\
& = &\frac{2}{N}\sum_{j=0}^{[Nt]-1}Y_N\left(\frac{j}{N}\right) + O\left(\frac{1}{N}\right).
\label{quad_var1}
\end{eqnarray}
Now suppose $Y_N(t)$ and $N^{-1}M_N(t)$ are tight in the sup topology, and $Y_N(t)\Rightarrow Z_t$, $N^{-1}M_N(t)\Rightarrow M(t)$ on subsequences. Then,
$M(t) = Z_t-Z_0 -\int_{0}^{t} (1-4cZ_{s})ds$ and $\langle M(t)\rangle = 2\int_0^t Z_s ds$.
Hence, by Levy's criterion for continuous martingales, we have that $Z_t$ is uniquely characterized by
$$Z_t \ = Z_{0} + \int_{0}^{t} (1-4cZ_{s})ds + \int_{0}^{t}\sqrt{2Z_{s}}dB_s$$
Similarly, for $[Nt]\geq [(1-\alpha) N]$, since now $\eta_{j+1}= \sum_{i=1}^{\eta_j}\xi_{j,i}$, the drift is not present, and we can write
\begin{eqnarray*}
\frac{1}{N}(M_N(t)- M_N(1-\alpha)) & = & Y_N(t) - Y_N(1-\alpha) + \sum_{j = [(1-\alpha)N]}^{[Nt] - 1} (E(\eta_{j+1} | \eta_{j}) - \eta_{j})\\
&=& Y_N(t) - Y_N(1-\alpha) + \frac{4c}{N +c}\sum_{j=[N(1-\alpha)]}^{[Nt]-1} Y_N\left(\frac{j}{N}\right)
\end{eqnarray*}
and also
\begin{eqnarray*}
\frac{1}{N}(\langle M_N(t)\rangle -\langle M_N(1-\alpha)\rangle)
&=& \frac{1}{N^2} \sum_{j=[N(1-\alpha)]}^{[Nt]-1} (\eta_j +1) \mbox{Var}(D_{N})\\
& =&
\frac{2}{N} \sum_{j=[N(1-\alpha)]}^{[Nt]-1} Y_N\left(\frac{j}{N}\right) + O\left(\frac{1}{N}\right). \nonumber\\
\end{eqnarray*}
Hence, as before, given tightness of $N^{-1}M_N(t)$, and subsequential convergences $Y_n(t)\Rightarrow Z(t)$ and $N^{-1}M_N(t) \Rightarrow M(t)$ on $[1-\alpha, 1]$ where $M(t) - M(1-\alpha) = Z_t-Z_{1-\alpha}$ and $\langle M(t) - M(1-\alpha)\rangle = 2\int_{1-\alpha}^t Z_s ds$, we conclude, for $t\in [1-\alpha,1]$, that
$$Z_t \ = Z_{1-\alpha} + \int_{1-\alpha}^{t }\sqrt{2Z_{s}}dB_s.$$
Consequently, it follows, putting the subsequential converges together, for $0 \leq t \leq 1$ that $Y_N(t)$ converges weakly to $Z_{t}$.
\vskip .1cm
{\it Tightness.} We now argue tightness of $Y_N(t)$ and $N^{-1}M_N(t)$ on $[0,1-\alpha]$. Tightness of $Y_N(t)$ follows from tightness of $N^{-1}M_N(t)$ in the sup topology which can be argued by a Kolmogorov-Centsov argument. First, for a general discrete time martingale $(M(l), \mathcal F_l)$ with difference $\delta(l) = M(l)-M(l-1)$, we have that
\begin{eqnarray*}
E\left[ (M(l)-M(k))^4\right] &=& 6\sum_{j=k+1}^l E\left[ \delta(j)^2(M(j-1)-M(k))^2\right]\\
&&\ \ \ + 4 \sum_{j=k+1}^l E\left[ \delta(j)^3(M(j-1)-M(k))\right] + \sum_{j=k+1}^l E\left[\delta^4_j\right]
\end{eqnarray*}
and by Jensen inequality,
\begin{eqnarray*}
&&E\left[ (M(l)-M(k))^4\right] \ \leq \ 6\sum_{j=k+1}^l E\left[ E[\delta(j)^2|\mathcal F_{j-1}](M(j-1)-M(k))^2\right]\\
&&\ \ \ \ \ \ \ +\ 4 \sum_{j=k+1}^l \left\{E\left[ E[\delta(j)^4|\mathcal F_{j-1}]^{3/2}(M(j-1)-M(k))^2\right]\right\}^{1/2}\\
&&\ \ \ \ \ \ \ +\ \sum_{j=k+1}^l E\left[E[\delta(j)^{4}|\mathcal F_{j-1}]\right].
\end{eqnarray*}
Now, in our context, define the martingale, for $l\leq [(1-\alpha)N]$,
$$M(l) \ = \ \eta_l -\eta_0 -\sum_{i=0}^{l-1} \left (E[\eta_{i+1}|\eta_i] - \eta_i\right )$$
so that $M_N(t) = M([Nt])$, and also the stopping time
$$\theta_{y,N} \ = \ \inf\{l\geq 0: \eta_l \geq Ny\}.$$
Compute, with respect to $M(l\wedge \theta_{y,N})$, that
\begin{eqnarray*}
\delta(l) & = & M(l\wedge \theta_{y,N})-M(l-1\wedge \theta_{y,N}) \\
& = & \eta_{l\wedge \theta_{N,y}} - E[\eta_{l\wedge \theta_{y,N}}|\eta_{l-1\wedge \theta_{y,N}}]
\ = \ \sum_{i=0}^{\eta_{l-1\wedge\theta_{N,y}}} (\xi_{l\wedge \theta_{y,N},i}-E(D_{N})).\end{eqnarray*}
Hence,
we have
$$E\Big[ \Big(\sum_{i=0}^{\eta_{l-1\wedge \theta_{y,N}}} (\xi_{l\wedge\theta_{y,N},i}-E(D_{N}))\Big)^2\Big |\mathcal F_{l-1\wedge\theta_{y,N}}\Big] \ \leq\ {\rm Var}(D_{N}) \eta_{l-1\wedge \theta_{y,N}}\ \leq\ c_{1}Ny$$
and
\begin{eqnarray*}
E\Big[\Big(\sum_{i=0}^{\eta_{l-1\wedge \theta_{y,N}}} (\xi_{l\wedge \theta_{y,N},i}-E(D_{N}) )\Big)^4|\mathcal F_{l-1\wedge\theta_{y,N}}\Big] &\leq& c_{2}ED_{N}^4 \eta^2_{l-1\wedge \theta_{y,N}} + \eta_{l-1\wedge \theta_{y,N}}\\
&\leq& (c_{2}ED^4_N + 1) ((Ny)^2 + Ny).\end{eqnarray*}
Also, noting the quadratic variation estimate (\ref{quad_var1}),
$$\frac{1}{N^2}E\left[\left(M(j-1\wedge \theta_{N,y}\right)-M\left(k\wedge\theta_{N,y})\right)^2\right] \ \leq\ c_{3}|j-k|\left( \frac{1}{N} + y\right).$$
Hence, we have, for some constant $c_4$ not depending on $N$ or $y$, that
\begin{eqnarray*}
\frac{1}{N^4}E [ (M([Nt]\wedge\theta_{y,N}) - M([Ns]\wedge \theta_{y,N}))^4] &\leq& c_{4} \max\{y^2,1\}(|t-s|^2\vee \frac{1}{N^{2}}).
\end{eqnarray*}
Then, by Theorem 12.3 Billingsley \cite{Billingsley}, $N^{-1}M([Nt]\wedge \theta_{y,N})$ is tight for any $y<\infty$. Hence, $N^{-1}M_N(t)$ is tight in the sup topology on $[0,1-\alpha]$.
Tightness with respect to the interval $[1-\alpha,1]$, and consequently the whole interval $[0,1]$ follows similarly. \hfill$\nabla$
\subsection{Asymmetric walks}
The situation is much different for asymmetric walks, in particular, the local times are
of order $O(1)$, and no scaling is required.
Given $p>q$, the walk starting from $x=[\alpha N]$ will exit to the right with probability tending to $1$ as $N\uparrow\infty$. The sequence $\eta_j$ for $1\leq j\leq [(1-\alpha) N]$ is a branching process with mean offspring $ED_N = q/p<1$ and immigration at each time of one individual. The initial population is $\eta_1$ with the distribution of $D=D_N$, a Geometric$(q)$ random variable minus $1$. Hence, this sequence is a positive recurrent Markov chain, and $\eta_{[(1-\alpha)N]}$ converges to the stationary distribution $\pi$.
On the other hand, the chain $\eta_j$ for $j\geq [(1-\alpha) N]$ is the usual Branching process with offspring distribution $D_N$ (and no immigration). Hence, it dies out in finite time.
The stationary distribution $\pi$ can be described by its probability generating function $\Psi(s) = \sum_{k\geq 0} \pi(k)s^k$. Let $\phi(s)$ be the probability generating function of $D$. Then, easy computations give that $\Psi(s)=\Psi(\phi(s))\phi(s)$.
Hence, since the distribution of $\eta_{[(1-\alpha)N]}$ converges to $\pi$, we can state a limit characterization in terms of a reversed process.
\begin{proposition}
\label{asym_limprop} Consider the asymmetric walk when $p>q$ starting from $[\alpha N]$.
For any $M\geq 1$,
the reversed process $\{\beta_k=\eta_{[(1-\alpha) N]-k}\}_{k= 0}^M$ converges in distribution to the reversed process starting from the stationary distribution $\pi$ of the chain.
However, $\{\beta_{-k}= \eta_{[(1-\alpha)N] +k}\}_{k=0}^M$ converges to a Branching process with offspring distribution $D$ starting from $\pi$.
\end{proposition}
\section{Question 3: Periodicity}
\label{independent}
We now address the parity of various well-separated locations visited by the walk before exiting. We remark different types of multiple point structures in other settings have been studied in Hamana \cite{Ham} and Pitt \cite{Pitt}.
Let $0<\alpha_1<\alpha_2<\cdots<\alpha_k<1$, and $e_i \in \{0,1\}$ for $1\leq i\leq k$.
\begin{proposition}
\label{thm1_question3}
With respect to symmetric or weakly asymmetric walks, for $\alpha\in (0,1)$, we have
$$\lim_{N\rightarrow\infty} P_{[\alpha N]}\left( \cap_{i=1}^k \{G([\alpha_i N]) = e_i \ {\rm mod}_2\}| \max_{1\leq i\leq k}T_{[\alpha_i N]}<\tau_N\right) \ = \ \frac{1}{2^k}.$$
\end{proposition}
In other words, in the symmetric or weakly asymmetric cases, given that the locations are visited, the parities at $\{[\alpha_i N]\}_{i=1}^k$ converge to i.i.d. fair Bernoulli random variables.
But, with respect to asymmetric walks when $p>q$, starting from $[\alpha N]$, unless $\alpha<\beta$, $[\beta N]$ is not visited with probability tending to $1$. So, it makes sense only to discuss parities of sites to the right of $[\alpha N]$.
\begin{proposition}
\label{thm2_question3}
With respect to asymmetric walks when $p>q$, suppose $0<\alpha<\alpha_1$. Then,
$$\lim_{N\rightarrow\infty} P_{[\alpha N]}\left( \cap_{i=1}^k \{G([\alpha_i N]) = 1 \ {\rm mod}_2\}\right) \ = \ \frac{1}{(2-(p-q))^k}.$$
\end{proposition}
By the inclusion-exclusion principle, one concludes, in the asymmetric situation, the parities at $\{[\alpha_i N]\}_{i=1}^k$ converge
to i.i.d. Bernoulli random variables with success probability $(2-(p-q))^{-1}$.
We remark, with respect to the `stochastic locker' intepretation, one concludes that the expected proportion of lockers left
closed is half or $(2-(p-q))^{-1}$ times the proportion of the range in the symmetric/weakly asymmetric, or asymmetric cases respectively.
\subsection{Proofs of Propositions \ref{thm1_question3} and \ref{thm2_question3}}
The proofs of the above propositions are similar. We first derive the chance a single site is left open, and then later use this development in an induction scheme.
Let $T^r_y$ be the $r$th hitting time of $y$, and $\tilde{T}_y = \inf\{n\geq 1: X_n = y\}$ be the return time to $y$.
The event that site $y$ is left open, with various prescribed exits, is expressed as
\begin{eqnarray*}
&&\{G(y) = 1 \ {\rm mod}_2, T_N<T_0\}\\
&&\ \ \ \ \ \ \ \ \ =\ \cup_{k\geq 0} \{T^1_y<\tau_N\}\cap \{T^{2k+1}_y<\tau_N\}\cap\{T_N<T^{2k+2}_y\wedge T_0\}.
\end{eqnarray*}
Similarly,
\begin{eqnarray*}
&&\{G(y) = 1 \ {\rm mod}_2, T_0<T_N\}\\
&&\ \ \ \ \ \ \ \ \ = \ \cup_{k\geq 0} \{T^1_y<\tau_N\}\cap \{T^{2k+1}_y<\tau_N\}\cap\{T_0<T^{2k+2}_y\wedge T_N\}\\
&&\{G(y) = 1 \ {\rm mod}_2\} \ = \ \cup_{k\geq 0} \{T^1_y<\tau_N\}\cap \{T^{2k+1}_y<\tau_N\}\cap\{\tau_N<T^{2k+2}_y\}.
\end{eqnarray*}
Then,
\begin{eqnarray*}
P_x(G(y) = 1 \ {\rm mod}_2, T_N<T_0) &=& P_x(T_y<\tau_N) P_y(T_N<\tilde T_y)\sum_{l\geq 0} P_y(\tilde T_y<\tau_N)^{2l}\\
&=&\frac{P_x(T_y<\tau_N) P_y(T_N<\tilde T_y)}{1-(1-P_y(\tau_N<\tilde T_y))^2}\\
&=&\frac{P_x(T_y<\tau_N)}{2-P_y(\tau_N<\tilde T_y)}\frac{P_y(T_N<\tilde T_y)}{P_y(\tau_N<\tilde T_y)}.
\end{eqnarray*}
Also,
\begin{eqnarray*}
P_x(G(y)=1\ {\rm mod}_2, T_0<T_N) &=&\frac{P_x(T_y<\tau_N)}{2-P_y(\tau_N<T_y)}\frac{P_y(T_0<\tilde T_y)}{P_y(\tau_N<\tilde T_y)}\\
P_x(G(y)=1\ {\rm mod}_2) &=&\frac{P_x(T_y<\tau_N)}{2-P_y(\tau_N<\tilde T_y)}.
\end{eqnarray*}
In this last expression $P_x(T_y<\tau_N)$ is the probability $y$ is visited starting from $x$, and $(2-P_y(\tau_N<\tilde T_y))^{-1}$ is the factor specifying that $y$ is left open. The quantity $P_y(\tau_N<\tilde T_y)$ can be viewed as an ``escape probability.''
Suppose now $x=[ \alpha N]$ and $y=[ \beta N]$. In the symmetric case, we compute
$$P_x(T_y<\tau_N) = \left\{\begin{array}{rl}
\frac{N-x}{N-y} & {\rm \ for \ }y<x<N, \\
\frac{x}{y} &\ {\rm for \ }0<x<y\end{array}\right.$$
and
$$
P_y(\tilde T_y<\tau_N) \ = \ \frac{1}{2}P_{y-1}(T_y<T_0) +
\frac{1}{2}P_{y+1}(T_y<T_N) \ = \ 1-\frac{N}{2y(N-y)}.$$
In the (weakly) asymmetric case, we have
$$
P_x(T_y<\tau_N) = \left\{\begin{array}{rl}
\frac{s_N^x-s_N^N}{s_N^y-s_N^N} & \ {\rm for \ }x>y\\
\frac{1-s_N^x}{1-s_N^y}&\ {\rm for \ }x<y\end{array}\right.$$
and
\begin{eqnarray*} P_y(\tau_N<\tilde T_y) &=& q_NP_{y-1}(T_0<T_y) + p_NP_{y+1}(T_N<T_y) \\
&=&\frac{p_N(1-s_N)(1-s_N^N)}{(1-s_N^y)(1-s_N^{N-y})}.\end{eqnarray*}
Then,
$$P_y(\tau_N<\tilde T_y) \ \rightarrow\ \left\{\begin{array}{rl}
0 & \ {\rm for \ symmetric/weakly \ asymmetric \ walks }\\
p-q& \ {\rm for \ asymmetric \ walks}.
\end{array}\right.
$$
Putting these observations together, we have the following result.
\begin{proposition}
\label{oddprop}
Under symmetric or weakly asymmetric motion,
\begin{eqnarray*}
&&\lim_{N\uparrow\infty}P_x(G(y)=1\ {\rm mod}_2| T_y<T_N<T_0)\\
&&\ \ \ \ \ \ \ \ \ =\ \lim_{N\uparrow\infty}P_x(G(y)=1\ {\rm mod}_2| T_y<T_0< T_N) \ =\ \frac{1}{2},
\end{eqnarray*}
and hence $\lim_{N\uparrow\infty}P_x(G(y)=1\ {\rm mod}_2| T_y<\tau_N) = 1/2$.
However, under asymmetric motion, for $x\leq y$,
$$\lim_{N\uparrow\infty}P_x(G(y)=1\ {\rm mod}_2) \ = \ \frac{1}{2-(p-q)}.$$
\end{proposition}
\vskip .1cm
{\it Proof of Proposition \ref{thm1_question3}.}
Let $G_n(y) = \sum_{l=0}^{n\wedge \tau_N} 1_y(X_l)$ be the number of visits to $y$ up to time $n\wedge \tau_N$. First, we write
\begin{eqnarray}
&&P_{[\alpha N]}(\cap_{i=1}^k \{T_{[\alpha_i N]}<\tau_N\},\cap_{i=1}^k G([\alpha_i N])=e_i \ {\rm mod}_2)\nonumber\\
&&\ \ = \ P_{[\alpha N]}(\cap_{i=1}^k \{G([\alpha_i N])=e_i \ {\rm mod}_2\}, T_{[\alpha_1 N]}<T_N<T_0)\nonumber\\
&&\ \ \ \ \ \
+ P_{[\alpha N]}(\cap_{i=1}^k G([\alpha_i N])=e_i \ {\rm mod}_2, T_{[\alpha_k N]}<T_0<T_N).
\label{sec_3_eqn0}
\end{eqnarray}
We now concentrate on the first term on the right when $T_N<T_0$, as the argument is similar for the second term.
Since, on the set $T_N<T_0$, the walk must leave $[\alpha_1 N]$ never to return, and is also nearest-neighbor, write
\begin{eqnarray}
&&P_{[\alpha N]}(T_{[\alpha_1 N]}<T_N<T_0, \cap_{i=1}^k\{G([\alpha_i N])=e_i \ {\rm mod}_2\})\nonumber\\
&&\ \ \ \ = \ \sum_{\stackrel{z_1,\ldots, z_k}{z_1 = e_1 \ {\rm mod}_2}} P_{[\alpha N]}(T^{z_1}_{[\alpha_1 N]}<\tau_N, \cap_{i=2}^k \{G_{T^{z_1}_{[\alpha 1 N]}}([\alpha_i N]) = z_i\})\nonumber\\
&&\ \ \ \ \ \ \ \ \ \cdot \ P_{[\alpha_1 N]}( T_{[\alpha_2 N]}<\tilde T_{[\alpha_1 N]}\wedge T_N)\nonumber\\
&&\ \ \ \ \ \ \ \ \ \cdot \ P_{[\alpha_2 N]}(T_N<T_{[\alpha_1 N]}, \cap_{i=2}^k \{G([\alpha_i N])=e(z_i)\})
\label{sec_3_eqn}
\end{eqnarray}
where $e(z_i)=e_i$ or $1-e_i$ if $z_i$ is even or odd respectively.
In the last factor, which deals with the parities of $k-1$ points, $[\alpha_1 N]$ can be translated to $x=0$. Treating the limit in Proposition \ref{oddprop} as a base step, we may conclude by induction, for fixed $e(z_i)$, that
$$\lim_{N\rightarrow \infty}P_{[\alpha_2 N]}(\cap_{i=2}^k \{G([\alpha_i N])=e(z_i)\}|T_N<T_{[\alpha_1 N]}) \ = \ {2}^{-(k-1)}.$$
Hence, by bounded convergence, we may replace the last factor of (\ref{sec_3_eqn}), by
$$2^{-(k-1)}P_{[\alpha_2N]}(T_N<T_{[\alpha_1N]}) + o(1)$$
as $N\uparrow\infty$. Summing over $z_2,\ldots, z_k$, we have
\begin{eqnarray*}
&&P_{[\alpha N]}(T_{[\alpha_1 N]}<T_N<T_0, \cap_{i=1}^k\{G([\alpha_i N])=e_i\})\\
&& \ \ = \ \left[\sum_{z_1 = e_1 \ {\rm mod}_2}P_{[\alpha N]}(T^{z_1}_{[\alpha_1 N]}<\tau_N)\right ]\cdot P_{[\alpha_1 N]}(T_{[\alpha_2 N]}<\tilde T_{[\alpha_1 N]}\wedge T_N)\\
&&\ \ \ \ \ \ \ \ \ \ \ \ \ \cdot [2^{-(k-1)}P_{[\alpha_2N]}(T_N<T_{[\alpha_1N]}) +o(1)] \\
&& \ \ = \ \frac{1}{2^{k-1}}P_{[\alpha N]}(G([\alpha_1 N])= e_1 \ {\rm mod}_2, T_{[\alpha_1 N]}<T_N<T_0) +o(1).
\end{eqnarray*}
Therefore, noting Proposition \ref{oddprop},
$$\lim_{N\rightarrow\infty}
P_{[\alpha N]}(\cap_{i=1}^k\{G([\alpha_i N])=e_i\}| T_{[\alpha_1 N]}<T_N<T_0) \ = \ \frac{1}{2^k}.
$$
A similar expression is derived when the conditioning event is $T_{[\alpha_k N]}<T_0<T_N$, and so the limit in Proposition \ref{thm1_question3} is recovered. \hfill$\nabla$
\vskip .1cm
{\it Proof of Proposition \ref{thm2_question3}.} The proof is easier than that for Proposition \ref{thm1_question3}.
Since the probability of backtracking, $P_{[\gamma N]}(T_{[\beta N]}<\tau_N)$ is exponentially small in $N$ for $\beta<\gamma$, and noting Proposition \ref{oddprop}, we have
\begin{eqnarray*}
P_{[\alpha N]}(\cap_{i=1}^k \{G([\alpha_i N]) = 1 \ {\rm mod}_2\}) &=&
o(1)+\prod_{i=1}^k P_{[\alpha_i N]}(G([\alpha_i N]) = 1 \ {\rm mod}_2)\\
&\rightarrow & (2-(p-q))^{-k}
\end{eqnarray*}
\hfill$\nabla$
|
{
"timestamp": "2010-09-22T02:01:03",
"yymm": "1009",
"arxiv_id": "1009.3999",
"language": "en",
"url": "https://arxiv.org/abs/1009.3999"
}
|
\section{Introduction}
Phenology, in agricultural science, is the study of periodic plant
developmental stages and their responses to climate (especially to seasonal
and interannual variations in climate) and other physical variables (e.g.
photoperiod). For example, an apple tree, in each of its development cycles,
may go through developmental events from bud-bursting, blooming to fruiting.
It is well known in the agricultural science community that climate variables,
especially daily average temperature, and possibly photoperiod are major
factors that influence the timings of phenological events. But one question remains: how to model their relationship so as to
predict the timings of future phenological events?
Many empirical biological models have been built for representing the
relationship between phenological events and climate variables (see
\citealp{Chuine2000a}, for a comprehensive review). These models are
deterministic and the values of parameters in the models are either determined
experimentally or obtained as point estimates given by least squares to yield
best fits to observed data. The uncertainties associated with these parameter estimates and predictions are often not assessed.
Statistical models have also been applied to phenology. Ordinary least square
(OLS) linear regression is widely used to study the linear association between
timings of phenological events and climate variables. Survival
data analysis techniques, such as the Cox model, have also been applied (e.g.
\citealp{Gienapp2005}). However statistical issues arising in phenological
data analysis are quite complicated. Firstly, the phenological events are
irreversible progressive events -- a tree cannot bloom without having gone
through bud-bursting and leafing, and once it blooms, it cannot repeat the
earlier stages in the same development cycle. Secondly, the climate variables
are external time-dependent covariates (i.e. covariates not influenced by the
occurrence of the events of central interest, \citealp{Kalbfleisch2002}). When
time-dependent covariates are present, OLS linear regression is not
suitable for prediction since the covariates values at future event times
are unknown. The Cox model is also not suitable for prediction because it does
not extrapolate beyond the last observation. Furthermore, the Cox model may be
subject to substantial loss of efficiency due to the strong trends in climate
covariates.
This paper presents an approach to prediction of the bloom--dates of perennial
crops based on time-dependent climate covariates using a regression model
developed by authors for progressive event history data \citep{Cai2010}. This
approach can incorporate all time-dependent covariate information, so there is no
loss of efficiency. Also, prediction is easily formulated in this framework.
Finally this approach can be applied in other areas of application, for example in medical research, to the analysis of survival data with progressive health outcomes.
The report is organized as follows. In Section \ref{sect:data} we describe the data used in our application along with its objectives. Then in Section \ref{sec:method} we describe the approach used in our analysis. That analysis is presented in Section \ref{sec:analysis}. Finally conclusions are given in Section \ref{sec:conclusions}.
\section{Data and objectives}\label{sect:data}
The data represents the bloom--dates of six high-value perennial
agricultural crops (apricot, cherry, peach, prune, pear, and apple) in
Summerland, British Columbia, Canada, recorded during the
years between 1937 and 1964 inclusive.
Each year, the blooming event occurs at most once for each crop. The bloom
date is then recorded as the number of days from the first day of a year to a
``representative'' bloom--date of all the trees in the area in that year. Daily
maximum and minimum temperatures in the same area in the corresponding years are also recorded.
Phenological studies (e.g. \citealp{Murray1989}) suggest that the occurrence
of a phenological event may be mainly influenced by the accumulation of the
so-called growing degree days (GDD),
\begin{equation}
GDD\left(t \right) = \left\{ \begin{array}{cc}
\frac{T_{min}\left(t \right)+T_{max}\left(t \right)}{2} - T_{base} &
\text{if} \enskip \frac{T_{min}\left(t \right)+T_{max}\left(t \right)}{2}
> T_{base} \\
0 & \text{otherwise} \end{array} \right., \ \text{for } t > t_0 \;,
\label{eq:GDDdef}
\end{equation}
where the time $t$ is recorded in days, $t_0$ is a well-defined temporal
origin, and $T_{min}\left(t \right)$ (respectively $T_{max}\left(t \right)$) is the daily
minimum (respectively maximum) temperature, The constant
$T_{base}$ is a unknown constant threshold
temperature. The time origin $t_0$ is usually chosen as the
start--date of a development stage \citep{Chuine2000a}. Here, for blooming, we choose $t_0$ as January $1^{st}$. Although this choice may have an impact on
the analysis we do not believe it to be significant. For as can be seen from
(\ref{eq:GDDdef}), the actual start--date is controlled by $T_{base}$. If we
choose some date earlier than that start--date, the daily average
temperatures, $\frac{T_{min}\left(t \right)+T_{max}\left(t \right)}{2}$, in
those earlier days will be smaller than $T_{base}$, and the corresponding GDDs
will be 0. Therefore, that choice will not impact the results of the analysis. For
the bloom--date, the choice of January $1^st$ may be far earlier than the
start--date of the blooming stage, and so its impact on results should be
negligible.
The results of this paper aim to answer the following questions: (1) In what form of aggregation does the GDD most influence the date of the blooming event. More specifically, is it the cumulative sum of the GDD, a weighted cumulative sum, or something else? (2) What is the value of $T_{base}$? (3) Most importantly, how should the future bloom--date be best predicted the and how should the uncertainty associated with the prediction be best assessed?
From a statistical scientist's perspective, the first question is a model selection
problem, the second, an estimation problem, and the last, a
prediction problem.
\section{Methodology} \label{sec:method}
This paper uses a regression method for a single
phenological event developed by authors \citep{Cai2010}. This method is based on a model that uses the observed (discrete) process that represents the state indicator of a phenological event. The blooming event is a single progressive event, i.e. in the development cycle of a plant, once it blooms, the plant stays in
the ``occurred'' state and cannot return to the ``not--occurred'' state. At each
time $t$, we denote the state of the event by an indicator $Y_t$, being
$0$ or $1$ according as the event has ``occurred'' or not. It can be shown
that for such an event, the process of the state indicator $Y_t$ is a Markov
chain. Let $\mathcal{X}_{t}$ be an associated time-dependent covariate vector,
and $\Prob_{t}\equiv\Prob\left(Y_t=1\lvert Y_t=0,\,\mathcal{X}_{t}\right)$,
where $\Prob\left(\cdot\right)$ is a probability set function. Then at each
time point $t$, we can consider a model for the binary event $Y_t$:
\begin{equation}
g\left(\Prob_{t}\right) = f\left(\mathcal{X}_{t};\,\beta\right) \ ,
\label{eq:regmodel}
\end{equation}
where $g: \left(0,\,1\right) \rightarrow \left( -\infty,\, \infty \right)$ is
a monotonic link function, and $f: \left( -\infty,\, \infty \right)
\rightarrow \left( -\infty,\, \infty \right)$ is a function of
$\mathcal{X}_{t}$ with parameter vector $\beta$, which encodes the
relationship between $\Prob_{t}$ and $\mathcal{X}_{t}$. Here we
take $g$ to be the logit funcion and restrict $f$ to be a linear function of $\beta$.
One then can derive the probability that a plant blooms at any time point $t$
after the time origin $t_0$. If there are $N$ independent observations, the
likelihood function of the data then can be easily written down accordingly.
Up to this point, the parameter vector $\beta$ can be estimated by the maximum
likelihood (ML) method.
After the model parameters have been estimated, the fitted model can be used to
predict future bloom--dates. For a new year, in which the bloom--date is
unknown, take the time origin $t_0$ as January $1^{st}$ and suppose the
current time is $t_c \ge t_0$, up to which the blooming event has not
occurred. Denote the unknown bloom--date of this new year as $T^*$, with
corresponding state indicator $Y^*_t$ at time $t\ge t_0$. Now, since the bloom
date of a plant usually is related to the associated climate covariates up to
the bloom--date itself, we need to predict the future values of the climate
variables first. For this purpose we fit a ARIMA time series model. Then at any
time $t\ge t_0$, we denote the covariate vector associated with $Y^*_{t}$ by
$\mathcal{X}^*_{t}$. Because we have observed the value of $\mathcal{X}^*_{t}$
up to the current time $t_c$, we may decompose $\mathcal{X}^*_{t}$ into two parts:
one part $\mathcal{X}^*_{t,\,obs}$ consists of covariates evaluated from time
0 to time $t_c$, which we have observed exactly, and the other part
$\mathcal{X}^*_{t,\,pred}$ consists of predicted covariates values from
$t_c+1$ to $t$, whose predictive distributions are given by the ARIMA model.
Treating the maximum likelihood estimate (MLE) of the parameter vector
$\hat{\beta}$ as if it were the true value of the parameter vector, we can
obtain a ``plug-in'' formula for the predictive probability that the blooming
event occurs at time $t_c+K$, for any $K>0$:
\begin{align}
&\Prob_{\hat{\beta}}\left( T^*=t_c+K \left |
\mathcal{X}^*_{t_c+K,\,obs} \right. \right)
\notag \\
=&
\int
\Prob_{\hat{\beta}}\left( T^*=t_c+K \left|
\mathcal{X}^*_{t_c+K,\,obs}, \; \mathcal{X}^*_{t_c+K,\,pred}
\right. \right)
d \Prob \left(\mathcal{X}^*_{t_c+K,\,pred} \right)
\notag \\
=&
\int g^{-1}\left(f \left(\mathcal{X}^*_{t_c+K}; \, \hat{\beta}\right)\right)
\prod_{s=1}^{K-1}
\left( 1-g^{-1}\left(f
\left(\mathcal{X}^*_{t_c+s}; \, \hat{\beta}\right) \right)\right)
d \Prob \left(\mathcal{X}^*_{t_c+K,\,pred} \right)
\ .
\label{eq:preddistplugin}
\end{align}
Generate a sample of large size $L$ (e.g. thousands or more) from the
predictive
distribution of $\mathcal{X}^*_{t_c+K,\,pred}$, and denote the sample points
as $\mathcal{X}^*_{t_c+K,\,pred}\left(l \right)$ ($l=1,\,\cdots,\,L$). The
above predictive probability then can be approximated by Monte Carlo (MC)
integration,
\begin{equation}
\Prob_{\hat{\beta}} \left( T^*=t_c+K \lvert \mathcal{X}^*_{t_c+K,\, obs}\right)
\approx \frac{1}{L}\sum_{l=1}^L \Prob_{\hat{\beta}}\left( T^*=t_c+K \left|
\mathcal{X}^*_{t_c+K,\,obs}, \; \mathcal{X}^*_{t_c+K,\,pred}\left(l
\right) \right. \right) \ .
\label{eq:predictivedMC}
\end{equation}
Note that in this ``plug-in'' approach, the uncertainties associated with the unknown parameters are not taken into account. However one may use a re-sampling method such as the bootstrap to assess their effect.
\citet{Cai2010} also provides a regression model for multiple phenological
events, which will not be needed here.
\section{Analysis}\label{sec:analysis}
In this section, we apply the method described above to the bloom--dates of the
six different crops separately, and present the results of the analysis.
\subsection{Assumptions}
For each crop, the bloom--date over years is a time series. However, sample
auto-correlations suggest that the auto-correlations of the bloom--dates over
years are negligible for all six crops. We therefore assume that for each
crop, the bloom--dates of different years are independent realizations from the
same population.
\subsection{The relationship between bloom--dates and GDD}
As mentioned above, scientists believe that the bloom--dates are related
to the accumulation of GDD. In particular, empirical results suggest
that it is the AGDD, the cumulative sum of GDD starting from the time
origin $t_0$, that most influences the timing of the blooming event
(e.g. \citealt{Chuine2000a}, \citealt{Murray1989}):
\begin{equation}
\text{AGDD} \left(t \right) = \sum_{k=t_0}^{t} \text{GDD}\left(k \right) \ ,
\label{eq:AGDD}
\end{equation}
where $t_0$ is time origin and $t\ge t_0$ is recorded in days, the same time scale as that of the GDD.
Here we seek by statistical means, the form of GDD aggregation that best
models our data. For example, one might conjecture that
the GDD evaluated at times near the bloom--date
would predict the blooming
event better than those in the past. For this model, using a weighted sum of GDDs with weights increasing over time as a covariate may plausibly yield a better model fit than using AGDD, an unweighted sum of GDDs over time, as a covariate. To investigate this conjecture, we fitted the regression model described in Section \ref{sec:method} to the data with bloom--date as the response,
and we consider the following alternative ways of incorporating GDD as a covariate
$\mathcal{X}_{t}$:
\begin{description}
\item [Model AGDD] Take $f \left( \mathcal{X}_{t}; \; \beta \right)$ in
Equation (\ref{eq:regmodel}) as a linear function of AGDD evaluated at the
current time $t$:
\begin{equation}
f \left( \mathcal{X}_{t}; \; \beta \right) =
a + b \text{AGDD}\left(t\right)
= a + b\sum_{k=t_0}^t \text{GDD}\left(k\right) \ ,
\end{equation}
where $\mathcal{X}_{t} = \text{AGDD}$ and the subscript $T$ stands
for the transpose of a vector or matrix.
\item [Model ExpSmooth] Take $f \left( \mathcal{X}_{t}; \; \beta \right)$ as
a linear function of a weighted sum of GDD from the time origin $t_0$ to
the current time $t$:
\begin{equation}
f \left( \mathcal{X}_{t}; \; \beta \right) = a +
b\sum_{k=0}^{t-t_0} \left(1 - \gamma\right)^k \text{GDD}\left(t-k\right)
\ ,
\label{eq:3expsmooth}
\end{equation}
where $\mathcal{X}_{t} = \sum_{k=0}^{t-t_0} \left(1 - \gamma\right)^k
\text{GDD}\left(t-k\right)$, $\beta=\left(a, \;b, \;\gamma,
\;T_{base}\right)^T$, and $0\le\gamma\le1$. We call this model
``ExpSmooth'' because the weighted average term is similar to the
exponential smoothing used in time series \citep{Chatfield2004}.
\item [Model GDD] Take $f \left( \mathcal{X}_{t}; \; \beta \right)$ as a linear function of GDD evaluated
at the current time $t$:
\begin{equation}
f \left( \mathcal{X}_{t}; \; \beta \right) =a + b
\text{GDD}\left(t\right) \ ,
\end{equation}
where $\mathcal{X}_{t} = \text{GDD}\left(t\right)$ and $\beta=\left(a,
\;b, \;T_{base}\right)^T$.
\item [Model 5Days] Take $f \left( \mathcal{X}_{t}; \; \beta \right)$ as a linear function of the GDD evaluated at the 5
most recent days:
\begin{equation}
f \left( \mathcal{X}_{t}; \; \beta \right) =a + \sum_{k=1}^5 b_k
\text{GDD}\left(t-k+1\right) \ ,
\end{equation}
where $\mathcal{X}_{t} = \big(\text{GDD}\left(t\right),
\;\text{GDD}\left(t-1\right), \;\text{GDD}\left(t-2\right),
\;\text{GDD}\left(t-3\right), \;\text{GDD}\left(t-4\right) \big)^T$ and
$\beta=\left(a, \;b_1, \;b_2, \;b_3, \;b_4, \;b_5, \;T_{base}\right)^T$
\end{description}
Note that in each of the above models, $T_{base}$ is a parameter
included in the expression of GDD.
Model AGDD incorporates the empirical results referred to above,
and it serves as a basis of
comparison. Model GDD is used to assess whether the probability of blooming is influenced mainly by the GDD evaluated at the current time. Model 5Days tests
the theory that the GDD evaluated at each of many time points prior to the
current time might be important predictors,
each having a different effect on $P_{t}$. In the latter model, we give each GDD evaluated at several days
prior to and at the current day a different regression coefficient. However
we consider only GDD evaluated at the five most recent days, because given 28
years of bloom--dates, we won't be able to get good estimates of model
parameters, if the number of parameters is too large.
We found the most promising model to be Model ExpSmooth. This model, $f \left(
\mathcal{X}_{t}; \; \beta \right)$ is a linear function of the weighted sum of
GDD evaluated from the time origin $t_0$ to the current time $t$. For a fixed
$\gamma$ ($0\le\gamma\le1$), $\left(1-\gamma\right)^k$, the weight on
the GDD at lag $k$, the number of days prior to the current date, decays as $k$
increases. This reflects the idea that the GDD evaluated at recent time points
contribute more to the probability of occurrence of the blooming event at the
current time than do the GDD evaluated at time points long before the current
time $t$.
In Model ExpSmooth, the value of $\gamma$ controls the speed of with which the weight decays. Figure \ref{fig:3expsmooth} shows how the weight decays when the lag
increases for different values of $\gamma$.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.6]{expsmooth.pdf}
\caption{The actual weights in the weighted sum in Model ExpSmooth for
different $\gamma$ parameter values. The weight decays when the lag
(number of days prior to the current date) increases. A larger $\gamma$
corresponds to a faster speed of decaying.}
\label{fig:3expsmooth}
\end{center}
\end{figure}
When $\gamma$ becomes larger, the weight decays faster. In the extreme case of
$\gamma=1$, the weighted sum is just GDD evaluated at the current time and so
Model ExpSmooth becomes Model GDD. If $\gamma$ becomes smaller, the weight
decays slower. In the extreme case of $\gamma=0$, i.e. no decay, Model
ExpSmooth becomes Model AGDD. In other words, Model GDD and AGDD are only
special cases of Model ExpSmooth. The value of $\gamma$, however, is not known
in advance. Thus, we treat it as a model parameter, and estimate it using the
maximum likelihood estimator (MLE).
\begin{table}[!ht]
\begin{center}
\caption{BICs of the fitted models}
\label{tab:BIC}
\begin{tabular}{|l|cccc|}
\hline
& AGDD & ExpSmooth & GDD & 5Days \\
\hline
Apricot &148.70 &147.53 &249.65 &239.01 \\
Cherry &163.69 &157.41 &255.72 &243.73 \\
Peach &146.68 &146.49 &255.06 &255.25\\
Prune &142.79 &141.91 &224.14 &232.26\\
Pear &132.57 &133.14 &231.19 &224.63\\
Apple &126.31 &128.79 &265.11 &254.44\\
\hline
\end{tabular}
\end{center}
\end{table}
For every crop, we fit all the above models to the data, and compare the
Bayesian information criterion (BIC) for each with the results in
Table \ref{tab:BIC}. Clearly, for all crops Model AGDD and
ExpSmooth are essentially equivalent and both are
much better than the other two models. The
estimated smoothing parameter, $\hat\gamma$, in Model ExpSmooth for different crops
are shown in Table \ref{tab:gamma}.
\begin{table}[!ht]
\begin{center}
\caption{Estimated smoothing parameter $\gamma$ of Model ExpSmooth for
different crops}
\label{tab:gamma}
\begin{tabular}{|cccccc|}
\hline
Apricot & Cherry & Peach & Prune & Pear & Apple \\
\hline
0.014 & 0.020 & 0.0083 & 0.016 & 0.023 & 0.0036 \\
\hline
\end{tabular}
\end{center}
\end{table}
All $\hat\gamma$'s are very small, which suggests that the
weights in the fitted Model ExpSmooth's decay very slow, and therefore that the
fitted Model ExpSmooth resembles the fitted Model AGDD for our data.
This statistical result supports scientists' experimental result: the
accumulation of GDD is roughly in the form of a sum with equal weights, in other words the AGDD.
Although the quality of the models, ExpSmooth and Model AGDD, seem roughly equivalent, we study only Model AGDD for the following reasons: (1) it has been the traditional choice; (2) it is more parsimonious, having one less parameter, making it preferable for the small samples we need to deal with.
\subsection{Estimating model parameters} \label{subsec:est}
The estimates of parameters in Model AGDD in the above discussion were obtained by maximum likelihood (ML) and these are shown in Table \ref{tab:estPar}.
\begin{table}[!ht]
\begin{center}
\caption{Estimated parameters of Model AGDD} \label{tab:estPar}
\begin{tabular}{|l|ccc|}
\hline
Model & $\hat{a}$ & $\hat{b}$ & $\hat{T}_{base}$ \\
\hline
Apricot & -13.49 & 0.061 & 2.65 \\
Cherry & -11.72 & 0.043 & 3.35 \\
Peach & -19.67 & 0.043 & 0.38 \\
Prune & -18.23 & 0.057 & 2.80 \\
Pear & -22.27 & 0.07 & 2.97 \\
Apple & -26.77 & 0.07 & 2.82 \\
\hline
\end{tabular}
\end{center}
\end{table}
Are these estimated parameters close to the true values of the
parameters? \citet{Wald1949} gave famous sufficient conditions that would ensure that at least as the sample size approaches infinity, the ML estimates
converge to their true counterparts, a property called consistency. These conditions in turn lead to others that ensure not only consistency but as well, other desirable properties, namely an approximately normal distribution and asymptotic efficiency. However, one of those conditions requires that the likelihood function be a continuous function of parameters while the definition
of the GDD (\ref{eq:GDDdef}) ensures that the likelihood function for Model AGDD is not a continuous function of $T_{base}$. Thus we cannot apply Wald's theory and we use a different approach to assess estimator quality, namely simulation.
In the simulation study, we generate data as follows. First, we get one year
of long term averaged daily average temperature series by taking the average of the daily average temperature from year 1916 to 2005 in
the Okanagan region of British Columbia for each day of a year.
We then add a noise process to this long term averaged series. This noise
process is generated from an $ARMA(3,\,1)$ model:
\begin{equation}
X_t= 1.83X_{t-1} - 0.96X_{t-2} + 0.12X_{t-3} + Z_t -0.96Z_{t-1}
\label{eq:arima}
\end{equation}
where the white noise $Z$ has a normal distribution with mean 0 and variance
5.253 for any $t$. This ARMA model is fitted to the same daily average
temperature used for extracting long term averaged series above. Now we get
one year of simulated daily average temperature data. We calculate the GDD of the generated temperature data with parameter
$T_{base}=3.5$. Now starting from day 1, we generate
a random number $Y_1$ from a Bernoulli distribution $Ber\left(p\right)$ with
parameter $p=logit^{-1}\left( -13 + 0.04\sum_{k=1}^1\text{GDD}_{k}\right)$. If $Y_1=0$,
we generate
\begin{equation}
Y_2 \sim Ber\Big(logit^{-1} \big( -13 + 0.04\sum_{k=1}^2X_{k} \big)
\Big) \ .
\end{equation}
Again, as long as $Y_2=0$, we will generate $Y_3$ similarly, and so on
until we get a 1 at time $t$. This $t$ is the simulated bloom--date. Using this procedure, we
generate one year of GDDs and a bloom--date for that year, as one year of
data.
Now we generate 30 years of data as one sample (i.e. a sample of size 30), and
we generate 1000 such samples. For each sample $i$ ($i=1,\,\cdots,\,1000$), we
apply Model AGDD, and calculate the MLEs of the model parameters: $\hat{a}_i$,
$\hat{b}_i$, and ${\hat{T}_{base}}_i$. For each parameter, say
$a$, we calculate the estimated mean of the MLE,
\begin{equation}
\bar{\hat{a}} = \frac{1}{1000}\sum_{i=1}^{1000} \hat{a}_i \ ,
\end{equation}
the estimated variance of the MLE,
\begin{equation}
\frac{1}{1000-1}\sum_{i=1}^{1000} \left(\hat{a}_i-\bar{\hat{a}} \right)^2 \ ,
\end{equation}
and the standard error of the mean of the MLE,
\begin{equation}
\sqrt{\frac{
\frac{1}{1000-1}\sum_{i=1}^{1000} \left(\hat{a}_i-\bar{\hat{a}} \right)^2
}{1000}} \ ,
\end{equation}
which characterize how well the estimated mean approximate the true mean of the
MLE.
We repeat the above procedure for sample sizes $S$ of 80, 150 and 400. If
the MLEs were consistent, we would be able to see as the sample size becomes larger, that for each parameter the estimated mean comes closer to the true
value of the parameter, while the estimated variance becomes smaller. Table
\ref{tab:consMean} shows that when the sample size increases, the estimated
means of the MLEs of $a$ and $b$ become closer to the true parameters values
$a=-13$ and $b=0.04$. When the sample size reaches 400, the estimated means
are basically the true values. For parameter $T_{base}$, the estimated means
using different sample sizes are all fairly close to the true value of
$T_{base}=3.5$. Standard errors of the means (Table \ref{tab:consErr}) show
that these estimated means are reliable. On the other hand, when the sample
size increases, the estimated variances (Table \ref{tab:consVar}) of all
parameters become smaller. These facts suggest that in Model AGDD, the MLEs of
all parameters might be consistent.
\begin{table}[ht]
\begin{center}
\caption{Estimated means of the MLEs. When the sample sizes increases, the
estimated means become closer to the true parameter values of $a=-13$,
$b=0.04$ and $T_{base}=3.5$}
\label{tab:consMean}
\begin{tabular}{ccccc}
\hline
& $S=30$ & $S=80$ & $S=150$ & $S=400$ \\
\hline
$\hat{a}$ & -13.82 & -13.23 & -13.20 & -13.07 \\
$\hat{b}$ & 0.043 & 0.041 & 0.041 & 0.040 \\
$\hat{T}_{base}$ ($\degree \text{C}$) & 3.50 & 3.50 & 3.48 & 3.51 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[ht]
\begin{center}
\caption{Standard errors of the means the MLEs. Small standard errors imply
that the estimated means of MLEs are reliable.}
\label{tab:consErr}
\begin{tabular}{ccccc}
\hline
& $S=30$ & $S=80$ & $S=150$ & $S=400$ \\
\hline
$\hat{a}$ & 0.066 & 0.035 & 0.026 & 0.015 \\
$\hat{b}$ & 0.0002 & 0.0001 & 0.0001 & 0.0001 \\
$\hat{T}_{base}$ ($\degree \text{C}$) & 0.027 & 0.015 & 0.010 & 0.0065 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[ht]
\begin{center}
\caption{Estimated variances of the MLEs. When the sample size increases,
the estimated variances become smaller.}
\label{tab:consVar}
\begin{tabular}{ccccc}
\hline
& $S=30$ & $S=80$ & $S=150$ & $S=400$ \\
\hline
$\hat{a}$ & 4.31 & 1.20 & 0.66 & 0.22 \\
$\hat{b}$ & 0.0001 & 0.0000 & 0.0000 & 0.0000 \\
$\hat{T}_{base}$ ($\degree \text{C}$) & 0.70 & 0.21 & 0.11 & 0.04 \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Assessing the uncertainty of the MLEs}
As noted above, we cannot use standard asymptotic results (e.g. in \citealp{Cox1979}) to find large sample approximations to the standard errors of parameter estimators, forcing us to use an alternative approach. The one we choose, the bootstrap \citep{Efron1994boot} if valid would allow us to not only estimate the standard error of the MLEs but as well to find quantile based confidence intervals for the model parameters.
However we know of know general theory that would imply that validity in this particular application, leading us to again resort to simulation to explore this issue.
Using the simulated data seen in section \ref{subsec:est},
for each different sample size $S$, we estimate the true variances of the
MLEs by the sample variances of the MLEs obtained using the 1000 samples. The
standard deviation of the MLEs is then estimated by the square root of these
sample variances. The results are shown in the ``Sim.'' fields in Table
\ref{tab:bootEstSd}. We can then see how the bootstrap estimates the
standard deviations of the MLEs compare with the corresponding estimates obtained from the simulated data.
To get these results, we randomly chose for each different sample size, one sample from the 1000 simulated samples and then 1000 bootstrap samples from this one sample of response and predictor pairs. For each such bootstrap sample, we calculated the MLEs of the parameters. For each parameter, we then took the
square root of the sample variance of the MLEs obtained from the 1000
bootstrap samples, as the bootstrap estimate of the standard error of the
MLE for that parameter. The results are shown in the ``Boot.'' fields in Table
\ref{tab:bootEstSd}. We can see that for each parameter, when the sample size
becomes large, the bootstrap estimates and the estimates obtained using the
simulated data both become small. The bootstrap estimates are always larger
than the estimates obtained from the simulated data, but when the sample size
gets large, the difference between them becomes small. In fact, for a sample size
of 400, the two estimates are fairly close. This may suggest the bootstrap estimates do converge to the true standard deviations of the MLEs,
although the rate of convergence seems low.
\begin{table}[ht]
\begin{center}
\caption{Comparison of bootstrap estimates of the standard deviations of the
MLEs and the estimated standard deviations using simulated data. ``Boot.''
stands for the bootstrap estimates; ``Sim.'' stands for the estimates
obtained using simulated data. As the sample size increases, the estimated
standard deviations calculated using the two different approaches become
smaller and also closer.}
\label{tab:bootEstSd}
\begin{tabular}{|l|cc|cc|cc|cc|}
\hline
& \multicolumn{2}{c|}{$S=30$} & \multicolumn{2}{c|}{$S=80$}
& \multicolumn{2}{c|}{$S=150$} & \multicolumn{2}{c|}{$S=400$} \\
\hline
& Boot. & Sim. & Boot. & Sim. & Boot. & Sim. & Boot. & Sim. \\
\hline
$\hat{a}$ & 2.23 & 2.08 & 1.55 & 1.10 & 0.71 & 0.81 & 0.54 & 0.46 \\
$\hat{b}$ & 0.0102 & 0.0076 & 0.0050 & 0.0041 & 0.0034 & 0.0031 & 0.0017 &
0.0018 \\
$\hat{T}_{base} ($\degree \text{C}$)$ & 1.36 & 0.84 & 0.49 & 0.46 & 0.27 & 0.33 & 0.25 & 0.21 \\
\hline
\end{tabular}
\end{center}
\end{table}
We also need to obtain 95\% confidence intervals for the model parameters.
Using the MLEs obtained from the simulated data, we can get quantile-based
confidence intervals for the model parameters. We also can calculate
quantile-based bootstrap confidence intervals using MLEs obtained from the
bootstrap samples of one simulated sample. The results are shown in Table
\ref{tab:bootEstCI}. We see that for each parameter, the lengths of confidence
intervals obtained by the two approaches are roughly the same, and as sample
size gets larger, they both become smaller. However, the confidence intervals
obtained using the two different approaches do not always agree -- the
bootstrap intervals seem to always have a bias. Fortunately, when the sample
size is large, the difference between the two kinds of intervals is pretty
small -- it is alway smaller than 1/20 of the length of the confidence
interval obtained from the simulated data when sample size is 400. We tried a
bias corrected version of quantile based bootstrap confidence interval (``BC''
method in \citealt{Efron1986}), but the results are even slightly worse than
this raw version. Overall, although a small bias may exist, use the quantile-based
bootstrap confidence interval seems reasonable in our application.
\begin{table}[ht]
\begin{center}
\caption{Comparison of quantile-based 95\% confidence intervals based on
bootstrap and simulated data. ``Boot.'' stands for the bootstrap estimates;
``Sim.'' stands for the estimates obtained using the simulated data. As the
sample size increases, the confidence intervals calculated using the two
different approaches both become smaller, but they do not always agree very
well.}
\label{tab:bootEstCI}
\small
\begin{tabular}{|l|c|c|c|c|}
\hline
& $S=30$ & $S=80$ & $S=150$ & $S=400$ \\
\hline
$a$ (Boot.) & (-17.44, -10.25) & (-17.94, -12.54) & (-13.96, -11.47) &
(-14.31, -12.51) \\
$a$ (Sim.) & (-17.60, -10.48) & (-15.16, -11.38) & (-14.63, -11.75) &
(-13.82, -12.20) \\
\hline
$b$ (Boot.) & (0.034, 0.066) & (0.036, 0.054) & (0.038, 0.050) & (0.036,
0.042) \\
$b$ (Sim.) & (0.033, 0.061) & (0.035, 0.050) & (0.036, 0.047) & (0.038,
0.044) \\
\hline
$T_{base} ($\degree \text{C}$)$ (Boot.) & (2.45, 5.43) & (2.02, 3.79) & (3.64, 4.60) & (2.82,
3.58) \\
$T_{base} ($\degree \text{C}$)$ (Sim.) & (2.14, 5.13) & (2.72, 4.39) & (2.95, 4.12) & (3.16,
3.90) \\
\hline
\end{tabular}
\end{center}
\end{table}
Table \ref{tab:bootEstRange} shows the observed range (minimum value to
maximum value) of the bootstrap MLEs. We see that for each parameter and all
the four choices of the sample sizes $S$, this range covers and is much larger
than the 95\% confidence interval obtained using the simulated data. Without
knowing the actual coverage probability, this range cannot be directly used as
a confidence interval. However, the usefulness of it is that if this range
does not contain a value, say $\theta_0$, then we get stronger evidence of
saying that the parameter value is not $\theta_0$ than the possibly biased
95\% bootstrap confidence interval not containing $\theta_0$.
\begin{table}[ht]
\begin{center}
\caption{Observed ranges of the bootstrap MLEs. These ranges always contain
the quantile-based 95\% confidence intervals based on the simulated data.}
\label{tab:bootEstRange}
\small
\begin{tabular}{|l|c|c|c|c|}
\hline
& $S=30$ & $S=80$ & $S=150$ & $S=400$ \\
\hline
$\hat{a}$ & (-36.65, -6.43) & (-21.67, -8.55) & (-15.50, -8.98) & (-15.17,
-7.68) \\
$\hat{b}$ & (0.0055, 0.1527) & (0.0290, 0.0652) & (0.0324, 0.0575) &
(0.0342, 0.0453) \\
$\hat{T}_{base} ($\degree \text{C}$)$ & (-19.00, 11.44) & (0.90, 6.71) & (3.11, 6.52) & (2.51,
7.09) \\
\hline
\end{tabular}
\end{center}
\end{table}
The bootstrap estimates of quantile-based 95\% confidence intervals of the
MLEs of Model AGDD for all crops are shown in Table \ref{tab:bootCI}. Given
the data, we are interested in knowing whether the regression coefficients $a$
and $b$ are significantly different from 0. Since none of the 95\% bootstrap
confidence intervals of $a$ and $b$ contains 0, we have a strong evidence that
neither $a$ and $b$ are 0 for all crops. The observed ranges of the bootstrap
MLEs (Table \ref{tab:bootRange}) also support this conclusion -- they do not
cover 0.
\begin{table}[ht]
\caption{Quantile-based 95\% bootstrap confidence intervals for the model
parameters}
\label{tab:bootCI}
\begin{center}
\begin{tabular}{lccc}
\hline
& $a$ & $b$ & $T_{base}$ \\
\hline
Apricot & (-22.43, -12.07) & (0.051, 0.096) & (0.95, 4.00) \\
Cherry & (-21.18, -10.72) & (0.030, 0.095) & (1.01, 5.15) \\
Peach & (-31.69, -16.37) & (0.030, 0.065) & (-2.51, 1.53) \\
Prune & (-31.04, -14.39) & (0.046, 0.093) & (0.18, 4.70) \\
Pear & (-37.95, -16.62) & (0.055, 0.122) & (1.93, 3.81) \\
Apple & (-39.54, -14.66) & (0.060, 0.111) & (1.84, 6.95) \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[ht]
\caption{Observed ranges of the bootstrap MLEs}
\label{tab:bootRange}
\begin{center}
\begin{tabular}{lcc}
\hline
& $\hat{a}$ & $\hat{b}$ \\
\hline
Apricot & (-31.46, -9.49) & (0.029, 0.142) \\
Cherry & (-36.13, -7.56) & (0.015, 0.150) \\
Peach & (-40.64, -6.86) & (0.0090, 0.0871) \\
Prune & (-40.48, -7.48) & (0.018, 0.178) \\
Pear & (-62.80, -6.85) & (0.023, 0.170) \\
Apple & (-50.27, -8.90) & (0.034, 0.169) \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Prediction}
To use Model AGDD to predict the future bloom--dates of a crop, we need to
first predict the future daily average temperature $\left(
T_{min}\left(t\right) + T_{max}\left(t \right)\right)/2$. Here, we use the
ARIMA model (\ref{eq:arima}) to generate future temperatures. The parameters in
the ARMIA model are estimated by fitting the model to the seasonality-removed
daily average temperature series from year 1916 to 2005 in the Okanagan region
of British Columbia. The order of this ARIMA model is selected by comparing
the BICs of fitted ARIMA$\left(p,\,d,\,q\right)$ models with different orders:
$p$ and $q$ range from 0 to 6, and $d$ ranges from 0 to 4.
To see how well the daily average temperatures generated from the
fitted ARIMA model approximate the truth, we use diagnostic plots. Figure \ref{fig:acfpacf} shows the plots of the sample
auto-correlation function (ACF) and the partial ACF \citep{Chatfield2004} of
the seasonality-removed series and simulated seasonality-removed series. These
two series have similar correlation structures. However, the ACF and partial
ACF do not uniquely determine a time series. The time series plots of the two
series are shown in Figure \ref{fig:tempts}. We see that, the magnitudes of
variations in the seasonality-removed series are not symmetric about 0. At
some time points, the seasonality-removed series have exceptionally low
values, which we do not observe in the simulated seasonality-removed series.
The cause of this difference might be that we didn't account for the periodic
signals other than seasonal variation in the ARMIA model, or may be that the
noise in the seasonality-removed series are not inherently normal, issues to
be addressed in future work.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.6]{acfpacf.pdf}
\caption{The sample ACF and PACF plots of the observed seasonality-removed
daily average temperature series and simulated seasonality-removed daily
average temperature series. The simulated seasonality-removed series have
similar sample ACF and PACF as the observed seasonality-removed series.}
\label{fig:acfpacf}
\end{center}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.6]{tempts.pdf}
\caption{Time series plots of the observed seasonality-removed daily
average temperature series and simulated seasonality-removed daily average
temperature series. The magnitudes of variations in the observed
seasonality-removed series do not match those in the simulated
seasonality-removed series very well.}
\label{fig:tempts}
\end{center}
\end{figure}
We now consider the prediction of bloom--dates. At the end of the current year, we generate 1000 series of the daily average temperatures of the whole next year
using the fitted ARIMA model. For each crop, we then use
(\ref{eq:predictivedMC}) to obtain the probability of blooming on each
successive day of the next year. This way we get a discrete predictive
distribution for bloom--date of the new year.
Now suppose that we are at the end of the first day of the new year with its
observed average daily temperature. We apply the fitted ARIMA model again to
generate 1000 series of temperatures starting from the second day of the new
year to the end of the year. We can then get another predictive distribution
for the bloom--date of the new year. We repeat this procedure on each
successive day, until the true bloom--date, at which time prediction ceases.
If the true bloom--date of the new year were day 120 for example, we would get
120 successive predictive distributions. What we expect to see are
increasingly more accurate predictions as the days progress toward the
bloom--date and more and more information about the daily averages
temperatures come to hand for that season. Growing confidence in that
prediction would provide an increasingly strong basis for management
decisions.
To see if our expectaions are realized, we perform a leave-one-out prediction
procedure -- at every step, for each crop, leave out one year of data for
prediction assessment and use the remaining years for training the model. Let's
consider apple as an example. For each left-out year, we follow the above
prediction scenario. We thus get 28 years (1937--1964) of assessments, with a
total of 3643 predictive distributions. For each of these predictive
distributions, we calculate the mean, median and mode as possible candidates
for point predictions of the new bloom--date. Also, we calculate a quantile based
95\% prediction interval (PI) for the new bloom--date. With all the 3643
predictive distributions, we can then estimate the root mean square error
(RMSE) and the mean abosolute errors (MAE) of the point predictions, and the
coverage probability and the average length of the 95\% PI. The results are
shown in Table \ref{tab:predscore}.
\begin{table}[!ht]
\begin{center}
\caption{Cross validation results: The RMSEs and MAEs for point predictions
using mode, median and mean are shown in column 2--7. The estimated
coverages and average lengths of the 95\% PIs are shown in the last two
column respectively. The units for RMSE, MAE and average length of the 95\%
PI are day. The estimated coverage probabilities of these 95\% PIs are
generally too high.}
\label{tab:predscore}
\begin{tabular}{|l|cc|cc|cc|cc|}
\hline
&\multicolumn{2}{c|}{Mode} &\multicolumn{2}{c|}{Median}
&\multicolumn{2}{c|}{Mean} &\multicolumn{2}{c|}{95\% PI} \\
\raisebox{1.0ex}[0cm][0cm]{Crop}
& RMSE & MAE & RMSE & MAE & RMSE & MAE
& Coverage & Ave. Len. \\
\hline
Apricot & 6.74 & 5.12 & 6.58 & 5.06 & 6.62 & 5.14 & 0.99 & 33.24 \\
Cherry & 6.76 & 4.97 & 6.59 & 4.88 & 6.59 & 4.92 & 0.99 & 34.29 \\
Peach & 5.43 & 4.09 & 5.33 & 4.04 & 5.34 & 4.05 & 0.99 & 28.41 \\
Prune & 5.82 & 4.31 & 5.45 & 4.11 & 5.46 & 4.16 & 0.99 & 30.55 \\
Pear & 5.60 & 4.19 & 5.65 & 4.36 & 5.69 & 4.40 & 0.99 & 29.99 \\
Apple & 5.39 & 4.07 & 5.44 & 4.19 & 5.45 & 4.23 & 0.99 & 28.86 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[ht]
\begin{center}
\caption{Maximum, minimum and range of the observed bloom--dates for each crop
in 1937--1964 in the Okanagan region}
\label{tab:datarange}
\begin{tabular}{|l|cccccc|}
\hline
& Apricot & Cherry & Peach & Prune & Pear & Apple \\
\hline
Maximum (day) & 126 & 136 & 135 & 138 & 139 & 146 \\
Minimum (day) & 94 & 102 & 105 & 111 & 110 & 115 \\
Range (day) & 32 & 34 & 30 & 27 & 29 & 31 \\
\hline
\end{tabular}
\end{center}
\end{table}
We also provide in Table \ref{tab:datarange} the observed range of the bloom
dates (the difference time between the maximal and minimal observed bloom
dates) of each crop as a measure of natural variation of the bloom--dates for
each crop. The mean, median and mode as point predictions perform roughly the
same in terms RMSE and MAE. The RMSEs for all crops fall between 5.3 and 6.8
days, and the MAEs fall between 4.0 to 5.2 days. Considering the observed
ranges of the bloom--dates, which vary from 27 to 34, our point predictions
provide more useful information about the future bloom--dates. The estimated
coverage probabilities of 95\% PIs are too high for all crops, relative to
the expected 95\%. For each crop, the average length of the 95\% PI
is roughly the same as the observed range of the bloom--dates, in accord with
the high estimated coverage probability.
These imply that our 95\% PIs
incorporate too much uncertainty,
possibly because that in the ARIMA model, we have included too much
variability caused by periodic signals other than seasonal variation as the
variability of the random noise.
We therefore reduce the variance of the white noise in the ARIMA model to half
the estimated value, while keeping all the other estimated parameter
unchanged. We use this new ARIMA model to generate daily average
temperatures, and perform the above cross validation procedure again. The
results (Table \ref{tab:predscoreInnoP4}) show that while the accuracy of the
point predictions is roughly the same as before, the estimated coverage
probabilities and average lengths of the 95\% PIs are reduced to reasonable
values. This result does not confirm that the high estimated coverage
probabilities are actually caused by the high uncertainty in the predicted
daily average temperatures, but it at least adds weight to this explanation.
\begin{table}[!ht]
\begin{center}
\caption{Cross validation results when using variance reduced simulated
daily average temperatures: The RMSEs and MAEs for point predictions using
mode, median and mean are shown in column 2--7. The estimated coverages and
average lengths of the 95\% PIs are shown in the last two column
respectively. The units for RMSE, MAE and average length of the 95\% PI are
day. The estimated coverage probabilities of these 95\% PIs are reasonable.}
\label{tab:predscoreInnoP4}
\begin{tabular}{|l|cc|cc|cc|cc|}
\hline
&\multicolumn{2}{c|}{Mode} &\multicolumn{2}{c|}{Median}
&\multicolumn{2}{c|}{Mean} &\multicolumn{2}{c|}{95\% PI} \\
\raisebox{1.0ex}[0cm][0cm]{Crop}
& RMSE & MAE & RMSE & MAE & RMSE & MAE
& Coverage & Ave. Len. \\
\hline
Apricot & 6.91 & 5.20 & 6.78 & 5.12 & 6.72 & 5.08 & 0.94 & 24.87 \\
Cherry & 6.62 & 5.06 & 6.64 & 5.05 & 6.58 & 4.99 & 0.93 & 26.46 \\
Peach & 5.56 & 4.03 & 5.51 & 3.95 & 5.49 & 3.98 & 0.95 & 21.17 \\
Prune & 5.48 & 4.16 & 5.46 & 4.04 & 5.46 & 4.07 & 0.98 & 22.38 \\
Pear & 5.98 & 4.46 & 5.79 & 4.33 & 5.75 & 4.31 & 0.94 & 21.45 \\
Apple & 5.76 & 4.36 & 5.53 & 4.17 & 5.48 & 4.15 & 0.95 & 20.36 \\
\hline
\end{tabular}
\end{center}
\end{table}
The above results for predictions derive from two models: Model AGDD for
blooming event and the ARIMA model for daily average temperature. To check the
performance of Model AGDD solely, we assume all the future daily average
temperatures are known, and then perform the above leave-one-out procedure
again. The results are reported in Table \ref{tab:predscoreknowntemp}. Note
that, in this case, since the uncertainty of the future daily average
temperatures is totally eliminated, we can only get one predictive
distribution for each test year. Therefore we cannot give a sensible estimate
for the coverage probability of the 95\% PIs. But we do see that the accuracies of
these point predictions are much higher than those of our previous
predictions, and the average lengths of the 95\% PIs are much smaller.
Although these are no longer real predictions, the results tend to validate
our Model AGDD for blooming event. Also, this finding shows the importance of
accurately modeling the covariate series and points to the need to improve the
temperature forecasting models.
\begin{table}[!ht]
\begin{center}
\caption{Cross validation results if future daily average temperatures were
known: The RMSEs and MAEs for point predictions using mode, median and
mean are shown in column 2--7. The average lengths of the 95\% PI are shown
in the last column. The units for RMSE, MAE and average length of the 95\%
PI are day. The point predictions are very accurate, and the average lengths
of the 95\% PIs are short.}
\label{tab:predscoreknowntemp}
\begin{tabular}{|l|cc|cc|cc|c|}
\hline
&\multicolumn{2}{c|}{Mode} &\multicolumn{2}{c|}{Median}
&\multicolumn{2}{c|}{Mean} & 95\% PI \\
\raisebox{1.0ex}[0cm][0cm]{Crop}
& RMSE & MAE & RMSE & MAE & RMSE & MAE
& Average Length \\
\hline
Apricot & 4.18 & 3.30 & 3.65 & 2.93 & 3.62 & 2.90 & 13.48 \\
Cherry & 3.74 & 2.93 & 3.39 & 2.43 & 3.46 & 2.42 & 18.43 \\
Peach & 3.43 & 2.85 & 3.35 & 2.78 & 3.27 & 2.76 & 12.67 \\
Prune & 3.06 & 2.54 & 2.98 & 2.50 & 2.93 & 2.36 & 11.46 \\
Pear & 2.93 & 2.11 & 2.64 & 1.89 & 2.67 & 2.00 & 9.21 \\
Apple & 2.12 & 1.86 & 2.04 & 1.71 & 1.97 & 1.61 & 8.29 \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{More about predictive uncertainties}
As noted above, we expected our point prediction to become more accurate and
predictive uncertainty to become smaller as time approaches the true bloom
date. To check this, for each crop, we calculate the MAE with median as point
prediction and the average lengths of 95\% PIs each day over the years of
interest, starting from 90 days prior to the bloom--date (call it "lag -90") to
1 day prior to the bloom--date ("lag -1"). The results are shown in Figure
\ref{fig:uncMAE} and \ref{fig:uncPI} respectively. It is clear that for all
crops, the MAE does become smaller and the average lenghth of 95\% PIs becomes
shorter as time approaches the true bloom--date.
In fact, by the time we reach
one month prior the bloom--date, the point prediction is quite accurate (the
MAE is 3.5--5 days).
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.6]{uncMAEMedianreduce.pdf}
\caption{Change of the MAE of median with the change of lag. The point
prediction becomes more accurate when time approaches the bloom
date.}
\label{fig:uncMAE}
\end{center}
\end{figure}
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.6]{uncPIreduce.pdf}
\caption{Change of the average length of 95\% PIs with the change of lag.
The predictive uncertainty decreases when time approaches the bloom
date.}
\label{fig:uncPI}
\end{center}
\end{figure}
We now compare our predictor with two naive predictors: the first one being
the 95\% confidence interval (CI) of a normal fit to the observed data, i.e.
$\pm1.96$ standard deviation around the sample mean; the second one being an
empirical quantile-based 95\% CI of the observed data, i.e. 2.5\%--97.5\%
sample quantiles. The length and percentage of the coverage of the normal fits
are shown in Table \ref{tab:PINormal}. Although the length of the normal CIs
are a few days shorter than our overall 95\% PIs reported in Table
\ref{tab:predscore}, the coverages of the normal CIs are too low, especially
for peach, prune and pear. Figure \ref{fig:3histogram} shows that the
histograms of the bloom dates of the crops are obviously skewed, which implies
that the normal fit may not be a good choice. The length and coverage of the
empirical quantile-based CIs are shown in Table \ref{tab:PIEmpirical}. These
CIs beat the Normal CIs in both length and coverage for most of the crops.
However, the coverages of them are still lower than 95\% except for Pear.
The results reported in Table \ref{tab:predscore} for our predictor are
calculated from the first day of a year to the
actual bloom date, which have incorporated too much uncertainty. If we look at
the predictions starting from one month prior to the bloom date to the actual
bloom date, the results (Table \ref{tab:PIonemonth}) are much improved -- the
average lengths of the 95\% PIs are much shorter and the estimated coverage
probabilities now range from 95\% to 98\%. These results apparently are much
better than those from both naive predictors.
\begin{table}[ht]
\begin{center}
\caption{The length and the coverage of 95\% CI of a normal fit to the observed
bloom--dates for each crop in 1937--1964 in the Okanagan region}
\label{tab:PINormal}
\begin{tabular}{|l|cccccc|}
\hline
&Apricot & Cherry & Peach & Prune & Pear & Apple \\
\hline
Length &30.65 & 29.23 & 27.20 & 25.62 & 26.76 & 25.57 \\
coverage &0.93 &0.93 &0.89 &0.89 &0.93 &0.93 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.6]{histogram.pdf}
\caption{Histograms of the bloom dates of the corps. Most of the
histograms are obviously skewed.}
\label{fig:3histogram}
\end{center}
\end{figure}
\begin{table}[ht]
\begin{center}
\caption{The length and the coverage of the empirical quantile-based 95\% CI of
the observed bloom--dates for each crop in 1937--1964 in the Okanagan
region}
\label{tab:PIEmpirical}
\begin{tabular}{|l|cccccc|}
\hline
&Apricot & Cherry & Peach & Prune & Pear & Apple \\
\hline
Length &28.10 &29.28 &24.15 &24.97 &27.65 &24.25 \\
coverage &0.93 &0.93 &0.93 &0.93 &0.96 &0.93 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[ht]
\begin{center}
\caption{The average length and the estimated coverage probability of 95\%
PIs of our predictions starting from one month prior to the bloom date to
the actual bloom date}
\label{tab:PIonemonth}
\begin{tabular}{|l|cccccc|}
\hline
&Apricot & Cherry & Peach & Prune & Pear & Apple \\
\hline
Length & 19.91 & 19.87 & 15.28 & 16.38 & 15.57 & 14.19 \\
coverage & 0.95 & 0.96 & 0.97 & 0.98 & 0.97 & 0.97 \\
\hline
\end{tabular}
\end{center}
\end{table}
Another thing that interests us is the shape of the predictive distributions.
To see this, we plot the predictive distribution of a ``randomly'' picked crop
and test year -- the predictive distribution of peach in year 1944 with daily
average temperatures of the first 60 days of that year known (Figure
\ref{fig:pdplot}). Note that the true bloom--date of peach in that year is day
125, and we have smoothed the discrete predictive distribution to a continuous
curve. We see that the predictive distribution (the solid curve) is nearly
bell-shaped which roughly looks like a normal distribution. Since this
predictive distribution is calculated by plugging in the MLEs as if they were
the true parameters, there is an uncertainty associated with this predictive
distribution. Just as before, we use the bootstrap to assess this
uncertainty -- we calculate a quantile based 95\% bootstrap confidence band
(shown as the shaded area in Figure \ref{fig:pdplot}) for this predictive
distribution. The plot shows that this confidence band is not too wide, so we
basically can ``trust'' this predictive distribution.
\begin{figure}[!ht]
\begin{center}
\includegraphics[scale=0.6]{predictiveDist.pdf}
\caption{The predictive distribution (solid curve) of peach in year 1944
with daily average temperatures of the first 60 days of that year known.
The shaded area is a 95\% confidence band for this predictive
distribution. The true bloom -- date of peach in that year is day 125.}
\label{fig:pdplot}
\end{center}
\end{figure}
The validity of the bootstrap procedure arises again.
Do the bootstrap estimates of the quantiles of the predictive probabilities
reflect the true quantiles of the predictive probabilities? Again, we conduct
a simulation study to answer this question. Take the same settings for the
simulation as described in Section \ref{subsec:est}. For each sample size $S$,
where $S \in \left\{ 30, \,80, \,150, \,400\right\}$, we now generate one more
year of data as test data. For a fixed sample size $S$, for each sample, we
estimate the model parameters, and we then use this set of of parameters to
predict the bloom--date of the test year by assuming the first 60 days of
temperatures are known. We then get 1000 predictive distributions for each
sample size. For each future day, we take the 2.5\% and 97.5\% sample
quantiles of the 1000 predictive probabilities to approximate a quantile based
95\% confidence interval for the predictive probability. Now, randomly pick
one sample, and then take 1000 bootstrap samples of this sample, and estimate
model parameters using each bootstrap sample. With each set of estimated
parameters obtained from the bootstrap, we can then make a prediction on the
test year. With 1000 bootstrap samples, we get 1000 predictive distributions.
For each future day, as with the simulated data, we can obtain a quantile
based 95\% bootstrap confidence interval for the predictive probability. We
now compare the confidence intervals obtained in these two ways. Randomly
picking one future day, the 95\% confidence intervals for the predictive
probability of blooming that day calculated using the simulated data and
bootstrap are shown in Table \ref{tab:predCIsim}. We can see that both types
of confidence intervals become narrower when sample size becomes larger. For
each sample size, the bootstrap interval is close to the interval obtained
using the simulated data. Moreover, when the sample size reaches 400, the two
types of intervals are basically identical. This result suggests that applying
bootstrap might be a reasonable way to estimate the uncertainty of the
predictive probabilities.
\begin{table}[ht]
\caption{Comparison of the 95\% confidence intervals for predictive
probabilities obtained using bootstrap and the simulated data.}
\label{tab:predCIsim}
\begin{center}
\small
\begin{tabular}{|l|c|c|c|c|}
\hline
& $S=30$ & $S=80$ & $S=150$ & $S=400$ \\
\hline
Bootstrap & (0.0300, 0.0350) & (0.0321, 0.0358) & (0.0300, 0.0320)
& (0.0305, 0.0322) \\
\hline
Simulation & (0.0288, 0.0348) & (0.0298, 0.0332) & (0.0302, 0.0326)
& (0.0307, 0.0320) \\
\hline
\end{tabular}
\end{center}
\end{table}
\section{Conclusions}\label{sec:conclusions}
In this paper, we presented an application of our regression model for
irreversible progressive events to a single phenological event -- blooming
event of six high-valued perennial crops. The approach we introduced here can
be also widely applicable to the analysis of survival data and other similar
time-to-event data.
For the blooming events of the six crops, our method provides a sensible way
to estimate the important parameter $T_{base}$ in the definition of growing degree day (GDD) and
our statistical analysis supports an earlier empirical finding -- that the timing
of a bloom event is related to AGDD, a cumulative unweighted sum of GDDs. Our method also provides useful point predictions of the future bloom--dates, as well as the assessment of the predictive uncertainties which is useful for risk
analysts and policy makers. Our 95\% PIs are excessively large however,
quite probably because we have used a crude ARIMA model for predicting the future daily average temperatures. After reducing and then total eliminating the
uncertainty about the daily average temperatures, we see increased accuracy of
point predictions and much shortened 95\% PIs. This observation validates our
regression model for phenological events.
In our analysis, we did not consider the possible auto-correlation of the bloom
date. In future, we will consider more complicated models to incorporate
this. On the other hand, to improve estimation and prediction, we may build a
multivariate model to encompass all crops as responses simultaneously.
Moreover, if crops of multiple locations are involved, we will consider a
spatial-temporal modeling approach for phenological events.
\afterpage{\clearpage}
\bibliographystyle{plainnat}
|
{
"timestamp": "2010-09-22T02:00:42",
"yymm": "1009",
"arxiv_id": "1009.3970",
"language": "en",
"url": "https://arxiv.org/abs/1009.3970"
}
|
\section{Introduction}
Recently the Babar collaboration observed four excited charmed
mesons $D(2550)$, $D(2600)$, $D(2750)$ and $D(2760)$ in the decay
channels $D^0(2550)\to D^{*+}\pi^-$, $D^{0}(2600)\to
D^{*+}\pi^-,\,D^{+}\pi^-$, $D^0(2750)\to D^{*+}\pi^-$,
$D^{0}(2760)\to D^{+}\pi^-$, $D^{+}(2600)\to D^{0}\pi^+$ and
$D^{+}(2760)\to D^{0}\pi^+$ respectively in the inclusive $e^+e^-
\rightarrow c\bar{c}$ interactions at the SLAC PEP-II
asymmetric-energy collider \cite{Babar2010}, see Table 1. The Babar
collaboration also analyzed the helicity distributions to determine
the spin-parity, and suggested that the $(D(2550),D(2600))$ (denoted
as $(D^{\prime},D^{*\prime})$ respectively in Table 2) may be
the $2S$ radial excitation of the $(D,D^*)$, and the
$D(2750)$ and $D(2760)$ may be the $D$-wave states. Furthermore, the
Babar collaboration measured the following ratios of the branching
fractions:
\begin{eqnarray}
\frac{{\rm{Br}}\left(D^*_2(2460)^0\to
D^+\pi^-\right)}{{\rm{Br}}\left(D^*_2(2460)^0\to
D^{*+}\pi^-\right)}&=&1.47\pm0.03\pm0.16 \, , \nonumber \\
\frac{{\rm{Br}}\left(D(2600)^0\to
D^+\pi^-\right)}{{\rm{Br}}\left(D(2600)^0\to
D^{*+}\pi^-\right)}&=&0.32\pm0.02\pm0.09\, , \nonumber \\
\frac{{\rm{Br}}\left(D(2760)^0\to
D^+\pi^-\right)}{{\rm{Br}}\left(D(2750)^0\to
D^{*+}\pi^-\right)}&=&0.42\pm0.05\pm0.11\, .
\end{eqnarray}
In the heavy quark limit $m_Q \to \infty$, the heavy-light mesons
$Q{\bar q}$ can be classified in doublets according to the total
angular momentum of the light degrees of freedom ${\vec s}_\ell$,
${\vec s}_\ell= {\vec s}_{\bar q}+{\vec L} $, where the ${\vec
s}_{\bar q}$ is the spin of the light antiquark $\bar q$ and the
${\vec L}$ is the orbital angular momentum of the light degrees of
freedom \cite{RevWise,RevNeubert}. In the quark models, we usually
use the $n$ to denote the radial quantum number. In the case $n=1$,
for $L=0$, the doublet $(P,P^*)$ have the spin-parity
$J^P_{s_\ell}=(0^-,1^-)_{\frac{1}{2}}$; $L=1$,
the two doublets $(P^*_0,P^{\prime}_1)$ and $(P_1,P^*_2)$ have the spin-parity
$J^P_{s_\ell}=(0^+,1^+)_{\frac{1}{2}}$ and $(1^+,2^+)_{\frac{3}{2}}$
respectively; $L=2$, the two doublets $(P^*_1,P_2)$ and
$(P_2^{\prime *},P_3)$ have the spin-parity
$J^P_{s_\ell}=(1^-,2^-)_{\frac{3}{2}}$ and $(2^-,3^-)_{\frac{5}{2}}$
respectively; where the superscript $P$ denotes the parity. The
$n=2,3,4, \cdots$ states are clarified by the analogous doublets,
for example, $n=2$, $L=0$, the doublet $(P^{\prime},P^{*\prime})$
have the spin-parity $J^P_{s_\ell}=(0^-,1^-)_{\frac{1}{2}}$; $n=2$,
$L=1$, the two doublets $(P^{*\prime}_0,{P^{\prime}_1}^\prime)$ and
$({P_1}^{\prime},P^{*\prime}_2)$ have the spin-parity
$J^P_{s_\ell}=(0^+,1^+)_{\frac{1}{2}}$ and $(1^+,2^+)_{\frac{3}{2}}$
respectively.
The helicity distributions favor identifying the $D^0(2550)$ as the
$0^-$ state, the $D^0(2600)$ as the $1^-$, $2^+$, $3^-$ state, and
the $D^0(2750)$ as the $1^+$, $2^-$ state \cite{Babar2010}. From
the Review of Particle Physics
\cite{PDG}, we can see that only six low-lying
states, $D$, $D^*$, $D_0(2400)$, $D_1(2430)$, $D_1(2420)$ and
$D_2(2460)$ are established, while the $2S$ and $1D$ states are
still absent. The newly observed charmed mesons $D(2550)$,
$D(2600)$, $D(2750)$ and $D(2760)$ may be tentatively identified as
the missing $2S$ and $1D$ states.
The mass is a fundamental parameter in describing a hadron, in Table
2, we present the predictions from some theoretical models, such
as the relativized quark model based on a universal
one-gluon-exchange-plus-linear-confinement potential
\cite{GI}, the semirelativistic quark potential model \cite{MMS}, the relativistic quark model includes the leading
order $1/M_h$ corrections \cite{PE}, the QCD-motivated relativistic
quark model based on the quasipotential approach \cite{EFG}, for
comparison. From the Table, we can see that the masses of the
$D(2550)$, $D(2600)$ and $D(2750)$, $D(2760)$ lie in the regions
of $2S$ and $1D$ states, respectively.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c }\hline\hline
& Mass [MeV] & Width [MeV] & Decay channel \\ \hline
$D^0(2550)$ & $2539.4 \pm 4.5 \pm 6.8$ & $130\pm12 \pm13$ &$D^{*+}\pi^-$ \\ \hline
$D^0(2600)$ & $2608.7\pm 2.4\pm 2.5$ & $93\pm 6\pm13$ &$D^+\pi^-$,\,$D^{*+}\pi^-$ \\ \hline
$D^0(2750)$ & $2752.4\pm 1.7\pm 2.7$ & $71\pm6\pm11$ &$D^{*+}\pi^-$ \\ \hline
$D^0(2760)$ & $2763.3\pm 2.3\pm 2.3$ & $60.9\pm5.1\pm3.6$ &$D^{+}\pi^-$ \\ \hline
$D^+(2600)$ & $2621.3\pm 3.7\pm 4.2$ & $93$ &$D^0\pi^+$ \\ \hline
$D^+(2760)$ & $2769.7\pm 3.8\pm 1.5$ & $60.9 $ &$D^0\pi^+$ \\ \hline\hline
\end{tabular}
\end{center}
\caption{The experimental results from the Babar collaboration.}
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c|c|}\hline\hline
& $n\,L\,s_\ell\,J^P$& Experiment \cite{Babar2010,PDG} &GI \cite{GI} & MMS \cite{MMS} & PE \cite{PE} & EFG \cite{EFG}\\ \hline
$D$ & $1\,S\,\frac{1}{2}\,0^-$ & 1867& 1880& 1869& 1868& 1871\\ \hline
$D^*$ & $1\,S\,\frac{1}{2}\,1^-$ & 2008 &2040 &2011& 2005& 2010\\ \hline
$D^*_0$ & $1\,P\,\frac{1}{2}\,0^+$ & 2400 &2400 &2283& 2377& 2406\\ \hline
$D^{\prime}_1$ & $1\,P\,\frac{1}{2}\,1^+$ & 2427& 2490& 2421& 2490& 2469\\ \hline
$D_1$ & $1\,P\,\frac{3}{2}\,1^+$ & 2420& 2440& 2425& 2417& 2426\\ \hline
$D_2^*$ & $1\,P\,\frac{3}{2}\,2^+$ & 2460 &2500& 2468& 2460& 2460\\ \hline
$D_1^*$ & $1\,D\,\frac{3}{2}\,1^-$ & ?\,2763 &2820 & 2762 & 2795 &2788 \\ \hline
$D_2$ & $1\,D\,\frac{3}{2}\,2^-$ & ?\,2752 & & 2800& 2833&2850 \\ \hline
$D_2^{\prime*}$ & $1\,D\,\frac{5}{2}\,2^-$ & ?\,2752 & & & 2775 & 2806 \\ \hline
$D_3$ & $1\,D\,\frac{5}{2}\,3^-$ & ?\,2763 & 2830& & 2799 &2863 \\ \hline
$D^{\prime}$ & $2\,S\,\frac{1}{2}\,0^-$ & ?\,2539 & 2580& & 2589& 2581\\ \hline
${D^*}^{\prime}$ & $2\,S\,\frac{1}{2}\,1^-$ & ?\,2609& 2640& & 2692& 2632\\ \hline
\hline
\end{tabular}
\end{center}
\caption{ The masses of the charmed mesons from different quark
models compared with experimental data, and the possible
identifications of the newly observed charmed mesons. }
\end{table}
In Ref.\cite{Sun1008}, Sun et al study the strong decays of the
$D(2550)$, $D(2600)$ and $D(2760)$ in the $^3P_0$ model, and
identify the $D(2600)$ as a mixture of the $2^3S_1-1^3D_1$ states
and the $D(2760)$ as either the orthogonal partner of the $D(2600)$
or the $1^3D_3$ state. In Ref.\cite{Zhong1009}, Zhong studies the
strong decays of the $D(2550)$, $D(2600)$ and $D(2760)$ in a chiral
quark model, and identifies the $D(2760)$ as the $1^3D_3$ state and
the $D(2600)$ as the low-mass mixing state of the $1^3D_1-2^3S_1$
states.
In this work, we study the strong decays of the newly observed
charmed mesons with the heavy quark effective theory in the leading
order approximation to distinguish the different identifications.
There have been several works using the heavy quark effective theory
to identify the excited $D_s$ mesons, such as the $D_s(3040)$,
$D_s(2700)$, $D_s(2860)$
\cite{Colangelo1001,Colangelo0710,Colangelo0607,Colangelo0511}.
The article is arranged as follows: we study the strong decays
of the newly observed charmed mesons with the heavy quark
effective theory in Sect.2; in Sect.3, we present the
numerical results and discussions; and Sect.4 is reserved for our
conclusions.
\section{ The strong decays with the heavy quark effective theory }
In the heavy quark effective theory, the spin doublets can be
described by the effective super-fields $H_a$, $S_a$, $T_a$, $X_a$
and $Y_a$, respectively \cite{Falk1992},
\begin{eqnarray}
H_a & =& \frac{1+{\rlap{v}/}}{2}\left\{P_{a\mu}^*\gamma^\mu-P_a\gamma_5\right\} \, , \nonumber \\
S_a &=& \frac{1+{\rlap{v}/}}{2} \left\{P_{1a}^{\prime \mu}\gamma_\mu\gamma_5-P_{0a}^*\right\} \, , \nonumber \\
T_a^\mu &=&\frac{1+{\rlap{v}/}}{2} \left\{ P^{\mu\nu}_{2a}\gamma_\nu
-P_{1a\nu} \sqrt{3 \over 2} \gamma_5 \left[ g^{\mu \nu}-{\gamma^\nu
(\gamma^\mu-v^\mu) \over 3} \right]\right\}\, , \nonumber\\
X_a^\mu &=&\frac{1+{\rlap{v}/}}{2} \Bigg\{ P^{*\mu\nu}_{2a}
\gamma_5\gamma_\nu -P^{\prime*}_{1a\nu} \sqrt{3 \over 2} \left[ g^{\mu \nu}-{\gamma^\nu (\gamma^\mu-v^\mu) \over 3} \right]\Bigg\} \, , \nonumber \\
Y_a^{ \mu\nu} &=&\frac{1+{\rlap{v}/}}{2} \left\{
P^{\mu\nu\sigma}_{3a} \gamma_\sigma -P^{\prime*\alpha\beta}_{2a}
\sqrt{5 \over 3} \gamma_5 \left[ g^\mu_\alpha g^\nu_\beta -
{\gamma_\alpha g^\nu_\beta (\gamma^\mu-v^\mu) \over 5}
- {\gamma_\beta g^\mu_\alpha (\gamma^\nu-v^\nu) \over 5} \right]
\right\}\, ,
\end{eqnarray}
where the heavy field operators contain a factor $\sqrt{M_P}$ and
have dimension of mass $\frac{3}{2}$. The ground state and radial
excited state heavy mesons with the same heavy flavor have the same
spin, parity, time-reversal and charge conjunction properties except
for the masses, and can be denoted by the super-fields: $H_a$,
$H_a'$, $H_a''$, $\cdots$; $S_a$, $S_a'$, $S_a''$, $\cdots$; $T_a$,
$T_a'$, $T_a''$, $\cdots$; etc, where the superscripts $\prime$,
$\prime\prime$ and $\prime\prime\prime$ denote the first, the second
and the third radial excited states, respectively. With a simple
replacement of the components $P_a$, $P_a^*$, $P_{0a}^*$, $\cdots$
to the corresponding radial excited states ${P_a}^\prime$,
${P_a^{*}}^\prime$, ${P_{0a}^{*}}^\prime$, $\cdots$, we can obtain
the corresponding super-fields $H_a^\prime$, $S_a^\prime$, $\cdots$.
The light pseudoscalar mesons are described by the fields
$\displaystyle \xi=e^{i {\cal M} \over
f_\pi}$, where
\begin{equation}
{\cal M}= \left(\begin{array}{ccc}
\sqrt{\frac{1}{2}}\pi^0+\sqrt{\frac{1}{6}}\eta & \pi^+ & K^+\nonumber\\
\pi^- & -\sqrt{\frac{1}{2}}\pi^0+\sqrt{\frac{1}{6}}\eta & K^0\\
K^- & {\bar K}^0 &-\sqrt{\frac{2}{3}}\eta
\end{array}\right) \, .
\end{equation}
At the leading order, the heavy meson chiral Lagrangians ${\cal
L}_H$, ${\cal L}_S$, ${\cal L}_T$, ${\cal L}_X$, ${\cal L}_Y$ for
the strong decays to $D^{(*)}\pi$, $D^{(*)}\eta$ and $D_s^{(*)}K$
are written as \cite{HL-1,HL-2,HL-3,HL-4,PRT1997}:
\begin{eqnarray}
{\cal L}_H &=& \, g_H {\rm Tr} \left\{{\bar H}_a H_b \gamma_\mu\gamma_5 {\cal A}_{ba}^\mu \right\} \, ,\nonumber \\
{\cal L}_S &=& \, g_S {\rm Tr} \left\{{\bar H}_a S_b \gamma_\mu \gamma_5 {\cal A}_{ba}^\mu \right\}\, + \, h.c. \, , \nonumber \\
{\cal L}_T &=& {g_T \over \Lambda_\chi}{\rm Tr}\left\{{\bar H}_a
T^\mu_b (i D_\mu {\not\! {\cal A} }+i{\not\! D }
{ \cal A}_\mu)_{ba} \gamma_5\right\} + h.c. \, , \nonumber \\
{\cal L}_X &=& {g_X \over \Lambda_\chi}{\rm Tr}\left\{{\bar H}_a
X^\mu_b(i D_\mu {\not\! {\cal A} }+i{\not\! D } { \cal A}_\mu)_{ba} \gamma_5\right\} + h.c. \, ,\nonumber \\
{\cal L}_{Y} &=& {1 \over {\Lambda_{\chi}^2}}{\rm Tr}\left\{ {\bar
H}_a Y^{\mu \nu}_b \left[k_1 \{D_\mu, D_\nu\} {\cal A}_\lambda
+ k_2 (D_\mu D_\lambda { \cal A}_\nu + D_\nu D_\lambda { \cal
A}_\mu)\right]_{ba} \gamma^\lambda \gamma_5\right\} + h.c. \, ,
\end{eqnarray}
where
\begin{eqnarray}
{\cal D}_{\mu}&=&\partial_\mu+{\cal V}_{\mu} \, , \nonumber \\
{\cal V}_{\mu }&=&\frac{1}{2}\left(\xi^\dagger\partial_\mu \xi+\xi\partial_\mu \xi^\dagger\right)\, , \nonumber \\
{\cal A}_{\mu }&=&\frac{1}{2}\left(\xi^\dagger\partial_\mu \xi-\xi\partial_\mu \xi^\dagger\right)\, ,
\end{eqnarray}
$\Lambda_\chi$ is the chiral symmetry-breaking scale and taken as
$\Lambda_\chi = 1 \, $ GeV \cite{Colangelo1001}, the strong coupling
constants $g_H$, $g_S$, $g_T$, $g_X$ and $g_Y=(k_1+k_2)$ can be
fitted phenomenologically if there are enough experimental data.
The subscript indexes $H$, $S$, $T$, $X$ and $Y$ denote the
interactions between the super-field $H$ and the super-fields $H$,
$S$, $T$, $X$ and $Y$, respectively. We have smeared the
superscripts $\prime$, $\prime\prime$, $\prime\prime\prime$,
$\cdots$ for simplicity, the notation $g_H$ denotes the strong
coupling constants in the vertexes $HH{\cal A}$, $H'H{\cal A}$,
$H'H'{\cal A}$, $H''H{\cal A}$, $\cdots$, the notations $g_S$,
$g_T$, $g_X$ and $g_Y$ should be understood in the same way. In
this article, we intend to study the ratios among different decay
channels, the strong coupling constants are canceled out with each
other, and cannot lead to confusion.
From the heavy meson chiral Lagrangians ${\cal L}_H$, ${\cal L}_S$,
${\cal L}_T$, ${\cal L}_X$, ${\cal L}_Y$, we can obtain the widths
$\Gamma$ for the strong decays to $D^{(*)}\pi$, $D^{(*)}\eta$ and
$D_s^{(*)}K$ easily,
\begin{eqnarray}
\Gamma&=&\frac{p_{cm}}{8\pi M^2 } |T|^2\, ,
\end{eqnarray}
where the $T$ denotes the scattering amplitudes, the $p_{cm}$ is the
momentum of the final states in the center of mass coordinate.
In calculations, we take the approximation ${\cal{A}}_\mu\approx
i\frac{\partial_\mu {\cal{M}}}{f_{\pi}} $. In the case that the
light pseudoscalar meson momenta are not very small, we should add
other terms and introduce new unknown coupling constants.
Furthermore, the flavor and spin violation corrections of order
$\mathcal {O}(1/m_Q)$ to the heavy quark limit may be sizable, again
we should introduce new unknown coupling constants, which will not
necessarily canceled out in the ratios of the decay widths. We
cannot estimate the role and the size of such corrections on
general grounds, however, we expect that they would not be larger
than (or as large as) the leading order contributions.
\section{Numerical Results}
The input parameters are taken from the particle data group
$M_{\pi^+}=139.57\,\rm{MeV}$, $M_{\pi^0}=134.9766\,\rm{MeV}$,
$M_{K^+}=493.677\,\rm{MeV}$, $M_{\eta}=547.853\,\rm{MeV}$,
$M_{D^+}=1869.60\,\rm{MeV}$, $M_{D^0}=1864.83\,\rm{MeV}$,
$M_{D_s^+}=1968.47\,\rm{MeV}$, $M_{D^{*+}}=2010.25\,\rm{MeV}$,
$M_{D^{*0}}=2006.96\,\rm{MeV}$, $M_{D_s^{*+}}=2112.3\,\rm{MeV}$,
$M_{D(2460)}=2460.1\,\rm{MeV}$ \cite{PDG}.
The numerical values for the widths of the strong
decays
\begin{eqnarray}
D_2^* &\to& D^{*+}\pi^-, \,D^{+}\pi^- \, , \nonumber \\
D^{\prime} & \to &D^{*+}\pi^- , \, D^{*0}\pi^0 \, , \nonumber \\
{D^*}^{\prime} (D_1^*,\,D_2,\,D_3) & \to & D^{*+}\pi^-, \, D^{+}\pi^-, \, D^{*0}\pi^0, \, D^{0}\pi^0,
\, D^{*0}\eta, \, D^{0}\eta, \,D^{*+}_sK^-, \,D_s^{+} K^-\, ,
\end{eqnarray}
are presented in Tables 3-4.
In Table 5, we present the experimental data for the ratio
$\frac{\Gamma(D^{+}\pi^-)}{\Gamma(D^{*+}\pi^-)}$ of the well
established meson $D_2^*(2460)$ from the Babar \cite{Babar2010},
CLEO \cite{CLEO1994,CLEO1990}, ARGUS \cite{ARGUS1989}, and ZEUS
\cite{ZEUS2009} collaborations, the prediction $2.30$ from the
heavy quark effective theory in the leading order approximation is
in excellent agreement with the average experimental value $2.35$.
Compared with the experimental data from the Babar collaboration
$\frac{\Gamma(D^{+}\pi^-)}{\Gamma(D^{*+}\pi^-)}=1.47\pm0.03\pm0.16$
\cite{Babar2010}, the heavy quark effective theory in the leading
order approximation leads to a larger ratio.
The total decay widths of the $(D(2550),D(2600))$ with the
spin-parity $(0^-,1^-)_{\frac{1}{2}}$ are
$\Gamma_{D^{\prime}}\approx 1.7g_H^2\,\rm{GeV}$ and
$\Gamma_{D^{*\prime}}\approx 2.0g_H^2\,\rm{GeV}$, the ratio
$\frac{\Gamma_{D^{\prime}}}{\Gamma_{D^{*\prime}}}\approx0.85$, which
is smaller than the experimental data
$\frac{\Gamma_{D^{\prime}}}{\Gamma_{D^{*\prime}}}=1.40$, where we
have used the central values of the widths
$\Gamma_{D^{\prime}}\approx (130\pm12\pm13)\,\rm{MeV}$ and
$\Gamma_{D^{*\prime}}= (93\pm6\pm13)\,\rm{MeV}$ from the Babar
collaboration \cite{Babar2010}. For the charmed mesons, the leading
power flavor and spin violation corrections (of order $\mathcal
{O}(1/m_Q)$) to the heavy quark limit may be sizable, we have to
introduce new unknown coupling constants, the discrepancy may be
smeared with the optimal parameters, furthermore, more precise
measurements are needed to make a reliable comparison. In the case
of the ratio $\frac{\Gamma_{D_1}}{\Gamma_{D_2^*}}$, the prediction
$0.30$ from the heavy quark effective theory in the leading order
approximation is also smaller than the experimental data $0.48$
from the Review of Particle Physics \cite{PDG}, if the leading
power spin corrections
to the heavy quark limit are taken into account,
the discrepancy can be smeared \cite{Falk1996}.
The ratio $\frac{\Gamma(D^{*\prime}\to
D^{+}\pi^-)}{\Gamma(D^{*\prime}\to D^{*+}\pi^-)}=0.82$ from the
heavy quark effective theory in the leading order approximation is
larger than the experimental data $0.32\pm0.02\pm0.09 $ from the
Babar collaboration \cite{Babar2010}, just like in the case of the
ratio $\frac{\Gamma(D^*_2 \to D^{+}\pi^-)}{\Gamma(D^*_2 \to
D^{*+}\pi^-)}$, and again more precise measurements are needed to
make a reliable comparison. The strong coupling constants
$g_{D^*D\pi}$ and $g_{D^*D^*\pi}$ receive sizable contributions
from the flavor and spin violation corrections
\cite{PRT1997,Grinstein1995}, in the present case, the strong
coupling constants $g_{D^{*\prime} D\pi}$ and $g_{D^{* \prime}
D^*\pi}$ also receive the flavor and spin violation corrections
besides the leading order strong coupling constant $g_H$, which
maybe account for the discrepancy. We can tentatively identify the
$(D(2550),D(2600))$ as the doublet $(0^-,1^-)_{\frac{1}{2}}$ with
$n=2$.
The existing theoretical estimations for the strong coupling
constant $g_H$ among the ground state heavy mesons ($n=1$) vary in
a large range $g_H=0.1-0.6$, it is difficult to select the ideal
value (one can consult Ref.\cite{Wang2007} for more literatures), we
usually use the value determined from the precise experimental data
on the decay $D^{*+} \to D^0 \pi^+$ from the CLEO collaboration
\cite{CLEO-gH1,CLEO-gH2}. In the present case, the strong coupling
constants involve the radial excited $S$-wave heavy mesons and
ground state $D$-wave heavy mesons, therefor the situation is more
involved, and it is impossible to determine the revelent parameters
with the heavy quark effective theory itself without enough
experimental data. The theoretical works focus on the strong
coupling constants $g_H$, $g_S$, $g_T$ of the ground state $S$-wave
and $P$-wave heavy mesons (one can consult
Refs.\cite{PRT1997,Wang2007,Wang2006} for more literatures), while
the works on the strong coupling constants $g_H$, $g_S$, $g_T$ of
the radial excited $S$-wave and $P$-wave heavy mesons and $g_X$,
$g_Y$ of the ground state $D$-wave heavy mesons are rare due to lack
experimental data \cite{Zhu1003}. In this article, we take the
strong coupling constants $g_H$, $g_T$, $g_X$ and $g_Y$ as unknown
parameters, and prefer the ratios of the decay widths in different
channels to compare with the experimental data.
From Table 4, we can see that if we identify the $(D(2760),D(2750))$
as the doublet $(1^-,2^-)_{\frac{3}{2}}$ with $n=1$, the ratio
$\frac{\Gamma(D_1^*\to D^{+}\pi^-)}{\Gamma(D_2\to
D^{*+}\pi^-)}=4.07$ from the leading order heavy quark effective
theory deviates from the experimental data $0.42\pm0.05\pm0.11$
greatly \cite{Babar2010}\footnote{We take the approximation
$\Gamma_{D(2760)}=\Gamma_{D(2750)}$.}, which requires the flavor and
spin violation corrections depressed by the inverse heavy quark mass
$1/m_Q$ are as large as the leading order contributions and have
opposite sign, it is impossible, as the heavy quark effective theory
has given many successful descriptions of the hadron properties
\cite{RevWise,RevNeubert,PRT1997}. On the other hand, if we identify
the $(D(2750),D(2760))$ as the doublet $(2^-,3^-)_{\frac{5}{2}}$
with $n=1$, the deviation of the ratio $\frac{\Gamma(D_3\to
D^{+}\pi^-)}{\Gamma(D_2^{\prime*}\to D^{*+}\pi^-)}=0.80$ from the
upper bound of the experimental data $0.42\pm0.05\pm0.11$ is not
large \cite{Babar2010}, the contributions from the flavor and spin
violation corrections maybe smear the discrepancy.
We also explore the possible identification of the $D(2760)$ and
$D(2750)$ as the same $3^-$ state with $n=1$, i.e. they are the
$D_3$ state, the ratio $\frac{ \Gamma(D_3\to
D^{+}\pi^-)}{\Gamma(D_3\to D^{*+}\pi^-)}=1.94$ from the heavy quark
effective theory in the leading order approximation is too large
compared with the experimental data $\frac{\Gamma\left(D(2760)^0\to
D^+\pi^-\right)}{\Gamma\left(D(2750)^0\to
D^{*+}\pi^-\right)}=0.42\pm0.05\pm0.11$ \cite{Babar2010}, which
again requires the flavor and spin violation corrections depressed
by the inverse heavy quark mass $1/m_Q$ are as large as the leading
order contributions and have opposite sign, such an identification
is disfavored. On the other hand, the helicity distribution
disfavors identifying the $D(2750)$ as the $3^-$ state
\cite{Babar2010}. We can tentatively identify the
$(D(2750),D(2760))$ as the doublet $(2^-,3^-)_{\frac{5}{2}}$ with
$n=1$.
In this article, we also present the widths for the $D_s^{(*)}K$ and
$D^{(*)}\eta$ decays, where the strong coupling constants are
retained, the predictions can be confronted with the experiential
data in the future at the BESIII, KEK-B, RHIC, $\rm{\bar{P}ANDA}$
and LHCb.
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c| }\hline\hline
& $n\,L\,s_\ell\,J^P$& Mass [MeV] &Decay channels & Width [GeV] \\ \hline
$D_2^*$ & $1\,P\,\frac{3}{2}\,2^+$ & 2460.1 & $D^{*+}\pi^-$; $D^{+}\pi^-$& $0.0543879g_T^2$; $0.124928g_T^2$ \\ \hline
$D^{\prime}$ & $2\,S\,\frac{1}{2}\,0^-$ & ?\,2539.4 &$D^{*+}\pi^-$; $D^{*0}\pi^0$& $1.13557g_H^2$; $0.583137g_H^2$ \\ \hline
${D^*}^{\prime}$ & $2\,S\,\frac{1}{2}\,1^-$ & ?\,2608.7& $D^{*+}\pi^-$; $D^{+}\pi^-$& $0.66068g_H^2$; $0.54317g_H^2$\\
& & & $D^{*+}_sK^-$; $D_s^{+} K^-$& $0.000518592g_H^2$; $0.106459g_H^2$\\
& & & $D^{*0}\pi^0$; $D^{0}\pi^0$& $0.336747g_H^2$; $0.276487g_H^2$\\
& & & $D^{*0}\eta$; $D^{0}\eta$& $0.00841286 g_H^2$; $0.029364g_H^2$\\ \hline
$D_1^*$ & $1\,D\,\frac{3}{2}\,1^-$ & ?\,2763.3 & $D^{*+}\pi^-$; $D^{+}\pi^-$ & $0.339606g_X^2$; $5.19392 g_X^2$ \\
& & & $D_s^{*+}K^-$; $D_s^{+}K^-$ & $0.0632191g_X^2$; $1.86912 g_X^2$ \\
& & & $D^{*0}\pi^0$; $D^{0}\pi^0$ & $0.173223g_X^2$; $2.65247 g_X^2$ \\
& & & $D^{*0}\eta$; $D^{0}\eta$ & $0.0226441g_X^2$; $0.508904 g_X^2$ \\\hline
$D_2$ & $1\,D\,\frac{3}{2}\,2^-$ & ?\,2752.4 & $D^{*+}\pi^-$; $D^{+}\pi^-$ & $1.27691 g_X^2$; 0 \\
& & & $D_s^{*+}K^-$; $D_s^{+}K^-$ & $0.180643 g_X^2$; 0 \\
& & & $D^{*0}\pi^0$; $D^{0}\pi^0$ & $0.653307 g_X^2$; 0 \\
& & & $D^{*0}\eta$; $D^{0}\eta$ & $0.069308 g_X^2$; 0 \\\hline
$D_2^{\prime*}$ & $1\,D\,\frac{5}{2}\,2^-$ & ?\,2752.4 & $D^{*+}\pi^-$; $D^{+}\pi^-$ & $0.221226 g_Y^2$; 0 \\
& & & $D_s^{*+}K^-$; $D_s^{+}K^-$ & $0.00413833g_Y^2$; 0 \\
& & & $D^{*0}\pi^0$; $D^{0}\pi^0$ & $0.114719 g_Y^2$; 0 \\
& & & $D^{*0}\eta$; $D^{0}\eta$ & $0.0027123 g_Y^2$; 0 \\ \hline
$D_3$ & $1\,D\,\frac{5}{2}\,3^-$ & ?\,2763.3 & $D^{*+}\pi^-$; $D^{+}\pi^-$ & $0.0907266g_Y^2$; $0.176388g_Y^2$ \\
& & & $D_s^{*+}K^-$; $D_s^{+}K^-$& $0.00218128 g_Y^2$; $0.018115 g_Y^2$ \\
& & & $D^{*0}\pi^0$; $D^{0}\pi^0$ & $0.0468994g_Y^2$; $0.0912646 g_Y^2$ \\
& & & $D^{*0}\eta$; $D^{0}\eta$ & $0.00124089g_Y^2$; $0.00618076 g_Y^2$ \\ \hline
\hline
\end{tabular}
\end{center}
\caption{ The strong decay widths of the newly observed charmed
mesons with possible identifications. }
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c| }\hline\hline
& $n\,L\,s_\ell\,J^P$& Mass [MeV] & Ratio \\ \hline
$D_2^*$ & $1\,P\,\frac{3}{2}\,2^+$ & 2460.1 & $\frac{\Gamma(D^{+}\pi^-)}{\Gamma(D^{*+}\pi^-)}=2.30$ \\ \hline
${D^*}^{\prime}$ & $2\,S\,\frac{1}{2}\,1^-$ & ?\,2608.7& $\frac{\Gamma(D^{+}\pi^-)}{\Gamma(D^{*+}\pi^-)}=0.82$;
$\frac{\Gamma(D^{*0}\pi^0)}{\Gamma(D^{*+}\pi^-)}=0.51$; $\frac{\Gamma(D^{0}\pi^0)}{\Gamma(D^{*+}\pi^-)}=0.42$;\\
& & & $\frac{\Gamma(D_s^{+} K^-)}{\Gamma(D^{*+}\pi^-)}=0.16$; $\frac{\Gamma(D^{0}\eta)}{\Gamma(D^{*+}\pi^-)}=0.044$; $\frac{\Gamma(D^{*0}\eta)}{\Gamma(D^{*+}\pi^-)}=0.013$; \\
& & & $\frac{\Gamma(D^{*+}_sK^-)}{\Gamma(D^{*+}\pi^-)}=0.001$ \\ \hline
$D_1^*$ & $1\,D\,\frac{3}{2}\,1^-$ & ?\,2763.3 & $\frac{\Gamma(D^{+}\pi^-)}{\Gamma(D^{*+}\pi^-)}=15.29$; $\frac{\Gamma(D^{0}\pi^0)}{\Gamma(D^{*+}\pi^-)}=7.81$; $\frac{\Gamma(D_s^{+}K^-)}{\Gamma(D^{*+}\pi^-)}=5.50$; \\
& & & $\frac{\Gamma(D^{0}\eta)}{\Gamma(D^{*+}\pi^-)}=1.50$; $\frac{\Gamma(D^{*0}\pi^0)}{\Gamma(D^{*+}\pi^-)}=0.51$; $\frac{\Gamma(D_s^{*+}K^-)}{\Gamma(D^{*+}\pi^-)}=0.19$; \\
& & & $\frac{\Gamma(D^{*0}\eta)}{\Gamma(D^{*+}\pi^-)}=0.067$ \\ \hline
$D_2$ & $1\,D\,\frac{3}{2}\,2^-$ & ?\,2752.4 & $\frac{\Gamma(D^{*0}\pi^0)}{\Gamma(D^{*+}\pi^-)}=0.51$; $\frac{\Gamma(D_s^{*+}K^-)}{\Gamma(D^{*+}\pi^-)}=0.14$;
$\frac{\Gamma(D^{*0}\eta)}{\Gamma(D^{*+}\pi^-)}=0.054$ \\\hline
$D_2^{\prime*}$ & $1\,D\,\frac{5}{2}\,2^-$ & ?\,2752.4 & $\frac{\Gamma(D^{*0}\pi^0)}{\Gamma(D^{*+}\pi^-)}=0.52$; $\frac{\Gamma(D_s^{*+}K^-)}{\Gamma(D^{*+}\pi^-)}=0.019$;
$\frac{\Gamma(D^{*0}\eta)}{\Gamma(D^{*+}\pi^-)}=0.012$ \\ \hline
$D_3$ & $1\,D\,\frac{5}{2}\,3^-$ & ?\,2763.3 & $\frac{\Gamma(D^{+}\pi^-)}{\Gamma(D^{*+}\pi^-)}=1.94$; $\frac{\Gamma(D^{0}\pi^0)}{\Gamma(D^{*+}\pi^-)}=1.01$; $\frac{\Gamma(D^{0*}\pi^0)}{\Gamma(D^{*+}\pi^-)}=0.52$; \\
& & & $\frac{\Gamma(D_s^{+}K^-)}{\Gamma(D^{*+}\pi^-)}=0.20$; $\frac{\Gamma(D^{0}\eta)}{\Gamma(D^{*+}\pi^-)}=0.068$; $\frac{\Gamma(D_s^{*+}K^-)}{\Gamma(D^{*+}\pi^-)}=0.024$; \\
& & & $\frac{\Gamma(D^{0*}\eta)}{\Gamma(D^{*+}\pi^-)}=0.014$ \\ \hline
$D_1^*$ & $1\,D\,\frac{3}{2}\,1^-$ & ?\,2763.3 & \\
$D_2$ & $1\,D\,\frac{3}{2}\,2^-$ & ?\,2752.4 & $\frac{\Gamma(D_1^*\to D^{+}\pi^-)}{\Gamma(D_2\to D^{*+}\pi^-)}=4.07$ \\ \hline
$D_2^{\prime*}$ & $1\,D\,\frac{5}{2}\,2^-$ & ?\,2752.4 & \\
$D_3$ & $1\,D\,\frac{5}{2}\,3^-$ & ?\,2763.3 & $\frac{\Gamma(D_3^*\to D^{+}\pi^-)}{\Gamma(D_2^{\prime*}\to D^{*+}\pi^-)}=0.80$ \\ \hline
\hline
\end{tabular}
\end{center}
\caption{ The ratios of the strong decay widths of the newly
observed charmed mesons with possible identifications. }
\end{table}
\begin{table}
\begin{center}
\begin{tabular}{|c|c|c|c|c|c|c| }\hline\hline
Babar& CLEO& CLEO & ARGUS&ZEUS &This work\\ \hline
$1.47\pm0.03\pm0.16$& $2.2\pm0.7\pm0.6$& $2.3\pm 0.8$ & $3.0\pm1.1\pm1.5$ &$2.8\pm 0.8^{+0.5}_{-0.6}$ &$2.30$\\ \hline
\hline
\end{tabular}
\end{center}
\caption{ The ratio of $\frac{\Gamma\left(D^*_2(2460)^0\to
D^+\pi^-\right)}{\Gamma\left(D^*_2(2460)^0\to D^{*+}\pi^-\right)} $
from the experimental data compared with the prediction from the
leading order heavy quark effective theory. }
\end{table}
\section{Conclusion}
In this article, we study the strong decays of the newly observed
charmed mesons $D(2550)$, $D(2600)$, $D(2750)$ and $D(2760)$ with
the heavy quark effective theory in the leading order approximation,
and tentatively identify the $(D(2550),D(2600))$ as the doublet
$(0^-,1^-)$ with $n=2$ and $(D(2750),D(2760))$ as the doublet
$(2^-,3^-)$ with $n=1$, respectively. The identification of the
$D(2750)$ and $D(2760)$ as the same particle with $J^P=3^-$ is
disfavored. The other predictions can be confronted with the
experimental data in the future at the BESIII, KEK-B, RHIC,
$\rm{\bar{P}ANDA}$ and LHCb.
\section*{Acknowledgment}
This work is supported by National Natural Science Foundation of
China, Grant Numbers 10775051, 11075053, and Program for New Century
Excellent Talents in University, Grant Number NCET-07-0282, and the
Fundamental Research Funds for the Central Universities.
|
{
"timestamp": "2010-11-30T02:02:56",
"yymm": "1009",
"arxiv_id": "1009.3605",
"language": "en",
"url": "https://arxiv.org/abs/1009.3605"
}
|
\section{Introduction}\label{sect: intro}
Massive stars have many fascinating aspects, which extend well
beyond stellar physics alone. One of their most striking
properties is conceptually very simple: their
high-degree of multiplicity. Most O- and early B-type stars are found
in binaries and multiple systems. Even single field stars are often
believed to have been part of a multiple system in the past, then ejected
by a supernova kick or by dynamical interaction. To ignore the
multiplicity of early-type stars is equivalent to neglecting one of
their most defining characteristics.
In this review we concern ourselves with the multiplicity of stars
more massive than 8~M$_\odot$ on the zero-age main sequence, which
have spectral types earlier than B3~V. Our approach is to focus on their
observational properties, with the emphasis on O-type binaries,
although early B-type binaries feature in some of the quoted
works. Despite the importance of detailed studies of individual
objects, our prime motivation here is to consider the broader results
from the literature, in an attempt to lift the veil on
some of the general properties of the binary population of early-type
stars.
The distributions of the orbital parameters of massive binaries, as a
population, are of fundamental importance to stellar evolution, yet
remain poorly constrained. These distributions trace the products of
star formation and the early dynamical evolution of the host systems,
and are necessary ingredients to population synthesis studies. Only
with an understanding of these distributions can we hope to recover
accurate predictions for some of the exotic late stages of binary
evolution.
This contribution is structured as follows. Section~\ref{sect: physic}
describes some of the physical processes and observational biases that
are present in multiple systems compared to single stars.
Section~\ref{sect: param} introduces the different parts of parameter
space occupied by massive binaries, and the observational means to
investigate them; Section~\ref{sect: fbin} then reviews the
multiplicity fraction of OB stars within each regime, and in
different astrophysical environments. Section~\ref{sect: CDF} attempts
to summarize our current understanding of the parameter distributions
of O\,$+$\,OB spectroscopic binaries. Finally, Section \ref{sect: ccl}
provides a summary.
\section{Physical processes and observational biases}\label{sect: physic}
Binaries are excellent astrophysical laboratories that provide us
with direct measurements of fundamental parameters such as stellar
masses and radii. Multiplicity induces new processes compared to
isolated single stars, offering the opportunity to confront our
understanding of a broad range of physics under the extreme conditions
found in, and close to, astrophysical objects. Moreover, if one fails to take
multiplicity into account, observations (and their analysis) can be significantly
biased or misleading. Most critically, early-type binaries with
orbital periods of up to 10 years follow significantly
different evolutionary paths, an aspect that can also impact the
outputs of population synthesis models \citep[e.g.,][]{Van09}. By way of additional motivation to
understand multiplicity in massive stars, some of the observational
and evolutionary impacts include:
{\underline{\it Different evolutionary paths:}} Binarity significantly affects
the evolutionary path of the components of the systems compared to
single stars. Tidal effects in close binaries modifies the evolution of
stellar rotation rates, thus also the induced rotational-mixing of
enriched material into their photospheres \citep{dMCL09}. Roche-lobe overflow will
result in mass and angular momentum transfer, spinning up the
secondary to its critical rotation rate \citep{Pac81,LCY08}. While the gaining star might
be rejuvenated by the increase in mass \citep{BrL95}, the primary will see a
reduction in the life-time of its red supergiant phase \citep{EIT08}. A
common-envelope phase and/or stellar mergers are other possible
outcomes of binary evolution. The impacts on observed stellar
populations are numerous, including modified surface abundances,
modified enrichment of the interstellar medium, the rate of supernova
and $\gamma$-ray burst explosions, and on the number of
evolved systems such as Wolf-Rayet stars and high-mass X-ray binaries
\citep[e.g.,][]{IDK06,BKD08}.
{\underline{\it Wind collisions:}} In binaries, the powerful
stellar wind from the stars may interact with one another or with the
surface of the star with the weaker wind \citep{Uso92}. The supersonic
collision heats the gas to temperature up to several 10$^7$~K
\citep{SBP92}. In several cases, the wind-wind interaction is also to
accelerate particles up to relativistic energies. The
signature of the wind collision can be observed throughout the
electromagnetic spectrum, through non-thermal radio (and possibly X-
and $\gamma$-ray) emission \citep{DeB07}, through X-ray thermal
emission \citep{PiP10} and via a contribution to the recombination lines
in the optical and infrared \citep{SRG01}. In massive binaries containing
evolved stars with very dense winds, the wind interaction region can act
as a nucleation site for dust particles, creating structures such as the
pinwheel nebulae \citep{TML08}. These effects can provide
indirect indiciations of multiplicity. However, if multiplicity is
not considered, wind collision can lead to erroneous estimates of fundamental
properties such as intrinsic X-ray luminosities \citep{SRN06},
spectral classifications, and stellar mass-loss rates (as measured
from the strength of, e.g., the H$\alpha$ line).
{\underline{\it Struve-Sahade effect:}} In its most generalized form,
the Struve-Sahade (S-S) effect can be described as the variation in
the apparent strength of the spectrum of one or both components when
the star is approaching/receding \citep[for an example, see e.g. ][]{SRG01}. Various physical effects can induce
a S-S signature: gaseous streams in the systems, ellipsoidal
variations, surface streams, and changes in the local surface
temperature due to, e.g., mutual illumination or heating from a
wind-wind collision \citep[e.g.,][]{BGR99, LRS07}.
{\underline{\it Cluster dynamical mass:}} Ignoring the contribution of
binaries to the stellar velocity dispersion in clusters (in both
integrated-light observations of distant systems and studies of
resolved clusters), can lead to a significant overestimate of their
dynamical mass \citep{BTT09,GSPZ10}. For example, some of the
disagreement in the mass-to-light ratio of young extragalactic
clusters might arise from the binary properties of their red
supergiant populations \citep{GSPZ10}.
{\underline{\it Supermassive stars:}} Unresolved multiple systems have
often been confused with very high mass stars due to their large
luminosity. Numerous objects have indeed seen their masses revised at
the light of improvements of the observing facilities \citep[e.g. the case of R136:][]{CMS81, WeB85, CSH10}.
\section{The parameter space}\label{sect: param}
Before discussing the multiplicity properties of populations of
massive stars, we attempt to give the reader a feel for
the typical parameter space that needs to be investigated. Our aim
is to provide a qualitative overview of the orders of magnitude
involved; the values and sketches should only be considered as
indicative!
While many more parameters are involved, it is useful to restrain our
discussion to a two-dimensional space. Indeed the detection efficiency
of most of the observing techniques can be discussed in terms of the
orbital separation (or, equivalently, of the orbital period) and of
the mass- or flux-ratio of the components. For a given evolutionary stage,
the mass-ratio can directly be related the flux ratio and we will therefore
assume a direct equivalence between these two values. This simplified approach assumes that
observations with sufficient time-sampling are available, and knowingly
neglects the second-order effects of eccentricity and orbital
inclination on the detection probabilities.
{\underline{\it Mass-ratio ($q=M_2/M_1$):}} In principle, the range of
possible mass-ratios spans equal-mass binaries ($q=1.0$) to a system
with a massive star with a light companion ($q<<1$). For example, an
O5\,$+$\,M8 system would have a mass ratio of only $q\sim0.002$. Of
course, a companion with such a low mass would be very hard to detect,
but the absence of observational clues does not preclude their
existence. There are other observational issues, such as the
likelihood that low-mass companions are still in the pre-main sequence
phase -- observations at longer wavelengths could provide crucial
information in this scenario. The range of flux-ratios that require
scrutiny can reach up to 10$^5$, providing a significant observational
challenge.
{\underline{\it Separations (d):}} An estimate of the minimal
separation can be adopted as the distance at which two main-sequence
stars would enter a contact phase. For typical O- and early B-type
primaries, this corresponds to rough separations of 20~R$_\odot$ or
0.1~AU, equating to periods of 1-2 days depending of the system
mass. The outer separation boundary is more of a grey zone that
depends on both the system environment and on the timescale involved. In
this context, we consider two arguments. The first makes the
distinction between {\it hard} and {\it soft} binary systems, i.e.,
between systems that have a large likelihood of surviving a three-body
interaction, versus systems that
will be easily disrupted. \citet{Heg75} defined {\it hard} binaries
as systems in which the binding energy ($E_b$) is larger than the kinetic
energy ($E_k$) brought about by an encounter :
\begin{equation}
|E_b|>E_k(encounter)=\frac{<m><v^2>}{2},
\end{equation} where $<m>$ and $<v^2>$ are the typical mass and velocity dispersions of stars in a given cluster. Following \citet{PZMMG10} and adopting an effective cluster radius of 1~pc and cluster masses in the range 2.5$\times$10$^3$ to 10$^5$~M$_\odot$, one estimates the maximum separation of {\it hard} binaries to be in the range of 10$^3$ to several 10$^4$~A.U.
A second more qualitative argument emphasized by \citet{MAp10} points
out that massive stars have short life-times. One could therefore limit
the parameter space to orbital periods of 10$^5$ to 10$^6$~yr as only
these systems would accomplish a significant number of orbits during
their life-time. Following the third Kepler law, this also
corresponds to typical separations of several 10$^4$ AU.
Interestingly, this means that most of the
massive binaries are hard binaries, that will be difficult to disrupt
over their life-time. The observed maximum range of separations considered
here is in line with the statement of \citet{Abt88}
that the more massive stars can sustain companions up to
several 10$^4$ AU\ or more.\\
{\underline{\it Observational techniques:}} Investigating such a
large parameter space requires a combination of techniques (Fig.~\ref{fig: paramspace}), each
characterized by their own sensitivities and observational biases.
Short-period close binaries are probed efficiently through
spectroscopy, while very wide binaries, with angular separations
larger than a couple of arcseconds can be detected by classical, high-contrast
imaging. Enhanced imaging techniques such as adaptive optics
(AO) and lucky imaging can provide about an order of magnitude in
terms of closer separation and can also reach large flux
contrasts. In principle, the gap between the spectroscopic and imaging
regimes can be bridged with speckle interferometry, and
ground-based and space interferometry. Speckle interferometry has
the potential for large surveys but, to date, its applications have been limited
to flux ratios of about ten \citep{MHG09}. Space and ground-based
interferometry can reach separations of milliarcsecond scales, at flux
ratios of up to 100, but are much more costly to operate and
no large survey has yet been attempted.
Combining these various methods allows us in principle to explore the full range
of separations for massive binaries out to a distance of
$\approx5$~kpc. In practise, these techniques are not equally
sensitive and do not offer the same detection probability in their
respective regions of parameter space. For example, spectroscopy is
very efficient for short-period binaries, with periods of up to a couple
of years. The detection probability however decreases dramatically for
long-period systems \citep[see, e.g., Fig.~2 of ][]{EBB10}, in part
due to the reduced radial velocity (RV) signal and also due to the
longer timescales involved. Moreover, eccentric systems are
harder to detect due the narrower window
(sometimes less than a tenth of the orbital cycle) during which the RV
variations are concentrated. Imaging techniques (classical, lucky, or
AO-corrected) share a common bias in which the achievable contrast
varies as a function of the separation \citep[see e.g., Fig.~2 of][]{MAp10}.
Detailed comprehension of the limitations of each technique and of
their observational bias is of prime importance in order to
retrieve the global multiplicity properties of massive star
populations.
\begin{figure}[t]
\begin{center}
\includegraphics[width=13cm]{review_sana_fig1.pdf}
\caption{Left-hand panel: typical parameter space for massive
binaries. A primary of 40~M$_\odot$ at a distance of 1~kpc has
been assumed to construct this sketch. The relevant regions for
various detection techniques have been overlaid. Right-hand panel:
measured multiplicity in those parts of parameter space (see text for
details).} \label{fig: paramspace}
\end{center}
\end{figure}
\section{The multiplicity fraction of O-type stars}\label{sect: fbin}
\subsection{Spectroscopic binary fraction in various separation regimes}
The right-hand panel of Fig.~\ref{fig: paramspace} gives an overview
of the results from recent surveys, including the minimum multiplicity
fraction obtained in each part of parameter space from the
relevant technique:
{\underline{\it Spectroscopy:}} The most comprehensive overview of the
spectroscopic binary (SB) fraction is provided by \citet{MHG09}. Based
on a review of the literature covering more than 300 O-type objects,
these authors found over half of the sample to be part of a SB
system. The systems are separated, almost equally, into single- (SB1)
and double-lined (SB2) systems.
{\underline{\it Speckle interferometry:}} In the same paper,
\citet{MHG09} provide speckle observations of 385 O-type stars, thus covering
almost all of the targets in the Galactic O star catalog
\citep{MAWG04}. 11\%\ of the objects in the \citeauthor{MHG09} sample are
found to have speckle companions.
{\underline{\it Enhanced imaging techniques:}} At larger separations,
AO-corrected and lucky imaging surveys (respectively, \citet{TBR08}
-- 138 O stars -- and \citet{MAp10} -- 128 O stars) found that 37\%\
of the O stars are part of wide multiple systems. These two studies are
mostly limited to the northern hemisphere and are thus missing
some of the richer massive star clusters and associations in the
southern sky. Part of this gap is filled by the AO campaigns of
\citet{DSE01} and \citet{SMG10} on, respectively, NGC~6611 and
Tr~14. Both studies revealed a lower multiplicity fraction of 18\%\
for their sample of OB stars. Yet, (part of) this difference results
from the fact that these two regions are dense clusters. In these
environments, disentangling the true pairs from chance alignment with
stars in the same clusters becomes more challenging and only a smaller
separation range can be investigated reliably. Interestingly, both
\citeauthor{DSE01} and \citeauthor{SMG10} concluded that
OB stars have more companions than lower mass-stars.
{\underline{\it Interferometry:}} As mentioned earlier, interferometry
is less suitable for surveys. To the best of our knowledge, only one
homogeneous survey has been attempted so far. \citet{NWW04} targeted a
limited sample of 23 O-type stars in the Carina region with the {\em Hubble
Space Telescope} fine guidance sensor, resolving close-by companions for
four stars.\\
Combining information from these various ranges, a minimum
multiplicity fraction close to 70\%\ for the population of Galactic
O-type stars is reached \citep{MHG09}. Given the detection limits of
these campaigns, there is ample scope for the true multiplicity
fraction to be even larger.
Despite the quality of the observations collected so far, improvements are still
needed in each of the ranges covered by the various observing techniques described above:
\begin{enumerate}
\item[-] Homogeneous AO and lucky imaging campaigns have been mostly limited to the northern sky. Extending such work to the rich and dense clusters and star-formation regions of the southern hemisphere is highly desirable,
\item[-] Higher flux contrasts are needed in the 10-100~mas separation regime. Techniques such as sparse-aperture masking coupled with AO could, in principle, bring some improvements,
\item[-] The separation range 5-100~AU remains almost unexplored,
\item[-] About half the known and suspected SBs lack an orbital solution. As a consequence, the distribution of the the orbital parameters remains largely uncertain (see also Section~\ref{sect: CDF}).
\end{enumerate}
\begin{table}[t]
\begin{center}
\caption{Overview of the spectroscopic binary fraction in clusters.}
\label{tab: clusters}
{\scriptsize
\begin{tabular}{|l c c c |l c c c|}\hline
{\bf Object} & {\bf\# O stars} & {\bf Binary fraction$^a$} & {\bf Ref} & {\bf Object} & {\bf\# O stars} & {\bf Binary fraction$^a$} & {\bf Ref.} \\ \hline
\multicolumn{4}{|c|}{\bf Nearby clusters} & \multicolumn{4}{c|}{\bf Distant/extragalactic clusters} \\
NGC 6611 & 9 & 0.44 & 1 & West 1 & 20 & 0.30 & 9 \\
NGC 6231 & 16 & 0.63 & 2 & 30 Dor & 54 & 0.45 & 10 \\
IC 2944 & 14 & 0.53 & 3 & NGC346 & 19 & 0.21 & 11 \\
Tr 16 & 24 & 0.48 & 4 & N11 & 44 & 0.43 & 11 \\
IC 1805 & 8 & 0.38 & 5 & NGC2004 & 4 & 0.25 & 11 \\
IC 1848 & 5 & 0.40 & 5 & NGC 330 & 6 & 0.00 & 11 \\
NGC 2244 & 6 & 0.17 & 6 & & & & \\
Tr 14 & 6 & 0.00 & 7 & \multicolumn{4}{c |}{\bf Milky Way O star population} \\
Col 228 & 15 & 0.33 & 8 & \begin{tabular}{c}Clusters \& \\ OB associations\end{tabular} & 305 & 0.57 & 12 \\
\hline
\end{tabular}
}
\end{center}
\vspace{1mm}
\scriptsize{ {\it Notes:} $^a$The quoted binary fraction is a lower
limit as each new detection will increase it. \\
{\it References:} 1. \citet{SGE09},
2. \citet{SGN08}, 3. \citet{SJG10}, 4. Literature review,
5. \citet{HGB06}, 6. \citet{MNR09}, 7. \citet{PGH93}, \citet{GML98}, 8. Sana et al. (in prep.), 9. \citet{RCN09}, 10. \citet{BTT09}, 11. \citet{ELS06},
12. \citet{MHG09} \\ }
\end{table}
\subsection{Spectroscopic binary fraction in clusters}
\citet{MHG09} investigated the dependence of the SB fraction on environment by
comparing stars from clusters and associations with runaway and field
stars, finding that the first category harbours many more binaries and
multiple systems. This picture is mostly consistent with an ejection
scenario for the field/runaway stars in which most of the multiple systems
would be disrupted. In this section, we take a different approach and
look for differences in the multiplicity fraction of various
clusters. Several authors have indeed proposed the SB fraction to be
related to the cluster density \citep[e.g.,][]{PGH93, GM01}.
To support our discussion, Table~\ref{tab: clusters} summarizes the SB
fraction of O-star rich clusters (i.e., clusters with at least five
O-type stars), with Fig.~\ref{fig: clusters} giving a graphical
comparison of the SB fractions in the various samples. Focusing on the
qualitatively homogeneous sample formed by the nearby clusters, we
calculate an average binary fraction of $f=0.44\pm0.05$.
While some deviations are observed
around this average value, each can be explained by statistical
fluctuations. Even the extreme case of Tr~14, with no known
spectroscopic companions to its six O-type stars, is not statistically
significant. For instance, the probability to have six single stars,
drawn from an underlying binomial distribution with a multiplicity
fraction of $f=0.44$ is 3\%.
Assuming that parent population is the same, the chance of obtaining zero binaries in any one of our nine clusters (given the size of their respective O star population) is 13\%, such that we cannot reject the null hypothesis.
Of course, the fact that Tr~14 is
the densest and possibly the youngest of the nine clusters in our
sample is intriguing.
\begin{figure}[t]
\begin{center}
\includegraphics[width=12cm,height=6cm]{review_sana_fig2.pdf}
\caption{Spectroscopic binary fraction of nearby (left) and distant/extragalactic (right) clusters. The plain line and dashed lines indicate the average fraction and 1$\sigma$ dispersion computed from the nearby cluster sample. }
\label{fig: clusters}
\end{center}
\end{figure}
The multiplicity properties from distant and extragalactic clusters
are less constrained and should be considered as lower limits, in part
because some of these works have a limited baseline and/or a limited
number of epochs. Aside from the case of NGC~330, there is again no
fundamental disagreement with the results from the nearby cluster
sample. With no companions seen for six O-type stars, the NGC~330
sample is similar, in terms of size and binary fraction, to
Tr~14. Sample size effects could be invoked (as for Tr~14), but the
fact that the much larger population of B-type stars in NGC~330 also show a
depleted binary population \citep{ELS06} is appealing. Interestingly,
NGC~330 is an older region with a very low surface density, in strong contrast with
the properties of Tr~14.
In summary, while some variations of the binary fraction might occur
in peculiar situations, the null hypothesis of a common parent
distribution cannot be rejected given the current data set. Adopting a uniform
binary fraction of $f\approx0.44$ is thus the most relevant
description of the current data. As a direct consequence of this
result, one can however reject with a very high confidence the null hypothesis
that all O stars are spectroscopic binaries.
\begin{table}[t]
\begin{center}
\caption{Overview of the two O star samples used to derive the distributions of the orbital parameters. The first part of the table indicates the number of O stars, the number of O-type binaries and the binary fraction of the two samples. The second part of the table provides the number and the fraction of systems with constraints on their periods, mass-ratios and eccentricities. }
\label{tab: sample}
{\scriptsize
\begin{tabular}{|l c c c |}\hline
{\bf } & {\bf\# Galactic O stars} && {\bf Nearby rich clusters} \\ \hline
\# O stars & 305 && 82 \\
\# binaries & 173 && 38 \\
Binary fraction & 0.57 && 0.46 \\
\hline
\# periods & 102 (59\%) && 33 (87\%) \\
\# mass-ratios & 76 (44\%) && 29 (76\%) \\
\# eccentricities & 86 (50\%) && 30 (79\%) \\
\hline
\end{tabular}
}
\end{center}
\vspace{1mm}
\scriptsize{
{\it Note:}
The sample of nearby clusters is formed by IC 1805, IC1848, IC 2944, NGC6231, NGC 6611 and Tr16.\\
}
\end{table}
\section{Distributions of the orbital parameters of spectroscopic binaries}\label{sect: CDF}
This section provides an overview of our current knowledge of the orbital parameter distributions for O-type spectroscopic binaries. In doing so, it is useful to define two samples (Table~\ref{tab: sample}):
\begin{enumerate}
\item[-] {\underline{\it The Galactic O-star sample:}} mostly based on the sample of \citet{MHG09}.
While \citeauthor{MHG09} only concentrate on the multiplicity aspect,
we perform our own literature review to search for estimates of
periods, mass-ratios and eccentricities. When no orbital solution was
available, we estimated the mass-ratios of SB2 systems by adopting
typical masses for the components as a function of their spectral
classification \citep{MSH05}. Compared to the review of \citet{MHG09}, we also
include information that became available in the last two years, as
well as preliminary results from our work.
\item[-] {\underline{\it The nearby O-star rich clusters:}} a subsample of
the Galactic O-star sample, focusing on the O-star rich clusters
within $\approx3$~kpc. These clusters have been more thoroughly
studied so that the scope for observational biases is more limited.
\end{enumerate}
The binary fraction of the two samples appear to be different, with
the Galactic O-star sample displaying more binaries. A possible
explanation for this is provided by \citet{GM01}, who noted that the O
stars in poor clusters (i.e., clusters with only one or two O-type stars)
were almost all multiple. These clusters are not included in our
second sample, which may pull the binary fraction to lower values.\
While the Galactic O-star sample is the most comprehensive, only about
50\%\ of the binaries have constraints on their orbital solution
(Fig.~\ref{fig: Pcdf_sample}), leaving a lot of room for observational
biases. For example, the orbital solutions are more difficult to obtain for
long-period high eccentricity systems. There might thus be an uneven
representation of various parameter ranges in the observed
distribution functions. The situation is much improved for the cluster
sample, as almost 80\%\ of the systems have proper orbital solutions
and 87\%\ have estimates of the orbital period. We therefore argue
that the distributions derived from the cluster sample are much less
affected by observational biases.
In the following, we will compare the parameter
distributions built from the two samples to one another and to
analytical distributions commonly used to represent the properties of
the massive star binary population.\\
\begin{figure}[t]
\begin{center}
\includegraphics[width=10cm]{review_sana_fig3.pdf}
\caption{Cumulative number function of orbital periods for the
complete sample (left) and for the nearby cluster sample
(right). This plot aims to give a graphical impression of the potential
biases affecting the two samples. Normalised cumulative
distribution functions for systems with solutions are given in Fig.~\ref{fig: CDF}.}
\label{fig: Pcdf_sample}
\end{center}
\end{figure}
{\underline{\it Period:}} Fig.~\ref{fig: Pcdf_sample} provides an
overview of the respective samples with the cumulative number
distributions of the orbital periods. It shows that the period
distribution function obtained from the cluster sample is almost fully
constrained, but that uncertainties could still affect the
Galactic sample. However, the cumulative distribution functions (CDFs)
are mostly in agreement (Fig.~\ref{fig: CDF}, left-hand panel). Both CDFs
show an overabundance of short periods, with 50 to 60\%\ of the systems
having a period shorter than 10 days. Consequently, the CDF of observed
periods in the spectroscopic regime can not be represented by the
traditional \"{O}pik Law\footnote{\"{O}pik's Law states that
the distribution of separations is flat in logarithmic space. The
corresponding period distribution should be flat in $\log P$ as
well.}. As already suggested by \citet{SGN08}, a much better
representation of the period CDF is provided by a bi-uniform
distribution in $\log P$ (which one could consider a `broken'
\"{O}pik Law) such that:
\begin{equation}
CDF(P) = \left\{
\begin{array}{l l}
\frac{F_\mathrm{break} \bigl( \log P-\log P_\mathrm{min} \bigl)}{\log P_\mathrm{break}-\log P_\mathrm{min}}, & \mathrm{for}\ \log P_\mathrm{min} \le \log P \le P_\mathrm{break} \\
\\
F_\mathrm{break}+\frac{ \bigl(1-F_\mathrm{break} \bigl) \bigl(\log P-\log P_\mathrm{break}\bigl)}{\log P_\mathrm{max}-\log P_\mathrm{break}} , & \mathrm{for}\ P_\mathrm{break}< \log P \le \log P_\mathrm{max} \\
\end{array}\right.
\label{eq: CDFp}
\end{equation}
where $P$ is expressed in days. Adopting a break-point at
$P_\mathrm{break}\approx10$~d, with upper and lower limits of $\log
P/\mathrm=0.3$ and 3.5\,d and considering that the binaries are evenly
spread in the short and long period regimes (i.e.,
$F_\mathrm{break}\approx0.5$), Eq.~\ref{eq: CDFp} becomes:
\begin{equation}
CDF(P) =\left\{
\begin{array}{l l}
\frac{5}{7} \log P-\frac{10.5}{7}, & \quad \mathrm{for}\ 0.3 \le \log P \le 1.0 \\
\\
\frac{1}{5} \log P-\frac{3}{10}, & \quad\mathrm{for}\ 1.0 < \log P \le 3.5 \\
\end{array}\right.
\label{eq: CDFp2}
\end{equation}
\begin{figure}[t]
\begin{center}
\includegraphics[width=13cm]{review_sana_fig4.pdf}
\caption{Cumulative distribution functions (CDFs) of the periods
($P$), mass-ratios ($q$) and eccentricities ($e$). The plain thin/magenta and thick/blue lines indicate the CDFs of, the Galactic O-star sample and
the nearby cluster sample, respectively. Left-hand panel: the dashed line shows an \"{O}pik
Law over this range of periods, while the dot-dashed line indicates the
alternative law given by Eq.~\ref{eq: CDFp2}. Middle panel: the
dashed line indicates a uniform distribution in the considered
range. Right-hand panel: the dashed line indicates a uniform distribution
for $e>0$.} \label{fig: CDF}
\end{center}
\end{figure}
Eqs. \ref{eq: CDFp} and \ref{eq: CDFp2} give an empirical description
of the CDF of the observed periods. The latter should still be
corrected for the detection probability (mostly affecting longer
periods) and for the systems lacking orbital solutions (also more
likely to affect the longer-period regime). The exact location of the
lower and upper limits and of the `break' still needs to be more
tightly constrained. That said, the general behaviour and the
overabundance of short-period spectroscopic binaries appear clear.
{\underline{\it Mass-ratio:}} The CDFs of the mass-ratios
(Fig.~\ref{fig: CDF}, middle panel) are well reproduced by a uniform
distribution in the range 0.2\,$<$\,$q$\,$<$\,1.0. The Galactic O-star sample
shows slightly fewer systems with $q<0.6$; this can be (partly)
explained by observational biases as the detection of the secondary
signature for systems with large mass differences (i.e., large flux contrasts)
requires very high-quality data that are not always available for the Galactic sample. SB1
binaries represent about 20-25\%\ of the cluster sample. For these
stars, one cannot directly estimate the mass-ratio. However, we note
that the fraction of SB1 is roughly compatible with an extension of
the uniform CDF towards $q<0.2$; testing this statement will require
detailed simulations.
As a direct consequence of a uniform mass-ratio CDF, the presence of a
twin population with $q>0.95$ proposed by \citet{PiS06} can be rejected.
Another implication resides in the fact that massive binaries cannot
be formed by random pairing from a Salpeter/Kroupa IMF. Our results
rather suggest the presence of a mechanism that favors the creation of
O\,$+$\,OB binaries. Such a mechnaism could find part of its origin in the
early dynamical evolution, where companion exchanges favor the capture
of more and more massive secondaries. It could also trace a
particular formation mechanism \citep{ZiY07}.
{\underline{\it Eccentricity:}} The CDF of the eccentricities
(Fig.\ref{fig: CDF}, right-hand panel) is characterized by an overabundance
of circular and low eccentricity systems. Indeed, 25-30\%\ of the systems
displayed a circular orbit, while another 30\%\ have $e<0.2$. This
behaviour contradicts the expected properties of a purely thermal
binary population, which can be qualitatively explained by the large
fraction of short-period systems for which tidal dissipation will tend
to circularize the orbit.
An analytical description of the observed CDF for eccentric systems
can be provided through $CDF(e>0) \propto e^{0.5}$ in the range
0.0\,$<$\,$e$\,$<$\,0.8. However, as 20\%\ of the cluster sample and
50\%\ of the Galactic sample are lacking robust eccentricities and as
biases are most likely to affect larger eccentricities, we cannot
consider this relation as definite. That said, one would expect that
$CDF(e)$ will remain overabundant towards low eccentricity systems.
\section{Summary}\label{sect: ccl}
We have attempted to provide an overview of our current knowledge of
the important multiplicity properties of massive stars. We described
some of the physical processes and observational biases that lead
to binaries behaving differently compared to single stars.
We then briefly described the observational parameter space that one
needs to explore to investigate massive binaries, and we discussed the
challenges of probing it homogeneously. Despite these difficulties,
it is now well established that the vast majority of O-type stars are part
of a multiple system. The typical separation between the multiple
components covers at least 4 order of magnitudes.
At least 45-55\%\ of the O star population in clusters and OB
associations is comprised by spectroscopic binaries, with a lower
fraction found for field and runaway stars \citep{MHG09}. Here we
have investigated possible variations of the multiplicity fraction
among clusters with a rich O star population. While room for small
variations remains due to our limited sample and due to the small O
star population of some clusters, the binary fraction can mostly be
considered as uniform with a value close to 44\%. Given the current data set,
one can hardly argue that the multiplicity fraction is significantly correlated
with the cluster density (at least not in the range covered in our sample)
. While density can still play a role, for example, to explain
the difference observed between O-star rich and O-star poor
clusters, its impact among rich clusters remain questionable in
light of the current data. It is well accepted that most
O-type stars are part of a multiple systems, but a similar statement does not
hold when limiting ourselves to spectroscopic companions. Given
the observed SB binary fraction and the sample sizes, it is
unlikely that the underlying fraction of SBs is larger than 70-75\%.
Finally, we have constructed CDFs for the periods, mass-ratios and
eccentricities for two samples of massive binaries. The Galactic
O-star sample is more extensive but has been studied less
homogeneously. The second sample, based on the O star binary
population in six rich nearby open clusters, is more homogeneous and
is less susceptible to detection biases. There are some differences
in the CDFs of the two samples (see Fig.~\ref{fig:
CDF}), but two-sided Kolmogorov-Smirnoff tests do not reveal
statistically significant deviations. These differences can be
qualitatively understood in terms of different observational effects.
Currently, the observed CDFs for $P$, $q$
and $e$ of spectroscopic O-type binaries can be analytically described
by the following functions:
\begin{enumerate}
\item[-] {\em Periods:} a broken \"{O}pik Law with a break point at $P\sim10$~days,
\item[-] {\em Mass-ratios:} a uniform distribution down to $q=0.2$, potentially extending in the SB1 domain (i.e., for $q<0.2$),
\item[-] {\em Eccentricities:} 25-30\%\ of the characterised systems have circular orbits. $CDF(e>0)$ shows a
square-root dependance with $e$, but detailed considerations of bias are lacking at present.
\end{enumerate}
A quantitative analysis of the effects of the detection limit and of
other observational biases would be highly desirable (although not
trivial) in order to: (i) assess the completness and the exactness of
the observed CDFs; (ii) retrieve the underlying distributions. \\
In conclusion, significant progress has been made in the past two
decades but uncertainties on the exact multiplicity properties of
massive stars remain numerous. In particular, an homogeneous
exploration of the parameter space, the distribution function of the
orbital parameters and the impact of the environment on the
multiplicity properties are likely the areas in which observational
progresses are the most crucially needed.
Fortunately, numerous projects are currently underway which aim at
improving our knowledge of these aspects. It is our hope to have drawn
attention to the importance of a proper understanding of the detection
limits and of the observational biases that affect each survey. These
are necessary information to consider in order to glue all the
pieces together toward a global view of the massive star properties
across the full reach of parameter space and in different
environements.
\section*{Acknowledgments}
The authors warmly thank the organizers for their invitation and for their flexibility. The authors also wished to express their thanks to M. De Becker, A.\ de Koter, S. de Mink, M. Gieles, E. Gosset, P. Massey and S. Portegies Zwart for useful discussion in the preparation and redaction of this review.
\bibliographystyle{aa.bst}
|
{
"timestamp": "2010-09-23T02:00:15",
"yymm": "1009",
"arxiv_id": "1009.4197",
"language": "en",
"url": "https://arxiv.org/abs/1009.4197"
}
|
\section{Introduction}
For heavy quarkonium production and decay, a naive perturbative QCD and nonrelativistic factorization treatment is applied straightforwardly. It is called color-singlet mechanism (CSM). To describe the huge discrepancy of the high-$p_t$ $J/\psi$ production between the theoretical prediction based on CSM and the experimental measurement at Tevatron, a color-octet mechanism~\cite{Braaten:1994vv} was proposed based on the non-relativistic QCD (NRQCD)~\cite{Bodwin:1994jh}. In the application, ${J/\psi}$ or $\Upsilon$ related productions or decays are very good places for two reasons, theoretically charm and bottom quarks are thought to be heavy enough, so that charmonium and bottomonium can be treated within the NRQCD framework, experimentally there is a very clear signal to detect ${J/\psi}$ and $\Upsilon$. The key point is that the color-octet mechanism depends on nonperturbative universal NRQCD matrix elements, which is obtained by fitting the data. Therefore various efforts have been made to confirm this mechanism, or to fix the magnitudes of the universal NRQCD matrix elements. Although it seems to show qualitative agreements with experimental data, there are certain difficulties. A
review of the situation could be found in Refs.~\cite{Kramer:2001hh,Lansberg:2006dh}.
To explain the experimental
measurements~\cite{Abe:2001za,Aubert:2005tj} of $J/\psi$ production at the B factories, a
series of calculations~\cite{Zhang:2005cha,Gong:2009ng} in the CSM reveal that the next-to-leading order (NLO) QCD corrections give
the main contribution to the related processes. Together with the relativistic correction~\cite{Bodwin:2006ke},
it seems that most experimental data for $J/\psi$ production at the B factories could be understood.
Recent studies show that the NLO QCD correction also plays an
important role in $J/\psi$ production at RHIC~\cite{Brodsky:2009cf} and the
hadroproduction of $\chi_c$~\cite{Ma:2010vd}.
For the $J/\psi$ photoproduction, the $p_t$ and $z$ distributions can be described by the NLO calculations
in CSM~\cite{Kramer:1995nb} by choosing a small renormalization scale, but recent NLO calculations
in CSM \cite{Artoisenet:2009xh} show that the $p_t$ distributions of the production and polarization for
$J/\psi$ can not be well described when choosing a proper renormalization scale.
Although the complete calculation at NLO in COM~\cite{Butenschoen:2009zy} can account for the
experimental measurements on the $p_t$ distribution, it cannot extend to $J/\psi$ polarization case.
To further study the heavy quarkonium production mechanism, there are
many other efforts performed, such as NLO QCD correction to $J/\psi$ production associated with
photon~\cite{Li:2008ym}, QED contributions in $J/\psi$ hadroproduction~\cite{He:2009cq},
inclusive $J/\psi$ production from $\Upsilon$ decay~\cite{He:2009by},
double heavy quarkonium hadronproduction~\cite{Li:2009ug}, and NLO QCD correction to $J/\psi$ production from $Z$ decay~\cite{Li:2010xu}.
For the polarized heavy quarkonium hadroproduction, the leading order (LO) NRQCD prediction gives a sizable transverse polarization for ${J/\psi}$ production at high $p_t$ at Tevatron while the experimental measurement~\cite{Abulencia:2007us} gives slight longitudinal polarized result. The discrepancy was also found in $\Upsilon$ production. In a recent paper~\cite{Abazov:2008za}, the measurement on polarization of $\Upsilon$ production at Tevatron is presented and the NRQCD prediction~\cite{Braaten:2000gw} is not coincide with it. Within the NRQCD framework, higher order correction is thought to be an important way towards the solution of such puzzles. Recently, NLO QCD corrections to ${J/\psi}$ and $\Upsilon$ hadroproduction have been calculated~\cite{Campbell:2007ws,Qiao:2003ba,Artoisenet:2007xi,Gong:2008sn,Gong:2008hk}, and the results show that the NLO QCD corrections give significant enhancement to both total cross section and momentum distribution for the color-singlet channel. This would reduce the contribution of color-octet channel in the production. Also, it is found in Ref.~\cite{Gong:2008sn} that the polarizations for ${J/\psi}$ and $\Upsilon$ hadroproduction via color-singlet channel would change drastically from transverse polarization dominant at LO into longitudinal polarization dominant in the whole range of the
transverse momentum $p_t$ at NLO. It seems that these results open a door to the solution of the problem. But things are not always going as expected. The NLO QCD corrections to the ${J/\psi}$ production via S-wave color-octet states were studied in our previous work~\cite{Gong:2008ft}. It was found that the effect of NLO QCD correction is small and the discrepancy holds on. For the color-singlet part, the partial next-to-next-to-leading order (NNLO) calculations for $\Upsilon$ and $J/\psi$ hadroproduction show that the uncertainty from higher order QCD
correction~\cite{Artoisenet:2008fc} is quite large, therefore no definite conclusion can be made.
As we know, the contribution from the color-octet states is smaller in the $\Upsilon$ production than that in ${J/\psi}$ production, thus things may be different. In this paper, we present our calculation on NLO QCD corrections to $\Upsilon$ hadroproduction via S-wave color-octet states. New matrix elements are fitted and new prediction for the polarization status is presented.
This paper is organized as follows. In Sec. II, we give the LO cross section for the process.
The calculation of NLO QCD corrections are described in Sec. III. In Sec. IV, we present the formula in final integration to obtain the transverse momentum distribution of $\Upsilon$ production. Sec. V. is devoted to the description about the calculation of $\Upsilon$ polarization. The treatment of ${J/\psi}$ is discussed in Sec. VI. The numerical results are presented in Sec. VII, while the summary and discussion are given in Sec. VIII. In the Appendix, several details of the calculation are presented.
\section{The LO cross section}
\begin{figure*}
\center{
\includegraphics*[scale=0.8]{lo
\caption {\label{fig:lo}Typical Feynman diagrams for LO processes. $a$) Feynman diagrams for process (\ref{prs:lo_ggg}); $b$) Feynman diagrams for processes (\ref{prs:lo_gqq}) and (\ref{prs:lo_qqg}). Diagrams in groups $(a_1),~(a_2),~(b_1)$ and $(b_2)$ are absent for the ${\bigl.^1\hspace{-1mm}S^{(8)}_0}$ state. }}
\end{figure*}
According to the NRQCD factorization formalism,
the inclusive cross section for direct $\Upsilon$ production in hadron-hadron collision is expressed as
\begin{eqnarray}
\sigma[pp\rightarrow \Upsilon+X]&=\sum\limits_{i,j,k,n}\int \mathrm{d}x_1\mathrm{d}x_2 G_{i/p}G_{j/p} \nonumber\\
&\times\hat{\sigma}[i+j\rightarrow (b\bar{b})_n +k]{\langle\mathcal{O}^H_n\rangle}, \end{eqnarray} where $p$
is either a proton or an anti-proton, the indices $i, j,k$ run over all
the partonic species and $n$ denotes the color, spin and angular
momentum states of the intermediate $b\bar{b}$ pair. The
short-distance contribution $\hat{\sigma}$ can be perturbatively
calculated order by order in $\alpha_s$. The hadronic matrix elements
${\langle\mathcal{O}^H_n\rangle}$ are related to the hadronization from the state
$(b\bar{b})_n$ into $\Upsilon$ which are fully governed by the
non-perturbative QCD effects. In the following, $\hat{\sigma}$ represents the corresponding partonic cross section.
At LO, there are three partonic processes:
\begin{align}
&&g(p_1)+ g(p_2) \rightarrow {\Upsilon\bigl[\bigl.^1\hspace{-1mm}S^{(8)}_0,\bigl.^3\hspace{-1mm}S^{(8)}_1\bigr]}(p_3) + g(p_4) \tag{L1},\label{prs:lo_ggg} \\
&&g(p_1)+ q(p_2) \rightarrow {\Upsilon\bigl[\bigl.^1\hspace{-1mm}S^{(8)}_0,\bigl.^3\hspace{-1mm}S^{(8)}_1\bigr]}(p_3) + q(p_4) ,\label{prs:lo_gqq} \tag{L2} \\
&&q(p_1)+ \overline{q}(p_2) \rightarrow {\Upsilon\bigl[\bigl.^1\hspace{-1mm}S^{(8)}_0,\bigl.^3\hspace{-1mm}S^{(8)}_1\bigr]}(p_3) + g(p_4) .\label{prs:lo_qqg} \tag{L3}
\end{align}
where $q$ represents a sum over all possible light quarks or
anti-quarks: $u,~d,~s,~c,~\overline{u},~\overline{d},~\overline{s}$ and $\bar{c}$. In our calculation of $\Upsilon$ production, we take charm quark as light quark as an approximation. Typical Feynman diagrams for these three processes are shown in Fig.~\ref{fig:lo}. And the partonic differential cross sections in $n=4-2\epsilon$ dimension for LO processes can be obtained as
\begin{widetext}
\begin{eqnarray}
\displaystyle\frac{\mathrm{d}\hat{\sigma}^{B}(q\overline{q}\rightarrow {\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]} g)}
{\mathrm{d}\hat{t}}&=&
\displaystyle\frac{\pi^2\alpha_s^3{\langle\mathcal{O}^\Upsilon_8(\bigl.^3\hspace{-1mm}S_1)\rangle} [(\hat{t}-1)^2+(\hat{u}-1)^2][4\hat{t}^2-\hat{t}\hat{u}+4\hat{u}^2]}
{324m_b^5\hat{s}^2(\hat{s}-1)^2\hat{t}\hat{u}} + {\cal O}(\epsilon), \nonumber\\
\displaystyle\frac{\mathrm{d}\hat{\sigma}^{B}(gq\rightarrow {\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]} q)}
{\mathrm{d}\hat{t}}&=&
\displaystyle\frac{-\pi^2\alpha_s^3{\langle\mathcal{O}^\Upsilon_8(\bigl.^3\hspace{-1mm}S_1)\rangle} [(\hat{s}-1)^2+(\hat{u}-1)^2][4\hat{s}^2-\hat{s}\hat{u}+4\hat{u}^2]}
{864m_b^5\hat{s}^3(\hat{t}-1)^2\hat{u}} + {\cal O}(\epsilon), \nonumber\\
\displaystyle\frac{\mathrm{d}\hat{\sigma}^{B}(gg\rightarrow {\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]} g)}
{\mathrm{d}\hat{t}}&=&
\displaystyle\frac{\pi^2\alpha_s^3{\langle\mathcal{O}^\Upsilon_8(\bigl.^3\hspace{-1mm}S_1)\rangle} [(\hat{s}^2-1)^2+(\hat{t}^2-1)^2+(\hat{u}^2-1)^2-6\hat{s}\hat{t}\hat{u}-2] [19-27(\hat{s}\hat{t}+\hat{t}\hat{u}+\hat{u}\hat{s})]}
{1152m_b^5\hat{s}^2(\hat{t}-1)^2(\hat{u}-1)^2(\hat{s}-1)^2} + {\cal O}(\epsilon), \nonumber\\
\displaystyle\frac{\mathrm{d}\hat{\sigma}^{B}(q\overline{q}\rightarrow {\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]} g)}
{\mathrm{d}\hat{t}}&=&
\displaystyle\frac{5\pi^2\alpha_s^3{\langle\mathcal{O}^\Upsilon_8(\bigl.^1\hspace{-1mm}S_0)\rangle} [\hat{t}^2+\hat{u}^2]}
{216m_b^5\hat{s}^3(\hat{s}-1)^2} + {\cal O}(\epsilon), \nonumber\\
\displaystyle\frac{\mathrm{d}\hat{\sigma}^{B}(gq\rightarrow {\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]} q)}
{\mathrm{d}\hat{t}}&=&
\displaystyle\frac{-5\pi^2\alpha_s^3{\langle\mathcal{O}^\Upsilon_8(\bigl.^1\hspace{-1mm}S_0)\rangle} [\hat{s}^2+\hat{u}^2]}
{576m_b^5\hat{s}^2\hat{t}(\hat{t}-1)^2} + {\cal O}(\epsilon), \nonumber\\
\displaystyle\frac{\mathrm{d}\hat{\sigma}^{B}(gg\rightarrow {\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]} g)}
{\mathrm{d}\hat{t}}&=&
\displaystyle\frac{5\pi^2\alpha_s^3{\langle\mathcal{O}^\Upsilon_8(\bigl.^1\hspace{-1mm}S_0)\rangle} [\hat{s}^2\hat{t}^2+\hat{s}^2\hat{u}^2+\hat{t}^2\hat{u}^2+\hat{s}\hat{t}\hat{u}] [\hat{s}^4+\hat{t}^4+\hat{u}^4+1]}
{256m_b^5\hat{s}^3\hat{t}\hat{u}(\hat{t}-1)^2(\hat{u}-1)^2(\hat{s}-1)^2} + {\cal O}(\epsilon), \nonumber\\
\end{eqnarray}
\end{widetext}
by introducing three dimensionless kinematic variables:
\begin{equation}
\hat{s}=\displaystyle\frac{(p_1+p_2)^2}{4m_b^2},\quad \hat{t}=\displaystyle\frac{(p_1-p_3)^2}{4m_b^2}, \quad \hat{u}=\displaystyle\frac{(p_1-p_4)^2}{4m_b^2},
\end{equation}
and the reasonable approximation $M_{\Upsilon}=2m_b$ is taken. Our LO results are consistent with those in Ref.~\cite{Cho:1995ce}.
The LO total cross section is obtained by convoluting the partonic cross section
with the parton distribution function (PDF) in the proton:
\begin{eqnarray}
\sigma^B[pp\rightarrow {\Upsilon^{(8)}}+X]&=&\sum\limits_{i,j,k}\int \hat{\sigma}^B[i+j\rightarrow {\Upsilon^{(8)}} +k] \\
&\times & G_{i/p}(x_1,\mu_f)G_{j/p}(x_2,\mu_f)\mathrm{d}x_1\mathrm{d}x_2, \nonumber
\end{eqnarray}
where ${\Upsilon^{(8)}}$ denotes certain color-octet ${\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]}$ or ${\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]}$, $\mu_f$ is the factorization scale.
\section{The NLO cross section}
The NLO contributions can be written as a sum of two
parts: first is the virtual corrections which arise from loop
diagrams, the other is the real corrections caused by radiation of a
real gluon, or a gluon splitting into a light quark-antiquark pair,
or a light (anti)quark splitting into a light (anti) quark and a
gluon.
\subsection{Virtual corrections}
There exist ultraviolet (UV), infrared (IR) and Coulomb singularities in the calculation of the
virtual corrections. UV divergences from self-energy and triangle
diagrams are canceled by introducing renormalization. Here we adopt
the renormalization scheme used in Ref.~\cite{Klasen:2004tz}. The
renormalization constants $Z_m$, $Z_2$, $Z_{2l}$ and $Z_3$ which
correspond to bottom quark mass $m_b$, bottom-field $\psi_b$, light
quark field $\psi_q$ and gluon field $A^a_{\mu}$ are defined in the
on-mass-shell (OS) scheme while $Z_g$ for the QCD gauge coupling constant
$\alpha_s$ is defined in the
modified-minimal-subtraction ($\overline{\mathrm{MS}}$) scheme: \begin{eqnarray}
\delta Z_m^{OS}&=&-3C_F\displaystyle\frac{\alpha_s}{4\pi}\left[\displaystyle\frac{1}{\epsilon_{UV}} -\gamma_E +\ln\displaystyle\frac{4\pi \mu_r^2}{m_b^2} +\frac{4}{3}\right] ,\nonumber \\
\delta Z_2^{OS}&=&-C_F\displaystyle\frac{\alpha_s}{4\pi}\left[\displaystyle\frac{1}{\epsilon_{UV}} +\displaystyle\frac{2}{\epsilon_{IR}} -3\gamma_E +3\ln\displaystyle\frac{4\pi
\mu_r^2}{m_b^2} +4 \right] ,\nonumber \\
\delta Z_{2l}^{OS}&=&-C_F\displaystyle\frac{\alpha_s}{4\pi}\left[
\displaystyle\frac{1}{\epsilon_{UV}} -\displaystyle\frac{1}{\epsilon_{IR}}
\right] ,\nonumber \\
\delta Z_3^{OS}&=&\displaystyle\frac{\alpha_s}{4\pi}\left[(\beta_0-2C_A)\left(\displaystyle\frac{1}{\epsilon_{UV}} -\displaystyle\frac{1}{\epsilon_{IR}}\right)
\right] , \\
\delta
Z_g^{\overline{\mathrm{MS}}}&=&-\displaystyle\frac{\beta_0}{2}\displaystyle\frac{\alpha_s}{4\pi}\left[\displaystyle\frac{1}{\epsilon_{UV}}
-\gamma_E +\ln(4\pi)\right] \nonumber, \end{eqnarray} where $\gamma_E$ is the Euler's
constant, $\beta_0=\frac{11}{3}C_A-\frac{4}{3}T_Fn_f$ is the one-loop
coefficient of the QCD beta function and $n_f$ is the number of
active quark flavors. We have four light quarks $u$, $d$, $s$ and $c$ in our calculation, so $n_f$=4. The
color factors are given by $T_F=1/2, C_F=4/3, C_A=3$
and $\mu_r$ is the renormalization scale.
\begin{figure*}
\center{
\includegraphics*[scale=0.7]{nlo
\caption {\label{fig:nlo}Typical one-loop diagrams. $a$) Feynman diagrams for $gq\rightarrow {\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]} q$ and $q\bar{q}\rightarrow {\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]} g$; $a+b$) Feynman diagrams for $gq\rightarrow {\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]} q$ and $q\bar{q}\rightarrow {\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]} g$; $c$) Feynman diagrams for $gg\rightarrow {\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]} g$; $c+d$) Feynman diagrams for $gg\rightarrow {\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]} g$. Counter-term diagrams, together with corresponding loop diagrams, are not shown here. }}
\end{figure*}
There are 267 (for the ${\bigl.^1\hspace{-1mm}S^{(8)}_0}$ state) and 413 (for the ${\bigl.^3\hspace{-1mm}S^{(8)}_1}$ state) NLO diagrams for process (\ref{prs:lo_ggg}), including counter-term
diagrams, while for both processes (\ref{prs:lo_gqq}) and
(\ref{prs:lo_qqg}), there are 49 (for the ${\bigl.^1\hspace{-1mm}S^{(8)}_0}$ state) and 111 (for
the ${\bigl.^3\hspace{-1mm}S^{(8)}_1}$ state) NLO diagrams. Part of the Feynman diagrams for these processes are shown in Fig.~\ref{fig:nlo}. The diagrams in which a virtual
gluon line connects the quark pair possess Coulomb singularities,
which can be isolated and attributed into renormalization of the
$b\bar{b}$ wave function.
For each process, by summing over contributions from all diagrams,
the virtual correction to the differential cross section can be
expressed as
\begin{equation}
\displaystyle\frac{\mathrm{d}\hat{\sigma}^{V}_{[{\mathrm{L}_i}]}}{\mathrm{d}t}
\propto 2\mathrm{Re}\left(M^B_{[\mathrm{L}_i]}M^{V^*}_{[\mathrm{L}_i]}\right),
\label{eqn:virtual_sme}
\end{equation} where $M^B_{[\mathrm{L}_i]}$ is the
amplitude of process ($\mathrm{L}_i$) at LO, and $M^V_{[\mathrm{L}_i]}$ is the renormalized
amplitude of corresponding process at NLO. $M^V_{[\mathrm{L}_i]}$ is UV and Coulomb
finite, but it still contains IR divergences.
And the total cross section of virtual contribution could be written as
\begin{eqnarray}
\sigma^V[pp\rightarrow {\Upsilon^{(8)}}+X]&=&\sum\limits_{i,j,k}\int \hat{\sigma}^V[i+j\rightarrow {\Upsilon^{(8)}} +k] \\
&\times & G_{i/p}(x_1,\mu_f)G_{j/p}(x_2,\mu_f)\mathrm{d}x_1\mathrm{d}x_2, \nonumber
\end{eqnarray}
\subsection{Real corrections}
There are eight processes involved in the real corrections:
\begin{align}
gg&\rightarrow {\Upsilon\bigl[\bigl.^1\hspace{-1mm}S^{(8)}_0,\bigl.^3\hspace{-1mm}S^{(8)}_1\bigr]} gg, \tag{R1} \label{prs:gggg} \\
gq&\rightarrow{\Upsilon\bigl[\bigl.^1\hspace{-1mm}S^{(8)}_0,\bigl.^3\hspace{-1mm}S^{(8)}_1\bigr]} gq,\tag{R2}\label{prs:qggq} \\
q\overline{q}&\rightarrow {\Upsilon\bigl[\bigl.^1\hspace{-1mm}S^{(8)}_0,\bigl.^3\hspace{-1mm}S^{(8)}_1\bigr]} gg ,\tag{R3}\label{prs:qqgg} \\
gg&\rightarrow {\Upsilon\bigl[\bigl.^1\hspace{-1mm}S^{(8)}_0,\bigl.^3\hspace{-1mm}S^{(8)}_1\bigr]} q\overline{q},\tag{R4}\label{prs:ggqq} \\
q\overline{q}&\rightarrow {\Upsilon\bigl[\bigl.^1\hspace{-1mm}S^{(8)}_0,\bigl.^3\hspace{-1mm}S^{(8)}_1\bigr]} q\overline{q} ,\tag{R5}\label{prs:qqqq0} \\
q\overline{q}&\rightarrow {\Upsilon\bigl[\bigl.^1\hspace{-1mm}S^{(8)}_0,\bigl.^3\hspace{-1mm}S^{(8)}_1\bigr]} q'\overline{q}' ,\tag{R6}\label{prs:qqqq1} \\
qq&\rightarrow {\Upsilon\bigl[\bigl.^1\hspace{-1mm}S^{(8)}_0,\bigl.^3\hspace{-1mm}S^{(8)}_1\bigr]} qq ,\tag{R7}\label{prs:qqqq2} \\
qq'&\rightarrow {\Upsilon\bigl[\bigl.^1\hspace{-1mm}S^{(8)}_0,\bigl.^3\hspace{-1mm}S^{(8)}_1\bigr]} qq' ,\tag{R8}\label{prs:qqqq3}
\end{align}
where $q, q'$ denote light quarks (anti-quarks) with different
flavors. Feynman diagrams for these processes are shown in Fig.~\ref{fig:real}.
\begin{figure*}
\center{
\includegraphics*[scale=0.8]{real
\caption {\label{fig:real}Feynman diagrams for real correction processes. $a$) R1 (${\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]}$); $a+b$) R1 (${\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]}$); $c$) R2$\sim$R4 (${\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]}$); $c+d$) R2$\sim$R4 (${\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]}$); $e$) R5$\sim$R8 (${\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]}$); $e+f$) R5$\sim$R8 (${\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]}$). R1 (${\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]}$) denotes process $gg\rightarrow {\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]} gg$, R1 (${\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]}$) denotes process $gg\rightarrow {\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]} gg$, and so on.}}
\end{figure*}
We have neglected the contributions from the another two processes, $gg\rightarrow {\Upsilon^{(8)}} b\bar{b}$ and $q\bar{q}\rightarrow {\Upsilon^{(8)}} b\bar{b}$, which are IR finite and small.
Phase space integrations of above eight processes generate IR
singularities, which are either soft or collinear and can be
conveniently isolated by slicing the phase space into different
regions. We use the two-cutoff phase space slicing method
\cite{Harris:2001sx}, which introduces two small cutoffs to
decompose the phase space into three parts. Then the real cross
section can be written as \begin{equation}
\sigma^R=\sigma^S+\sigma^{HC}+\sigma^{H\overline{C}} .\end{equation}
It is easy
to observe that different parts of IR singularities from one real
process may be factorized and each part should be added into the
cross sections of different LO processes. This is the reason why we
have to calculate the NLO corrections to the three LO processes
together.
\subsubsection{soft}
Soft singularities arise from real gluon emission. Thus only real processes (\ref{prs:gggg}), (\ref{prs:qggq}) and (\ref{prs:qqgg}) contain soft singularities, corresponding to the three LO processes. One should notice that, unlike
color-singlet case, the soft singularities caused by emitting a soft gluon from the quark pair in the S-wave color octet exists and we find that the factorized matrix element is the same as the case of emitting a soft gluon from a gluon.
Suppose $p_5$ is the momentum of the emitted gluon. If we define the
Mandelstam invariants as $s_{ij}=(p_i+p_j)^2$ and $t_{ij}=(p_i-p_j)^2$,
the soft region is defined in term of the energy of $p_5$ in the $p_1+p_2$ rest frame
by $0 \leq E_5 \leq \delta_s \sqrt{s_{12}}/2$.
For each of the three real processes,
$\hat{\sigma}^S$ from the soft regions is calculated analytically under the soft approximation.
Following the similar factorization procedure as applied in the calculation of color-singlet case~\cite{Gong:2008hk}, the matrix elements for a certain real process ($\mathrm{R}_i$) in the soft region can be written as
\begin{equation}
|M_{[\mathrm{R}_i]}|^2|_{\rm soft} \simeq -4\pi\alpha_s \mu_r^{2\epsilon} \sum_{j,k=1}^4
\frac{-p_j \cdot p_k}{(p_j \cdot p_5)(p_k\cdot p_5)} M_{[\mathrm{L}_i]}^{jk} \, ,
\label{eqn:sme_soft}
\end{equation}
with
\begin{equation}
M_{[\mathrm{L}_i]}^{jk} =
\left[{\bf T}^a(j) {\bf M}^{b_1\cdots b_{j^\prime}\cdots b_4}_{[\mathrm{L}_i]}\right]^{\sp\dagger} \left[{\bf T}^a(k)
{\bf M}^{b_1\cdots b_{k^\prime}\cdots b_4}_{[\mathrm{L}_i]} \right] \,
\label{eqn:me0_cc}
\end{equation}
where ${\bf M}^{b_1\cdots b_4}_{[\mathrm{L}_i]}$ is the color connected Born matrix element for LO processes ($\mathrm{L}_i)$. If the emitting parton $j$ is an initial state quark or a final state antiquark, ${\bf T}^a(j)=T^a_{b_{j^\prime}b_j}$. For an initial state antiquark or a final state quark ${\bf T}^a(j)=-T^a_{b_jb_{j^\prime}}$. If the emitting parton $j$ is a gluon or the color-octet state, ${\bf T}^a(j)=if_{ab_jb_{j^\prime}}$.
And the corresponding parton level differential cross section can be expressed as
\begin{equation}
d\hat{\sigma}^S_{[\mathrm{R}_i]} =
\left[ \frac{\alpha_s}{2\pi} \frac{\Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)} \left( \frac{4\pi\mu_r^2}{s_{12}} \right)^\epsilon \right]
\sum_{j,k=1}^4 d\hat{\sigma}^{jk}_{[\mathrm{L}_i]} I^{jk} \, ,
\label{eqn:soft_final}
\end{equation}
with
\begin{equation}
d\hat{\sigma}_{[\mathrm{L}_i]}^{jk} = \frac{1}{2\Phi} \overline{\sum} M_{[\mathrm{L}_i]}^{jk} d\Gamma_2 \, .
\label{eqn:soft_lo_c_o}
\end{equation}
The factor $I^{jk}$ is universal for all three real processes, and is given in Appendix.~(\ref{chapter:I_jk}). Sometimes
$d\hat{\sigma}_{[\mathrm{L}_i]}^{jk}$ may be written in a more compact form as
\begin{equation}
d\hat{\sigma}_{[\mathrm{L}_i]}^{jk}=C^{jk}_{[\mathrm{L}_i]} d\hat{\sigma}^B_{[\mathrm{L}_i]},
\label{eqn:soft_lo_c}
\end{equation}
where $C^{jk}_{[\mathrm{L}_i]}$ is a constant. This is always true if the LO process ($\mathrm{L}_i$) contain only one independent color factor in the matrix element. But for processes with two or more than two independent color factors, there seems no sure reason for it to be or not to be true. Of course, no matter Eq.~(\ref{eqn:soft_lo_c}) is true or not, we can always obtain $d\hat{\sigma}_{[\mathrm{L}_i]}^{jk}$ through Eq.~(\ref{eqn:soft_lo_c_o}). Most processes involved in this calculation have more than one independent color factors, and they are listed in Appendix.~(\ref{chapter:color_factors_lo}).
\subsubsection{hard collinear}
The hard collinear regions of the phase space are those where any invariant
($s_{ij}$ or $t_{ij}$) becomes smaller in magnitude than $\delta_c s_{12}$.
It is treated according to whether the singularities are from initial or final
state emitting or splitting in the origin.
\paragraph{final state collinear}
For real processes (R1) $\sim$ (R6), which contain final state collinear singularities, the final state collinear
region is defined by $0 \le s_{45} \le \delta_c s_{12}$. Again following the similar factorization procedure described in
Ref~\cite{Harris:2001sx}, the parton level cross section in the hard final state collinear region can be expressed as
\begin{equation}
\hat{\sigma}^{HC}_{f}[\mathrm{R}_i]=\hat{\sigma}^B[\mathrm{L}_i^\prime]\left[ \frac{\alpha_s}{2\pi} \frac{\Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)} \left( \frac{4\pi\mu_r^2}{s_{12}} \right)^\epsilon \right]A^{HC}_{i}.
\end{equation}
For a certain real process ($\mathrm{R}_i$), ($\mathrm{L}_i^\prime$) is the corresponding LO process it factorizes into. And the coefficient $A^{HC}_{i}$ are listed in Table.~\ref{table:final_coll}, with
\begin{eqnarray}
A_1^{g \rightarrow gg} &=& N \left( 11/6 + 2 \ln\delta'_s \right) \nonumber\\
A_0^{g \rightarrow gg} &=& N \left[ 67/18 - \pi^2/3 - \ln^2\delta'_s
- \ln\delta_c \left( 11/6 + 2 \ln\delta'_s \right) \right] \nonumber\\
A_1^{q \rightarrow qg} &=& C_F \left( 3/2+2\ln\delta'_s \right) \nonumber\\
A_0^{q \rightarrow qg} &=& C_F \left[ 7/2 - \pi^2/3 - \ln^2\delta'_s - \ln\delta_c \left(3/2+2\ln\delta'_s \right) \right] \nonumber\\
A_1^{g \rightarrow q\overline{q}} &=& -n_f/3 \nonumber\\
A_0^{g \rightarrow q\overline{q}} &=& n_f/3 \left( \ln\delta_c-5/3 \right) \, ,
\end{eqnarray}
and
\begin{equation}
\delta'_s = \frac{s_{12}}{s_{12}+s_{45}-M^2_{\Upsilon}} \simeq \frac{\hat{s}}{\hat{s}-1} \delta_s \, .
\end{equation}
Thus the total cross section for real correction processes in hard final state
collinear region can be written as:
\begin{eqnarray}
\sigma^{HC}_f&=&\sum\limits_{i,j,k_1,k_2}\int \hat{\sigma}^{HC}_f[i+j\rightarrow {\Upsilon^{(8)}} +k_1 +k_2] \nonumber\\
&&\times G_{i/p}(x_1,\mu_f)G_{j/p}(x_2,\mu_f)\mathrm{d}x_1\mathrm{d}x_2 \nonumber\\
&=&\sum\limits_{i,j,k}\int \hat{\sigma}^B[i+j\rightarrow {\Upsilon^{(8)}} +k]B^{HC}(k) \nonumber\\
&&\times G_{i/p}(x_1,\mu_f)G_{j/p}(x_2,\mu_f)\mathrm{d}x_1\mathrm{d}x_2 ,
\end{eqnarray}
where
\begin{eqnarray}
B^{HC}(g)&=&\left[ \frac{\alpha_s}{2\pi} \frac{\Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)} \left( \frac{4\pi\mu_r^2}{s_{12}} \right)^\epsilon\right]
\\&&
\times\biggl( \frac{A_1^{g \rightarrow gg}+A_1^{g \rightarrow q\overline{q}}}{\epsilon} + A_0^{g \rightarrow gg}+A_0^{g \rightarrow q\overline{q}} \biggr) \, ,\nonumber\\
B^{HC}(q)&=&\left[ \frac{\alpha_s}{2\pi} \frac{\Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)} \left( \frac{4\pi\mu_r^2}{s_{12}} \right)^\epsilon\right]
\biggl( \frac{A_1^{q \rightarrow qg}}{\epsilon} + A_0^{q \rightarrow qg} \biggr) \, .\nonumber
\end{eqnarray}
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|c|c|c|}
\hline\hline
$\mathrm{R}_i$&$\mathrm{L}_i^\prime$&$A^{HC}_{i}$\\
\hline
$gg\rightarrow {\Upsilon^{(8)}} gg$&$gg\rightarrow {\Upsilon^{(8)}} g$ &$\displaystyle\frac{1}{\epsilon}A_1^{g\rightarrow gg}+A_0^{g\rightarrow gg}$ \\
\hline
$gq\rightarrow {\Upsilon^{(8)}} gq$ &$gq\rightarrow {\Upsilon^{(8)}} q$ &$\displaystyle\frac{1}{\epsilon}A_1^{q\rightarrow qg}+A_0^{q\rightarrow qg}$\\
\hline
$gg\rightarrow {\Upsilon^{(8)}} q\bar{q}$ &$gg\rightarrow {\Upsilon^{(8)}} g$&$\displaystyle\frac{1}{\epsilon}A_1^{g\rightarrow q\bar{q}}+A_0^{g\rightarrow q\bar{q}}$ \\
\hline
$q\bar{q}\rightarrow {\Upsilon^{(8)}} gg$ &$q\bar{q}\rightarrow {\Upsilon^{(8)}} g$&$\displaystyle\frac{1}{\epsilon}A_1^{g\rightarrow gg}+A_0^{g\rightarrow gg}$\\
\hline
$q\bar{q}\rightarrow {\Upsilon^{(8)}} q\bar{q}$ &$q\bar{q}\rightarrow {\Upsilon^{(8)}} g$ &$\displaystyle\frac{1}{n_f}\left(\displaystyle\frac{1}{\epsilon}A_1^{g\rightarrow q\bar{q}}+A_0^{g\rightarrow q\bar{q}}\right)$\\
\hline
$q\bar{q}\rightarrow {\Upsilon^{(8)}} q'\bar{q}'$ &$q\bar{q}\rightarrow {\Upsilon^{(8)}} g$ &$\left(1-\displaystyle\frac{1}{n_f}\right)\left(\displaystyle\frac{1}{\epsilon}A_1^{g\rightarrow q\bar{q}}+A_0^{g\rightarrow q\bar{q}}\right)$\\
\hline\hline
\end{tabular}
\caption{The hard final state collinear factors for real correction processes and the corresponding LO processes.}
\label{table:final_coll}
\end{center}
\end{table}
\paragraph{initial state collinear}
Almost all real processes, except process (R6), contain hard initial state collinear singularities. These singularities are partly absorbed into the redefinition of the parton distribution
function (PDF) of the concerned hadrons (usually it is called as the
mass factorization \cite{Altarelli:1979ub}). Here we adopt the scale
dependent PDF using the $\overline{\rm MS}$ convention given in
Ref~\cite{Harris:2001sx}.
\begin{eqnarray}
G_{b/p}(x,\mu_f)&=&G_{b/p}(x)-\frac{1}{\epsilon} \left[
\frac{\alpha_s}{2\pi} \frac{\Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)}
\left(\frac{4\pi \mu_r^2}{\mu_f^2}\right)^{\epsilon}\right] \nonumber\\&&\times
\int_x^1 \frac{\mathrm{d} z}{z} P_{bb'}(z)G_{b'/p}(x/z) \, . \label{eqn:PDF}
\end{eqnarray}
The second term is sometimes referred as the mass factorization counter-term. There is still something remaining after the cancellation, which can be expressed in two terms. The first one, which only exists in the real processes with final state gluon, can be expressed as
\begin{eqnarray}
\hat{\sigma}^{HC}_{i}[\mathrm{R}_i]&=&\hat{\sigma}^B[\mathrm{L}_i]\left[ \frac{\alpha_s}{2\pi} \frac{\Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)} \left( \frac{4\pi\mu_r^2}{\mu_f} \right)^\epsilon \right]A^{SC}_i , \nonumber
\end{eqnarray}
with
\begin{eqnarray}
A^{SC}_1&=& 2A^{SC}(g\rightarrow gg) \nonumber\\
A^{SC}_2&=& A^{SC}(q\rightarrow qg)+A^{SC}(g\rightarrow gg) \nonumber\\
A^{SC}_3&=& 2A^{SC}(q\rightarrow qg) ,
\end{eqnarray}
and
\begin{eqnarray}
A^{SC}(q\rightarrow qg) &=& \displaystyle\frac{1}{\epsilon}C_F\left[3/2+2\ln(\delta_s) \right] \nonumber\\
A^{SC}(g\rightarrow gg) &=& \displaystyle\frac{1}{\epsilon}\left[2N \ln \delta_s + (11N-2 n_f)/6\right] \,.
\end{eqnarray}
The corresponding hadronic total cross section is
\begin{eqnarray}
\sigma^{HC}_i&=&\sum\limits_{i,j,k}\int \hat{\sigma}^{HC}_i[i+j\rightarrow {\Upsilon^{(8)}} +k+g] \nonumber\\
&&\times G_{i/p}(x_1,\mu_f)G_{j/p}(x_2,\mu_f)\mathrm{d}x_1\mathrm{d}x_2 \nonumber\\
&=&\sum\limits_{i,j,k}\int \hat{\sigma}^B[i+j\rightarrow {\Upsilon^{(8)}} +k][B^{SC}(i)+B^{SC}(j)] \nonumber\\
&&\times G_{i/p}(x_1,\mu_f)G_{j/p}(x_2,\mu_f)\mathrm{d}x_1\mathrm{d}x_2 ,
\end{eqnarray}
with
\begin{eqnarray}
B^{SC}(g)&=&\left[ \frac{\alpha_s}{2\pi} \frac{\Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)} \left( \frac{4\pi\mu_r^2}{\mu_f} \right)^\epsilon \right] A^{SC}(g\rightarrow gg) \nonumber\\
B^{SC}(q)&=&\left[ \frac{\alpha_s}{2\pi} \frac{\Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)} \left( \frac{4\pi\mu_r^2}{\mu_f} \right)^\epsilon \right] A^{SC}(q\rightarrow qg) .\nonumber
\end{eqnarray}
The other term is obtained by summing up the remaining contributions from all the real correction processes. It can be written as
\begin{eqnarray}
&&\sigma^{HC}_{add}\left[pp\rightarrow {\Upsilon^{(8)}}+X\right]\\
&\equiv&\sum\limits_{i,j,k}\int \hat\sigma^B\left[ij\rightarrow {\Upsilon^{(8)}}+k\right] \left[
\frac{\alpha_s}{2\pi} \frac{\Gamma(1-\epsilon)}{\Gamma(1-2\epsilon)} \left(\frac{4
\pi \mu_r^2}{s_{12}}\right)^{\epsilon}\right] \nonumber
\\&&\times
\biggl[G_{i/p}(x_1,\mu_f)\widetilde{G}_{j/p}(x_2,\mu_f) +(x_1\leftrightarrow x_2)\biggr] \mathrm{d} x_1 \mathrm{d} x_2, \nonumber
\end{eqnarray}
with
\begin{equation} \widetilde{G}_{c/p}(x,\mu_f) =
\sum_{c'} \int_x^{1-\delta_s\delta_{cc'}} \frac{dy}{y} G_{c'/p}(x/y,\mu_f)
\widetilde{P}_{cc'}(y) \, , \label{eqn:g_tilde} \end{equation}
and
\begin{equation}
\widetilde{P}_{ij}(y) = P_{ij}(y)\ln\left(\delta_c\frac{1-y}{y}
\frac{s_{12}}{\mu_f^2}\right) - P_{ij}^{\prime}(y) \, . \end{equation}
The $n$-dimensional unregulated ($y<1$) splitting functions $P_{ij}(y,\epsilon)$ has been written as $P_{ij}(y,\epsilon)=P_{ij}(y)+\epsilon P_{ij}^\prime(y)$ with
\begin{eqnarray}
P_{qq}(y) &=& C_F \frac{1+y^2}{1-y} ,\nonumber \\
P_{qq}^{\prime}(y) &=& -C_F(1-y) ,\nonumber\\
P_{gq}(y) &=& C_F \frac{1+(1-y)^2}{y} ,\nonumber \\
P_{qq}^{\prime}(y) &=& -C_Fy ,\nonumber\\
P_{gg}(y) &=& 2N\left[ \frac{y}{1-y}+\frac{1-y}{y}+y(1-y)\right] ,\nonumber\\
P_{gg}^{\prime}(y) &=& 0 ,\nonumber\\
P_{qg}(y) &=& \frac{1}{2} \left[ y^2+(1-y)^2 \right] ,\nonumber\\
P_{qg}^{\prime}(y) &=& -y(1-y) \, .
\end{eqnarray}
\subsection{Cross section of all NLO contributions}
The hard noncollinear part $\sigma^{H\overline{C}}$ is IR finite and can be numerically computed using the standard Monte-Carlo integration techniques. Now the real cross section can be expressed as
\begin{equation}
\sigma^R=\sigma^S+\sigma^{HC}_f+\sigma^{HC}_i+\sigma^{HC}_{add}+\sigma^{H\bar{C}}.
\end{equation}
And we have
\begin{equation}
\sigma^{NLO} =\sigma^B+\sigma^V+\sigma^R.
\end{equation}
\section{Transverse momentum distribution}
To obtain the transverse momentum $p_t$ distribution of $\Upsilon$, a similar
transformation for integration variables ($\mathrm{d} x_2 \mathrm{d} t \rightarrow
J\mathrm{d} p_t \mathrm{d} y$) which we introduced in our previous work \cite{Gong:2008hk} is applied. Therefore we have \begin{eqnarray} \displaystyle\frac{\mathrm{d} \sigma}{\mathrm{d}
p_t}= \sum_{i,j} \int J \mathrm{d} x_1 \mathrm{d} y G_{i/p}(x_1,\mu_f)G_{j/p}(x_2,\mu_f)
\displaystyle\frac{\mathrm{d} \hat \sigma}{\mathrm{d} t}, \end{eqnarray}
with
\begin{eqnarray}
&p_1=x_1\displaystyle\frac{\sqrt{S}}{2}(1,0,0,1),
&p_2=x_2\displaystyle\frac{\sqrt{S}}{2}(1,0,0,-1), \nonumber\\[3mm]
&m_t=\sqrt{M_{\Upsilon}^2+p_t^2},
&p_3=(m_t \cosh y,p_t,0,m_t \sinh y),\nonumber\\[3mm]
&x_t=\displaystyle\frac{2m_t}{\sqrt{S}},
&\tau=\displaystyle\frac{m_4^2-M_{\Upsilon}^2}{\sqrt{S}},\\[3mm]
&J=\displaystyle\frac{4 x_1 x_2 p_t}{2x_1-x_t e^y},
&x_2=\displaystyle\frac{2\tau+x_1~x_t e^{-y}}{2 x_1-x_t e^y}, \nonumber\\[3mm]
&x_1|_{min}=\displaystyle\frac{2 \tau + x_t e^y}{2- x_t e^{-y}},&\nonumber
\end{eqnarray}
where $\sqrt{S}$ is the center-of-mass energy of $p\bar{p}(p)$ at Tevatron or LHC,
$m_4$ is the invariant mass of all the final state particles except $\Upsilon$, and
$y$ and $p_t$ are the rapidity and transverse momentum of $\Upsilon$ in the laboratory frame respectively.
\section{Polarization}
The polarization parameter $\alpha$ is defined as:
\begin{equation}
\alpha(p_t)=\frac{{\mathrm{d}\sigma_T}/{\mathrm{d} p_t}-2 {\mathrm{d}\sigma_L}/{\mathrm{d} p_t}}
{{\mathrm{d}\sigma_T}/{\mathrm{d} p_t}+2 {\mathrm{d}\sigma_L}/{\mathrm{d} p_t}}.
\end{equation}
It represents the measurement of $\Upsilon$ polarization as
function of $\Upsilon$ transverse momentum $p_t$ when calculated at
each point in $p_t$ distribution. To evaluate $\alpha(p_t)$, the polarization of $\Upsilon$ must
be explicitly retained in the calculation. The partonic differential
cross section for a polarized $\Upsilon$ is expressed as: \begin{equation}
\displaystyle\frac{\mathrm{d} \hat{\sigma}_{\lambda}}{\mathrm{d} t}= a~\epsilon(\lambda) \cdot
\epsilon^*(\lambda) + \sum_{i,j=1,2} a_{ij} ~p_i \cdot
\epsilon(\lambda) ~p_j \cdot \epsilon^*(\lambda), \label{eqn:polar}
\end{equation} where $\lambda=T_1,T_2,L$.
$\epsilon(T_1),~\epsilon(T_2),~\epsilon(L)$ are the two transverse
and longitudinal polarization vectors of $\Upsilon$ respectively, and
the polarizations of all the other particles are summed over in
n-dimension. One can find that $a$ and $a_{ij}$ are finite when the
virtual corrections and real corrections are properly handled as
aforementioned. Therefore there is no difference
in the differential cross section ${\mathrm{d} \hat{\sigma}_{\lambda}}/{\mathrm{d} t}$
whether the polarization of $\Upsilon$ is summed over in 4 or $n$
dimensions. Thus we can just treat the polarization vectors of
$\Upsilon$ in 4-dimension, and also the spin average factor goes back
to 4-dimension. The gauge invariance is explicitly checked by
replacing the gluon polarization vector into its 4-momentum in the
final numerical calculation.
\section{Treatment of ${J/\psi}$}
The production mechanism of ${J/\psi}$ at Tevatron and LHC is much
similar to that of $\Upsilon$ except that, color-octet states
contribute much more in ${J/\psi}$ production according to the
experimental data and LO theoretical predictions. The results of above calculation can also be applied to the case of ${J/\psi}$ by doing the
substitutions: \begin{eqnarray}
m_b&\leftrightarrow& m_c \nonumber\\
M_{\Upsilon}&\leftrightarrow& M_{J/\psi} \nonumber\\
R_s(0)^{\Upsilon}&\leftrightarrow& R_s(0)^{J/\psi} \\
n_f=4 &\leftrightarrow& n_f=3 \nonumber \end{eqnarray}
Note that in ${J/\psi}$ production, charm quark is no longer treated as light quark.
\section{numerical result}
In our numerical computations, the CTEQ6L1 and CTEQ6M
PDFs \cite{cteq}, and the corresponding fitted value $\alpha_s(M_Z)=0.130$ and
$\alpha_s(M_Z)=0.118$ are used for LO and NLO calculations respectively.
The bottom quark mass is set as $4.75 \mathrm{~GeV}$.
The choice of the renormalization scale $\mu_r$ and factorization scale $\mu_f$ is an important issue in the calculations, and it causes uncertainties. We choose $\mu=\mu_r=\mu_f=\sqrt{(2m_b)^2+p_t^2}$ as our default choice.
And the center-of-mass energies are chosen as 1.96 TeV at Tevatron and 14 TeV at LHC.
At First, different values of the two cutoffs, $\delta_s$ and $\delta_c$, are used to to check the independence of the final results on the cutoffs and the invariance is observed within the error tolerance. Then the two phase space cutoffs are fixed as $\delta_s=10^{-3}$ and $\delta_c=\delta_s/50$ in the following calculations.
It is known that the QCD perturbative expansion is not good in the regions
of small transverse momentum or large rapidity of $\Upsilon$. Therefore, the results are
restricted in the region $p_t>3$. For the rapidity cut, $|y_{\Upsilon}|<1.8$ is chosen at the Tevatron,
the same cut condition as the experiments~\cite{Abazov:2005yc}, and at the LHC, it is chosen to be $|y|<3$.
To fix the NRQCD matrix elements for color-octet states of $\Upsilon(1S)$, the D0 data~\cite{Abazov:2005yc} is used, and the fitting starts from Eq.(4) of Ref.~\cite{Braaten:2000cm} where the contributions from spin-singlet states $\eta_b(nS)$ and $h_b(nS)$ are not included. And we have to take a few approximations in our fitting procedure:
\begin{itemize}
\item For the S-wave color-singlet part, only the direct color-singlet $\Upsilon(1S)$ and feed-down from $\Upsilon(2S)$ are considered, while other contributions have been neglected. The contribution from the feed-down of $\Upsilon(2S)$ can be included to the direct $\Upsilon(1S)$ production by multiplying a factor of $Br[\Upsilon(2S)\rightarrow\Upsilon(1S)+X]\times\ME{\Upsilon}{1}{(2S)}/\ME{\Upsilon}{1}{(1S)}$, which results in a factor of 1.127 after a short calculation with PDG data~\cite{Amsler:2008zzb}. And the results for direct $\Upsilon(1S)$ of color-singlet contribution are extracted from our previous work~\cite{Gong:2008hk}.
\item The contributions from P-wave color-singlet states $\chi_{bJ}(nP)$ are estimated by multiplying a decay fraction
$F^{\Upsilon(1S)}_{\chi_b(nP)}\approx F^{\Upsilon(1S)}_{\chi_b(1P)}+F^{\Upsilon(1S)}_{\chi_b(2P)}$, where $F^{\Upsilon(1S)}_{\chi_b(1P)}$ and $F^{\Upsilon(1S)}_{\chi_b(2P)}$ can be obtained from an older sample with the cuts
$p_t>8$ and $|y_{\Upsilon}|<0.4$~\cite{Affolder:1999wm}. As pointed out in Ref.~\cite{Artoisenet:2008fc}, the fraction should not depend very strongly on $p_t$ according to Fig.2 of Ref.~\cite{Acosta:2001gv}. Also, from Fig.4 of Ref.~\cite{Abazov:2005yc} we can see it should not depend very strongly on the rapidity cut either. Thus $F^{\Upsilon(1S)}_{\chi_b(1P)}=27.1\pm 6.9\pm4.4\%$ and $F^{\Upsilon(1S)}_{\chi_b(2P)}=10.5\pm4.4\pm1.4\%$ are taken in our calculation, which result in $F^{\Upsilon(1S)}_{\chi_b(nP)}\approx37.6\pm9.4\%$.
\item The contribution from P-wave color-octet states $\Upsilon[{\bigl.^3\hspace{-1mm}P^{(8)}_J}]$ at NLO are still not available. As shown below, the NLO QCD corrections to ${\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]}$ don't change the cross section very much. If we assume that the NLO QCD corrections to $\Upsilon[{\bigl.^3\hspace{-1mm}P^{(8)}_J}]$ are also small, we can mix it with ${\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]}$ again, like what we have done at LO. Thus, the value of our fitted ${\langle\mathcal{O}^\Upsilon_8(\bigl.^1\hspace{-1mm}S_0)\rangle}_{\mathrm{inc}}$ includes the contributions from $\Upsilon[{\bigl.^3\hspace{-1mm}P^{(8)}_J}]$ as well.
\end{itemize}
With these approximations, the formula we used for the fitting of inclusive color matrix elements becomes
\begin{eqnarray}
d\sigma[\Upsilon]_{\mathrm{inc}}&=&1.127\times d\sigma[(b\bar{b})_1(\OP{3}S{1})]\ME{\Upsilon}{1}{(\OP{3}S{1})} \nonumber\\
&+&F^{\Upsilon(1S)}_{\chi_b(nP)}d\sigma[\Upsilon]_{\mathrm{inc}}
+d\sigma[(b\bar{b})_8(\OP{1}S{0})]\ME{\Upsilon}{8}{(\OP{1}S{0})}_{\mathrm{inc}} \nonumber\\
&+&d\sigma[(b\bar{b})_8(\OP{3}S{1})]\ME{\Upsilon}{8}{(\OP{3}S{1})}_{\mathrm{inc}} ,
\label{eqn:fit}
\end{eqnarray}
and the NRQCD matrix elements for color-octet states $\ME{\Upsilon}{8}{}_{\mathrm{inc}}$ are determined as
\begin{eqnarray}
&&{\langle\mathcal{O}^\Upsilon_8(\bigl.^1\hspace{-1mm}S_0)\rangle}_{\mathrm{inc}}=(0.948\pm0.444)\times 10^{-2} \mathrm{~GeV}^3 \nonumber\\
&&{\langle\mathcal{O}^\Upsilon_8(\bigl.^3\hspace{-1mm}S_1)\rangle}_{\mathrm{inc}}=(4.834\pm0.719)\times 10^{-2} \mathrm{~GeV}^3,
\end{eqnarray}
where only the uncertainty in $F^{\Upsilon(1S)}_{\chi_b(nP)}$ has been considered.
The fitting is shown in Fig.~\ref{fig:fit_inc}, together with our prediction for inclusive $\Upsilon$ production at the LHC.
The direct fraction of direct $\Upsilon$ production can also obtained from Ref.~\cite{Affolder:1999wm} as $F^{\Upsilon(1S)}_{\mathrm{dir}}=50.9\pm 12.2 \%$. Thus we can use the formula
\begin{eqnarray}
F^{\Upsilon(1S)}_{\mathrm{dir}}d\sigma[\Upsilon]_{\mathrm{inc}}&=&d\sigma[(b\bar{b})_1(\OP{3}S{1})]\ME{\Upsilon}{1}{(\OP{3}S{1})} \nonumber\\
&+&d\sigma[(b\bar{b})_8(\OP{1}S{0})]\ME{\Upsilon}{8}{(\OP{1}S{0})} \nonumber\\
&+&d\sigma[(b\bar{b})_8(\OP{3}S{1})]\ME{\Upsilon}{8}{(\OP{3}S{1})} ,
\label{eqn:fit2}
\end{eqnarray}
to fit the direct color-octet matrix elements. The matrix elements are obtained as
\begin{eqnarray}
&&{\langle\mathcal{O}^\Upsilon_8(\bigl.^1\hspace{-1mm}S_0)\rangle}=(0.630\pm0.576)\times 10^{-2} \mathrm{~GeV}^3 \nonumber\\
&&{\langle\mathcal{O}^\Upsilon_8(\bigl.^3\hspace{-1mm}S_1)\rangle}=(3.900\pm1.063)\times 10^{-2} \mathrm{~GeV}^3,
\end{eqnarray}
where the uncertainty comes only from $F^{\Upsilon(1S)}_{\mathrm{dir}}$. Again the value of our fitted ${\langle\mathcal{O}^\Upsilon_8(\bigl.^1\hspace{-1mm}S_0)\rangle}$ includes the contribution from $\Upsilon[{\bigl.^3\hspace{-1mm}P^{(8)}_J}]$. This fitting is shown in Fig.~\ref{fig:fit_direct}, together with our prediction for direct $\Upsilon$ production at the LHC. The band in the figure is obtained from the uncertainty of $F^{\Upsilon(1S)}_{\mathrm{dir}}$.
\begin{figure}
\center{
\includegraphics*[scale=0.45]{fit_inc
\caption {\label{fig:fit_inc}Transverse momentum distribution of inclusive $\Upsilon$ production at Tevatron and LHC.
The D0 data is from Ref.~\cite{Abazov:2005yc}.}}
\end{figure}
\begin{figure}
\center{
\includegraphics*[scale=0.45]{fit_direct
\caption {\label{fig:fit_direct}Transverse momentum distribution of direct $\Upsilon$ production at Tevatron and LHC.
The D0 data is from Ref.~\cite{Abazov:2005yc}.}}
\end{figure}
The dependence of the total cross section on the renormalization
scale $\mu_r$ and factorization scale $\mu_f$ are shown in
Fig.~\ref{fig:total}. It is obvious that the NLO QCD corrections
make such dependence milder. We can also see that the NLO QCD corrections effect the cross section lesser at the LHC than at the Tevatron.
\begin{figure}
\center{
\includegraphics*[scale=0.45]{total_1s0
\\
\includegraphics*[scale=0.45]{total_3s1
\caption {\label{fig:total}Total cross section of $\Upsilon$ hadroproduction at LHC (upper curves) and Tevatron (lower curves), as function of $\mu$ with $\mu_r=\mu_f=\mu$ and $\mu_0=\sqrt{(2m_b)^2+p_t^2}$.
}}
\end{figure}
The $p_t$ distributions of $\Upsilon$ production via S-wave color-octet states are presented in Figs.~\ref{fig:pt_t} and ~\ref{fig:pt_l}, where only slight changes appear when the NLO QCD corrections are included.
\begin{figure}
\center{
\includegraphics*[scale=0.45]{pt1
\caption {\label{fig:pt_t}Transverse momentum distribution of $\Upsilon$ production
with $\mu_r=\mu_f=\mu_0$ at the Tevatron.}}
\end{figure}
\begin{figure}
\center{
\includegraphics*[scale=0.45]{pt2
\caption {\label{fig:pt_l}Transverse momentum distribution of $\Upsilon$ production
with $\mu_r=\mu_f=\mu_0$ at the LHC.}}
\end{figure}
${\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]}$ produces unpolarized $\Upsilon$, so it contributes to
$\alpha=0$ for both LO and NLO. The $p_t$ distributions of $\Upsilon$
polarization parameter $\alpha$ from ${\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]}$ are shown in
Fig.~\ref{fig:polar_dir} and there is slight change when the NLO
corrections are taken into account. Our predictions for the polarization of direct $\Upsilon$ production are also presented in the figure as a "total" result. In Fig.~\ref{fig:polar_inc}, the polarization of inclusive $\Upsilon$ production at the Tevatron is shown. As the polarization of $\Upsilon$ from the feed-down of $\chi_b(nP)$ is not available yet, a huge band is obtained by verifying the polarization of this part between -1 to 1. The experimental data from the D0 is also shown in the same figure. We can see that, there is still some distance between the theoretical prediction and experimental measurement, even with such a large band.
\begin{figure}
\center{
\includegraphics*[scale=0.45]{polt_dir}\\% Here is how to import EPS art
\includegraphics*[scale=0.45]{poll_dir
\caption {\label{fig:polar_dir}Transverse momentum distribution of polarization parameter
$\alpha$ for direct $\Upsilon$ production at the Tevatron (upper) and LHC (lower).}}
\end{figure}
\begin{figure}
\center{
\includegraphics*[scale=0.45]{polar_inc
\caption {\label{fig:polar_inc}Transverse momentum distribution of polarization parameter
$\alpha$ for inclusive $\Upsilon$ production at the Tevatron. The D0 data is from ref~\cite{Abazov:2008za}.}}
\end{figure}
\section{Summary and Discussion}
As a summary, in this work, we have calculated the NLO QCD
corrections to $\Upsilon$ production via S-wave color-octet states ${\Upsilon\bigl[\bigl.^1\hspace{-1mm}S^{(8)}_0,\bigl.^3\hspace{-1mm}S^{(8)}_1\bigr]}$ at the
Tevatron and LHC. With $\mu_r=\mu_f=\mu_0$, the K factors of total
cross section (ratio of NLO to LO) are 1.313 and 1.379 for ${\Upsilon\bigl[^1\hspace{-1mm}S^{(8)}_0\bigr]}$
and ${\Upsilon\bigl[^3\hspace{-1mm}S^{(8)}_1\bigr]}$ at Tevatron, while at LHC they are 1.044 and 1.182
respectively. Unlike for the color-singlet case, there are only
slight changes to the transverse momentum distributions of $\Upsilon$
production and the $\Upsilon$ polarization when the NLO QCD corrections
are taken into account. All the results imply that the perturbative
QCD expansion quickly converges for $\Upsilon$ production via the
S-wave color-octet states, in contrast with that via color-singlet,
where the NLO contributions are too large to hint a good convergence at
the NNLO.
By fitting the experimental data from the D0 at the Tevatron, the matrix elements for S-wave color-octet states are obtained. And new predictions for the $p_t$ distributions of the $\Upsilon$ production and polarization at the Tevatron and LHC are presented. The prediction for the polarization of inclusive $\Upsilon$ contains large uncertainty rising from the polarization of $\Upsilon$ from feed-down of $\chi_b$. Even with such a large uncertainty, there are still some distance between the prediction and experiment data.
Also, the errors of fractions used in the fitting, $F^{\Upsilon(1S)}_{\chi_b(1P)}$, $F^{\Upsilon(1S)}_{\chi_b(2P)}$ and $F^{\Upsilon(1S)}_{\mathrm{dir}}$, are quite large and result large uncertainty in the matrix elements. New measurements on the production and also polarization for direct $\Upsilon$ are expected. This would make the matrix elements more precise, and get rid of the large uncertainty from $\chi_b$.
This work is supported by the National Natural Science Foundation of
China (No.~10475083, 10979056 and 10935012), by the Chinese Academy of Science under
Project No. INFO-115-B01, and by the China Postdoctoral Science foundation
No.~20090460535.
|
{
"timestamp": "2010-09-21T02:03:33",
"yymm": "1009",
"arxiv_id": "1009.3839",
"language": "en",
"url": "https://arxiv.org/abs/1009.3839"
}
|
\section{Introduction} \label{sec:intro}
The molecular gas in the Galaxy exists in giant complexes ({\it giant
molecular clouds}, or GMCs) of masses $\sim 10^{5-6} M_\odot$, sizes of
several tens of parsecs, and mean densities $n \sim 100 {\rm ~cm}^{-3}$
\citep[see, e.g., the review by][]{1993prpl.conf..125B}, and contain a
large amount of substructure (parsec-scale {\it
clumps} of densities $n \sim 10^3 {\rm ~cm}^{-3}$, and sub-parsec-scale {\it
cores} of densities $n > 10^4 {\rm ~cm}^{-3}$). Molecular
clouds (MCs) contain roughly half the gaseous mass in the
Galaxy, and are the sites of all present-day star formation (SF) in the
Galaxy. Thus, the study of their origin and evolution is crucial for our
understanding of SF, besides the importance, on its own right, of
understanding this fundamental component of the ISM.
The seminal paper by \citet{1954BAN....12..177O} postulated the
existence of a cycle (now known as the {\it Oort cycle}), in which the
expanding HII regions around newly formed massive stars form shells of
cold gas around them. The shells subsequently fragment and produce a
population of cloudlets, which then grow by coagulation until they
become gravitationally unstable, at which point they proceed to collapse
and form a new generation of stars, starting the cycle all over
again. In this model, which was later developed by
\citet{1965ApJ...142..568F}, clouds were assumed to grow exclusively by
coagulation of randomly-moving small cloudlets. The same cloud growth
process was assumed in later models, such as that by
\citet{1980ApJ...238..158N}, which differed from the Oort model mainly
in that the driving agent was considered to be winds from low-mass,
T-Tauri stars rather than the ionizing radiation from massive stars, and
the model by \citet{1989ApJ...345..782M} for the SF rate (SFR) regulated
by the background photo-ionizing radiation. The coagulation process
implied very long ($\gtrsim 10^8$ yr) cloud growth times
\citep{1979ApJ...229..578S, 1979ApJ...229..567K}, which were however
ruled out on the basis of observational evidence by
\citet{1980ApJ...238..148B}. These authors proposed instead that MCs
form and grow by a \citet{1966ApJ...145..811P} instability triggered
by spiral-arm shocks, and have lifetimes $\sim 10^7$ yr.
Ever since the times of those early models, MCs have been an odd
component of the ISM. Because they are known to be strongly
self-gravitating \citep{1981MNRAS.194..809L, 1988ApJ...326L..27M} and at
significantly higher thermal pressures than the mean ISM pressure
\citep[e.g.,][]{1978ApJ...225..380M}, they did not fit in
thermal-pressure balance models of the ISM, such as that by
\citet{1977ApJ...218..148M}. Instead, they have traditionally been
considered to be in approximate virial equilibrium
\citep{1981MNRAS.194..809L, 1988ApJ...326L..27M}, supported against
their self-gravity by either the magnetic field \citep[the so-called
``standard'' model of magnetically regulated SF; see, e.g., the reviews
by][]{1987ARA&A..25...23S, 1991psfe.conf..449M}, or by turbulence driven
by stellar feedback \citep[the so-called ``turbulent'' model of SF; see
the reviews by][]{2000prpl.conf....3V, 2004RvMP...76..125M,
2007ARA&A..45..565M, 2007prpl.conf...63B}. In both cases, the
gravitational contraction was assumed to be halted by either of the two
mechanisms, and the clouds were assumed to reach near virial
equilibrium. In the last decade or so, however, the paradigm about the
formation, evolution and structure of MCs has changed significantly, and
in the remainder of this review I will discuss this emerging new view.
\section{Birth and infancy} \label{sec:formation}
\subsection{Observational and numerical evidence on the clouds' origin}
\label{sec:evidence}
As mentioned above, GMCs and their substructures had traditionally been
thought to be in virial equilibrium. However, recent observations of
GMCs in the LMC \citep{2009ApJS..184....1K} suggest that the clouds are
undergoing an evolutionary process, in which both their mass and their
SF activity increase in time, going in $\sim 25$ Myr from masses $M \sim
10^{4.5-5} M_\odot$ and virtually no massive-SF to $M \sim 10^{5.5-6}
M_\odot$ and a population of clusters and HII regions. This is consistent
with the conclusion by \citet{2003ApJS..149..343E} that the GMCs in M33
are being assembled rapidly from the atomic component, with a prompt
onset of SF.
Similar conclusions had been reached previously from numerical
studies. Numerical simulations of the ISM at the kpc scale with
turbulence driven by stellar feedback \citep[supernova explosions,
expanding HII regions;][]{1980ApJ...239..173B, 1988ApJ...328..427C,
1993ApJ...413..137R, 1995ApJ...440..634R, 1995ApJ...441..702V,
1995ApJ...455..536P, 1999ApJ...514L..99K, 1999ApJ...527..285B,
2000MNRAS.315..479D, 2004A&A...425..899D, 2005A&A...436..585D} showed
that compressive motions driven by large-scale gravitaional
instabilities in the diffuse ISM or by the global turbulence
are able to form clouds on short timescales, essentially given by the
turbulent crossing time accross the distance necessary to collect the
material that eventually reaches the cloud. This is facilitated by the
presence of cooling, which renders the medium highly compressible, even
when no thermal instability (cf.\ \S \ref{sec:instab}) is present
\citep{1996ApJ...473..881V}. Within such a dynamic scenario,
\citet{1999ApJ...515..286B} remarked that the clouds should not be
considered as isolated objects, because a significant mass flux is
expected to exist accross their boundaries, since the clouds are being
assembled from material from the outside.
\subsection{The physical processes}
\label{sec:phys_proc}
\subsubsection{Instabilities galore!} \label{sec:instab}
The scenario of cloud assembly by convergent motions in the diffuse ISM
(either driven by turbulence or by large-scale instabilities) was
formulated analytically by \citet{2001ApJ...562..852H}, who pointed out
that the column density of cold atomic hydrogen necessary for H$_2$ and
CO molecules to form is \citep{1988ApJ...334..771V, 1998ARA&A..36..317V}
\begin{equation}
N_{\rm H} \sim 1\hbox{--}2 \times 10^{21} {\rm ~cm}^{-2},
\label{eq:N_molec_form}
\end{equation}
corresponding to $A_V \sim 0.5$--1, is very similar to the value
necessary for the same gas to become gravitationally unstable \citep[see
also][]{1986PASP...98.1076F}, and to the column density necessary for
the gas to become magnetically supercritical, at a typical interstellar
magnetic field strength of $B\sim 5 \mu$G. This implies that when an
initially atomic cloud is assembled by a convergent velocity field in
the diffuse ISM, it should become molecular, self-gravitating,
and magnetically supercritical at roughly the same time. We now discuss
this phenomenology in some more detail.
A fundamental physical ingredient aiding the formation of dense atomic
and molecular clouds is thermal instability \citep{1965ApJ...142..531F},
which is a consequence of the various radiative heating and
cooling processes operating on the atomic ISM
\citep[for a modern discussion, see][]{1995ApJ...443..152W}. The atomic
ISM is subject to the so-called {\it isobaric} mode of this instability:
for densities in the range $\sim 1$--10 ${\rm ~cm}^{-3}$ ($T \sim 5000$--500 K),
the gas {\it loses thermal pressure} upon a dynamic compression
\citep[for a pedagogical discussion, see the review
by][]{2009arXiv0902.0820V}, implying that after the compression it will
be underpressured with respect to its surroundings, and will continue to
be squeezed by them until it exits the thermally unstable range, at
which point any further compression again increases its thermal
pressure, until it eventually reaches pressure equilibrium with its
surroundings, but at a much higher density and lower temperature. This
is the basis of the well-known two-phase model of the atomic ISM
\citep{1969ApJ...155L.149F}, originating the warm-diffuse and cold-dense
phases of this ISM component, which are respectively referred to as the {\it
warm} and {\it cold neutral media} (in turn, respectively, WNM and CNM).
The above discussion assumes that the gas is in the thermally unstable
range to begin with. However, \citet{1999A&A...351..309H} showed that
transonic compressions (i.e., of Mach number ${\cal M}_{\rm s} \gtrsim 1$) in the
{\it stable} warm phase can nonlinearly trigger a transition to the cold
phase, so that cold clouds can be formed out of the stable WNM in the
presence of transonic turbulence, which is indeed observed in this
medium \citep{1987ASSL..134...87K, 2003ApJ...586.1067H}. Alternatively,
large-scale ($\gtrsim 1$ kpc) instabilities in the diffuse medium, such
as gravitational \citep[e.g.,][]{1994ApJ...433...39E}, Parker
\citep{1966ApJ...145..811P} or magneto-Jeans
\citep{2002ApJ...581.1080K}, can provide the driving forces for these
motions.
The gas cooled and compressed by this process to form a cloud is subject
to a large number of dynamical instabilities. It has been long been
known that compressed layers formed by the collision of gas streams are
nonlinearly unstable, meaning that the layers become turbulent when the
colliding flows have sufficiently large velocities
\citep{1986ApJ...305..309H, 1992ApJ...386..265S, 1994ApJ...428..186V}.
This process is known as the {\it nonlinear thin-shell instability}
(NTSI). Furthermore, the presence of cooling lowers the required inflow
velocities to destabilize the layers \citep{2005A&A...438...11P}.
Finally, the Kelvin-Helmholz and Rayleigh-Taylor instabilities are also
expected to operate during the formation of a cloud. The interplay of
all these instabilities has been investigated numerically by
\citet{2005ApJ...633L.113H, 2006ApJ...648.1052H}. In summary, {\it
convergent motions in the WNM are expected to produce turbulent CNM
clouds}.
\subsubsection{Evolution of the mass-to-magnetic flux ratio}
\label{sec:M2FR}
Another crucial ingredient in MC dynamics is the magnetic field, and in
particular, the mass-to-magnetic flux ratio (M2FR). As it is well known,
the magnetic field can support a cloud against the latter's self-gravity
if the M2FR, or, equivalently, the ratio of column density to field
strength, exceeds some critical value which, for a cylindrical
configuration, is given by \citep{1978PASJ...30..671N}
\begin{equation}
\Sigma/B \approx (4 \pi^2 G)^{-1/2},
\label{eq:NN78}
\end{equation}
where $\Sigma$ is the mass column density. We denote by $\mu$ the value
of the M2FR normalized to the critical value. Thus, a {\it magnetically
subcritical} cloud has $\mu < 1$, and can be supported by the magnetic
field, while a {\it supercritical} one has $\mu > 1$, and cannot be
magnetically supported \citep[e.g.,][]{1987ARA&A..25...23S,
1991psfe.conf..449M}. In the ``standard'' model of SF, most clouds were
assumed to be strongly magnetically subcritical, and thus globally
supported by the field. SF was thought to occur on long timescales and
involving small fractions of the clouds' mass because, in the dense
cores, the process known as {\it ambipolar diffusion} (AD) allows a
redistribution of the magnetic flux, so that they can continue to
contract quasi-statically, until their M2FR finally becomes
supercritical, and then the cores collapse.
It is a very common practice to assume that the M2FR is a conserved
quantity as long as the flow behaves ideally (i.e., AD is negligible).
After all, for an isolated cloud of fixed mass, the mass is constant by
construction, and the magnetic flux is conserved by the flux-freezing
condition \citep[see, e.g.][]{1992phas.book.....S}. However, the M2FR
refers to the mass within flux tubes, and in general, field lines do not
end at the ``edge'' of a cloud, but rather continue out to arbitrarily
long distances.
In fact, it is well possible that magnetic field lines circle around the whole
Galaxy. Now, eq.\ (\ref{eq:NN78}) implies that a flux tube is
supercritical beyond an {\it accumulation length} given by
\citep{1985prpl.conf..320M, 2001ApJ...562..852H}
\begin{equation}
L_{\rm acc} \approx 470 \left(\frac{B}{5 \mu{\rm G}} \right) \left(\frac{n}{1
{\rm ~cm}^{-3}} \right)^{-1}~{\rm pc},
\label{eq:acc_length}
\end{equation}
so that, in principle, the entire ISM is supercritical, at least near
the midplane. However, in the case of a forming CNM cloud, what is
relevant is the M2FR {\it of the dense gas that makes up the cloud},
since the cloud is up to $100\times$ denser than its surroundings, and
thus it is the main source of the self-gravity that the field has to
oppose. Thus, in this problem, natural boundaries for up to where to
measure the M2FR are provided by the bounding surface of the dense gas.
Accumulation of material is {\it not} opposed by the magnetic field if it
occurs along field lines, and so this can occur freely in
the ISM. When compressions occur at an angle with the field, it has been
shown by \citet{2000A&A...359.1124H} that, up to a certain angle that
depends on the Mach number of the inflows and the field strength, the
inflow is reoriented along the field lines, and it behaves as if the
compression were parallel to the lines. Beyond that angle, the
compression behaves essentially as if it were perpendicular to the
field, and the flow bounces off, not forming any cloud. Thus, in what
follows, we consider the case of a cloud forming along field
lines. Equation (\ref{eq:NN78}) can be rewritten as
\begin{equation}
N_{\rm cr} = 1.45 \times 10^{21} \left(\frac{B}{5 \mu {\rm G}}
\right) {\rm ~cm}^{-2},
\label{eq:Sigma_crit}
\end{equation}
implying that the column density for a cloud to become supercritical is
very similar to that for molecule formation to begin, and for the cloud
to become gravitationally unstable (cf.\ eq.\ \ref{eq:N_molec_form}).
Thus, we can infer that, {\it as a cloud forms and grows out of a
compression in the WNM, it should become molecular, supercritical, and
gravitationally unstable at roughly the same time.}
Figure \ref{fig:mu_evol} shows the evolution of the M2FR in two
numerical simulations of cloud formation by V\'azquez-Semadeni et al.\
(2011 in prep.; see also \citeauthor{2009MNRAS.398.1082B}
\citeyear{2009MNRAS.398.1082B}), illustrating the growth of
the M2FR. In these simulations, two cylindrical streams of radius 32 pc
and length 112 pc each, are set to collide against each other at the
center of a 256-pc numerical box along the $x$-direction, so that the
cloud is a thin cylindrical layer perpendicular to the inflows. The M2FR
is measured for each line of sight (LOS) perpendicular to the cloud,
which is defined as the cylindrical volume of radius 32 pc and length 20
pc centered at the numerical box center. For each LOS, the M2FR is
measured as the ratio $\Sigma/B_{\|}$, where $B_{\|}$ is the field
component parallel to the LOS and $\Sigma$ is the mass column density
along each LOS. Thus, these measurements constitute {\it upper limits}
to the real M2FR (for a detailed discussion, see V\'azquez-Semadeni et
al.\ 2011). The simulations have mean magnetic field strengths of 3 and
4 $\mu$G, and $\mu = 0.91$ and 0.68, respectively. That is, both
simulations are globally subcritical, implying that no subregion of it
can be supercritical \citep{2005ApJ...618..344V} as long as the flow
remains ideal. Values of $\mu$ greater than those of the whole box
indicate that the measured values overestimate the actual
$\mu$. Nevertheless, for a real cloud not limited by the box size, the
magnetic criticality is expected to eventually become $>1$.
\articlefigure{VazquezS_E_f2.eps}{fig:mu_evol}{Evolution of the
normalized M2FR, $\mu$, for
two numerical simulations by V\'azquez-Semadeni et al.\
(2011, in prep.). See text for details on the simulations. The {\it top
panels} refer to the simulation with $\langle B \rangle = 3 \mu$G, while
the {\it bottom panels} refer to the simulation with $\langle B \rangle = 4
\mu$G. The {\it left panels} refer to measurements performed on the
entire ``cloud'' volume, while the {\it right} panels refer to
measurements perform only on those lines of sight for which the number
column density satisfies $N > 10^{21} {\rm ~cm}^{-2}$. The {\it solid lines}
denote the average of $\mu$ over all LOSs, while the {\it dotted lines}
denote the $3 \sigma$ deviations from the mean. The growth of $\mu$ from
very small values to near unity during the first 10 Myr of evolution is
clearly seen, especially in the plots for the whole cloud.}
These conclusions are consistent with
observational results showing that the CNM clouds are in general
magnetically subcritical \citep[][sec. 7]{2005ApJ...624..773H}, while
molecular structures appear to be critical or supercritical in general
\citep{2001ApJ...554..916B, 2008ApJ...680..457T}
\subsection{The early stages: thin CNM sheets} \label{sec:early}
The early stages of cloud formation were investigated analytically and
numerically by \citet{2006ApJ...643..245V}, determining the structure
and physical conditions (density, temperature, and expansion velocity of
the phase transition front) in the incipient cloud as a function of the
Mach number of the converging gas streams, before the dynamical
instabilities have time to grow. Figure \ref{fig:shock_str} shows the
structure of the cloud ({\it left panel}) and the dependence of the
physical properties of the cloud on the Mach number of the inflowing
streams ({\it right panel}).
\articlefiguretwo{VazquezS_E_f1a.eps}{VazquezS_E_f1b.eps}
{fig:shock_str} {{\it Left panel:} Schematic illustration of the structure of a
cloud formed by the collision of oppositely-directed WNM streams. The
streams are assumed to collide along the horizontal axis. Only the right
half of the system is shown, with the left stream replaced by a
wall. The inflowing gas approaches from the right, and shocks at the
interface between regions 1 and 2. The shock heating throws the
post-shock gas out of thermal balance between heating and cooling, and
so the gas cools as it flows towards the 2-3 interface. At the latter, the
gas undergoes a phase transition (condenses) to the CNM, and settles
into a dense, cold layer (region 3). The phase transition front moves
outwards from the collision wall at speed $v_{\rm f}$. {\it Right:}
Pressure $P$, number density $n$, outwards speed of the phase transition
front $v_{\rm f}$, and column density $N$ after 1 Myr of evolution, of
the thin layer, all as a funtion of the Mach number of the inflow
stream. From \cite{2006ApJ...643..245V}.}
One important implication of the study by \citet{2006ApJ...643..245V} is
that, in its initial stages, the forming cloud is expected to be a thin
CNM sheet, since it forms at the essentially two-dimensional interface
between the colliding streams. After a few Myr of evolution, the
predicted thin sheet has column densities that agree very well with the
CNM sheets reported by \citet{2003ApJ...586.1067H}. Thus, it is
suggested that such thin CNM sheets may constitute the earliest phases
of MC evolution. Note, however, that a GMC may never form if the
mass involved in the streaming flows is not high enough to attain the
column densities necessary for molecule formation.
\section{Maturity: Gravitational contraction and star formation (child
bearing)} \label{sec:onset_SF}
\subsection{A distribution of collapse timescales} \label{sec:tff_distr}
As discussed in \S \ref{sec:M2FR}, as a cloud grows, it should become
molecular, supercritical, and gravitationally unstable at roughly the
same time. This result implies that by the time a GMC forms, it should
be contracting gravitationally, since it cannot be supported by the
magnetic field because it is already supercritical, and it is not
forming stars yet, so no additional turbulence that can support the
cloud can be injected into it yet.
However, the initial turbulence produced by the convergent gas streams
has a crucial role in the subsequent development of the cloud and its SF
activity. Numerical simulations by various groups
\citep{2002ApJ...564L..97K, 2005A&A...433....1A, 2005ApJ...633L.113H,
2006ApJ...643..245V} have shown that the ``clouds'' are actually a
mixture of warm and cold gas \citep[see
also][]{2006ApJ...647..404H}. This is illustrated in Fig.\
\ref{fig:MC_struc} ({\it left panel}), which shows the granular, fractal
density structure observed in a high-resolution ($10000^2$ zones)
numerical simulation by \citet{2007A&A...465..431H}. The density field
has a wide probability density distribution function PDF (Fig.\
\ref{fig:MC_struc}, {\it right panel}), and thus, when gravity is
included, this density PDF implies the existence of a wide distribution
of free-fall times \citep[Fig.\
\ref{fig:tff_distr};][]{2008ApJ...689..290H}, with the bulk of the mass
remaining at relatively low densities . This implies that a small
fraction of the mass in the cloud will collapse, and thus SF will begin,
{\it before} the global collapse of cloud as a whole is completed, on
timescales $\sim 10$ Myr.
\articlefiguretwo{VazquezS_E_f3a.eps}{VazquezS_E_f3b.eps}{fig:MC_struc}
{{\it Left:} Density and velocity fields in a two-dimensional numerical
simulation of colliding flows in a 20-pc numerical box, with a
resolution down to a scale of $2 \times 10^{-3}$ pc. Note the extremely
granular texture of the density field. {\it Right:} Volume-weighted
probability density distribution (PDF) of the density field for this
simulation ({\it solid line}) and for a similar one with lower (half)
the resolution ({\it dotted line}). From \citet{2007A&A...465..431H}. }
\articlefigure{VazquezS_E_f4.eps}{fig:tff_distr}{Mass at a given
free-fall time as a funtion of time in three different numerical
simulations of cloud formation by colliding streams by
\citet{2008ApJ...689..290H}, with the free-fall time axis binned
linearly in the {\it top} panels, and logarithmically in the {\it
bottom} ones. It is seen that the bulk of the mass remains at low
densities, with long free-fall times. From \citet{2008ApJ...689..290H}.}
This kind of evolution, which we refer to as {\it hierarchical
gravitational collapse} was first observed numerically in the
simulations by \citet{2007ApJ...657..870V}, and is very similar to the
notion of hierarchical gravitational fragmentation introduced long ago
by \citet{1953ApJ...118..513H}, except that in that proposal, the
density fluctuations were assumed to be linear, and so they all had
essentially the same free-fall time as the large-scale cloud, while here
they are nonlinear, with shorter timescales than the cloud, due to the
initial turbulence induced by the very process of assembly of the
cloud.
\subsection{Low- and high-mass star-forming regions}
\label{sec:SF_masses}
The distribution of density fluctuations produced by the initial
turbulence has another implication: a clump mass spectrum is produced.
From high-resolution simulations, \citet{2007A&A...465..431H} reported a
power-law shape of the spectrum at high masses of the form $d N/dM
\propto M^{-\beta}$, with $\beta \sim 1.7$, in good agreement with
observational determinations of the mass spectrum for CO clumps
\citep{1998A&A...336..150M, 1998A&A...329..249K,
1998A&A...331L..65H}. It is well known that this form of the mass
spectrum implies that most of the mass is in the more massive clumps,
even though they are less numerous.
Of course, the less massive structures are expected to be embedded in the
larger-mass, lower-density ones in a hierarchical fashion
\citep{1994ApJ...423..681V}. This implies that the low-mass structures
should be expected to terminate their collapse and form stars earlier
than the more massive ones, and so a prediction from the present
scenario is that a large molecular complex should contain a relatively
large number of low-mass, somewhat older (by a few Myr) star-forming
regions, and a smaller number of massive regions, formed later, yet
containing most of the mass. The latter is consistent with the well
known result that most stars are formed in massive, cluster-forming
regions \citep{2003ARA&A..41...57L}. Of course, this picture
neglects triggering of secondary SF by previous generations of stars,
which complicates the picture.
This scenario has been quantified in a numerical simulation by
\citet{2009ApJ...707.1023V}, who selected a typical example of the
early-forming, low-mass regions, and the most massive cloud formed by
collapse of the cloud complex at large from a numerical simulation of
cloud formation by \citet{2007ApJ...657..870V}. This simulation used a
similar setup to that of the simulations by \citet{2009MNRAS.398.1082B},
discussed in \S \ref{sec:M2FR}, but with no magnetic
fields. \citet{2009ApJ...707.1023V} showed that indeed the masses,
sizes, velocity dispersions, and SFRs of the two regions were
respectively consistent with observations of such types of regions.
Moreover, they compared the distributions of masses, sizes and densities
of the dense cores within the massive region, showing that they were
very similar to the corresponding distributions for the set of cores in
the Cygnus-X region by \citet{2007A&A...476.1243M}. In summary, this
analysis suggested that the formation of low- and high-mass star-forming
regions by this mechanism is viable.
\section{Old age: Stellar feedback (popping all over)}
\label{sec:feedback}
Once SF begins in a cloud, the newly formed stars feedback on it via
either low-mass star outflows, which inject momentum, or ionizing
radiation and supernova explosions from massive stars. The effect of
this feedback on the parent cloud has been extensively studied by
numerous groups. For a detailed discussion, see the review by
V\'azquez-Semadeni (2011, in preparation). Here we focus only on whether
the feedback is able to bring the parent cloud to a quasi-static
equilibrium or whether, instead, it ends up dispersing the cloud. This
is an issue about which much effort is currently being devoted, and
because of that, no conclusive answer is yet available.
In recent years,
simulations of multiple jets in parsec-scale clumps
\citep{2007ApJ...662..395N, 2009ApJ...695.1376C} have suggested that
bipolar outflows are sufficient to drive and maintain the turbulence in
parsec-scale clumps, and to maintain the latter in a near-hydrostatic
equilibrium. However, in those works the clumps occupied the entire
numerical box, and therefore the simulations lacked the contracting
motions of the rest of the MC in which they are embedded.
A simplified model of the effect of massive-star feedback through
HII-region expansion on the {\it global} evolution of GMCs was performed by
\citet{2006ApJ...653..361K}. These authors computed the time-dependent
virial {\it balance} (not necessarily equilibrium) of a spherical GMC
under the effect of its self-gravity and the energy injection of its
embedded HII regions. In their simplified model, the SFR self-regulates,
and causes oscillations of the clouds, which are finally dispersed after
a few crossing times.
Full numerical simulations of HII-region feedback in the context of a
fully-evolving and contracting GMC have been recently performed by
\citet{2010ApJ...715.1302V}, who again used the setup described in \S
\ref{sec:M2FR}, now in adaptive mesh refinement (AMR) simulations which
allowed an effective resolution of $\sim 0.03$ pc, and that included
feedback from a single-mass population of stars. These authors found
that the feedback affects the immediate surroundings of the recently
formed stars, thus reducing the SFR, but is not capable of reverting the
global contraction of the GMC. Similar results have been obtained in
high-resolution simulations at the clump-scale with outflow feedback by
\citet{2010ApJ...709...27W}. Those authors have found that the accretion
that feeds the most massive star is not restricted to the core in which
the star is being formed directly, but rather can be traced out to the
scale of the whole clump containing the core, in spite of the fact that
the outflows distort this flow, reducing the SFE of the system.
In summary, the available numerical evidence suggests that the scenario
of gravitationally contracting MCs is maintained even in the presence of
stellar feedback. The latter may be an important source of energy for
driving the turbulence at the clump (parsec) scale, but has a harder
time halting the gravitational contraction at the scale of the whole GMC
(tens of parsecs). The calculations by \citet{2006ApJ...653..361K} and
\citet{2010ApJ...715.1302V} suggest that low-mass clouds are more
readily destroyed by the feedback than more massive ones. Nevertheless,
the numerical experiments performed so far are relatively scarce, and
have used a limited set of initial conditions. A more complete coverage
of parameter space, such as that conducted by
\citet{2010MNRAS.406.1875R} for the variability of the SFR with the
initial conditions, and with a more accurate modeling of the stellar
feedback, is necessary to better understand the details of cloud dispersal.
\section{Conclusions} \label{sec:conclusions}
\subsection{Summary} \label{sec:summary}
The scenario of MC formation and evolution (under solar neighborhood
conditions) discussed in the previous sections can be summarized as
follows:
\begin{itemize}
\item The route to the formation of a GMC starts with a large-scale,
moderately supersonic converging motion in the WNM, which may be driven
either by large-scale instabilities, the passage of a spiral-arm or
supernova shock, or by intermediate-scale generic turbulence in
the WNM.
\item The compression nonlinearly triggers a phase transition to the
CNM, forming a large, though not very dense, cold atomic cloud. The
earliest stages of these clouds may constitute the thin CNM clouds
reported recently by \citet{2003ApJ...586.1067H}. At later times, a
combination of the Kelvin-Helmholz, Rayleigh-Taylor, nonlinear
thin-shell and thermal instabilities destabilizes the cloud, rendering
it turbulent and clumpy.
\item If sufficient mass is available in the converging flow that a
column density $\sim 1.5 \times 10^{21} {\rm ~cm}^{-2}$ is reached in the newly
formed cloud, then the cloud begins to be dominated by self-gravity
(rather than by the confining pressure [thermal+ram] of its
surroundings) and also reaches a high enough extinction ($A_V \sim 1$)
to allow the formation of CO molecules. Moreover, for a fiducial value
of the mean Galactic magnetic field of $5 \mu$G, such a cloud should be
near to becoming magnetically supercritical as well. So, for solar
neighborhood conditions, a growing cloud should become molecular,
self-gravitating and magnetically supercritical at approximately the
same time.
\item A cloud that has reached such a column density then begins to
contract gravitationally. However, the clumps produced by the various
instabilities, which have shorter free-fall times than the bulk of the
cloud, culminate their collapses before the bulk of the cloud does. Star
formation thus begins a few Myr after the cloud's global contraction has
started, but a few Myr before the bulk of the cloud cuminates its own
collapse.
\item The collapse of isolated, low-mass clumps produces scattered
low-mas star-form\-ing regions, while the collapse of the bulk of the
cloud produces high-mass star-form\-ing regions.
\item The termination of a SF episode in a cloud is still not fully
understood from the avilable numerical simulations. More complete
coverage of parameter space and with more detailed modeling of the
feedback is necessary. In any case, the attainment of a nearly
hydrostatic equilibrium appears very difficult. Instead, it appears that
a cloud may begin to be destroyed or dispersed locally, while the outer
layers may still be falling in, establishing a complicated flow with
both infall and outflow.
\end{itemize}
\subsection{Implications: hierarchical gravitational contraction}
\label{sec:implications}
The evolutionary scenario for MCs described here strongly suggests the
ubiquitous existence of generalized gravitational contraction in MCs and
their substructure, given the approximate simultaneity of the onset of
gravitational contraction with the onset of molecule formation and the
attainement of magnetic supercriticality.
This implication is in fact consistent with a growing body
of observational results. In particular, principal component analysis of
the contributions to the velocity dispersion in MCs and their
substructure invariably show a ``dipolar'' main component, indicating
that the dominant contribution comes from a large-scale (i.e., at the
scale of the full structure observed) velocity gradient
\citep{2007IAUS..237....9H, 2009A&A...504..883B} which is consistent
with a global contraction or shear of the cloud, but inconsistent with
solid-body rotation (M. Heyer, priv. comm., 2010). Also, studies
comparing specific regions with numerical simulations
\citep{2007ApJ...654..988H, 2007A&A...464..983P} have shown that those
regions are well modeled by gravitationally contracting structures.
Finally, recent studies of massive star-forming regions by
\citep{2009ApJ...706.1036G, 2010arXiv1009.0598C} have provided evidence
of the existence of {\it hierarchical accretion flows} at multiple
scales in the regions. The notion of hierarchical gravitational
fragmentation has recently been formulated analytically by
\citet{2008MNRAS.385..181F}, who have proposed the existence of a
gravitationally driven cascade from the large to the small scales in
MCs, analogous to a turbulent cascade, except that the quantity being
cascaded is mass rather than energy.
Moreover, recent observations by
\citet{2009ApJ...699.1092H} having much higher angular and spectral
resolution and
higher dynamic range in column density than earlier studies
\citep[e.g.,][]{1987ApJ...319..730S}, suggest that MCs do not, after all,
have a roughly constant column density $\Sigma$, but rather span up to
two orders of magnitude in this variable, and that the velocity
dispersion actually scales with size {\it and} column density as
\begin{equation}
\sigma_v \propto (\Sigma L)^{1/2}.
\label{eq:Heyer_rel}
\end{equation}
Recently, \citet{2010arXiv1009.1583B} have proposed that this scaling is
exactly what is expected from a hierarchical gravitational cascade, in
which, rather than virial {\it equilibrium}, the governing relation is
simply energy conservation during the contraction, so that the
gravitational and kinetic energies satisfy
\begin{equation}
|E_{\rm g}| = E_{\rm k},
\label{eq:ener_cons}
\end{equation}
from which eq.\ (\ref{eq:Heyer_rel}) follows directly. It is noteworthy
that, at face value, eq.\ (\ref{eq:ener_cons}) seems to fit the data
better than the virial equilibrium condition $|E_{\rm g}| = 2 E_{\rm
k}$, as shown in Fig.\ \ref{fig:Heyer_law}, although large
uncertainties, especially in the mass determinations, prevent any firm
conclusions. Interestingly, note that free-fall implies {\it larger}
velocities than virial equilibrium, contrary to the standard notion that
velocities higher than virial imply lack of gravitational binding.
\articlefigure{VazquezS_E_f5.eps}{fig:Heyer_law}{Massive dense cores
from \citet[][labeled ``G'']{2009ApJ...705..123G}, and clouds and
clumps from \citet[][labeled ``H'']{2009ApJ...699.1092H} in the
$\sigma_v/R^{1/2}$ vs.\ $\Sigma$ plane, where $\sigma_v$ is the velocity
dispersion, $R$ is the region size, and $\Sigma$ is the mass column
density. The straight lines show the loci of virial equilibrium,
$|E_{\rm g}| = 2 E_{\rm k}$, and of energy conservation under free-fall,
$|E_{\rm g}| = E_{\rm k}$. From \citet{2010arXiv1009.1583B}.}
It is important to remark that the hypothesis that MCs might be undergoing
generalized gravitational contraction is not new. It was first proposed
by \citet{1974ApJ...189..441G}. However, it was soon deemed untenable by
\citet{1974ARA&A..12..279Z} who argued that, if the clouds were in
free-fall, then a simple estimate obtained by dividing the total
molecular mass in the Galaxy by the mean free-fall time would imply a
SFR roughly two orders of magnitude larger than observed. However, this
conundrum may not pose a serious problem if the star-forming activity of
the clouds is terminated prematurely by the feedback from the first
stars formed. More work to understand the details of this process is
still needed.
\subsection{Final remarks} \label{sec:final_rem}
The picture of MC evolution described in this review appears promising
as a self-consistent scenario that connects the dynamics of the ISM at
large with the physical properties of MCs and their star-forming
properties. However, several features still need to be worked out in
more detail, such as the precise form in which stellar feedback
terminates a SF episode and disperses a cloud locally, and at what
scales can this process be considered in an averaged sense, if at all.
Also, more quantitative statistical issues, such as the fraction of the
time, and of the stellar production of the clouds, is spent under the
magnetically subcritical and supercritical regimes, respectively.
Finally, it is important to remark that the present scenario is likely to
only apply directly to solar neighborhood-like conditions. In
particular, it may need modification for regions like the molecular ring
of the Galaxy, or to galaxies where the disk is mostly molecular, as in
those cases the atomic component may be absent, at least locally. In
those cases, it is possible that the entire molecular ring or disk is
the equivalent of the isolated GMCs we have discussed here, and that the
phase transition occurs at the boundaries of these regions, both
radially and vertically. More work is needed in order to assess this.
\acknowledgements
I warmfully thank all the colleagues who have helped shaping my ideas
about the subject of this review over the years, especially Javier
Ballesteros-Paredes, Lee Hartmann, Patrick Hennebelle, and Mordecai Mac
Low. Gilberto G\'omez and Pedro Col\'\i n have provided invaluable
expertise in translating those ideas into numerical simulations. This
work has been funded in part by CONACYT grant 102488 to E.V.-S.
|
{
"timestamp": "2010-09-22T02:00:38",
"yymm": "1009",
"arxiv_id": "1009.3962",
"language": "en",
"url": "https://arxiv.org/abs/1009.3962"
}
|
\section{Introduction}
Groups of galaxies contain about half of all galaxies in the
Universe (e.g., Huchra \& Geller 1982; Geller \&
Huchra 1983; Nolthenius \& White 1987; Ramella et al. 1989).
They represent the link between galaxies and large-scale structures
and play an important role to galaxy formation and evolution.
One of the most important questions about galaxy systems is related to
segregation phenomena. The study of segregation effects is important
to understand how system environment is transforming galaxies at the
present epoch. Evidence for different loci in position and velocity
spaces according to luminosity, spectral type and color of galaxies suggests ongoing evolution
of clusters through the process of mergers, dynamical friction and secondary infall
(e.g. Adami, Biviano \& Mazure 1998, Biviano et al. 2002). Segregation has also been observed
in galaxy groups (e.g. Mahdavi et al. 1999, Carlberg et al. 2001), suggesting a
continuum of segregation properties of galaxies from low-to-high mass systems
(Girardi et al. 2003). However, the dynamical state of galaxy groups
is not taken into account in these studies. Differences in segregation phenomena
may emerge if one divides groups according to their evolutionary stage. Recently, Hou et al. (2009) have examined three goodness-of-fit tests (Anderson-Darling, Kolmogorov and $\chi^2$ tests) to find which statistical tool is best able to distinguish between relaxed and non-relaxed galaxy groups. Using Monte Carlo simulations and
a sample of groups selected from the CNOC2, they found that the Anderson-Darling (AD) test
is far more reliable at detecting real departures from normality. Their results show
that Gaussian and non-Gaussian groups present distinct velocity dispersion profiles, suggesting that discrimination
of groups according to their velocity distributions may be a promising way to access
the dynamics of galaxy systems. Extending up this kind of analysis to the outermost edge of groups
one can probe the regions where they might not be in dynamical equilibrium.
In this letter, we look for segregation effects in galaxy groups selected from the 2PIGG catalog (Eke et al. 2004), using 2dF data out to 4$R_{200}$, and taking into account the evolutionary stage of the groups resulting from the AD test.
\section{Data and Methodology}
\subsection{2PIGG sample}
We use a subset of the 2PIGG catalog,
corresponding to groups located in areas
of at least 80\% redshift coverage in 2dF data out to 10 times the
radius of the systems, roughly estimated from the projected harmonic mean (Eke et al. 2004).
The idea of working with such large areas is to probe the effect of
secondary infall onto groups. Members and interlopers were redefined after the
identification of gaps in the redshift distribution according
to the technique described by Lopes et al. (2009).
Before selecting group members and rejecting interlopers we first
refine the spectroscopic redshift of each group and identify its
velocity limits. For this purpose, we employ the gap-technique described
in Katgert et al. (1996) and Olsen et al. (2005) to identify
gaps in the redshift distribution. A variable gap, called
{\it density gap} (Adami et al. 1998), is
considered. To determine the group redshift, only galaxies within
0.50 h$^{-1}$ Mpc are considered. Details about this procedure are found
in Lopes et al. (2009); see also Ribeiro et al. (2009) for applications
of this technique to 2dF galaxy groups. With the new redshift and
velocity limits, we apply an algorithm for interloper
rejection to define the final list of group members. We use the
``shifting gapper'' technique (Fadda et al. 1996), which consists of the
application of the gap-technique to radial bins from the group center.
We consider a bin size of 0.42 h$^{-1}$ Mpc (0.60 Mpc for h = 0.7) or
larger to ensure that at least 15 galaxies are selected. Galaxies not
associated with the main body of the group are discarded. This procedure
is repeated until the number of group members is stable and no further galaxies
are eliminated as intruders.
In the present work,
we have sampled galaxies out to 10 times the hamonic mean radius of the systems, including
galaxies whose distances to the centers can reach $\sim$8 Mpc. To avoid contamination
of nearby structures, we select galaxies within the maximum radius $R_{max}=$ 4.0 Mpc
(see La Barbera et al. 2010). After applying the shifting gapper procedure we have a list of group members and we call $R_A$ the aperture equivalent to the radial offset of the most distant member (normally close to $R_{max}$). We estimate the velocity dispersion ($\sigma$) within $R_A$ and then the physical radius (R$_{200}$) of each group.
Finally, a virial
analysis is perfomed for mass estimation (M$_{200}$). Further
details regarding the interloper removal and estimation of global properties
($\sigma$, physical radius and mass) are found in Lopes et al. (2009).
\subsection{Classifying groups}
The first step in our analysis is to apply the AD test (see Hou et al. 2009 for
a good description of the test) to the velocity
distributions of galaxies in groups. This is done for different distances, producing the following
ratios of non-Gaussian groups: 6\% ($R \leq 1R_{200}$), 9\% ($R\leq 2R_{200}$),
and 16\% ($R\leq 3R_{200}$ and $R\leq 4R_{200}$).
Approximately 90\% of all galaxies
in our sample have distances $\leq 4R_{200}$. This is the natural cutoff in space we have made in this work. Some properties of galaxy groups are presented in Table 1. We have classified groups according to the AD test (at 0.05 significance level)
done at $R\leq 4R_{200}$ , encompassing all groups with evidence for normality deviations.
Properties of non-Gaussian (NG) groups in Table 1 were computed twice, with and without
a correction based on iterative removal
of galaxies whose absence in the sample cause the groups become Gaussian, following a procedure
similar to Perea, del Olmo \& Moles (1990). The corrected properties are just those the system would have if it was made only with galaxies
consistent with the normal velocity distribution. This correction allows one to honestly compare typical
properties of G and NG groups. Not doing that, NG groups could have their properties overestimated by a factor of $\sim 1.5$, taking
all members within $4R_{200}$. After this procedure, we see in Table 1 that G and corrected NG groups
have similar properties.
\subsection{Composite groups}
A suitable way to investigate galaxies in multiple galaxy systems is to combine them
in stacked objects (Biviano et al. 1992). Thus, we built two composite groups, Gaussian--G (composed of 48 systems) and non-Gaussian--NG (composed of 9 systems). Galaxies in theses composite groups have distances to group centers normalized by $R_{200}$ and their velocities refers to the group median velocities and are scaled by the group velocity dispersions
\begin{equation}
u_i={{v_i - \langle v \rangle_j}\over \sigma_j}
\end{equation}
\noindent where $i$ and $j$ are, respectively, the galaxy and the group indices. Velocity
dispersions of the composite groups refer to the dimensionless quantity $u_i$.
Absolute magnitudes, $M_R$, are obtained from Super-COSMOS R band, a 2dF photometric information.
Cosmology is defined by $\Omega_m$ = 0.3, $\Omega_\lambda$ = 0.7, and $H_0 = 100~h~{\rm km~s^{-1}Mpc^{-1}}$
Distance-dependent quantities are calculated using $h=0.7$. All figures presented in the next section correspond to cumulative data in $R/R_{200}$ or $M_R$.
Error-bars in our analysis are obtained from a bootstrap technique with 1000 resamplings.
\begin{table}
\caption{Mean properties of groups}
\label{tab1}
\tiny{
\begin{tabular}{l c c c c c}
\hline\hline
Type & $R_{200}$ (Mpc) & $M_{200}$ ($10^{14}~M_{\sun}$)& $\sigma $ (km/s) & $N_{200}$ & $N_T$ \\
\hline
G & $0.94\pm 0.31$ & $0.88\pm 0.79$ & $223 \pm 89$ & $10\pm 4$ & $24\pm 11$ \\
NG & $1.32\pm 0.27$ & $1.41\pm 0.83$ & $363 \pm 99$ & $12\pm 5$ & $40\pm 12$ \\
${\rm NG_{c}}$ & $0.97\pm 0.23$ & $0.95\pm 0.95$ & $257\pm 76$ & $10\pm 4$ & $31\pm 10$ \\
\hline
\end{tabular}
}
\end{table}
\section{Segregation Analysis}
Segregation analysis is a powerful tool to evaluate galaxy evolution in
galaxy systems (e.g. Goto, 2005).
We probe segregation phenomena out to $4R_{200}$, looking for differences in galaxies
with respect to the dynamical state of the groups.
First, we test the presence of luminosity segregation in the velocity space
by computing the normalized velocity dispersion, $\sigma_u$, of the stacked G and NG groups.
In Figure 1, we plot $\sigma_u$ of the composite groups as a function of the absolute magnitude in the R band. We clearly see that, at $M_R \leq -21.5$, the velocity dispersions decreases
towards brighter absolute magnitudes. On the other hand, for fainter absolute magnitudes,
the velocity dispersions are approximately constant. More interestingly, although the
result is similar for both stacked groups, for the NG group we see a steeper correlation
in the bright end than that we see for the G group. If one assumes a constant galaxy mass-to-light ratio, energy equipartition implies $\sigma_u \propto 10^{0.2M_R}$ (e.g. Adami, Biviano \& Mazure, 1998). The regression lines between $\log{\sigma_u}$ and $M_R$ have slopes 0.18$\pm$0.05 and 0.38$\pm$0.03, for G and NG groups, respectively. That is, the brightest galaxies are moving more slowly than other group galaxies. Such a segregation in the velocity space may be interpreted as evidence that these galaxies have reached energy equipartition, as a consequence of dynamical friction (e.g Capelato et al. 1981). In fact, the slope we found for galaxies in the G group is consistent with this interpretation.
However, the steeper relation between $\sigma_u$ and
$M_R$ probably indicates a departure from equipartition state for galaxies in the NG group.
We also should note that, for $M_R\geq -21.5$, velocity dispersions are larger in the NG group.
Therefore, although fainter galaxies both in G and NG groups seem to lie in the
velocity equipartition state generated by violent relaxation,
these galaxies in the NG group have more kinetic energy.
A complementary view of this scenario follows from what is seen in Figure 2. Note that
the velocity dispersion profiles show declining and rising trends, for G and NG groups, respectively.
They approximately cross each other at $2.5R_{200}$ and then separate more and more
for larger radii. This is consistent with the results of Hou et al. (2009), for the CNOC2
galaxy groups sample. Rising profiles are generally interpreted as a possible signature
of mergers (Menci \& Fusco-Femiano, 1996), which suggests a current intense phase of environmental influence
on galaxies in the inner parts of non-Gaussian groups.
Looking for a counterpart of these effects in color,
we plot in Figure 3 the color profiles for the G and NG groups. They clearly reveal a stronger reddenning towards the center for galaxies
in the G group. Also, note that the profiles turn flat approximately at 3$R_{200}$, but
galaxies are still redder in Gaussian groups out to 4$R_{200}$. This result indicates that non-Gaussian groups contain less evolved galaxies at the present epoch even in the outskirts. In fact, galaxies in the NG group are fainter than those
in the G group for all radii, with luminosities presenting rising
profiles in both cases (see Figure 4). Spearman tests indicate significant increasing trends up to 1$R_{200}$ and
2.3$R_{200}$ for the G and NG stacked systems, respectively.
\begin{figure}
\includegraphics[width=84mm]{f01.pdf}
\caption{Composite groups velocity dispersion as a function of the absolute magnitude in the R band. Filled circles denote galaxies in G groups, while open circles denote galaxies in NG groups. Dashed lines indicate the regression fits for galaxies with $M_R\leq -21.5$.}
\label{}
\end{figure}
\begin{figure}
\includegraphics[width=84mm]{f02.pdf}
\caption{Composite groups velocity dispersion as a function of the normalized
radial distances to the group centers. Filled circles denote galaxies in G groups, while open circles denote galaxies in NG groups.}
\label{}
\end{figure}
\begin{figure}
\includegraphics[width=84mm]{f03.pdf}
\caption{B-R color of galaxies in the composite groups as a function of the normalized
radial distances to the group centers. Filled circles denote galaxies in G groups, while open circles denote galaxies in NG groups.}
\label{}
\end{figure}
\begin{figure}
\includegraphics[width=84mm]{f04.pdf}
\caption{Absolute magnitude in R band as a function of the normalized
radial distances to the group centers. Filled circles denote galaxies in G groups, while open circles denote galaxies in NG groups.}
\label{}
\end{figure}
\section{Discussion}
We have studied segregation effects in 57 galaxy groups selected from the 2PIGG catalog (Eke et al. 2004) using 2dF data out to $4R_{200}$. This means we probe galaxy distribution near to the
turnaround radius, thus probably taking into account all members in the infall pattern around the groups
(e.g. Rines \& Diaferio 2006; Cupani, Mezzetti \& Mardirossian 2008).
Instead of focusing our analysis on choosing specific galaxy types to study
segregation, we have used the dynamical state of galaxy systems to test for different
levels of environmental influence on galaxies. The theoretical expectation
is that the underlying velocity distribution is normal for systems in dynamical equilibrium.
Using the AD test, we divided the sample in Gaussian and non-Gaussian groups. These were used
to build the composite G and NG groups. Some general results we found were expected:
segregation in velocity space (galaxies brighter than $M_R=-21.5$ are moving more slowly than other group galaxies);
and color and luminosity gradients towards the center of the groups.
However, important differences emerge when we compare the behaviour of galaxies in G and NG groups.
For instance, color gradient and overall reddening are stronger in the case of the G group out to large distances,
showing a significant raise of more evolved galaxies from non-relaxed to relaxed systems (Figure 3).
This is consistent with the luminosity profiles, indicating that galaxies in the G group are
significantly brighter than those in the NG group (Figure 4).
On the other hand, the rising velocity dispersion profile for galaxies in the NG group indicate that, though less
evolved now, galaxies in non-Gaussian systems may be undergoing a more intense phase of interactions
in their inner parts at the present epoch (Figure 2). These results
are in agreement with the work of Popesso et al. (2007), in which Abell clusters with an abnormally low X-ray luminosity for their
mass have a higher fraction of blue galaxies, and
are characterized by leptokurtic (more centrally concentrated than a Gaussian) velocity distribution of their member
galaxies in the outskirts ($1.5 < R/R_{200} \leq 3.5$), as expected for systems undergoing a phase of mass accretion.
This also fairly agrees with Osmond \& Ponman (2004) who have found that groups with an abnormally low velocity dispersion
relative to their X-ray properties have a higher fraction of
spirals and could be dynamically unrelaxed. The low velocity dispersions are probably consequence of the
interactions in the inner part of the groups. Since they only considered the central group regions and the
brightest galaxies, our analysis suggests that they may have found unrelaxed systems,
but could have understimated the global velocity dispersions of
these groups (see Table 1 and Figure 2).
Taken together, these facts point out a scenario where young systems have galaxies bluer and fainter up to large radii ($\sim 4R_{200}$),
possessing lower velocity dispersions in the inner parts (and higher velocity dispersions in the
outer parts) in comparison to more evolved systems.
This latter result is also related to the segregation detected in the velocity space.
Galaxies brighter than $M_R=-21.5$ are moving more slowly than other group galaxies,
but the relation $\sigma_u-M_R$ is steeper for non-Gaussian groups, indicating a departure
from the energy equipartition expectation -- $\sigma_u \propto 10^{0.2M_R}$ (see Figure 1).
Our work suggests that the slope of the relation $\sigma_u-M_R$ could be
used to determine the evolutionary
stage of galaxy groups.
\section*{Acknowledgments}
We thank the referee for very useful suggestions.
We also thank S. Rembold for interesting discussions.
ALBR thanks the support of CNPq, grants 201322/2007-2 and 471254/2008-8.
PAAL thanks the support of FAPERJ, process 110.237/2010. MT thanks the support of FAPESP, process 2008/50198-3.
|
{
"timestamp": "2010-09-23T02:02:53",
"yymm": "1009",
"arxiv_id": "1009.4452",
"language": "en",
"url": "https://arxiv.org/abs/1009.4452"
}
|
\chapter{Einleitung}
Dem Wesen theoretischer Arbeiten im Bereich der mathematischen Physik
entsprechend, bereichert auch diese Diplomarbeit auf zweierlei Weisen.
Zum einen bietet sie eine Fülle an mathematischem
Abwechslungsreichtum, da hier sowohl algebraische als auch
topologischen Methoden zum Einsatz kommen, die das Erreichen
der im Rahmen dieser Arbeit erstrebten Ziele durch ihr elegantes
Zusammenspiel überhaupt erst ermöglichen. Die Kombination von
abtrakter homologischer Algebra mit Funktional-Analytischen Konzepten
wird unter Ausnutzung explizit konstruierter Kettenabbildungen tiefere Erkenntnisse
über die gewünschten Hochschild-Kohomologien liefern, die im Rahmen des
allgemeinen algebraischen Formalismuses nicht greifbar wären und über
reine Isomorphieaussagen weit hinaus gehen.
Die physikalische Motivation ist dabei von dem innigen Wunsch getragen,
eine der grundlegendsten physikalischen Theorien, die
Quantenfeldtheorie, in ihrer Natur zu ergründen und ihr einzigartiges Zusammenwirken
mit den altbewährten klassichen Theorien besser zu verstehen. Der
wichtige Beitrag soll hierbei im Rahmen Deformationsquantisierung
geleistet werden, die als Bindeglied zwischen den klassischen und
den moderneren Quantenfeldtheorien anzusehen ist.
\section*{Motivation}
Die Quantisierung einer klassischen Theorie ist eine
oftmals schwer zu fassende Prozedur, die sich eher
formalen Argumenten bedient, als wirklich die grundlegenden
Zusammenhänge aufzuzeigen. Hierbei erweist sich die quantisierte Theorie
im Allgemeinen als die fundamentalere von beiden, jedoch ist nicht
ignorierbar, dass auch die klassiche Theorie die Natur im Rahmen ihres
Gültigkeitsbereiches vortrefflich beschreibt. Insofern ist es eine interessante und zudem
absolut nicht-triviale Frage, unter
welchen Bedingungen und aus welchem Grund die klassiche Theorie der allgemeineren
Quantentheorie vorzuziehen ist. Eng damit verbunden ist Frage,
inwiefern wir unsere bisherigen Ansichten über die täglich erlebte
Realität in Frage stellen müssen. Denn es ist sicher nicht klar, ob
die Quantenfeldtheorie oder spezieller die Quantenmechanik nicht auch in anderen Bereichen als dem Mikrokosmos
anwendbar ist. Imposante Beispiele sind hierbei sicher das
Konzept der Superposition von Zuständen und die Rolle des Messprozesses an sich.
Diese haben in der klassichen Mechanik keine Bedeutung, sorgen jedoch in der
Quantenmechanik dafür, dass sich die physikalische Realität
eines ganzen Quantensystems allein durch eine Messung vollkommen
verändert werden kann.
\subsection*{Deformation von Observablenalgebren}
In der klassischen Mechanik besteht der Konfigurationsraum aus Orts-
und-Impulsko\-or\-dinaten $q$ und $p$ und kann bei einem $n$-Teilchensystem als
$\mathbb{R}^{2n}$ aufgefasst werden. Geläufige assoziativen,
kommutative Observablenalgebren
sind hierbei die Polynome
$\Pol\left(\mathbb{R}^{n}\right)$ oder die glatten Funktionen
$C^{\infty}\left(\mathbb{R}^{2n}\right)$, die vermöge der total antisymmetrischen
Poisson-Klammer
\begin{equation*}
\{f,g\}=
\displaystyle\sum_{k=1}^{n}\frac{\pt f}{\pt q^{k}}\frac{\pt g}{\pt
p_{k}}-\frac{\pt g}{\pt q^{k}}\frac{\pt f}{\pt p_{k}}\qquad \forall\:f,g
\in \Pol\left(\mathbb{R}^{2n}\right)\text{ oder } C^{\infty}\left(\mathbb{R}^{2n}\right),
\end{equation*} welche die Leibnizregel und die Jacobiidentität erfüllt,
zu einer Poisson-Algebra werden. Auf der anderen Seite besteht der
Konfigurationsraum der Quantenmechanik aus einem Hilbertraum
$\mathcal{H}$ und die Observablen aus der $^{*}$-Unteralgebra der
beschränkten Operatoren $\mathcal{B}(\mathcal{H})$, wobei die hierin
enthaltenen selbstadjungierten
Operatoren die tatsächlich physikalisch messbaren Observablen sind.
Die Deformationstheorie assoziativer Algebren legt uns nun ein effektives
Werkzeug in die Hand, Quantenobservablen aus denen der klassichen
Theorie derart zu konstruieren, dass die Nichtkommutativität der
neuen Algebramultiplikation $\star$ bereits gewährleistet ist. Hierbei
betrachtet man für
eine klassiche Observablenalgebra $(\mathcal{A},*)$
den Raum $\mathcal{A}\llbracket
\hbar\rrbracket$
der formalen Potenzreihen in dem formalen Parameter $\hbar$ mit
Koeffizienten in $\mathcal{A}$, in welchen man sich $\mathcal{A}$ im
Vektorraum-Sinne als
Monome $0$-ter Ordnung eingebettet vorstellen kann.
Eine formale
Deformation $\mu$ der Ordnung $k$ ist dann (vgl. \cite{bayen.et.al:1978a})
eine
$\mathbb{C}\nk{\hbar}$-bilineare Multiplikation
$\mu \colon \mathcal{A}\nk{\hbar}\times
\mathcal{A}\nk{\hbar}\longrightarrow \mathcal{A}\nk{\hbar}$
der Form
\begin{equation*}
\mu(a, b) = \sum_{r=0}^{k}\hbar^{r}
\mu_{r}(a,b)\qquad\forall\:a,b\in \mathcal{A}
\end{equation*}mit $\mathbb{C}$-bilinearen Abbildungen
$\mu_{r}\colon \mathcal{A}\times \mathcal{A}\longrightarrow \mathcal{A}$,
so dass $\mu$ folgende Eigenschaften besitzt:
\begin{enumerate}
\item
$\mu$ ist assoziativ bis zur Ordnung $k$.
\item
$\mu_{0}(a,b)=a*b$.
\item
$\mu_{1}(a,b)-\mu_{1}(b,a)=i\{a,b\}$.
\end{enumerate}
Hierbei ist \textit{iii.)} auch als
\emph{Korrespondezprinzip} bekannt und garantiert, dass
der total antisymmetrische Teil des Quanten-Kommutators $[a,b]:=\mu(a, b)- \mu(b, a)$ in
der Ordnung $\hbar$ mit $i\{\cdot,\cdot\}$ übereinstimmt. Ist $\hbar$
in der physikalischen Situation eine dimensions-behaftete Größe, so
sind die $\mu_{r}$ als Größen der Dimension $[(Js)^{-r}]$ zu verstehen,
womit $\mu$ eine Multiplikation
$\mathcal{A}\times \mathcal{A}\rightarrow \mathcal{A}$ definiert.
Im Falle $k=\infty$ spricht man von einem Sternprodukt und schreibt
$\star$ anstelle von $\mu$. Diese stellen die eigentlich interessanten
Objekte dar, wobei hier die physikalische Wohldefiniertheit von $\mu$, also die
Konvergenz der Summe, ein im
Allgemeinen ungelöstes Problem darstellt. Die Existenz und Klassifikation solcher Sternprodukte sind
für die klassiche Situation wohlverstandene Probleme und wurde für die Algebra $C^{\infty}(M)$ einer
endlich-dimensionalen symplektischen Mannigfaltigkeit $M$ erstmals von Lecomte und de Wilde
\cite{dewilde.lecomte:1983b, dewilde.lecomte:1988a} unter
Verwendung kohomologischer Überlegungen gelöst. Einen sehr
einfachen und geometrischen Existenzbeweis liefert zudem die
Fedosov-Konstruktion, die ohne kohomologische Überlegungen auskommt
und sich lediglich "`konventioneller"' Techniken wie kovarianter
Ableitungen und dem Tensorkalkül bedient, vgl. \cite{fedosov:1996a}. Ganz allgemein für endlich-dimensionale
Poisson-Mannigfaltigkeiten wurde das Existenz- und Klassifikations-Problem von
Kontsevich (vgl. \cite{kontsevich:2003a}) gelöst.
Ein einfache Verfahren um Sternprodukte zu erhalten, ist, sich diese
Ordnung für Ordnung zu konstruieren. dabeibei sind die Assoziativität
von $\mathcal{A}$ und $\mathcal{A}\nk{\hbar}$ die entscheidenden
Faktoren. Hierfür betrachten wir eine formale Deformation $\mu=\mu_{0}+…+\mu_{k}$
der Ordnung $k$, die wir durch ein $\mathbb{C}$-bilineares $\mu_{k+1}$ zu einer formalen
Deformation $\circ=\mu + \hbar^{k+1}\mu_{k+1}$ der Ordnung $k+1$ fortsetzen wollen. Dann muss die
Bedingung $a\circ(b\circ c)=(a\circ b)\circ c$ insbesondere für alle $a,b,c \in
\mathcal{A}$ bis zur Ordnung $k+1$
erfüllt sein, in welcher wir erhalten, dass\footnote{vgl. \cite[Kapitel 2]{Weissarbeit}}
\begin{align*}
a*\mu_{k+1}(b,c) - \mu_{k+1}(a*b,c) + &\:\mu_{k+1}(a,b*c)-
\mu_{k+1}(a,b)*c
\\ &=\underbrace{\sum_{r=1}^{k}\big[\mu_{r}(\mu_{k+1-r}(a,b),c)-\mu_{r}(a,\mu_{k+1-r}(b,c))\big]}_{R_{k}}
\end{align*}gilt, was mit Hilfe des Hochschild-Differentials auch in der
Form $\delta\mu_{k+1}=R_{k}$ geschrieben werden kann. Eine
längere Rechnung unter Ausnutzung der Assoziativität von $\mathcal{A}$
zeigt zudem $\delta R_{k}=0$, womit ein derartiges $\mu_{k+1}$ nur
dann gefunden werden kann, wenn $[R_{k}]$ die $0$-Klasse ist.
In diesem Sinne bilden die Elemente der dritten
Hochschild-Kohomologie $HH^{3}(\mathcal{A},\mathcal{A})$ der Algebra $\mathcal{A}$, die
Quelle von Obstruktionen für die Fortsetzbarkeit
formaler Deformationen zu Sternprodukten und es
lässt sich zeigen, dass die zweite Hochschild-Kohomologie
die Äquivalenzklassen von infinitisimalen Deformationen,
also solchen bis zur Ordnung $1$ klassifiziert. Ist hingegen ein
Sternprodukt $\left(\mathcal{A}\nk{\hbar},\star\right)$ für eine
assoziative und
kommutative Algebra $\mathcal{A}$ vorgegeben und hat man einen $\mathcal{A}$-Modul $\mathcal{M}$, dessen Modulstruktur man
auf $\mathcal{A}\nk{\hbar}$ fortsetzen möchte, so sind
die Hoch\-schild-Ko\-ho\-mologien
$HH^{2}(\mathcal{A},\End_{\mathbb{K}}(\mathcal{M}))$ und
$HH^{1}(\mathcal{A},\End_{\mathbb{K}}(\mathcal{M}))$ von
entscheidender
Wichtigkeit\footnote{$\End_{\mathbb{K}}(\mathcal{M})$ ist
hier als $\mathcal{A}-\mathcal{A}$-Bimodul aufzufassen} für das
Deformationsproblem.
Die Berechnung dieser Hochschild-Kohomologien ist ein für Observablenalgebren auf endlich-dimensionalen
Vektorräume, wie es beispielsweise der Konfigurationsraum der klassichen
Mechanik ist, gut verstandenes Problem. Die Kohomologie-Gruppen
$HH^{k}\left(\Pol\left(\mathbb{R}^{n}\right),\Pol\left(\mathbb{R}^{n}\right)\right)$
wurden erstmals von Hochschild, Kostant und Rosenberg im Rahmen des
Hochschild-Kostant-Rosenberg-Theoremes bestimmt
(vgl. \cite{hochschild.kostant.rosenberg:1962a}), welches inbesondere
besagt, dass jede Kohomologieklasse $[\eta]\in
HH^{k}\left(\Pol\left(\mathbb{R}^{n}\right),\Pol\left(\mathbb{R}^{n}\right)\right)$
genau einem $k$-Multivektorfeld entspricht. Analoge Aussagen wurden ebenfalls
für lokalen, stetigen und
die differentiellen Hochschild-Koho\-mo\-lo\-gien
\begin{align*}
&HH^{k}_{\operatorname{\mathrm{diff}}}(C^{\infty}(M),C^{\infty}(M)),\\
&HH^{k}_{\mathrm{loc}}(C^{\infty}(M), C^{\infty}(M))\qquad\text{und}\\
&HH^{k}_{\operatorname{\mathrm{cont}}}(C^{\infty}(M), C^{\infty}(M))
\end{align*}
gezeigt
\cite{pflaum:1998a, connes:1994a ,cahen.gutt.dewilde:1980a}. In
\cite{Weissarbeit} wurden zudem die stetig-differentielle
Hoch\-schild-Kohomologie $HH_{\mathrm{c,d}}^{k}(C^{\infty}(V),\mathcal{M})$ für
differentielle $C^{\infty}(V)-C^{\infty}(V)$-Bimoduln $\mathcal{M}$ mit einer
konvexen Teilmenge $V\subseteq \mathbb{R}^{n}$ berechnet.
\subsection*{Observablenalgebren in der Feldtheorie, die symmetrische Algebra}
In jeder klassichen Feldtheorie
besteht der Konfigurationsraum aus
Feldern, die im Speziellen selbst die glatten Funktionen auf einer Mannigfaltigkeit, aber in jedem Falle unendlich-dimensionale
$\mathbb{K}$-Vektorr"aume\footnote{Hier und im Folgenden bedeutet
$\mathbb{K}$ immer $\mathbb{R}$ oder
$\mathbb{C}$.} sind. Ist zum Beispiel $X\subseteq \mathbb{R}^{n}$ eine
offene Teilmenge und $\mathcal{D}(X)$ der Testfunktionen-Raum
der glatten Funktionen von $X\longrightarrow\mathbb{R}$ mit Kompaktem Träger
in $X$. Dann ist man insbesondere an den Observablen der Form
\begin{equation*}
p(\psi)=\sum_{k=0}^{n}\int_{X_{1}\times…\times X_{k}}\phi_{k}(x_{1},…,x_{k})\:\psi(x_{1})…\psi(x_{k})\:dx_{1}…dx_{k}
\end{equation*}mit $n\in \mathbb{N}$, $\psi\in \mathcal{D}(X)$, $X_{i}=X$ für alle $1\leq
i\leq k$ und $\phi_{k}\in \mathcal{E}^{\mathrm{sym}}_{\mathrm{sep}}\left(X^{k}\right)$
interessiert. Hierbei bezeichnet
{\small\begin{equation*}
\mathcal{E}_{\mathrm{sep}}\left(X^{k}\right)=\left\{\phi\in
\mathcal{E}\left(X^{k}\right)\:\Bigg|\:\phi(x_{1},…,x_{k})=\sum_{i=1}^{n}\phi_{1}(x_{1})…\phi_{k}(x_{k})\:\:\forall\:(x_{1},…,x_{k})\in
X^{k}\right\}
\end{equation*}}mit $\phi_{1},…,\phi_{k}\in \mathcal{E}(X)=C^{\infty}(X)$ für alle $1\leq
i\leq k$ den Vektorraum aller Abbildungen $X^{k}\longrightarrow \mathbb{R}$, die als endliche Summe
faktorisierender, glatter Funktionen geschrieben werden können und
$\mathcal{E}^{\mathrm{sym}}_{\mathrm{sep}}\left(X^{k}\right)$ den
Unterraum der
total symmetrische Elemente
von $\mathcal{E}_{\mathrm{sep}}$. Jedes
$\mathcal{E}_{\mathrm{sep}}\left(X^{k}\right)$ ist eine Realisierung
des $k$-fachen Tensorproduktes $\Tt^{k}(\mathcal{E}(X))$ und
$\mathcal{E}^{\mathrm{sym}}_{\mathrm{sep}}\left(X^{k}\right)$ eine
Realisierung des symmetrischen Tensorproduktes
$\Ss^{k}(\mathcal{E}(X))$. Mit Hilfe der Abbildung
\begin{equation*}
\tau\colon (\phi,\psi)\longrightarrow \int_{X}\phi(x)\psi(x)dx
\end{equation*}und linearer Fortsetzung von
\begin{align*}
\Delta\colon \Tt^{k}(\mathcal{E}(X))\times \mathcal{D}(X)&\longrightarrow
\mathbb{R}\\
(\phi_{1}\ot…\ot \phi_{k},\psi)&\longmapsto \tau(\phi_{1},\psi)…\tau(\phi_{k},\psi)
\end{align*}auf die gesamte symmetrische Algebra
$\Ss^{\bullet}(\mathcal{E}(X))$, entspricht diese gerade den
Observablen der obigen Form. Möchte man hingegen auch
unendliche Summen und symmetrische $\phi_{k}$ in ganz
$\mathcal{E}\left(X^{k}\right)$ zulassen, so ist dies möglich, indem
man $\mathcal{E}(X)$
mit der üblichen Fr\'echet-Topologie versieht und die symmetrische Algebra mit
Hilfe des Konzeptes des $\pi$-Tensorproduktes lokalkonvex
topologisiert. Durch Vervollständigung von
$\Ss^{\bullet}(\mathcal{E}(X))$
erhält man eine Algebra $\mathrm{Hol}(\mathcal{E}(X))$, welche die
gewünschten Observablen\footnote{Unter der Vorraussetzung, dass diese
gewisse "`Konvergenzbedingungen"' erfüllen.} induziert.
Sind
in diesem Rahmen Feldgleichungen auf $\mathcal{D}(X)$ durch einen
linearen Operator $\Lambda$ gegeben, so lässt sich unter gewissen
Voraussetzungen zeigen (vgl. \cite{baer.ginoux.pfaeffle:2007a}), dass
zu $\Lambda$ gehörige avancierte und retardierte Greensche
Funktionen $G_{x,y}^{+}$ und $G^{-}_{x,y}$ existieren mit denen man
durch derivative Fortsetzung von
\begin{align*}
\{\phi,\psi\}=\Delta(\phi,\psi)\qquad \forall\:\phi,\psi\in
\mathcal{E}(X)\quad\text{mit}\quad \Delta=G_{x,y}^{+}-G^{-}_{x,y}
\end{align*} auf ganz $\Ss^{\bullet}(\mathcal{E}(X))$, eine stetige Poisson-Klammer erhält, die stetig bilinear auf
$\mathrm{Hol}(\mathcal{E}(X))$ fortgesetzt werden kann. Hierfür existieren bereits
eine Fülle an Beispielen für Sternprodukte (vgl,
\cite{duetsch.fredenhagen:2001a},\cite{duetsch.fredenhagen:2003a}), die
mit den Resultaten dieser Arbeit nun in einem formalen Rahmen
behandelbar sind.
Das eben behandelte Beispiel ist lediglich der Spezialfall eines
allgemeinen Konzeptes, das es erlaubt, sich Observablenalgebren mit
Hilfe der symmetrischen Algebra zu konstruieren. Hierfür benötigt man lediglich einen
Konfigurationsraum $\V$ und einen hausdorffschen, lokalkonvexen Vektorraum
$(\mathbb{U},P)$ mit einer $\mathbb{K}$-bilineare Abbildung
$\tau\colon \V\times \mathbb{U}\colon \longrightarrow \mathbb{K}$, deren Bild sich für festes $v\in \V$ für alle $u\in
\mathbb{U}$ durch eine Halbnorm aus $P$ abschätzen lässt. Dann definiert jedes Element aus
$\mathrm{Hol}(\mathbb{U})$ eine auf ganz $\V$ konvergente
Potenzreihenfunktion. Ist beispielsweise $\V=\mathbb{C}^{n}$ und
$\mathbb{U}=\mathbb{C}^{n*}$ schwach*-topologisiert, so entspricht
$\mathrm{Hol}(\mathbb{C}^{n*})$ gerade den ganz holomorphen Funktionen auf
$\mathbb{C}^{n}$, was gleichzeitig der Grund für die
Namensgebung $\mathrm{Hol}$ ist. Im Falle $\V=\mathcal{E}(X)$ mit schwach*-topologisiertem
$\mathbb{U}=\mathcal{E}'(X)$ ist dann beispielsweise die Exponentialfunktion
\begin{equation*}
p(\phi)=\sum_{k=0}^{\infty}\frac{1}{k!}\overbrace{\delta_{z}(\phi)…\delta_{z}(\phi)}^{k-mal}=\sum_{k=0}^{\infty}\frac{1}{k!}\phi(z)^{k}
\end{equation*}in $\mathrm{Hol}(\mathcal{E}'(X))$ enthalten und die
angeführten Beispiele bilden nur einen Bruchteil der Kombinationen, die möglich sind.
\section*{Ziele dieser Arbeit}
Die symmetrische Algebra über einem beliebigen $\mathbb{K}$-Vektorraum
$\V$ besitzt die Eigenschaft, dass sie eine Fülle an wichtigen
Observablenalgebren der klassichen Feldtheorie als Spezialfall enthält.
Das Ziel dieser Arbeit soll es daher sein, die
Elementarbausteine der Deformationsquantisierung, die
Hochschild-Kohomologien, dieser reichhaltigen Algebra zu berechnen um
hiermit die Deformationstheorie dieser Observablenalgebren
auf ein festes Fundament zu stellen. Zudem wollen wir in den wichtigen
Spezialfällen lokalkonvexer Vektorräume $\V$ die interessantere stetige
Hochschild-Kohomologien von $\SsV$ und im hausdorffschen
Fall ebenfalls die ihrer noch umfassenderen Vervollständigung $\Hol$ berechnen, welche unter anderem
solch wichtige Observablen wie die Exponentialfunktion
enthält. Hierbei ist der Nutzen der stetigen Hochschild-Kohomologien
darin zu sehen, als dass Sternprodukte, die aus stetigen bilinearen
Komponenten $\mu_{r}$ bestehen, im Allgemeinen reguläreres Verhalten
zeigen und somit leichter zu handhaben sind.
\section*{Resultate}
Sei im Folgenden $\V$ ein beliebig-dimensionaler $\mathbb{K}$-Vektorraum mit
$\mathbb{K}=\mathbb{R}$ oder $\mathbb{C}$. Dann bezeichnet
$(\SsV,\vee)$ die symmetrische Algebra über $\V$ und $\mathcal{M}$ einen $\SsV-\SsV$-Bimodul, der
ebenfalls ein $\mathbb{K}$-Vektorraum ist.
Befinden wir uns im lokalkonvexen Rahmen, so verstehen wir $(\V,\T_{P})$ als
lokalkonvexen Vektorraum mit erzeugendem Halbnormensystem $P$ und $\SsV$
denken wir uns dann, vermöge dem submultiplikativen Halbnormensystem
$\Pp$, bestehen aus den Elementen
\begin{equation*}
\p(\omega)=\sum_{k=0}^{\infty}p^{k}(\omega_{k})\qquad
p\in \tilde{P},\:\SsV\ni\omega=\sum_{k}\omega_{k}\text{ mit }\omega_{k}\in \Ss^{k}(\V),
\end{equation*} lokalkonvex topologisiert. Hierbei bezeichnet
$\tilde{P}$ das filtrierende System aller bezüglich $\T_{P}$ stetigen
Halbnormen und wegen der Submultiplikativität ist $\vee$
stetig. Ist $(\V,\T_{P})$ hausdorffsch, so bezeichnet $(\Hol,*)$ die lokalkonvexe Algebra mit
submultiplikativem Halbnormensystem $\hat{\Pp}$, die durch
Vervollständigung von $(\SsV,\vee)$ erhalten wird. In diesem Rahmen
ist $\mathcal{M}$ zudem als lokalkonvexer Vektorraum derart zu
verstehen, dass die Modul-Multiplikationen stetig sind.\\\\
In jedem Fall verlangen wir, dass die Modul-Multipliaktionen
$\mathbb{K}$-bilinear sind und dass $\mathcal{M}$ verträglich ist,
dass also $1*_{L}m=m=m*_{R} 1$ für alle $m\in \mathcal{M}$ gilt.\\\\
\emph{Die Resultate dieser Arbeit sind:}
\begin{itemize}
\item
\textbf{Satz}\\
\begin{enumerate}
\item
Gegeben ein $\SsV-\SsV$-Bimodul $\mathcal{M}$, dann gilt:
\begin{equation*}
HH^{k}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong H^{k}\left(KC(\V,\mathcal{M}),\Delta\right).
\end{equation*}
Ist $\mathcal{M}$ zudem symmetrisch, so ist:
\begin{equation*}
HH^{k}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong \Hom^{a}_{\mathbb{K}}\big(\V^{k},\mathcal{M}\big).
\end{equation*}
\item
Gegeben ein lokalkonvexer, $\SsV-\SsV$-Bimodul $\mathcal{M}$, dann gilt:
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}\left(\Ss^{\bullet}(\mathbb{V}),\mathcal{M}\right)\cong
H^{k}\big(KC^{\operatorname{\mathrm{cont}}}(\V,\mathcal{M}),\Delta^{c}\big)
\end{equation*}
Ist $\mathcal{M}$ zudem symmetrisch, so ist:
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong\Hom_{\mathbb{K}}^{a,\operatorname{\mathrm{cont}}}(\V^{k},\mathcal{M}).
\end{equation*}
\item
Gegeben ein vollständiger, hausdorffscher, lokalkonvexer
$\Hol-\Hol$-Bimodul $\mathcal{M}$, dann gilt:
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}\big(\Hol,\mathcal{M}\big)\cong HH^{k}_{\operatorname{\mathrm{cont}}}\big(\Ss^{\bullet}(\V),\mathcal{M}\big).
\end{equation*} Ist $\mathcal{M}$ zudem symmetrisch, so ist:
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}(\Hol,\mathcal{M})\cong \Hom_{\mathbb{K}}^{a,\operatorname{\mathrm{cont}}}(\V^{k},\mathcal{M}).
\end{equation*}
\end{enumerate}
Hierbei bezeichnet $\Hom_{\mathbb{K}}^{a}(\V^{k},\mathcal{M})$ die
total antisymmetrischen, $\mathbb{K}$-multilinearen Abbildungen von
$\V^{k}$ nach $\mathcal{M}$ und $\V^{k}$ das $k$-fache
kartesische Produkt von $\V$. Mit
$\left(KC(\V,\mathcal{M}),\Delta\right)$ ist der Kokettenkomplex
mit Kettengliedern $KC^{k}(\V,\mathcal{M})=\Hom_{\mathbb{K}}^{a}(\V^{k},\mathcal{M})$
und Kettendifferentialen
\begin{equation*}
(\Delta^{k}\phi)(v_{1},\dots, v_{k+1})=
\sum_{l=1}^{k+1}(-1)^{l-1}\:[u_{l}*_{L}-\:u_{l}\:*_{R}]\:\phi(v_{1},\dots,\blacktriangle^{l},\dots,v_{k+1})
\end{equation*}
gemeint. Schließlich bezeichnet $\left(KC_{cont.}(\V,\mathcal{M}),\Delta^{c}\right)$
den stetigen Unterkomplex von
$\left(KC(\V,\mathcal{M}),\Delta\right)$ mit
Kettengliedern
$\Hom_{\mathbb{K}}^{a,\operatorname{\mathrm{cont}}}(\V^{k},\mathcal{M})\subseteq\Hom_{\mathbb{K}}^{a}(\V^{k},\mathcal{M})$. Die
Isomorphien in Teil \textit{iii.)} werden hierbei durch die
Einschränkungs-Abbildungen $\tau^{k}\colon \hat{\phi}\longrightarrow
\hat{\phi}\big|_{\SsV^{k}}$ induziert, die einen
Kettenisomorphismus $\tau$ zwischen dem Hochschild-Komplex von
$\Hol$ und dem von $\SsV$ definieren. Die Isomorphie in Teil
\textit{i.)} ist abstrakter Natur, kann aber auch durch explizite
Kettenabbildungen $F$ und $G$ oder genauer durch die aus ihnen durch
Anwendung des
$\hom_{\mathcal{A}^{e}}(\cdot,\mathcal{M})$-Funktors\footnote{$\mathcal{A}=\SsV$}
gewonnenen Kettenabbildungen $F^{*}$ und $G^{*}$ erhalten
werden. Besagte $F$ und $G$ haben wir hierbei durch Abstraktion
von solchen
gewonnen\footnote{vgl. \cite{bordemann.et.al:2003a:pre},
\cite[Sect. III.2$\alpha$]{connes:1994a}}, die
für den endlich-dimensionalen Fall existieren. Diese
Kettenabbildungen induzieren nun auch die Isomorphismen in
Teil \textit{ii.)}, wobei hier unter anderem deren
Stetigkeit zu zeigen und explizite stetige Homotopieabbildungen $s_{k}$ zu konstruieren waren.
\item
Eine weitergehende Analyse mit der Hilfe von $F$ und $G$
sowie der Homotopie $s$, liefert in den symmetrischen Fällen die folgenden
unend\-lich-dimen\-sio\-nalen Verallgemeinerungen der klassischen
Hochschild-Kostant-Ro\-sen\-berg-The\-oreme, die nun insbesondere für
den physikalisch relevanteren Spezialfall $\mathcal{M}=\SsV,\Hol$
gelten.\\\\
\textbf{Satz (Hochschild-Kostant-Rosenberg)}
\begin{enumerate}
\item
Gegeben ein symmetrischer $\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul $\mathcal{M}$.
Dann besitzt jede Kohomologieklasse
$\left[\eta\right]\in HH^{k}(\Ss^{\bullet}(\V),\mathcal{M})$ genau einen total
antisymmetrischen Repräsentanten $\phi^{a,\eta}_{D}$.
Dieser ist derivativ in jedem Argument und gegeben durch
$\phi^{a,\eta}_{D}=\mathrm{Alt}_{k}(\phi)$ f"ur beliebiges $\phi\in
[\eta]$ mit $\phi^{a,0}_{D}=0$ für
die $0$-Klasse $[0]$.
Insgesamt gilt für alle $\phi\in [\eta]$:
\begin{equation*}
\phi=\underbrace{\phi^{a,\eta}_{D}}_{\mathrm{Alt}_{k}(\phi)}+\underbrace{\delta^{k-1}\big(\zeta^{k-1}_{-1}s^{*}_{k-1}\zeta^{k}\phi\big)}_{\phi-\mathrm{Alt}_{k}(\phi)}.
\end{equation*}
\item
Gegeben ein symmetrischer, lokalkonvexer
$\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul $\mathcal{M}$.
Dann besitzt jedes
$\left[\eta_{c}\right]\in HH^{k}_{\operatorname{\mathrm{cont}}}(\Ss^{\bullet}(\V),\mathcal{M})$ genau einen total
antisymmetrischen, stetigen Repräsentanten
$\phi^{a,\eta}_{c,D}$. Dieser ist derivativ in jedem Argument und gegeben durch
$\phi^{a,\eta}_{c,D}=\mathrm{Alt}_{k}(\phi_{c})$ f"ur beliebiges $\phi_{c}\in
[\eta_{c}]$ mit $\phi^{a,0}_{c,D}=0$ für
die $0$-Klasse $[0_{c}]$.
Insgesamt gilt für alle $\phi_{c}\in [\eta_{c}]$:
\begin{equation*}
\phi_{c}=\underbrace{\phi^{a,\eta}_{c,D}}_{\mathrm{Alt}_{k}(\phi_{c})}+\underbrace{\delta_{c}^{k-1}\big(\zeta^{k-1}_{-1}s^{*}_{k-1}\zeta^{k}\phi_{c}\big)}_{\phi_{c}-\mathrm{Alt}_{k}(\phi_{c})}.
\end{equation*}
\item
Gegeben ein vollständiger, symmetrischer, hausdorffscher, lokalkonvexer\\
$\Hol-\Hol$-Bimodul $\mathcal{M}$. Dann besitzt jedes
$\left[\hat{\eta}_{c}\right]\in HH^{k}_{\operatorname{\mathrm{cont}}}(\Hol,\mathcal{M})$ genau einen total
antisymmetrischen, stetigen Repräsentanten
$\hat{\phi}^{a,\eta}_{c,D}$. Dieser ist derivativ in
jedem Argument und gegeben durch $\hat{\phi}^{a,\eta}_{c,D}=\mathrm{Alt}_{k}\big(\hat{\phi}_{c}\big)$ f"ur beliebiges $\hat{\phi}_{c}\in
[\hat{\eta}_{c}]$ mit $\hat{\phi}^{a,0}_{c,D}=0$ für
die $0$-Klasse $[\hat{0}_{c}]$.
Insgesamt gilt für alle $\hat{\phi}_{c}\in [\hat{\eta}_{c}]$:
\begin{equation*}
\hat{\phi}_{c}=\underbrace{\hat{\phi}^{a,\eta}_{c,D}}_{\mathrm{Alt}_{k}(\hat{\phi}_{c})}+\underbrace{\hat{\delta}_{c}^{k-1}\widehat{\Big(\zeta^{k-1}_{-1}s^{*}_{k-1}\zeta^{k}\phi_{c}\Big)}}_{\hat{\phi}_{c}-\mathrm{Alt}_{k}(\hat{\phi}_{c})}\quad\text{
mit }\quad \phi_{c}=\hat{\phi}_{c}\big|_{\SsV^{k}}.
\end{equation*}
\end{enumerate}
Hierbei ist $\zeta$ ein sehr einfacher Kettenisomorphismus, zwischen dem Hochschild-Komplex und einem Hilfkomplex, in
den auch $G^{*}$ abbildet. Mit
$s^{*}_{k}=\hom_{\mathcal{A}^{e}}s_{k}$ ist der Pullback mit der rekursiv definierten
Homotopieabbildung $s_{k}$ gemeint. In Teil \textit{iii.)} bezeichnet
$\widehat{\Big(\zeta^{k-1}_{-1}s^{*}_{k-1}\zeta^{k}\phi_{c}\Big)}$
die stetige Fortsetzung von
$\Big(\zeta^{k-1}_{-1}s^{*}_{k-1}\zeta^{k}\phi_{c}\Big)$
und $\phi_{c}=\hat{\phi}_{c}\big|_{\SsV}$ die Einschränkung der
stetige Abbildung $\hat{\phi}_{c}$ auf $\SsV\subseteq \Hol$. Eine
genauere
Analyse liefert für die erste Hochschild-Kohomologie
\begin{align*}
[\eta]&=\phi_{D}^{a,\eta}\quad \text{für alle}\quad[\eta]\in HH^{1}(\SsV,\mathcal{M}),\\
[\eta_{c}]&=\phi_{c,D}^{a,\eta}\quad\text{für
alle}\quad[\eta]\in HH_{\operatorname{\mathrm{cont}}}^{1}(\SsV,\mathcal{M}),\\
[\hat{\eta}_{c}]&=\hat{\phi}_{c,D}^{a,\eta}\quad\text{für alle}\quad[\hat{\eta}]\in HH_{\operatorname{\mathrm{cont}}}^{1}(\Hol,\mathcal{M})
\end{align*}und für die zweite die explizite Formel
\begin{align*}
\left(\zeta^{1}_{-1}s^{*}_{1}\zeta^{2}\phi\right)(x)=&\:\phi(1,x)+\sum_{p=1}^{l}\Bigg[\frac{1}{p}\phi\left(x^{p},x_{p}\right)+…\nonumber
\\ &+\binom{p}{l}^{-1}\sum_{j_{1},…j_{l}}^{p-1}\phi\left((x^{p})^{j_{1},…,j_{l}},x_{p}\right)*_{R}(x^{p})_{j_{1},…,j_{l}}+…\nonumber
\\ &+\frac{1}{p}\phi(1,x_{p})*_{R}x^{p}\Bigg]
\end{align*}für alle $x\in \SsV$ mit $\deg(x)=l$. Analoge, wenn
auch weniger konkrete Aussagen für den
nichtsymmetrischen Fall sind
\begin{align*}
\phi&=\overbrace{\zeta_{-1}^{k}\Omega^{*}_{k}\zeta^{k}\phi}^{=\:\tilde{\phi}\in
[\eta]} +
\left(\phi-\zeta_{-1}^{k}\Omega^{*}_{k}\zeta^{k}\phi\right)=\tilde{\phi}+
\delta^{k-1}\big(\zeta_{-1}^{k-1}s^{*}_{k-1}\zeta^{k}\phi\big),\\
\phi_{c}&=\overbrace{\zeta_{-1}^{k}\Omega^{*}_{k}\zeta^{k}\phi_{c}}^{=\:\tilde{\phi_{c}}\in
[\eta_{c}]} +
\left(\phi_{c}-\zeta_{-1}^{k}\Omega^{*}_{k}\zeta^{k}\phi_{c}\right)=\tilde{\phi_{c}}+
\delta_{c}^{k-1}\big(\zeta_{-1}^{k-1}s^{*}_{k-1}\zeta^{k}\phi_{c}\big)
\end{align*}für $\phi\in[\eta]\in HH^{k}(\SsV,\mathcal{M})$,
$\phi_{c}\in[\eta_{c}]\in HH_{\operatorname{\mathrm{cont}}}^{k}(\SsV,\mathcal{M})$ und
$\hat{\phi}_{c}\in[\hat{\eta}_{c}]\in
HH_{\operatorname{\mathrm{cont}}}^{k}(\Hol,\mathcal{M})$ sowie
\begin{equation*}
\hat{\phi}_{c}=\overbrace{\widehat{\Big(\zeta_{-1}^{k}\Omega^{*}_{k}\zeta^{k}\phi_{c}\Big)}}^{\in [\hat{\eta}_{c}]} +\:
\hat{\delta}_{c}^{k-1}\widehat{\left(\zeta_{-1}^{k-1}s^{*}_{k-1}\zeta^{k}\phi_{c}\right)}\quad\text{
mit }\quad [\hat{\eta}_{c}]\ni\phi_{c}=\hat{\phi}_{c}\big|_{\SsV^{k}}
\end{equation*}und $\Omega_{k}=F_{k}\cp G_{k}$.
Hierfür beachte man, dass wir in obiger Formel f"ur
$\zeta^{1}_{-1}s^{*}_{1}\zeta^{2}\phi$ explizit zwischen $*_{R}$
und $*_{L}$ unterschieden haben, diese also auch für
nicht\--\-sym\-me\-trischen Fall gültig ist. Besagte Formel lässt
sich dann in der
Situation der Deformation einer Modul-Struktur zur rekursiven
Konstruktion nutzbringend
einsetzen, sofern die zweite Hochschild-Kohomologie
$HH^{2}(\mathcal{A},\End_{\mathbb{K}}(\mathcal{M}))$ verschwindet.
Die Bezeichnungsweise
"`Hochschild-Kostant-Rosen\-berg-Theoreme"' ist hierbei auch
insofern gerechtfertigt als dass jedes, in allen Argumenten
derivative $\phi \in
HC^{k}(\SsV,\mathcal{M})$,
die algebraische Definition eines Differentialoperators erster
Ordnung erfüllt, also $\phi_{D}^{a,\eta}\in \DiffOpS{k}{1}$ gilt. Weiterhin ist jedes derartige Element
$\phi\in HC^{k}_{\operatorname{\mathrm{cont}}}(\SsV,\mathcal{M})$ ein stetiger
Differentialoperator erster Ordnung und ebenso verhält es sich mit
solchen Elementen aus
$HC^{k}_{\operatorname{\mathrm{cont}}}(\Hol,\mathcal{M})$. In diesem Sinne nehmen die
Repräsentanten $\phi_{D}^{a,\eta}$ den Platz ein, der den Multivektorfelder
im endlich-dimensionalen Rahmen gebührt, vgl. \cite[Prop
6.2.8]{waldmann:2007a}.
\item
Motiviert durch den Fakt, dass $G^{*}$ in beiden Fällen eine
Kettenabbildung
\begin{align*}
\xi\colon&\left(KC(\V,\mathcal{M}),\Delta\right)\longrightarrow\left(HC(\SsV,\mathcal{M}),
\delta\right)\qquad\qquad\quad\text{bzw.}
\\
\xi\colon&\left(KC_{\operatorname{\mathrm{cont}}}(\V,\mathcal{M}),\Delta^{c}\right)\longrightarrow\left(HC_{\operatorname{\mathrm{cont}}}(\SsV,\mathcal{M}),\delta_{c}\right)
\end{align*}
definiert, die Isomorphismen $\wt{\xi}^{k}$ auf Kohomologie-Niveau
induziert und zudem differentielle Bilder hat, sind wir der
Frage nachgegangen, ob sich mit dieser auch die
Hochschild-Kohomologien der differen\-tiellen und
der stetig-diffe\-rentiellen Unterkomoplexe
$\left(HC_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M}),\delta_{\operatorname{\mathrm{diff}}}\right)$ und
$\left(HC_{\mathrm{c,d}}(\SsV,\mathcal{M}),\delta_{\mathrm{c,d}}\right)$
berechnen lassen. Diese sind zunächst nur im Falle symmetrischer Bimoduln
wohldefiniert, für den wir folgendes Korollar erhielten:\\\\
\textbf{Korollar}
\begin{enumerate}
\item
Sei $\mathcal{M}$ ein symmetrischer $\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul. Dann induzieren $\xi$ und $\hat{\xi}$ Kettenabbildungen $\wt{\xi^{k}}$ und $\wt{\hat{\xi}^{k}}$ zwischen $(HC_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M}),\delta_{\operatorname{\mathrm{diff}}})$ und\\ $(KC(\V,\mathcal{M}),\Delta)$.
Des Weiteren ist $\wt{\xi^{k}}$ injektiv und $\wt{\hat{\xi}^{k}}$ surjektiv.
\item
Sei $\mathcal{M}$ ein symmetrischer, lokalkonvexer
$\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul. Dann induzieren $\xi$
und $\hat{\xi}$ Kettenabbildungen zwischen
$(HC_{\mathrm{c,d}}(\SsV,\mathcal{M}),\delta_{\mathrm{c,d}})$
und $(KC^{\operatorname{\mathrm{cont}}}(\V,\mathcal{M}),\Delta)$. Des Weiteren ist
$\wt{\xi^{k}}$ injektiv und $\wt{\hat{\xi}^{k}}$ surjektiv.
\item
Gegeben ein vollständiger, symmetrischer,
hausdorffscher, lokalkonvexer\\ $\Hol-\Hol$-Bimodul $\mathcal{M}$, so
induzieren die Einschränkungs-Abbildung\-en einen Kettenisomorphismus:
\begin{equation*}
\big(HC_{\mathrm{c,d}}(\Hol,\mathcal{M}),\hat{\delta}_{\mathrm{c,d}}\big)\cong
\big(HC_{\mathrm{c,d}}(\SsV,\mathcal{M}),\delta_{\mathrm{c,d}}\big)
\end{equation*}und es gilt die Isomorphie:
\begin{equation*}
HH^{k}_{\mathrm{c,d}}(\Hol,\mathcal{M})\cong HH^{k}_{\mathrm{c,d}}(\SsV,M).
\end{equation*}
\end{enumerate}Hierbei bezeichnet $\hat{\xi}$ die mit Hilfe
von $F^{*}$ definierte Kettenabbildung, welche im
nicht-differentiellen Falle die auf Kohomologie-Niveau
zu $\widetilde{\xi^{k}}$ inversen Isomorphismen
$\wt{\hat{\xi}^{k}}$ induziert.
Eine analoge Aussage ist auch für die sogenannten differentiellen
$\SsV-\SsV$-Bimoduln $k$-ter Ordnung herleitbar,
deren Rechtsmodul-Multiplikation in der Form
\begin{equation*}
*_{R}=*_{L}+ D_{1}+…+D_{k}
\end{equation*}mit $\mathbb{K}$-bilinearen Abbildungen
$D_{l}:\SsV\times \mathcal{M}\longrightarrow \mathcal{M}$ derart geschrieben
werden kann, dass zusätzlich folgende Bedingungen erfüllt sind:
\begin{itemize}
\item[\textbf{a.)}]
Jedes $D_{l}$ ist $*_{L}$-linear im zweiten Argument.
\item[\textbf{b.)}]
Für $D_{l_{1},\dots,l_{p}}^{a_{1},\dots,a_{p}}=D_{l_{1}}^{a_{1}}\cp
\dots \cp D_{l_{p}}^{a_{p}}$ mit $D_{l}^{a}(m):= D_{l}(a,m)$
und $a\in\SsV$ ist
$D_{l_{1},\dots,l_{p}}^{a_{1},\dots,a_{p}}= 0$, falls $\sum_{i=1}^{p}l_{i}> k$.
\item[\textbf{c.)}]
F"ur alle $1\leq l\leq k$ gilt:
\begin{align*}
D_{l}(a*b,m)=&\:b*_{L}D_{l}(a,m)+ D_{1}(b,D_{l-1}(a,m))+D_{2}(b,D_{l-2}(a,m))+\dots\\ &\qquad+D_{l-2}(b,D_{2}(a,m))+ D_{l-1}(b,D_{1}(a,m))+a*_{L}D_{l}(b,m).
\end{align*}
\item[\textbf{d.)}]
F"ur jedes $m\in \mathcal{M}$ ist $D_{l}(\cdot,m)\in
\DiffOp{1}{l}$ sowie $D_{l}(\operatorname{\mathrm{v}},m)=0$, falls $l\geq 2$ und $\deg(\operatorname{\mathrm{v}})=1$.
\end{itemize}
\end{itemize}
Für derartige Bimoduln sind die differentiellen Unterkomplexe ebenfalls
wohldefiniert und es gilt folgender Satz:\\\\
\textbf{Satz}
\begin{enumerate}
\item
Sei $\mathcal{M}$ ein differentieller
$\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul. Dann besitzt
jede Kohomologieklasse $[\eta]\in HH^{k}(\SsV,\mathcal{M})$
mindestens einen differentiellen Repr"asentanten $\phi\in \DiffOpS{k}{s+1}$. Des Weiteren induzieren $\xi$ und $\hat{\xi}$ Kettenabbildungen zwischen $\left(HC_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M}),\delta_{\operatorname{\mathrm{diff}}}\right)$ und $(KC(\V,\mathcal{M}),\Delta)$.
Hierbei ist $\wt{\xi^{k}}$ injektiv und $\wt{\hat{\xi}^{k}}$ surjektiv.
\item
Sei $\mathcal{M}$ ein differentieller, lokalkonvexer
$\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul. Dann besitzt
jede Kohomologieklasse $[\eta]\in
HH_{\operatorname{\mathrm{cont}}}^{k}(\SsV,\mathcal{M})$ mindestens einen
differentiellen Repr"asentanten $\phi\in \DiffOpS{k}{s+1,\operatorname{\mathrm{cont}}}$. Des Weiteren induzieren $\xi$
und $\hat{\xi}$ wohldefinierte Kettenabbildungen zwischen
$(HC_{\mathrm{c,d}}(\SsV,\mathcal{M}),\delta_{\mathrm{c,d}})$
und $(KC^{\operatorname{\mathrm{cont}}}(\V,\mathcal{M}),\Delta)$. Hierbei ist
$\wt{\xi^{k}}$ injektiv und $\wt{\hat{\xi}^{k}}$ surjektiv.
\end{enumerate}
Ein Beispiel für einen solchen Bimodul ist hierbei
der Unterraum $\mathcal{M}$ aller
Differentialoperatoren
$m\in\mathrm{DiffOp}_{1}^{s}(\SsV,\SsV)$, die als eine endliche
Summe der Form
\begin{equation*}
m=\sum_{l=0}^{s}\sum_{|\alpha|=l}\delta_{\alpha_{1}}^{|\alpha_{1}|}…\delta_{\alpha_{k}}^{|\alpha_{k}|}
\end{equation*}mit Derivationen
$\delta_{\alpha_{i}}\in\mathrm{DiffOp}_{1}{1}(\SsV,\SsV)$
geschrieben werden können. Hierbei ist in der zweiten Summe
$\alpha \in \mathbb{N}^{k}$, wobei $k$ für jeden Summanden
variieren darf. Mit $\delta^{|\alpha_{i}|}_{\alpha_{i}}$ ist
die $|\alpha_{i}|$-fache Anwendung von
$\delta_{\alpha_{i}}$ gemeint, und wegen der
Derivationseigenschaft ist die Reihenfolge Verkettungen
unwichtig. Der Summand für $l=0$ soll dann lediglich aus einem
Element $m_{0}\in \SsV$ bestehen.\\\\
Obiger Satz und obiges Korollar stellen nun die unendlich-dimensionalen
Verallgemeinerungen der in \cite[Kapitel 5]{Weissarbeit} behandelten
Zusammenhänge dar, in welchen es sogar möglich ist, die Isomorphie
besagter Kohomologiegruppen für die Algebra $C^{\infty}(V)$ mit einer
konvexen Teilmengen $V\subseteq \mathbb{R}^{n}$ zu zeigen. Dies ist im
wesentlichen dem Fakt geschuldet, dass die Differentialoperatoren hier
besonders einfach mit Hilfe partieller Ableitungen
geschrieben werden können. Es gelten dann Kettenregeln der Form
$\pt_{y}f(tx+(1-t)y)=f'(tx +(1-t)y)(1-t)$, womit die Homotopie $s$
bzw. $s^{*}$
auch im
differentiellen Fall gewinnbringend eingesetzt werden kann,
siehe \cite[Prop 5.6.6]{Weissarbeit}. In unserem
unendlich-dimensionalen Rahmen bleibt jedoch zu hoffen, dass
eine andere Homotopie als $s^{*}$ existiert, die letztlich
die Surjektivität von $\wt{\xi^{k}}$ und die Injektivität von
$\wt{\hat{\xi}^{k}}$ zeigt.
\section*{Aufbau}
Diese Arbeit ist wie folgt gegliedert:
\begin{itemize}
\item
In Kapitel 1 definieren wir den zentralen Begriff der
Hochschild-Kohomologie ganz allgemein f"ur Hochschild-Koketten mit
Werten in Bimoduln und verwenden die im Anhang A bereitgestellten Grundlagen, um besagte Kohomologiegruppen exemplarisch
f"ur die Polynom-Algebra $\Pol(\mathbb{R}^{n})$ zu
berechnen. Hierbei bedienen wir uns den Methoden aus \cite[Kapitel
5]{Weissarbeit}. Durch Abstraktion der hier benutzten Homotopien und Kettenabbildungen sind wir schließlich in der Lage, die Hochschild-Kohomologien f"ur die symmetrische Algebra eines beliebigen $\mathbb{K}$-Vektorraumes $\V$ zu bestimmen.
\item
Im 2. Kapitel führen wir den Begriff des topologischen Komplexes und den der stetigen Hochschild-Kohomologie ein. Aufbauend auf Kapitel 1 und mit Hilfe der in Anhang B bereitgestellten funktional-analytischen Mittel werden die stetigen
Hochschild-Kohomologien der geeignet topologisierten symmetrischen
Algebra unter expliziter Verwendung der im vorherigen Kapitel
definierten Kettenabbildungen f"ur beliebige lokalkonvexe
Vektorr"aume $(\V,P)$ sowie lokalkonvexe Bimoduln berechnet.
Im letzten Abschnitt dieses Kapitels betrachten wir speziell
lokalkonvexe Vektorräume $(\V,\T_{P})$ mit hausdorffschen
Topologien. In diesem Fall ist die symmetrische Algebra ebenfalls
hausdorffsch topologisiert und wir dürfen deren Vervollst"andigung $\big(\Hol,\hat{\Pp}\big)$ betrachten.
Wir geben hier zun"achst eine detaillierte Beschreibung dieses Raumes
und mit Dichtheitsargumenten werden wir in der Lage sein, auch die
Hochschild-Kohomologie dieser Algebra für hausdorffsche und zudem
vollständige $\Hol-\Hol$-Bimoduln zu charakterisieren. Ein
essentielles Beispiel f"ur einen solchen Bimodul wird dann immer
$\Hol$ selbst darstellen.
\item
Das 3. Kapitel ist ganz der Verallgemeinerung der
Hochschild-Kostant-Rosenberg-Theoreme auf den
unendlich-dimensionalen Fall gewidmet, welche wir für symmetrische
$\SsV-\SsV$-Bimoduln sowohl im rein algebraischen als auch im
lokalkonvexen Fall und für symmetrische lokalkonvexe
$\Hol-\Hol$-Bimoduln formulieren werden.
\item
Im letzten Kapitel beschäftigen wir uns mit dem algebraischen
Konzept des Multidifferentialoperators und gehen der Frage nach,
für welche $\SsV-\SsV$- bzw. $\Hol-\Hol$-Bimoduln es möglich ist,
den Begriff des Hochschild-Komplexes zu definieren und dessen
Kohomologien mit Hilfe der uns zur Verfügung stehenden Mittel zu berechnen.
\end{itemize}
\clearpage
\thispagestyle{empty}
\cleardoublepage
\pagestyle{fancy}
\fancyhf{}
\fancyhead[OR]{\rightmark}
\fancyhead[EL]{\leftmark}
\fancyfoot[C]{\thepage}
\mainmatter
\chapter{Hochschild-Kohomologien}
\label{sec:Hochschkohoms}
In diesem Kapitel geben wir die zentrale Definition dieser Arbeit, die
der Hochschild-Kohomologie und berechnen diese exemplarisch für die
Polynomalgebra $\Pol(\mathbb{R}^{n})$, sowie als Verallgemeinerung für die symmetrische
Algebra über einem beliebigen $\mathbb{K}$-Vektorraum $\V$. Dabei wollen wir
hier und für den Rest dieser Arbeit $\mathbb{K}$ immer als $\mathbb{R}$ oder
$\mathbb{C}$ annehmen. Unter einer $\mathbb{K}$-Algebra $\mathcal{A}$ verstehen
wir im Folgenden einen $\mathbb{K}$-Vektorraum $\mathcal{A}$ mit
assoziativer, $\mathbb{K}$-bilinearer Algebramultiplikation.
Sprechen wir von einem $\mathcal{A}-\mathcal{A}$-Bimodul $\mathcal{M}$, so ist
stets ein $\mathbb{K}$-Vektorraum mit
$\mathcal{A}-\mathcal{A}$-Bimodulstruktur derart gemeint, dass sowohl
die Linksmodul-Multiplikation $*_{L}$ als auch die
Rechtsmodul-Multiplikation $*_{R}$ $\mathbb{K}$-bilineare Abbildungen sind. Ist
$\mathcal{A}$ unitär, so setzen wir immer die Verträglichkeit von
$\mathcal{M}$, also $1_{\mathcal{A}}*_{L} m=m$ und
$m*_{R}1_{\mathcal{A}}=m$ für alle $m\in \mathcal{M}$ voraus.
Der Verständlichkeit halber werden wir jedoch die an gegebener
Stelle wichtigen Eigenschaften nochmals explizit erwähnen.
Alle im Folgenden auftretenden Tensorprodukte sind als solche
über dem jeweils verwendeten Körper $\mathbb{K}$ zu verstehen.
\section{Einführung}
\label{sec:Einf}
Gegeben eine $\mathbb{K}$-Algebra $\mathcal{A}$ und ein
$\mathcal{A}-\mathcal{A}$-Bimodul $\mathcal{M}$, so betrachten wir für $k\in \mathbb{Z}$ die $\mathbb{K}$-Vektorräume
\begin{equation*}
HC^{k}(\mathcal{A},\mathcal{M}):=
\begin{cases} \{0\} & k<0\\
\mathcal{M} & k=0\\
\Hom_{\mathbb{K}}(\underbrace{\mathcal{A}\times…\times
\mathcal{A}}_{k-mal},\mathcal{M})& k\geq 1,
\end{cases}
\end{equation*} die $\mathbb{K}$-multilinearen Abbildungen von
$\mathcal{A}\times…\times \mathcal{A}$ nach $\mathcal{M}$.
Vermöge der gegebenen Links- und Rechtsmodulstruktur
definieren wir $\mathbb{K}$-lineare Abbildungen
\begin{equation*}
\delta^{k}\colon HC^{k}(\mathcal{A},\mathcal{M})\longrightarrow
HC^{k+1}(\mathcal{A},\mathcal{M})
\end{equation*}
durch
\begin{equation}
\label{eq:Hochschilddelta}
\begin{split}
(\delta^{k}\phi)(a_{1},…,a_{k+1})=a_{1}*_{L}\phi(a_{2},…,a_{k+1})&+\sum_{i=1}^{k}(-1)^{i}\phi(a_{1},…,a_{i}*a_{i+1},…,a_{k+1})\\
&+(-1)^{k+1}\phi(a_{1},…,a_{k})*_{R}a_{k+1}.
\end{split}
\end{equation}
Eine elementare Rechnung zeigt $\delta^{k+1}\cp\delta^{k}=0$, und wir erhalten einen
Kokettenkomplex $(HC^{\bullet}(\mathcal{A},\mathcal{M}),\delta)$ mit
$HC^{k}(\mathcal{A},\mathcal{M})$ als $\mathbb{K}$-Moduln und
$\delta^{k}$ als $\mathbb{K}$-Homomorphismen. Wir kommen nun zu der
für diese Arbeit zentralen Definition.\\
\begin{definition}[Hochschild-Kohomologie]
Wir definieren die $k$-te Hochschild-Kohomologie durch:
\begin{equation*}
HH^{k}(\mathcal{A},\mathcal{M}):=
\begin{cases}
\ker\left(\delta^{0}\right) & k=0\\
HH^{k}(\mathcal{A},\mathcal{M})=\ker\left(\delta^{k}\right)/\im\left(\delta^{k-1}\right)&
k\geq 1.
\end{cases}
\end{equation*}
Ungeachtet ihrer $\mathbb{K}$-Vektorraum Struktur wollen wir
diese im Folgenden entweder als Kohomologiegruppen oder einfach als
Kohomologien bezeichnen.
\end{definition}
Vermöge der universellen Eigenschaft des Tensorproduktes erhalten wir
einen Isomorphismus
\begin{equation*}
\ot_{k*}\colon
\Hom_{\mathbb{K}}(\mathcal{A}\times…\times
\mathcal{A},\mathcal{M})\longrightarrow \Hom_{\mathbb{K}}(\mathcal{A}\ot…\ot
\mathcal{A},\mathcal{M}),
\end{equation*}
dessen Inverses $\ot_{k}^{*}=\ot_{k*}^{-1}$ einfach der Pullback mit $\ot_{k}$ ist.
Die Tensorvariante von \eqref{eq:Hochschilddelta} ist dann gegeben
durch lineare Fortsetzung von
\begin{equation*}
\label{eq:THochschilddelta}
\begin{split}
(\delta^{k}\phi)(a_{1}\ot…\ot a_{k})=&\:a_{1}*_{L}\phi(a_{2}\ot…\ot
a_{k+1})\\ &+\sum_{i=1}^{k}(-1)^{i}\phi(a_{1}\ot…\ot
a_{i}a_{i+1}\ot…\ot a_{k+1})\\
&+(-1)^{k+1}\phi(a_{1}\ot …\ot a_{k})*_{R}a_{k+1}
\end{split}
\end{equation*}vermöge Korollar \ref{kor:WohldefTensorprodabbildungen}
und wir erhalten
\begin{equation}
\label{eq:TensorglKetteniso}
\delta^{k}_{\ot}\cp \ot_{k*}= \ot_{k+1*}\cp \delta^{k}_{\times},
\end{equation}
womit $\ot_{*}$ ein Kettenisomorphismus zwischen diesen
beiden Kokettenkomplexen ist. Dies bedeutet insbesondere die
Isomorphie derer Kohomologien (vgl. Lemma \ref{lemma:kettenabzu}~\textit{ii.)}), und wir dürfen uns im Folgenden
darauf beschränken, die einfacher handhabbare Tensorvariante des
Hochschild-Komplexes zu betrachten.\\\\
Wir wollen nun zunächst einsehen, dass die Kohomologiegruppen $HH^{k}(\mathcal{A},\mathcal{M})$ durch
Anwendung eines $\mathrm{Ext}$-Funktors auf die Algebra $\mathcal{A}$ erhalten
werden können, dass also
\begin{equation*}
HH^{k}(\mathcal{A},\mathcal{M})\cong \mathrm{Ext}_{R}^{k}(\cdot,\mathcal{M})(\mathcal{A})=H^{k}(\hom_{R}(\cdot,\mathcal{M})C)=H^{k}(\Hom_{R}(C,\mathcal{M}),d^{*})
\end{equation*}gilt. Dabei bezeichnet $C$ eine projektive Auflösung
$(C,d,\epsilon)$ von $\mathcal{A}$ und $R$ einen geschickt zu
wählenden Ring. Mit Beispiel \ref{bsp:ExtBeisp} folgt dann bereits
$HH^{0}(\mathcal{A},\mathcal{M})\cong \Hom_{R}(\mathcal{A},\mathcal{M})$.
\begin{lemma}
\label{lemma:AewirdzuunitRing}
Gegeben eine assoziative $\mathbb{K}$-Algebra $(\mathcal{A},*)$.
\begin{enumerate}
\item
Die Menge
$\mathcal{A}^{e}=\mathcal{A}\ot \mathcal{A}$ wird vermöge
der Multiplikation
\begin{equation}
\label{eq:AeRingmultdef}
(a\ot b) *_{e} (\widetilde{a}\ot
\widetilde{b}):=(a*\widetilde{a})\ot
(b*^{opp}\widetilde{b})=(a*\widetilde{a})\ot
(\widetilde{b}*b)
\end{equation} zu einer assoziativen $\mathbb{K}$-Algebra. Ist $\mathcal{A}$
unitär, so auch $\mathcal{A}^{e}$ vermöge $1_{\mathcal{A}^{e}}=1_{\mathcal{A}}\ot 1_{\mathcal{A}}$.
\item
Jeder $\mathcal{A}-\mathcal{A}$-Bimodul $\mathcal{M}$ wird durch
\begin{equation*}
a\ot b*_{e} m=a*_{L}( m*_{R} b)=(a*_{L}m)*_{R}b\qquad\quad \forall\:a\ot b\in \mathcal{A}^{e},\: m\in \mathcal{M}
\end{equation*} zu einem $\mathcal{A}^{e}$-Linksmodul.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Die Assoziativität folgt unmittelbar aus der Assoziativität von
$\mathcal{A}$. Der Rest ist ebenfalls klar. Für
die Wohldefiniertheit von $*_{e}$ definieren
wir die Abbildung $*'_{e}$ vermöge
Korollar \ref{kor:WohldefTensorprodabbildungen} durch lineare
Fortsetzung von
\begin{align*}
*'_{e}\colon \mathcal{A}\ot \mathcal{A} \ot \mathcal{A} \ot
\mathcal{A}&\longrightarrow \mathcal{A}\ot \mathcal{A}\\
a\ot b \ot \tilde{a}\ot \tilde{b} & \longmapsto
a\tilde{a}\ot b\tilde{b}
\end{align*}und setzen
$*_{e}= *'_{e}\cp \cong \cp\ot_{2}$ mit $\ot_{2}\colon \mathcal{A}^{e}\times
\Ae\longrightarrow \Ae\ot \Ae$ und $\cong$ der Isomorphismus $\Ae\ot \Ae\cong\mathcal{A}\ot
\mathcal{A}\ot \mathcal{A}\ot \mathcal{A}$.
\item
Für die Wohldefiniertheit beachte man, dass für festes
$m\in \mathcal{M}$ die Abbildung $*_{m}\colon a\ot b\rightarrow
amb$ die Bedingungen von Korollar \ref{kor:WohldefTensorprodabbildungen} erfüllt, mithin linear auf ganz
$\mathcal{A}^{e}$ fortsetzt. Mit \textit{i.)}
folgt
\begin{align*}
[(a\ot b) *_{e} (\widetilde{a}\ot\widetilde{b})]*_{e}m=a\widetilde{a}m\widetilde{b}b=(a\ot b)*_{e} (\widetilde{a}m\widetilde{b})=(a\ot b)*_{e}[(\widetilde{a}\ot\widetilde{b})*_{e}m],
\end{align*} was die Behauptung zeigt.
\end{enumerate}
\end{beweis}
\end{lemma}
\begin{definition}[Bar-Komplex]
Sei $\mathcal{A}$ eine assoziative $\mathbb{K}$-Algebra. Wir betrachten die
$\mathbb{K}$-Vektorräume
\begin{align*}
\C_{k}=\mathcal{A}\ot \underbrace{\mathcal{A}\ot …
\ot \mathcal{A}}_{k-mal} \ot \mathcal{A}
\end{align*} $\qquad\qquad\qquad\qquad \C_{0}=\mathcal{A}\ot \mathcal{A},\quad\quad
\C_{1}=\mathcal{A}\ot\mathcal{A}\ot \mathcal{A},\quad\quad
\C_{2}=\mathcal{A}\ot\mathcal{A}\ot \mathcal{A}\ot \mathcal{A}$,
\\\\ die wie in Lemma \ref{lemma:AewirdzuunitRing} durch
\begin{equation}
\label{eq:ModMultiplBarkom}
a\ot b *_{e}\left(x_{0}\ot x_{1}\ot … \ot x_{k}\ot
x_{k+1}\right)=(a x_{0})\ot x_{1}\ot … \ot x_{k}\ot\:
(x_{k+1}b)
\end{equation}zu $\mathcal{A}^{e}$-Linksmoduln werden.
Des Weiteren seien $\mathcal{A}^{e}$-Homomorphismen durch lineare
Fortsetzung von
\begin{align*}
d_{k}\colon\C_{k}&\longrightarrow \C_{k-1}\\
(x_{0}\ot … \ot x_{k+1})&\longmapsto
\sum_{j=0}^{k}(-1)^{j}x_{0}\ot…\ot
x_{j}x_{j+1}\ot…\ot x_{k+1}
\end{align*} für $k\geq 1$ definiert.
Dann gilt $d_{k}\cp d_{k+1}=0$ und wir erhalten einen wohldefinierten
Kettenkomplex $(\C,d)$, den wir im Folgenden als den zu $\mathcal{A}$
gehörigen Bar-Komplex bezeichnen wollen.
\end{definition}
Für unitäres $\mathcal{A}$ ist $\mathcal{A}^{e}$ unitär und
Lemma \ref{lemma:unitarereRingModulnhabenfreieAufloesung} besagt, dass es dann
rein abstrakt eine projektive (sogar freie) Auflösung von
$\mathcal{A}$ als $\mathcal{A}^{e}$-Modul geben
muss. Ein essentielles Beispiel liefert folgendes Lemma.
\begin{lemma}[Bar-Auflösung für unitäre $\mathbb{K}$-Algebren]
\label{lemma:unitAlhabeBarAufloesProj}
Gegeben eine unitäre, assoziative $\mathbb{K}$-Algebra $\mathcal{A}$, so wird
der Bar-Komplex vermöge der Abbildung
\begin{align*}
\epsilon\colon\C_{0}&\longrightarrow \mathcal{A}\\
a\ot b&\longmapsto ab
\end{align*} zu einer projektiven Auflösung $(\C,d,\epsilon)$ von
$\mathcal{A}$.
\begin{beweis}
Zunächst ist $\epsilon$ ein wohldefinierter $\mathcal{A}^{e}$-Homomorphismus
\begin{equation*}
\epsilon(a\ot b *_{e}x_{0}\ot
x_{1})\glna{\eqref{eq:ModMultiplBarkom}}\epsilon\:(ax_{0}\ot
x_{1}b)=ax_{0}x_{1}b\glna{\eqref{eq:ModMultiplBarkom}}a\ot b*_{e}x_{0}x_{1}=a\ot
b*_{e}\epsilon(x_{0}\ot x_{1}).
\end{equation*} Des Weiteren ist $\epsilon$ surjektiv, da
$\epsilon\:(1\ot a)=a\in \mathcal{A}$, und es gilt zudem
\begin{equation*}
(\epsilon\circ d_{1})(x_{0}\ot x_{1}\ot
x_{2})=\epsilon(x_{0}x_{1}\ot x_{2}-x_{0}\ot x_{1}x_{2})=0.
\end{equation*} Für die Projektivität reicht es, die
$\mathcal{A}^{e}$-Freiheit jedes $\C_{k}$ zu zeigen. Dafür
beachte man, dass
$\C_{0}\cong \mathcal{A}^{e}$, $\C_{1}\cong
\mathcal{A}^{e}\ot \mathcal{A}$, $…$, $\C_{k}\cong
\mathcal{A}^{e}\bigotimes^{k-2} \mathcal{A}$, womit $\C_{k}\cong \mathcal{A}^{e}\ot V_{k}$
für $\mathbb{K}$-Vektorräume $V_{k}$. Dann liefert
die $\mathcal{A}^{e}$-lineare Fortsetzung der Abbildung
\begin{align*}
\tau_{k}\colon\mathcal{A}^{e}\ot V_{k}&\longrightarrow
(\mathcal{A}^{e})^{\dim(V_{k})}\\
a^{e}\ot\vec{e}_{\alpha}&\longmapsto \oplus_{\alpha} a^{e}\qquad\qquad \forall\:a^{e}\in \mathcal{A}^{e}
\end{align*} mit $\{\vec{e}_{\alpha}\}_{\alpha\in I}$ eine Basis von
$V_{k}$, einen Isomorphismus
$\C_{k}\longrightarrow(\mathcal{A}^{e})^{\dim(V_{k})}$.\\\\
Es bleibt die Exaktheit
des Komplexes nachzuweisen. Hierfür betrachten wir die
Kettenabbildungen
\begin{align*}
h_{k}\colon\C_{k}&\longrightarrow \C_{k+1}\\
x_{0}\ot…\ot x_{k+1}&\longmapsto 1\ot
x_{0}\ot…\ot x_{k+1}\qquad k\geq -1
\end{align*}für welche wir erhalten, dass
\begin{equation}
\label{eq:Homotbar}
\begin{split}
\epsilon\circ h_{-1}&=\id_{A},\\
d_{1}\cp h_{0}+h_{-1}\cp\epsilon &=\id_{\C_{0}}, \text{ sowie}\\
d_{k+1}\cp h_{k}+h_{k-1}\cp d_{k}&=\id_{\C_{k}} \text{ für
}k\geq 1.
\end{split}
\end{equation} Hiermit folgt für $\alpha\in \ker(d_{k})$ und
$k\geq 1$:
\begin{equation*}
\alpha=(d_{k+1}\cp h_{k})(\alpha)+(h_{k-1}\cp
d_{k})(\alpha)=(d_{k+1}\cp h_{k})(\alpha)\in \im(d_{k+1}),
\end{equation*} analog für $\alpha\in \ker(\epsilon)$. Dies
zeigt die Exaktheit.
\end{beweis}
\end{lemma}
\begin{proposition}
\label{prop:barauffuerunitalgebraIsomozuHochschildkohomo}
Gegeben eine unitäre, assoziative $\mathbb{K}$-Algebra
$\mathcal{A}$ und ein $\mathcal{A}-\mathcal{A}$-Bimodul
$\mathcal{M}$. Bezeichne
$(\C,d,\epsilon)$ die Bar-Auflösung über $\mathcal{A}$. Dann gilt:
\begin{equation*}
HH^{k}(\mathcal{A},\mathcal{M})\cong \mathrm{Ext}^{k}_{\mathcal{A}^{e}}(\cdot,
\mathcal{M})(\mathcal{A})=H^{k}(\mathrm{hom}_{\mathcal{A}^{e}}(\cdot,\mathcal{M})\C)=H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}),d^{*}).
\end{equation*}
\begin{beweis}
Sowohl $(HC^{\bullet}(\mathcal{A},\mathcal{M}),\delta)$, als auch
der durch Anwendung von $\hom_{\mathcal{A}^{e}}$ auf $(\C,d)$ ge\-wonnene
Kokettenkomplex $(\C^{*},d^{*})$ mit Kokettengliedern
$\C^{k}=\C_{k}^{*}=\Hom_{\mathcal{A}^{e}}(\C_{k},\mathcal{M})$ und Ko-Differentialen
$d^{k}=d_{k+1}^{*}\colon\C^{k}\longrightarrow \C^{k+1}$, sind Komplexe
von $\mathbb{K}$-Vektorräumen. Mit den Abbildungen
\begin{align*}
\Xi^{k}\colon\Hom_{\mathcal{A}^{e}}(\C_{k},\mathcal{M})&\longrightarrow
HC^{k}(\mathcal{A},\mathcal{M})\\
\psi &\longmapsto \left(\widetilde{\psi}\colon(x_{1}\ot…\ot x_{k})\mapsto
\psi(1\ot x_{1}\ot…\ot
x_{k}\ot1)\right),
\end{align*} erhalten wir aus der Verträglichkeit von
$\mathcal{M}$ sowie der Bilinearität der Modul-Multi\-pli\-ka\-tio\-nen:
\begin{align*}
\Xi^{k}(\psi)(\lambda\: x_{1}\ot…\ot
x_{k})&=\psi\:(1\ot x_{1}\ot…\ot \lambda x_{i}\ot…\ot
x_{k}\ot1)\\ &=\psi\:(\lambda 1\ot x_{1}\ot…\ot
x_{k}\ot1)\\ & =\lambda 1\ot 1 *_{e}\psi\:(1\ot x_{1}\ot…\ot
x_{k}\ot1)\\ &=\lambda\: \psi\:(1\ot x_{1}\ot…\ot
x_{k}\ot1)\\ &=\lambda\: \Xi^{k}(\psi)(x_{1}\ot…\ot
x_{k}).
\end{align*} Damit bilden die $\Xi^{k}$ in der Tat in die behauptete
Menge ab, und da
\begin{equation*}
\Xi^{k}(\lambda \psi+\phi)=\lambda\:
\Xi^{k}(\psi)+ \Xi(\phi),
\end{equation*}sind diese zudem
$\mathbb{K}$-Homomorphismen.
Nun folgt die Injektivität obiger
Abbildung unmittelbar aus der $\mathcal{A}^{e}$-Linearität der
Urbilder. Für die Surjektivität betrachten wir ein
$\widetilde{\psi}\in HC^{k}(\mathcal{A},\mathcal{M})$ und
definieren $\Xi^{k}_{-1}\colon HC^{k}(\mathcal{A},\mathcal{M})\longrightarrow
\Hom_{\mathcal{A}^{e}}(\C_{k},\mathcal{M})$ durch
\begin{align*}
\Xi^{k}_{-1}(\psi)(x_{0}\ot
x_{1}\ot…\ot x_{k}\ot
x_{k+1})=x_{0}\ot x_{k+1}*_{e}\psi(x_{1}\ot…\ot
x_{k}),
\end{align*}
womit $\Xi^{k} \cp\Xi^{k}_{-1} =\id_{HC^{k}(\mathcal{A},\mathcal{M})}$ , also die
$\Xi^{k}$ Isomorphismen sind.
Nun folgt
\begin{equation}
\label{eq:isobarHsch}
\Xi^{k+1}d^{*}_{k+1}=\delta^{k}\:\Xi^{k},
\end{equation}da
{\allowdisplaybreaks\small
\begin{align*}
\left(\Xi^{k+1}\cp d^{*}_{k+1}\right)&(\psi)(x_{1}\ot…\ot
x_{k+1})=(d^{*}_{k+1}\psi)(1\ot x_{1}\ot…\ot x_{k+1}\ot 1)
\\ =&\: \psi(x_{1}\ot…\ot x_{k+1}\ot 1)
+\sum_{j=1}^{k}(-1)^{j}
\psi(1\ot x_{1}\ot…\ot x_{j}x_{j+1}\ot…\ot x_{k+1}\ot 1)
\\ &\qquad\qquad\qquad\qquad\qquad+ (-1)^{k+1}\psi(1\ot…\ot x_{k+1})
\\=&\: x_{1}\psi(1\ot x_{2}\ot…\ot x_{k+1}\ot 1)
+\sum_{j=1}^{k}(-1)^{j}
\psi(1\ot x_{1}\ot…\ot x_{j}x_{j+1}\ot…\ot x_{k+1}\ot 1)
\\ &\qquad\qquad\qquad\qquad\qquad\qquad\:\:\:+ (-1)^{k+1}\psi(1\ot x_{1}\ot…\ot x_{k}\ot 1)\:x_{k+1}
\\=&\: x_{1}\left(\Xi^{k}\cp\psi\right)(x_{2}\ot…\ot x_{k+1})
+ \sum_{j=1}^{k}(-1)^{k}\left(\Xi^{k}\cp\psi\right)(x_{1}\ot…\ot
x_{j}x_{j+1}\ot…\ot x_{k+1})
\\ &\qquad\qquad\qquad\qquad\qquad\qquad\:\:\:+ (-1)^{k+1}\left(\Xi^{k}\cp \psi\right)(x_{1}\ot…\ot x_{k})\:x_{k+1}
\\ =& \left(\delta^{k}\cp\:\Xi^{k}\right)(\psi)(x_{1}\ot…\ot x_{k+1}),
\end{align*}}und mit der $\mathbb{K}$-Linearität der $ \Xi^{k}$
zeigt Lemma \ref{lemma:kettenabzu}~\textit{iii)} die
Isomorphismen-Eigenschaft der $\wt{\Xi^{k}}$.
\end{beweis}
\end{proposition}
\section{Die Hochschild-Kohomologie der Algebra
$\Poly$}
\label{subsec:HschKPol}
Wir wollen als erstes einfaches Beispiel die
Hochschild-Kohomologie der $\mathbb{R}$-Algebra $\mathcal{A}=\Poly$, der Polynome
auf $\mathbb{R}^{n}$, berechnen. Diese ist sicher unitär und assoziativ und wir haben gemäß
Lemma \ref{lemma:unitAlhabeBarAufloesProj} und
Proposition \ref{prop:barauffuerunitalgebraIsomozuHochschildkohomo} bereits eine
projektive Auflösung, deren Kohomologiegruppen zu den
gesuchten Hochschild-Kohomologien isomorph sind.
Als nächstes wollen wir uns eine
weitere projektive Auflösung $(C',d',\epsilon')$ von
$\mathcal{A}$ verschaffen und wissen bereits, dass dann:
\begin{equation*}
H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}))\cong \mathrm{Ext}^{k}_{\mathcal{A}^{e}}(\cdot,
\mathcal{M})(\mathcal{A})\cong
H^{k}(\Hom_{\mathcal{A}^{e}}(C',\mathcal{M})).
\end{equation*}
\begin{definition}[Koszul-Komplex]
Gegeben die Algebra $\mathcal{A}=\Poly$, so definieren wir den Koszul-Komplex
$(\K,\partial)$ durch die $\mathcal{A}^{e}$-Moduln
\begin{equation*}
\K_{0}=\mathcal{A}^{e}\qquad\text{ sowie }\qquad
\K_{k}=\mathcal{A}^{e}\ot
\Lambda^{k}(\mathbb{R}^{n*})\quad\forall\:k\geq 1
\end{equation*}mit der offensichtlichen
$\mathcal{A}^{e}$-Multiplikation im ersten Faktor. Dies bedeutet insbesondere
$\K_{k}=0$ falls $k>n$. Für $0< k\leq n$ definieren wir die
$\mathcal{A}^{e}$-Homomorphismen:
\begin{align*}
\partial_{k}\colon\K_{k}&\longrightarrow \K_{k-1}\\
\omega&\longmapsto \left[(v,w)(x_{1},…,x_{k-1})\mapsto \omega(v,w)((v-w),x_{1},…,x_{k-1})\right].
\end{align*}Es folgt unmittelbar $\pt_{k}\cp
\pt_{k+1}=0$, und man beachte zudem, dass die $\pt_{k}$ mit Hilfe
der Einsetzabbildung $i_{a}(v,\omega)(x_{2},…,x_{k})=\omega(v,x_{2},…,x_{k})$:
\begin{align*}
&i_{a}\colon \mathbb{R}^{n}\times \Lambda^{k}(\mathbb{R}^{n*})\longrightarrow
\Lambda^{k-1}(\mathbb{R}^{n*})\\
&(v, \omega^{1}\wedge…\wedge \omega^{k})\longmapsto
\sum_{l=1}^{k}(-1)^{l-1}\omega^{l}(v)\:\omega^{1}\wedge…\blacktriangle^{l}…\wedge \omega^{k}
\end{align*}
auch als
$\pt_{k}=\displaystyle\sum_{j=1}^{n}\xi^{j}i_{a}(\vec{e}_{j},\cdot)$
mit $\mathcal{A}^{e}\ni\xi^{j}=x^{j}\ot 1-1\ot x^{j}$ geschrieben werden können.
\end{definition}
\begin{lemma}[Koszul-Auflösung]
Sei $\mathcal{A}=\Poly$, so wird der Koszul-Komplex
vermöge der Abbildung
\begin{align*}
\epsilon\colon\K_{0}&\longrightarrow \mathcal{A}\\
a\ot b&\longmapsto ab
\end{align*} zu einer projektiven und sogar freien Auflösung $(\K,\partial,\epsilon)$
von $\Poly$.
\begin{beweis}
Wir hatten bereits gesehen, dass $\epsilon$ ein surjektiver
$\mathcal{A}^{e}$-Homomorphismus ist. Die
Freiheit der $\K_{k}$ folgt ebenso wie für die
Bar-Auflösung, da die $\Lambda^{k}(\mathbb{R}^{n*})$
ebenfalls Vektorräume $V$ mit Basen sind.
Für die Exaktheit definieren wir die Abbildungen
\begin{align*}
h_{-1}\colon\mathcal{A}&\longrightarrow
\K_{0}\\
p&\longmapsto [(v,w)\mapsto p(w)]
\end{align*} und $h_{k}\colon\K_{k}\longrightarrow
\K_{k+1}$ für $k\geq 0$, durch:
\begin{equation}
\label{eq:exakthAbbHvonPol}
\begin{split}
h_{k}(\omega)(v,w)&=\sum_{j=1}^{n}e^{j}\wedge\int_{0}^{1}dt\:
t^{k}\frac{\partial\omega}{\partial v^{j}}(tv+(1-t)w,w)\\ &=\frac{1}{k!}
\sum_{i_{1},…,i_{k},j=1}^{n}\int_{0}^{1}dt\:
t^{k}\frac{\partial\omega_{i_{1},…,i_{k}}}{\partial
v^{j}}(tv+(1-t)w,w)\:e^{j}\wedge e^{i_{1}}\wedge
…\wedge e^{i_{k}}\nonumber,
\end{split}
\end{equation}wobei
\begin{equation*}
\omega=\frac{1}{k!}
\sum_{i_{1},…,i_{k}}^{n}\omega_{i_{1},…,i_{k}}\ot
e^{i_{1}}\wedge…\wedge e^{i_{k}}\quad\text{ und }\quad
\omega_{i_{1},…,i_{k}}\in \mathcal{A}^{e}.
\end{equation*}Zunächst überzeugt man sich, dass besagte
Abbildungen in der Tat nach
\begin{equation*}
\mathrm{Pol}(\mathbb{R}^{n})\ot\mathrm{Pol}(\mathbb{R}^{n})\ot
\Lambda^{k}(\mathbb{R}^{n*}) =
\mathrm{Pol}(\mathbb{R}^{n}\times\mathbb{R}^{n})\ot \Lambda^{k}(\mathbb{R}^{n*})
\end{equation*}
abbilden, denn es ist ja jedes
$\frac{\partial\omega_{i_{1},…,i_{k}}}{\partial v^{j}}$ als Ableitung eines
Polynoms nach den ersten Argumenten wieder ein
Polynom auf $\mathbb{R}^{n}\times \mathbb{R}^{n}$.
Ebenso haben wir $p(t\vec{x}+(1-t)\vec{y})\in
\mathrm{Pol}(\mathbb{R}\times \mathbb{R}^{n}\times \mathbb{R}^{n})$ für $p\in \mathrm{Pol}(\mathbb{R}^{n}\times \mathbb{R}^{n})$,
und die Integration ist nichts weiter, als die
$\mathcal{\mathbb{R}}$-lineare Fortsetzung der Abbildung
\begin{equation*}
\int_{0}^{1}dt\:t^{k}\colon t^{l}x^{n}\longmapsto
\frac{1}{l+k+1}x^{n}.
\end{equation*}
Behändiges Rechnen unter Verwendung der Derivationseigenschaft
\begin{equation*}
i_{a}(v)(\phi\wedge \psi )=i_{a}(v)(\phi)\wedge \psi +(-1)^{deg(\phi)}\phi\wedge i_{a}(v)(\psi)
\end{equation*}
zeigt:
\begin{align*}
\epsilon\circ h_{-1}&=\id_{\mathcal{A}}, \\
h_{-1}\circ\epsilon+\partial_{1}\circ h_{0}&=\id_{\K_{0}}\quad\quad\text{und}\\
h_{k-1}\circ \partial_{k}+\partial_{k+1}\circ h_{k}&=\id_{\K_{k}}\quad\quad k\geq1,
\end{align*}
mithin die Exaktheit von
$(\K,\partial,\epsilon)$, vgl. \cite[Kapitel 5]{Weissarbeit}.
\end{beweis}
\end{lemma}
Mit obigem Lemma erhalten wir nun umgehend folgenden Satz.
\begin{satz}
\label{satz:PolsatzHochsch}
Sei $\mathcal{A}=\Poly$ und $\mathcal{M}$ ein
$\mathcal{A}-\mathcal{A}$-Bimodul, dann gilt:
\begin{equation}
\label{eq:HochschPol11}
HH^{k}(\mathcal{A},\mathcal{M})\cong
H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}),d^{*})\cong
H^{k}(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{M}),\pt^{*}).
\end{equation}
Ist $\mathcal{M}$ zudem symmetrisch, so ist:
\begin{equation}
\label{eq:HochschPol22}
HH^{k}(\mathcal{A},\mathcal{M})\cong \Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})=\mathcal{M}\ot\Lambda^{k}(\mathbb{R}^{n}).
\end{equation}
\begin{beweis}
\eqref{eq:HochschPol11} ist wegen$H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}))\cong \mathrm{Ext}^{k}_{\mathcal{A}^{e}}(\cdot,
\mathcal{M})(\mathcal{A})\cong
H^{k}(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{M}))$ klar und für \eqref{eq:HochschPol22} betrachten wir den Komplex
$(\K^{*},\partial^{*})$,
der durch Anwendung des
$\mathrm{hom}_{\mathcal{A}^{e}}(\cdot,\mathcal{M})$-Funktors auf $(\K,\partial)$
gewonnen wird.
Seien weiter $\phi\in
\K^{*}_{k}=\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})$ und
$\omega=\sum_{i_{1},…,i_{k+1}}\omega_{i_{1},…,i_{k+1}}\ot e^{i_{1}}\wedge…\wedge
e^{i_{k+1}}\in \K_{k+1}$.\\ Dann folgt für
$\pt^{*}_{k+1}\colon\K^{*}_{k}\longrightarrow \K^{*}_{k+1}$:
\begin{align*}
(\partial^{*}_{k+1}\phi)(\omega)\glna{\eqref{eq:homfktrMor}}&(\phi\circ
\partial_{k+1})(\omega)\\=&\:\phi\left(\partial_{k+1}\left(\sum_{i_{1},…,i_{k+1}}\omega_{i_{1},…,i_{k}}\ot
u^{i_{1}}\wedge…\wedge
u^{i_{k+1}}\right)\right)\\ &=
\sum_{i_{1},…,i_{k+1}}\phi\left(\sum_{j=1}^{n}\xi^{j}*_{e}\omega_{i_{1},…,i_{k+1}}\ot
i_{a}\left(\vec{e}_{j},u^{i_{1}}\wedge…\wedge
u^{i_{k+1}}\right)\right)\\
&=\sum_{i_{1},…,i_{k+1}}\sum_{j=1}^{n}\xi^{j}*_{e}\phi\left(\omega_{i_{1},…,i_{k+1}}\ot
i_{a}\left(\vec{e}_{j},u^{i_{1}}\wedge…\wedge
u^{i_{k+1}}\right)\right) \\ &= \sum_{i_{1},…,i_{k+1}}\sum_{j=1}^{n}\:(x^{j}\ot
1-1\ot x^{j})*_{e}\phi\left(\omega_{i_{1},…,i_{k+1}}\:i_{a}\left(\vec{e}_{j},u^{i_{1}}\wedge…\wedge
u^{i_{k+1}}\right)\right)\\ &=0.
\end{align*}Dabei gilt die letzte Gleichheit wegen der
Symmetrie von $\mathcal{M}$.
Dies zeigt $\ker(\partial^{*}_{k+1})= \K^{*}_{k}$ und $\im(\partial^{*}_{k})=0$, mithin
$H^{k}(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{A}))=\ker(\pt_{k+1}^{*})/\im(\pt_{k}^{*})=\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{A})$.\\\\
Für die letzte Gleichheit in \eqref{eq:HochschPol22} erinnern wir, dass
$\Lambda^{k}(\mathbb{R}^{n*})^{*}=\Lambda^{k}(\mathbb{R}^{n})$
und erhalten für $\phi \in \Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})$:
{\allowdisplaybreaks\small
\begin{align*}
\phi(a^{e}\ot \omega)=\:&a^{e}*_{e}\phi\left(1^{e} \ot
\sum_{j_{1},…,j_{k}=1}^{n}\omega_{j_{1},…,j_{k}}e^{j_{1}}\wedge…\wedge
e^{j_{k}}\right)
\\=&\: a^{e}*_{e}
\sum^{n}_{j_{1},…,j_{k}}\omega_{j_{1},…,\omega_{k}}\ot
1*_{e}\phi\:(1^{e}\ot e^{j_{1},…,j_{k}})
\\=&\:
a^{e}*_{e}\sum_{j_{1},…,j_{k}=1}^{n}\omega_{j_{1},…,j_{k}}\:\phi^{j_{1},…,j_{k}}
\\=&\: a^{e}*_{e} \left(\sum_{j_{1},…,j_{k}=1}^{n} \phi^{j_{1},…,j_{k}}\ot e_{j_{1}}\wedge…\wedge
e_{j_{k}}\right)(1^{e}\ot\omega)
\\=&\: \left(\sum_{j_{1},…,j_{k}=1}^{n} \phi^{j_{1},…,j_{k}}\ot e_{j_{1}}\wedge…\wedge
e_{j_{k}}\right)(a^{e}\ot\omega),
\end{align*}} und mit der Endlichkeit der Summe in der Tat
{\small\begin{equation*}
\left(\sum_{j_{1},…,j_{k}=1}^{n} \phi^{j_{1},…,j_{k}}\ot e_{j_{1}}\wedge…\wedge
e_{j_{k}}\right)\in \mathcal{M}\ot \Lambda^{k}(\mathbb{R}^{n}).
\end{equation*}}Dabei haben wir im zweiten Schritt wieder
$1*_{L}m=m=m*_{R}1$ für alle $m\in \mathcal{M}$ und die Bilinearität der Modul-Multiplikationen benutzt. Die letzten beiden Schritte folgen mit
der Konvention
\begin{equation}
\label{eq:Multi}
m\ot \lambda\:(a^{e}\ot
\omega):=a^{e}*_{e}m\cdot
\omega(\lambda)\qquad\text{ mit }\qquad \lambda\in \Lambda^{k}(\mathbb{R}^{n}).
\end{equation}
Umgekehrt ist klar, dass jedes Element aus
$\mathcal{M}\ot \Lambda^{k}(\mathbb{R}^{n})$ vermöge \eqref{eq:Multi} ein Element in
$\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})$ definiert.
\end{beweis}
\end{satz}
Für $M=\Pol(\mathbb{R}^{n})$ wurde dieser Satz ursprünglich von
Hochschild, Kostant und Rosenberg bewiesen, siehe
\cite{hochschild.kostant.rosenberg:1962a}. Eine Behandlung des Falles
$\mathcal{M}=\mathcal{A}=C^{\infty}(M)$ findet man in
\cite{cahen.gutt.dewilde:1980a}.
\begin{bemerkung}
Für einen expliziten Isomorphismus benötigen wir zunächst eine zu $\mu=\id_{\mathcal{A}^{e}}$ gehörige Kettenabbildung
\begin{equation}
\label{eq:Gpol}
G\colon(\C,d)\rightarrow (\K,\partial)
\end{equation} oder
\begin{equation}
\label{eq:FPol}
F\colon(\K,\partial)\rightarrow (\C,d).
\end{equation}
Nach dem Beweis von
Lemma \ref{lemma:GruppenKOhomsausprojaufloesundFunktoren}~\textit{ii.)}
erhalten wir durch Anwenden des
$\mathrm{hom}_{\mathcal{A}^{e}}(\cdot,\mathcal{M})$-Funktors
Abbildungen $F_{k}^{*}$ und $G_{k}^{*}$, die Isomorphismen
\begin{align*}
\widetilde{G_{k}^{*}}\colon&H^{k}(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{M}))\longrightarrow
H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}))\\
\widetilde{F_{k}^{*}}\colon&H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}))\longrightarrow
H^{k}(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{M}))
\end{align*}auf Kohomologie-Niveau induzieren. Nach selbigem Beweis
sind $\widetilde{F_{k}^{*}}$ und $\widetilde{G_{k}^{*}}$ sogar zueinander invers.
Die Verkettung $\wt{\ot_{k}^{*}} \cp\wt{\Xi^{k}}\cp\wt{G_{k}^{*}}$
ist dann der gewünschten Isomorphismus nach
$HH^{k}(\mathcal{A},\mathcal{M})$.
Explizite Kettenabbildungen sind beispielsweise gegeben durch \cite{bordemann.et.al:2003a:pre}:
\begin{align*}
G_{k}&\colon \bigotimes^{k+2}\Poly \longrightarrow
\mathrm{Pol}(\mathbb{R}^{n} \times \mathbb{R}^{n})\ot
\Lambda^{k}(\mathbb{R}^{n*})\\ (G_{k}p)(v,w)&=
\sum_{i_{1},…,i_{k}}^{n}e^{i_{1}}\wedge…\wedge
e^{i_{k}}\int_{0}^{1}dt_{1}\int_{0}^{t_{1}}dt_{2}...\int_{0}^{t_{k-1}}dt_{k}\\
&\frac{\partial
p}{\partial q_{1}...\partial q_{k}}(v,t_{1}v+(1-t_{1})w,…,t_{k}v+(1-t_{k})w,w)
\end{align*} mit $\partial_{k+1}\cp G_{k+1}=G_{k}\cp d_{k+1}$, $G_{k}=0$
für $k>n$ und $G_{0}=\id_{\mathcal{A}^{e}}$, sowie durch die
Abbildung aus \cite[Sect. III.2$\alpha$]{connes:1994a}:
\begin{align*}
F_{k}\colon \mathrm{Pol}(\mathbb{R}^{n} \times \mathbb{R}^{n})\ot
\Lambda^{k}(\mathbb{R}^{n*})&\longrightarrow \bigotimes^{k+2}\Poly
\\ \omega & \longmapsto [(v,w)(x_{1},…,x_{k})\mapsto \omega(v,w)(x_{1}-v,…,x_{k}-v)],
\end{align*} mit $d_{k+1}\cp F_{k+1}=F_{k}\cp \partial_{k+1}$,
$F_{k}=0$ für $k>n$ und $F_{0}=\id_{\mathcal{A}^{e}}$.\\\\
Eine einfache Rechnung zeigt dann $G_{k}\cp F_{k}= \id_{\K_{k}}$,
also $F_{k}^{*}\cp
G_{k}^{*}=\id_{\Hom_{\mathcal{A}^{e}}}(\K_{k},\mathcal{M})$ und
somit $\wt{F_{k}^{*}}\cp
\wt{G_{k}^{*}}=\id_{H^{k}(\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M}))}$.
Zusammen mit der Isomorphismus-Eigenschaft von $\wt{F_{k}^{*}}$
und $\wt{G_{k}^{*}}$ bestätigt dies $\wt{G_{k}^{*}}^{-1}=\wt{F_{k}^{*}}$.
\end{bemerkung}
\section{Die Hochschild-Kohomologie der Algebra
$\Ss^{\bullet}(\mathbb{V})$}
\label{subsec:HochschKohSym}
\subsection{Die symmetrische und die Graßmann Algebra}
Als abstrakte Variante des Polynom-Begriffes betrachten wir für einen
gegebenen $\mathbb{K}$-Vektorraum $\V$, die unitäre, assoziative $\mathbb{K}$-Algebra
$\left(\sym,\vee\right)$.
Dies ist der gradierte Vektorraum
$\displaystyle\Ss^{\bullet}(\V)=\bigoplus_{k=0}^{\infty}\mathrm{S}^{k}(\mathbb{V})$
mit Untervektorräumen
$\mathrm{S}^{k}(\mathbb{V})=\im\left(\mathrm{Sym}_{k}\right)\subseteq
\Tt^{k}(\V)$ und $S^{0}(\V)=\mathbb{K}$. Hierbei bezeichnet $\mathrm{Sym}_{k}\colon\Tt^{k}(\V)\longrightarrow \Tt^{k}(\V)$ die mit
Korollar \ref{kor:WohldefTensorprodabbildungen} wohldefinierte lineare
Fortsetzung von
\begin{align*}
v_{1}\ot\dots\ot v_{k}\longmapsto \frac{1}{k!}\sum_{\sigma\in S_{k}}v_{\sigma(1)}\ot\dots\ot v_{\sigma(k)}.
\end{align*}
Die bis auf kanonische Isomorphie kommutative, assoziative
Algebramultiplikation ist dabei definiert durch $\vee=S\cp
\ot^{\bullet}$. Hierbei bezeichnet $S\colon\Tt^{\bullet}(\V)\longrightarrow \SsV$
die Abbildung:
\begin{align*}
\SsV\ni\sum_{l}\alpha_{l}\longmapsto \sum_{l}\mathrm{Sym}_{l}(\alpha_{l})\quad\text{ mit }\quad\mathrm{Sym}_{0/1}=\id_{\Ss^{0/1}(\V)},
\end{align*}
und
$\ot^{\bullet}\colon \Ss^{\bullet}(\V)\times
\Ss^{\bullet}(\V)\longrightarrow \mathrm{T}^{\bullet}(\V)$ ist gegeben durch
\begin{align*}
\ot^{\bullet}\colon\left(\sum_{l}\alpha_{l},\sum_{m}\beta_{m}\right)\longmapsto\sum_{l,m}\alpha_{l}\ot\beta_{m}\qquad\forall\:\alpha_{l}\in \Ss^{l}(\V), \beta_{m}\in \Ss^{m}(\V).
\end{align*}
Hierbei haben wir stillschweigend Lemma \ref{lemma:assTenprod}~\textit{ii.)}, also $\Tt^{l}(\V)\ot \Tt^{m}(\V)\cong \Tt^{l+m}(\V)$ benutzt.
Weiter beachte man, dass $ \mathbb{K}\ot \mathbb{W} \cong \mathbb{W} \cong
\mathbb{W}\ot \mathbb{K}$
für beliebigen $\mathbb{K}$-Vektorraum
$\mathbb{W}$ gilt, da $\mathbb{K}$ selbst ein eindimensionaler
$\mathbb{K}$-Vektorraum ist.
Man setzt dann
\begin{equation*}
\ot^{\bullet}(\mathbb{k},\alpha_{l})=\left(\:\cong_{\Ss^{l}(\V)}\cp\ot\right)(\mathbb{k},\alpha_{l})=\mathbb{k}\cdot
\alpha_{l}\qquad\forall\: \alpha_{l}\in \Ss^{l}(\V)
\end{equation*}
und erhält insbesondere $1_{\SsV}=1_{\mathbb{K}}$ als Einselement.
Obige Definition hat dabei den Vorteil, dass
\begin{equation*}
(\alpha_{1}\vee\dots\vee\alpha_{l})\vee (\alpha_{l+1}\vee\dots\vee \alpha_{k})=\alpha_{1}\vee\dots\vee\alpha_{k}
\end{equation*}
f"ur $\alpha_{1}\vee\dots\vee \alpha_{k}:=\mathrm{Sym}_{k}(\alpha_{1}\ot…\ot
\alpha_{k})$ gilt.
Analog definieren wir die Gra"smann-Algebra $\Lambda^{\bullet}(\V)=\displaystyle\bigoplus_{k=0}^{\infty}\Lambda^{k}(\V)$ mit $\Lambda^{k}(\V)=\im\left(\mathrm{Alt}_{k}\right)$, und $\mathrm{Alt}_{k}$ die lineare Fortsetzung von $u_{1}\ot…\ot u_{k} \longmapsto \frac{1}{k!}\displaystyle\sum_{\sigma\in
S^{k}}\sign(\sigma)\:u_{\sigma(1)}\ot…\ot u_{\sigma(k)}$. Mit
$A=\displaystyle\sum_{k=1}^{\infty}\mathrm{Alt}_{k}$ ist dann die zugeh"orige
Algebramultiplikation durch $\wedge=A\cp\ot^{\bullet}$ definiert.
Wie f"ur den symmetrischen Fall, folgt $(\alpha_{1}\wedge\dots\wedge\alpha_{l})\wedge
(\alpha_{l+1}\wedge\dots\wedge \alpha_{k})=\alpha_{1}\wedge\dots
\wedge\alpha_{k}$
mit
$\alpha_{1}\wedge\dots\wedge \alpha_{k}:=\mathrm{Alt}_{k}(\alpha_{1}\ot\dots\ot
\alpha_{k})$.
\subsection{Bestimmung der Hochschild-Kohomologie von
$\Ss^{\bullet}(\mathbb{V})$}
Mit der Unitarität von $\mathcal{A}=\SsV$ ist die Existenz einer Bar-Auflösung $(\C,d,\epsilon)$ von
$\sym$ gesichert, und es gilt die Isomorphie $HH^{k}(\mathcal{A},\mathcal{M})\cong
H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}))$ für die von uns betrachteten $\mathcal{A}-\mathcal{A}$-Bimodul $\mathcal{M}$. Was wir nun noch benötigen, um die Hochschild-Kohomologie von $\SsV$ zu bestimmen, ist
lediglich eine Koszul-Auflösung $(\K,\partial,\epsilon)$ von $\Ss^{\bullet}(\V)$, da dann wieder
\begin{equation*}
H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}))\cong \mathrm{Ext}^{k}_{\mathcal{A}^{e}}(\cdot,
\mathcal{M})(\mathcal{A})\cong
H^{k}(\Hom_{\mathcal{A}^{e}}(C',\mathcal{M}))
\end{equation*}gilt.
Zu diesem Zwecke seien die $\K_{k}$ wie in Abschnitt
\ref{subsec:HschKPol} gegeben durch
$\mathcal{A}^{e}$-Moduln
\begin{equation*}
\K_{0}=\mathcal{A}^{e}\qquad\text{ und }\qquad
\K_{k}=\mathcal{A}^{e}\ot \Lambda^{k}(\mathbb{V})\:\text{ f"ur }k\geq 1
\end{equation*} mit der bekannten $\mathcal{A}^{e}$-Multiplikation im
ersten Faktor.\\\\
F"ur $\alpha \in \mathrm{S}^{l}(\mathbb{V})$ mit $\alpha=\alpha_{1}\vee…\vee \alpha_{l}$
bezeichne im Folgenden
$\alpha^{j}\in \Ss^{l-1}(\mathbb{V})$ das Element
$\alpha_{1}\vee…\blacktriangle^{j}…\vee \alpha_{l}$, welches durch Weglassen von
$\alpha_{j}$ aus $\alpha$ entsteht. Ebenso sei $\alpha^{j_{1},…,j_{s}}\in
\Ss^{l-s}(\mathbb{V})$ das Element, welches durch Weglassen der $\alpha_{j_{1}},\dots,\alpha_{j_{l}}$ aus $\alpha$ hervorgeht. Ist $\deg(\alpha)=l$, so setzen wir $\alpha^{1,…,l}=1_{\SsV}$. Sinngem"a"s benutzen wir diese Konventionen f"ur die Elemente in $\Lambda^{k}(\V)$.\\
\begin{definition}
\label{def:partialdef}
F"ur obige $\mathcal{A}^{e}$-Moduln definieren wir Kettendifferentiale durch:
\begin{align*}
\partial_{k}\colon \mathcal{A}^{e}\ot
\Lambda^{k}(\mathbb{V})&\longmapsto \mathcal{A}^{e}\ot \Lambda^{k-1}(\mathbb{V})\\
\omega&\longmapsto
\big[\pt^{k}_{1}-\pt^{k}_{2}\big](\omega)
\end{align*}
mit
\begin{align*}
\pt_{1}^{k}\colon\Ss^{\bullet}(\mathbb{V})\ot
\Ss^{\bullet}(\mathbb{V})\ot\Lambda^{k}(\mathbb{V})&\longrightarrow \Ss^{\bullet+1}(\mathbb{V})\ot
\Ss^{\bullet}(\mathbb{V})\ot\Lambda^{k-1}(\mathbb{V})\\
\KE{\alpha}{\beta}{u}&\longmapsto
\sum_{j=1}^{k}(-1)^{j-1}\:\KE{u_{j}\vee
\alpha}{\beta}{u^{j}}\\
\pt_{2}^{k}\colon\Ss^{\bullet}(\mathbb{V})\ot
\Ss^{\bullet}(\mathbb{V})\ot\Lambda^{k}(\mathbb{V})&\longrightarrow \Ss^{\bullet}(\mathbb{V})\ot
\Ss^{\bullet+1}(\mathbb{V})\ot\Lambda^{k-1}(\mathbb{V})\\
\KE{\alpha}{\beta}{u}&\longmapsto
\sum_{j=1}^{k}(-1)^{j-1}\:\KE{\alpha}{u_{j}\vee\beta}{u^{j}},
\end{align*}
wobei wir hier und im Folgenden $\mathcal{A}^{e}\ot \Lambda^{0}(\V)$ mit $\mathcal{A}^{e}$ identifizieren wollen.
\end{definition}
\begin{bemerkung}
Die Wohldefiniertheit obiger Abbildungen folgt wieder mit Korollar \ref{kor:WohldefTensorprodabbildungen}, da wir diese auch schreiben können, als
\begin{align*}
\pt^{k}_{1}&=(S \ot \id\ot \id)\cp
\tilde{\pt}^{k}_{1}\Big|_{\K_{k}}\\
\pt^{k}_{2}&=(\id \ot S \ot \id)\cp
\tilde{\pt}^{k}_{2}\Big|_{\K_{k}}
\end{align*} mit
\begin{align*}
\tilde{\pt}^{k}_{1}\colon\Tt^{\bullet}(\mathbb{V})\bbot
\Tt^{\bullet}(\mathbb{V})\bbot
\Tt^{k}(\mathbb{V})&\longrightarrow \Tt^{\bullet+1}(\mathbb{V})\bbot
\Tt^{\bullet}(\mathbb{V})\bbot
\Tt^{k-1}(\mathbb{V})\\
\alpha\bbot \beta\bbot u&\longmapsto k\left(u_{1}\ot \alpha\bbot\beta\bbot u^{1}\right)\\
\tilde{\pt}^{k}_{2}\colon\Tt^{\bullet}(\mathbb{V})\bbot
\Tt^{\bullet}(\mathbb{V})\bbot
\Tt^{k}(\mathbb{V})&\longrightarrow \Tt^{\bullet}(\mathbb{V})\bbot
\Tt^{\bullet+1}(\mathbb{V})\bbot
\Tt^{k-1}(\mathbb{V})\\
\alpha\bbot \beta\bbot u&\longmapsto k\left(\alpha\bbot u_{1}\ot\beta\bbot u^{1}\right).
\end{align*}
Dabei liegt der Faktor $k$ an unserer Konvention
\begin{equation*}
u_{1}\wedge\dots \wedge
u_{k}=\frac{1}{k!}\displaystyle\sum_{\sigma\in
S_{k}}\sign(\sigma)(u_{\sigma(1)}\ot\dots\ot u_{\sigma(k)}).
\end{equation*} In der Tat erhalten wir f"ur $\alpha, \beta \in \Ss^{\bullet}(\V)$:
{\allowdisplaybreaks\begin{align*}
\wt{\pt}^{k}_{1}(\alpha\ot\beta\ot&\: u_{1}\wedge\dots\wedge u_{k})=\frac{1}{k!}\sum_{\sigma\in S_{k}}\sign(\sigma)\wt{\pt}^{k}_{1}\left(\alpha\:\boldsymbol{\ot}\:\beta\:\boldsymbol{\ot}\: u_{\sigma(1)}\ot\dots\ot u_{\sigma(k)}\right)\\
=&\:\frac{1}{(k-1)!}\sum_{\sigma\in S_{k}}\sign(\sigma)\left(u_{\sigma(1)}\ot\alpha\:\boldsymbol{\ot}\:\beta\:\boldsymbol{\ot}\:u_{\sigma(2)}\ot\dots\ot u_{\sigma(k)}\right)
\\=&\:\frac{1}{(k-1)!}\sum_{j=1}^{k}\sum_{\substack{\sigma\in
S_{k}\\\sigma(1)=j}}\sign(\sigma)\left(u_{j}\ot\alpha\:\boldsymbol{\ot}\:\beta\:\boldsymbol{\ot}\:u_{\sigma(2)}\ot\dots\ot u_{\sigma(k)}\right)
\\=&\:\frac{1}{(k-1)!}\sum_{j=1}^{k}(-1)^{j-1}\sum_{\sigma\in
S_{k-1}}\sign(\sigma)\left(u_{j}\ot\alpha\:\boldsymbol{\ot}\:\beta\:\boldsymbol{\ot}\:\sigma^{*}\left[u_{1}\ot\dots\blacktriangle^{j}\dots\ot u_{k}\right]\right)
\\=&\:\sum_{j=1}^{k}(-1)^{j-1}\left(u_{j}\ot\alpha\:\boldsymbol{\ot}\:\beta\:\boldsymbol{\ot}\:u_{1}\wedge\dots\blacktriangle^{j}\dots\wedge u_{k}\right),
\end{align*}}wobei $\sigma^{*}\left[u_{1}\ot\dots\ot u_{k-1}\right]=u_{\sigma(1)}\ot\dots\ot
u_{\sigma(k-1)}$ bedeutet.
Symmetrisieren im ersten Argument liefert dann die gew"unschte Gleichheit. Analog folgt die Behauptung f"ur $\wt{\pt}^{k}_{2}$.
\end{bemerkung}
Folgendes Lemma zeigt, dass $(\K,\pt)$ ein Kettenkomplex ist.
\begin{lemma}
Es gilt $\partial_{k-1}\cp\pt_{k}=0$.
\begin{beweis}
Seien hierf"ur abk"urzend
\begin{align*}
u *_{L}\: (\alpha\ot \beta \ot \omega)&= u\vee \alpha\ot \beta \ot \omega \quad\text{ sowie }\\
u *_{R} \:(\alpha\ot \beta \ot \omega)&= \alpha\ot u\vee\beta \ot \omega,
\end{align*}
dann folgt:
{\allowdisplaybreaks
\begin{align*}
\left(\partial_{k-1}\cp
\pt_{k}\right)(\alpha\ot&\:\beta\ot u)=\:\pt_{k-1}\left(\sum_{j=1}^{k}(-1)^{j-1}[u_{j}*_{L}-u_{j}\:*_{R}]\: \alpha\ot\beta\ot u^{j}\right)
\\=&\: \sum_{j=1}^{k}(-1)^{j-1}\Bigg[\sum_{i=1}^{j-1}(-1)^{i-1}[u_{i}*_{L}-u_{i}\:*_{R}][u_{j}*_{L}-u_{j}\:*_{R}]\:\alpha\ot\beta\ot u^{i,j}
\\ &\qquad\qquad\qquad +\sum_{i=j+1}^{k}(-1)^{i-2}[u_{i}*_{L}-u_{i}\:*_{R}][u_{j}*_{L}-u_{j}\:*_{R}]\:\alpha\ot\beta\ot u^{j,i}\Bigg]
\\=&\:\sum_{j=2}^{k}\sum_{i<j}(-1)^{j+i}\:[u_{i}*_{L}-u_{i}\:*_{R}][u_{j}*_{L}-u_{j}\:*_{R}]\:\alpha\ot\beta\ot u^{i,j}
\\ &-\sum_{j=1}^{k-1}\sum_{i>j}\overbrace{(-1)^{j+i}\:[u_{i}*_{L}-u_{i}\:*_{R}][u_{j}*_{L}-u_{j}\:*_{R}]}^{\tau_{i,j}}\alpha\ot\beta\ot u^{j,i}
\\=&\:0,
\end{align*}}da
$\tau_{i,j}=\tau_{j,i}$ mit der Kommutativit"at von $\vee$, und somit
\begin{align*}
\sum_{j=1}^{k-1}\sum_{i>j}\tau_{i,j}\alpha\ot\beta\ot
u^{j,i}\:\glna{\mathit{i}\leftrightarrow \mathit{j}}\:\sum_{i=1}^{k-1}\sum_{i<j}\tau_{i,j}\:\alpha\ot\beta\ot
u^{i,j}=\:\sum_{j=2}^{k}\sum_{i<j}\tau_{i,j}\:\alpha\ot\beta\ot
u^{i,j}.
\end{align*}
Dabei ist letzte Gleichheit rein kombinatorischer Natur.
\end{beweis}
\end{lemma}
Zusammen mit der Abbildung
\begin{align*}
\epsilon\colon\mathcal{A}\ot \mathcal{A}&\longrightarrow \mathcal{A}\\
\alpha\ot \beta&\longmapsto \alpha\vee \beta
\end{align*} wird $(\K,\partial,\epsilon)$ zu einem
projektiven Komplex über $\SsV$, und es bleibt nun lediglich dessen Exaktheit
nachzuweisen. Hierfür definieren wir die abstrakte Variante von
\eqref{eq:exakthAbbHvonPol} durch:
\begin{definition}
\label{def:KosSymExakthHomothidelta}
Sei
\begin{align*}
h_{-1}\colon \mathcal{A}&\longrightarrow \K_{0}\\
\alpha &\longmapsto 1\ot\alpha
\end{align*}und
\begin{align*}
h_{k}\colon \Ss^{\bullet}(\mathbb{V})\ot
\Ss^{\bullet}(\mathbb{V})\ot\Lambda^{k}(\mathbb{V})&\longrightarrow \Ss^{\bullet}(\mathbb{V})\ot
\Ss^{\bullet}(\mathbb{V})\ot\Lambda^{k+1}(\mathbb{V})\\
\mu
&\longmapsto\int_{0}^{1}dt\:t^{k}(i_{t}\circ\delta)(\mu),
\end{align*}für $k\geq 0$. Hierbei haben wir die folgenden Abbildungen benutzt:
\begin{enumerate}
\item
$i_{t}:\SsV\ot\SsV\ot \Lambda^{k}(\V)\longrightarrow
\big[\SsV\ot\SsV\ot \Lambda^{k}(\V)\big][t]$ definiert durch
\begin{align*}
i_{t}\colon\KE{\alpha}{\beta}{u}\longmapsto&\:
\:t^{l}\KE{\alpha}{\beta}{u}\:+\:t^{l-1}(1-t)\sum_{j=1}^{l}\KE{\alpha^{j}}{\alpha_{j}\vee\beta}{u} +…
\\ &+t^{l-s}(1-t)^{s}\sum_{j_{1},…,j_{s}}^{l}\KE{\alpha^{j_{1},…,j_{s}}}{\alpha_{j_{1},…,j_{s}}\vee\beta}{u} +…
\\ &+(1-t)^{l}\KE{1}{\alpha\vee \beta}{u}
\end{align*}für $\deg(\alpha)=l$ und $i_{t}(1\ot \beta\ot
u)=(1\ot \beta\ot u)$. Hierbei ist mit $\displaystyle\sum_{j_{1},…,j_{s}}$
die Summe über alle $j_{1}\neq\dots \neq j_{s}$
gemeint. Das Bild unter $i_{i}$ ist dann
als Polynom in $t$ mit Werten in $\SsV\ot\SsV\ot
\Lambda^{k}(\V)$ zu verstehen.
\item
\begin{align*}
\delta\colon\Ss^{\bullet}(\mathbb{V})\ot \Ss^{\bullet}(\mathbb{V})\ot
\Lambda^{k}(\mathbb{V})&\longrightarrow \Ss^{\bullet-1}(\mathbb{V})\ot \Ss^{\bullet}(\mathbb{V})\ot \Lambda^{k+1}(\mathbb{V})\\
\alpha_{l}\ot \beta \ot u&\longmapsto
\sum_{j=1}^{l}\alpha_{l}^{j}\ot \beta\ot \:(\alpha_{l})_{j}\wedge u
\end{align*}
für $\alpha_{l}\in \Ss^{l}(\V)$ und $\delta(1\ot\beta\ot u)=0$.\\
\end{enumerate}
\end{definition}
\begin{bemerkung}
Obige Abbildungen sind wohldefiniert, da zum einen $\delta=(\id\ot \id\ot A) \cp \tilde{\delta}\big|_{\K_{k}}$ mit
\begin{align*}
\tilde{\delta}\colon\Tt^{\bullet}(\mathbb{V})\bbot \Tt^{\bullet}(\mathbb{V})\bbot
\Tt^{k}(\mathbb{V})&\longrightarrow
\Tt^{\bullet-1}(\mathbb{V})\bbot \Tt^{\bullet}(\mathbb{V})\bbot
\Tt^{k+1}(\mathbb{V})\\
\alpha_{l}\bbot\beta\bbot u&\longmapsto l\left(\alpha_{l}^{1}\bbot\: \beta\bbot\: (\alpha_{l})_{1}\ot u\right),
\end{align*}wobei der Faktor $l$ der Konvention f"ur $v_{1}\vee\dots\vee v_{l}=\frac{1}{l!}\displaystyle\sum_{\sigma\in S_{l}}v_{\sigma(1)}\ot\dots\ot v_{\sigma(l)}$\\ geschuldet ist, und da zum anderen
\begin{equation*}
i_{t}=(\id\ot S\ot \id)\cp \left[\sum_{l=0}^{\infty}\sum_{s=0}^{l}\eta_{t}^{l,s}\right]\Bigg|_{\K_{k}}
\end{equation*} mit
\begin{align*}
\eta^{l,s}_{t}\colon\Tt^{l}(\mathbb{V})\bbot
\Tt^{\bullet}(\mathbb{V})\bbot
\Tt^{k}(\mathbb{V})&\longrightarrow \Tt^{\bullet}(\mathbb{V})\bbot
\Tt^{\bullet}(\mathbb{V})\bbot
\Tt^{k}(\mathbb{V})\\
\alpha_{l}\bbot \beta\bbot u&\longmapsto
\binom{l}{s}\:t^{l-s}(1-t)^{s}\alpha_{l}^{1,…,s}\bbot\:
(\alpha_{l})_{1,…,s}\ot \beta\bbot u,
\end{align*}wobei $\eta_{t}^{0,0}(1\ot \beta\ot u)=(1\ot \beta\ot
u)$.
In der Tat erhalten wir f"ur $\alpha_{l}\in \Ss^{l}(\V)$, $\beta \in \SsV$ und $u\in \Lambda^{k}(\V)$:
{\begin{align*}
\eta_{t}^{l,s}(\alpha_{l}\ot \beta \ot u)
=&\:\frac{1}{s!(l-s)!}t^{l-s}(1-t)^{s}\sum_{\sigma\in
S_{l}}(\alpha_{l})_{\sigma(s+1)}\ot\dots
\ot\:(\alpha_{l})_{\sigma(l)}\bbot
\\ &\qquad\qquad\qquad\qquad\qquad\qquad\:(\alpha_{l})_{\sigma(1)}\ot\dots \ot\:(\alpha_{l})_{\sigma(s)}\ot \beta \bbot u
\\=&\:t^{l-s}(1-t)^{s}\sum_{j_{1},\dots,j_{s}}^{l}\alpha_{l}^{j_{1},\dots,j_{l}}\bbot\: (\alpha_{l})_{j_{1},\dots,j_{l}}\ot\beta \bbot\:u.
\end{align*}}
\end{bemerkung}
Um nun die gew"unschte Homotopieeigenschaft f"ur von $h$ nachzuweisen, ben"otigen wir zun"achst einige Rechenregeln. Sei hierf"ur $\cdot$ die $\mathbb{K}$-bilineare Abbildung:
\begin{align*}
\cdot\colon \mathcal{A}^{e} \ot \Lambda^{k}(\V)\times \mathcal{A}^{e} \ot \Lambda^{k'}(\V) &\longrightarrow \mathcal{A}^{e} \ot \Lambda^{k+k'}(\V)\\
\big(\KE{\alpha}{\beta}{u},\KE{\wt{\alpha}}{\wt{\beta}}{\wt{u}}\big)&\longmapsto \KE{\alpha\vee \wt{\alpha}}{\:\beta\vee\wt{\beta}}{\:u\wedge \wt{u}}
\end{align*} und $\pt$ die Abbildung, die durch die
$\pt_{k}$ auf ganz $\Ss^{\bullet}(\V)\ot\Ss^{\bullet}(\V)\ot
\Lambda^{\bullet}(\V)$, verm"oge der Konvention $\pt|_{\mathcal{A}^{e}\ot\Lambda^{0}(\V)}=0$, induziert wird.
\begin{lemma}
\label{lemma:EigenschHomotBaukloetze}
\begin{enumerate}
\item
\label{item:deltaDerivat}
\begin{align*}
\delta(\KE{\alpha}{\beta}{u}\cdot
\KE{\wt{\alpha}}{\wt{\beta}}{\wt{u}})=&\:\delta\KE{\alpha}{\beta}{u}\cdot\KE{\wt{\alpha}}{\wt{\beta}}{\wt{u}}\\
&+
(-1)^{\deg(u)}\KE{\alpha}{\beta}{u}\cdot\: \delta\KE{\wt{\alpha}}{\wt{\beta}}{\wt{u}};
\end{align*}
\item
\label{item:partialDerivat}
\begin{align*}
\partial(\KE{\alpha}{\beta}{u}\cdot
\KE{\wt{\alpha}}{\wt{\beta}}{\wt{u}})=&\:\partial\KE{\alpha}{\beta}{u}\cdot\KE{\wt{\alpha}}{\wt{\beta}}{\wt{u}}\\
&+
(-1)^{\deg(u)}\KE{\alpha}{\beta}{u}\cdot\: \partial\KE{\wt{\alpha}}{\wt{\beta}}{\wt{u}};
\end{align*}
\item
\label{item:iFaktorisation}
\begin{equation*}
i_{t}(\mu\cdot\nu)=i_{t}(\mu)\cdot
i_{t}(\nu)\quad\quad\forall\:\mu,\nu\in \Ss^{\bullet}(\mathbb{V})\ot \Ss^{\bullet}(\mathbb{V})\ot\Lambda^{\bullet}(\mathbb{V});
\end{equation*}
\item
\label{item:special_tAbleitunsRel}
\begin{equation*}
\frac{d}{dt}i_{t}(\alpha\ot \beta\ot 1)=(\partial_{1}\circ i_{t}\circ
\delta)(\alpha\ot \beta\ot 1).
\end{equation*}
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Wir erhalten mit $\alpha\wedge
\beta=(-1)^{\deg(\alpha)\deg(\beta)}\:\beta\wedge \alpha$
sowie der Assoziativität von $\wedge$:
{\allowdisplaybreaks\begin{align*}
\delta&\:(\alpha\vee\wt{\alpha}\ot \beta\vee\wt{\beta}\ot u\wedge\wt{u})\\
&=\sum_{j=1}^{l}\KE{\alpha^{j}\vee\wt{\alpha}}{\:\beta\vee\wt{\beta}}{\:\alpha_{j}\wedge
u\wedge\wt{u}}
+\sum_{j=1}^{\wt{l}}\KE{\alpha\vee\wt{\alpha}^{j}}{\beta\vee\wt{\beta}}{\wt{\alpha}_{j}\wedge
u\wedge\wt{u}}
\\ &= \sum_{j=1}^{l}\KE{\alpha^{j}}{\beta}{\alpha_{j}\wedge
u}\cdot\KE{\wt{\alpha}}{\wt{\beta}}{\wt{u}}
+(-1)^{\deg(u)}\KE{\alpha}{\beta}{u}\cdot \sum_{j=1}^{\wt{l}}\KE{\wt{\alpha}^{j}}{\wt{\beta}}{\wt{\alpha}_{j}\wedge
\wt{u}}
\\ &= \delta\KE{\alpha}{\beta}{u}\cdot\KE{\wt{\alpha}}{\wt{\beta}}{\wt{u}}+(-1)^{\deg(u)}\KE{\alpha}{\beta}{u}\cdot\:\delta\KE{\wt{\alpha}}{\wt{\beta}}{\wt{u}}.
\end{align*}}
\item
Sei $\deg(u)=k$ und $\deg(\wt{u})=\wt{k}$, dann folgt:
{\begin{align*}
\partial^{1}_{k+\wt{k}}(\alpha\vee\wt{\alpha}\ot\beta\vee\wt{\beta}\ot u&\:\wedge\wt{u})\\=&\:\sum_{j=1}^{k}(-1)^{j-1}\KE{u_{j}\vee\alpha\vee\wt{\alpha}\:}{\:\beta\vee\wt{\beta}}{\:u^{j}\wedge\wt{u}}
\\ &\:+\sum_{j=1}^{\wt{k}}(-1)^{j+k-1}\KE{u_{j}\vee\alpha\vee\wt{\alpha}\:}{\:\beta\vee\wt{\beta}}{\:u\wedge\wt{u}^{j}}
\\
=&\:\sum_{j=1}^{k}(-1)^{j-1}\KE{u_{j}\vee\alpha}{\beta}{u^{j}}\cdot\KE{\wt{\alpha}}{\wt{\beta}}{\wt{u}}
\\ &\: +(-1)^{[k=\deg(u)]}\KE{\alpha}{\beta}{u}\cdot
\sum_{j=1}^{\wt{k}}(-1)^{j-1}\KE{\wt{u}_{j}\vee\wt{\alpha}}{\wt{\beta}}{\wt{u}^{j}}\\
=&\: \partial^{1}_{k}\KE{\alpha}{\beta}{u}\cdot\KE{\wt{\alpha}}{\wt{\beta}}{\wt{u}}
\\ &\:+\:(-1)^{\deg(u)}\KE{\alpha}{\beta}{u}\cdot\:\partial^{1}_{\wt{k}}\KE{\wt{\alpha}}{\wt{\beta}}{\wt{u}}.
\end{align*}}Analog folgt dies für $\pt^{2}_{k+\wt{k}}$, was die Behauptung zeigt.
\item
Dies folgt unmittelbar daraus, dass jeder Summand aus $i_{t}(\mu\cdot\nu)$ eindeutig als Produkt zweier
Summanden aus $i_{t}(\mu)$ und $i_{t}(\nu)$ geschrieben werden kann.
\item
Zunächst reicht es, die Aussage für Elemente
$\alpha\ot 1\ot 1$ zu zeigen, da man auf beiden Seiten
der zu zeigenden Gleichung $1\ot \beta\ot 1$
herausziehen kann. Es folgt
{\begin{align*}
\frac{d}{dt}i_{t}(1\ot 1\ot1)=
\frac{d}{dt}(1\ot 1\ot1)=0=(\partial_{1}\circ i_{t}\circ
\delta)(1\ot1\ot1)
\end{align*} und weiter für $\deg(\operatorname{\mathrm{v}})=1$:
\begin{align*}
\frac{d}{dt}i_{t}(\operatorname{\mathrm{v}}\ot 1\ot1)=&\:\frac{d}{dt}\Big[t(\operatorname{\mathrm{v}}\ot 1\ot1)+(1-t)(1\ot\operatorname{\mathrm{v}}\ot1)\Big]
=(\operatorname{\mathrm{v}}\ot 1\ot1)-(1\ot\operatorname{\mathrm{v}}\ot1)
\\=&\: \partial_{1}(1\ot1\ot \operatorname{\mathrm{v}})
=(\partial_{1}\circ i_{t}\ot1)\KE{1}{1}{v}
\\=&\:(\partial_{1}\circ i_{t}\circ \delta)(\operatorname{\mathrm{v}}\ot 1\ot1).
\end{align*}}Angenommen, obige Aussage gelte für
$\deg(\alpha)=k$, dann erhalten wir:
{\allowdisplaybreaks
\begin{align*}
\frac{d}{dt}i_{t}(\operatorname{\mathrm{v}}&\vee\alpha\ot1\ot1)=
\frac{d}{dt}\Big[i_{t}(\operatorname{\mathrm{v}}\ot 1\ot1)\cdot\:i_{t}(\alpha\ot1\ot1)\Big]\\
=&\:\frac{d}{dt}i_{t}(\operatorname{\mathrm{v}}\ot1\ot1)\cdot
\:i_{t}(\alpha\ot1\ot1)
+\:i_{t}(\operatorname{\mathrm{v}}\ot1\ot1)\cdot\:\frac{d}{dt}i_{t}(\alpha\ot1\ot1)\\
=&\:(\partial_{1}\circ i_{t}\circ\delta)(\operatorname{\mathrm{v}}\ot1\ot1)\cdot\:
i_{t}(\alpha\ot1\ot1)
+\:i_{t}(\operatorname{\mathrm{v}}\ot1\ot1)\cdot\:(\partial_{1}\circ
i_{t}\circ\delta)(\alpha\ot1\ot1)\\
=&\:\partial_{1}\Big[(i_{t}\circ\delta)(\operatorname{\mathrm{v}}\ot1\ot1)\cdot\:i_{t}(\alpha\ot1\ot1)
+\:i_{t}(\operatorname{\mathrm{v}}\ot1\ot1)\cdot\:(i_{t}\circ\delta)(\alpha\ot1\ot1)\Big]
\\ =&\:(\partial_{1}\circ
i_{t})\Big[\:\delta(\operatorname{\mathrm{v}}\ot1\ot1)\cdot(\alpha\ot1\ot1)
+(\operatorname{\mathrm{v}}\ot1\ot1)\cdot\:\delta(\alpha\ot1\ot1)\Big]
\\ =&\:(\partial_{1}\circ i_{t}\circ\delta\ot1)\Big[(\operatorname{\mathrm{v}}\ot1\ot1)\cdot(\alpha\ot1\ot1)\Big]\\=&\: (\partial_{1}\circ i_{t}\circ\delta\ot1)(\operatorname{\mathrm{v}}\vee\:\alpha\ot1\ot1).
\end{align*}}
\end{enumerate}
\end{beweis}
\end{lemma}
Folgende Proposition liefert schließlich die Exaktheit von
$(\K,\partial,\epsilon)$.
\begin{proposition}
\label{prop:ExaktheitsbewSym}
Es gilt
\begin{align*}
\epsilon\circ h_{-1}&=\id_{\mathcal{A}}, \\
h_{-1}\circ\epsilon+\partial_{1}\circ h_{0}&=\id_{\K_{0}}\quad\quad\text{und}\\
h_{k-1}\circ \partial_{k}+\partial_{k+1}\circ
h_{k}&=\id_{\K_{k}}\quad\quad \text{für }k\geq1.
\end{align*}
\begin{beweis}
Zunächst folgt
\begin{equation*}
(\epsilon\cp h_{-1})(\alpha)=\epsilon(1\ot \alpha)=\alpha,
\end{equation*} sowie mit Lemma \ref{lemma:EigenschHomotBaukloetze}~\textit{iv.)}:
\begin{align*}
(h_{-1}\cp \epsilon + \partial_{1}\cp h_{0})(\alpha\ot
\beta)=&\:1\ot \alpha\vee\beta+ \int_{0}^{1}dt\:(\pt_{1}\cp i_{t}\cp\delta)(\alpha\ot
\beta)\\\glna{\textit{iv.)}}&\: 1\ot \alpha\vee\beta + \alpha\ot \beta -1\ot
\alpha\vee \beta\\=&\: \alpha\ot\beta.
\end{align*}
Das zeigt die ersten beiden Behauptungen. Für die
dritte sei $\mu=\alpha\ot\beta\ot u\in \K_{k}$, dann folgt:
\begin{align*}
(\partial_{k+1}\circ
h_{k})(\mu)=\partial_{k+1}\left[\int_{0}^{1}dt\:t^{k}\:(i_{t}\circ\delta)(\mu)\right]=\int_{0}^{1}dt\:t^{k}\:(\partial_{k+1}\circ
i_{t}\circ\delta)(\mu),
\end{align*} und für den Integranden mit
Lemma \ref{lemma:EigenschHomotBaukloetze}~\textit{ii.)}
\begin{align*}
(\pt_{k+1}\circ \:i_{t}\circ \delta)(\mu)=&\:(\pt_{k+1}\circ\:
i_{t})\big[\delta\KE{\alpha}{1}{1}\cdot\KE{1}{\beta}{u}\big]\\
=& \:\pt_{k+1}\big[(i_{t}\circ\delta)\KE{\alpha}{1}{1}\cdot\KE{1}{\beta}{u}\big]\\
\glna{\textit{ii.)}}& \:(\pt_{k+1}\circ\:
i_{t}\circ\delta)\KE{\alpha}{1}{1}\cdot\KE{1}{\beta}{u}\\ &+(-1)^{1}(i_{t}\circ\delta)\KE{\alpha}{1}{1}\cdot\:\pt_{k}\KE{1}{\beta}{u},
\end{align*} womit insgesamt
\begin{equation}
\label{eq:TermvonGlHomotdersichschnellweghebt}
\begin{split}
(\partial_{k+1}\circ
h_{k})(\mu)=&\int_{0}^{1}dt\:t^{k}\:(\partial_{k+1}\circ
i_{t}\circ\delta)\KE{\alpha}{1}{1}\cdot\KE{1}{\beta}{u} \\ &
-\int_{0}^{1}dt\:t^{k}\:(i_{t}\circ\delta)\KE{\alpha}{1}{1}\cdot\:\partial_{k}\KE{1}{\beta}{u}.
\end{split}
\end{equation}
Für
\begin{align*}
(h_{k-1}\circ\partial_{k})(\mu)=\int_{0}^{1}dt\:t^{k-1}\:(i_{t}\circ\delta\circ\partial_{k})(\mu)
\end{align*} rechnen wir zunächst:
{\begin{align*}
(\delta\circ\partial_{k})(\mu)=&\:(\delta\circ\partial_{k})\big[\KE{1}{\beta}{u}\cdot\KE{\alpha}{1}{1}\big]\\
=&\:\delta\big[\pt_{k}\KE{1}{\beta}{u}\cdot \KE{\alpha}{1}{1}\big]\\
=& \:(\delta\circ\partial_{k})\KE{1}{\beta}{u}\cdot
\KE{\alpha}{1}{1}+\:
(-1)^{k-1}\partial_{k}\KE{1}{\beta}{u}\cdot\:\delta\KE{\alpha}{1}{1}\\
= & \:(\delta\circ\partial_{k})\KE{1}{\beta}{u}\cdot
\KE{\alpha}{1}{1}+\:
\delta\KE{\alpha}{1}{1}\cdot\:\partial_{k}\KE{1}{\beta}{u}.
\end{align*}} Anwenden von $i_{t}$ liefert:
\begin{align*}
(i_{t}\circ\delta\circ\partial_{k})(\mu)=&\:(i_{t}\circ\delta\circ \partial_{k})\KE{1}{\beta}{u}\cdot\:i_{t}\KE{\alpha}{1}{1}\\
&+
(i_{t}\circ\delta)\KE{\alpha}{1}{1}\cdot\:(i_{t}\circ\partial_{k})\KE{1}{\beta}{u}\\
= &\: k \KE{1}{\beta}{u}\cdot\: i_{t}\KE{\alpha}{1}{1}\\&\: +t\:(i_{t}\circ\delta)\KE{\alpha}{1}{1}\cdot\:\partial_{k}\KE{1}{\beta}{u}.
\end{align*} In der Tat erhalten wir für den ersten Term in
der letzten Gleichheit:
\begin{align*}
(i_{t}\circ\delta\circ\partial_{k})\KE{1}{\beta}{u}=&\:(i_{t}\circ\delta)\left(\sum_{j=1}^{k}(-1)^{j-1}\:u_{j}\ot
\beta\ot u^{j} -\sum_{j=1}^{k}(-1)^{j-1}\:1\ot
u_{j}\vee\beta\ot
u^{j}\right)
\\=&\:i_{t}\left(\sum_{j=1}^{k}(-1)^{j-1}1\ot\beta\ot u_{j}\wedge u^{j}\right)
\\=&\:k\KE{1}{\beta}{u}
\end{align*} und für den zweiten:
{\allowdisplaybreaks
\begin{align*}
(i_{t}\circ\partial_{k})(1\ot\beta&\:\ot u)=i_{t}\left(\sum_{j}^{k}(-1)^{j-1}\left[u_{j}\ot\beta\ot u^{j}-1\ot u_{j}\vee\beta\ot u^{j}\right]\right)\\
=&\:\sum_{j}^{k}(-1)^{j-1}\left[t\KE{u_{j}}{\beta}{u^{j}}+(1-t)\KE{1}{u_{j}\vee\beta}{u^{j}}-\KE{1}{u_{j}\vee\beta}{u^{j}}\right]
\\ =&\: t\:\partial_{k}\KE{1}{\beta}{u}.
\end{align*}}Das Zwischenergebnis lautet
\begin{equation}
\label{eq:zwischenergebnis}
\begin{split}
(h_{k-1}\circ\pt_{k})(\mu)=&\int_{0}^{1}dt\:k\:t^{k-1}i_{t}\KE{\alpha}{1}{1}\cdot\KE{1}{\beta}{u}\\
&
+\int_{0}^{1}dt\:t^{k}(i_{t}\circ\delta)\KE{\alpha}{1}{1}\cdot\:\partial_{k}\KE{1}{\beta}{u},
\end{split}
\end{equation} wobei der zweite Summand bereits das Negative
vom zweiten Summanden in
\eqref{eq:TermvonGlHomotdersichschnellweghebt} ist.\\\\ Für
den ersten erhalten wir weiter:
\begin{align*}
\int_{0}^{1}dt\:k\:t^{k-1}i_{t}\KE{\alpha}{1}{1}=&\int_{0}^{1}dt\frac{d}{dt}\:\big[t^{k}i_{t}\KE{\alpha}{1}{1}\big]
-\int_{0}^{1}dt\:t^{k}\frac{d}{dt}i_{t}\KE{\alpha}{1}{1}\\
=&\:\KE{\alpha}{1}{1}-\int_{0}^{1}dt\:t^{k}(\partial_{k}\circ i_{t}\circ\delta)\KE{\alpha}{1}{1},
\end{align*} dabei folgt die letzte Gleichheit mit Lemma \ref{lemma:EigenschHomotBaukloetze}~\textit{iv.)} und
\begin{align*}
\int_{0}^{1}dt\frac{d}{dt}\:\big[t^{k}i_{t}\KE{\alpha}{1}{1}\big]=&\:\big[t^{k}
i_{t}\KE{\alpha}{1}{1}\big]_{0}^{1}=\:i_{t}\KE{\alpha}{1}{1}\big|_{t=1}=
\KE{\alpha}{1}{1}.
\end{align*}
Aus \eqref{eq:zwischenergebnis} wird dann
\begin{equation}
\label{eq:rueckrichtwegheb}
\begin{split}
(h_{k-1}\circ\pt_{k})(\mu)=&\KE{\alpha}{\beta}{u}-
\int_{0}^{1}dt\:t^{k}(\partial_{k}\circ i_{t}\circ\delta)\KE{\alpha}{1}{1}\cdot\KE{1}{\beta}{u}\\
&
+\int_{0}^{1}dt\:t^{k}(i_{t}\circ\delta)\KE{\alpha}{1}{1}\cdot\:\partial_{k}\KE{1}{\beta}{u},
\end{split}
\end{equation}
und Addition von
\eqref{eq:TermvonGlHomotdersichschnellweghebt} und
\eqref{eq:rueckrichtwegheb} zeigt schließlich die Behauptung.
\end{beweis}
\end{proposition}
Hiermit erhalten wir umgehend folgenden Satz:
\begin{satz}
\label{satz:HochschkohmvonSym}
Sei $\mathcal{A}=\SsV$ und $\mathcal{M}$ ein $\SsV-\SsV$-Bimodul, dann gilt:
\begin{equation*}
HH^{k}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong
H^{k}\left(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M})\right)\cong
H^{k}\left(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{M})\right).
\end{equation*}
Ist $\mathcal{M}$ zudem symmetrisch, so ist:
\begin{equation*}
HH^{k}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong \Hom_{\mathcal{A}^{e}}\left(\K_{k},\mathcal{M}\right).
\end{equation*}
\begin{beweis}
Die erste Isomorphie hatten wir bereits eingesehen, und die
zweite folgt mit
Lemma \ref{lemma:GruppenKOhomsausprojaufloesundFunktoren}~\textit{ii.)} unmittelbar aus dem Fakt, dass sowohl $(\C,d,\epsilon)$, als auch $(\K,\pt,\epsilon)$ projektive Aufl"osungen von $\SsV$ sind.
F"ur die zweite Behauptung sei $\phi\in
\K^{*}_{k}=\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})$ und
$\omega=\alpha\ot \beta \ot u \in \K_{k+1}$. Dann folgt:
\begin{align*}
(\partial^{*}_{k+1}\phi)(\omega)=&\:\phi\left(\partial_{k+1}(\alpha\ot
\beta \ot u)\right)\\=&\:
\phi\left(\sum_{j=1}^{n}(-1)^{j-1}\left[u_{j}\vee \alpha
\ot \beta \ot u^{j}- \alpha
\ot u_{j}\vee\beta \ot u^{j} \right]\right)
\\=&\sum_{j=1}^{n}(-1)^{j-1} [u_{j}\ot 1 - 1\ot u_{j}]*_{e} \phi\left(\alpha
\ot \beta \ot u^{j}\right)
\\=&\:0,
\end{align*}}womit
$\ker(\pt_{k+1}^{*})=\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})$ und
$\im(\pt_{k}^{*})=0$, also
\begin{equation*}
H^{k}(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{A}))=\ker(\pt_{k+1}^{*})/\im(\pt_{k}^{*})=\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{A}).
\end{equation*}
\end{beweis}
\end{satz}
Ohne zusätzliche Annahmen über $\V$ und
$\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})$ erhalten wir jedoch im
Allgemeinen kein Analogon zu \eqref{eq:HochschPol22}. Es gilt jedoch:
\begin{korollar}
Sei $\V$ ein endlich-dimensionaler $\mathbb{K}$-Vektorraum und
$\mathcal{M}$ ein symmetrischer $\SsV-\SsV$-Bimodul, dann ist:
\begin{equation*}
HH^{k}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong
\mathcal{M}\ot \Lambda^{k}(\V).
\end{equation*}
\begin{beweis}
Dies folgt analog zum zweiten Teil von
Satz \ref{satz:PolsatzHochsch}, da wegen der endlichen Dimension $n$
von $\V$ mit $\V^{*}\cong\V$ ebenfalls
$\Lambda^{k}(\V)^{*}\cong\Lambda^{k}(\V^{*})\cong\Lambda^{k}(\V)$
gilt.
Es folgt dann zunächst für $\phi\in
\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})$, dass
\begin{equation*}
\phi(a^{e}\ot
\omega)=\left(\sum_{j_{1},…,j_{k}}^{n}\phi^{j_{1},…,j_{k}}\ot
e^{j_{1}}\wedge…\wedge e^{j_{k}}\right)(a^{e}\ot \omega)
\end{equation*}mit $\big\{e_{i}\big\}_{1\leq i\leq n}$ eine Basis von
$\V$ und $\left\{e^{i}\right\}_{1\leq i\leq n}$ die hierzu duale Basis
von $\V^{*}$. Dies zeigt
$\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})\cong
\mathcal{M}\ot \Lambda^{k}(\V^{*})\cong \mathcal{M}\ot
\Lambda^{k}(\V)$, wobei die zweite Isomorphie vermöge $\Lambda^{k}(\V^{*})\cong\Lambda^{k}(\V)$ am leichtesten mit einem Basis-Argument und
Bemerkung \ref{bem:TenprodBasis} folgt.
\end{beweis}
\end{korollar}
\subsection{Explizite Kettenabbildungen}
Wir wollen nun explizite Kettenabbildungen für die Bar- und
Koszulauflösung angeben. Seien hierf"ur abstrakte $\mathcal{A}^{e}$-lineare Varianten von \eqref{eq:Gpol} und
\eqref{eq:FPol} gegeben durch:
\begin{equation}
\label{eq:SymF}
\begin{split}
F_{k}\colon\Ss^{\bullet}(\mathbb{V})\ot \Ss^{\bullet}(\mathbb{V})\ot \Lambda^{k}(\mathbb{V})&\longrightarrow \bigotimes^{k+2}\Ss^{\bullet}(\mathbb{V})\\
\KE{\alpha}{\beta}{u}&\longmapsto\sum_{\sigma\in
S_{k}}\mathrm{sign}(\sigma)\: (\alpha\ot
u_{\sigma(1)}\ot…\ot u_{\sigma(k)}\ot\beta)
\end{split}
\end{equation}für $u=u_{1}\wedge\dots\wedge u_{k}$ sowie
\begin{equation}
\label{eq:SymG}
\begin{split}
G_{k}\colon\bigotimes^{k+2}\Ss^{\bullet}(\mathbb{V})&\longrightarrow \Ss^{\bullet}(\mathbb{V})\ot
\Ss^{\bullet}(\mathbb{V})\ot \Lambda^{k}(\mathbb{V})\\
\omega&\longmapsto
\int_{0}^{1}dt_{1}\int_{0}^{t_{1}}dt_{2}…\int_{0}^{t_{k-1}}dt_{k}\:(i\circ\delta)(\omega)
\end{split}
\end{equation}mit $G_{0}=F_{0}=\id_{\mathcal{A}^{e}}$.
Die beteiligten Komponenten sind dabei wie folgt definiert:
\begin{definition}
\label{def:GAbb}
\begin{enumerate}
\item
Seien $\mu=(\alpha\ot
u_{1}\ot…\ot u_{m}\ot\beta\ot \omega)$ und $\nu=(\alpha'\ot
u'_{1}\ot…\ot u'_{m}\ot\beta'\ot \omega')$, so definieren wir das
komponentenweise Produkt
\begin{align*}
\cdot\colon \bigotimes^{m+2}\Ss^{\bullet}(\mathbb{V})\ot \Lambda^{k}(\V)\times
\bigotimes^{m+2}\Ss^{\bullet}(\mathbb{V})\ot \Lambda^{k}(\V)\longrightarrow
\bigotimes^{m+2}\Ss^{\bullet}(\mathbb{V})\ot
\Lambda^{k}(\V)
\end{align*} durch
\begin{align*}
\cdot\colon (\mu,\nu)\longmapsto
\alpha\vee\alpha'\ot u_{1}\vee u'_{1}\ot…\ot u_{m}\vee u'_{m}\ot
\beta\vee \beta'\ot \omega\wedge \omega'.
\end{align*}
Im Spezialfall $m=0$ stimmt dieses mit unserer alten Definition
überein. Sinngemäß sei diese Abbildung auch für $k=0$ definiert.
\item
\begin{align*}
\hat{\circ}\colon\bigotimes^{l+2}\Ss^{\bullet}(\mathbb{V})\ot \Lambda^{\bullet}(\V)\times\bigotimes^{l'+2}\Ss^{\bullet}(\mathbb{V})\ot
\Lambda^{\bullet}(\V)&\longrightarrow
\bigotimes^{l+l'+2}\Ss^{\bullet}(\mathbb{V})\ot \Lambda^{\bullet}(\V)\\
\Big((\alpha\ot \ovl{u}\ot \beta\ot \omega), (\alpha'\ot \ovl{u}'\ot
\beta'\ot \omega')\Big)&\longmapsto \alpha\vee \alpha'\ot \ovl{u}\ot
\ovl{u}'\ot \beta\vee \beta'\ot \omega\wedge \omega'.
\end{align*}
\item
\begin{align*}
i\colon\bigotimes^{k+2}\Ss^{\bullet}(\mathbb{V})\ot
\Lambda^{\bullet}(\mathbb{V})&\longrightarrow
\Big[\Ss^{\bullet}(\V)\ot \Ss^{\bullet}(\V)\ot
\Lambda^{\bullet}(\V)\Big] \big[ t_{1},…,t_{k}\big]\\
\alpha\ot u_{1}\ot…\ot u_{k}\ot \beta \ot\omega&\longmapsto
(\alpha\ot\beta)*_{e} \left[\prod_{s=1}^{k}\hat{i}_{s}(1\ot u_{s}\ot1)\right]\ot\omega
\end{align*}mit $\prod$ das Produkt $\cdot$ für den Spezialfall
$k=0$ und $\hat{i}_{s}$ die $\mathcal{A}^{e}$-lineare
Abbildung
\begin{align*}
\hat{i}_{s}\colon\bigotimes^{3}\Ss^{\bullet}(\mathbb{V}) \longrightarrow&\: \Big[\bigotimes^{2}\Ss^{\bullet}(\mathbb{V})\Big]\big[ t_{s}\big]\\
\alpha\ot u\ot\beta\longmapsto&\: t_{s}^{m}u\vee\alpha\ot\beta\:+\:
t_{s}^{m-1}(1-t_{s})\sum_{j=1}^{m}u^{j}\vee\alpha\ot
u_{j}\vee\beta\:+…\\
&+
t_{s}^{m-l}(1-t_{s})^{l}\sum_{j_{1},…,j_{l}}^{m}u^{j_{1},…,j_{l}}\vee
\alpha\ot u_{j_{1},…,j_{l}}\vee\beta\:+\:…\:\\ &+
(1-t_{s})^{m}\alpha\ot u\vee\beta,
\end{align*} mit $\deg(u)=m$ und $\hat{i}_{s}(\alpha\ot 1\ot
\beta)=\alpha \ot \beta$.\\\\
Für Elemente $\alpha\ot u \ot\beta\ot \omega$ schreiben wir im
Folgenden auch $i_{s}(\alpha\ot u \ot\beta\ot \omega)$ anstelle
$i(\alpha\ot u \ot\beta \ot\omega)$, um zu verdeutlichen, dass
das Bild dieses Elementes nur von einer Variablen $t_{s}$
abhängt. Ist es an gegebener Stelle angebracht, so schreiben
wir der Deutlichkeit halber auch $i_{t_{1},…,t_{k}}$ anstatt $i$.
\item
\begin{align*}
\delta\colon\bigotimes^{\bullet+2}\Ss^{\bullet}(\mathbb{V})\longrightarrow&\:
\bigotimes^{\bullet+2}\Ss^{\bullet}(\mathbb{V})\ot \Lambda^{\bullet}(\mathbb{V})\\
\alpha\ot u_{1}\ot…\ot u_{k}\ot \beta\longmapsto&\:
\sum_{j_{1}}^{n_{1}}…\sum_{j_{k}}^{n_{k}}\alpha\ot
u_{1}^{j_{1}}\ot…\ot u_{k}^{j_{k}}\ot \beta\ot\:
(u_{1})_{j_{1}}\wedge…\wedge (u_{k})_{j_{k}}\\ \longmapsto &\:
(\alpha\ot\beta)*_{e}\widehat{\bigodot}_{s=1}^{k}\wt{\delta}(1\ot u_{s}\ot1)
\end{align*}
mit $\deg(u_{i})=n_{i}$ und $\widehat{\bigodot}$
das Produkt $\hat{\circ}$. Dabei bezeichnet $\wt{\delta}$ die $\mathcal{A}^{e}$-lineare Abbildung
\begin{align*}
\wt{\delta}\colon\Ss^{\bullet}(\mathbb{V})\ot \Ss^{\bullet}(\mathbb{V})\ot
\Ss^{\bullet}(\mathbb{V})&\longrightarrow \Ss^{\bullet}\ot
\Ss^{\bullet-1}(\mathbb{V})\ot \Ss^{\bullet}(\mathbb{V})\ot \Lambda^{1}(\mathbb{V})\\
\alpha\ot u\ot\beta&\longmapsto \sum_{j=1}^{k}\alpha\ot
u^{j}\ot\beta\ot u_{j}
\end{align*} mit $\deg(u)=k\:$
und $\:\wt{\delta}(\alpha\ot 1\ot \beta)=0$.
\end{enumerate}
\end{definition}
\begin{bemerkung}
\begin{enumerate}
\item
Der Wohldefiniertheit wegen sei angemerkt, dass auch hier die beteiligten Komponenten als Einschr"ankungen von
symmetrisierten und antisymmetrisierten Abbildungen auf die
jeweiligen Unterr"aume geschrieben werden k"onnen. Besagte
Abbildungen werden in Kapitel \ref{sec:SvonV} nachgeliefert, da wir sie dort
auch explizit ben"otigen.
\item
In dem Moment, in dem wir die Kettenabbildungs-Eigenschaft von $F$ und $G$
nachgewiesen haben zeigt der Beweis von
Lemma \ref{lemma:GruppenKOhomsausprojaufloesundFunktoren}~\textit{ii.)},
dass $F^{*}$ und $G^{*}$
zueinander inverse Isomorphismen $\wt{F^{*}}$ und $\wt{G^{*}}$ auf
Kohomologie-Niveau induzieren.
\end{enumerate}
\end{bemerkung}
Der zweite Teil des folgenden Lemmas liefert uns die Kettenabbildungs-Eigenschaft von
$F$. Die Bedeutung des ersten Teils wird am Ende
dieses Kapitels klar werden.
\begin{lemma}
\label{lemma:Fkettenabb}
Es gilt:
\begin{enumerate}
\item
$G_{k}\cp F_{k}= \id_{\K_{k}}$
\item
$d_{k}\circ F_{k}=F_{k-1}\circ \partial_{k}$.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Mit $\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}=\frac{1}{k!}$
und $\alpha\ot\beta\ot u \in \K_{k}$
folgt:
{\allowdisplaybreaks\small\begin{align*}
(G_{k}\circ F_{k})(\alpha\ot\beta\ot
u)
=&\:\int_{0}^{1}dt…\int_{0}^{t_{k-1}}dt_{k}\:(i\cp\delta)(F_{k}(\alpha\ot
\beta\ot u))
\\=&\int_{0}^{1}dt…\int_{0}^{t_{k-1}}dt_{k}\:(i\cp\delta)\left(\sum_{\sigma\in
S_{k}}\mathrm{sign}(\sigma)(\alpha\ot u_{\sigma(1)}\ot…\ot
u_{\sigma(k)}\ot\beta)\right)
\\=&\int_{0}^{1}dt…\int_{0}^{t_{k-1}}dt_{k}\:i\left(\sum_{\sigma\in
S_{k}}\mathrm{sign}(\sigma)(\alpha\ot\beta\ot u_{\sigma(1)}\wedge…\wedge
u_{\sigma(k)})\right)
\\=& \int_{0}^{1}dt…\int_{0}^{t_{k-1}}dt_{k}\:i\left(\sum_{\sigma\in
S_{k}}\mathrm{sign}(\sigma)\:\mathrm{sign}(\sigma)(\alpha\ot\beta\ot
u)\right)
\\=&\: k! \int_{0}^{1}dt…\int_{0}^{t_{k-1}}dt_{k}\:(\alpha\ot\beta\ot
u)
\\=&\: (\alpha\ot\beta\ot u).
\end{align*}}
\item
Sei zunächst $\mu=\alpha\ot \beta\ot \mathrm{v}$ mit
$\deg(\mathrm{v})=1$, dann folgt mit $F_{0}=\id_{\mathcal{A}^{e}}$
\begin{align*}
(d_{1}\cp F_{1})(\alpha\ot\beta\ot \mathrm{v})=\alpha\vee \mathrm{v}\ot\beta
-\alpha\ot \mathrm{v}\vee \beta=(F_{0}\cp \pt_{1})(\alpha\ot\beta\ot \mathrm{v}),
\end{align*}und für $k>1$
erhalten wir
{\small\allowdisplaybreaks
\begin{align*}
(d_{k}\circ F_{k})(\mu)=&\:(-1)^{0}\sum_{\sigma\in
S_{k}}\mathrm{sign}(\sigma)\:
(\alpha\vee u_{\sigma(1)}\ot u_{\sigma(2)}\ot…\ot u_{\sigma(k)}\ot\beta)\\ &+ (-1)^{k}\sum_{\sigma\in
S_{k}}\mathrm{sign}(\sigma)\:
(\alpha\ot u_{\sigma(1)}\ot…\ot
u_{\sigma(k-1)} \ot
u_{\sigma(k)}\vee\beta)\\ &+\:
\underbrace{\sum_{j=1}^{k-1}(-1)^{j}\sum_{\sigma\in
S_{k}}\mathrm{sign}(\sigma)\:(\alpha\ot
u_{\sigma(1)}\ot…\ot u_{\sigma(j)}\vee
u_{\sigma(j+1)}\ot…\ot
u_{\sigma(k)}\ot\beta}_{0})
\\= &\:\sum_{j=1}^{k}\sum_{\substack{\sigma\in
S_{k}\\\sigma(1)=j}}\mathrm{sign}(\sigma)\:
(\alpha\vee u_{j}\ot u_{\sigma(2)}\ot…\ot
u_{\sigma(k)}\ot\beta)\\ &\quad+ (-1)^{k}\sum_{j=1}^{k}\sum_{\substack{\sigma\in
S_{k}\\\sigma(k)=j}}\mathrm{sign}(\sigma)\:
(\alpha\ot u_{\sigma(1)}\ot…\ot u_{\sigma(k-1)}\ot
u_{j}\vee\beta) \\ = &\:
\sum_{j=1}^{k}\:\mathrm{sign}(\pi_{1\shortleftarrow j})\sum_{\sigma\in
S_{k-1}}\mathrm{sign}(\sigma)\:
(\alpha\vee u_{j}\ot\sigma^{*}u^{j}\ot\beta)\\ &\quad+ (-1)^{k}\sum_{j=1}^{k}\mathrm{sign}(\pi_{j\shortrightarrow k})\sum_{\sigma\in
S_{k-1}}\mathrm{sign}(\sigma)\:
(\alpha\ot \sigma^{*}u^{j}\ot
u_{j}\vee\beta).
\end{align*}}Dabei bedeutet $\sigma^{*}u^{j}$ lediglich die Permutation
$\sigma$, angewandt auf das Element
\begin{equation*}
\Tt^{k-1}(\V)\ni u^{j}=u_{1}\ot…\blacktriangle^{j}…\ot u_{k}.
\end{equation*}
$\pi_{1\shortleftarrow j}$ bezeichnet die Permutation, die
$u_{j}$ sukzessive durch Transpositionen an die erste Stelle schiebt, sinngemäß für $\pi_{j\shortrightarrow
k}$. Es folgt
{\allowdisplaybreaks\small
\begin{align*}
(d_{k}\circ F_{k})(\mu)=
&\:\sum_{j=1}^{k}(-1)^{j-1}\sum_{\sigma\in
S_{k-1}}\mathrm{sign}(\sigma)\:
(\alpha\vee u_{j}\ot\sigma^{*}u^{j}\ot\beta)\\ &+ \sum_{j=1}^{k}(-1)^{k}(-1)^{k-j}\sum_{\sigma\in
S_{k-1}}\mathrm{sign}(\sigma)\:
(\alpha\ot \sigma^{*}u^{j}\ot
u_{j}\vee\beta)\\ =
&\:\sum_{j=1}^{k}(-1)^{j-1}\sum_{\sigma\in
S_{k-1}}\mathrm{sign}(\sigma)\:
(\alpha\vee u_{j}\ot\sigma^{*}u^{j}\ot\beta)\\ &\quad-\sum_{j=1}^{k}(-1)^{j-1}\sum_{\sigma\in
S_{k-1}}\mathrm{sign}(\sigma)\:
(\alpha\ot \sigma^{*}u^{j}\ot
u_{j}\vee\beta)\\ =&\:
F_{k-1}\left(\sum_{j=1}^{k}(-1)^{j-1}\Big[u_{j}\vee\alpha\ot\beta\ot
u^{j}-\alpha\ot u_{j}\vee\beta\ot
u^{j}\Big]\right)\\ = &\: (F_{k-1}\circ\partial_{k})(\mu).
\end{align*}}
\end{enumerate}
\end{beweis}
\end{lemma}
Für die Kettenabbildungs-Eigenschaft von $G$ benötigen wir zunächst
einige Rechen-regeln.
\begin{lemma}
\begin{enumerate}
\item
\begin{align}
\label{eq:faktIs}
\hat{i}_{s}(v\cdot w)&=\hat{i}_{s}(v)\cdot \hat{i}_{s}(w) \\
\label{eq:InormalesProdFaktor}
i(\mu\cdot\nu)&=i(\mu)\cdot i(\nu),
\end{align} für $v,w \in \bigotimes^{3}\Ss^{\bullet}(\mathbb{V})$
sowie $\mu,\nu\in \displaystyle\bigotimes^{m+2}\Ss^{\bullet}(\V)\ot \Lambda^{\bullet}$.
\item
\begin{equation}
\label{eq:DeltaDerivaufuElem}
\delta(v\cdot w)=\delta(v)\cdot w\ot1 +
v\ot 1\cdot\delta(w),
\end{equation} für $v,w \in \bigotimes^{3}\Ss^{\bullet}(\mathbb{V})$.
\item
\begin{align}
\label{eq:iFaktor}
i(\mu\:\hat{\circ}\:\nu)&=i(\mu)\cdot i(\nu)\\
\label{eq:deltaFakt}
\delta(\mu\:\hat{\circ}\:\nu)&=\delta(\mu)\:\hat{\circ}\:\delta(\nu)
\end{align}
\item
\begin{align}
\label{eq:tAbli}
\frac{d}{ds} \hat{i}_{s}(\alpha\ot u\ot\beta)&=(\partial_{1}
\cp i_{s}\circ \delta)(\alpha\ot u\ot \beta),\\
\label{eq:partialmultisuperderiv}
\pt_{k}\left[\prod_{i=1}^{k}(1\ot1\ot u_{i})\right]&=\sum_{j=1}^{k}(-1)^{j-1}\pt_{1}(1\ot
1\ot u_{j})\cdot
\prod_{i\neq j}(1\ot 1\ot u_{i})
\end{align} für $u_{i}\in \Lambda^{1}(\mathbb{V})$.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
\eqref{eq:faktIs} folgt wie
Lemma \ref{lemma:EigenschHomotBaukloetze}~\textit{iv)} mit der
Kommutativität von $\vee$.
Für \eqref{eq:InormalesProdFaktor} seien
$\mu=(1\ot u_{1}\ot…\ot u_{m}\ot1\wedge \omega)$ und $\nu=(1\ot
u'_{1}\ot…\ot u'_{m}\ot 1\wedge \omega')$, dann erhalten
wi
\begin{align*}
i(\mu\cdot\nu)=&
\:\left[\prod_{s=1}^{m}\hat{i}_{s}(1\ot u_{s}\vee u'_{s}\ot
1)\right]\ot
\omega\wedge\omega'\\
=& \:\left[\prod_{s=1}^{m}\hat{i}_{s}(1\ot u_{s}\ot 1)\cdot \hat{i}_{s}(1\ot u'_{s}\ot
1)\right]\ot
\omega\wedge\omega'\\
=& \left(\left[\prod_{s=1}^{m}\hat{i}_{s}(1\ot u_{s}\ot 1)\right]\ot
\omega \right)\cdot \left(\left[\prod_{s=1}^{m}\hat{i}_{s}(1\ot u'_{s}\ot 1)\right]\ot
\omega' \right)\\
= &\: i(\mu)\cdot i(\nu).
\end{align*}
\item
Wir rechnen
{\small\allowdisplaybreaks\begin{align*}
\delta\big[(1\ot u\ot1)&\:\cdot(1\ot u'\ot1)\big]
\\=&\:\sum_{j=1}^{m+m'}\left(1\ot\: [u\vee u']^{j}\ot1\ot\: [u\vee u']_{j}\right)
\\=&\:\sum_{j=1}^{m}(1\ot u^{j}\ot1\ot u_{j})\cdot(1\ot u'\ot1\ot 1)+(1\ot
u\ot1\ot 1)\cdot\sum_{j=1}^{m'}(1\ot u'^{j}\ot1\ot u'_{j})
\\=&\:\delta(1\ot u\ot1)\cdot (1\ot u'\ot1\ot 1)+(1\ot u\ot1\ot 1) \cdot
\delta(1\ot u'\ot1).
\end{align*}}Zusammen mit der
$\mathcal{A}^{e}$-Linearität von $\delta$ zeigt dies \eqref{eq:DeltaDerivaufuElem}.
\item
Für \eqref{eq:iFaktor} seien
$\mu=(\alpha\ot\ovl{u}\ot \beta\ot \omega)$ und
$\nu=(\alpha'\ot\ovl{u}'\ot \beta'\ot\omega')$,
dann folgt:
{\small\allowdisplaybreaks
\begin{align*}
i(\mu\:
\hat{\cp}\:\nu)=&\:(\alpha\vee\alpha'\ot\beta\vee\beta')*_{e}\left[\prod_{s=1}^{m+m'}\hat{i}_{s}(1\ot\:(\ovl{u}\ot\ovl{u}')_{s}\ot
1)\right]\ot\:
\omega\wedge\omega'\\
=&\:(\alpha\ot\beta)*_{e} (\alpha'\ot\beta')*_{e}\left[\prod_{s=1}^{m}\hat{i}_{s}(1\ot\:\ovl{u}_{s}\ot
1)\cdot \prod_{s=1}^{m'}\hat{i}_{s}(1\ot\:\ovl{u}'_{s}\ot
1)\right]\ot\:
\omega\wedge\omega'\\
=& \left((\alpha\ot
\beta)*_{e}\left[\prod_{s=1}^{m}\hat{i}_{s}(1\ot\ovl{u}\ot1)\right]\ot \omega\right)
\cdot \left((\alpha'\ot
\beta')*_{e}\left[\prod_{s=1}^{m'}\hat{i}_{s}(1\ot\ovl{u}'\ot1)\right]\ot
\omega'\right)\\ =&\: i(\mu)\cdot i(\nu).
\end{align*}}
\eqref{eq:deltaFakt} folgt mit $\mu=(\alpha\ot\ovl{u}\ot \beta)$ und
$\nu=(\alpha'\ot\ovl{u}'\ot \beta')$ analog zu \eqref{eq:iFaktor} f"ur $\widehat{\bigodot}$ anstelle von $\prod$ , $i$ anstelle $\delta$ und $\wt{\delta}$ anstelle $i_{s}$:
{\small\allowdisplaybreaks
\begin{align*}
\delta(\mu\:
\hat{\cp}\:\nu)=&\:(\alpha\vee\alpha'\ot\beta\vee\beta')*_{e}\left[\widehat{\bigodot}_{l=1}^{m+m'}\wt{\delta}(1\ot\:(\ovl{u}\ot\ovl{u}')_{l}\ot
1)\right]\\
=&\:(\alpha\ot\beta)*_{e} (\alpha'\ot\beta')*_{e}\left[\widehat{\bigodot}_{l=1}^{m}\wt{\delta}(1\ot\:\ovl{u}_{l}\ot
1)\:\hat{\circ}\widehat{\bigodot}_{l=1}^{m'}\wt{\delta}(1\ot\:\ovl{u}'_{l}\ot
1)\right]\\
=& \left((\alpha\ot
\beta)*_{e}\left[\widehat{\bigodot}_{l=1}^{m}\wt{\delta}(1\ot\ovl{u}\ot1)\right]\right)\hat{\circ} \left((\alpha'\ot
\beta')*_{e}\left[\widehat{\bigodot}_{l=1}^{m'}\wt{\delta}(1\ot\ovl{u}'\ot1)\right]\right)\\ =&\: \delta(\mu)\:\hat{\circ}\: \delta(\nu).
\end{align*}}
\item
F"ur \eqref{eq:tAbli} erhalten wir analog zu
Lemma \ref{lemma:EigenschHomotBaukloetze}~\textit{iv)}:
{\begin{equation*}
\frac{d}{ds}\hat{i}_{s}(\alpha\ot 1\ot \beta)=0=(\pt_{1}\cp
i_{s}\cp \delta)(\alpha\ot1\ot \beta)
\end{equation*}}sowie
{\begin{align*}
\frac{d}{ds}\hat{i}_{s}(\alpha\ot \mathrm{v}
\ot\beta)=&\:\frac{d}{ds}\big[s(\mathrm{v}\vee\alpha\ot1)+(1-s)(1\ot\mathrm{v}\vee\beta)\big]
\\=&\: (\mathrm{v}\vee\alpha\ot1-1\ot\mathrm{v}\vee\beta)
\\=&\:\pt_{1}\:(\alpha\ot\beta\ot\mathrm{v} )
\\=&\:(\pt_{1}\cp i_{s})(\alpha\ot1\ot\beta\ot\mathrm{v})
\\=&\:(\pt_{1}\cp i_{s}\cp\delta)(\alpha\ot\mathrm{v}\ot\beta),
\end{align*}}und mit der
$\mathcal{A}^{e}$-Linearität beider Seiten induktiv:
{\allowdisplaybreaks
\begin{align*}
\frac{d}{ds}\hat{i}_{s}&(1\ot \mathrm{v}\vee u\ot
1)=\:\frac{d}{ds}\big[\hat{i}_{s}(1\ot \mathrm{v}\ot 1)\cdot
\hat{i}_{s}(1\ot u\ot 1)\big]\\
=&\:\frac{d}{ds}\hat{i}_{s}(1\ot \mathrm{v}\ot 1)\cdot
\hat{i}_{s}(1\ot u\ot 1)+\hat{i}_{s}(1\ot \mathrm{v}\ot 1)\cdot
\frac{d}{ds}\hat{i}_{s}(1\ot u\ot 1)
\\ =&\:(\pt_{1}\cp
i_{s}\cp \delta)(1\ot \mathrm{v}\ot 1)\cdot \hat{i}_{s}(1\ot u\ot 1)+\hat{i}_{s}(1\ot \mathrm{v}\ot 1)\cdot(\pt_{1}\cp
i_{s}\cp \delta)(1\ot u\ot 1)\\
=& \:\pt_{1}\Big[(i_{s}\cp \delta)(1\ot \mathrm{v}\ot
1)\cdot i_{s}(1\ot u\ot 1\ot 1) +i_{s}(1\ot \mathrm{v}\ot 1\ot 1)\cdot( i_{s}\cp
\delta)(1\ot u\ot 1)\Big]\\
=&\:(\pt_{1}\cp i_{s})\Big[\delta(1\ot
\mathrm{v}\ot 1)\cdot(1\ot u\ot 1\ot1)+(1\ot \mathrm{v}\ot 1\ot1)\cdot\delta\:(1\ot u\ot 1)\Big]
\\=&\: (\pt_{1}\cp i_{s}\cp\delta)(1\ot\mathrm{v}\vee u\ot1).
\end{align*}}
\eqref{eq:partialmultisuperderiv} erhält man mit
{\allowdisplaybreaks\small\begin{align*}
\pt_{k}\left[\prod_{i=1}^{k}(1\ot 1\ot u_{i}
)\right]=& \sum_{j=1}^{k}(-1)^{j-1}\left[\prod_{i=1}^{j-1}(1\ot 1\ot u_{i})\cdot \pt_{1}(1\ot 1\ot u_{i})\cdot \prod_{i=j+1}^{k}(1\ot 1\ot u_{i})\right]
\\=&\sum_{j=1}^{k}(-1)^{j-1}\pt_{1}(1\ot 1\ot u_{i})\cdot
\prod_{i\neq j}(1\ot 1\ot u_{i}).
\end{align*}}
\end{enumerate}
\end{beweis}
\end{lemma}
Folgende Proposition zeigt schließlich die gewünschte Eigenschaft von
$G$.
\begin{proposition}
\label{prop:GcirFistidundGKettenabb}
Es gilt
\begin{equation*}
\pt_{k}\cp G_{k}=G_{k-1}\cp d_{k}.
\end{equation*}
\begin{beweis}
Wir beginnen mit {\allowdisplaybreaks{\footnotesize
\begin{align*}
(\pt_{k}\cp&\: G_{k})(1\ot\ovl{u}\ot 1)
\\\glna{\substack{\eqref{eq:iFaktor}\\\eqref{eq:deltaFakt}}}&\:
\pt_{k}\left[\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}\:(i_{1}\cp\delta)(1\ot
u_{1}\ot 1)\cdot…\cdot \ck{k}\right]
\\\glna{\eqref{eq:partialmultisuperderiv}}&\:\sum_{j=1}^{k}(-1)^{j-1}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}\:
(\pt_{1}\cp i_{j}\cp\delta)(1\ot u_{j}\ot
1)\cdot\prod_{i\neq j}\ck{i}
\\\glna{\eqref{eq:tAbli}}&\:\sum_{j=1}^{k}\:(-1)^{j-1}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}\:
\frac{d}{dt_{j}}i_{j}(1\ot u_{j}\ot 1\ot1)\cdot\prod_{i\neq
j}\ck{i}
\\=&\:\sum_{j=1}^{k-1}\:\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}\:
\frac{d}{dt_{j}}\left[i_{j}(1\ot u_{j}\ot
1\ot1)\cdot\ck{j+1}\right]\cdot\prod_{\substack{i\neq
j\\ i\neq j+1}}\ck{i}
\\ &+ (-1)^{k-1} \int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}\:
\frac{d}{dt_{k}}i_{k}(1\ot u_{k}\ot 1\ot1)\cdot\prod_{i=1}^{k-1}\ck{i}.
\end{align*}}} Nun folgt durch Anwendung von {\footnotesize$\displaystyle\int_{0}^{t_{j-1}}dt_{j}$} auf {\footnotesize
\begin{align*}
\int_{0}^{t_{j}}dt_{j+1}\frac{d}{dt_{j}}f(t_{j},t_{j+1})=\frac{d}{dt_{j}}\left[\int_{0}^{t_{j}}dt_{j+1}f(t_{j},t_{j+1})\right]-f(t_{j},t_{j}),
\end{align*}} dass{\footnotesize
\begin{equation*}
\int_{0}^{t_{j-1}}dt_{j}\int_{0}^{t_{j}}dt_{j+1}\frac{d}{dt_{j}}f(t_{j},t_{j+1})=\int_{0}^{t_{j-1}}dt_{j+1}f(t_{j-1},t_{j+1})-\int_{0}^{t_{j-1}}dt_{j}f(t_{j},t_{j}),
\end{equation*}} mithin für $2\leq j\leq k-1$: {\footnotesize
\begin{align*}
\int_{0}^{t_{j-1}}dt_{j}\int_{0}^{t_{j}}dt_{j+1}&\:\frac{d}{dt_{j}}\left[i_{j}(1\ot
u_{j}\ot 1\ot 1)\cdot \ck{j+1}\right]\int_{0}^{t_{j+1}}dt_{j+2}…
\\=&\int_{0}^{t_{j-1}}dt_{j+1}\left[i_{j-1}(1\ot
u_{j}\ot 1\ot 1)\cdot \ck{j+1}\right]\int_{0}^{t_{j+1}}dt_{j+2}…\\
&-\int_{0}^{t_{j-1}}dt_{j}\left[i_{j}(1\ot
u_{j}\ot 1\ot 1)\cdot (i_{j}\cp\delta)(1\ot
u_{j+1}\ot 1)\right]\int_{0}^{t_{j}}dt_{j+2}…\:.
\end{align*}}Für $j=1$ gilt nun obige Formel ebenfalls
mit $t_{0}\simeq t_{j-1}=1$, und wir erhalten
{\allowdisplaybreaks {\footnotesize
\begin{align*}
(\pt_{k}\cp G_{k})&(1\ot\ovl{u}\ot 1)
\\=&\: \overbrace{i_{0}(1\ot u_{1}\ot 1\ot
1)\big|_{t_{0}=1}}^{u_{1}\ot1\ot1}\cdot\int_{0}^{1}dt_{2}\int_{0}^{t_{2}}dt_{3}…\int_{0}^{t_{k-1}}dt_{k}\prod_{2\leq
i\leq k}\ck{i}
\\ &- \int_{0}^{1}dt_{1}\:i_{1}(1\ot u_{1}\ot1\ot1)\cdot
(i_{1}\cp \delta)(1\ot u_{2}\ot
1)\cdot\int_{0}^{t_{1}}dt_{3}…\int_{0}^{t_{k-1}}dt_{k}\prod_{3\leq i\leq k}\ck{i}
\\ &+\sum_{j=2}^{k-1}\:(-1)^{j-1}\int_{0}^{1}dt_{1}…\int_{0}^{t_{j-2}}dt_{j-1}\bold{\int_{0}^{t_{j-1}}dt_{j+1}}\int_{0}^{t_{j+1}}dt_{j+2}…\int_{0}^{t_{k-1}}dt_{k}\\
&\quad\quad \left[i_{j-1}(1\ot u_{j}\ot1\ot1)\cdot\prod_{i\neq
j}\ck{i}\right]
\\ &-\sum_{j=2}^{k-1}\:\int_{0}^{1}dt_{1}…\int_{0}^{t_{j-2}}dt_{j-1}\int_{0}^{t_{j-1}}dt_{j}\bold{\int_{0}^{t_{j}}}dt_{j+2}\int_{0}^{t_{j+2}}dt_{j+3}…\int_{0}^{t_{k-1}}dt_{k}
\\ &\quad\quad \left[i_{j}(1\ot u_{j}\ot1\ot1)\cdot
(i_{j}\cp\delta)(1\ot u_{j+1}\ot1)\cdot\prod_{\substack{i\neq
j\\ i\neq j+1}}\ck{i}\right]
\\ &+ (-1)^{k-1}
\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\prod_{i=1}^{k-1}\ck{i}\:\:\cdot
\underbrace{\Big[i_{k}(1\ot u_{k}\ot
1\ot1)\Big]_{0}^{t_{k-1}}}_{i_{k-1}(1\ot
u_{k}\ot1\ot1)-1\ot u_{k}\ot1}.
\end{align*}}Durch Umbenennung der
$t_{j}$-Variablen in jedem Summanden zu $t_{1},…,t_{k-1}$,
folgt: {\allowdisplaybreaks\footnotesize
\begin{align*}
(\pt_{k}&\:\cp G_{k})(1\ot\ovl{u}\ot 1)
\\=& \int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:(i\cp\delta)(u_{1}\ot
u_{2}\ot…\ot u_{k}\ot 1)
\\ &- \int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:i_{1}(1\ot
u_{1}\ot 1\ot 1)\:\cdot \prod_{i=1}^{k-1}(i_{i}\cp\delta)(1\ot u_{i+1}\ot 1)
\\
&+\sum_{j=2}^{k-1}\:(-1)^{j-1}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:\Bigg[i_{j-1}(1\ot
u_{j}\ot1\ot1)\:\cdot\prod_{i=1}^{j-1}\ck{i}\:\cdot
\\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\prod_{i=j}^{k-1}(i_{i}\cp\delta)(1\ot
u_{i+1}\ot1)\Bigg]\displaybreak
\\ & - \sum_{j=2}^{k-1}\:\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:\Bigg[i_{j}(1\ot u_{j}\ot1\ot1)\cdot
(i_{j}\cp\delta)(1\ot
u_{j+1}\ot1)\cdot\prod_{i=1}^{j-1}\ck{i}\cdot
\\ & \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\prod_{i=j+1}^{k-1}(i_{i}\cp\delta)(1\ot
u_{i+1}\ot1)\Bigg]
\\ &+(-1)^{k-1}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:\prod_{i=1}^{k-1}\ck{i}\cdot\:
i_{k-1}(1\ot u_{k}\ot 1\ot1)
\\ &+ (-1)^{k}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:(i\cp\delta)(1\ot
u_{1}\ot…\ot u_{k})
\\=& \int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:(i\cp\delta)(u_{1}\ot
u_{2}\ot…\ot u_{k}\ot 1)
\\ &+ (-1)^{1} \int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:i_{1}(1\ot
u_{1}\ot 1\ot1)\cdot (i_{t_{1},…,t_{k}}\cp\delta)(1\ot u_{2}\ot…\ot u_{k}\ot1)
\\
&-\sum_{j=2}^{k-1}(-1)^{j}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:\left[\prod_{i=1}^{j-1}\ck{i}\cdot\:
i_{j-1}(1\ot
u_{j}\ot1\ot1)\cdot\prod_{i=j}^{k-1}(i_{i}\cp\delta)(1\ot
u_{i+1}\ot1)\right]
\\ & +
\sum_{j=2}^{k-1}(-1)^{j}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:\Bigg[\prod_{i=1}^{j-1}\ck{i}\cdot\:
i_{j}(1\ot u_{j}\ot1\ot1)\cdot \prod_{i=j}^{k-1}(i_{i}\cp\delta)(1\ot
u_{i+1}\ot1)\Bigg]
\\&-(-1)^{k}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:\prod_{i=1}^{k-1}\ck{i}\cdot\:
i_{k-1}(1\ot u_{k}\ot 1\ot1)
\\ &+ (-1)^{k}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:(i\cp\delta)(1\ot
u_{1}\ot…\ot u_{k})
\\=&\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:(i\cp\delta)(u_{1}\ot
u_{2}\ot…\ot u_{k}\ot 1)
\\ & + \sum_{j=1}^{k-1}(-1)^{j}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:\Bigg[\prod_{i=1}^{j-1}\ck{i}\cdot\:
i_{j}(1\ot u_{j}\ot1\ot1)\cdot \prod_{i=j}^{k-1}(i_{i}\cp\delta)(1\ot
u_{i+1}\ot1)\Bigg]
\\ &-\sum_{j=2}^{k}(-1)^{j}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:\left[\prod_{i=1}^{j-1}\ck{i}\cdot\:
i_{j-1}(1\ot
u_{j}\ot1\ot1)\cdot\prod_{i=j}^{k-1}(i_{i}\cp\delta)(1\ot
u_{i+1}\ot1)\right]
\\ & +(-1)^{k}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:(i\cp\delta)(1\ot
u_{1}\ot…\ot u_{k})
\\=&\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:(i\cp\delta)(u_{1}\ot
u_{2}\ot…\ot u_{k}\ot 1)
\\ & + \sum_{j=1}^{k-1}(-1)^{j}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:\Bigg[\prod_{i=1}^{j-1}\ck{i}\cdot\:
i_{j}(1\ot u_{j}\ot1\ot1)\cdot \prod_{i=j}^{k-1}(i_{i}\cp\delta)(1\ot
u_{i+1}\ot1)\Bigg]
\\ &+\sum_{j=1}^{k-1}(-1)^{j}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:\left[\prod_{i=1}^{j}\ck{i}\cdot\:
i_{j}(1\ot
u_{j+1}\ot1\ot1)\cdot\prod_{i=j+1}^{k-1}(i_{i}\cp\delta)(1\ot
u_{i+1}\ot1)\right]
\\ & +(-1)^{k}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:(i\cp\delta)(1\ot
u_{1}\ot…\ot u_{k})
\\=&\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:(i\cp\delta)(u_{1}\ot
u_{2}\ot…\ot u_{k}\ot 1)\displaybreak
\\ & +
\sum_{j=1}^{k-1}(-1)^{j}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:\Bigg[\prod_{i=1}^{j-1}\ck{i}\cdot
\\ &\quad\quad\quad\quad
\left[i_{j}(1\ot u_{j}\ot1\ot1)\cdot(i_{j}\cp\delta)(1\ot
u_{j+1}\ot1)+ \ck{j} \cdot\: i_{j}(1\ot
u_{j+1}\ot1\ot1)\right]\cdot
\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
\prod_{i=j+1}^{k-1}(i_{i}\cp\delta)(1\ot
u_{i+1}\ot1)\Bigg]
\\ & +(-1)^{k}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:(i\cp\delta)(1\ot
u_{1}\ot…\ot u_{k})
\\=& \int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:(i\cp\delta)(u_{1}\ot
u_{2}\ot…\ot u_{k}\ot 1)
\\ & +
\sum_{j=1}^{k-1}(-1)^{j}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:\Bigg[\prod_{i=1}^{j-1}\ck{j}\cdot\:(i_{j}\cp
\delta)(1\ot
u_{j}\vee
u_{j+1}\ot 1)\cdot
\\ &\qquad\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad\qquad \qquad \qquad \qquad \qquad\qquad
\prod_{i=j+1}^{k-1}(i_{i}\cp\delta_{k})(1\ot
u_{i+1}\ot1)\Bigg]
\\ &+ (-1)^{k}\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}\:(i\cp\delta)(1\ot
u_{1}\ot…\ot u_{k})
\\=& \int_{0}^{1}dt_{1}…\int_{0}^{t_{k-2}}dt_{k-1}(i\cp\delta\cp d)(1\ot
\ovl{u}\ot 1)
\\=&\: (G_{k-1}\cp d_{k})(1\ot \ovl{u}\ot 1).
\end{align*}}}Ebenso folgt für den Fall
$\alpha\ot u\ot \beta\in \C_{1}$, dass
\begin{align*}
(\pt_{1}\cp G_{1})(\alpha\ot u\ot \beta)=&\:\int_{0}^{1}dt\: (\pt_{1}\cp i_{t}\cp\delta)(\alpha\ot u\ot \beta)=\:\hat{i}_{t}(\alpha\ot u\ot \beta)\big|_{0}^{1}
\\=&\:d_{1}(\alpha\ot u\ot
\beta)=(G_{0}\cp d_{1})(\alpha\ot u\ot \beta).
\end{align*}
\end{beweis}
\end{proposition}
\begin{bemerkung}[Die Kettenabbildungen $F$ und $G$]
\label{bem:HoimotAbstrIsosUnterk}
Es gibt auch durchaus abstraktere Möglichkeiten, die Hochschild-Kohmologie
der symmetrischen Algebra zu berechnen, vgl.
\cite{cartan.eilenberg:1999a}.
Der von uns beschrittene Weg bietet
jedoch den großen Vorteil, dass uns nun explizite Kettenabbildungen
$F$ und $G$ zwischen Bar- und Koszul-Komplex zur Verfügung
stehen, die zueinander inverse Isomorphismen
\begin{align*}
\wt{F_{k}^{*}}\colon H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}))&\longrightarrow H^{k}(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{M}))\qquad\text{und}\\
\wt{G_{k}^{*}}\colon
H^{k}(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{M}))&\longrightarrow
H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}))
\end{align*}
induzieren. Diese können auf verschiedene Arten nutzbringend
eingesetzt werden:
\begin{enumerate}
\item
Mit obigen
Kettenabbildungen ist es möglich, die Isomorphien
der Ko\-ho\-mo\-lo\-gie-Grup\-pen von Unterkomplexen $(\mathcal{X},d^{*})$ von
$(\C^{*},d^{*})$ und $(\mathit{K},\pt^{*})$ von $(\K^{*},\pt^{*})$
zu zeigen. Dabei ist ein Unterkomplex
$(\mathcal{X},d^{*})$ von $(\C^{*},d^{*})$ ein
Kokettenkomplex derart, dass $\mathcal{X}^{k}\subseteq \C^{*}_{k}$
und $d_{k+1}^{*}: \mathcal{X}^{k}\longrightarrow \mathcal{X}^{k+1}$
gilt. Analog für $(\mathit{K},\pt^{*})$ und $(\K^{*},\pt^{*})$.
Hierfür ist zunächst nachzuweisen, dass
\begin{align*}
F_{k}^{*}\colon \mathcal{X}^{k}&\longrightarrow \mathit{K}^{k}\qquad\text{und}\\
G_{k}^{*}\colon \mathit{K}^{k}&\longrightarrow \mathcal{X}^{k},
\end{align*} also
$\mathrm{F}^{k}=F^{*}_{k}\big|_{\mathcal{X}^{k}}$ gilt und somit $\mathrm{G}^{k}=G_{k}^{*}\big|_{\mathit{K}^{k}}$ wohldefinierte
Kettenabbildungen zwischen besagten Unterkomplexen
sind. Lemma \ref{lemma:Fkettenabb}~\textit{i.)} zeigt dann, dass
\begin{equation*}
F_{k}^{*}\cp
G_{k}^{*}=\hom_{\mathcal{A}^{e}}(\cdot,\mathcal{M})(G_{k}\cp
F_{k})=\id_{\K^{*}_{k}},
\end{equation*}also $\mathrm{F}^{k}\cp
\mathrm{G}^{k}=\id_{\mathrm{K}^{k}}$ und somit
$\wt{\mathrm{F}^{k}}\cp\wt{\mathrm{G}^{k}}=\wt{\mathrm{F}^{k}\cp
\mathrm{G}^{k}}=\id_{H^{k}(\mathrm{K},d^{*})}$ gilt. Hiermit
folgt die
Injektivität von $\wt{\mathrm{G}^{k}}$ und die Surjektivität von
$\wt{\mathrm{F}^{k}}$.
Für die umgekehrte Aussage beachten man, dass
$\mathrm{G}\cp \mathrm{F}\colon
(\mathcal{X},d^{*})\longrightarrow (\mathcal{X},d^{*})$
eine Kettenabbildung ist. Können wir dann $\mathrm{G}\cp
\mathrm{F}\sim\id_{\mathcal{X}^{k}}$ vermöge Homotopieabbildungen $\mathrm{s}^{k}\colon
\mathcal{X}^{k}\longrightarrow\mathcal{X}^{k-1}$ nachweisen, so
zeigt Lemma \ref{lemma:tildeabbeind}, dass
$\wt{\mathrm{G}^{k}}\cp\wt{\mathrm{F}^{k}}=\wt{\mathrm{G}^{k}\cp\mathrm{F}^{k}}=\id_{H^{k}(\mathcal{X},d^{*})}$
gilt. Dies liefert die Surjektivität von $\wt{\mathrm{G}^{k}}$ und die Injektivität von
$\wt{\mathrm{F}^{k}}$, also $H^{k}(\mathcal{X},d^{*})\cong
H^{k}(\mathrm{K},\pt^{*})$ vermöge den zueinander inversen
Isomorphismen $\wt{\mathrm{G}^{k}}$ und
$\wt{\mathrm{F}^{k}}$. Hierbei beachte man, dass die Existenz einer derartigen Homotopie
nicht offensichtlich ist. In der Tat besagt zwar
Satz \ref{satz:AufluProjKompKettab}, dass $F\cp G\sim
\id_{\C_{k}}$ vermöge $\mathcal{A}^{e}$-linearen Homotopieabbildungen
$s_{k}\colon \C_{k}\longrightarrow \C_{k+1}$ und somit $G^{*}\cp F^{*}\sim
\id_{\K^{*}_{k}}$ vermöge $s_{k}^{*}$. Jedoch ist in keiner Weise
gewährleistet, dass sich die $s_{k}^{*}$ auf Abbildungen zwischen
den Unterkomplexen einschränken lassen und die gewünschte Homotopie
$\mathrm{s}$ liefern.
Ist man schließlich an den Kohomologie-Gruppen eines speziellen Unterkom-plexes
$HC_{\circ}(\SsV,\mathcal{M})$ von
$HC(\SsV,\mathcal{M})$ interessiert, so muss jetzt nur noch
sichergestellt werden, dass sich der Kettenisomorphismus $\Xi$ auf eine Isomorphismus
zwischen $(\mathcal{X},d^{*})$ und $HC_{\circ}(\SsV,\mathcal{M})$
einschränken lässt. Ein essentielles Beispiel ist hierbei der
stetige Hochschild-Komplex $HC_{\operatorname{\mathrm{cont}}}(\SsV,\mathcal{M})$,
den wir im nächsten Kapitel kennen lernen werden.
\item
Obige Kettenabbildungen erlauben es, tiefere Erkenntnisse
über die Natur von $HH^{k}(\SsV,\mathcal{M})$ und
$HH^{k}_{\operatorname{\mathrm{cont}}}(\SsV,\mathcal{M})$ zu gewinnen. Im Falle
symmetrischer Bimoduln beispielsweise erhalten wir Analoga zu dem bekannten
Hochschild-Kostant-Rosenberg-Theorem\footnote{vgl. \cite[Prop
6.2.48]{waldmann:2007a}, \cite{cahen.gutt.dewilde:1980a}} für
$HH^{k}(\SsV,\mathcal{M})$ und
$HH^{k}_{\operatorname{\mathrm{cont}}}(\SsV,\mathcal{M})$, siehe Kapitel
\ref{sec:HKRTheos}. Auch für die Klasse der differentiellen
Bimoduln, welche die symmetrischen als Spezialfall enthalten, lassen sich
mit Hilfe der Abbildung $G$ ähnliche Aussagen ableiten, siehe
Kapitel \ref{cha:DiffHochK}. Selbst für den Fall, dass in der
Situation von \textit{i.)} nicht klar ist, dass eine Homotopie $\mathrm{s}$
existiert, sind $\wt{\mathrm{G}^{k}}$ injektiv und
$\wt{\mathrm{F}^{k}}$ surjektiv und liefern nützliche
Informationen über $HH^{k}_{\circ}(\SsV,\mathcal{M})$. Dies
wird beispielsweise für den differentiellen
$HH^{k}_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})$ und den
stetig-differentiellen Unterkomplex
$HH^{k}_{\mathrm{c,d}}(\SsV,\mathcal{M})$ der Fall sein, welchen
wir in Kapitel \ref{cha:DiffHochK} begegnen werden.
\item
Die Kettenabbildungen $F$ und $G$ werden es uns erlauben, die
Hochschild-Koho-mologie des stetigen Unterkomplexes
$HC_{\operatorname{\mathrm{cont}}}(\SsV,\mathcal{M})$, den wir im nächsten Kapitel
für beliebige lokalkonvexe Algebren definieren werden, zu berechnen. Hieraus erhalten
wir die Hochschild-Kohomologien
$HH_{\operatorname{\mathrm{cont}}}^{k}(\Hol,\mathcal{M})$ des stetigen Unterkomplexes
$HC_{\operatorname{\mathrm{cont}}}(\Hol,\mathcal{M})$ für vollständige
lokalkonvexe\footnote{vgl. Kapitel \ref{cha:TopKompl}}
Bimoduln $\mathcal{M}$. Hierbei ist $(\Hol,*)$, die durch
Vervollständigung von $(\SsV,\vee)$ erhaltene Algebra (vgl.
Kapitel \ref{sec:StetHKHol}), für welche wir ebenfalls ein
Hochschild-Kostant-Rosenberg-Theorem erhalten werden. Hierfür beachte man, dass die Berechnung der Hochschild-Kohomologie des Kokettenkomplexes
$HH^{k}(\Hol,\mathcal{M})$ ein im Allgemeinen schwieriges
Problem darstellt, die in Rahmen der Deformationsquantisierung
weitaus interessantere, stetige Hochschild-Kohomologie mit dem
hier gewählten Zugang aber ausgesprochen einfach zu erhalten ist.
\end{enumerate}
\end{bemerkung}
\chapter{Topologische Komplexe und stetige Hochschild-Kohomologien}
\label{cha:TopKompl}
Wie bereits in Bemerkung \ref{bem:HoimotAbstrIsosUnterk} erwähnt, wollen wir
in diesem Kapitel die stetigen Hochschild-Kohomologien
$HC_{\operatorname{\mathrm{cont}}}(\SsV,\mathcal{M})$ und $HC_{\operatorname{\mathrm{cont}}}(\Hol,\mathcal{M})$ der
lokalkonvex topologisierten Algebren $\SsV$ und $\Hol$ für
lokalkonvexe Bimoduln berechnen.
Hierbei heißt ein $\mathbb{K}$-Vektorraum lokalkonvex, wenn er ein topologischer
Vektorraum ist (Vektorraumoperationen sind stetig) und
seine Topologie durch ein Halbnormensystem $P$ erzeugt wird. Dabei
soll $P$ im Folgenden immer als filtrierend\footnote{vgl. Definition \ref{def:Halbnormensysteme}~\textit{ii.)}} voraussetzen werden. Eine
lokalkonvexe $\mathbb{K}$-Algebra ist dann ein lokalkonvexer
$\mathbb{K}$-Vektorraum mit stetiger Algebramultiplikation. Dies ist
gleichbedeutend damit (vgl. Satz \ref{satz:stetmultabb}), dass für jede
Halbnorm $q\in P$ eine Konstante
$c_{*}>0$ und Halbnormen $p_{*1},p_{*2}\in P$ derart existieren, dass für
alle $a,b\in \mathcal{A}$ die Abschätzung $p(a*b)\leq c_{*}\:
p_{*1}(v)\:p_{*2}(w)$ erfüllt ist. Sei $\mathcal{A}$ lokalkonvex, so
verstehen wir unter einem lokalkonvexen
$\mathcal{A}-\mathcal{A}$-Bimodul $(\mathcal{M},*_{L},*_{R})$ einen
lokalkonvexen Vektorraum $(\mathcal{M},Q)$ mit stetigen, $\mathbb{K}$-bilinearen
Modul-Multiplikationen. Dies bedeutet, dass für
jedes $q\in Q$
Konstanten $c_{L},c_{R}>0$ sowie Halbnormen $p_{L},p_{R}\in P$ und
$q_{L},q_{R}\in Q$ existieren, so dass für alle $a\in \mathcal{A}$ und
alle $m\in \mathcal{M}$ die Abschätzungen $q(a*_{L}m)\leq c_{L}\:
p_{L}(a)\:q_{L}(m)$ und $q(m*_{R}a)\leq c_{R}\: p_{R}(a)\:q_{R}(m)$
gelten. Ist $\mathcal{A}$ unitär, so wollen wir wieder
$1_{\mathcal{A}}*_{L}m=m=m*_{R}1_{\mathcal{A}}$ für alle $m\in
\mathcal{M}$ voraussetzen.
Nach Bemerkung \ref{bem:HoimotAbstrIsosUnterk}~\textit{i.)} besteht die
Aufgabe nun zunächst darin, einen
Unterkomplex $(\mathcal{X},d^{*})$ von $(\C^{*},d^{*})$ für
$\mathcal{A}=\SsV$ derart zu
finden, dass $\Xi$ einen Kettenisomorphismus
$HC_{\operatorname{\mathrm{cont}}}(\mathcal{A},\mathcal{M})\longrightarrow (\mathcal{X},d^{*})$
induziert. Dies wird mit Hilfe des folgenden Abschnittes sogar für beliebige
lokalkonvexe Algebren $\mathcal{A}$ erreichbar sein.
\section{Vorbereitung}
\label{subsec:Vorber}
Gegeben eine lokalkonvexe Algebra $(\mathcal{A},*)$ und ein lokal
konvexer $\mathcal{A}-\mathcal{A}$-Bimodul $\mathcal{M}$, so betrachten wir die $\mathbb{K}$-Vektorräume
\begin{equation}
\label{eq:stetHSCHK}
HC_{\operatorname{\mathrm{cont}}}^{k}(\mathcal{A},\mathcal{M}):=
\begin{cases} \{0\} & k<0\\
\mathcal{M} & k=0\\
\Hom^{\operatorname{\mathrm{cont}}}_{\mathbb{K}}(\underbrace{\mathcal{A}\times…\times
\mathcal{A}}_{k-mal},\mathcal{M})& k\geq 1,
\end{cases}
\end{equation} die stetigen $\mathbb{K}$-multilinearen Abbildungen von
$\mathcal{A}^{k}$ nach $\mathcal{M}$. Mit
Satz \ref{satz:stetmultabb} sind dies wieder gerade die Elemente $\phi\in
\Hom_{\mathbb{K}}(\mathcal{A}\times…\times
\mathcal{A},\mathcal{M})$ für welche $q\in Q$
vorgegeben, eine Konstante $c>0$ und Halbnormen
$p_{1},…,p_{k}\in P$ derart existieren, dass\\
\begin{equation}
\label{eq:phiAbsch}
q(\phi(a_{1},…a_{k}))\leq c\:
p_{1}(a_{1})…p_{k}(a_{k})\qquad\qquad\forall\:a_{1},…,a_{k}\in \mathcal{A}
\end{equation}gilt.
Vermöge \eqref{eq:Hochschilddelta} seien $\mathbb{K}$-lineare Abbildungen
\begin{equation*}
\delta^{k}_{c}\colon HC_{\operatorname{\mathrm{cont}}}^{k}(\mathcal{A},\mathcal{M})\longrightarrow HC_{\operatorname{\mathrm{cont}}}^{k+1}(\mathcal{A},\mathcal{M})
\end{equation*} definiert, und es ist zunächst zu zeigen, dass besagtes
Bild unter $\delta^{k}_{c}$ stetig ist.\\ Wir haben:
\begin{align*}
(\delta_{c}^{k}\phi)(a_{1},…,a_{k+1})=a_{1}*_{L}\phi(a_{2},…,a_{k+1})&+\sum_{i=1}^{k}(-1)^{i}\phi(a_{1},…,a_{i}*a_{i+1},…,a_{k+1})\\
&+(-1)^{k+1}\phi(a_{1},…,a_{k})*_{R}a_{k+1}.
\end{align*}
Für die Stetigkeit des ersten Summanden rechnen wir mit den Abschätzungen für $*_{L}$
und $\phi$:
\begin{align*}
q\left(a_{1}*_{L}\phi(a_{2},…,a_{k+1})\right)\leq&
\:c_{L}\:p_{L}(a_{1})\:q_{L}(\phi(a_{2},…,a_{k+1}))
\\\leq&\: \hat{c}\:p_{L}(a_{1})\:p_{2}(a_{2})…p_{k+1}(a_{k+1}).
\end{align*}mit $\hat{c}=c_{L}c$ und $c$, $p_{2},…,p_{k+1}$ die zu
$q_{L}$ gehörigen Halbnormen aus \eqref{eq:phiAbsch}.
Satz \ref{satz:stetmultabb} zeigt dann die Stetigkeit, und die des letzten
Summanden folgt analog. Ebenso erhalten wir für den mittleren
Summanden, dass
\begin{align*}
q(\phi(a_{1},…,a_{i}a_{i+1}&,…,a_{k+1}))
\\\leq&\:c\:
p_{1}(a_{1})…p_{i-1}(a_{i-1})\:p_{i}(a_{i}*a_{i+1})\:p_{i+1}(a_{i+2})…p_{k}(a_{k+1})
\\\leq &
\:\hat{c}\:p_{1}(a_{1})…p_{i-1}(a_{i-1})\:p_{*1}(a_{i})\:p_{*2}(a_{i+1})\:p_{i+1}(a_{i+2})…p_{k}(a_{k+1})
\end{align*}mit $\hat{c}=c_{*}c$ und $p_{i}(a_{i}*a_{i+1})\leq
c_{*}p_{*1}(a_{i})\:p_{*2}(a_{i+1})$. Die Stetigkeit von $\delta_{c}^{k}(\phi)$ folgt
nun unmittelbar mit der Stetigkeit der Vektorraumaddition in
$\mathcal{M}$, da kartesische Produkte stetiger Funktionen bezüglich
den zugehörigen Produkttopologien ebenfalls stetig sind.
Mit $\delta_{c}^{k+1}\cp\delta^{k}_{c}=0$ liefert uns dies einen
Koketten-Unterkomplex $(HC_{\operatorname{\mathrm{cont}}}^{\bullet}(\mathcal{A},\mathcal{M}),\delta_{c})$ von
$(HC^{\bullet}(\mathcal{A},\mathcal{M}),\delta)$ und definieren die
$k$-te stetige Hochschild-Kohomologien durch
\begin{equation*}
HH_{\operatorname{\mathrm{cont}}}^{k}(\mathcal{A},\mathcal{M}):=
\begin{cases}
\ker\left(\delta_{c}^{0}\right) & k=0\\
HH_{\operatorname{\mathrm{cont}}}^{k}(\mathcal{A},\mathcal{M})=\ker\left(\delta_{c}^{k}\right)/\im\left(\delta_{c}^{k-1}\right)&
k\geq 1.
\end{cases}
\end{equation*}
Für die Tensorvariante des Hochschild-Komplexes erhalten wir analoge
Aussagen. Dabei folgt die Stetigkeit des Bildes unter
$\delta_{c\ot}^{k}$ zum einen durch elementare Rechnung, oder aber auch
durch Rechtskomposition \eqref{eq:TensorglKetteniso} mit $\ot_{k}^{*}$, da die $\ot_{k*}$ vermöge
der Definition der $\pi$-Topologie, also insbesondere der Stetigkeit
der Abbildungen $\ot_{k}$, bijektiv stetige auf stetige Elemente abbilden.
Insbesondere bedeutet dies, dass $\ot_{*}$ ein Ketten-isomorphismus
zwischen diesen beiden Unterkomplexen ist, was die Isomorphie derer
Kohomologiegruppen impliziert.
Wir dürfen uns also wieder auf die Tensorvariante des besagten
stetigen Hochschild-Komplexes beschränken, und es soll nun unter anderem darum gehen, die Isomorphie
\begin{equation*}
HH_{\operatorname{\mathrm{cont}}}^{k}(\mathcal{A},\mathcal{M})\cong H^{k}\left(\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}(\C_{c},\mathcal{M}),d_{c}^{*}\right)
\end{equation*}einzusehen. Dabei bezeichnet $(\C_{c},d_{c})$ den topologische
Bar-Komplex, welchen wir bald kennen lernen werden.\\
\begin{bemerkung}
Gegeben lokalkonvexe Vektorräume $(\V_{1},P_{1}),…,(\V_{k},P_{k}),(\mathbb{W},Q)$
und eine stetige $\mathbb{K}$-multilineare
Abbildung $\phi\colon \V_{1}\times…\times\V_{k}\longrightarrow \mathbb{W}$, so sind im Folgenden bei Stetigkeitsabschätzungen
$q(\phi\:(v_{1},…,v_{k}))\leq c\:p(v_{1})…p(v_{k})$ mit $c$ und
$p_{1},…,p_{k}$ immer die nach Satz \ref{satz:stetmultabb} zu $q\in
Q$ gehörige Konstante und die zu $q$ gehörigen Halbnormen
gemeint. Hierfür mache man sich noch einmal explizit klar, dass
wir Halbnormensysteme am Anfang dieses Kapitels immer als
filtrierend vorausgesetzt haben
\end{bemerkung}
Wir wollen an dieser Stelle an die
$\pi_{k}$-Topologie erinnern:
\begin{definition}[$\pi$-Topologie]
Gegeben lokalkonvexe Vektorräume
$(\V_{1},P_{1}),…,(\V_{k},P_{k})$, so ist die $\pi_{k}$-Topologie
die durch das System
$\Pi_{P_{1}\times…\times
P_{k}}$, bestehend aus Halbnormen
\begin{equation*}
p_{1}\ot…\ot p_{k} (z):=\inf\left\{\sum_{i=1}^{n}p_{1}(x^{i}_{1})…p_{k}(x^{i}_{k})\right\},
\end{equation*}auf $\V_{1}\ot…\ot\V_{k}$ induzierte
lokalkonvexe Topologie.
Hierbei ist das Infimum über alle Zerlegungen
$z=\displaystyle\sum_{i=1}^{n}x^{i}_{1}\ot…\ot
x^{i}_{k}$ von $z$ zu nehmen.
Der so gewonnenen lokalkonvexen
Vektorraum sei im Folgenden mit
$\V_{1}\pite…\pite\V_{k}$ bezeichnet. Ist es an
gegebener Stelle der Lesbarkeit zuträglich, so
benutzen wir das Symbol $\pi_{p_{1},…,p_{k}}$ anstelle von
$p_{1}\ot…\ot p_{k}$.
\end{definition}
\begin{bemerkung}
\label{bem:PiTopArBem}
Die $\pi_{k}$-Topologie besitzt die folgenden wichtigen
Eigenschaften, siehe Kapitel \ref{subsec:TenprodLkvVr}:
\begin{enumerate}
\item
Es gilt $p_{1}\ot…\ot
p_{k}\:(x_{1}\ot…\ot x_{k})=p_{1}(x_{1})…p_{k}(x_{k})$ für alle
separablen Elemente
$x_{1}\ot…\ot x_{k}\in \V_{1}\pite…\pite \V_{k}$.
\item
$\pi_{k}$ ist genau dann hausdorffsch, wenn alle $(\V_{i},P_{i})$
mit $1\leq i\leq k$ hausdorffsch sind.
\item
Sind $P_{1},\dots, P_{k}$ filtrierend, so auch $\prod_{P_{1},…,P_{k}}$.
\item
Eine lineare Abbildung $\phi\colon \mathbb{V}_{1} \pite…\pite
\mathbb{V}_{k}\longrightarrow \mathbb{M}$ ist genau dann stetig, wenn die Abbildung $\phi\cp \ot_{k}$ bezüglich der Produkttopologie auf $\mathbb{V}_{1}\times…\times
\mathbb{V}_{k}$ stetig ist, also für jedes $q\in Q$ ein $c>0$ und
$p_{i}\in P_{i}$ mit $1\leq i\leq k$ existieren, so dass
\begin{equation*}
q(\phi(v_{1}\ot…\ot v_{k}))\leq\:c\:p(v_{1})…p(v_{k})\qquad\forall\:v_{i}\in \V_{i},1\leq i\leq k
\end{equation*}gilt.
\end{enumerate}
\end{bemerkung}
\begin{lemma}
\label{lemma:AezuunittopRing}
Gegeben eine assoziative, lokalkonvexe $\mathbb{K}$-Algebra $(\mathcal{A},*)$.
\begin{enumerate}
\item
Dann wird die Menge
$\mathcal{A}^{e}=\mathcal{A}\pite \mathcal{A}$, versehen mit der
distributiven Fortsetzung der Multiplikation
\begin{equation*}
(a\ot b) *_{e} (\tilde{a}\ot
\tilde{b})=(a*\tilde{a})\ot
(b*^{opp}\tilde{b})=(a*\tilde{a})\ot
(\tilde{b}*b)
\end{equation*} auf ganz
$\mathcal{A}\pite \mathcal{A}$, zu einer lokalkonvexen
Algebra. Ist $\mathcal{A}$ unitär, so auch $\mathcal{A}^{e}$.
\item
Jeder lokalkonvexe $\mathcal{A}-\mathcal{A}$-Bimodul $\mathcal{M}$ wird vermöge
\begin{equation*}
a\ot b*_{e} m=a*_{L}(m *_{R}b)=(a*_{L} m)*_{R}b\quad\quad a\ot b\in \mathcal{A}^{e},\: m\in \mathcal{M}
\end{equation*} zu einem lokalkonvexen $\mathcal{A}^{e}$-Linksmodul.
\end{enumerate}
\begin{beweis}
Die Algebra- und Moduleigenschaften folgen wie in Lemma \ref{lemma:AewirdzuunitRing}, die Stetigkeit
der Ringaddition ist die Stetigkeit der Vektorraumaddition in
$(\mathcal{A}^{e},\pi_{2})$ als lkVR und
die Stetigkeit der Algebra-Multiplikation erhalten wir mit der
Stetigkeit von $*$, da
\begin{align*}
p_{1}\ot p_{2}\:(z*_{e}\tilde{z})=\:& p_{1}\ot p_{2}\:\bigg(\sum_{i}a_{i}\ot b_{i}*_{e}\sum_{j}\tilde{a}_{j}\ot \tilde{b}_{j}\bigg)
\leq\: \sum_{i,j}p_{1}\ot p_{2}\:\left(a_{i}*\tilde{a}_{j}\ot
\tilde{b}_{j}*b_{i}\right)
\\=&\: \sum_{i,j}p_{1}\big(a_{i}*\tilde{a}_{j}\big)\:p_{2}\big(b_{i}*\tilde{b}_{j}\big)
\leq\:c\sum_{i,j}p'_{1}\big(a_{i}\big)\:p''_{1}\big(\tilde{a}_{j}\big)\:p'_{2}\big(b_{i}\big)\:p''_{2}\big(\tilde{b}_{j}\big)
\\=&\:c\:\bigg(\sum_{i}p'_{1}(a_{i})p'_{2}(b_{i})\bigg)\bigg(\sum_{j}p''_{1}(\tilde{a}_{j})p''_{2}(\tilde{b}_{j})\bigg)
\end{align*} für alle Zerlegungen von $z,\tilde{z}\in \mathcal{A}^{e}$ und somit
\begin{align*}
\pi_{p,q}\:(z*_{e}\tilde{z})\leq&\: c\: \inf\bigg(\sum_{i}p'_{1}(a_{i})\:p'_{2}(b_{i})\bigg)\:\inf\bigg(\sum_{j}p''_{1}(\wt{a}_{j})\:p''_{2}(\wt{b}_{j})\bigg)
=\:c\: \pi_{p'_{1},p'_{2}}\:(z)\:p''_{1}\ot p''_{2}\:(\tilde{z}).
\end{align*}
Dies zeigt die Stetigkeit von $*$ und folglich \textit{i.)}.\\\\
Für \textit{ii.)} sei $m\in \mathcal{M}$ und $\mathcal{A}^{e}\ni z=
\sum_{i}a_{i}\ot b_{i}$. Dann folgt für alle
Zerlegungen $\sum_{i}a_{i}\ot b_{i}$ von $z$, dass
\begin{align*}
q(z*_{e}m) =q\bigg(\sum_{i}a_{i}(m b_{i})\bigg)
\leq&\:\sum_{i}q(a_{i}(m b_{i}))\leq c\sum_{i}p_{1}(a_{i})\:q'(m
b_{i})
\\\leq&\:\hat{c}\sum_{i}p_{1}(a_{i})\:q''(m)\:p_{2}(b_{i})
=\: \hat{c}\: q''(m)\sum_{i}p_{1}(a_{i})\:p_{2}(b_{i})
\end{align*} und somit $p(zm)\leq \hat{c}\:\pi_{p_{1},p_{2}}(z)\: q''(m)$.
\end{beweis}
\end{lemma}
\begin{definition}[Topologischer Bar-Komplex]
\label{def:topBarkompl}
Gegeben eine lokalkonvexe, assoziative Algebra $\mathcal{A}$, so definieren wir
den topologischen Bar-Komplex $(\C_{c},d_{c})$ durch die
$\mathcal{A}^{e}$-Moduln
\begin{align*}
\C^{c}_{k}=\mathcal{A}\pite \underbrace{\mathcal{A}\pite …
\pite \mathcal{A}}_{k-mal} \pite \mathcal{A}
\end{align*} $\qquad\qquad\qquad\quad \C_{0}=\mathcal{A}\pite \mathcal{A},\quad\quad
\C_{1}=\mathcal{A}\pite\mathcal{A}\pite \mathcal{A},\quad\quad \C_{2}=\mathcal{A}\pite\mathcal{A}\pite \mathcal{A}\pite \mathcal{A}$ \\\\ mit $\mathcal{A}^{e}$-Multiplikation
\begin{equation*}
(a\ot b)(x_{0}\ot x_{1}\ot … \ot x_{k}\ot
x_{k+1}):=(ax_{0})\ot x_{1}\ot … \ot x_{k}\ot
(x_{k+1}b)
\end{equation*} und $\mathcal{A}^{e}$-Homomorphismen
\begin{align*}
d^{c}_{k}\colon\C^{c}_{k}&\longrightarrow \C^{c}_{k-1}\\
(x_{0}\ot … \ot x_{k+1})&\longmapsto
\sum_{j=0}^{k}(-1)^{k}x_{0}\ot…\ot
x_{j}x_{j+1}\ot…\ot x_{k+1}
\end{align*} für $k\geq1$ mit $d^{c}_{k}\cp d^{c}_{k+1}=0$.
\end{definition}
Die relevanten Eigenschaften klärt folgende Proposition:
\begin{proposition}
\label{prop:topBarKomplexprop}
\begin{enumerate}
\item
Die $X^{c}_{k}$ sind lokalkonvexe $\mathcal{A}^{e}$-Moduln.
\item
Die $d^{c}_{k}$ sind stetig, ebenso die
exaktheitsliefernde Homotopieabbildungen:
\begin{equation*}
h^{c}_{k}\colon x_{0}\ot…\ot x_{k+1}\longmapsto 1\ot
x_{0}\ot…\ot x_{k+1}.
\end{equation*}
\item
Ist $\mathcal{A}$ unitär und $\mathcal{M}$ ein lokalkonvexer
$\mathcal{A}-\mathcal{A}$-Bimodul, so gilt:
\begin{equation*}
HH_{\operatorname{\mathrm{cont}}}^{k}(\mathcal{A},\mathcal{M})\cong H^{k}\left(\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}(\C_{c},\mathcal{M}),d^{*}_{c}\right).
\end{equation*}
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Sei $z\in \mathcal{A}^{e}$ und $x\in X^{c}_{k}$, dann folgt:
\begin{align*}
\pi_{p_{0},…,p_{k+1}}&\:(zx)=\pi_{p_{0},…,p_{k+1}}\bigg(\sum_{i}a_{i}\ot
b_{i}\cdot\sum_{j}x_{0}^{j}\ot…\ot x^{j}_{k+1}\bigg)
\\ \leq&\:
\sum_{i,j}p_{0}\big(a_{i}x_{0}^{j}\big)\:p_{k+1}\big(x^{j}_{k+1}b_{i}\big)\:\pi_{p_{1},…,p_{k}}\big(x_{1}^{j}\ot…\ot
x_{k}^{j}\big)
\\\leq&\:
c\sum_{i,j}p'_{0}(a_{i})\:p''_{0}(x_{0}^{j})\:\:p'_{k+1}(x_{k+1}^{j})\:p''_{k+1}(b_{i})\:\:\pi_{p_{1},…,p_{k}}(x_{1}^{j}\ot…\ot
x_{k}^{j})
\\=&\:c\:\sum_{i}p'_{0}(a_{i})\:p''_{k+1}(b_{i})\:\: \sum_{j}
p''_{0}(x_{0}^{j})\:\pi_{p_{1},…,p_{k}}\big(x_{1}^{j}\ot…\ot
x_{k}^{j}\big)\:p'_{k+1}(x_{k+1}^{j}).
\end{align*} Dies zeigt
\begin{equation*}
\pi_{p_{0},…,p_{k+1}}(zx)\leq c\: \pi_{p'_{0},p''_{k+1}}(z)\:\pi_{p''_{0},…,p'_{k+1}}(x)
\end{equation*} und somit die Stetigkeit der
Modul-Multiplikation. Die Stetigkeit der Addition in den
$\C^{c}_{k}$ ist klar, da diese vermöge $\pi_{k+2}$
topologische Vektorräume sind.
\item
Mit der Stetigkeit von $+$ sind wieder Summen stetiger Funktionen stetig und es reicht
daher, die Stetigkeit der Abbildungen:
\begin{equation*}
x_{0}\ot…\ot
x_{k+1}\longmapsto x_{0}\ot…\ot x_{i}x_{i+1}\ot…\ot
x_{k+1}
\end{equation*}
nachzuweisen. Wir erhalten diese mit dem üblichen
Infimums-Argument aus
\begin{align*}
p_{0}\ot…\ot p_{k}\:(x_{0}\ot…\ot x_{i}x_{i+1}&\ot…\ot
x_{k+1})\\ &\leq c\: p_{0}(x_{0})…p'_{i}(x_{i})\:p''_{i}(x_{i+1})…p_{k}(x_{k+1}),
\end{align*}oder unmittelbar mit
Bemerkung \ref{bem:PiTopArBem}. Die Stetigkeit der $h^{c}_{k}$
folgt auf die gleiche
Weise vermöge:
\begin{align*}
p\ot p_{0}\ot…\ot p_{k}\:(1\ot x_{0}\ot x_{k+1}) =
p(1)\: p_{0}(x_{0})…p_{k+1}(x_{k+1}).
\end{align*}
\item
Mit der Stetigkeit der $d_{k}^{c}$ ist sofort
einsichtig, dass in der Tat
\begin{equation*}
d_{k+1}^{c*}\colon
\Hom^{cont}_{\mathcal{A}^{e}}(\C^{c}_{k},\mathcal{M})\longrightarrow \Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}(\C^{c}_{k+1},\mathcal{M})
\end{equation*} und somit $(\C^{*}_{c},d^{*}_{c})$ ein
wohldefinierter Kokettenkomplex ist.
Es bleibt dann lediglich nachzuweisen, dass die Isomorphismen
$\Xi^{k}$ aus
Proposition \ref{prop:barauffuerunitalgebraIsomozuHochschildkohomo},
in beide Richtungen stetige auf stetige Homomorphismen
abbilden. In der Tat sind dann deren Einschränkungen
$\Xi^{k}_{c}=\Xi^{k}|_{HC^{k}_{\operatorname{\mathrm{cont}}}(\mathcal{A},\mathcal{M})}$
ebenfalls Isomorphismen und mit
\begin{equation*}
\Xi^{k+1}_{c}d^{c*}_{k+1}=\delta^{k}_{c}\:\Xi^{k}_{c}
\end{equation*}zudem Kettenabbildungen. Dies zeigt, dass
die $\wt{\Xi^{k}_{c}}$ Isomorphismen sind.
Für $\Xi^{k}$ folgt die gewünschte Eigenschaft mit
stetigem $\psi$ aus:
\begin{align*}
q\left(\Big(\Xi^{k}\psi\Big)(\omega_{k})\right)=&\:
q\big(\psi(1\ot \omega_{k}\ot 1)\big)\leq c\:
\pi_{p_{0},…,p_{k+1}}(1\ot\omega_{k}\ot 1)
\\=&\:\underbrace{c\:
p_{0}(1)\:p_{k+1}(1)}_{\hat{c}}\:\pi_{p_{1},…,p_{k}}(\omega_{k}).
\end{align*}
Umgekehrt erhalten wir für $\phi\in
HC^{k}_{\operatorname{\mathrm{cont}}}(\mathcal{A},\mathcal{M})$ mit Lemma \ref{lemma:AezuunittopRing}~\textit{ii.)}:
\begin{align*}
q(x_{0}\ot x_{k+1}*_{e}\phi(x_{1}\ot…\ot x_{k}))\leq&\: c\:
\pi_{p_{0},p_{k+1}}(x_{0}\ot x_{k+1})\:q'(\phi(x_{1}\ot…\ot x_{k}))
\\\leq& \:\hat{c}\:\prod_{i=0}^{k+1}p_{i}(x_{i}).
\end{align*}
\end{enumerate}
\end{beweis}
\end{proposition}
\section{Die stetige Hochschild-Kohomologie der Algebra
$\Ss^{\bullet}(\mathbb{V})$}
\label{sec:SvonV}
In diesem Abschnitt wollen wir die stetige Hochschild-Kohomologie der Algebra $\SsV$ berechnen. Hierf"ur ist es zun"achst notwendig, diese mit einer lokalkonvexen Topologie derart auszustatten, dass die Algebramultiplikation $\vee$ stetig ist.
Sei hierfür $(\mathbb{V},P)$ ein lokalkonvexer Vektorraum und bezeichne
$\tilde{P}$ das filtrierende System aller bezüglich $\T_{P}$
stetigen Halbnormen\footnote{vgl. Korollar \ref{kor:HNTop}~\textit{ii.)}}. F"ur jedes $p\in \tilde{P}$ und jede positive Konstante
$|c|>0$ ist dann insbesondere die Halbnorm $|c|\:p$ in $\tilde{P}$ enthalten.\\
Jedes $\Ss^{l}(\mathbb{V})$ sei nun $\pi_{l}$-topologisiert bez"uglich des Halbnormensystemes $\tilde{P}$. Dann ist es insbesondere ausreichend, das Teilsystem $\{p^{l}\}_{p\in
\tilde{P}}=\{\pi_{p,…,p}\}_{p\in \hat{P}}\subseteq \tilde{P}$
zu betrachten. Denn mit der Filtrationseigenschaft existiert zu jedem Satz von Halbnormen $p_{1},\dots,p_{l}\in \tilde{P}$ ein $p\in \tilde{P}$ derart, dass $p \geq
p_{i}\:\forall\:1\leq i\leq k$ und folglich
\begin{equation*}
p_{1}\ot…\ot p_{l}\leq \overbrace{p\ot…\ot p}^{l-mal}=p^{l}
\end{equation*}
gilt. Ebenso zeigt man die umgekehrte Absch"atzbarkeit, und da besagtes Teilsystem ebenfalls filtrierend ist, zeigt Korollar \ref{kor:HNTop}~\textit{iv.)}, dass beide
Halbnormensysteme die selbe Topologie auf $\Ss^{l}(\mathbb{V})$
definieren. Diese ist gerade die durch $(\Tt^{l}(\V),\pi_{l})$ auf
$\Ss^{l}(\V)$ induzierte Teilraumtopologie.\\\\
Um nun die direkte Summe, also $\SsV$ lokalkonvex zu topologisieren,
betrachten wir das auf $\Tt^{\bullet}(\V)$ und somit auch auf $\SsV$
definierte System $\Pp$, bestehend aus den Halbnormen
\begin{align*}
\p\colon\sum_{l}\omega_{l}\longmapsto \sum_{l=0}^{\infty}p^{l}(\omega_{l})
\end{align*}mit $p^{0}=||_{\mathbb{K}}$. Dieses ist ebenfalls
filtrierend und insbesondere ist klar, dass dann die Teilraumtopologien auf den $\Ss^{l}(\mathbb{V})$
gerade mit den $\pi_{l}$-Topologien übereinstimmen.
Des Weiteren sei darauf hingewiesen, dass die Halbnormen
$\tilde{\p}=\displaystyle\sum_{l=0}^{\infty}p_{i}^{l}$
mit paarweise verschiedenen $p_{i}\in \tilde{P}$ in der Tat
eine andere Topologie definieren, da die Summe nicht endlich ist und
wir
im Allgemeinen kein $p\in \tilde{P}$ derart finden, dass $p\geq
p_{i}\:\forall\:i\in \mathbb{N}$. Den Hauptgrund für unsere Wahl liefert das n"achste Lemma.
Essentiell an besagtem Halbnormensystem ist zudem, dass mit $\p \in \Pp$, per
Konstruktion, ebenfalls die Halbnorm
\begin{align*}
\p_{c}(v)=\sum_{l=0}^{\infty}|c|^{l}p^{l}
\end{align*} in $\Pp$ enthalten ist. Dies wird für spätere Stetigkeitsabschätzungen von hohem Nutzen sein. Das folgende Lemma macht $(\SsV,\vee)$ schlie"slich zu einer lokalkonvexen Algebra:
\begin{lemma}
Vermöge $\vee$ wird $(\Ss^{\bullet}(\mathbb{V}),\Pp)$ zu einer
assoziativen, unitären, lokalkonvexen Algebra mit
submultiplikativem Halbnormensystem.
\begin{beweis}
Zun"achst erinnern wir, dass eine lokalkonvexe Algebra $(\mathcal{A},*,P)$ submultiplikativ gennant wird, falls $p(a*b)\leq p(a)\:p(b)$ f"ur alle $a,b\in \mathcal{A}$ und alle $p\in P$.
Da dies insbesondere die Stetigkeit von $*$ impliziert, reicht es, diese Relation f"ur $(\SsV,\vee, \Pp)$ nachzuweisen.
Zun"achst erhalten wir
\begin{equation}
\label{eq:fopliu}
\begin{split}
\p\big(\ot^{\bullet}(\alpha,\beta)\big)=&\p\left(\sum_{l,m}\alpha_{l}\ot\beta_{m}\right)=\sum_{k}p^{k}\left(\sum_{l+m=k}\alpha_{l}\ot\beta_{m}\right)
\\ \leq&\: \sum_{l,m}p^{l+m}\left(\alpha_{l}\ot\beta_{m}\right)
\leq \sum_{l,m,i_{l},j_{m}}p^{l+m}\left(\alpha^{i_{l}}_{l}\ot\beta^{j_{m}}_{m}\right)
\\=&\: \sum_{l,m,i_{l},j_{m}}p^{l}(\alpha^{i_{l}}_{l})\:p^{m}(\beta^{j_{m}}_{m})
=
\left(\sum_{l,i_{l}}p^{l}\big(\alpha_{l}^{i_{l}}\big)\right)\left(\sum_{m,j_{m}}p^{m}\left(\beta_{m}^{j_{m}}\right)\right)
\end{split}
\end{equation} für alle Zerlegungen
$\Tt^{l}(\V)\ni\alpha_{l}=\displaystyle\sum_{i_{l}}\alpha^{i_{l}}_{l}$
in separable $\alpha_{l}^{i_{l}}$ und
$\Tt^{m}(\V)\ni\beta_{m}=\displaystyle\sum_{j_{m}}\beta^{j_{m}}_{m}$
in separable $\beta_{m}^{j_{m}}$. Dies zeigt
\begin{align*}
\p\left(\ot^{\bullet}(\alpha,\beta)\right)\leq&
\:\Bigg[\sum_{l}\inf\bigg(\sum_{i}p^{l}\big(\alpha^{i}_{l}\big)\bigg)\Bigg]\:\Bigg[\sum_{m}\inf\bigg(\sum_{j}p^{m}\big(\beta^{j}_{m}\big)\bigg)\Bigg]
\\=&\: \p\Bigg(\sum_{l}\alpha_{l}\Bigg)\:\p\Bigg(\sum_{m}\beta_{m}\Bigg),
\end{align*}wobei wieder das Infimum über alle Zerlegungen von
$\alpha_{l}$ und $\beta_{m}$ gemeint ist.\\\\
F"ur $S\colon\Tt^{\bullet}(\mathbb{V})\longrightarrow
\Ss^{\bullet}(\mathbb{V})\subseteq \Tt^{\bullet}(\mathbb{V})$ folgt
\begin{equation*}
p^{l}(\mathrm{Sym}_{l}(\alpha_{l}))=p^{l}\Bigg(\frac{1}{l!}\sum_{\sigma\in S_{l}}\sigma^{*}\alpha_{l}\Bigg)\leq
\frac{1}{l!}\sum_{\sigma\in S_{l}}\:p^{l}(\alpha_{l})=p^{l}(\alpha_{l}),
\end{equation*}
womit
\begin{align*}
\p\left(S(\alpha)\right)=\p\left(\sum_{l}\mathrm{Sym}_{l}(\alpha_{l})\right)=\sum_{l}p^{l}(\mathrm{Sym}_{l}(\alpha_{l}))\leq \sum_{l}p^{l}(\alpha_{l})=\p(\alpha).
\end{align*}
Mit $\vee=S\cp \ot^{\bullet}$ (vgl. Abschnitt
\ref{subsec:HochschKohSym}) zeigt dies
\begin{align*}
\p(\alpha\vee
\beta)=\p((S\cp\ot^{\bullet})(\alpha,\beta))\leq
\p(\ot^{\bullet}(\alpha,\beta))\leq \p(\alpha)\p(\beta)
\end{align*}f"ur alle $\alpha,\beta \in \Tt^{\bullet}(\V)$ und somit die Behauptung, da obige Ungleichung dann insbesondere f"ur alle $\alpha,\beta \in\SsV\subseteq \Tt^{\bullet}(\V)$ korrekt ist.
\end{beweis}
\end{lemma}
Als Resultat dieses Lemmas erhalten wir mit Proposition \ref{prop:topBarKomplexprop}~\textit{iii.}), dass
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong H^{k}\left(\Hom_{\mathcal{A}^{e}}^{\operatorname{\mathrm{cont}}}(\C_{c},\mathcal{M}),d^{*}_{c}\right)
\end{equation*} für jeden lokalkonvexen $\SsV-\SsV$-
Bimodul $\mathcal{M}$. Hierbei bezeichnet $\C_{c}$ den zu $\Ss^{\bullet}(\mathbb{V})$
gehörigen, topologischen Bar-Komplex mit
$\mathcal{A}^{e}=\Ss^{\bullet}(\mathbb{V})\pite \Ss^{\bullet}(\mathbb{V})$.\\\\
Wir wollen nun den in Abschnitt \ref{subsec:HochschKohSym} betrachteten
Koszul-Komplex in geeigneter Weise derart topologisieren, dass die
Kettendifferentiale $\partial_{k}$ stetige Abbildungen sind und wir somit in
wohlbegründeter Weise vom topologischen Koszul-Komplex
$(\K_{c},\pt_{c})$ und folglich auch vom stetigen Kokettenkomplex
$(\Hom_{\mathcal{A}^{e}}^{\operatorname{\mathrm{cont}}}(\K_{c},\mathcal{M}),\pt_{c}^{*})$
sprechen dürfen. Des Weiteren werden wir nachweisen, dass dann unsere
Kettenabbildungen $F$ und $G$ in den gegebenen Topologien ebenfalls
stetig sind und somit Kettenabbildungen zwischen
$(\C_{c}^{*},d_{c}^{*})$ und $(\K_{c}^{*},\pt_{c}^{*})$ induzieren.
\begin{definition}[Topologischer Koszul-Komplex]
Den Koszul-Komplex aus \ref{subsec:HochschKohSym} vor Augen,
definieren wir die topologischen Räume
\begin{equation*}
\K_{k}^{c}=\Ss^{\bullet}(\mathbb{V})\pite
\Ss^{\bullet}(\mathbb{V})\pite \Lambda^{k}(\mathbb{V})
\end{equation*} und erhalten ein erzeugendes
System $\Pp_{k}$, vermöge den Halbnormen:
\begin{equation*}
\p_{k}=\p\ot\p\ot p^{k}=\p^{2}\ot p^{k}.
\end{equation*}
Mit $\pt^{c}_{k}$ bezeichnen wir die durch
Definition \ref{def:partialdef} auf den $\K_{k}^{c}$ induzierten
Homomorphismen, von denen wir im Folgenden nachweisen werden, dass
sie stetig sind.
$\Tt^{c}_{k}\supseteq \K_{k}^{c}$ sei der mit selbigem
Halbnormensystem ausgestatteten Raum
\begin{equation*}
\Tt^{\bullet}(\mathbb{V})\pite
\Tt^{\bullet}(\mathbb{V})\pite \Tt^{k}(\mathbb{V}),
\end{equation*}
der obige Topologie als
Teilraumtopologie auf $\K_{k}^{c}$ induziert.
\end{definition}
\begin{bemerkung}
\label{bem:Teilraumargument}
Will man die Stetigkeit einer Abbildung $\phi\colon\K^{c}_{k}\longrightarrow
\K^{c}_{k'}$ nachweisen, so reicht es, diese für eine Abbildung
$\tilde{\phi}\colon\Tt^{c}_{k}\longrightarrow \Tt^{c}_{k'}$ zu zeigen,
die $\phi$ auf $\K^{c}_{k}$ induziert, f"ur die also
$\tilde{\phi}\big|_{\K^{c}_{k}}=\phi$ gilt.
Dies sieht man sofort
daran, dass die Stetigkeitsabschätzungen für $\tilde{\phi}$
insbesondere f"ur die besagten Unterräume g"ultig sind.
\end{bemerkung}
\begin{proposition}
\label{prop:wichpropKoszStetSym}
\begin{enumerate}
\item
Eine lineare Abbildung $\phi\colon\Tt^{c}_{k}\rightarrow
\Tt^{c}_{k'}$ ist genau dann stetig, wenn für jedes
$\q_{k'}\in \Pp_{k'}$ ein $\p_{k}\in \Pp_{k}$ derart
existiert, dass
\begin{equation}
\label{eq:stetrelSummenKos}
\q_{k'}(\phi(\alpha_{l}\ot \beta_{m}\ot u))\leq c\: \p_{k}(\alpha_{l}\ot \beta_{m}\ot u)
\end{equation}für alle separablen $\alpha_{l}\in \Tt^{l}(\mathbb{V})$, $\beta_{m}\in
\Tt^{m}(\mathbb{V})$ und $u\in \Tt^{k}(\mathbb{V})$ gilt.
\item
\label{item:partkStetig}
Es sind alle $\pt^{c}_{k}$ stetig.
\item
Es sind alle $h_{k}$ stetig.
\item
Es sind alle $F_{k}$ stetig.
\item
Es sind alle $G_{k}$ stetig.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Ist $\phi$ stetig, so gilt besagte Relation sogar f"ur alle Elemente in $\Tt_{c}^{k}$.\\
Für die umgekehrte Richtung sei $\alpha\ot \beta\ot u \in
\Tt^{c}_{k}$. Dann folgt
\begin{align*}
\q_{k'}(\phi(\alpha\ot\beta\ot
u))=&\:\q_{k'}\left(\sum_{l,m,i_{l},j_{m},s}\phi\left(\alpha^{i_{l}}_{l}\ot\beta^{j_{m}}_{m}\ot
u^{s}\right)\right)
\\\leq&\sum_{l,m,i_{l},j_{m},s}\q_{k'}\left(\phi\left(\alpha_{l}^{i_{l}}\ot\beta_{m}^{j_{m}}\ot
u^{s}\right)\right)
\\\leq&\sum_{l,m,i_{l},j_{m},s}c\:\p_{k}\left(\alpha_{l}^{i_{l}}\ot\beta_{m}^{j_{m}}\ot
u^{s}\right)
\\=&\:c\sum_{l,m,i_{l},j_{m},s}p^{l}\big(\alpha_{l}^{i_{l}}\big)\:p^{m}\left(\beta_{m}^{j_{m}}\right)\:p^{k}(u^{s})
\end{align*} für alle Zerlegungen von
$\alpha_{l}$, $\beta_{m}$ und $u$ in separable Summanden
$\alpha_{l}^{i_{l}}$, $\beta_{m}^{j_{m}}$, $u^{s}$ Dies
zeigt:
{\begin{align*}
\label{eq:stetmuhmuh}
\q_{k'}(\phi(\alpha\ot&\: \beta\ot u))
\\\leq&\:c\sum_{l}\inf\left(\sum_{i}
p^{l}\big(\alpha_{l}^{i}\big)\right)\sum_{m}\inf\left(\sum_{i}p^{m}\left(\beta_{m}^{i}\right)\right)\inf\left(\sum_{s}\left(p^{k}(u^{s})\right)\right)\\=&\:c\:
\p(\alpha)\p(\beta)\:p^{k}(u),
\end{align*}}also die Stetigkeit von $\phi\cp
\ot_{3}$ und somit die von $\phi$ in $\pi_{3}$.
\item
In Abschnitt \ref{subsec:HochschKohSym} hatten wir eingesehen, dass
\begin{align*}
(S\ot S\ot A)\cp
\tilde{\pt}^{k}_{1}\Big|_{\K^{c}_{k}}&=\pt^{k}_{1}\\
(S\ot S\ot A)\cp
\tilde{\pt}^{k}_{2}\Big|_{\K^{c}_{k}}&=\pt^{k}_{2}.
\end{align*} Hierbei haben wir die zus"atzlichen Symmetrisierungen und
Antisymmetrisierungen aus reiner Bequemlichkeit
eingef"ugt, was wegen $S|_{\SsV}=\id_{\SsV}$ und
$A|_{\Lambda^{\bullet}(V)}=\id_{\Lambda^{\bullet}(V)}$
ohne weiteres möglich ist.
Nun ist $S\ot S\ot A$ stetig in $\Tt^{c}_{k}$
nach \textit{i.)}, denn für
$\alpha_{l}\ot\beta_{m}\ot u\in \Tt_{k}^{c}$
mit
separablen Faktoren folgt:
\begin{align*}
\p_{k}((S\ot S\ot A)(\alpha_{l}\ot \beta_{m} \ot
u))=&\:\p_{k}\left(S(\alpha_{l})\ot
S(\beta_{m})\ot A(u)\right)\\
=&\:p^{l}(S(\alpha_{l}))\:p^{m}(S(\beta_{m}))\:p^{k}(A(u))
\\\leq&\:p^{l}(\alpha_{l})\:p^{m}(\beta_{m})\:p^{k}(u)
\\=&\: \p_{k}(\alpha_{l}\ot \beta_{m} \ot u).
\end{align*}
Es bleiben die Stetigkeiten von
$\tilde{\pt}^{k}_{1}$ und $\tilde{\pt}^{k}_{2}$ zu zeigen. Diese
folgen mit \textit{i.)} und
\begin{align*}
\p_{k-1}\left(\tilde{\pt}^{k}_{1}\left(\alpha_{l}\bbot\beta_{m}\bbot
u\right)\right)
=&\:k\p_{k-1}\left(u_{1}\ot\alpha_{l}\bbot\beta_{m}\bbot
u^{1}\right)
\\=&\:k\:p^{l+1}(u_{1}\ot
\alpha_{l})\:p^{m}(\beta_{m})\:p^{k-1}(u^{1})
\\=&\:k\:p^{l}(\alpha_{l})\:p^{m}(\beta_{m})\:p^{k}(u)
\\=&\:k\p_{k}(\alpha_{l}\bbot\beta_{m}\bbot u)
\end{align*}
sowie einer analogen Rechnung für $\tilde{\pt}^{k}_{2}$.
Bemerkung \ref{bem:Teilraumargument} zeigt dann die Stetigkeit von
$\pt^{k}_{1}$ und $\pt^{k}_{2}$ und folglich die von $\pt^{c}_{k}$.
\item
Wir erinnern, dass $h_{k}=\int_{0}^{1}dt\:t^{k}\:
i_{t}\cp\delta$, und zeigen zunächst die Stetigkeit von
$\delta$. Nun war $(S\ot S\ot A) \cp
\tilde{\delta}\big|_{\K^{c}_{k}}=\delta$ mit
\begin{align*}
\tilde{\delta}\colon\Tt^{\bullet}(\mathbb{V})\ot \Tt^{\bullet}(\mathbb{V})\ot
\Tt^{k}(\mathbb{V})&\longrightarrow
\Tt^{\bullet-1}(\mathbb{V})\ot \Tt^{\bullet}(\mathbb{V})\ot
\Tt^{k+1}(\mathbb{V})\\
\alpha_{l}\bbot\beta\bbot u&\longmapsto l\left(\alpha_{l}^{1}\ot \beta\bbot\: (\alpha_{l})_{1}\bbot u\right),
\end{align*}und für separable $\alpha_{l}\in \Tt^{l}(\mathbb{V})$, $\beta_{m}\in
\Tt^{m}(\mathbb{V})$ sowie $u\in \Tt^{k}(\mathbb{V})$ folgt:
\begin{align*}
\p_{k+1}\left(\:\tilde{\delta}(\alpha_{l}\ot\beta_{m}\ot
u)\right)=&\p^{2}\left(l\:\alpha^{1}_{l}\ot
\beta_{m}\right)p^{k+1}((\alpha_{l})_{1}\ot u)
\\=&\:l\:p^{l}(\alpha_{l})\:p^{m}(\beta_{m})\:p^{k}(u)
\\\leq&\:
2^{l}2^{m}2^{k}\:p^{l}(\alpha_{l})\:p^{m}(\beta_{m})\:p^{k}(u)
\\=&\:\hat{p}^{l}(\alpha_{l})\:\hat{p}^{m}(\beta_{m})\:\hat{p}^{k}(u)
\\=&\:\hat{\p}_{k}(\alpha_{l}\ot\beta_{m}\ot u)
\end{align*}
mit $\hat{p}=2p$.
Bemerkung \ref{bem:Teilraumargument} sowie \textit{i.)} zeigen die Stetigkeit von
$\delta$.\\\\
Um die Stetigkeit von $\int_{0}^{1}dt\:t^{k}i_{t}$
nachzuweisen, erinnern wir daran, dass
\begin{equation*}
\int_{0}^{1}dt\:t^{k}\:i_{t}=\left.\left[(S\ot S\ot A)\cp \int_{0}^{1}dt\:t^{k}\underbrace{\left[\sum_{l=0}^{\infty}\sum_{s=0}^{l}\eta_{t}^{l,s}\right]}_{\hat{i}_{t}}\right]\right|_{\K^{c}_{k}}
\end{equation*}
mit
\begin{align*}
\eta^{l,s}_{t}\colon\Tt^{\bullet}(\mathbb{V})\ot
\Tt^{\bullet}(\mathbb{V})\ot
\Tt^{k}(\mathbb{V})&\longrightarrow \Tt^{\bullet}(\mathbb{V})\ot
\Tt^{\bullet}(\mathbb{V})\ot
\Tt^{k}(\mathbb{V})\\
\alpha_{l}\ot \beta\ot u&\longmapsto
\binom{l}{s}\:t^{l-s}(1-t)^{s}\alpha_{l}^{1,…,s}\ot\:
(\alpha_{l})_{1,…,s}\ot \beta\ot u.
\end{align*}
Wir rechnen für separable $\alpha_{l},\:\beta_{m}$ und
$u$:
{\allowdisplaybreaks
\begin{align*}
\p_{k}\Bigg(\int_{0}^{1}dt\:t^{k}\:&\hat{i}_{t}\left(\alpha_{l}\ot\beta_{m} \ot
u\right)\Bigg)=\:\p_{k}\left(\int_{0}^{1}dt\:t^{k}\sum_{l'=0}^{\infty}\sum_{s=0}^{l'}\eta_{t}^{l',s}(\alpha_{l}\ot
\beta_{m}\ot
u)\right)
\\=&\:\p_{k}\left(\int_{0}^{1}dt\:t^{k}\sum_{s=0}^{l}\eta_{t}^{l,s}(\alpha_{l}\ot\beta_{m}\ot
u)\right)
\\=&\:\p_{k}\left(\sum_{s=0}^{l}\underbrace{\int_{0}^{1}dt\:t^{k}\:t^{l-s}(1-t)^{s}\binom{l}{s}}_{\tau^{l}_{s}}
\left(\alpha_{l}^{1,…,s}\ot\: (\alpha_{l})_{1,…,s}\ot\beta_{m}\ot
u\right)\right)
\\=&\:\p^{2}\left(\sum_{s=0}^{l}\tau^{l}_{s}\:\alpha_{l}^{1,…,s}\ot\:
(\alpha_{l})_{1,…,s}\ot\beta_{m}\right)p^{k}(u)
\\\leq&\:\sum_{s=0}^{l}\tau^{l}_{s}\:\p^{2}\left(\alpha_{l}^{1,…,s}\ot\:
(\alpha_{l})_{1,…,s}\ot\beta_{m}\right)p^{k}(u)
\\=&\:\sum_{s=0}^{l}\tau^{l}_{s}\:\:p^{l-s}\left(\alpha_{l}^{1,…,s}\right)\:p^{s}\Big((\alpha_{l})_{1,…,s}\ot\beta_{m}\Big)\:p^{k}(u)
\\=&\:\left[\sum_{s=0}^{l}\tau^{l}_{s}\right]\:p^{l}(\alpha_{l})\:p^{m}(\beta_{m})\:p^{k}(u)
\\=&\:\frac{1}{k+1}\p_{k}(\alpha_{l}\ot \beta_{m}\ot u).
\end{align*}}In der Tat erhalten wir die letzte Gleichheit mit
{\allowdisplaybreaks
\begin{align*}
\sum_{s=0}^{l}\tau^{l}_{s}=&
\int_{0}^{1}dt\:t^{k}\sum_{s=0}^{l}\binom{l}{s}t^{l-s}(1-t)^{s}=\int_{0}^{1}dt\:t^{k}[t+(1-t)]^{l}=\int_{0}^{1}dt\:t^{k}=\frac{1}{k+1}.
\end{align*}}Die Stetigkeit von $\int_{0}^{1}dt\:t^{k}\:
i_{t}$ folgt abermals mit \textit{i.)} und
Bemerkung \ref{bem:Teilraumargument} liefert schließlich die Behauptung.
\item
Es ist $F_{k}=\tilde{F}_{k}\big|_{\K^{c}_{k}}$ f"ur
\begin{align*}
\tilde{F}_{k}\colon\Tt^{c}_{k}&\longrightarrow \bigotimes^{k+2}\Tt^{\bullet}(\mathbb{V})\\
\alpha\bbot \beta \bbot u_{1}\ot…\ot u_{k}&\longmapsto k!\left(\alpha\ot
u_{1}\ot…\ot u_{k} \ot \beta\right)
\end{align*} und
\allowdisplaybreaks{
\begin{align*}
\p^{k+2}\left(\tilde{F}_{k}(\alpha \ot \beta\ot
u)\right)=&\:k!\:\p^{k+2}\left(\sum_{l,m,i_{l},j_{m},i}\alpha^{i_{l}}_{l}\ot u^{i}_{1}\ot…\ot
u^{i}_{k}\ot\beta^{j_{m}}_{m}\right)
\\\leq&\:k! \sum_{l,m,i_{l},j_{m},i}\p^{k+2}\left(\alpha^{i_{l}}_{l}\ot u^{i}_{1}\ot…\ot
u^{i}_{k}\ot\beta^{j_{m}}_{m}\right)
\\=&\:k!\sum_{l,m,i_{l},j_{m},i} p^{l}\left(\alpha^{i_{l}}_{l}\right)\:p\Big(u^{i}_{1}\Big)…p\Big(u^{i}_{k}\Big)\:p^{m}\bigg(\beta^{j_{m}}_{m}\Big)
\\=&\:k! \left(\sum_{l,i_{l}}p^{l}\left(\alpha^{i_{l}}_{l}\right)\right)\left(\sum_{m,j_{m}}p^{m}\Big(\beta^{j_{m}}_{m}\Big)\right)\:\left(\sum_{i}p^{k}\Big(u^{i}\Big)\right)
\end{align*}}f"ur alle Zerlegungen der $\alpha_{l}$, $\beta_{m}$ und von $u$. Hiermit folgt
\begin{align*}
\p^{k+2}\Big(\tilde{F}_{k}(\alpha \ot \beta&\:\ot
u)\Big)
\\\leq&
\:k!\:\sum_{l}\inf\left(\sum_{i}p^{l}\left(\alpha_{l}^{i}\right)\right)\:\sum_{m}\inf\left(\sum_{i}p^{l}\left(\beta_{m}^{i}\right)\right)\:\inf\left(\sum_{i}p^{k}\left(u^{i}\right)\right)
\\=&\:k!\:\p(\alpha)\p(\beta)\:p^{k}(u),
\end{align*}
also die Stetigkeit von $\tilde{F}_{k}$ in $\pi_{3}$. Die von
$F_{k}$ folgt analog zu Bemerkung \ref{bem:Teilraumargument}, da obige Ungleichung insbesondere f"ur alle Elemente aus $\K_{k}^{c}$ korrekt ist.
\item
Wir zeigen dies wieder schrittweise. Sei hierfür
$\bigotimes^{k+2}\Tt^{\bullet}(\mathbb{V})\ot
\Tt^{k}(\mathbb{V})$ topologisiert vermöge den Halbnormen
$\p^{k+2}\ot p^{k}$ und
$\bigotimes^{k+2}\Tt^{\bullet}(\mathbb{V})$ vermöge
$\p^{k+2}$. Wir definieren sinngemäß zu Definition \ref{def:GAbb}~\textit{iv)}:
\begin{align*}
\tilde{\delta}_{k}\colon\bigotimes^{k+2}\Tt^{\bullet}(\V)&\longrightarrow
\bigotimes^{k+2}\Tt^{\bullet}(\V)\bbot\Tt^{k}(\mathbb{V})\\
\alpha\ot u_{1}\ot…\ot u_{k}\ot\beta &\longmapsto \left[\prod_{i=1}^{k}n_{i}\right]\:
\alpha\ot u_{1}^{1}\ot…\ot u_{k}^{1}\ot\beta\bbot \:(u_{1})_{1}\ot…\ot (u_{k})_{1}
\end{align*}für $\deg(u_{i})=n_{i}$ und $\tilde{\delta}_{k}(\alpha \ot u_{1}\ot\dots \ot u_{j-1}\ot
1 \ot u_{j+1} \ot …u_{k}\ot \beta)=0$, womit
\begin{equation*}
\delta=\left(\bigotimes^{k+2} \id\ot A\cp
\tilde{\delta}_{k}\right)\Bigg|_{\C^{c}_{k}}.
\end{equation*}
Für die Stetigkeit von
$\tilde{\delta}_{k}$ sei $\alpha\ot u_{1}\ot…\ot
u_{k}\ot\beta\in
\Tt^{\bullet}(\mathbb{V})\ot\displaystyle\bigotimes^{k}_{n=1}\Tt^{n_{k}}(\mathbb{V})\ot
\Tt^{\bullet}(\mathbb{V})$ mit
$u_{i}$ für $1\leq i \leq k$ separabel. Dann folgt
{\allowdisplaybreaks
\begin{align*}
\big(\p^{k+2}\ot p^{k}\big)&\left(\tilde{\delta}(\alpha\ot u_{1}\ot…\ot
u_{k}\ot \beta)\right)
\\ &=\left(\p^{k+2}\ot
p^{k}\right)\left(\left[\prod_{i=1}^{k}n_{i}\right]\alpha\ot\:u_{1}^{1}\ot…\ot\:
u_{k}^{1}\ot \beta\:\bbot\:
(u_{1})_{1}\ot…\ot\:(u_{k})_{1}\right)
\\ &= \left[\prod_{i=1}^{k}n_{i}\right]\p(\alpha)\p\left(u_{1}^{1}\right)…\p\left(u_{k}^{1}\right)\p(\beta)\:p^{k}((u_{1})_{1}\ot…\ot\:(u_{k})_{1})
\\ &=\left[\prod_{i=1}^{k}n_{i}\right]\p(\alpha)\:p^{n_{1}-1}\left(u_{1}^{1}\right)…p^{n_{k}-1}\left(u_{k}^{1}\right)\p(\beta)\:p^{k}((u_{1})_{1}\ot…\ot\:(u_{k})_{1})
\\ &=\left[\prod_{i=1}^{k}n_{i}\right]\p(\alpha)\:p^{n_{1}}\left(u_{1}\right)…p^{n_{k}}\left(u_{k}\right)\p(\beta)
\\ &\leq\tilde{\p}(\alpha)\:\tilde{p}^{n_{1}}\left(u_{1}\right)…\tilde{p}^{n_{k}}\left(u_{k}\right)\tilde{\p}(\beta)
\\ &= \tilde{\p}^{k+2}(\alpha\ot u_{1}\ot…\ot
u_{k}\ot \beta)
\end{align*}}mit $\tilde{p}=2p$.
Den allgemeinen Fall
erhalten wir vermöge obiger Ungleichung mit
\begin{align*}
\big(\p^{k+2}\ot p^{k}\big)\Big(\tilde{\delta}(\alpha\ot&\: u_{1}\ot…\ot
u_{k}\ot \beta)\Big)\\=&\:\big(\p^{k+2}\ot
p^{k}\big)\left(\sum_{\substack{n_{1},…,n_{k}\\i_{n_{1}},…,i_{n_{k}}}}\tilde{\delta}\left(\alpha\ot\:
(u_{1})^{i_{n_{1}}}_{n_{1}}\ot…\ot\:(u_{k})^{i_{n_{k}}}_{n_{k}}\ot
\beta\right)\right)
\\\leq&\sum_{\substack{n_{1},…,n_{k}\\i_{n_{1}},…,i_{n_{k}}}}\big(\p^{k+2}\ot p^{k}\big)\left(\tilde{\delta}\left(\alpha\ot\:
(u_{1})^{i_{n_{1}}}_{n_{1}}\ot…\ot\:(u_{k})^{i_{n_{k}}}_{n_{k}}\ot
\beta\right)\right)
\\\leq&\sum_{\substack{n_{1},…,n_{k}\\i_{n_{1}},…,i_{n_{k}}}}\tilde{\p}^{k+2}\left(\alpha\ot\:
(u_{1})^{i_{n_{1}}}_{n_{1}}\ot…\ot\:(u_{k})^{i_{n_{k}}}_{n_{k}}\ot
\beta\right)
\\=&\:
\tilde{\p}(\alpha)\sum_{\substack{n_{1},…,n_{k}\\i_{n_{1}},…,i_{n_{k}}}}\tilde{p}^{n_{1}}\left((u_{1})_{n_{1}}^{i_{n_{1}}}\right)…\tilde{p}^{n_{k}}\left((u_{k})_{n_{k}}^{i_{n_{k}}}\right)\tilde{\p}(\beta)
\\=&\:\tilde{\p}(\alpha)\left(\sum_{n_{1}}\sum_{i_{n_{1}}}\tilde{p}^{n_{1}}\left((u_{1})^{i_{n_{1}}}_{n_{1}}\right)\right)…\left(\sum_{n_{k}}\sum_{i_{n_{k}}}\tilde{p}^{n_{k}}\left((u_{1})^{i_{n_{k}}}_{n_{k}}\right)\right)\tilde{\p}(\beta)
\end{align*}
für alle Zerlegungen der $(u_{i})_{n_{i}}\in \Tt^{n_{i}}(\V)$, und es
folgt
\begin{equation*}
\big(\p^{k+2}\ot p^{k}\big)\left(\tilde{\delta}(\alpha\ot u_{1}\ot…\ot
u_{k}\ot \beta)\right)\leq \tilde{\p}(\alpha)\:\tilde{\p}(u_{1})…\tilde{\p}(u_{k})\:\tilde{\p}(\beta).
\end{equation*}Dies zeigt die Stetigkeit von
$\tilde{\delta}$ in $\pi_{k+2}$, und mit dem üblichen
Teilraumargument ebenso die von $\delta$.\\\\
Um die Stetigkeit von
$\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}\:i_{t_{1},…,t_{k}}$
nachzuweisen, definieren wir
\begin{align*}
\eta^{m,l}\colon\Tt^{\bullet}(\mathbb{V})\ot \Tt^{m}(\mathbb{V})\ot \Tt^{\bullet}(\mathbb{V})&\longrightarrow
\bigotimes^{2}\Tt^{\bullet}(\mathbb{V})\\
\alpha\ot u \ot \beta&\longmapsto u^{1,…,l}\ot \alpha \ot u_{1,…,l}\ot\beta
\end{align*} mit $l\leq m$ sowie $\eta^{0,0}(\alpha\ot
1\ot \beta)=(\alpha\ot \beta)$. Dann folgt für
\begin{align*}
\hat{i}'_{t}=\sum_{m=0}^{\infty}\sum_{l=0}^{m}\binom{m}{l}t^{m-l}(1-t)^{l}\eta^{m,l}
\end{align*}
sowie
\begin{align*}
\tilde{i}_{t_{1},…,i_{t_{k}}}(\alpha\ot u_{1}\ot…\ot
u_{k}\ot\beta\ot \omega)=\alpha\ot\beta \:*_{e}\:\left[
\prod_{s=1}^{k}\hat{i}'_{t_{s}}(1\ot u_{s}\ot 1)\right]\ot \omega
\end{align*} mit $\prod$ das Produkt
$(\alpha\ot u\ot \beta)\cdot (\alpha'\ot u'\ot \beta')
\alpha\ot \alpha'\bbot u\ot u'\bbot \beta\ot \beta'$
und
$\alpha \ot \beta *_{e} \hat{\alpha}\bbot \hat{\beta}\bbot\omega=\alpha\ot\hat{\alpha}\bbot \hat{\beta}\ot\beta\bbot\omega$, dass
\begin{align*}
i_{t_{1},…,t_{k}}=\Big[(S\ot S\ot \id)\cp
\tilde{i}_{t_{1},…,t_{k}}\Big]\Big|_{\bigotimes^{k+2}\Ss^{\bullet}(\mathbb{V})\ot
\Lambda^{k}(\mathbb{V})}.
\end{align*}
Sei nun $\alpha_{p}\ot u_{1}\ot…\ot u_{k}\ot
\beta_{q}\ot \omega\in
\displaystyle\bigotimes^{k+2}\Tt^{\bullet}(\mathbb{V})\ot
\Tt^{k}(\mathbb{V})$ mit $\deg(u_{i})=p_{i}$,\\
$\deg(\alpha_{p})=p$ und $\deg(\beta_{q})=q$.
Für
$\alpha_{p},\:\beta_{q},\:u_{i}$ separabel und $\omega\in
\Tt^{k}(\V)$ beliebig erhalten wir:
{\footnotesize
\begin{align*}
\p_{k}\bigg(&\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}\:\tilde{i}_{t_{1},…,t_{k}}(\alpha_{p}\ot u_{1}\ot…\ot u_{k}\ot
\beta_{q}\ot \omega)\bigg)
\\=&\p^{2}\left(\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}\:\alpha_{p}\ot\beta_{q}*_{e}\prod_{i=1}^{k}\hat{i}'_{t_{i}}(1\ot
u_{i}\ot1)\right)p^{k}(\omega)
\\=&\:p^{k}(\omega)\p^{2}\left(\alpha_{p}\ot\beta_{q}*_{e}\left[\sum_{\substack{m_{i}=0\\1\leq
i\leq
k}}^{\infty}\:\sum_{\substack{l_{i}=0\\1\leq i\leq
k}}^{m_{i}}\underbrace{\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}\prod_{i=1}^{k}\binom{m_{i}}{l_{i}}t_{i}^{m_{i}-l_{i}}(1-t_{i})^{l_{i}}
}_{\tau^{m_{1},…,m_{k}}_{l_{1},…,l_{k}}}\cdot\right.\right.
\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \left.\left. \phantom{\underbrace{\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}}_{\tau^{m_{1},…,m_{k}}_{l_{1},…,l_{k}}}}
\eta^{m_{i},l_{i}}(1\ot u_{i}\ot 1)\right]\right)
\\=&\:p^{k}(\omega)\p^{2}\left(\alpha_{p}\ot\beta_{q}*_{e}\left[\sum_{\substack{l_{i}=0\\1\leq i\leq
k}}^{p_{i}} \tau^{p_{1},…,p_{k}}_{l_{1},…,l_{k}}\prod_{i=1}^{k}u_{i}^{1,…,l_{i}}\ot
\:(u_{i})_{1,\dots,l_{i}}\right]\right)
\\=&\:p^{k}(\omega)\p^{2}\left(\sum_{\substack{l_{i}=0\\1\leq i\leq
k}}^{p_{i}}
\tau^{p_{1},…,p_{k}}_{l_{1},…,l_{k}}\alpha_{p}\ot u_{1}^{1,…,l_{1}}\ot…\ot
u_{k}^{1,…,l_{k}} \bbot
\beta_{q}\ot
\:(u_{1})_{1,…,l_{1}}\ot…\ot
\:(u_{k})_{1,…,l_{k}}\right)
\\\leq&\:p^{k}(\omega)\sum_{\substack{l_{i}=0\\1\leq i\leq
k}}^{p_{i}}
\tau^{p_{1},…,p_{k}}_{l_{1},…,l_{k}}\p^{2}\left(\alpha_{p}\ot u_{1}^{1,…,l_{1}}\ot…\ot
u_{k}^{1,…,l_{k}} \bbot
\beta_{q}\ot
\:(u_{1})_{1,…,l_{1}}\ot…\ot
\:(u_{k})_{1,…,l_{k}}\right)
\\=&\:p^{k}(\omega)\sum_{\substack{l_{i}=0\\1\leq i\leq
k}}^{p_{i}}
\tau^{p_{1},…,p_{k}}_{l_{1},…,l_{k}}p^{p}\Big(\alpha_{p}\Big)\:p^{p_{1}-l_{1}}\left(u_{1}^{1,…,l_{1}}\right)…p^{p_{k}-l_{k}}\left(u_{k}^{1,…,l_{k}}\right)\cdot \\
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\:
p^{q}\Big(\beta_{q}\Big)\:p^{l_{1}}\Big((u_{1})_{1,…,l_{1}}\Big)…p^{l_{k}}\Big((u_{k})_{1,…,l_{k}}\Big)\displaybreak
\\=&\:p^{k}(\omega)\left[\sum_{\substack{l_{i}=0\\1\leq i\leq
k}}^{p_{i}}
\tau^{p_{1},…,p_{k}}_{l_{1},…,l_{k}}\right]p^{p}(\alpha_{p})\:p^{p_{1}}(u_{1})…p^{p_{k}}(u_{k})\:p^{q}(\beta_{q})
\\ =&\:\frac{1}{k!}\big(\p^{k+2}\ot p^{k}\big)(\alpha_{p}\ot u_{1}\ot…\ot
u_{k}\ot\beta_{q}\ot \omega).
\end{align*}} In der Tat erhalten wir die letzte Gleichheit mit:
\begin{align*}
\sum_{\substack{l_{i}=0\\1\leq i\leq k}}^{p_{i}}\tau^{l_{1},…,l_{k}}_{p_{1},…,p_{k}}=&\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}\prod_{i=1}^{k}\sum_{l_{i}=0}^{p_{i}}\binom{p_{i}}{l_{i}}t^{p_{i}-l_{i}}(1-t_{i})^{l_{i}}
\\=&\:\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}=\frac{1}{k!}.
\end{align*}
Sei $\phi=\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}\:\tilde{i}_{t_{1},…,t_{k}}$, so folgt f"ur beliebige $\alpha=\sum_{p}\alpha_{p}$, $\beta=\sum_{q}\beta_{q}$
und
$u_{i}=\sum_{i}(u_{i})_{p_{i}}$ mit $\alpha_{p}\in \Tt^{p}(\V)$, $\beta_{q}\in \Tt^{p}(\V)$ sowie$(u_{i})_{p_{i}}\in \Tt^{p_{i}}(\V)$:
{\small\begin{align*}
\p_{k}(&\phi(\alpha\ot u_{1}\ot…\ot
u_{k}\ot\beta\ot \omega))\\=&\:\p_{k}\left(\sum_{\substack{p,q,p_{j}\\i_{0},\dots,i_{k+1}}}\phi\left(\alpha^{i_{0}}_{p}\ot\:
(u_{1})^{i_{1}}_{p_{1}}\ot…\ot\:(u_{k})^{i_{k}}_{p_{k}}\ot
\beta^{i_{k+1}}_{q}\ot \omega\right)\right)
\\\leq&\sum_{\substack{p,q,p_{j}\\i_{0},\dots,i_{k+1}}}\p_{k}\left(\phi\left(\alpha^{i_{0}}_{p}\ot\:
(u_{1})^{i_{1}}_{p_{1}}\ot…\ot\:(u_{k})^{i_{k}}_{p_{k}}\ot
\beta^{i_{k+1}}_{q}\ot \omega\right)\right)
\\\leq&\sum_{\substack{p,q,p_{j}\\i_{0},\dots,i_{k+1}}}\left(\p^{k+2}\ot p^{k}\right)\left(\alpha^{i_{0}}_{p}\ot\:
(u_{1})^{i_{1}}_{p_{1}}\ot…\ot\:(u_{k})^{i_{k}}_{p_{k}}\ot
\beta^{i_{k+1}}_{q}\ot \omega\right)
\\=&\:\Bigg(\sum_{p,i_{0}}p^{p}\Big(\alpha^{i_{0}}_{p}\Big)\Bigg)\Bigg(\sum_{p_{1},i_{1}} p^{p_{1}}\Big((u_{1})^{i_{1}}_{p_{1}}\Big)\Bigg)…\Bigg(\sum_{p_{k},i_{k}}p^{p_{k}}\Big((u_{k})^{i_{k}}_{p_{k}}\Big)\Bigg)\Bigg(\sum_{q,i_{k+1}}p^{q}\left(\beta^{i_{k+1}}_{q}\right)\Bigg)\:p^{k}(\omega),
\end{align*}}und somit
\begin{equation*}
\p_{k}(\phi(\alpha\ot u_{1}\ot…\ot
u_{k}\ot\beta\ot \omega))\leq \p(\alpha)\p(u_{1})…\p(u_{k})\p(\beta)\:p^{k}(\omega).
\end{equation*}
\end{enumerate}
\end{beweis}
\end{proposition}
Wir befinden uns nun in folgender Situation:
\begin{bemerkung}
\label{bem:StetIso}
Mit Proposition \ref{prop:wichpropKoszStetSym}~\textit{ii.)} ist
der Kokettenkomplex
$(\Hom_{\mathcal{A}^{e}}^{\operatorname{\mathrm{cont}}}(\K_{c},\mathcal{M}),\pt_{c}^{*})$ ein Unterkomplex von
$(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{M}),\pt^{*})$, und ebenso war
$(\Hom_{\mathcal{A}^{e}}^{\operatorname{\mathrm{cont}}}(\C_{c},\mathcal{A}),d_{c}^{*})$
ein Unterkomplex von $(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{A}),d^{*})$ . Mit
Proposition \ref{prop:wichpropKoszStetSym}
sind $F$ und $G$ stetige Kettenabbildungen. Folglich bilden
$F^{*}$ und $G^{*}$ stetige auf stetige Homomorphismen ab und
definieren somit Kettenabbildungen
$\mathrm{F}^{k}=F_{k}^{*}\big|_{\C^{*}_{k}}$ und $\mathrm{G}^{k}=G_{k}^{*}\big|_{\K^{*}_{k}}$ zwischen obigen stetigen
Unterkomplexen. Nach
Bemerkung \ref{bem:HoimotAbstrIsosUnterk}~\textit{i.)} bedeutet dies die
Injektivität von $\wt{\mathrm{F}^{k}}$ und die
Surjektivität von $\mathrm{G}^{k}$.\\
Für die
umgekehrte Aussage nehmen wir an, die Kettenabbildung $\Omega =F\cp
G$ mit $\Omega_{k} =F_{k}\cp
G_{k}\colon\C_{k}\longmapsto \C_{k}$ und
\begin{equation}
\label{eq:thetarel}
\Omega_{k}\cp d_{k+1}=d_{k+1}\cp \Omega_{k+1}
\end{equation}
wäre homotop zu der
Identität auf $\C$, vermöge einer stetigen, $\mathcal{A}^{e}$-linearen
Homotopie $s$.
Dies war eine
Familie von stetigen $\mathcal{A}^{e}$-Homomorphismen $\{s_{k}\}_{k\in \mathbb{N}}$ mit $s_{k}\colon \C_{k}^{c}\longrightarrow \C_{k+1}^{c}$ derart, dass:
\begin{align}
\label{eq:Homotopa}
F_{k}\cp G_{k}- \id_{\C_{k}}=d_{k+1}s_{k}+s_{k-1}d_{k}\qquad\qquad \forall\: k\in \mathbb{N}
\end{align}gilt.
Mit der $\mathcal{A}^{e}$-Linearit"at liefert
Anwenden des $\hom_{\mathcal{A}^{e}}(\cdot,\mathcal{M})$-Funktors, dass
\begin{align*}
G^{*}_{k}\cp
F^{*}_{k}-\id_{\Hom_{\mathcal{A}^{e}}\left(\C_{k},\mathcal{M}\right)}=d_{k}^{*}s^{*}_{k-1}+s^{*}_{k}d^{*}_{k+1}
\end{align*}und die Stetigkeit zeigt:
\begin{align*}
\mathrm{G}^{k}\cp
\mathrm{F}^{k}-\id_{\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}\left(\C^{c}_{k},\mathcal{M}\right)}=d_{k}^{*}s^{*}_{k-1}+s^{*}_{k}d^{*}_{k+1}.
\end{align*}
Mit den Definitionen $d^{k}= d_{k+1}^{*}$ und $s^{k}=s^{*}_{k-1}$
bedeutet dies
\begin{equation}
\label{eq:homotalg}
\mathrm{G}^{k}\cp
\mathrm{F}^{k}-\id_{\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}(\C^{c}_{k},\mathcal{M})}=d^{k-1}s^{k}+s^{k+1}d^{k},
\end{equation}
also $\mathrm{G}^{k}\cp \mathrm{F}^{k}\sim
\id_{\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}(\C^{c}_{k},\mathcal{M})}$ und folglich $\wt{\mathrm{G}^{k}}\cp
\wt{\mathrm{F}^{k}}=
\id_{H^{k}\left(\left(\Hom^{cont}_{\mathcal{A}^{e}}(\C_{c},\mathcal{M}),d_{c}^{*}\right)\right)}$.\\\\
Dies bedeutet $\wt{\mathrm{G}^{k}}\cp \wt{\mathrm{F}^{k}}=
\id_{H^{k}\left(\left(\Hom_{\mathcal{A}^{e}}^{\operatorname{\mathrm{cont}}}(\C_{c},\mathcal{M},d_{c}^{*}\right))\right)}$, also die Surjektivität von $\wt{\mathrm{G}^{k}}$ und die
Injektivität von $\wt{\mathrm{F}^{k}}$.
\end{bemerkung}
Um besagte Homotopie $s$ zu konstruieren gehen wir den in
\cite[Kapitel 5]{Weissarbeit} beschrittenen Weg. Hierf"ur ben"otigen wir das Konzept der $\mathcal{A}^{e}$-Linearisierung von
$\mathbb{K}$-linearen Abbildungen $\phi\colon\C_{k}\longrightarrow \C_{k'}$
zwischen Bar-Moduln.\\
\begin{definition}
Gegeben eine $\mathbb{K}$-lineare Abbildung $\phi\colon\C_{s}\rightarrow \C_{r}$, so
ist die $\mathcal{A}^{e}$-Linearisierung von $\phi$ definiert durch:
\begin{equation}
\begin{split}
\label{eq:AelinausClin}
\ovl{\phi}\colon \C_{s}&\longrightarrow \C_{r}\\
v\ot \alpha_{s}\ot w&\longmapsto v\ot w *_{e} \phi(1\ot
\alpha_{s}\ot 1).
\end{split}
\end{equation}Diese ist $\mathbb{K}$-linear, also mit
Korollar \ref{kor:WohldefTensorprodabbildungen} durch \eqref{eq:AelinausClin}
wohldefiniert. Sei $\C'_{s}=\mathcal{A}\ot \C_{s}\ot \mathcal{A}$,
dann definieren wir die $\mathcal{A}^{e}$- sowie $\mathbb{K}$-linearen Abbildungen:
\begin{align*}
\phi'\colon\C'_{s}&\longrightarrow \C'_{r}\\
v\ot v'\ot \alpha_{s}\ot w'\ot w&\longmapsto v\ot \phi(v'\ot
\alpha_{s}\ot w')\ot w,
\end{align*}
\begin{align*}
p_{s}\colon\C_{s}&\longrightarrow \C'_{s}\\
v\ot \alpha_{s}\ot w&\longmapsto v\ot 1\ot
\alpha_{s}\ot 1\ot w,
\end{align*}
\begin{align*}
i_{r}\colon\C'_{r}&\longrightarrow \C_{r}\\
v\ot v'\ot \alpha_{s}\ot w'\ot w&\longmapsto v\ot w *_{e} v'\ot
\alpha_{s}\ot w'.
\end{align*}Hiermit l"asst sich $\ovl{\phi}$ auch schreiben als $\ovl{\phi}=i_{r}\cp \phi'\cp p_{s}$. Weiterhin folgt unmittelbar
$i_{t}\cp p_{t}= \id_{\C_{t}}$.
\end{definition}
Folgende Proposition liefert uns einige wichtige Eigenschaften.
\begin{proposition}
\label{prop:gedoens}
Gegeben seien $\mathbb{K}$-lineare Abbildungen
$\psi\colon\C_{s}\longrightarrow \C_{t}$ und
$\phi\colon\C_{t}\longrightarrow \C_{r}$. Dann gilt:
\begin{enumerate}
\item
Es gilt $\ovl{\id}_{\C_{s}}=\id_{\C_{s}}$, zudem ist die
$\mathcal{A}^{e}$-Linearisierung aufgefasst als Abbildung
\begin{equation*}
\ovl{\phantom{u}}\colon
\Hom_{\mathbb{K}}(\C_{s},\C_{t})\longrightarrow
\Hom_{\mathcal{A}^{e}}(\C_{s},\C_{t})
\end{equation*}$\mathbb{K}$-linear.
\item
Für $\mathcal{A}^{e}$-lineares $\phi$ ist
$\ovl{\phi}=\phi$, also insbesondere
$\ovl{d}_{k}=d_{k}$ und
$\ovl{\Omega}_{k}=\Omega_{k}$.
\item
Ist $i_{r}\cp \phi'=\phi\cp i_{t}$, so gilt $\ovl{\phi}=\phi$ und $\phi\cp
\ovl{\psi}=\ovl{\phi\cp \psi}$. Im Speziellen ist dies für
alle $d_{k}$ der Fall.
\item
Im lokalkonvexen Fall gilt: Ist $\phi$ stetig, so auch $\ovl{\phi}$.
\end{enumerate}
\begin{beweis}
\textit{i.)} und \textit{ii.)} sind unmittelbar klar und
\textit{iii.)} folgt mit
\begin{equation*}
\ovl{\phi}=i_{r}\cp \phi' \cp p_{t}=\phi \cp i_{t}\cp p_{t}=\phi\cp \id_{\C_{t}}=\phi.
\end{equation*}Um die zweite Aussage zu zeigen, rechnen wir
\begin{align*}
(\phi\cp\psi)'(v\ot v'\ot \alpha_{s}\ot w'\ot
w)=&\:v\ot \big[(\phi\cp\psi)(v'\ot \alpha_{s}\ot w')\big]\ot
w
\\=&\:\phi'(v\ot \psi(v'\ot \alpha_{s}\ot w')\ot
w)
\\=&\:(\phi'\cp \psi')\:(v\ot v'\ot \alpha_{s}\ot w'\ot
w),
\end{align*} und erhalten
\begin{equation*}
\ovl{\phi\cp \psi}= i_{r}\cp
\phi'\cp \psi'\cp p_{s}=\phi\cp i_{t}\cp \psi'\cp
p_{s}=\phi\cp \ovl{\psi}.
\end{equation*}
Die letzte Behauptung folgt mit
\begin{align*}
(i_{k-1}\cp d'_{k})(v\ot x_{0}\ot…\ot x_{k+1}\ot w)=&\: i_{k-1}\left(v\ot d_{k}(x_{0}\ot…\ot
x_{k+1})\ot w\right)
\\=&\:v\ot w *_{e}\sum_{j=0}^{k}(-1)^{k}\: x_{0}\ot…\ot
x_{j}x_{j+1}\ot…\ot x_{k+1}
\\=&\:(d_{k}\cp i_{k})(v\ot x_{0}\ot…\ot x_{k+1}\ot
w).
\end{align*}
Für \textit{iv.)} sei $\phi$ stetig, dann folgt mit der Stetigkeit von $*_{e}$:
\begin{align*}
q^{k+2}\left(\ovl{\phi}\:(v\ot \alpha_{1}\ot…\ot \alpha_{k}\ot
w)\right)=&\:q^{k+2}(v\ot w *_{e} \phi(1\ot\alpha_{1}\ot…\ot \alpha_{k}\ot1))
\\\leq&\:c\: p_{1}^{2}(v\ot w)\:
p_{2}^{k+2}(1\ot\alpha_{1}\ot…\ot \alpha_{k}\ot 1)
\\=&\:cp_{2}(1)^{2}\:p_{1}(v)\:p_{1}(w)\:p_{2}(\alpha_{1})…p_{2}(\alpha_{k}).
\end{align*}Dies zeigt die Stetigkeit in $\pi_{k+2}$ und beendet den Beweis.
\end{beweis}
\end{proposition}
Folgendes Lemma liefert uns schließlich die erwünschte Homotopie
$s$.
\begin{lemma}
\label{lemma:Homotopiejdfgjkf}
Es ist $\id_{\C_{k}}-\:\Omega_{k}=d_{k+1} s_{k}+s_{k-1} d_{k}$, also $\Omega\sim\id_{\C}$ vermöge der $\mathcal{A}^{e}$-linearen Homotopie
$s_{k}\colon \C_{k}\longrightarrow \C_{k+1}$, die für $k\geq 0$ rekursiv definiert ist durch:
\begin{equation*}
s_{k}=\ovl{h_{k}
(\id_{\C_{k}}-\:\Omega_{k}\:-\:s_{k-1}
d_{k})}\qquad\text{mit}\qquad s_{0}=0.
\end{equation*}
Hierbei bezeichnet $h$ die exaktheitsliefernde Homotopie aus
Proposition \ref{prop:topBarKomplexprop}~\textit{ii.)}. Zudem ist
$s_{k}:\C^{c}_{k}\longrightarrow \C^{c}_{k+1}$, aufgefasst als Abbildung zwischen den
lokalkonvexen Vektorräumen $\C^{c}_{k}$ und $\C^{c}_{k+1}$ stetig.
\begin{beweis}
Die Stetigkeit der $s_{k}$ folgt unmittelbar aus der
Stetigkeit der definierenden Abbildungen\footnote{vgl. Proposition \ref{prop:wichpropKoszStetSym}} und
Proposition \ref{prop:gedoens}~\textit{iv.)}. Die $\mathcal{A}^{e}$-Linearität
ist ebenfalls klar.\\ Für den Induktionsanfang rechnen wir mit
Proposition \ref{prop:gedoens}:
{\large\begin{align*}
d_{2}s_{1}-\cancel{s_{0} d_{1}}=&\:d_{2}
\ovl{h_{1}\left(\id_{\C_{1}}-\:\Omega_{1}\:-\:\cancel{s_{0}
d_{1}}\right)}\glna{\textit{iii.)}}
\ovl{(d_{2}h_{1})
\left(\id_{\C_{1}}-\:\Omega_{1}\right)}
\\\glna{\eqref{eq:Homotbar}}&\:\ovl{(\id_{\C_{1}}-\:h_{0}d_{1})
\left(\id_{\C_{1}}-\:\Omega_{1}\right)}\glna{\textit{i.),ii.)}}\id_{\C_{1}}-\:\Omega_{1}-\ovl{h_{0}d_{1}}+\ovl{h_{0}d_{1}\Omega_{1}}
\\\glna{\eqref{eq:thetarel}}&\id_{\C_{1}}-\:\Omega_{1},
\end{align*}}wobei wir im letzten Schritt zudem
$\Omega_{0}=\id_{\mathcal{A}^{e}}$ benutzt haben. Für die
höheren Grade folgt:
{\large\begin{align*}
d_{k+1} s_{k}=&\:\ovl{(d_{k+1}
h_{k})
(\id_{\C_{k}}-\:\Omega_{k}\:-\:s_{k-1}
d_{k})}
\\=&\:\ovl{(\id_{\C_{k}}-h_{k-1} d_{k})
(\id_{\C_{k}}-\:\Omega_{k}\:-\:s_{k-1}
d_{k})}
\\=&\:\id_{\C_{k}}- \:\Omega_{k} - \ovl{s_{k-1}
d_{k}} -
\ovl{h_{k-1}(d_{k}\:-\:d_{k}\Omega_{k}\:-\:(d_{k}
s_{k-1}) d_{k})}
\\=&\:\id_{\C_{k}}- \:\Omega_{k} - s_{k-1}
d_{k} -
\ovl{h_{k-1}(d_{k}\:-\:\Omega_{k-1} d_{k}\:-\:(\id_{\C_{k}}-\:\Omega_{k-1}\:-\:s_{k-2}
d_{k-1}) d_{k})}
\\=&\:\id_{\C_{k}}- \:\Omega_{k} - s_{k-1}
d_{k}.
\end{align*}}
\end{beweis}
\end{lemma}
Dies zeigt schließlich folgenden Satz:
\begin{satz}
\label{satz:stetigHochschSym}
Gegeben ein lokalkonvexer $\SsV-\SsV$-Bimodul $\mathcal{M}$, dann gilt:
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}\Big(\Ss^{\bullet}(\mathbb{V}),\mathcal{M}\Big)\cong
H^{k}\Big(\Hom_{\mathcal{A}^{e}}^{\operatorname{\mathrm{cont}}}(\C_{c},\mathcal{M}),d_{c}^{*}\Big)\cong
H^{k}\Big(\Hom_{\mathcal{A}^{e}}^{\operatorname{\mathrm{cont}}}(\K_{c},\mathcal{M}),\pt_{c}^{*}\Big).
\end{equation*} Ist $\mathcal{M}$ zudem symmetrisch, so ist:
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong \Hom_{\mathcal{A}^{e}}^{\operatorname{\mathrm{cont}}}\big(\K^{c}_{k},\mathcal{M}\big).
\end{equation*}
\begin{beweis}
Die erste Isomorphie hatten wir bereits eingesehen. Die zweite
folgt nun unmittelbar mit Bemerkung \ref{bem:StetIso}, da wir mit
Lemma \ref{lemma:Homotopiejdfgjkf} die benötigte stetige Homotopie
gefunden haben. Die letzte Aussage folgt wie für
Satz \ref{satz:HochschkohmvonSym}, da auch hier
$\ker(\partial^{*}_{k+1})= \K^{*}_{k}$ und
$\im(\partial^{*}_{k})=0$ erfüllt ist.
\end{beweis}
\end{satz}
\section{Die stetige Hochschild-Kohomologie der Algebra $\Hol$}
\label{sec:StetHKHol}
Sei im Folgenden $\V$ ein hausdorffscher, lokalkonvexer
Vektorraum. Wir beginnen mit der folgenden, klärenden Proposition:
\begin{proposition}
\label{prop:HollkAlgebra}
\begin{enumerate}
\item
Gegeben ein hlkVR $\V$, so existiert eine bis auf lineare
Homöomorphie eindeutig bestimmte vollständige, hausdorffsche, submultiplikative, lokalkonvexe
Algebra $(\Hol,\Pp_{H},*)$, die $(\Ss^{\bullet}(\V),\Pp,\vee)$ im
isometrischen Sinne als dichte Unteralgebra enthält. Diese ist zudem
assoziativ und unitär.
\item
Jeder hausdorffsche, lokalkonvexe $\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul $(\mathcal{M},*_{L},*_{R})$ vervollständigt
zu einem hausdorffschen, lokalkonvexen $\Hol-\Hol$-Bimodul $(\hat{\mathcal{M}},\hat{*}_{L},\hat{*}_{R})$.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Zunächst ist nach Satz \ref{satz:PiTopsatz}~\textit{v.)} mit $\V$
ebenfalls jedes $\left(\Ss^{k}(\V),\pi_{k}\right)$ hausdorffsch und es ist
offensichtlich, dass dies dann ebenfalls für
$(\Ss^{\bullet}(V),\Pp)$ der Fall ist.
Es folgt mit Satz \ref{satz:vervollsthlkVR}, dass die bis auf lineare Homöomorphie eindeutig
bestimmte Vervollständigung
$\Big(\Hol,\Pp_{H}\Big)=\left(\widehat{\Ss^{\bullet}(\V)},\hat{\Pp}\right)$
existiert und ebenfalls hausdorffsch ist.
Weiter folgt, dass die bilineare Abbildung
\begin{align*}
\tilde{\vee}\colon i(\Ss^{\bullet}(\V))\times i(\Ss^{\bullet}(\V))&\longrightarrow
i(\Ss^{\bullet}(\V))\subseteq\Hol\\
(x,y)&\longmapsto i(i^{-1}(x) \vee i^{-1}(y))
\end{align*} als Verkettung stetiger Funktionen stetig
ist. Hierbei haben wir benutzt, dass die stetige Isometrie
$i$ aus Satz \ref{satz:vervollsthlkVR} ein Homöomorphismus
zwischen $(\SsV,\Pp)$ und $(i(\SsV),\hat{\Pp})$ ist.
Mit
$\ovl{i(\Ss^{\bullet}(\V))}=\Hol$ liefert uns
Satz \ref{satz:stetfortsBillphi} eine eindeutig bestimmte stetige, bilineare Fortsetzung
\begin{align*}
*\colon \Hol\times \Hol\longrightarrow \Hol,
\end{align*}
und wegen
\begin{equation*}
\hat{\p}(x * y)=\displaystyle\lim_{\alpha\times\beta}\p\left(x_{\alpha}\tilde{\vee}
y_{\beta}\right)\leq
\displaystyle\lim_{\alpha\times\beta}\p(x_{\alpha})\p(y_{\beta})=\hat{\p}(x)\:\hat{\p}(y)
\end{equation*}ist $(\Hol,\hat{\Pp},*)$ zudem
submultiplikativ. Hierbei ist $x,y\in \Hol$ mit Netzen
$i(\SsV)\supseteq\net{x}{I}\rightarrow x$,
$i(\SsV)\supseteq\nettt{y}{\beta}{J}\rightarrow y$.\\\\
Für die Unitarität betrachten wir das Element $\Hol\ni
\tilde{1}:= i(1_{\Ss^{\bullet}(\V)})$ und erhalten für ein
Netz $i(\Ss^{\bullet}(\V))\supseteq\net{x}{I}\rightarrow x\in \Hol$ sowie
$\hat{1}=\left\{\tilde{1}\right\}$ die konstante Folge $\tilde{1}$, dass:
\begin{align*}
\hat{1}* x = & \lim_{n\times \alpha}
\{i(1_{\Ss^{\bullet}(\V)})\}\:\tilde{\vee}\: x_{\alpha}=\lim_{\alpha}\:
x_{\alpha}=x.
\end{align*}
Spätestens hier ist nun auch klar, dass wir
vermöge $i$ die Räume $(\Ss^{\bullet}(\V),\vee)$ und
$i(\Ss^{\bullet}(\V),\tilde{\vee})$ identifizieren dürfen.
Für die
Assoziativität rechnen wir daher in Kurzschreibweise mit
$x,y,z\in \Hol$:
\begin{align*}
x*(y*z)=\lim_{\alpha\times (\beta\times
\gamma)}x_{\alpha}\vee (y_{\beta}\vee z_{\gamma})=\lim_{(\alpha\times \beta)\times
\gamma}(x_{\alpha}\vee y_{\beta})\vee z_{\gamma}=(x*y)*z,
\end{align*}da definitionsgemäß $\{y_{\beta}\vee
z_{\gamma}\}_{\beta\times \gamma\in J\times L}\rightarrow y*z$
und $\{x_{\alpha}\vee
y_{\beta}\}_{\alpha\times \beta\in I\times J}\rightarrow x*y$.
\item
Zunächst ist wieder klar, dass für jeden solchen Bimodul
$\mathcal{M}$ eine Vervollständigung $\hat{\mathcal{M}}$
existiert. Des Weiteren induzieren $*_{L}$ und $*_{R}$ stetige, bilineare Abbildungen
auf den dichten Teilräumen $i(\Ss^{\bullet}(\V))\subseteq
\Hol$ und $i'(M)\subseteq \hat{M}$, womit stetige
bilineare Fortsetzungen $\hat{*}_{L}$ und $\hat{*}_{R}$ existieren. Die
$\Hol$-Verträglichkeit, also $\hat{1}\hat{*}_{L}
\hat{m}=\hat{m}=\hat{m}\hat{*}_{R}\hat{1}$ für alle
$\hat{m}\in \hat{\mathcal{M}}$ folgt dann wie
die Unitarität in \textit{i.)}, und die Vererbung der
Bimoduleigenschaft wie die Assoziativität in \textit{i.)}.
\end{enumerate}
\end{beweis}
\end{proposition}
Punkt \textit{ii.)} ist unter anderem als Motivation dafür gedacht, dass
überhaupt derartige $\Hol-\Hol$-Bimoduln existieren. Als wichtiges
Resultat aus \textit{i.)} erhalten wir umgehend:
\begin{korollar}
\label{kor:HolBarBimodul}
Gegeben ein hlkVR $\V$ und ein lokalkonvexer
$\mathrm{Hol}(\V)-\mathrm{Hol}(\V)$-Bimodul $\mathcal{M}$, so gilt:
\begin{align*}
HH_{\operatorname{\mathrm{cont}}}^{k}(\Hol,\mathcal{M})\cong
H^{k}\left(\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}(\C_{c},\mathcal{M}),d^{*}_{c}\right).
\end{align*}
Hierbei bezeichnet $(\C_{c},d_{c})$ den zu $\mathcal{A}=\Hol$ gehörigen,
topologischen Bar-Komplex.
\begin{beweis}
Dies folgt schon wie im letzten Abschnitt aus
Proposition \ref{prop:topBarKomplexprop}~\textit{iii.)}, da\\ $(\Hol,\Pp_{H},*)$ mit
Proposition \ref{prop:HollkAlgebra} eine
lokalkonvexe, unitäre und assoziative $\mathbb{K}$-Algebra ist.
\end{beweis}
\end{korollar}
Wir wollen den Raum $\Hol$ zunächst mit ein wenig Anschauung füllen. Hierfür definieren wir:
\begin{definition}
\label{def:Potenzr}
Gegeben ein hlkVR $(\V,P)$ sowie die topologischen Räume $\left(\widehat{\Ss^{k}(\V)}, \hat{p^{k}}\right)$.\\
Wir bezeichnen mit $\left(\prod_{\widehat{\Ss^{\bullet}(\V)}}, \Pp_{\times}\right)$ den hlkVR
\begin{equation*}
\textstyle\prod_{\widehat{\Ss^{\bullet}(\V)}}=\left\{\displaystyle\prod_{k=0}^{\infty}\widehat{\Ss^{k}(\V)}\ni\hat{\omega}= (\hat{\omega}_{0}\:,\hat{\omega}_{1}\:,\hat{\omega}_{2}\:,\dots)\:\Bigg| \p_{\times}(\hat{\omega})=\sum_{k=0}^{\infty}\hat{p^{k}}(\hat{\omega}_{k})<\infty, \:\forall\: p\in \tilde{P}\right\}.
\end{equation*}
\end{definition}
Zusammen mit Satz \ref{satz:vervollsthlkVR} zeigt der Folgende, dass wir $\left(\Hol,\Pp_{H}\right)$ mit dem Potenzreihenraum $\left(\prod_{\widehat{\Ss^{\bullet}(\V)}}, \Pp_{\times}\right)$ identifizieren dürfen.
\begin{satz}[Potenzreihen]
\label{satz:Potenz}
Gegeben ein hlkVR $(\V,P)$, dann gilt:
\begin{enumerate}
\item
$\left(\prod_{\widehat{\Ss^{\bullet}(\V)}}, \Pp_{\times}\right)$ ist vollständig.
\item
$\displaystyle\bigoplus_{k=0}^{\infty}\widehat{\Ss^{k}(\V)}$, topologisiert vermöge $\Pp_{\times}$, ist folgendicht in $\left(\prod_{\widehat{\Ss^{\bullet}(\V)}}, \Pp_{\times}\right)$.
\item
$\left(\Ss^{\bullet}(\V),\Pp\right)$ ist dicht in
$\left(\prod_{\widehat{\Ss^{\bullet}(\V)}},\Pp_{\times}\right)$
vermöge isometrischer Einbettung:
\begin{align*}
i\colon\sum_{k}\omega_{k}&\longmapsto \prod_{k=0}^{\infty}i_{k}(\omega_{k})\\
\sum_{k=0}^{\infty}p^{k}&\longmapsto \sum_{k=0}^{\infty}\hat{p^{k}}.
\end{align*}
Dabei bezeichnen die $i_{k}$ die Isometrien
$i_{k}\colon\Ss^{k}(\V)\hookrightarrow
\widehat{\Ss^{k}(\V)}$.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Sei $\{\hat{\omega}_{\alpha}\}_{\alpha\in I}\subseteq
\prod_{\widehat{\Ss^{\bullet}(\V)}}$ ein Cauchynetz, dann
existiert für jedes $\epsilon\geq 0$ ein $\alpha_{\epsilon}\in I$, so dass:
\begin{equation*}
\sum_{k=0}^{\infty}\hat{p^{k}}\left(\hat{\omega}_{\alpha}^{k}-\hat{\omega}_{\beta}^{k}\right)< \epsilon\qquad\quad\forall\:\alpha,\beta \geq \alpha_{\epsilon}\in I.
\end{equation*}
Damit ist insbesondere
$\left\{\hat{\omega}_{\alpha}^{k}\right\}_{\alpha\in I}\subseteq
\widehat{\Ss^{k}(\V)}$ für jedes $k\in \mathbb{N}$ ein Cauchynetz, womit
$\left\{\hat{\omega}_{\alpha}^{k}\right\}_{\alpha\in
I}\rightarrow \hat{\omega}^{k}$ mit eindeutigem
$\hat{\omega}^{k}\in \widehat{\Ss^{k}(\V)}$ gilt.
Wir definieren
$\hat{\omega}=\displaystyle\prod_{k=0}^{\infty}\hat{\omega}^{k}$
und behaupten, dass dann
$\p_{\times}(\hat{\omega})<\infty$ für alle
$p\in\tilde{P}$ sowie
$\left\{\hat{\omega}_{\alpha}\right\}_{\alpha\in
I}\rightarrow \hat{\omega}$ erfüllt ist.
Nun gilt
\begin{equation*}
\p_{\times}(\hat{\omega}_{\alpha})\leq \p_{\times}(\hat{\omega}_{\alpha}-\hat{\omega}_{\alpha_{\epsilon}})+\p_{\times}(\hat{\omega}_{\alpha_{\epsilon}})<\epsilon + \p_{\times}(\hat{\omega}_{\alpha_{\epsilon}})= \wt{c}\qquad\forall\:\alpha\geq \alpha_{\epsilon}\in I
\end{equation*}
und folglich:
\begin{equation*}
\sum_{k=0}^{n}\hat{p^{k}}\left(\hat{\omega}_{\alpha}^{k}\right)
\leq \p_{\times}(\hat{\omega}_{\alpha})
< \wt{c}\qquad\quad\forall\: \alpha\geq \alpha_{\epsilon}\in I,\:\forall \: n\in \mathbb{N}.
\end{equation*}
Hiermit erhalten wir
\begin{equation*}
\tau_{n}=\sum_{k=0}^{n}\hat{p^{k}}\left(\hat{\omega}^{k}\right)= \sum_{k=0}^{n}\lim_{\alpha}\:\hat{p^{k}}\left(\hat{\omega}_{\alpha}^{k}\right)= \lim_{\alpha}\:\sum_{k=0}^{n}\hat{p^{k}}\left(\hat{\omega}_{\alpha}^{k}\right)\leq \hat{c}\qquad\quad\forall\:n\in \mathbb{N},
\end{equation*} womit
$\left\{\tau_{n}\right\}_{n\in \mathbb{N}}$ eine
Cauchyfolge ist, da alle Summanden positiv sind.
Die zeigt die Existenz des Limes $n\rightarrow \infty$ und es folgt:
\begin{equation*}
\p_{\times}(\hat{\omega})=\lim_{n}\:\sum_{k=0}^{n}\hat{p^{k}}\left(\hat{\omega}^{k}\right)=\lim_{n}\tau_{n}\leq \hat{c}<\infty.
\end{equation*}
Für die Konvergenzaussage beachten wir, dass
\begin{equation*}
\mu_{n}=\sum_{k=0}^{n}\hat{p^{k}}\left(\hat{\omega}_{\alpha}^{k}-\hat{\omega}_{\beta}^{k}\right)
\leq\p_{\times}\Big(\hat{\omega}_{\alpha}-\hat{\omega}_{\beta}\Big)< \epsilon\qquad\forall\:\alpha,\:\beta\geq \alpha_{\epsilon},
\end{equation*}
womit $\mu_{n}$ eine Cauchyfolge ist und der Limes
existiert. Es folgt:
\begin{equation*}
\sum_{k=0}^{n}\hat{p^{k}}\left(\hat{\omega}^{k}-\hat{\omega}_{\beta}^{k}\right)
=\lim_{\alpha}\:\sum_{k=0}^{n}\hat{p^{k}}\left(\hat{\omega}_{\alpha}^{k}-\hat{\omega}_{\beta}^{k}\right)\leq \epsilon\qquad\forall\:\beta\geq \alpha_{\epsilon},\:\forall\:n\in \mathbb{N}.
\end{equation*
Dies zeigt
\begin{equation*}
\p_{\times}\Big(\hat{\omega}-\hat{\omega}_{\beta}\Big)=\lim_{n}\sum_{k=0}^{n}\hat{p^{k}}\left(\hat{\omega}^{k}-\hat{\omega}_{\beta}^{k}\right)
\leq \epsilon\qquad\forall\:\beta\geq \alpha_{\epsilon},
\end{equation*}
also $\{\hat{\omega}_{\alpha}\}_{\alpha\in I}\rightarrow \hat{\omega}\in \prod_{\widehat{\Ss^{\bullet}(\V)}}$.
\item
Sei $\prod_{\widehat{\Ss^{\bullet}(\V)}}\ni\hat{\omega}=\left(\hat{\omega}^{0},\:\hat{\omega}^{1},\:\hat{\omega}^{2},\dots\right)$ und
\begin{equation*}
\displaystyle\bigoplus_{k=0}^{\infty}\widehat{\Ss^{k}(\V)}\ni\hat{\omega}_{n}=\left(\hat{\omega}^{0},\:\hat{\omega}^{1},\dots,\:\hat{\omega}^{n},0,0,0,\dots\right).
\end{equation*}
Dann ist
$\displaystyle\lim_{n}\p_{\times}(\hat{\omega}_{n})=\displaystyle\lim_{n}\:\sum_{k=0}^{n}\hat{p^{k}}(\hat{\omega}_{k})=\p_{\times}(\hat{\omega})=c<\infty$,
also $\left\{\p_{\times}(\hat{\omega}_{n})\right\}_{n\in
\mathbb{N}}$ eine Cauchyfolge, womit
$\p_{\times}(\hat{\omega}_{m}-\hat{\omega}_{n})=\displaystyle\sum_{k=n+1}^{m}\hat{p^{k}}(\hat{\omega}^{k})
=|\p_{\times}(\hat{\omega}_{m})-\p_{\times}(\hat{\omega}_{n})|<\epsilon\:$ f"ur alle $m\geq n\geq N_{\epsilon}$.
Es folgt
\begin{equation*}
\p_{\times}(\hat{\omega}-\hat{\omega}_{n})=\lim_{m}\p_{\times}(\hat{\omega}_{m}-\hat{\omega}_{n})=\lim_{m}|\p_{\times}(\hat{\omega}_{m})-\p_{\times}(\hat{\omega}_{n})|\leq\epsilon\qquad\quad\forall\:n\geq N_{\epsilon}
\end{equation*}
und somit $\left\{\hat{\omega}_{n}\right\}_{n\in \mathbb{N}}\rightarrow \hat{\omega}$.
\item
Seien $\hat{\omega}$ und $\hat{\omega}_{n}$ wie in
\textit{ii.)}. Wir fassen $P\times \mathbb{N}$ als
gerichtete Menge auf\footnote{vgl. Definition \ref{def:kanNetzIsoetc}~\textit{ii.)}} und wählen für jedes Element $(p,n)$ ein $k_{p,n}\in \mathbb{N}$ derart, dass
\begin{equation*}
\p_{\times}(\hat{\omega}-\hat{\omega}_{k})< \frac{1}{2n}\qquad\quad\forall\:k\geq k_{p,n}
\end{equation*}
gilt, was nach \textit{ii.)} ohne Einschränkung möglich ist.
Ferner denken wir uns $\Ss^{\bullet}(\V)\subseteq \displaystyle\bigoplus_{k=0}^{\infty}\widehat{\Ss^{k}(\V)}$ isometrisch eingebettet und finden für jedes $k\in \mathbb{N}$ ein Netz $\left\{\omega^{k}_{\alpha_{k}}\right\}_{\alpha_{k} \in I_{k}}\subseteq \Ss^{k}(\V)$ mit $\left\{\omega^{k}_{\alpha_{k}}\right\}_{\alpha_{k} \in I_{k}}\rightarrow \hat{\omega}^{k}\in \widehat{\Ss^{k}(\V)}$.\\\\
Für besagtes $(p,n)$ und $0\leq k \leq k_{p,n}$ bedeutet dies die Existez von Indizes $\alpha_{k}\in I_{k}$ derart, dass
\begin{equation*}
\hat{p^{k}}\left(\hat{\omega}^{k}-\omega_{\alpha_{k}}^{k}\right)<\frac{1}{2n(k_{p,n}+1)}.
\end{equation*}
Wir definieren $\omega_{p,n}=\left(\omega^{0}_{\alpha_{1}},\:\omega^{1}_{\alpha_{2}},\dots,\: \omega^{k_{p,n}}_{\alpha_{k_{p,n}}},0,0,0,\dots\right)$, womit
\begin{equation*}
\p_{\times}\left(\hat{\omega}_{k_{p,n}}-\omega_{p,n}\right)=\:\sum_{k=0}^{k_{p,n}}\:\hat{p}^{k}\left(\hat{\omega}^{k}-\omega^{k}_{\alpha_{k}}\right)< \frac{1}{2n}
\end{equation*}
und folglich für alle $(p',n')\geq (p,n)$:
\begin{align*}
\p_{\times}(\hat{\omega}-\omega_{p',n'})\leq&\: \p_{\times}\left(\hat{\omega}-\hat{\omega}_{k_{p',n'}}\right)+\p_{\times}\left(\hat{\omega}_{k_{p',n'}}-\omega_{p',n'}\right)
\\\leq&\: \p'_{\times}\left(\hat{\omega}-\hat{\omega}_{k_{p',n'}}\right)+\p'_{\times}\left(\hat{\omega}_{k_{p',n'}}-\omega_{p',n'}\right)
\\<& \:\frac{1}{2n'}+\frac{1}{2n'}\:\leq\: \frac{1}{n}.
\end{align*}Dies zeigt $\{\omega_{p,n}\}_{P\times
\mathbb{N}}\rightarrow \hat{\omega}$ und mit Bemerkung \ref{bem:Netzbem}~\textit{iii.)} die Behauptung.
\end{enumerate}
\end{beweis}
\end{satz}
\begin{bemerkung}
Obiger Satz besagt also insbesondere, dass die Vervollständigung
von $(\SsV,\Pp)$ bereits durch die Vervollständigungen der
$\left(\Ss^{k},\pi_{k}\right)$ festgelegt ist. Des Weiteren ist es sogar
möglich, jedes $\hat{\omega}\in\widehat{\mathrm{S}^{\bullet}(\mathbb{V})}$ durch eine Folge in
$\{\omega_{n}\}_{n\in
\mathbb{N}}\subseteq\displaystyle\bigoplus_{k=0}^{\infty}\widehat{\Ss^{k}(\V)}$
zu approximieren. Die Schwierigkeit liegt hierbei also in der
Vervollständigung der $\left(\Ss^{k},\pi_{k}\right)$, die für
unendlich-dimensionales $\V$ im Allgemeinen alles andere als trivial
ist. Für endlich-dimensionales $\V$ hingegen ist
$\left(\Ss^{k},\pi_{k}\right)$ bereits vollständig, vgl. Beispiel \ref{bsp:holomorpheFunkts}~\textit{i.)}.
\end{bemerkung}
\begin{beispiel}[Holomorphe Funktionen]
\label{bsp:holomorpheFunkts}
\begin{enumerate}
\item
Wir versehen den Vektorraum $\mathbb{C}^{n*}$
mit der üblichen euklidischen Normtopologie. Mit der
Äquvivalenz aller Normen auf $\mathbb{C}^{n*}$ können wir uns
wahlweise auf das System, bestehend aus allen bezüglich der
Maximumsnorm
\begin{equation*}
p_{\mathrm{max}}(x)=\sum_{i=1}^{n}|x_{i}|\quad \text{mit}\quad
x=\sum_{i=1}^{n}x_{i}\: e^{i}
\end{equation*}
stetigen Halbnormen festlegen, und verschaffen uns so
ein filtrierendes System $\tilde{P}$ auf
$\mathbb{C}^{n*}$. Dieses enthält dann insbesondere wieder alle
Normen der Form $|c|\:p_{\mathrm{max}}$ für positive Konstanten $|c|$. Die
symmetrische Algebra sei wie in Abschnitt \ref{subsec:Vorber}
topologisiert vermöge $\Pp$.
Wir behaupten zunächst, dass
\begin{equation}
\label{eq:pitopendlCn}
p^{k}_{\mathrm{max}}(z)=\sum_{i_{1},…,i_{k}}^{n}|a_{i_{1}},…,a_{i_{k}}|
\end{equation}
für alle $\Tt^{\bullet}(\mathbb{C}^{n*})\ni
z=\displaystyle\sum_{i_{1},…,i_{k}}^{n}
a_{i_{1},…,i_{k}}e^{i_{1}}\ot…\ot e^{i_{k}}$ gilt. Hierbei beachte
man, dass dann \eqref{eq:pitopendlCn} mit unserer
Konvention $e^{i_{1}}\vee…\vee e^{i_{k}}=\frac{1}{k!}\sum_{\sigma\in
S_{k}}e^{\sigma(i_{1})}\ot…\ot e^{\sigma(i_{k})}$ ebenso für alle $\Ss^{\bullet}(\mathbb{C}^{n*})\ni
z=\displaystyle\sum_{i_{1},…,i_{k}}^{n}
a_{i_{1},…,i_{k}}e^{i_{1}}\vee…\vee e^{i_{k}}$
richtig ist.
Bezeichne hierfür $p_{\ot}^{k}$ die durch
\eqref{eq:pitopendlCn} charakterisierte Norm, dann gilt
\begin{equation*}
p_{\mathrm{max}}^{k}(z)\leq
\sum_{i_{1},…,i_{k}}^{n}|a_{i_{1},…,a_{i_{k}}}e^{i_{1}}|\cdot |e^{i_{2}}|…|e^{i_{k}}|=p_{\ot}^{k}(z)
\end{equation*}per Definition von $p_{\mathrm{max}}^{k}$. Mit der Normeigenschaft von $p^{\ot k}$ folgt weiter, dass
\begin{equation*}
p_{\ot}^{k}(z)\leq \sum_{i}p_{\ot}^{k}(z^{i})=\sum_{i}p_{\mathrm{max}}^{k}(z^{i})
\end{equation*}
für alle Zerlegungen $z=\displaystyle\sum_{i}
z^{i}$ in separable $z^{i}=x_{1}^{i}\ot…\ot x_{k}^{i}$ richtig
ist. Dabei
folgt die zweite Gleichheit mit $x_{j}=\displaystyle\sum_{i_{j}}(x_{j})_{i_{j}}e^{i_{j}}$ aus:
\begin{align*}
p_{\ot}^{k}(x_{1}\ot…\ot x_{k})=&\sum_{i_{1},…,i_{k}}^{n}|(x_{1})_{i_{1}}\cdot…\cdot(x_{k})_{i_{k}}|=\sum_{i_{1},…,i_{k}}^{n}|(x_{1})_{i_{1}}|\cdot…\cdot|(x_{k})_{i_{k}}|
\\=&\:p_{\mathrm{max}}(x_{1})\cdot…\cdot p_{\mathrm{max}}(x_{k})=\:p_{\mathrm{max}}^{k}(x_{1}\ot…\ot x_{k}).
\end{align*}
\\
Um die Vervollständigung
$(\mathrm{Hol}(\mathbb{C}^{n*}),\hat{\Pp})$ von
$(\Ss^{\bullet}(\mathbb{C}^{n*}),\Pp)$ zu charakterisieren beachten wir, dass
für festes $k\in \mathbb{N}$ die Topologie auf
$\Ss^{k}(\mathbb{C}^{n*})$ bereits durch die Norm $p^{k}_{\max}$
erzeugt wird. Dies folgt unmittelbar aus
Korollar \ref{kor:HNTop}~\textit{iii.)}, da mit
Satz \ref{satz:wichtigerSatzueberHalbnormentopologien}~\textit{vi.)}
$p\leq |c|\: p_{\max}$ für alle $p\in \wt{P}$ und somit
ebenfalls $p^{k}\leq |c|^{k} p_{\max}^{k}$ für alle $p\in
\wt{P}$ gilt. Bemerkung \ref{bem:Netzbem}~\textit{i.)} zeigt dann,
dass wir lediglich Folgenvollständigkeit nachweisen müssen,
wenn wir
$\widehat{\Ss^{k}(\mathbb{C}^{n*})}=\Ss^{k}(\mathbb{C}^{n*})$
zeigen wollen. Sei hierfür $\{z_{n}\}_{n\in \mathbb{N}}\subseteq
\Ss^{k}(\mathbb{C}^{n*})$ eine Cauchyfolge, dann ist
\begin{align*}
\sum_{i_{1},…,i_{k}}^{n}|a^{m}_{i_{1},…,i_{k}}-a^{n}_{i_{1},…,i_{k}}|=p^{k}_{\max}(z_{m}-z_{n})< \epsilon\qquad\forall\:m,n\geq N_{\epsilon},
\end{align*}und mit der Vollständigkeit von $\mathbb{C}$ zeigt
dies, dass $a^{n}_{i_{1},…,i_{k}}\longrightarrow a_{i_{1},…,i_{k}}\in
\mathbb{C}$. Eine analoge Abschätzung liefert $z_{n}\longrightarrow
\displaystyle\sum_{i_{1},…,i_{k}}^{n}a_{i_{1},…,i_{k}}e^{i_{1}}\vee…\vee
e^{i_{k}}\in \Ss^{k}(\mathbb{C}^{n*})$, was die
Folgenvollständigkeit beweist. Hierbei ist wesentlich
eingegangen, dass $\Ss^{k}(\mathbb{C}^{n*})$ eine endliche
Basis besitzt.
Nach Satz \ref{satz:Potenz} ist dann
\begin{equation*}
\mathrm{Hol}(\mathbb{C}^{n*})=\left\{\displaystyle\prod_{k=0}^{\infty}\Ss^{k}(\mathbb{C}^{n*})\ni\hat{\omega}= (\omega_{0}\:,\omega_{1}\:,\omega_{2}\:,\dots)\:\Bigg| \sum_{k=0}^{\infty}{p}^{k}(\omega_{k})<\infty, \:\forall\: p\in \tilde{P}\right\},
\end{equation*}und für $p_{z}=|z|p_{\max}$ bedeutet dies:
\begin{equation}
\label{eq:Potreiendl}
\begin{split}
\p_{z}(\omega)=&\:\p_{z}\left(\prod_{k=0}^{\infty}\sum_{i_{1},…,i_{k}}^{n}a_{i_{1},…,i_{k}}e^{i_{1}}\vee…\vee e^{i_{k}}\right)
\\=&\:\sum_{k=0}^{\infty}p_{z}^{k}\left(\sum_{i_{1},…,i_{k}}^{n}a_{i_{1},…,i_{k}}e^{i_{1}}\vee…\vee e^{i_{k}}\right)
\\=&\:\sum_{k=0}^{\infty}\sum_{i_{1},…,i_{k}}^{n}|a_{i_{1},…,i_{k}}||z|^{k}
\\<&\:\infty.
\end{split}
\end{equation}
Vermöge der bijektiven Zuordnung
\begin{equation}
\label{eq:isomPolSym}
\begin{split}
\Pol^{k}(\mathbb{C}^{n})&\longleftrightarrow \Ss^{k}(\mathbb{C}^{n*})\\
a_{i_{1},…,i_{k}}z^{i_{1}}…z^{i_{k}}&\longleftrightarrow
a_{i_{1},…,i_{k}} e^{i_{1}}\vee…\vee e^{i_{k}},
\end{split}
\end{equation}ist jedes $\Ss^{k}(\mathbb{C}^{n*})$
isomorph zu $\Pol^{k}(\mathbb{C}^{n})$, dem Raum der Polynome
$k$-ten Grades auf $\mathbb{C}^{n}$. Für $\left(\begin{array}{c} z_{1} \\ \vdots\\
z_{n} \end{array}\right)\in \mathbb{C}^{n}$ sei
$|z|=\displaystyle\max_{0\leq i\leq n}|z_{i}|$. Dann zeigen
\eqref{eq:Potreiendl} und \eqref{eq:isomPolSym}, dass jedes
Element aus $\mathrm{Hol(\mathbb{C}^{n*})}$ einer absolut
konvergenten Potenzreihe auf $\mathbb{C}^{n}$
entspricht. Umgekehrt ist für jede derartige Potenzreihe
\begin{equation*}^{}
\sum_{k=0}^{\infty}\sum_{i_{1},…,i_{k}}^{n}|a_{i_{1},…,i_{k}}||z|^{k}<
\infty\qquad\forall\:|z|\geq 0,
\end{equation*}und da $p\leq |z|\:p_{max}$ für alle $p\in
\wt{P}$, folgt:
\begin{equation*}
\p\left(\sum_{k=0}^{\infty}\sum_{i_{1},…,i_{k}}^{n}a_{i_{1},…,i_{k}}e^{i_{1}}\vee…\vee
e^{i_{k}}\right)\leq \sum_{k=0}^{\infty}\sum_{i_{1},…,i_{k}}^{n}|a_{i_{1},…,i_{k}}||z|^{k}
< \infty.
\end{equation*}Insgesamt zeigt dies, dass wir
$\big(\mathrm{Hol}(\mathbb{C}^{n*}),\hat{\Pp}\big)$ mit den
auf $\mathbb{C}^{n}$ absolut konvergenten Potenzreihen, also
mit dem Raum der ganz holomorphen Funktionen $\mathit{Hol}(\mathbb{C}^{n})$
identifizieren dürfen und liefert die Begründung für die
Wahl der Bezeichnung $\mathrm{Hol}$.
Die stetige Fortsetzung
der Halbnorm $\p_{z}$ ist dann in
Multiindexschreibweise auch darstellbar in der Form:
\begin{equation*}
\hat{\p}_{z}(\phi)=\sum_{k=0}^{\infty}\frac{|z|^{k}}{\alpha!}\left|\frac{\pt^{k}\phi}{\pt
x^{\alpha_{1}}…\pt x^{\alpha_{n}}}(0)\right|\quad\quad
\forall\:\phi\in \mathit{Hol}(\mathbb{C}^{n}).
\end{equation*}
\item
Sei $\mathbb{V}=\mathbb{K}^{|\mathbb{N}|}$ oder ein anderer
unendlichdimensionaler $\mathbb{K}$-Vektorraum und
$\V^{*}$ schwach* topologisiert,
vermöge den Halbnormen $P^{*}=\left\{p_{v}\right\}_{v\in \V}$ mit
\begin{equation*}
p_{v}(\phi)=|\phi(v)|\quad\quad \forall\:\phi\in
\V^{*}.
\end{equation*}
Insbesondere ist dann bereits $|c|\:p_{v}=p_{cv}$ in $P^{*}$
enthalten und
$p_{\max}=\displaystyle\sum_{i=1}^{n}p_{e_{i}}$ zeigt, dass wir es hier in
der Tat mit einer Verallgemeinerung von \textit{i.)} zu tun
haben. Bezeichne wieder $\tilde{P}^{*}$ das filtrierende und
separierende System aller bezüglich
dieser Topologie stetigen Halbnormen.
Wegen Satz \ref{satz:Potenz} dürfen wir uns
$\mathrm{Hol}(\V^{*})$ als unendliche Potenzreihen mit
Summanden in den $\widehat{\Ss^{k}(\V^{*})}$ vorstellen, und
wir wollen im Folgenden zeigen, dass jedes $h\in
\mathrm{Hol}(\V^{*})$ sogar eine komplexwertige Potenzreihe im
Funktionensinne auf $\V$ definiert.
Sei hierfür $\left\{h_{\alpha}\right\}_{\alpha\in I}\subseteq\Ss^{\bullet}(\V^{*})$ mit
$\left\{h_{\alpha}\right\}_{\alpha\in I}\rightarrow h$. Dann
ist $\left\{h_{\alpha}\right\}_{\alpha\in I}$ insbesondere ein
Cauchynetz und somit $\p_{v}\left(h_{\alpha}-h_{\beta}\right)<
\epsilon$ für alle $\alpha,\beta\geq \alpha_{\epsilon}$.
Für $u_{1}\ot…\ot u_{k}\in \Tt^{k}(\V^{*})$ und $v\in \V$ ist $\tau_{v}\colon
u_{1}\ot…\ot u_{k} \longmapsto u_{1}(v)…u_{k}(v)$, vermöge linearer
Fortsetzung durch Korollar \ref{kor:WohldefTensorprodabbildungen}, auf
ganz $\Tt^{k}(\V^{*})$ wohldefiniert. Sei weiter
$\left(\Delta_{k}^{*}u^{k}\right)(v):=\tau_{v}\left(u^{k}\right)$,
so folgt $\left(\Delta_{k}^{*}u_{1}\vee…\vee u_{k}\right)(v)=u_{1}(v)…u_{k}(v)$.
Nun ist $h_{\alpha}=\displaystyle\sum_{k}u_{\alpha}^{k}$ mit
$u_{\alpha}^{k}\in \Ss^{k}(\V^{*})$ und endlicher Summe, und
wir definieren:
\begin{align*}
h(v)=\lim_{\alpha}\:\sum_{k}\left(\Delta^{*}_{k}u_{\alpha}^{k}\right)(v)\qquad \forall\:v\in \V
\end{align*}
Um
die Wohldefiniertheit dieser Abbildung zu zeigen, sei $u^{k}\in \Ss^{k}(\V^{*})$, dann folgt
\begin{equation}
\label{eq:abskonv}
\begin{split}
\left(\Delta_{k}^{*}u^{k}\right)(v)=&\sum_{i=1}^{n}u^{i}_{1}(v)…u^{i}_{k}(v)\leq \left|\sum_{i=1}^{n}u^{i}_{1}(v)…u^{i}_{k}(v)\right|
\\\leq&
\sum_{i=1}^{n}\left|u^{i}_{1}(v)…u^{i}_{k}(v)\right|=
\sum_{i=1}^{n}p_{v}(u^{i}_{1})…p_{v}(u^{i}_{k})
\end{split}
\end{equation}
für alle Zerlegungen
$\displaystyle\sum_{i=1}^{n}u^{i}_{1}\ot…\ot u^{i}_{k}$ von
$u^{k}$ und somit $\left(\Delta_{k}^{*}u^{k}\right)(v)\leq
p_{v}^{k}(u)$.\\
Dies bedeutet
\begin{align*}
|h_{\alpha}(v)-h_{\beta}(v)|=&\:\left|\:\sum_{k}\left(\Delta^{*}_{k}\left[u_{\alpha}^{k}-u_{\beta}^{k}\right]\right)(v)\:\right|
\leq \:\sum_{k}p_{v}^{k}\left(u_{\alpha}^{k}-u_{\beta}^{k}\right)
\\=&\:\p_{v}(h_{\alpha}-h_{\beta})<\:\epsilon
\end{align*}
für alle $\alpha,\beta\geq \alpha_{\epsilon}$. Damit ist
$\left\{h_{\alpha}(v)\right\}_{\alpha\in I}$ ein Cauchynetz in $\mathbb{C}$
und $h(v)=\displaystyle\lim_{\alpha} h_{\alpha}(v)$
existiert. Für die Unabhängigkeit obiger
Definition von der Wahl des Netzes sei
$\Ss^{\bullet}(\V^{*})\supseteq\big\{h'_{\beta}\big\}_{\beta\in J}$ ein weiteres Netz
mit $\big\{h'_{\beta}\big\}_{\beta\in J}\rightarrow
h$. Dann gilt
\begin{align*}
|h(v)-h'_{\beta}(v)|\leq&\:
|h(v)-h_{\alpha}(v)|+|h_{\alpha}(v)-h'_{\beta}(v)|
\\=&\:|h(v)-h_{\alpha}(v)|+\p_{v}(h_{\alpha}-h'_{\beta}).
\end{align*}Wegen $\left\{h_{\alpha}(v)\right\}_{\alpha\in
I}\rightarrow h(v)$ existiert ein $\alpha_{\epsilon}\in I$, so
dass $|h(v)-h_{\alpha}(v)|\leq \frac{\epsilon}{2}$ für alle
$\alpha\geq \alpha_{\epsilon}$ und wegen
$\left\{h_{\alpha}(v)\right\}_{\alpha\in
I}\sim\big\{h'_{\beta}\big\}_{\beta\in J}$
(vgl. Definition \ref{def:kanNetzIsoetc}~\textit{i.)}) ein $(\alpha',\beta')\in
I\times J$, so dass $\p_{v}(h_{\alpha}-h'_{\beta})\leq
\frac{\epsilon}{2}$ für alle $(\alpha,\beta)\geq
(\alpha',\beta')$. Insgesamt zeigt dies
$|h(v)-h'_{\beta}(v)|\leq \epsilon$ für alle $\alpha\geq
\wt{\alpha}$ mit $\wt{\alpha}\geq \alpha_{\epsilon},\alpha'$, also $\big\{h'_{\beta}\big\}_{\beta\in J}\rightarrow
h$.
Als Beispiel sei $\hat{u}\in \widehat{\V^{*}}$, dann ist
$\overbrace{\hat{u}\vee\dots\vee \hat{u}}^{k}\in
\widehat{\Ss^{k}(\V^{*})}$ wegen $\widehat{\Ss^{k}\left(\V^{*}\right)}\cong
\widehat{\Ss^{k}\big(\widehat{\V^{*}}\big)}$ nach Satz \ref{satz:PiTopsatz}~\textit{vi.)}, und wir definieren
$\exp(\hat{u})=\displaystyle\prod_{k=0}^{\infty}\frac{1}{k!}\:\overbrace{\hat{u}\vee\dots\vee
\hat{u}}^{k}$.\\ Dann folgt für $q=p_{v}$\::
\begin{align*}
\q_{\times}(\exp(\hat{u}))=&\:\sum_{k=0}^{\infty}\frac{1}{k!}\:\hat{p_{v}^{k}}\left(\hat{u}\vee\dots\vee\hat{u}\right)=\:\sum_{k=0}^{\infty}\frac{1}{k!}\:\hat{p}_{v}^{k}\left(\hat{u}*\dots*\hat{u}\right)
\\\leq& \:\sum_{k=0}^{\infty}\frac{1}{k!}\:\left(\hat{p}_{v}\left(\hat{u}\right)\right)^{k}<\infty,
\end{align*}
also $\exp(\hat{u})\in \mathrm{Hol}(\V^{*})$. Dabei folgt die letzte Ungleichung mit der
Submultiplikativität von $(\mathrm{Hol},\hat{\Pp},*)$ nach
Proposition \ref{prop:HollkAlgebra}~\textit{i.)}.
Allgemein funktioniert diese Konstruktion für alle absolut konvergenten Potenz-reihen
$\displaystyle\sum_{k=0}^{\infty}a_{k}z^{k}$ auf
$\mathbb{C}$.
Ganz allgemein können wir also die Elemente aus $\Hol$ immer als so
etwas, wie ganz holomorphen Funktionen auf dem
Prädualraum $\V_{*}$\footnote{$(\V_{*})^{*}=\V$} auffassen, sofern er
existiert.
\item
Das Resultat aus \textit{ii.)} lässt sich auch allgemeiner
formulieren. Seien hierfür $\V$ ein $\mathbb{K}$-Vektorraum
und $(\mathbb{U},P)$ ein hausdorffscher, lokalkonvexer $\mathbb{K}$-Vektorraum
derart, dass eine $\mathbb{K}$-bilineare Abbildung
$\tau\colon \V\times \mathbb{U}\colon \longrightarrow \mathbb{K}$
existiert, deren Bild sich für festes $v\in \V$
in der Form
\begin{equation*}
\tau(v,u) \leq p_{v}(u)\qquad p_{v}\in P,\:\forall\:u\in \mathbb{U}
\end{equation*}abschätzen lässt. Sei
$\Ss^{\bullet}(\mathbb{U})$, in gewohnter Weise, durch das System $\tilde{P}$ aller bezüglich $P$ stetigen Halbnormen
topologisiert und $\left(\Delta_{k}^{*}u^{k}\right)\colon
v\longrightarrow \mathbb{K}$ durch lineare Fortsetzung von
\begin{equation*}
\left(\Delta_{k}^{*}u_{1}\ot…\ot u_{k}\right)(v)=\tau(v,u_{1})…\tau(v,u_{k})\qquad\forall\: u_{1}\ot…\ot u_{k} \in \Tt^{k}(\mathbb{U})
\end{equation*}auf ganz $\Tt^{k}(\mathbb{U})$ definiert.
Dann folgt wie in \textit{ii.)}, dass
$\left(\Delta_{k}^{*}u^{k}\right)(v)\leq p_{v}^{k}(u^{k})$ für
alle $u^{k}\in \Ss^{k}(\mathbb{U})$ gilt und wir erhalten durch
\begin{align*}
h(v)=\lim_{\alpha}\:\sum_{k}\left(\Delta^{*}_{k}u_{\alpha}^{k}\right)(v)\qquad \forall\:v\in \V
\end{align*}mit $h\in \mathrm{Hol}(\mathbb{U})$ und
$\Ss^{\bullet}(\mathbb{U})\supseteq \{h_{\alpha}\}_{\alpha\in
I}\rightarrow h$ wieder eine wohldefinierte $\mathbb{K}$-wertige
Potenzreihenfunktion auf $\V$. Die restlichen Aussagen aus
\textit{ii.)} gelten dann analog.
Physikalisch relevant sind beispielsweise die Kombinationen
\begin{table*}[h]
\centering
\begin{tabular}{|c|c|}
$\V$ & $\mathbb{U}$\\\hline
$\mathcal{D}(X)$ &
$\big(\mathcal{E}(X),\T_{\mathcal{E}}\big)$, $\big(\mathcal{D}(X),\T_{\mathcal{E}}\big)$, $\big(\mathcal{D}(X),\T_{\mathcal{D}}\big)$\\
$\mathcal{D}'(X)$ &
$\big(\mathcal{D}(X),\T_{\mathcal{D}}\big)$, $\big(\mathcal{D}_{K}(X),\T_{\mathcal{D}_{K}}\big)$\\
$\mathcal{E}'(X)$ &
$\big(\mathcal{E}(X),\T_{\mathcal{E}}\big)$,
$\big(\mathcal{D}(X),\T_{\mathcal{D}}\big)$, $\big(\mathcal{D}(X),\T_{\mathcal{E}}\big)$\\
\end{tabular}
\end{table*}
mit einer offenen Teilmenge $X\subseteq
\mathbb{R}^{n}$.
Hierbei bezeichnet $\mathcal{E}(X)$ die
glatten Funktionen $X\longrightarrow \mathbb{R}$,
$\mathcal{D}(X)\subseteq \mathcal{E}(X)$ die glatten
Funktionen $X\longrightarrow \mathbb{R}$ mit kompakten Träger
$K\subseteq X$ und $\mathcal{D}_{K}(X)\subseteq \mathcal{D}(X)$ die glatten Funktionen
$X\longrightarrow \mathbb{R}$ mit kompaktem Träger
$K'\subseteq K\subseteq X$, wobei hier $K$ ein fest gewähltes
Kompaktum ist.
$\T_{\mathcal{E}}$ ist die durch das filtrierende,
abzählbare System $P_{\mathcal{E}}$, bestehend aus den Halbnormen
\begin{equation*}
p_{K,l}(\phi)=\sup_{\substack{|\alpha|\leq l\\ x\in
K}}\left|\pt^{\alpha}\phi(x)\right|\qquad
\forall\:\phi\in \mathcal{E}(X)
\end{equation*}mit $\alpha\in \mathbb{N}^{n}$ ein Multiindex, $l\in \mathbb{N}$, $K\subseteq X$ kompakt sowie $\displaystyle\pt^{\alpha}=\frac{\pt^{|\alpha|}}{\pt^{\alpha_{1}}
x_{1}…\pt^{\alpha_{n}}x_{n}}$, erzeugte lokalkonvexe
Topologie.
$\T_{\mathcal{D}}$ ist die lokalkonvexe
$\mathcal{D}(X)$-Raum Topologie (vgl. \cite[Def
6.3]{rudin:1991a}), deren erzeugendes Halbnormensystem
eher formaler Natur
ist\footnote{vgl. Minkowski-Funktional:
Bemerkung \ref{bem:Minkowski}}.
$\T_{\mathcal{D}_{K}}$ ist die durch dass
Halbnormensystem $P_{K,l}=\{p_{K,l}\}_{l\in
\mathbb{N}}$ erzeugte, lokalkonvexe Topologie, wobei
hier $K$ wieder fest vorgegeben ist.
$\mathcal{D}'(X)$ ist der
zu $(\mathcal{D}(X),\T_{\mathcal{D}})$
topologische Dualraum, weshalb nach Satz \ref{satz:stetmultabb}
\begin{equation*}
\tau(v,u):=v(u)\leq|v(u)|\leq
\overbrace{|c|\:\tilde{p}(u)}^{\tilde{p}'}\qquad
v\in \mathcal{D}'(X),\:
\forall\:u\in \mathcal{D}(X)
\end{equation*}für eine bezüglich $\T_{\mathcal{D}}$
stetigen Halbnorm $\tilde{p}\in
\tilde{P_{\mathcal{D}}}$ und ein $|c|> 0$ gilt. Man beachte, dass dann $\tilde{p}'$ ebenfalls
wieder stetig ist. Nun lässt sich zeigen (vgl. \cite[Thm
6.6]{rudin:1991a}), dass eine lineare Abbildung
$\phi\colon \mathcal{D}(X)\longrightarrow \mathbb{M}$
in einen weiteren lokalkonvexen Vektorraum
$(\mathbb{M},Q)$ genau dann stetig bezüglich
$\T_{\mathcal{D}}$ ist, wenn für jedes Kompaktum
$K\subseteq X$ die Einschränkung
$\phi\big|_{\mathcal{D}_{K}}$ stetig bezüglich
$\T_{\mathcal{D}_{K}}$ ist. Dies zeigt
\begin{equation*}
\tau(v,u):=v(u)\leq|v(u)|\leq p_{K,l}(u)\qquad\forall\: u\in
\mathcal{D}_{K}(X)
\end{equation*}und somit die zweite Zeile obiger
Tabelle.
$\mathcal{E}'(X)$ ist der topologische
Dualraum von $\mathcal{E}(X)$, womit wir
Definitionsgemäß die Abschätzbarkeit
\begin{equation*}
\tau(v,u):=v(u)\leq |v(u)|\leq |c|\:p_{K,l}(u)\qquad\forall\: u\in \mathcal{E}(X)
\end{equation*}mit $v\in \mathcal{E}'(X)$ und
$|c|\:p_{K,l}\in \tilde{P}_{\epsilon}$ erhalten. Der
Rest der dritten Zeile folgt dann unmittelbar aus dem
bereits gezeigten sowie
$\mathcal{E}'(X)\subseteq
\mathcal{D}'(X)$\footnote{Dies sind gerade die
Elemente aus $\mathcal{D}'(X)$ mit kompakten Träger,
wie bsp. $\delta_{z}\colon \phi\longmapsto \phi(z)$.} und
$\mathcal{D}(X)\subseteq \mathcal{E}(X)$. Dabei
beachte man, dass
$\big(\mathcal{D}(X),\T_{\mathcal{E}}\big)$ im
Gegensatz zu
$\big(\mathcal{E}(X),\T_{\mathcal{E}}\big)$ und\\
$\big(\mathcal{D}(X),\T_{\mathcal{D}}\big)$ weder
vollständig noch Folgen vollständig ist.
Für die erste Zeile erhalten wir mit
$\mathcal{D}(X)\ni v\neq0$ und $\supp(v)=K\subseteq X$, dass
\begin{equation}
\label{eq:stetab}
\begin{split}
\tau_{\alpha}(v,u):=&\:\int_{X}
v(x)\pt^{\alpha}u(x)dx
\\=&\:\int_{K}
v(x)\pt^{\alpha}u(x)\leq \overbrace{\sup_{x\in
K}|v(x)|\mathrm{Vol}(K)}^{|c|>0}\:p_{K,|\alpha|}(u)
\end{split}
\end{equation}für alle $u\in \mathcal{E}(X)$ und somit
auch für alle $u\in \mathcal{D}(X)$ gilt. Wegen
$|c|\:p_{K,l}\in \tilde{P}_{\mathcal{E}}$ begründet dies
die ersten beiden Kombinationen. Für die letzte
überlegt man sich, dass die Halbnormen $p_{l}(u)=
\sup_{\substack{x\in X\\ |\alpha|\leq
l}}|\pt^{\alpha}u(x)|$ mit $l\in \mathbb{N}$ und
somit auch $c\: p_{l}$ in $\tilde{P}_{\mathcal{D}}$
enthalten ist.
Nun gilt
$\mathcal{D}'_{K}(X)\subseteq
\mathcal{D}^{*}_{K}(X)$, $\mathcal{D}'(X)\subseteq
\mathcal{D}^{*}(X)$,
$\mathcal{E}'(X)\subseteq \mathcal{E}^{*}(X)$ und mit
\eqref{eq:stetab} erhalten wir die Stetigkeitsabschätzung:
\begin{equation*}
\phi\colon \psi\longmapsto \tau(\phi,\psi)\leq |c|\:
p_{K,l}(\psi)\qquad\forall\:\psi\in \mathcal{E}(X).
\end{equation*} Hiermit ist jedes $\phi\in \mathcal{D}(X)$ auch als Element
in $\mathcal{E}'(X)$ auffassbar. Umgekehrt
überlegt man sich, dass auch jedes $\psi\in
\mathcal{E}(X)$ als Element in $\mathcal{D}_{K}'(X)$ und mit dem Stetigkeitskriterium der
$\mathcal{D}(X)$-Raum Topologie dann ebenfalls als Element in $\mathcal{D}'(X)$ aufgefasst
werden kann. Zusammen mit \textit{ii.)} liefert dies Kombinationen der Form:
\begin{table}[h]
\centering
\begin{tabular}{|c|c|}
$\V$ & $\mathbb{U}$\\\hline
$\mathcal{D}_{K}(X)$ &
$\big(\mathcal{D}'_{K}(X),\T^{*}\big)$, $\big(\mathcal{E}(X),\T^{*}\big)$ \\
$\mathcal{D}(X)$ &
$\big(\mathcal{D}'(X),\T^{*}\big)$, $\big(\mathcal{E}(X),\T^{*}\big)$\\
$\mathcal{D}(X)$, $\mathcal{E}(X)$ &
$\big(\mathcal{E}'(X),\T^{*}\big)$, $\big(\mathcal{D}(X),\T^{*}\big)$ \\
\end{tabular}
\end{table}
Hierbei bezeichnen $\T^{*}$ die
jeweiligen schwach*-Topologien. In der letzten Zeile haben wir
$\mathcal{E}'(X)\subseteq\mathcal{D}'(X)$
benutzt und insgesamt sind hier natürlich noch sehr viel mehr
Kombinationsmöglichkeiten erlaubt.
Ein einfaches Beispiel für $\V=\mathcal{D}(X)$ und
$\mathbb{U}=\big(\mathcal{E}(X), \T_{\mathcal{E}}\big)$ ist
dann nach \textit{ii.)}
\begin{align*}
\exp(\phi)(\psi)=&\:\sum_{k=0}^{\infty}\frac{1}{k!}\big(\Delta_{k}^{*}\overbrace{\phi\vee…\vee
\phi}^{k-mal}\big)(\psi)
\\=&\:\sum_{k=0}^{\infty}\frac{1}{k!}\overbrace{u(\psi,\phi)…u(\psi,\phi)}^{k-mal}
=\sum_{k=0}^{\infty}\frac{1}{k!}\left(\int_{X}\phi(x)\psi(x)dx\right)^{k}
\end{align*}mit $\phi\in \mathcal{E}(X)$ und $\psi\in
\mathcal{D}(X)$ oder für $\V=\mathcal{D}(X),\mathcal{E}(X)$ und $\mathbb{U}=\big(\mathcal{E}'(X), \T^{*}\big)$:
\begin{equation*}
\exp(\delta_{z})(\psi)=\sum_{k=0}^{\infty}\frac{1}{k!}\left(\delta_{z}(\psi)\right)^{k}=\sum_{k=0}^{\infty}\frac{1}{k!}\psi(z)^{k}
\end{equation*}mit $\psi\in \mathcal{E}(X),\mathcal{D}(X)$ und
$\delta_{z}\in \mathcal{E}'(X)\subseteq \mathcal{D}'(X)$.
Im Allgemeinen sind natürlich auch sehr viel komplexere
Summanden erlaubt. Wie diese im konkreten aussehen dürfen,
hängt dann natürlich auch stark davon ab, ob $\mathbb{U}$
vollständig ist oder nicht. In den vollständigen Fällen
$\big(\mathcal{E}(X),\T_{\mathcal{E}}\big)$,
$\big(\mathcal{D}(X),\T_{\mathcal{D}}\big)$,
$\big(\mathcal{D}_{K}(X),\T_{\mathcal{D}_{K}}\big)$
beispielsweise, braucht man $\Ss^{k}(\mathcal{E}(X))$,
$\Ss^{k}(\mathcal{D}(X))$ und $\Ss^{k}(\mathcal{D}_{K}(X))$ nach
Satz \ref{satz:PiTopsatz}~\textit{vii.)} nur in den $\pi_{k}$-Topologien
vervollständigen und man kann sich überlegen, dass diese als
Teilräum von $\mathcal{E}\left(X^{k}\right)$, $\mathcal{D}\left(X^{k}\right)$
bzw. $\mathcal{D}_{K}\left(X^{k}\right)$ auffassbar sind. Als
Realisierung des Tensorproduktes nimmt man dann beispielsweise
den Teilraum
{\small\begin{equation*}
\mathcal{E}_{\mathrm{sep}}\left(X^{k}\right)=\left\{\phi\in
\mathcal{E}\left(X^{k}\right)\:\Bigg|\:\phi(x_{1},…,x_{k})=\sum_{i=1}^{n}\phi_{1}(x_{1})…\phi_{k}(x_{k})\:\:\forall\:(x_{1},…,x_{k})\in
X^{k}\right\}
\end{equation*}}und rechnet für separable $\phi_{1}\dots \phi_{k}\in
\mathcal{E}(X)$ nach, dass
\begin{equation*}
p_{K_{1}\times…\times
K_{k},l}(\phi_{1}…\phi_{k})\leq
p_{K_{1},l}(\phi_{1})…p_{K_{k},l}(\phi_{k})
\end{equation*}
und somit ebenfalls $p_{K_{1}\times…\times
K_{k},l}(\phi_{\mathrm{sep}})\leq
p^{k}(\phi_{\mathrm{sep}})$ für alle $\phi_{\mathrm{sep}}\in\mathcal{E}_{\mathrm{sep}}(X)$ gilt. Dies bedeutet, dass die
$\pi_{k}$-Topologie auf
$\mathcal{E}_{\mathrm{sep}}\left(X^{k}\right)$ feiner ist,
als die durch $\left(\mathcal{E}\left(X^{k}\right)
\T_{\mathcal{E}}\right)$ auf
$\mathcal{E}_{\mathrm{sep}}\left(X^{k}\right)$ induzierte
Teilraumtopologie. Da $\left(\mathcal{E}\left(X^{k}\right),
\T_{\mathcal{E}}\right)$ vollständig ist, bedeutet dies,
dass die Vervollständigung von
$\mathcal{E}_{\mathrm{sep}}\left(X^{k}\right)$ bezüglich
$\pi_{k}$ in $\mathcal{E}\left(X^{k}\right)$ enthalten sein muss.
In der Tat lässt sich sogar zeigen, dass diese wegen der
Nuklearität von $\mathcal{E}(X)$
übereinstimmen, siehe \cite[Thm 51.6]{treves:1967a}. Die
Vervollständigung von $\Ss^{k}(\mathcal{E}(X))$
besteht dann gerade aus allen total symmetrischen $\phi\in
\mathcal{E}\left(X^{k}\right)$. Hiermit lässt sich zeigen,
dass dann ebenfalls alle
Potenzreihen der Form
\begin{equation*}
p(\psi)=\sum_{k=0}^{\infty}\int_{X_{1}\times…\times X_{k}}\phi_{k}(x_{1},…,x_{k})\psi(x_{1})…\psi(x_{k})d_{x_{1}}…d_{x_{k}}
\end{equation*}
mit total symmetrischen Elementen $\phi_{k}\in
\mathcal{E}\left(X^{k}\right)$ und
$\sum_{k=0}^{\infty}\hat{p^{k}}(\phi_{k})<\infty$
durch Elemente aus
$\Hol(\mathcal{E}(X))$ induziert werden können.
Die
gleiche Aussage erhalten wir ebenfalls für $(\mathcal{D}_{K}(X),\T_{\mathcal{D}_{K}})$.
Im Falle $(\mathcal{D}(X),\T_{\mathcal{E}})$
ist allerdings auch die Vervollständigung $\hat{\mathcal{D}}(X)$ von
$\mathcal{D}(X)$ bezüglich $\T_{\mathcal{E}}$
zu berücksichtigen und dies gilt natürlich auch für die
obigen schwach*-topologisierten Varianten.
\end{enumerate}
\end{beispiel}
Folgender Satz klärt die Gestalt der
Hochschild-Kohomologien von $\Hol$ für vollständige, hausdorffsche,
lokalkonvexe $\Hol-\Hol$-Bimoduln $\mathcal{M}$. Dabei stellt
die Vollständigkeit für uns in der Tat eine unverzichtbare
Grundvoraussetzung dar. Man beachte, dass dann der wichtige
symmetrische Spezialfall $\mathcal{M}=\Hol$ in diesem Rahmen komplett
behandelbar sein wird.\\
\begin{satz}
\label{satz:HochschildHol}
Sei $\mathcal{M}$ ein vollständiger, hausdorffscher, lokalkonvexer
$\Hol-\Hol$-Bimodul und
\begin{align*}
\tau^{k}\colon HC^{k}_{\operatorname{\mathrm{cont}}}\big(\Hol,\mathcal{M}\big)&\longrightarrow HC^{k}_{\operatorname{\mathrm{cont}}}\big(\Ss^{\bullet}(\V),\mathcal{M}\big)\\
\hat{\phi}&\longmapsto \hat{\phi}\big|_{HC^{k}_{\operatorname{\mathrm{cont}}}\big(\Ss^{\bullet}(\V),\mathcal{M}\big)}
\end{align*}die durch (\ref{eq:Hochschilddelta}) für
$\mathcal{A}=\Hol$ bzw. $\mathcal{A}=\SsV$ definierten Kettendifferentiale.
Dann induzieren die Abbildungen
\begin{align*}
\tau^{k}\colon HC^{k}_{\operatorname{\mathrm{cont}}}\big(\Hol,\mathcal{M}\big)&\longrightarrow HC^{k}_{\operatorname{\mathrm{cont}}}\big(\Ss^{\bullet}(\V),\mathcal{M}\big)\\
\hat{\phi}&\longmapsto \hat{\phi}\big|_{HC^{k}_{\operatorname{\mathrm{cont}}}\big(\Ss^{\bullet}(\V),\mathcal{M}\big)}
\end{align*} einen Kettenisomorphismus
$\tau:\big(HC_{\operatorname{\mathrm{cont}}}\big(\Hol,\mathcal{M}\big),\hat{\delta}_{c}^{k}\big)\longrightarrow
\big(HC_{\operatorname{\mathrm{cont}}}\big(\Ss^{\bullet}(\V),\mathcal{M}\big),\delta_{c}^{k}\big)$
und es gilt:
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}\big(\Hol,\mathcal{M}\big)\cong HH^{k}_{\operatorname{\mathrm{cont}}}\big(\Ss^{\bullet}(\V),\mathcal{M}\big).
\end{equation*} Ist $\mathcal{M}$ zudem symmetrisch, so ist
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}(\Hol,\mathcal{M})\cong \Hom_{\mathcal{A'}^{e}}^{\operatorname{\mathrm{cont}}}\big(\K^{c}_{k},\mathcal{M}\big)
\end{equation*} mit $(\K_{c},\pt_{c})$ der Koszul-Komplex über $A'=\Ss^{\bullet}(\V)$.
\begin{beweis}
Da jeder $\Hol-\Hol$-Bimodul insbesondere ein
$\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul ist und sich alle
angeführten Eigenschaften auf die Unteralgebra übertragen, folgt
die zweite Isomorphie mit Satz \ref{satz:stetigHochschSym} aus
der ersten, und für diese reicht es nach Lemma \ref{lemma:kettenabzu}~\textit{ii.)}, die
Kettenisomorphismus-Eigenschaft von $\tau$ nachzuweisen.\\\\
Zunächst folgt mit Satz \ref{satz:stetfortsBillphi} und Bemerkung \ref{bem:stetfortmult}~\textit{ii.)}, dass die $\tau^{k}$ Isomorphismen mit
stetiger Fortsetzung $\tau^{k}_{-1}$ als Umkehrabbildung sind.
Dabei folgt $\widehat{\phi+\psi}=\hat{\phi}+\hat{\psi}$, also die
Linearität von $\tau_{-1}^{k}$, sofort mit der Stetigkeit der
Addition.
Die Kettenabbildungs-Eigenschaft erhalten wir unmittelbar aus der Definition der
$\Hol$-Algebramultiplikation $*$, da hiermit
\begin{equation}
\label{eq:muhkuhmilch}
\hat{\delta}^{k}_{c}\Big(\hat{\phi}\Big)\Big|_{\SsV^{k+1}}=\delta^{k}_{c}\left(\hat{\phi}\big|_{\SsV^{k}}\right),
\end{equation}
also $\tau^{k+1}\cp \hat{\delta}^{k}_{c}= \delta^{k}_{c}\cp
\tau^{k}$ gilt.
Dies zeigt die Behauptung.
\end{beweis}
\end{satz}
Wir wollen die Isomorphien in Satz \ref{satz:HochschildHol} noch ein
wenig n"aher betrachten.\\% Hierfür benötigen wir:\\
\begin{definition}[Koszul-Komplex und Vervollständigter Koszul-Komplex]
\label{def:vervollstBarKoszulkompl}
\begin{enumerate}
\item
Bezeichne $\left(\K'_{c},\pt'_{c}\right)$ den topologischen Koszul-Komplex der Algebra
$\mathcal{A}'=\Ss^{\bullet}(\V)$ und $*_{S}$ die zugeh"orige $\Ss^{\bullet}(\V)\pite
\Ss^{\bullet}(\V)$-Modul-Multiplikation.
Mit Hilfe von Satz \ref{satz:PiTopsatz}~\textit{vi.)} definieren wir:
{\allowdisplaybreaks
\begin{align*}
\cK_{k}^{c}=\widehat{\K'^{c}_{k}}=&\widehat{\Bigg(\Ss^{\bullet}(\V)\pite
\Ss^{\bullet}(\V) \pite
\Lambda^{k}(\V)\Bigg)}=\:\widehat{\Bigg(\widehat{\Ss^{\bullet}(\V)}\pite
\widehat{\Ss^{\bullet}(\V)}\pite
\widehat{\Lambda^{k}(\V)}\Bigg)}
\\=& \:\widehat{\Bigg(\Hol\pite \Hol \pite
\widehat{\Lambda^{k}(\V)}\Bigg)}
=\:\widehat{\Bigg(\Hol\pite \Hol \pite
\Lambda^{k}(\V)\Bigg)}
\end{align*}}sowie
\begin{equation*}
\cK_{0}^{c}=\widehat{\Big(\Ss^{\bullet}(\V)\pite
\Ss^{\bullet}(\V)\Big)}=\widehat{\Big(\Hol\pite \Hol\Big)}= \hat{\mathcal{A}^{e}}.
\end{equation*}
Wie in Proposition \ref{prop:HollkAlgebra}~\textit{ii.)} werden diese,
vermöge stetiger Fortsetzung $\hat{*}_{S}$ von $*_{S}$, zu
hausdorffschen, lokalkonvexen
$\hat{\mathcal{A}^{e}}$-Linksmoduln. Zudem erhalten wir stetige Fortsetzungen
$\hat{\pt}^{c}_{k}$ der Kettendifferentiale $\pt'^{c}_{k}$ , die
wegen
\begin{align*}
\hat{\pt}^{c}_{k}\:\left(\hat{a}^{e}\:\hat{*}_{S}\:
\hat{\kappa}^{k}\right)=&
\:\hat{\pt}^{c}_{k}\left(\lim_{\alpha\times\beta}\left[a^{e}_{\alpha}\:*_{S}\:
\kappa^{k}_{\beta}\right]\right)=\lim_{\alpha\times\beta}\pt'^{c}_{k}\left(a^{e}_{\alpha}\:*_{S}\:
\kappa^{k}_{\beta}\right)
\\=&\lim_{\alpha\times\beta}\left[a^{e}_{\alpha}\:*_{S}\:
\pt'^{c}_{k}\big( \kappa^{k}_{\beta}\big)\right]=\hat{a}^{e}\:\hat{*}_{S}\: \hat{\pt}^{c}_{k}\big(\hat{\kappa}^{k}\big)
\end{align*}$\hat{\mathcal{A}}^{e}$-linear sind.
Den so gewonnenen topologischen Kettenkomplex
$\big(\cK_{c},\cpt_{c}\big)$ bezeichnen wir als vervollständigten
Koszul-Komplex über $\mathcal{A}=\Hol$.
\item
Mit $\K^{c}_{k}$ benennen wir die
hausdorffschen, lokalkonvexen $\mathcal{A}^{e}$-Linksmoduln
\begin{equation*}
\K^{c}_{0}=\Hol\pite\Hol \quad\text{ sowie }\quad \K^{c}_{k}=\Hol\pite\Hol\pite \Lambda^{k}(\V)
\end{equation*}
mit der offensichtlichen
$\mathcal{A}^{e}$-Multiplikation $*_{Hol}$ in den ersten beiden
Faktoren. Diese induziert dann ebenfalls eine stetige
$\hat{\mathcal{A}}^{e}$-Multiplikation $\hat{*}_{Hol}$ auf
$\cK_{k}^{c}$, und da
\begin{equation*}
*_{Hol}\big|_{\Ss^{\bullet}(\V)\pite \Ss^{\bullet}(\V)\times
\Lambda^{k}(\V)}=*_{S}=\hat{*}_{S}\big|_{\Ss^{\bullet}(\V)\pite \Ss^{\bullet}(\V)\times
\Lambda^{k}(\V)},
\end{equation*} stimmen beide auf einer dichten Teilmenge
von $\cK_{c}^{k}$ überein. Mit der Eindeutigkeit der
stetigen Fortsetzung von $*_{S}$ ist
\begin{equation}
\label{eq:Modulmultsallegleich}
\hat{*}_{S}=\hat{*}_{Hol}\qquad\text{und somit}\qquad\hat{*}_{S}\big|_{\K^{c}_{k}}=*_{Hol}.
\end{equation}
$(\K^{c}_{k},\pt_{c})$ bezeichne dann den
Kettenkomplex mit Kettendifferentialen
$\pt^{c}_{k}=\hat{\pt}^{c}_{k}\big|_{\K^{c}_{k}}$, für dessen
Wohldefiniertheit wir zeigen müssen, dass die $\pt^{c}_{k}$
ausschließlich nach $\K^{c}_{k-1}\subseteq \hat{\K}^{c}_{k-1}$
abbilden, und nicht in $\hat{\K}^{c}_{k-1}\backslash \K^{c}_{k-1}$
landen. Hierbei bedeutet $\backslash$ die mengentheoretische Differenz.
Sei hierfür
$\Ss^{\bullet}(\V)\supseteq\net{x}{I}\rightarrow x\in \Hol$ und
$\Ss^{\bullet}(\V)\supseteq\{y_{\beta}\}_{\beta\in
J}\rightarrow y\in \Hol$. Dann folgt $\{x_{\alpha}\pite
y_{\beta}\pite u\}_{\alpha\times \beta \in I\times J}\rightarrow
x\pite y\pite u \in \K^{c}_{k}$, vermöge zweimaliger Anwendung
der Dreiecksungleichung und der Definition der $\pi_{3}$-Halbnormen
(siehe auch Beweis zu
Satz \ref{satz:PiTopsatz}~\textit{vi.)}). Wir erhalten
\begin{align*}
\pt^{c}_{k}(x&\pite y \pite u)=\lim_{\alpha\times
\beta}\pt'^{c}_{k}\:(x_{\alpha}\pite y_{\beta}\pite u)\\=&\:\lim_{\alpha\times
\beta}\:\sum_{j=1}^{k}(-1)^{j-1}(u_{j}\vee x_{\alpha}\pite
y_{\beta}\pite u^{j}) - \lim_{\alpha\times
\beta}\:\sum_{j=1}^{k}(-1)^{j-1}(x_{\alpha}\pite
u_{j}\vee y_{\beta}\pite u^{j})
\\=&\: \sum_{j=1}^{k}(-1)^{j-1}(u_{j}* x \pite
y \pite u^{j}) - \sum_{j=1}^{k}(-1)^{j-1}(x\pite
u_{j}* y\pite u^{j})\in \K^{c}_{k-1}
\end{align*}
mit gleicher Argumentation wie oben, da definitionsgemäß $\{u_{j}\vee
x_{\alpha}\}_{\alpha\in I}\rightarrow u_{j}* x$ und $\{u_{j}\vee
y_{\beta}\}_{\beta\in J}\rightarrow u_{j}* y$.
\end{enumerate}
\end{definition}
\begin{lemma}
\label{lemma:fgh}
Gegeben ein vollständiger, hausdorffscher, lokalkonvexer,
$\Hol-\Hol$-Bimodul $\mathcal{M}$, so ist
\begin{equation*}
\Big(\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}(\K_{c},\mathcal{M}),\pt^{*}_{c}\Big)\cong \Big(\Hom^{\operatorname{\mathrm{cont}}}_{\hat{\mathcal{A}}^{e}}(\cK_{c},\mathcal{M}),\hat{\pt}^{*}_{c}\Big)\cong \Big(\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}'^{e}}(\K'_{c},\mathcal{M}),\pt'^{*}_{c}\Big),
\end{equation*}wobei $\cong$ Kettenisomorphie vermöge
Einschränkung und stetiger Fortsetzung bedeutet. Des Weiteren gilt:
\begin{equation*}
H^{k}\Big(\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}(\K_{c},\mathcal{M}),\pt^{*}_{c}\Big)\cong H^{k}\Big(\Hom^{\operatorname{\mathrm{cont}}}_{\hat{\mathcal{A}}^{e}}(\cK_{c},\mathcal{M}),\hat{\pt}^{*}_{c}\Big)\cong H^{k}\Big(\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}'^{e}}(\K'_{c},\mathcal{M}),\pt'^{*}_{c}\Big).
\end{equation*}
\begin{beweis}
Für jedes $\phi \in
\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}(\K^{c}_{k},\mathcal{M})$ existiert eine
eindeutige lineare Fortsetzung $\hat{\phi}\in
\Hom^{\operatorname{\mathrm{cont}}}_{\hat{\mathcal{A}}^{e}}(\hat{\mathcal{K}}_{k}^{c},\mathcal{M})$
mit $\hat{\phi}\big|_{\K^{c}_{k}}=\phi$.
Umgekehrt erhalten wir nach \eqref{eq:Modulmultsallegleich} aus jedem $\hat{\phi}\in
\Hom^{\operatorname{\mathrm{cont}}}_{\hat{\mathcal{A}}^{e}}(\hat{\mathcal{K}}_{k}^{c},\mathcal{M})$,
vermöge Einschränkung, ein $\phi\in
\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}(\K^{c}_{k},\mathcal{M})$,
dessen stetige lineare Fortsetzung $\hat{\phi}$ ist. Dies zeigt
$\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}(\K_{c},\mathcal{M})\cong
\Hom^{\operatorname{\mathrm{cont}}}_{\hat{\mathcal{A}}^{e}}(\hat{\mathcal{K}}_{c},\mathcal{M})$, und wegen
\begin{equation*}
\hat{\pt}^{c*}_{k+1}\Big(\hat{\phi}\Big)\Big|_{\mathcal{K}^{c}_{k+1}}=
\left(\hat{\phi}\:\cp\:
\hat{\pt}^{c}_{k+1}\right)\Big|_{\mathcal{K}^{c}_{k+1}}=
\hat{\phi}\big|_{\mathcal{K}^{c}_{k}}\cp\: \hat{\pt}^{c}_{k+1}\big|_{\mathcal{K}^{c}_{k+1}}=
\hat{\phi}\big|_{\mathcal{K}^{c}_{k}}\cp\: \pt^{c}_{k+1} =\pt^{c*}_{k+1}\left(\hat{\phi}\big|_{\mathcal{K}^{c}_{k}}\right),
\end{equation*}haben wir es hierbei wieder mit einem
Kettenisomorphismus zu tun. Dies zeigt die jeweils erste Isomorphie,
und die zweite folgt ganz analog.
\end{beweis}
\end{lemma}
\begin{korollar}
\label{kor:hgf}
Mit den Voraussetzungen aus Satz \ref{satz:HochschildHol} gilt:
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}\Big(\Hol,\mathcal{M}\Big)\cong
H^{k}\Big(\Hom_{\mathcal{A}'^{e}}^{\operatorname{\mathrm{cont}}}(\K'_{c},\mathcal{M}),\pt'^{*}_{c}\Big)\cong
H^{k}\Big(\Hom_{\mathcal{A}^{e}}^{\operatorname{\mathrm{cont}}}(\K_{c},\mathcal{M}),\pt^{*}_{c}\Big).
\end{equation*} Ist $\mathcal{M}$ symmetrisch, so ist:
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}\big(\Hol,\mathcal{M}\big)\cong \Hom_{\mathcal{A}'^{e}}^{\operatorname{\mathrm{cont}}}\big(\K'^{c}_{k},\mathcal{M}\big)\cong \Hom_{\mathcal{A}^{e}}^{\operatorname{\mathrm{cont}}}\big(\K^{c}_{k},\mathcal{M}\big).
\end{equation*}
\end{korollar}
Lemma \ref{lemma:fgh} und Korollar \ref{kor:hgf} kann man auch so auffassen, dass
es für vollständige, hausdorffsche $\Hol-\Hol$-Bimoduln es egal ist, ob wir $\hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}'^{e}}(\cdot,\mathcal{M})$
auf $(\K'_{c},\pt'^{*}_{c})$, $\hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}(\cdot,\mathcal{M})$
auf $(\K_{c},\pt^{*}_{c})$ oder $\hom^{\operatorname{\mathrm{cont}}}_{\hat{\mathcal{A}^{e}}}(\cdot,\mathcal{M})$
auf $(\hat{\K}_{c},\hat{\pt}^{*}_{c})$ anwenden. Wir erhalten in jedem
Fall den "`gleichen"' Kokettenkomplex mit
den "`gleichen"' Kohomologien-Gruppen.
\begin{Bemerkung}[Nicht vollständige Bimoduln]
Abschließend wollen wir noch erklären, warum die Berechnung der
Hochschild-Koho-mologien für nicht vollständige Bimoduln $\mathcal{M}$ mit Hilfe
der uns in diesem Rahmen zur Verfügung stehenden Mittel
fehlschlägt.
Zunächst ist der zu $\mathcal{A}=\Hol$
gehörige Bar-Komplex sowohl projektiv, als auch exakt. Für
$(\mathcal{K}_{c},\pt_{c})$ ist jedoch nur die Projektivität
unmittelbar einsichtig, da die Einschränkungen
$\hat{h}^{c}_{k}\big|_{\K^{c}_{k}}$ der stetigen Fortsetzungen der exaktheitsliefernden
Homotopieabbildungen $h^{c}_{k}$ des topologischen Koszul-Komplexes
$(\K_{c},\pt_{c})$, im Gegensatz zu den Einschränkungen
der Kettendifferentiale
$\hat{\pt}^{c}_{k}\big|_{\K^{c}_{k}}$ im Allgemeinen nicht
ausschließlich in die Unterräume $\K^{c}_{k}\subseteq \hat{\mathcal{K}}^{c}_{k}$
abbilden, sondern in der Tat in den Vervollständigungen
$\hat{\mathcal{K}}^{c}_{k}$ landen.
Algebraisch gesehen haben wir damit nichts in der Hand, um besagte Isomorphie zu begründen. Nun
könnte man versuchen mit Hilfe der Einschränkungen der stetigen
Fortsetzungen $\hat{F}$ und $\hat{G}$
zu argumentieren. Jedoch bildet auch $\hat{G}$ im Allgemeinen nicht in die
Unterräume, sondern in die jeweilige Vervollständigung
ab. Mit Hilfe von Lemma \ref{lemma:Homotopiejdfgjkf}, der stetigen Fortsetzung der Homotopie $s$ aus
Lemma \ref{lemma:Fkettenabb}~\textit{i.)} sowie $\widehat{\big(G_{k}\cp F_{k}\big)}=\hat{G}_{k}\cp \hat{F}_{k}$ und
$\widehat{\big(d_{k+1}s_{k}+s_{k-1}d_{k}\big)}=\hat{d}_{k+1}\hat{s}_{k}+\hat{s}_{k-1}\hat{d}_{k}$
folgt dann zwar unmittelbar:
\begin{equation*}
H^{k}\left(\Hom_{\hat{\mathcal{A}^{e}}}\big(\hat{\C}_{c},\mathcal{M}\big),
\hat{d}_{c}^{*}\right)\cong H^{k}\left(\Hom_{\hat{\mathcal{A}^{e}}}\big(\hat{\K}_{c},\mathcal{M}\big),\hat{\pt}^{*}_{c}\right).
\end{equation*}
Um jedoch die Isomorphie zu der gewünschten
Hochschild-Kohomologie herzustellen, also beispielsweise
\begin{equation*}
H^{k}\Big(\Hom_{\mathcal{A}^{e}}(\C_{c},\mathcal{M}),d^{*}_{c}\Big)\cong
H^{k}\Big(\Hom_{\mathcal{A}^{e}}\big(\hat{\C}_{c},\mathcal{M}\big),\hat{d}^{*}_{c}\Big)
\end{equation*}nachzuweisen,
benötigt man wieder Fortsetzungsargumente und hierzu die Vollständigkeit von
$\mathcal{M}$. Dies liegt im wesentlichen daran, dass der Raum
$\Hom_{\mathcal{A}^{e}}(\C_{c},\mathcal{M})$ für nicht vollständige
Bimoduln $\mathcal{M}$ im Allgemeinen gehaltvoller als
$\Hom_{\mathcal{A}^{e}}\big(\hat{\C}_{c},\mathcal{M}\big)$
ist. Dies sieht man sofort daran, dass jedes $\psi\in
\Hom_{\mathcal{A}^{e}}\big(\hat{\C}_{c},\mathcal{M}\big)$, vermöge
Einschränkung, ein Element in
$\Hom_{\mathcal{A}^{e}}(\C_{c},\mathcal{M})$ definiert, dessen
stetige Fortsetzung es ist. Hierbei haben wir uns
$\mathcal{M}\subseteq\hat{\mathcal{M}}$ kanonisch eingebettet
gedacht. Umgekehrt können aber durchaus
$\phi\in\Hom_{\mathcal{A}^{e}}(\C_{c},\mathcal{M})$ existieren,
deren stetige Fortsetzung nicht ausschließlich nach $\mathcal{M}$
abbildet, sondern auch Bilder in $\hat{\mathcal{M}}\backslash \mathcal{M}$ besitzt.
\end{Bemerkung}
\chapter{Hochschild-Kostant-Rosenberg-Theoreme}
\label{sec:HKRTheos}
In diesem Kapitel soll es darum gehen, Hochschild-Kostant-Rosenberg-Theoreme (vgl. \cite[Kapitel 6]{waldmann:2007a}) im Falle symmetrischer Bimoduln $\mathcal{M}$, f"ur die Kohomologie-Gruppen $HH^{k}(\SsV, \mathcal{M})$, $HH_{\operatorname{\mathrm{cont}}}^{k}(\SsV, \mathcal{M})$ und $HH_{\operatorname{\mathrm{cont}}}^{k}(\Hol, \mathcal{M})$ zu beweisen. Die Basis hierf"ur bilden die Sätze
\ref{satz:HochschkohmvonSym}, \ref{satz:stetigHochschSym} und
\ref{satz:HochschildHol}, die wir zun"achst etwas umformulieren wollen.
\section{Vorbereitung}
Wir benötigen die folgenden Kokettenkomplexe und Kettenabbildungen:
\begin{definition}
\begin{enumerate}
\item
Sei $\mathcal{M}$ ein$\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul und
\begin{equation*}
KC_{\Lambda}^{k}(\V,\mathcal{M}):=
\begin{cases} \{0\} & k<0\\
\mathcal{M} & k=0\\
\Hom_{\mathbb{K}}(\Lambda^{k}(\V),\mathcal{M})& k\geq 1.
\end{cases}
\end{equation*}
Vermöge der Links- und Rechtsmodulstruktur auf $\mathcal{M}$
definieren wir $\mathbb{K}$-lineare Abbildungen $ \Delta_{\Lambda}^{k}\colon KC_{\Lambda}^{k}(\V, \mathcal{M})\longrightarrow KC_{\Lambda}^{k+1}(\V, \mathcal{M})$ durch:
\begin{equation*}
(\Delta_{\Lambda}^{k}\phi)(u_{1}\wedge…\wedge u_{k+1})=
\sum_{l=1}^{k+1}(-1)^{l-1}\:[u_{l}*_{L}-\:u_{l}\:*_{R}]\:\phi(u_{1}\wedge…\blacktriangle^{l}…\wedge
u_{k+1}).
\end{equation*}
Des Weiteren definieren wir die Isomorphismen
\begin{align*}
\Upsilon^{k}\colon \Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})&\longrightarrow KC_{\Lambda}^{k}(\V, \mathcal{M})\\
\wt{\phi}&\longmapsto \left[\phi: \omega \mapsto \wt{\phi}\:(1_{e}\ot \omega)\right]
\end{align*} mit Umkehrabbildungen
\begin{align*}
\Upsilon^{k}_{-1}\colon KC^{k}(\V, \mathcal{M})&\longrightarrow \Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})\\
\phi &\longmapsto \left[\wt{\phi}:a^{e}\ot \omega \mapsto a^{e}*_{e}\phi(\omega)\right].
\end{align*}
Nun ist
{\allowdisplaybreaks\begin{align*}
\left(\Upsilon^{k+1}\pt_{k}^{*}\right)\big(\wt{\phi}\big)(u_{1}&\:\wedge\dots \wedge u_{k+1})
=\big(\pt_{k}^{*}\wt{\phi}\big)(1_{e}\ot u_{1}\wedge\dots \wedge u_{k+1})
\\=&\:\sum_{l=1}^{k+1}(-1)^{l-1}\wt{\phi}\left([u_{l}\ot 1-1\ot u_{l}]\ot u_{1}\wedge\dots\blacktriangle^{l}\dots\wedge u_{k+1}\right)
\\=&\:\sum_{l=1}^{k+1}(-1)^{l-1}[u_{l}\ot 1-1\ot u_{l}]*_{e}\wt{\phi}\left(1_{e}\ot u_{1}\wedge\dots\blacktriangle^{l}\dots\wedge u_{k+1}\right)
\\=&\:\sum_{l=1}^{k+1}(-1)^{l-1}[u_{l}*_{L}- u_{l}\:*_{R}]\left(\Upsilon^{k}\wt{\phi}\right)\left(u_{1}\wedge\dots\blacktriangle^{l}\dots\wedge u_{k+1}\right)
\\=&\: \left(\Delta_{\Lambda}^{k}\Upsilon^{k}\right)\big(\wt{\phi}\big)(u_{1}\wedge\dots\wedge u_{k+1}),
\end{align*}}also insbesondere
\begin{equation*}
\Delta_{\Lambda}^{k+1}\Delta^{k}_{\Lambda}=\Upsilon^{k+2}\pt^{*}_{k+1}\pt^{*}_{k}\Upsilon^{k}_{-1}=0.
\end{equation*}
Hiermit
induzieren die $\Upsilon^{k}$ einen Kettenisomorphismus zwischen den Kokettenkomplexen $\left(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{M}),\pt^{*}\right)$ und $\left(KC_{\Lambda}(\V,\mathcal{M}),\Delta_{\Lambda}\right)$.
\item
Wir definieren
\begin{equation*}
KC^{k}(\V,\mathcal{M}):=
\begin{cases} \{0\} & k<0\\
\mathcal{M} & k=0\\
\Hom^{a}_{\mathbb{K}}(\V^{k},\mathcal{M})& k\geq 1,
\end{cases}
\end{equation*} die total antisymmetrischen, $\mathbb{K}$-multilinearen Abbildungen von
$\V^{k}$ nach $\mathcal{M}$, sowie die $\mathbb{K}$-lineare Abbildungen $ \Delta^{k}:KC^{k}(\V, \mathcal{M})\longrightarrow KC^{k+1}(\V, \mathcal{M})$
durch
\begin{equation*}
(\Delta^{k}\phi)(v_{1},\dots, v_{k+1})=
\sum_{l=1}^{k+1}(-1)^{l-1}\:[u_{l}*_{L}-\:u_{l}\:*_{R}]\:\phi(v_{1},\dots,\blacktriangle^{l},\dots,v_{k+1}).
\end{equation*}
Weiter definieren wir die Isomorphismen
\begin{align*}
\Theta^{k}\colon KC_{\Lambda}^{k}(\V, \mathcal{M})&\longrightarrow KC^{k}(\V, \mathcal{M})\\
\phi'&\longmapsto \left[\phi\colon (v_{1},\dots,v_{k}) \mapsto \phi'(v_{1}\wedge\dots \wedge v_{k})\right]
\end{align*}mit Umkehrabbildungen
\begin{align*}
\Theta^{k}_{-1}\colon KC^{k}(\V, \mathcal{M})&\longrightarrow KC_{\Lambda}^{k}(\V, \mathcal{M})\\
\phi &\longmapsto [\phi'\colon v_{1}\wedge\dots\wedge v_{k}\mapsto \phi(v_{1},\dots,v_{k})].
\end{align*}
Dabei entsprechen die letzteren gerade den Einschr"ankungen der durch die universelle Eigenschaft induzierten linearen Abbildungen $\phi_{\ot}$ auf die total antisymmetrischen Tensorelemente, denn mit der totalen Antisymmetrie von $\phi$ ist:
\begin{equation*}
\phi_{\ot}(v_{1}\wedge\dots\wedge v_{k})=\frac{1}{k!}\sum_{\sigma\in S_{k}}\sign(\sigma)\:\phi(v_{\sigma(1)},\dots,v_{\sigma(k)})=\phi(v_{1},\dots,v_{k}).
\end{equation*}Umgekehrt ist $\Theta^{k}(\phi')=\phi' \cp \mathrm{Alt}_{k} \cp\ot_{k}$.
Auch hier erhalten wir
\begin{align*}
\left(\Theta^{k+1}\Delta_{\Lambda}^{k}\right)\big(\phi'\big)(v_{1},\dots,v_{k+1})=&\: \left(\Delta^{k}_{\Lambda}\phi'\right)(v_{1}\wedge\dots\wedge v_{k+1})
\\=&\:\sum_{l=1}^{k+1}(-1)^{l-1}[u_{l}*_{L}-u_{l}\:*_{R}]\:\phi'(v_{1}\wedge\dots\blacktriangle^{l}\dots\wedge v_{k+1})
\\=&\:\sum_{l=1}^{k+1}(-1)^{l-1}[u_{l}*_{L}-u_{l}\:*_{R}]\:\big(\Theta^{k}\phi'\big)(v_{1},\dots,\blacktriangle^{l},\dots,v_{k+1})
\\=&\:\left(\Delta^{k}\Theta^{k}\right)(\phi')(v_{1},\dots,v_{k+1}),
\end{align*}also insbesondere wieder $\Delta^{k+1}\cp\Delta^{k}=0$, womit $\Theta$ ein Kettenisomorphismus zwischen den Kokettenkomplexen $\left(KC_{\Lambda}(\V,\mathcal{M}),\Delta_{\Lambda}\right)$ und $\left(KC(\V,\mathcal{M}),\Delta\right)$ ist.
\end{enumerate}
\end{definition}
Folgendes Lemma kl"art weitere wichtige Eigenschaften obiger Definitionen.\\
\begin{lemma}
\label{lemma:stethhh}
\begin{enumerate}
\item
Gegeben ein symmetrischer $\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul $\mathcal{M}$, dann gilt:
\begin{equation*}
H^{k}\big(KC(\V,\mathcal{M}),\Delta\big)=\Hom^{a}_{\mathbb{K}}\big(\V^{k},\mathcal{M}\big).
\end{equation*}
\item
Seien $\V$ und $\mathcal{M}$ lokalkonvex und $\Lambda^{k}(\V)$ $\pi_{k}$
topologisiert. Dann bilden sowohl die $\Upsilon^{k}$ als auch
die $\Theta^{k}$, in beide Richtungen stetige Homomorphismen auf stetige Homomorphismen ab.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Für alle $\phi\in KC^{k}(\V,\mathcal{M})$ ist
\begin{align*}
\left(\Delta^{k}\phi\right)(v_{1},…,v_{k+1})=&\:\sum_{l=1}^{k+1}(-1)^{l-1}\:[u_{l}*_{L}-\:u_{l}\:*_{R}]\:\phi(v_{1},\dots,\blacktriangle^{l},\dots,v_{k+1})
\\=&\:\sum_{l=1}^{k+1}(-1)^{l-1}\:[u_{l}*_{L}-\:u_{l}\:*_{L}]\:\phi(v_{1},\dots,\blacktriangle^{l},\dots,v_{k+1})
\\=&\:0.
\end{align*}
\item
Sei $\wt{\phi}\in
\Hom^{\operatorname{\mathrm{cont}}}_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})$ und
$\omega\in \Lambda^{k}(\V)$, dann folgt
\begin{align*}
q\left(\Big(\Upsilon^{k}\wt{\phi}\Big)(\omega)\right)=&\:q\left(\wt{\phi}\left(1_{e}\ot
\omega\right)\right)\leq \:c \p^{2}\ot p^{k}\left(1_{e}\ot\omega\right)=\:c\: p^{k}\left(\omega\right),
\end{align*}was die Stetigkeit von $\Upsilon^{k}\big(\wt{\phi}\big)$
zeigt. Sei umgekehrt $\phi\in
\Hom^{\operatorname{\mathrm{cont}}}_{\mathbb{K}}(\Lambda^{k}(\V),\mathcal{M})$, so
zeigt Lemma \ref{lemma:AezuunittopRing}~\textit{ii.)}, dass
\begin{align*}
q\left(\left(\Upsilon^{k}_{-1}\phi\right)(a_{e}\ot
\omega)\right)=&\:q\left(a_{e}*_{e}\phi(\omega)\right)\leq c\:
\p^{2}(a^{e})\:q'\left(\phi(\omega)\right)
\\\leq&\: c'\: \p^{2} (a^{e})\:
p'^{k}(\omega)\leq c'
\p''_{k}(a^{e}\ot\omega)
\end{align*}mit einer Halbnorm $p''\geq p,p'$ gilt.
Für $\Theta^{k}$ sei $\phi'\in \Hom^{\operatorname{\mathrm{cont}}}_{\mathbb{K}}(\Lambda^{k}(\V),\mathcal{M})$ wie eben, dann folgt:
\begin{align*}
q\left(\left(\Theta^{k}\phi\right)(v_{1},…,v_{k})\right)=&\:q\big(\phi(v_{1}\wedge…\wedge
v_{k})\big)\leq c\: p^{k}(v_{1}\wedge…\wedge
v_{k})
\\\leq&\:c\:p^{k}(v_{1}\ot…\ot v_{k})=c\:p(v_{1})…p(v_{k}).
\end{align*}
Sei umgekehrt $\phi
\in\Hom^{a,\operatorname{\mathrm{cont}}}_{\mathbb{K}}(\V^{k},\mathcal{M})$, dann
sind
$\Theta^{k}_{-1}\phi=\phi_{\ot}\Big|_{\Lambda^{k}(\V)}$
und $\phi_{\ot}$ stetig mit der Charakterisierung von
$\pi_{k}$ und es gilt:
\begin{align*}
q\left(\left(\Theta^{k}_{-1}\phi\right)(v_{1}\wedge…\wedge
v_{k})\right)=&\:
q\big(\phi_{\ot}(v_{1}\wedge…\wedge v_{k})\big)\leq
c \:p^{k}(v_{1}\wedge…\wedge v_{k}).
\end{align*}
\end{enumerate}
\end{beweis}
\end{lemma}
Hiermit erhalten wir folgende Umformulierungen der S"atze
\ref{satz:HochschkohmvonSym}, \ref{satz:stetigHochschSym} und \ref{satz:HochschildHol}:\\
\begin{korollar}
\label{kor:UmformSatz}
\begin{enumerate}
\item
Gegeben ein $\SsV-\SsV$-Bimodul $\mathcal{M}$, dann gilt:
\begin{equation*}
HH^{k}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong H^{k}\left(KC(\V,\mathcal{M}),\Delta\right).
\end{equation*}
Ist $\mathcal{M}$ zudem symmetrisch, so ist:
\begin{equation*}
HH^{k}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong \Hom^{a}_{\mathbb{K}}\big(\V^{k},\mathcal{M}\big).
\end{equation*}
\item
Gegeben ein lokalkonvexer, $\SsV-\SsV$-Bimodul $\mathcal{M}$, dann gilt:
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}\left(\Ss^{\bullet}(\mathbb{V}),\mathcal{M}\right)\cong
H^{k}\big(KC^{\operatorname{\mathrm{cont}}}(\V,\mathcal{M}),\Delta^{c}\big)
\end{equation*}
Ist $\mathcal{M}$ zudem symmetrisch, so ist:
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong\Hom_{\mathbb{K}}^{a,\operatorname{\mathrm{cont}}}(\V^{k},\mathcal{M}).
\end{equation*}
\item
Gegeben ein vollständiger, hausdorffscher, lokalkonvexer
$\Hol-\Hol$-Bimodul $\mathcal{M}$, dann gilt:
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}\big(\Hol,\mathcal{M}\big)\cong HH^{k}_{\operatorname{\mathrm{cont}}}\big(\Ss^{\bullet}(\V),\mathcal{M}\big).
\end{equation*} Ist $\mathcal{M}$ zudem symmetrisch, so ist:
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{cont}}}(\Hol,\mathcal{M})\cong \Hom_{\mathbb{K}}^{a,\operatorname{\mathrm{cont}}}(\V^{k},\mathcal{M}).
\end{equation*}
\end{enumerate}
\begin{beweis}
Wegen
$\Delta_{\Lambda}^{k}=\Upsilon^{k+1}\pt_{k}^{*}\Upsilon^{k}_{-1}$
sowie
$\Delta^{k}=\Theta^{k+1}\Delta_{\Lambda}^{k}\Theta^{k}_{-1}$
und Lemma \ref{lemma:stethhh}~\textit{ii.)} bilden sowohl
$\Delta_{\Lambda}^{k}$ als auch $\Delta^{k}$ stetige Elemente
auf stetige Elemente ab. Dies zeigt die Wohldefiniertheit der
Kokettenkomplexe $(KC^{\operatorname{\mathrm{cont}}}_{\Lambda},\Delta^{c}_{\Lambda})$
und $(KC^{\operatorname{\mathrm{cont}}},\Delta^{c})$ sowie die Isomorphie ihrer
Kohomologien. Die jeweils letzten Aussagen folgen mit Lemma \ref{lemma:stethhh}~\textit{i.)}.
\end{beweis}
\end{korollar}
\section{Hochschild-Kostant-Rosenberg-Theoreme}
In diesem Abschnitt werden wir die jeweils zweite Isomorphie in Korollar \ref{kor:UmformSatz}
explizit ausformulieren und erhalten Analoga zu dem bekannten
Hochschild-Kostant-Rosenberg-Theorem, siehe \cite[Prop~6.2.48 ]{waldmann:2007a},\cite{cahen.gutt.dewilde:1980a}.
\begin{proposition}
\label{prop:wichEizuHKR}
\begin{enumerate}
\item
Seien $\V$ und $\mathcal{M}$ $\mathbb{K}$-Vektorräume, wobei
$(\mathcal{M},*)$ zusätzlich ein $\Ss^{\bullet}(\V)$-Modul ist. Dann besitzt jede
$\mathbb{K}$-multilineare Abbildung $\phi:\V^{k}\longrightarrow
\mathcal{M}$
eine eindeutig bestimmte, in jedem Argument derivative, $\mathbb{K}$-multilineare
Fortsetzung
$\phi_{D}\colon\SsV^{k}\longrightarrow \mathcal{M}$.
Diese ist gegeben durch multilineare Fortsetzung von
\begin{equation*}
\label{eq:derFortsetz}
\phi_{D}(\omega_{1},…,\omega_{k})=\sum_{m_{1}=1}^{n_{1}}…\sum_{m_{k}=1}^{n_{k}}\omega_{1}^{m_{1}}\vee…\vee\omega_{k}^{m_{k}}
* \phi\big((\omega_{1})_{m_{1}},…,(\omega_{k})_{m_{k}}\big)
\end{equation*}
auf ganz $\SsV^{k}$ mit $\omega_{i}\in \Ss^{n_{i}}(\V)$ und
$\phi_{D}\left(\omega_{1},…,1_{i},…,\omega_{k}\right)=0$ für
$1\leq i\leq k$.
Des Weiteren ist $\phi_{D}$ genau dann total
antisymmetrisch, wenn $\phi$ total
antisymmetrisch ist.
\item
Gegeben eine kommutative Algebra $\mathcal{A}$ und ein
symmetrischer $\mathcal{A}-\mathcal{A}$-Bimodul $\mathcal{M}$.
Sei weiter $\mathrm{Alt}_{k}\colon HC^{k}(\mathcal{A},\mathcal{M})\longrightarrow
HC^{k}(\mathcal{A},\mathcal{M})$ definiert durch
\begin{align*}
\mathrm{Alt}_{k}(\phi)(a_{1},…,a_{k})=\frac{1}{k!}\sum_{\sigma\in S_{k}}\sign(\sigma)\:\phi(a_{\sigma(1)},…a_{\sigma(k)}).
\end{align*}Dann gilt $\mathrm{Alt}_{k}\cp\delta^{k-1}=0$. Wegen
$\mathrm{Alt}_{k}\cp \mathrm{Alt}_{k}=\mathrm{Alt}_{k}$ bedeutet dies insbesondere,
dass ein total antisymmetrischer Hochschild-Kozyklus $\phi$ nur dann
auch ein Hochschild-Korand sein kann, wenn bereits $\phi=0$ gilt.
\item
Sei $\phi\in \Hom^{a}_{\mathbb{K}}(\V^{k},\mathcal{M})$ und
$\xi^{k}= \left(\ot_{k}^{*}\cp\Xi^{k}\cp G_{k}^{*} \cp
\Upsilon^{k}_{-1}\cp \Theta^{k}_{-1}\right)$. Sei weiter
$\mathcal{M}$ ein symmetrischer $\SsV-\SsV$-Bimodul, dann gilt:
\begin{equation*}
\xi^{k} (\phi)= \frac{1}{k!}\:\phi_{D}.
\end{equation*}
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Mit der Symmetrie von
$\omega_{1}^{m_{1}}\vee…\vee\omega_{k}^{m_{k}}$
ist $\phi_{D}$ total antisymmetrisch, falls $\phi$ total antisymmetrisch
ist. Dies zeigt die letzte Aussage, da die umgekehrte
Implikation trivial ist.
Für die Wohldefiniertheit sei $\V^{n_{i}}\ni
\omega^{\times}_{i}=\big((\omega_{i})_{1},…,(\omega_{i})_{n_{i}}\big)$
mit $1\leq i\leq k$ und $\omega_{i}^{m_{i}}\in
\Ss^{n_{i-1}}(\V)$ das Element
$(\omega_{i})_{1}\vee…\blacktriangle^{m_{i}}…\vee(\omega_{i})_{n_{i}}$.
Dann ist
\begin{align*}
\phi^{n_{1},…,n_{k}}\left(\omega^{\times}_{1},…,\omega^{\times}_{k}\right)=\sum_{m_{1}=1}^{n_{1}}…\sum_{m_{k}=1}^{n_{k}}
\omega_{1}^{m_{1}}\vee…\vee\omega_{k}^{m_{k}}
*
\phi\big((\omega_{1})_{m_{1}},…,(\omega_{k})_{m_{k}}\big)
\end{align*}
eine wohldefinierte $\mathbb{K}$-multilineare Abbildung von
$\V^{[n_{1}+…+n_{k}]}$ nach $\mathcal{M}$. Mit
Lemma \ref{lemma:assTenprod}~\textit{ii.)} sowie der universellen
Eigenschaft des Tensorproduktes erhalten wir eine lineare Fortsetzung
\begin{equation*}
\phi_{\ot}^{n_{1},…,n_{k}}\colon\Tt^{n_{1}}(\V)\ot…\ot\Tt^{n_{k}}(\V)\longmapsto
\mathcal{M},
\end{equation*}
die verkettet mit $\ot_{n_{1},…,n_{k}}\colon\Tt^{n_{1}}(\V)\times…\times\Tt^{n_{k}}(\V)\longrightarrow
\Tt^{n_{1}}\ot…\ot\Tt^{n_{k}}$ die Eigenschaft
\begin{equation*}
\phi_{\ot}^{n_{1},…n_{k}}\cp \ot_{n_{1},…,n_{k}}\big|_{\Ss^{n_{1}}(\V)\times…\times\Ss^{n_{k}}(\V)}=\phi_{D}\big|_{\Ss^{n_{1}}(\V)\times…\times\Ss^{n_{k}}(\V)}
\end{equation*}besitzt.
Insgesamt folgt
\begin{equation*}
\phi_{D}=\sum_{n_{1},…,n_{k}}\big[\phi_{\ot}^{n_{1},…,n_{k}}\cp\ot_{n_{1},…,n_{k}}\big]\big|_{\Ss^{n_{1}}(\V)\times…\times\Ss^{n_{k}}(\V)},
\end{equation*}also die Wohldefiniertheit von $\phi_{D}$.
Für die Derivativität rechnen wir
{\allowdisplaybreaks{\small\begin{align*}
\phi&_{D}(\omega_{1},…,\omega_{l}\vee \omega'_{l}
,…,\omega_{k})
\\ &=\sum_{\substack{m_{i}=1\\i\neq
l}}^{n_{i}}\sum_{\wt{m}_{l}=1}^{\deg(\omega_{l}\vee\omega'_{l})}\omega_{1}^{m_{1}}\vee…\vee\left(\omega_{l}\vee\omega'_{l}\right)^{\wt{m}_{l}}\vee…\vee\omega_{k}^{m_{k}}*
\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\:\:\:\phi\big((\omega_{1})_{m_{1}},…,\left(\omega_{l}\vee\omega'_{l}\right)_{\wt{m}_{l}},…,(\omega_{k})_{m_{k}}\big)
\\ &=\omega_{l}\vee\left[\sum_{\substack{m_{i}=1\\i\neq
l}}^{n_{i}}\sum_{m'_{l}=1}^{\deg(\hat{\omega}{l})}\omega_{1}^{m_{1}}\vee…\vee\omega_{l}'^{m'_{l}}\vee…\vee\omega_{k}^{m_{k}}*\phi\big((\omega_{1})_{m_{1}},…,(\omega'_{l})_{m'_{l}},…,(\omega_{k})_{m_{k}}\big)\right]
\\ &\:\:\:+\omega'_{l}\vee\left[\sum_{\substack{m_{i}=1\\i\neq
l}}^{n_{i}}\sum_{m_{l}=1}^{\deg(\omega_{l})}\omega_{1}^{m_{1}}\vee…\vee\omega_{l}^{m_{l}}\vee…\vee\omega_{k}^{m_{k}}*\phi\big((\omega_{1})_{m_{1}},…,(\omega_{l})_{m_{l}},…,(\omega_{k})_{m_{k}}\big)\right]
\\ &=\omega_{l}\vee \phi_{D}(\omega_{1},…,\omega'_{l},…,\omega_{k})+ \omega'_{l}\vee \phi_{D}(\omega_{1},…,\omega_{l},…,\omega_{k}),
\end{align*}}}wobei wir im zweiten Schritt die
Moduleigenschaft von $\mathcal{M}$ benutzt haben.
Für die Eindeutigkeit sei $\wt{\phi}$ eine weitere
derivative Fortsetzung von $\phi$. Im Falle
$k=1$ und mit $\omega=\omega_{1}\vee…\vee\omega_{l}$ erhalten
wir sukzessive:
\begin{align*}
\wt{\phi}\:(\omega)=&\:\omega^{1}*\wt{\phi}\:(\omega_{1})+\omega_{1}*\wt{\phi}\:(\omega^{1})
\\=&\:\omega^{1}*\wt{\phi}\:(\omega_{1})+
\omega^{2}*\wt{\phi}\:(\omega_{2})+\omega_{1,2}*\wt{\phi}\:(\omega^{1,2})
\\=&\:\omega^{1}*\wt{\phi}\:(\omega_{1})+…+\omega^{l-1}*\wt{\phi}\:(\omega_{l-1})+\omega_{1,…,l-1}*\wt{\phi}\:(\omega^{1,…,l-1})
\\=&\:\omega^{1}*\wt{\phi}\:(\omega_{1})+…+\omega^{l}*\wt{\phi}\:(\omega_{l}).
\end{align*}
Für $k>1$ zeigt induktives Anwenden obiger Relation, dass
{\allowdisplaybreaks\begin{align*}
\wt{\phi}(\omega)=&\:\sum_{m_{1}=1}^{n_{1}}…\sum_{m_{k}=1}^{n_{k}}\omega_{1}^{m_{1}}\vee…\vee\omega_{k}^{m_{k}}
*
\wt{\phi}\big((\omega_{1})_{m_{1}},…,(\omega_{k})_{m_{k}}\big)
\\=&\:\sum_{m_{1}=1}^{n_{1}}…\sum_{m_{k}=1}^{n_{k}}\omega_{1}^{m_{1}}\vee…\vee\omega_{k}^{m_{k}}
*
\phi\big((\omega_{1})_{m_{1}},…,(\omega_{k})_{m_{k}}\big)
\\=&\:\phi_{D}(\omega),
\end{align*}}und die Derivativität erzwingt
\begin{align*}
\wt{\phi}(\omega_{1},…,1_{i},…,\omega_{k})=\wt{\phi}(\omega_{1},…,1\vee
1_{i},…,\omega_{k})=2\wt{\phi}(\omega_{1},…,1_{i},…,\omega_{k}),
\end{align*}
also $\wt{\phi}(\omega_{1},…,1_{i},…,\omega_{k})=0$. Dies zeigt die Eindeutigkeit besagter
derivativer Fortsetzung.
\item
Wir erhalten:
\begin{align*}
\left(\mathrm{Alt}_{k}\cp
\delta^{k-1}\right)&\big(\phi\big)(a_{1},…,a_{k})
\\=&\: \mathrm{Alt}_{k}\left(a_{1}*_{L}\phi(a_{2},…,a_{k})\right)
+\sum_{i=1}^{k-1}(-1)^{i}\mathrm{Alt}_{k}\left(\phi(a_{1},…,a_{i}a_{i+1},…,a_{k+1})\right)
\\ &
\qquad\qquad\qquad\qquad\qquad\qquad+(-1)^{k}\mathrm{Alt}_{k}(\phi(a_{1},…,a_{k-1})*_{R}a_{k})
\\=&\: \mathrm{Alt}_{k}\left(a_{1}*_{L}\phi(a_{2},…,a_{k})\right) +(-1)^{k}(-1)^{k-1}\mathrm{Alt}_{k}(\phi(a_{2},…,a_{k})*_{R}a_{1})
\\
&\qquad\qquad\qquad\qquad\qquad\qquad+\sum_{i=1}^{k-1}(-1)^{i}\mathrm{Alt}_{k}\left(\phi(a_{1},…,a_{i}a_{i+1},…,a_{k})\right)
\\=&\:0.
\end{align*}
Dabei verschwindet die letzte Summe wegen
der Kommutativität von $\mathcal{A}$.
\item
Sei $\phi\in \Hom^{a}_{\mathbb{K}}(\V^{k},\mathcal{M})$. Dann ist
\begin{equation*}
\begin{split}
\left(\xi^{k}\phi\right)(u_{1},\dots,u_{k})=&\:\bigg(\ot_{k}^{*}\cp\Xi^{k}\cp G_{k}^{*} \cp \Upsilon^{k}_{-1}\cp
\Theta^{k}_{-1}\bigg) \big(\phi\big)(u_{1},\dots,u_{k})
\\=&\: \left(\Xi^{k}\cp G_{k}^{*} \cp \Upsilon^{k}_{-1}\cp
\Theta^{k}_{-1}\right)\big(\phi\big)(u_{1}\ot\dots\ot u_{k})
\\=&\: \left(G_{k}^{*} \cp \Upsilon^{k}_{-1}\cp
\Theta^{k}_{-1}\right)\big(\phi\big)(1\ot u_{1}\ot\dots\ot u_{k}\ot 1)
\\=&\: \left(\Upsilon^{k}_{-1}\cp
\Theta^{k}_{-1}\right)\big(\phi\big)\big(G_{k}(1\ot u_{1}\ot\dots\ot u_{k}\ot 1)\big).
\end{split}
\end{equation*}Für $v_{1},…,v_{k}\in \V$ folgt
{\allowdisplaybreaks\begin{align*}
G_{k}(1\ot v_{1}\ot\dots\ot v_{k}\ot
1)=&\:\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k}i\big(1\ot1\bbot
\overbrace{1\ot…\ot 1}^{k-mal} \bbot
v_{1}\wedge…\wedge v_{k}\big)
\\=&\:\int_{0}^{1}dt_{1}…\int_{0}^{t_{k-1}}dt_{k} 1\ot1\ot
v_{1}\wedge…\wedge v_{k}
\\=&\:\frac{1}{k!} 1\ot1\ot
v_{1}\wedge…\wedge v_{k},
\end{align*}}also
{\allowdisplaybreaks\begin{align*}
\left(\xi^{k}\phi\right)(v_{1},…,v_{k})=&\:\frac{1}{k!}\left(\Upsilon^{k}_{-1}\cp
\Theta^{k}_{-1}\right)\big(\phi\big)(1\ot
1\ot v_{1}\wedge\dots\wedge v_{k})
\\=&\:\frac{1}{k!} \Theta^{k}_{-1}\big(1\ot 1 *_{e}\phi\big)(1\ot
1\ot v_{1}\wedge\dots\wedge v_{k})
\\=&\:\frac{1}{k!}\: \phi(v_{1},…,v_{k}).
\end{align*}}Des Weiteren ist
$\left(\xi^{k}\phi\right)(u_{1},…,u_{k})=0$,
falls $u_{i}=1$ für ein $1\leq i\leq k$, da dann
$\delta(1\ot u_{1}\ot…\ot u_{k}\ot 1)=0$ gilt.
Sei abkürzend $\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})
\ni\wt{\phi}=\left(\Upsilon^{k}_{-1}\cp\Theta^{k}_{-1}\right)\big(\phi\big)$, dann
folgt mit \eqref{eq:DeltaDerivaufuElem} und sukzessiver
Anwendung von
\begin{equation}
\label{eq:aeTrickt}
\begin{split}
\hat{i}(1\ot \mathrm{v}\ot 1) *_{e} m =& \:\big[t (\mathrm{v}\ot 1) + (1-t) (1\ot \mathrm{v}) \big] *_{e} m
\\=&\: t \:\mathrm{v}*_{L} m + \mathrm{v}*_{R}m - t\: \mathrm{v}*_{R} m
\\=&\: \mathrm{v} *_{L} m,
\end{split}
\end{equation}
dass
{\allowdisplaybreaks\begin{align*}
\left(\xi^{k}\phi\right)(u_{1},…,&\:u_{j}\vee
u'_{j},…,u_{k})
\\ =&\:\Big(\wt{\phi} \cp G_{k}\Big)(1\ot u_{1}\ot\dots u_{j}\vee u'_{j}\ot\dots\ot u_{k}\ot 1)
\\ =&\int_{0}^{1}dt_{1}\dots
\int_{0}^{t_{k-1}}dt_{k}\:\hat{i}_{j}(1\ot u_{j}\ot
1)*_{e}
\\ &\qquad\qquad\qquad\qquad \wt{\phi}\left(\prod_{s\neq j}(i_{s}\cp\delta)(1\ot u_{s}\ot 1) \cdot (i_{j}\cp\delta)(1\ot u'_{j}\ot 1)\right)
\\ & +\int_{0}^{1}dt_{1}\dots
\int_{0}^{t_{k-1}}dt_{k}\:\hat{i}_{j}(1\ot u'_{j}\ot
1)*_{e}
\\ &\qquad\qquad\qquad\qquad\: \wt{\phi}\left(\prod_{s\neq
j}(i_{s}\cp\delta)(1\ot u_{s}\ot 1) \cdot
(i_{j}\cp\delta)(1\ot u_{j}\ot 1)\right)
\\ =&\:u_{j}*_{L} \left(\wt{\phi}\cp
G_{k}\right)(1\ot u_{1}\ot\dots
u'_{j}\ot\dots\ot u_{k}\ot 1)\:+
\\ &\qquad\qquad\qquad\qquad\quad u'_{j}*_{L} \left(\wt{\phi}\cp G_{k}\right)(1\ot u_{1}\ot\dots
u_{j}\ot\dots\ot u_{k}\ot 1)
\\ =&\:u_{j}*_{L} \left(\xi^{k}\phi\right)(u_{1},…,u'_{j},…,u_{k})
+u'_{j}*_{L} \left(\xi^{k}\phi\right)(u_{1},…,u_{j},…,u_{k}).
\end{align*}}Mit \textit{i.)} zeigt dies die
Behauptung. Hierfür beachte man, dass wir $\wt{\phi}$ mit den
Integralen vertauschen dürfen, da die Intergrationen lediglich
den verschiedenen $t$-Faktoren reelle Zahlen zuordnen und
$\wt{\phi}$ nach Voraussetzung $\mathbb{K}$-linear ist.
\end{enumerate}
\end{beweis}
\end{proposition}
Bevor wir zu dem Hauptresultat dieses Kapitels kommen, erinnern wir
an folgendes kommutatives Diagramm:
\[\begin{xy}
\xymatrix{
...\ar[r]^{\delta^{k-2}}
&HC^{k-1} \ar[d]_{\ot_{k-1*}}\ar[r]^{\delta^{k-1}}\ar@{->}@/^ 0.6cm/[dd]^/.6em/{\zeta^{k-1}}
&HC^{k} \ar[d]_{\ot_{k*}}\ar[r]^{\delta^{k}}\ar@{->}@/^ 0.6cm/[dd]^/.6em/{\zeta^{k}}
&HC^{k+1}
\ar[d]_{\ot_{k+1*}}\ar[r]^{\delta^{k+1}}\ar@{->}@/^ 0.6cm/[dd]^/.6em/{\zeta^{k+1}} &\dots \\
... \ar[r]^{\delta^{k-2}_{\ot}}
&HC^{k-1}_{\ot}\ar[d]_{\Xi^{k-1}_{-1}} \ar[r]^{\delta^{k-1}_{\ot}}
&HC^{k}_{\ot}\ar[d]_{\Xi^{k}_{-1}} \ar[r]^{\delta^{k}_{\ot}} &HC^{k+1}_{\ot}\ar[d]_{\Xi^{k+1}_{-1}} \ar[r]^{\delta^{k+1}_{\ot}}&\dots\\
... \ar[r]^{d_{k-1}^{*}}
&\C_{k-1}^{*} \ar[r]^{d_{k}^{*}}
&\C_{k}^{*} \ar[r]^{d_{k+1}^{*}}\ar@{.>}@/^ 0.5cm/[l]^{s^{*}_{k-1}} &\C_{k+1}^{*} \ar[r]^{d_{k+2}^{*}}\ar@{.>}@/^ 0.5cm/[l]^{s^{*}_{k}}&\dots\:.
}
\end{xy}\]
Hierbei ist $HC^{k}=HC^{k}(\SsV,\mathcal{M})$, sowie
$HC^{k}_{\ot}=HC^{k}_{\ot}(\SsV,\mathcal{M})$ die
Tensorvariante des Hochschild-Komplexes.
Der untere Komplex
ist der durch Anwendung des
$\hom_{\mathcal{A}^{e}}(\cdot,\mathcal{M})$-Funktors
erhaltenen Kokettenkomplex $(\C^{*},d^{*})$ mit
$\C_{k}^{*}=\Hom_{\mathcal{A}^{e}}(\C_{k},\mathcal{M})$ und
$d_{k+1}^{*}\phi_{k}=\phi_{k}\cp d_{k+1}$.
Die $s_{k}$
bezeichnen die in Lemma \ref{lemma:Homotopiejdfgjkf} definierten
Homotopieabbildungen und $\zeta^{k}\colon
HC^{k}(\SsV,\mathcal{M})\longrightarrow
\Hom_{\mathcal{A}^{e}}(\C_{k},\mathcal{M})$ den
Kettenisomorphismus $\zeta^{k}=\Xi^{k}_{-1}\cp \ot_{k*}$.
\begin{satz}[Hochschild-Kostant-Rosenberg]
\label{satz:HKR}
\begin{enumerate}
\item
Gegeben ein symmetrischer $\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul $\mathcal{M}$.
Dann besitzt jede Kohomologieklasse
$\left[\eta\right]\in HH^{k}(\Ss^{\bullet}(\V),\mathcal{M})$ genau einen total
antisymmetrischen Repräsentanten $\phi^{a,\eta}_{D}$.
Dieser ist derivativ in jedem Argument und gegeben durch
$\phi^{a,\eta}_{D}=\mathrm{Alt}_{k}(\phi)$ f"ur beliebiges $\phi\in
[\eta]$ mit $\phi^{a,0}_{D}=0$ für
die $0$-Klasse $[0]$.
Insgesamt gilt für alle $\phi\in [\eta]$:
\begin{equation}
\label{eq:RepHKR}
\phi=\underbrace{\phi^{a,\eta}_{D}}_{\mathrm{Alt}_{k}(\phi)}+\underbrace{\delta^{k-1}\big(\zeta^{k-1}_{-1}s^{*}_{k-1}\zeta^{k}\phi\big)}_{\phi-\mathrm{Alt}_{k}(\phi)}.
\end{equation}
\item
Gegeben ein symmetrischer, lokalkonvexer
$\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul $\mathcal{M}$.
Dann besitzt jedes
$\left[\eta_{c}\right]\in HH^{k}_{\operatorname{\mathrm{cont}}}(\Ss^{\bullet}(\V),\mathcal{M})$ genau einen total
antisymmetrischen, stetigen Repräsentanten
$\phi^{a,\eta}_{c,D}$. Dieser ist derivativ in jedem Argument und gegeben durch
$\phi^{a,\eta}_{c,D}=\mathrm{Alt}_{k}(\phi_{c})$ f"ur beliebiges $\phi_{c}\in
[\eta_{c}]$ mit $\phi^{a,0}_{c,D}=0$ für
die $0$-Klasse $[0_{c}]$.
Insgesamt gilt für alle $\phi_{c}\in [\eta_{c}]$:
\begin{equation}
\label{eq:RepHKRstet}
\phi_{c}=\underbrace{\phi^{a,\eta}_{c,D}}_{\mathrm{Alt}_{k}(\phi_{c})}+\underbrace{\delta_{c}^{k-1}\big(\zeta^{k-1}_{-1}s^{*}_{k-1}\zeta^{k}\phi_{c}\big)}_{\phi_{c}-\mathrm{Alt}_{k}(\phi_{c})}.
\end{equation}
\item
Gegeben ein vollständiger, symmetrischer, hausdorffscher, lokalkonvexer
$\mathrm{Hol}(\V)-\Hol$-Bimodul $\mathcal{M}$. Dann besitzt jedes
$\left[\hat{\eta}_{c}\right]\in HH^{k}_{\operatorname{\mathrm{cont}}}(\Hol,\mathcal{M})$ genau einen total
antisymmetrischen, stetigen Repräsentanten
$\hat{\phi}^{a,\eta}_{c,D}$. Dieser ist derivativ in
jedem Argument und gegeben durch $\hat{\phi}^{a,\eta}_{c,D}=\mathrm{Alt}_{k}\big(\hat{\phi}_{c}\big)$ f"ur beliebiges $\hat{\phi}_{c}\in
[\hat{\eta}_{c}]$ mit $\hat{\phi}^{a,0}_{c,D}=0$ für
die $0$-Klasse $[\hat{0}_{c}]$.
Insgesamt gilt für alle $\hat{\phi}_{c}\in [\hat{\eta}_{c}]$:
\begin{equation}
\label{eq:RepHKRstethol}
\hat{\phi}_{c}=\underbrace{\hat{\phi}^{a,\eta}_{c,D}}_{\mathrm{Alt}_{k}(\hat{\phi}_{c})}+\underbrace{\hat{\delta}_{c}^{k-1}\widehat{\Big(\zeta^{k-1}_{-1}s^{*}_{k-1}\zeta^{k}\phi_{c}\Big)}}_{\hat{\phi}_{c}-\mathrm{Alt}_{k}(\hat{\phi}_{c})}\quad\text{
mit }\quad \phi_{c}=\hat{\phi}_{c}\big|_{\SsV^{k}}.
\end{equation
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Zunächst ist die Existenz eines total antisymmetrischen
Repräsentanten $\phi_{D}^{a,\eta}$ gesichert, da die
Abbildung $\wt{\xi^{k}}:\Hom_{\mathbb{K}}^{a}(\V^{k},\mathcal{M})\longrightarrow
HH^{k}(\Ss^{\bullet}(\V),\mathcal{M})$
ein Isomorphismus war und
$\xi^{k}(\phi)$ nach
Proposition \ref{prop:wichEizuHKR}~\textit{iii.)} total
antisymmetrisch und derivativ ist. Mit
Proposition \ref{prop:wichEizuHKR}~\textit{ii.)} folgt
\begin{equation}
\label{eq:Altdelta}
\mathrm{Alt}_{k}(\phi)=\mathrm{Alt}_{k}\left(\phi^{a,\eta}_{D}+\delta^{k-1}(\psi)\right)
=\phi^{a,\eta}_{D}\qquad\quad\forall\:\phi\in[\eta],
\end{equation}und ebenso die Eindeutigkeit. Denn für total
antisymmetrische $\phi^{a},\hat{\phi}^{a}\in
\left[\eta\right]$ ist
$\phi^{a}-\hat{\phi}^{a}$ zudem exakt und somit
$(\phi^{a}-\hat{\phi}^{a})=\mathrm{Alt}_{k}(\phi^{a}-\hat{\phi}^{a})=0$.
Die Aussage $\phi_{D}^{a,0}=0$ ist dann wegen
der Linearität von $ \wt{\xi^{k}}$ trivial.
Für \eqref{eq:RepHKR}
sei $\hat{\xi}^{k}=\Theta^{k}\cp
\Upsilon^{k}\cp F_{k}^{*}\cp \Xi^{k}_{-1}\cp \ot_{k*}$ , womit
\begin{equation*}
\wt{\hat{\xi}^{k}}\colon HH^{k}(\Ss^{\bullet}(\V),\mathcal{M})\longrightarrow
\Hom_{\mathbb{K}}^{a}(\V^{k},\mathcal{M})
\end{equation*}
der zu $\wt{\xi^{k}}$ inverse Isomorphismus ist. Dann folgt
\begin{equation*}
\left(\xi^{k}\cp\hat{\xi}^{k}\right)(\phi)=\left(\zeta^{k}_{-1}\Omega^{*}_{k}\zeta^{k}\right)(\phi)=\phi_{D}^{a,\eta}=\mathrm{Alt}_{k}(\phi),
\end{equation*}
was man unmittelbar daran sieht, dass jedes $\phi\in [\eta]$ unter $\hat{\xi}^{k}$
das gleiche Bildelement haben muss. Alternativ rechnet man
$\xi^{k}\cp\hat{\xi}^{k}=\mathrm{Alt}_{k}$ auch explizit
nach (vgl. Bemerkung \ref{bem:HRKBem}~\textit{ii.)}).
Nach
Lemma \ref{lemma:Homotopiejdfgjkf} haben wir nun
\begin{equation}
\id_{\C^{*}_{k}}-\:\Omega^{*}_{k}=s^{*}_{k}d^{*}_{k+1}+d^{*}_{k}s^{*}_{k-1},
\end{equation}
also
\begin{equation}
\label{eq:1}
\id_{HC^{k}}-\:\zeta_{-1}^{k}\Omega^{*}_{k}\zeta^{k}=\zeta_{-1}^{k}s^{*}_{k}d^{*}_{k+1}\zeta^{k}+\zeta_{-1}^{k}d^{*}_{k}s^{*}_{k-1}\zeta^{k}
\end{equation}
und somit:
\begin{equation}
\label{eq:dgh}
\id_{HC^{k}}-\mathrm{Alt}_{k}=\zeta_{-1}^{k}s^{*}_{k}\zeta^{k+1}\delta^{k}+\delta^{k-1}\big(\zeta_{-1}^{k-1}s^{*}_{k-1}\zeta^{k}\big).
\end{equation}Anwenden von \eqref{eq:dgh} auf einen Korand
$\phi\in [\eta]$ liefert
\begin{equation*}
\phi-\mathrm{Alt}_{k}(\phi)=\delta^{k-1}\big(\zeta_{-1}^{k-1}s^{*}_{k-1}\zeta^{k}\phi\big)
\end{equation*}
und zeigt somit die Behauptung.
\item
Alle Isomorphismen aus obigem Diagramm sind ebenfalls
Isomorphismen auf den stetigen Unterkomplexen. Nach
Lemma \ref{lemma:Homotopiejdfgjkf} sind die $s_{k}$
\emph{stetig} und somit besagtes Diagramm auch auf die stetige Situation anwendbar.
Des Weiteren sind
\begin{align*}
\wt{\xi^{k}_{c}}\colon\Hom^{a,\operatorname{\mathrm{cont}}}_{\mathbb{K}}(\V^{k},\mathcal{M})\longrightarrow
HH^{k}_{\operatorname{\mathrm{cont}}}(\Ss^{\bullet}(\V),\mathcal{M})
\end{align*}
und
\begin{equation*}
\wt{\hat{\xi}_{c}^{k}}\colon HH_{\operatorname{\mathrm{cont}}}^{k}(\Ss^{\bullet}(\V),\mathcal{M})\longrightarrow
\Hom_{\mathbb{K}}^{a,\operatorname{\mathrm{cont}}}(\V^{k},\mathcal{M})
\end{equation*}
mit
$\xi^{k}_{c}=\xi^{k}\big|_{\Hom^{a,\operatorname{\mathrm{cont}}}_{\mathbb{K}}(\V^{k},\mathcal{M})}$ und
$\hat{\xi}_{c}^{k}=\hat{\xi}^{k}\big|_{HC^{k}_{\operatorname{\mathrm{cont}}}(\SsV,\mathcal{M})}$
zueinander inverse Isomorphismen. Hiermit folgen alle
Behauptungen analog zu \textit{i.)}.
\item
Die Eindeutigkeit folgt unmittelbar aus
Proposition \ref{prop:wichEizuHKR}~\textit{ii.)}. Des Weiteren haben
wir nach Satz \ref{satz:HochschildHol}, vermöge Einschr"ankung und stetiger
Fortsetzung, eine Isomorphie
$HC_{\operatorname{\mathrm{cont}}}^{k}(\Hol,\mathcal{M})\cong
HC_{\operatorname{\mathrm{cont}}}^{k}(\SsV,\mathcal{M})$, welche die Isomorphie $HH^{k}_{\operatorname{\mathrm{cont}}}(\Hol,
\mathcal{M})\cong
HH^{k}_{\operatorname{\mathrm{cont}}}(\Ss^{\bullet}(\V),\mathcal{M})$
induziert. Mit der Linearität besagter Isomorphismen folgt
dann unmittelbar:
\begin{equation*}
\widehat{\mathrm{Alt}_{k}(\phi_{c})}=\mathrm{Alt}_{k}\big(\hat{\phi}_{c}\big).
\end{equation*}
Nach \textit{ii.)} ist für jedes $\hat{\phi}_{c}\in
[\hat{\eta}_{c}]\in HH^{k}(\Hol,\mathcal{M})$ die Einschr"ankung,\\
$\phi_{c}=\hat{\phi}_{c}\big|_{\Ss^{\bullet}(\V)^{k}}$, darstellbar in der Form:
\begin{equation*}
\phi_{c}=\phi^{a,\eta}_{c,D}+\delta_{c}^{k-1}\left(\zeta^{k-1}_{-1}s^{*}_{k-1}\zeta^{k}\phi_{c}\right).
\end{equation*}
Hieraus folgt durch stetige Fortsetzung beider Seiten von
\eqref{eq:muhkuhmilch}, dass
\begin{equation*}
\hat{\phi}_{c}= \hat{\phi}^{a,\eta}_{c,D}+\hat{\delta}^{k-1}_{c}\left(\widehat{\zeta^{k-1}_{-1}s^{*}_{k-1}\zeta^{k}\phi_{c}}\right),
\end{equation*}
wobei der erste Summand wegen
$\mathrm{Alt}_{k}\left(\hat{\phi}^{a,\eta}_{c,D}\right)=\widehat{\mathrm{Alt}_{k}\left(\phi^{a,\eta}_{c,D}\right)}=\hat{\phi}^{a,\eta}_{c,D}$
total antisymmetrisch ist. Mit \eqref{eq:Altdelta}
zeigt dies die Zuweisungen unter den geschweiften Klammern, da die Zerlegung
$\hat{\phi_{c}}=\mathrm{Alt_{k}}(\hat{\phi_{c}})+\hat{\phi_{c}}-\mathrm{Alt_{k}}(\hat{\phi}_{c})$
offenbar trivial ist.
Es bleibt nun lediglich die Derivationseigenschaft von
$\hat{\phi}^{a,\eta}_{c,D}$ nachzuweisen. Hierfür rechnen
wir mit den Stetigkeiten von $*$ und $*_{L}$ , der
Definition von $\hat{\phi}^{a,\eta}_{c,D}$ sowie der
Derivativit"at von $\phi^{a,\eta}_{c,D}$:
\begin{align*}
\hat{\phi}^{a,\eta}_{c,D}(\omega_{1},\dots, \omega_{l}*\omega'_{l},&\dots,\omega_{k})
\\=&\:\lim_{\Lambda}\: \phi^{a,\eta}_{c,D}\left((\omega_{1})_{\alpha_{1}},\dots,(\omega_{l})_{\alpha_{l}}\vee(\omega'_{l})_{\alpha'_{l}} ,\dots,(\omega_{k})_{\alpha_{k}}\right)
\\=&\: \lim_{\Lambda}\: (\omega_{l})_{\alpha_{l}}*_{L}\phi^{a,\eta}_{c,D}\left((\omega_{1})_{\alpha_{1}},\dots,(\omega'_{l})_{\alpha'_{l}} ,\dots,(\omega_{k})_{\alpha_{k}}\right)
\\ &+\lim_{\Lambda}\:(\omega'_{l})_{\alpha'_{l}} *_{L}\phi^{a,\eta}_{c,D}\Big((\omega_{1})_{\alpha_{1}},\dots,(\omega_{l})_{\alpha_{l}} ,\dots,(\omega_{k})_{\alpha_{k}}\Big)
\\=&\:\omega_{l}*_{L}\hat{\phi}^{a,\eta}_{c,D}(\omega_{1},\dots,
\omega'_{l},\dots,\omega_{k})
+\omega'_{l}*_{L}\hat{\phi}^{a,\eta}_{c,D}(\omega_{1},\dots, \omega_{l},\dots,\omega_{k})
\end{align*}
mit Netzen $\SsV\supseteq\{\omega_{i}\}_{\alpha_{i}\in
J_{i}}\rightarrow \omega_{i}\in \Hol$ $\forall$
$1\leq i\leq k$ und $\{\omega'_{l}\}_{\alpha'_{l}\in
J'_{l}}\rightarrow \omega'_{l}$ sowie $\Lambda=\alpha_{1}\times\dots\times
(\alpha_{l}\times\alpha'_{l})\times\dots\times \alpha_{k}$.
\end{enumerate}
\end{beweis}
\end{satz}
\begin{bemerkung}
\label{bem:HRKBem}
\begin{enumerate}
\item
Obiger Satz besagt nun nicht nur, dass jedes
$\phi\in[\eta]\in HH^{k}(\SsV,\mathcal{M})$ in der Form
$\phi=\phi_{D}^{a,\eta}+ \delta^{k-1}(\psi)$ geschrieben werden kann, sondern legt
uns sogar eine explizite Formel für die Berechnung eines
derartigen $\psi\in HC^{k-1}(\SsV,\mathcal{M})$ in die
Hand\footnote{Dieses ist wegen $\ker(\delta^{k-1})\neq
\{0\}$ nicht eindeutig bestimmt.}. Nun ist die
Berechnung wegen der rekursiven Definition von $s_{k}$ im
Allgemeinen recht kompliziert, jedoch im Rahmen der
Deformationsquantisierung, bei der man zunächst sowieso nur
an den ersten drei Hochschild-Kohomologien interessiert ist,
durchaus ausführbar:
\begin{itemize}
\item[$k=1$:]
Hier ist $s^{*}_{0}=0$, also $[\eta]=\phi_{D}^{a,\eta}$ für alle $[\eta]\in
HH^{1}(\SsV,\mathcal{M})$. Dies ist auch konsistent damit, dass
wegen
\begin{equation*}
(\delta^{0}m)(a)=a*_{L}m-m*_{R}a=0 \quad\text{für alle}\quad m\in
\mathcal{M}= HC^{0}(\SsV,\mathcal{M})
\end{equation*}$\im(\delta^{0})=0$ und somit
\begin{equation*}
HH^{1}(\SsV,\mathcal{M})=\left\{\phi \in
\Hom_{\mathbb{K}}(\SsV,\mathcal{M})\:\big|\:\phi \text{ ist derivativ}\right\}
\end{equation*}gilt
\item[$k=2$:]
In diesem Fall gilt
$s_{1}=\ovl{h}_{1}-\ovl{h_{1}\Omega}_{1}$ mit
\begin{equation*}
\ovl{h}_{1}(x_{0}\ot x_{1} \ot x_{2})=x_{0}\ot 1\ot
x_{1}\ot x_{2}
\end{equation*}für $x_{0},x_{1},x_{2}\in \SsV$ und
\begin{equation*}
\ovl{h_{1}\Omega}_{1}(x_{0}\ot x_{1} \ot x_{2})=\sum_{p=1}^{l}x_{0}\ot\left(\left[\int_{0}^{1}dt_{1}\hat{i}_{1}\left(1\ot
x_{1}^{p}\ot 1\right)\right]*_{e}1\ot (x_{1})_{p}\ot x_{2}\right).
\end{equation*}für $\deg(x_{1})=p$, da
\begin{align*}
\Omega_{1}(x_{0}\ot x_{1} \ot
x_{2})=&\:F_{1}\left(\sum_{p=1}^{l}\int_{0}^{1}dt_{1}i_{1}\left(x_{0}\ot
x_{1}^{p}\ot x_{2}\bbot\: (x_{1})_{p}\right)\right)
\\=&\:\sum_{p=1}^{l}\left[\int_{0}^{1}dt_{1}\hat{i}_{1}\left(1\ot
x_{1}^{p}\ot 1\right)\right]*_{e}x_{0}\ot (x_{1})_{p}\ot
x_{2}.
\end{align*}
Sei nun $\phi\in HC^{2}(\SsV,\mathcal{M})$ und
$x\in \SsV$ mit $\deg(x)=p$, so folgt:
{\allowdisplaybreaks\begin{align*}
\left(\zeta^{1}_{-1}s^{*}_{1}\zeta^{2}\phi\right)(x)=&\:\left(s^{*}_{1}\zeta^{k}\phi\right)(1\ot
x \ot 1)
\\=&\:\overbrace{\big(\zeta^{2}\phi\big)(1\ot 1\ot x\ot 1)}^{\phi(1,x)}
\\ &+\sum_{p=1}^{l}\big(\zeta^{2}\phi\big)\left(1\ot\left(\left[\int_{0}^{1}dt_{1}\hat{i}_{1}\left(1\ot
x^{p}\ot 1\right)\right]*_{e}1\ot x_{p}\ot 1\right)\right).
\end{align*}}Für den letzten Summanden beachte man, dass
\begin{align*}
\hat{i}_{1}\left(1\ot x^{p}\ot 1\right)=&\: t_{1}^{p-1}x^{p}\ot 1 +…
\\ &+ t_{1}^{(p-1)-l}(1-t_{1})^{l}\sum_{j_{1},…j_{l}}^{p-1}(x^{p})^{j_{1},…,j_{l}}\ot
\:(x^{p})_{j_{1},…,j_{l}}+…
\\ &+ (1-t_{1})^{p-1}1\ot x^{p}
\end{align*}
$\displaystyle\int_{0}^{1}dt_{1}t_{1}^{(p-1)-l}(1-t_{1})^{l}=\binom{p}{l}^{-1}$
sowie $\big(\zeta^{2}\phi\big)(1\ot x
\ot x_{p} \ot y)=\phi(x, x_{p})*_{R} y$
gilt. Hiermit folgt:
\begin{align}
\label{eq:vbv}
\left(\zeta^{1}_{-1}s^{*}_{1}\zeta^{2}\phi\right)(x)=&\:\phi(1,x)+\sum_{p=1}^{l}\Bigg[\frac{1}{p}\phi\left(x^{p},x_{p}\right)+…\nonumber
\\ &+\binom{p}{l}^{-1}\sum_{j_{1},…j_{l}}^{p-1}\phi\left((x^{p})^{j_{1},…,j_{l}},x_{p}\right)*_{R}(x^{p})_{j_{1},…,j_{l}}+…\nonumber
\\ &+\frac{1}{p}\phi(1,x_{p})*_{R}x^{p}\Bigg].
\end{align}
\item[$k=n$:]Wir haben
\begin{equation*}
s_{n}=\ovl{h}_{n}-\ovl{h_{n}\Omega}_{n}-\ovl{h_{n}s_{n-1}d}_{n}
\end{equation*}
und
\begin{equation*}
\ovl{h}_{n}(x_{0}\ot x_{1}\ot…\ot x_{n+1})= x_{0}\ot
1 \ot x_{1}\ot…\ot x_{n+1}.
\end{equation*}Den zweiten Summanden berechnet man wie im
Falle $k=2$, wobei hier sehr viel mehr Kombinatorik zu
berücksichtigen ist.
Im Falle $k=3$ ist der letzte Summand gleich $\ovl{h_{2}\ovl{h}_{1}d_{2}}-\ovl{h_{2}\ovl{h_{1}\Omega}_{1}d}_{2}$,
was ebenfalls noch berechenbar ist.
\end{itemize}
Analoge Aussagen gelten nun natürlich auch für die anderen
beiden Fälle, wobei f"ur $\Hol$ natürlich die
Vervollständigung von \eqref{eq:vbv} zu nehmen ist. Des
Weiteren ist $s^{*}$ auch für nicht
symmetrische Bimoduln $\mathcal{M}$ gewinnbringend einsetzbar. Hier erhalten wir mit \eqref{eq:1} f"ur $\phi\in [\eta]\in HH^{k}(\SsV,\mathcal{M})$, dass
\begin{equation*}
\phi=\zeta_{-1}^{k}\Omega^{*}_{k}\zeta^{k}\phi + \left(\phi-\zeta_{-1}^{k}\Omega^{*}_{k}\zeta^{k}\phi\right)=\tilde{\phi}+ \delta^{k-1}\big(\zeta_{-1}^{k-1}s^{*}_{k-1}\zeta^{k}\phi\big)
\end{equation*}mit $\tilde{\phi}=\zeta_{-1}^{k}\Omega^{*}_{k}\zeta^{k}\phi\in [\eta]$.
Hierfür beachte man, dass wir in obiger Formel f"ur $\zeta^{1}_{-1}s^{*}_{1}\zeta^{2}\phi$
explizit zwischen $*_{R}$ und $*_{L}$ unterschieden haben.
\item
Für endlich-dimensionales $\V$ ist
Satz \ref{satz:HKR}~\textit{i.)}, von der expliziten Formel für
den Korand, ein bereits wohl bekanntes
Resultat. Ebenso für den Fall, dass $\mathcal{A}$ die Algebra
der $C^{\infty}(M)$ der glatten Funktionen auf einer endlich-dimensionalen Mannigfaltigkeit $M$ ist,
vgl. \cite{waldmann:2007a},\cite{cahen.gutt.dewilde:1980a}. Die
Multivektorfelder nehmen hierbei den Platz der total
antisymmetrischen, in jedem Argument derivativen
Repräsentanten ein und in der Tat liefert dies eine zutreffende
Analogie, da jeder derartige Repräsentant $\phi_{D}^{a,\eta}$
ein total antisymmetrisches Element in $\DiffOpS{k}{1}$, den
Differentialoperator der Ordnung $1$, ist\footnote{vgl. Proposition \ref{prop:MultidiffOps}}.
\item
Betrachtet man (\ref{eq:RepHKRstethol}), so k"onnte man den Wunsch versp"uren, $\widehat{\zeta^{k-1}_{-1}s^{*}_{k-1}\zeta^{k}}$ durch $\hat{\zeta}^{k-1}_{-1}\hat{s}^{*}_{k-1}\hat{\zeta}^{k}$ zu ersetzen. Hierbei bezeichnen $\hat{\zeta}^{k}$ und $\hat{\zeta}^{k}_{-1}$ die f"ur $\Hol$ analog zu $\zeta^{k}$ und $\zeta^{k}_{-1}$ definierten Isomorphismen.
Dies ist jedoch ohne weiteres nicht m"oglich, da wir f"ur die Definition von $s$ explizit die Kettenabbildung $G$ benutzt haben und somit die Einschr"ankung $\hat{s}_{k-1}|_{\C_{c}^{k-1}}$ im Allgemeinen nach $\hat{\C}_{c}^{k}$ und nicht ausschlie"slich nach $\C_{c}^{k}$ abbildet. Hierbei bezeichnet $(\C_{c},d_{c})$ den zu $\Hol$ geh"origen Bar-Komplex und $(\hat{\C}_{c},\hat{d}_{c})$ dessen Vervollst"andigung.
\item
Wir m"ochten f"ur Satz \ref{satz:HKR}~\textit{i.)} noch einmal auf anderen Weise argumentieren. Hierf"ur seien Eindeutigkeit und Existenz bereits gezeigt. Dann ist mit $\delta^{k}\phi=0$ ebenfalls $\left(\delta^{k}\cp \mathrm{Alt}_{k}\right)(\phi)=0$, also für einen Kozyklus
$\phi\in [\eta]$ auch $\mathrm{Alt}_{k}({\phi})$ ein Kozyklus.
Dies bedeutet $\mathrm{Alt}_{k}(\phi)=\phi^{a,\eta'}_{D}\in \left[\eta'\right]$ mit der
Eindeutigkeit des total antisymmetrischen Repräsentanten in
$\left[\eta'\right]$. Insbesondere ist dann $\mathrm{Alt}_{k}(\phi)$ derivativ, und die Aufgabe besteht nun darin, $\left[\eta'\right]=\left[\eta\right]$ nachzuweisen.
Hierf"ur beachten wir, dass
{\allowdisplaybreaks\begin{align*}
\Big(\hat{\xi}^{k}\phi\Big)(v_{1},\dots,v_{k})=&\: \left(\Upsilon^{k}\cp F_{k}^{*}\cp
\Xi^{k}_{-1}\cp \ot_{k*}\right)\big(\phi\big)(v_{1}\wedge\dots\wedge v_{k})
\\=&\:\left(F_{k}^{*}\cp
\Xi^{k}_{-1}\cp \ot_{k*}\right)\big(\phi\big)(1\ot 1\ot v_{1}\wedge\dots\wedge v_{k})
\\=&\:\left(\Xi^{k}_{-1}\cp \ot_{k*}\right)\big(\phi\big)\big(F_{k}(1\ot 1\ot v_{1}\wedge\dots\wedge v_{k})\big)
\\=&\:\sum_{\sigma\in S_{k}}\sign(\sigma)\left(\Xi^{k}_{-1}\cp \ot_{k*}\right)\big(\phi\big)(1\ot v_{\sigma(1)}\ot\dots\ot v_{\sigma(k)}\ot 1)
\\=&\:\sum_{\sigma\in S_{k}}\sign(\sigma)\big(\ot_{k*}\phi\big)(v_{\sigma(1)}\ot\dots\ot v_{\sigma(k)})
\\=&\:\sum_{\sigma\in S_{k}}\sign(\sigma)\:\phi\:(v_{\sigma(1)},\dots, v_{\sigma(k)})
\\=&\: k!\: \mathrm{Alt}_{k}(\phi)\:(v_{\sigma(1)},\dots, v_{\sigma(k)}),
\end{align*}}also $\Big(\hat{\xi}^{k}\phi\Big)=k!\:
\mathrm{Alt}_{k}\left(\phi\big|_{\V^{k}}\right)=k!\:
\mathrm{Alt}_{k}(\phi)\big|_{\V^{k}}$. Proposition \ref{prop:wichEizuHKR}~\textit{iii.)} zeigt dann $\left(\xi^{k}\cp\hat{\xi}^{k}\right)(\phi)=\left(\mathrm{Alt}_{k}(\phi)\big|_{\V^{k}}\right)_{D}$,
und mit der Derivativität von $\mathrm{Alt}_{k}(\phi)$ zeigt Proposition \ref{prop:wichEizuHKR}~\textit{i.)}, dass
$\left(\xi^{k}\cp\hat{\xi}^{k}\right)(\phi)=\mathrm{Alt}_{k}(\phi)$. Nun gilt
$\xi^{k}\cp \hat{\xi}^{k}\colon [\eta]\longrightarrow[\eta]$,
also $\mathrm{Alt}_{k}(\phi)\in [\eta]$.
\end{enumerate}
\end{bemerkung}
\newpage
\chapter{Differentielle Hochschild-Kohomologien}
\label{cha:DiffHochK}
In diesem Kapitel wollen wir uns dem Begriff der differentiellen
Hochschild-Koketten zuwenden und die Rolle der Kettenabbildungen $F^{*}$
und $G^{*}$ in diesem Rahmen diskutieren. Im Großen und Ganzen soll es hier
darum gehen, nützliche Relationen und Ideen zusammenzutragen, die als
Basis für weitergehende Analysen benutzt werden können. Wir werden
dabei besonderen Wert auf die Diskussion etwaiger
Fallstricke legen, die aus der zu naiven Betrachtung des
Unterkomplex-Begriffes resultieren.
\section{Multidifferentialoperatoren und symmetrische Bimoduln}
In diesem Abschnitt soll es zunächst darum gehen, den Begriff des
Multidifferentialoperators in voller algebraischer Allgemeinheit
kennenzulernen (vgl.\cite[Anhang A]{waldmann:2007a}), um hiermit die
differentiellen Hochschildkomplexe sowie die stetigen, differentiellen
Hochschildkomplexe für symmetrische Bimoduln zu definieren.
Hierf"ur sei daran erinnert, dass wir unter einer $\mathbb{K}$-Algebra immer
einen $\mathbb{K}$-Vektor-raum mit assoziativer,
$\mathbb{K}$-bilinearer Algebramultiplikation verstehen. Ist von einem $\mathcal{A}$-Modul $\mathcal{M}$ die Rede,
so meinen wir einen $\mathbb{K}$-Vektorraum mit $\mathbb{K}$-bilinearer
$\mathcal{A}$-Modul-Multiplikation, wobei die Linearität im
Modul-Element für die Wohldefiniertheit des
Multi-differentialoperator-Begriffes unabdingbar ist.
Befinden wir uns im Folgenden in der Situation einer kommutativen
Algebra, so behandeln wir $\mathcal{M}$ als $\mathcal{A}$-Linksmodul
und bezeichnen die Modul-Multiplikation mit $*_{L}$, wohlwissend, dass
dies f"ur derartige Algebren keine Einschr"ankung bedeutet, siehe
Anhang \ref{sec:ringe-moduln-und}. Ist $\mathcal{A}$ kommutativ, so
setzen wir $1_{\mathcal{A}}*_{L}m=m$ für alle
$m\in \mathcal{M}$ voraus.
Als Teilresultat dieses Abschnittes erhalten wir dann die Isomorphie der stetigen,
differentiellen Kohomologien der Algebren $\Hol$ und $\SsV$ für derartige vollständige,
lokalkonvexe, symmetrischen $\Hol-\Hol$-Bimoduln.
\begin{definition}[Multidifferentialoperator]
\label{def:MultidiffOps}
Gegeben eine assoziative, kommutative $\mathbb{K}$-Algebra $(\mathcal{A},*)$ und ein
$\mathcal{A}$-Modul $(\mathcal{M},*_{L})$. Dann sind die
Multidifferentialoperatoren
$\mathrm{DiffOp}_{k}^{L}(\mathcal{A},\mathcal{M})$ mit Argumenten
im $k$-fachen kartesischen Produkt $\mathcal{A}^{k}$ von
$\mathcal{A}$ und Werten in $\mathcal{M}$, der Multiordnung
$L=(l_{1},\dots,l_{k})\in \mathbb{Z}^{k}$, induktiv definiert durch
\begin{equation*}
\mathrm{DiffOp}_{k}^{L}(\mathcal{A},\mathcal{M})=\{0\}\qquad\quad \forall\: L\in \mathbb{Z}^{k} \text{ mit }l_{i}< 0 \text{ f"ur ein }1\leq i\leq k
\end{equation*}
sowie
\begin{align*}
\mathrm{DiffOp}_{k}^{L}(\mathcal{A},\mathcal{M})=&\Big\{D\in \Hom_{\mathbb{K}}\big(\mathcal{A}^{k},\mathcal{M}\big)\:\Big|\:\forall\: a\in \mathcal{A},\:\forall\: 1\leq i\leq k \text{ gilt}:
\\ &\qquad\qquad\qquad L_{a}\cp D - D\cp L_{a}^{i} \in \mathrm{DiffOp}_{k}^{L-e_{i}}(\mathcal{A},\mathcal{M})\Big\}.
\end{align*}
Dabei bedeutet $L-m\cdot e_{i}=(l_{1},\dots,l_{i}-m,\dots, l_{k})$,
$L_{a}\cp D (a_{1},\dots,a_{k})=a*_{L}D (a_{1},\dots,a_{k})$ und
$\left(D\cp
L^{i}_{a}\right)(a_{1},\dots,a_{k})=D(a_{1},\dots,a_{i}*a,\dots,a_{k})$.
Weiterhin setzen wir $|L|=\sum_{i}l_{i}$ mit $|L|=-1$ falls $l_{i}< 0$ f"ur ein $1\leq i\leq k$ und schreiben
$L\leq L'$ falls $\:l_{i}\leq l'_{i}\:\:\forall\:1\leq i\leq k$ sowie $L=n$ falls $\:l_{i}=n \:\:\forall\:1\leq i\leq k$.
\end{definition}
\begin{proposition}
\label{prop:MultidiffOps}
Unter den Voraussetzungen obiger Definition gilt:
\begin{enumerate}
\item
Sei
\begin{align*}
[]_{i}^{a}\colon\Hom_{\mathbb{K}}(\mathcal{A}^{k},\mathcal{M})&\longrightarrow \Hom_{\mathbb{K}}(\mathcal{A}^{k},\mathcal{M})\\
D &\longmapsto L_{a}\cp D - D\cp L_{a}^{i}.
\end{align*}
Dann gilt $[]^{a}_{i}\cp\: []^{b}_{j}=[]_{j}^{b}\cp\: []^{a}_{i}\:$
f"ur alle $a,b\in \mathcal{A}$ und alle $i,j\in\{1,\dots,k\}$.
\item
Sei $D\in
\Hom_{\mathbb{K}}(\mathcal{A}^{k},\mathcal{M})$. Dann ist
genau dann $D\in \DiffOp{k}{L}$, wenn:
\begin{equation}
\label{eq:DiffopCharack}
[][]_{i}^{a_{1},\dots,a_{l_{i}+1}}D=0\qquad\forall\:1\leq i\leq k,\:\forall\:a_{1},\dots,a_{l_{i}+1}\in \mathcal{A}.
\end{equation}
Hierbei steht $[]_{i}^{a_{1},\dots,a_{l_{i}+1}}\colon\DiffOp{k}{L}\longrightarrow 0\:$
abk"urzend f"ur $[]_{i}^{a_{1}}\cp\dots\cp
\:[]_{i}^{a_{l_{i}+1}}$.
\item
F"ur $K\leq L$ ist
\begin{equation*}
\DiffOp{k}{K}\subseteq \DiffOp{k}{L}
\end{equation*}
und
\begin{equation*}
\DiffOp{k}{\bullet}=\bigcup_{L\geq 0} \DiffOp{k}{L}
\end{equation*}
ein filtrierter Untervektorraum von $\Hom_{\mathbb{K}}(\mathcal{A}^{k},\mathcal{M})$.
\item
Jedes $\DiffOp{k}{L}$ und somit $\DiffOp{k}{\bullet}$ wird verm"oge $a*_{L}D=L_{a}\cp D$ zu einem $\mathcal{A}$-Linksmodul. Des Weiteren ist $D*^{i}_{R}a=D\cp L_{a}^{i}\:$
f"ur alle $1\leq i\leq k$
eine Rechtsmodul-Multiplikation auf jedem $\DiffOp{k}{L}$ und somit auf $\DiffOp{k}{\bullet}$.
\item
Sei $D\in \DiffOp{k}{L}$ und
$L\prec_{i}
P:=(l_{1},…,l_{i-1},l_{i}+p_{1},…,l_{i}+p_{m},l_{i+1},…,l_{k})$.
Dann ist:
\begin{enumerate}
\item
$D_{\mathcal{A}}*_{L}D\in\DiffOp{k+m}{(P,L)}$ für $D_{\mathcal{A}}\in
\mathrm{DiffOp}_{m}^{P}(\mathcal{A},\mathcal{A})$,
\item
$D\cp_{i}D_{\mathcal{A}}\in \DiffOp{k+m\:-1}{L\prec_{i}P}$
für $D_{\mathcal{A}}\in
\mathrm{DiffOp}_{m}^{P}(\mathcal{A},\mathcal{A})$.
\end{enumerate}
Insbesondere ist $\id_{\mathcal{A}}*_{L}D\in\DiffOp{k+1}{(0,L)}$
und $D\cp_{i}*\in \DiffOp{k+1}{L\prec_{i}(0,0)}$. Hierf"ur beachte man, dass jede assoziative Algebra $\mathcal{A}$ insbesondere ein $\mathcal{A}$-Linksmodul ist.
\item
Sei $D\in \Hom_{\mathbb{K}}(\mathcal{A}^{k},\mathcal{M})$ derivativ in jedem Argument. Dann ist $D\in \DiffOp{k}{1}$.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Zun"achst ist das Bild unter $[]_{i}^{a}$ in der Tat $\mathbb{K}$-multilinear, da $*_{L}$ linear im $\mathcal{M}$-Argument und $*$ bilinear ist. Die behauptete Vertauschungsrelation folgt unmittelbar aus der Kommutativit"at von $\mathcal{A}$.
\item
Sei $D\in \DiffOp{k}{L}$. Dann gilt per Definition:
\begin{equation*}
[]_{i}^{a_{1},\dots,a_{l_{i}+1}}D\in \DiffOp{k}{(l_{1},\dots,-1_{i},\dots,l_{k})}=0.
\end{equation*}
F"ur die umgekehrte Implikation sei $[]_{j}^{a}D=0$ f"ur alle $1\leq j\leq k$. Dann folgt unter Ber"ucksichtigung von $a*_{L} 0=0_{\mathbb{K}}\cdot [a*_{L} 0]=0$, dass
\begin{equation*}
D'=[]^{a_{1}}_{1}\cp \dots \cp []^{a_{k}}_{k}D=0\in \DiffOp{k}{-1},
\end{equation*}
also $[]^{a_{2}}_{2}\cp \dots\cp []^{a_{k}}_{k}D\in \DiffOp{k}{(0,-1,\dots,-1)}$ und induktiv $D\in \DiffOp{k}{0}$.
Sei nun die Aussage f"ur $L$ korrekt und \eqref{eq:DiffopCharack} f"ur $L'=L+e_{i}$ erf"ullt. Sei weiter $D'= []_{i}^{a}D$, so ist nach Voraussetzung $[]_{i}^{a_{1},\dots,a_{l_{i}+1}}D'=0$ und f"ur $j\neq i$ folgt:
\begin{align*}
[]_{j}^{a_{1},\dots,a_{l_{j}+1}}D' \glna{\textit{i.)}}\:[]_{i}^{a}\cp\: []_{j}^{a_{1},\dots,a_{l_{j}+1}}D'=[]_{i}^{a}0=0.
\end{align*}
Dies zeigt $D'\in \DiffOp{k}{L}$ und somit $D\in \DiffOp{k}{L'}$.
\item
Die erste Inklusion folgt unmittelbar aus \textit{ii.)} und $[]_{i}^{a}0=0$. F"ur die zweite Behauptung beachte man, dass die $[]_{i}^{a}$ lineare Abbildungen sind und somit \eqref{eq:DiffopCharack} ein lineares Kriterium ist. Hiermit ist $\DiffOp{k}{\bullet}$ ein Untervektorraum von $\Hom_{\mathbb{K}}(\mathcal{A}^{k},\mathcal{M})$ und besagte Inklusion liefert die Filtrationseigenschaft.
\item
Dies folgt sofort aus \textit{ii.)}, $[]_{i}^{a}0=0$ sowie den Vertauschungsrelationen $ []_{i}^{b}\left[L_{a} D\right] =L_{a} \left[[]_{i}^{b}D\right]$ und $[]_{i}^{b}\left[D\cp L^{i}_{a}\right] =\left[[]_{i}^{b}D\right]\cp L^{i}_{a}$.
\item
Zunächst sind alle Kompositionen $\mathbb{K}$-multilineare Abbildungen
nach $\mathcal{M}$.
\begin{enumerate}
\item
Mit der Kommutativität von $\mathcal{A}$ und den Modul-Multiplikationsregeln gilt:
\begin{equation*}
[]_{j}^{a}\big[D_{\mathcal{A}}*_{L}D\big]=\big[a*D_{\mathcal{A}}\big]*_{L}D-\big[D_{\mathcal{A}}\cp L_{a}^{j}\big]*_{L}D =\big[[]_{j}^{a}D_{\mathcal{A}}\big]*_{L}D
\end{equation*}
für $1\leq j\leq m\:$ und
\begin{equation*}
[]_{j}^{a}\big[D_{\mathcal{A}}*_{L}D\big]=D_{\mathcal{A}}*_{L}\big[a*_{L}D\big]-D_{\mathcal{A}}*_{L}\big[D\cp
L_{a}^{j}\big]=D_{\mathcal{A}}*_{L}\big[[]_{j}^{a}D\big]
\end{equation*}
f"ur $m+1\leq j\leq k+m$.
Die Behauptung folgt dann unmittelbar aus \textit{ii.)}.\\
\item
Seien $|L|,|P|\neq -1$, andernfalls ist die Aussage trivial. Wir zeigen diese per Induktion über $|L|+|P|$.
Sei hierf"ur $L=m=0$. Dann ist $[]_{j}^{a}D=0$ für
$1\leq j\leq k$ und
$[]_{j}^{a}D_{\mathcal{A}}=0$ für $1\leq
j\leq m$. In den Fällen $1\leq j< i$ und $i+m-1< j\leq
k+m-1$ folgt
{\small\begin{align*}
[]_{j}^{a}\big[D\cp_{i} D_{\mathcal{A}}\big]=\big[[]_{j}^{a}D\big]\cp_{i} D_{\mathcal{A}}=0
\end{align*}}und für $i\leq j\leq i+m-1$ gilt
{\small\begin{align*}
\big[D\cp_{i}D_{\mathcal{A}}\big]\cp L_{a}^{j}=&\:D\cp_{i}\big[D_{\mathcal{A}}\cp
L^{j-i+1}_{a}\big]=D\cp_{i}\big[a*_{L}D_{\mathcal{A}}\big]
=\big[D\cp L_{a}^{i}\big]\cp
D_{\mathcal{A}}\\=&\:a*_{L}\big[D\cp_{i}D_{\mathcal{A}}\big],
\end{align*}}also $[]_{j}^{a}\big[D\cp_{i}D_{\mathcal{A}}\big]=0$.
Sei nun die Aussage für $|L|+|P|-1$ korrekt. Dann \- gilt in den Fällen
$1\leq j< i$ und $i+m-1< j\leq k+m-1$:
\begin{equation*}
[]_{j}^{a}\big[D\cp_{i}D_{\mathcal{A}}\big]=\big[[]_{j}^{a}D\big]\cp_{i}D_{\mathcal{A}}\in
\DiffOp{k+m\:-1}{[L-e_{j}]\prec_{i}P}
\end{equation*}
nach Induktionsvorraussetzung und für $i\leq j\leq i+m-1$ erhalten wir:
{\small\begin{align*}
[]_{j}^{a}\big[D\cp_{i}D_{\mathcal{A}}&\:\big]=a*_{L}\big[D\cp_{i}D_{\mathcal{A}}\big]
\overbrace{-\big[D\cp L_{a}^{i}\big]\cp_{i} D_{\mathcal{A}} + \big[D\cp
L_{a}^{i}\big]\cp_{i} D_{\mathcal{A}}}^{0}-
\big[D\cp_{i}D_{\mathcal{A}}\big]\cp L^{j}_{a}
\\=&\:\big[a*_{L}D\big]\cp_{i}D_{\mathcal{A}}
-\big[D\cp L_{a}^{i}\big]\cp_{i} D_{\mathcal{A}} + D\cp_{i} \big[a*D_{\mathcal{A}}\big]- D\cp_{i}\big[D_{\mathcal{A}}\cp L^{j-i+1}_{a}\big]
\\=&\:\underbrace{\big[[]_{i}^{a}D\big]\cp_{i}D_{\mathcal{A}}}_{\DiffOp{k+m\:-1}{[L-
e_{i}]\prec_{i} P}}+\underbrace{D\cp_{i}\big[[]_{j-i+1}^{a}D_{\mathcal{A}}\big]}_{\DiffOp{k+m\:-1}{L\prec_{i}[P-e_{(j-i+1)}]}}.
\end{align*}}Hierbei gelten die Zugeh"origkeiten unter den geschweiften Klammern nach Induktionsvorraussetzung.
Im ersten Fall: $j\neq \{i,\dots,i+m-1\}$ ist
$[L-e_{j}]\prec_{i}P=[L\prec_{i}P]-e_{j}$ und für $i\leq
j\leq i+m-1$ gilt $L\prec_{i}[P-e_{(j-i+1)}]=
[L\prec_{i}P]-e_{j}$ sowie $[L-e_{i}]\prec_{i}P=[L\prec_{i}
P]-e_{i}-…-e_{i+m-1}\leq [L\prec_{i}P]-e_{j}$.
Mit \textit{iii.)} zeigt dies $[]_{j}^{a}\big[D\cp_{i}D_{\mathcal{A}}\big]\in
\DiffOp{k+m-1}{[L\prec_{i}P]-e_{j}}$ und Definition \ref{def:MultidiffOps} liefert schlie"slich $D\cp_{i} D_{\mathcal{A}}\in \DiffOp{k+m-1}{L\prec_{i} P}$.
\end{enumerate}
Die letzte Behauptung folgt mit dem bereits Gezeigten und mit $\id_{\mathcal{A}}\in
\mathrm{DiffOp}_{1}^{0}(\mathcal{A},\mathcal{A})$ sowie
$*\in\mathrm{DiffOp}_{2}^{0}(\mathcal{A},\mathcal{A})$.
\item
F"ur jedes $1\leq j\leq k$ ist
$\big[[]_{j}^{a}D\big](a_{1},\dots,a_{k})=-\:a_{i}*_{L}D(a_{1},\dots,a,\dots,a_{k})$,
also $[]_{j}^{a,b}D=0$.
\end{enumerate}
\end{beweis}
\end{proposition}
\begin{bemerkung}
\label{bem:DiffOpCRn}
Die obige rein algebraische Definition der
Multidifferentialoperatoren scheint zun"achst etwas befremdlich. Es
l"asst sich jedoch zeigen (vgl.\cite[Anhang A]{waldmann:2007a}), dass diese f"ur die assoziative, kommutative Algebra $\mathcal{A}=C^{\infty}(M)$ auf einer glatten Mannigfaltigkeit $M$ gerade mit der "ublichen analytischen Definition "ubereinstimmt. Beispielsweise ist genau dann $D\in \mathrm{DiffOp}_{1}^{n}(\mathcal{A},\mathcal{A})$, wenn ein mit der auf $M$ gegebenen differenzierbaren Struktur vertr"aglicher $C^{\infty}$-Atlas $\mathbf{A}$ von $M$ derart existiert, dass f"ur
jede Karte $(U,x)\in \mathbf{A}$, in den Indizes $i_{1},\dots,i_{k}$ symmetrische Funktionen $D_{U}^{i_{1},\dots,i_{k}}\in C^{\infty}(U)$ existieren, so dass:
\begin{equation*}
D\left(f\big|_{U}\right)=\sum_{r=0}^{n}\sum_{i_{1},\dots,i_{r}}\frac{1}{r!}D_{U}^{i_{1},\dots,i_{k}}\frac{\pt^{r}f\big|_{U}}{\pt x^{i_{1}}\dots\pt x^{i_{r}}}.
\end{equation*}
\end{bemerkung}
\begin{definition}[Stetige Multidifferentialoperatoren]
\label{def:MultidiffOpsstet}
Gegeben eine assoziative, kommutative, lokalkonvexe $\mathbb{K}$-Algebra $(\mathcal{A},*)$ und ein
lokal-konvexer $\mathcal{A}$-Modul $(\mathcal{M},*_{L})$, so sind die
stetigen Multidifferentialoperatoren induktiv definiert durch
\begin{equation*}
\mathrm{DiffOp}_{k}^{L,\operatorname{\mathrm{cont}}}(\mathcal{A},\mathcal{M})=\{0\}\qquad\quad \forall\: L\in \mathbb{Z}^{k} \text{ mit }l_{i}< 0 \text{ f"ur ein }1< i\leq k
\end{equation*}
sowie
\begin{align*}
\mathrm{DiffOp}_{k}^{L,\operatorname{\mathrm{cont}}}(\mathcal{A},\mathcal{M})=&\Big\{D\in \Hom^{\operatorname{\mathrm{cont}}}_{\mathbb{K}}\big(\mathcal{A}^{k},\mathcal{M}\big)\:\Big|\:\forall\: a\in \mathcal{A},\:\forall\: 1\leq i\leq k \text{ gilt}:
\\ &\qquad\qquad\qquad\qquad\quad L_{a}\cp D - D\cp L_{a}^{i} \in \mathrm{DiffOp}_{k}^{L-e_{i},\operatorname{\mathrm{cont}}}(\mathcal{A},\mathcal{M})\Big\}.
\end{align*}
\end{definition}
\begin{bemerkung}
\label{bem:DeriBeiStet}
\begin{enumerate}
\item
Proposition \ref{prop:MultidiffOps} überträgt sich sinngemäß auf $\DiffOpc{k}{\bullet}$,
da in der Situation von Definition \ref{def:MultidiffOpsstet} sowohl $*$ als auch $*_{L}$ stetige Abbildungen sind und somit $[]_{i}^{a}\colon\DiffOpc{k}{L}\longrightarrow
\DiffOpc{k}{L-e_{i}}$ gilt.
In der Tat gewährleistet dies im
Induktionsschritt zu Proposition \ref{prop:MultidiffOps}~\textit{ii.)}, dass $D' \in
\Hom_{\mathbb{K}}^{\operatorname{\mathrm{cont}}}(\mathcal{A}^{k},\mathcal{M})$ stetig
ist. Proposition \ref{prop:MultidiffOps}~\textit{ii.)} zeigt dann
insbesondere, dass genau dann $\phi\in
\DiffOpc{k}{L}$ gilt, wenn $\phi\in
\Hom_{\mathbb{K}}^{\operatorname{\mathrm{cont}}}(\mathcal{A}^{k},\mathcal{M})$ und
$\phi\in\DiffOp{k}{L}$ ist.
\item
Man beachte, dass die Forderung der Stetigkeit der Differentialoperatoren in der Tat eine echte Zusatzbedingung liefert. F"ur den Fall $\mathcal{M}=\mathcal{A}=C^{\infty}(\mathbb{R}^{n},\mathbb{R})$, in welchem $\mathcal{A}$ durch die "ublichen Halbnormen
\begin{equation*}
p_{K,l}:\phi\mapsto \sup_{\substack{x\in K\\ |\alpha|\leq l}}\left|\frac{\pt^{\alpha}\phi}{\pt x^{\alpha}}\right|
\end{equation*}
mit $K\subseteq \mathbb{R}^{n}$ kompakt und $l\in \mathbb{N}$ topologisiert ist, sind Multidifferentialoperatoren eben nur wegen ihrer analytischen Form, also ihrer unmittelbaren "Ahnlichkeit zu obigen Halbnormen stetig. W"ahlt man hier ein anderes Halbnormensystem, so ist deren Stetigkeit auch f"ur diese Algebra im Allgemeinen nicht gew"ahrleistet. Um dies noch deutlicher zu machen, sei dem Leser nahegelegt zu versuchen, die Stetigkeit der Derivation $i_{u}\in \mathrm{DiffOp}_{1}^{1}(\Ss^{\bullet}(\V),\Ss^{\bullet}(\V))$ mit
\begin{align*}
i_{u}:v_{1}\vee\dots\vee v_{k}\longmapsto \sum_{i=1}^{k} |u(v_{i})|\cdot v_{1}\vee\dots\blacktriangle^{i}\dots\vee v_{k}
\end{align*}und $i_{u}(1)=0$ f"ur beliebiges $u\in \V^{*}$
nachzuweisen. Im Falle $u\in V'$ ist dies allerdings kein Problem.
\end{enumerate}
\end{bemerkung}
\begin{lemma}
\label{lemma:stetfortDiffOp}
Gegeben ein vollständiger, hausdorffscher, lokalkonvexer $\Hol$-Modul
$\mathcal{M}$. Dann gilt
\begin{equation*}
\mathrm{DiffOp}_{k}^{\bullet,\operatorname{\mathrm{cont}}}(\Hol,\mathcal{M})\cong\DiffOpS{k}{\bullet,\operatorname{\mathrm{cont}}}
\end{equation*}
vermöge Einschränkung und
stetiger Fortsetzung.
\begin{beweis}
Mit Proposition \ref{prop:MultidiffOps}~\textit{ii.)} folgt unmittelbar $\phi\big|_{\SsV^{k}}\in \DiffOpS{k}{\bullet,\operatorname{\mathrm{cont}}}$
für alle $\phi\in\mathrm{DiffOp}_{k}^{\bullet,\operatorname{\mathrm{cont}}}(\Hol,\mathcal{M})$.
Für den Rest der Behauptung reicht es zu zeigen, dass $\hat{\phi}\in\mathrm{DiffOp}_{k}^{\bullet,
\operatorname{\mathrm{cont}}}(\Hol,\mathcal{M})$, falls $\phi \in
\DiffOpS{k}{\bullet, \operatorname{\mathrm{cont}}}$. Hierzu beachten wir, dass
\begin{align*}
[]_{i_{1},…,i_{l}}^{\bullet_{1},…,\bullet_{l}}\phi:(a_{1},…,a_{l+k})&\longmapsto
[]_{i_{1},…,i_{l}}^{a_{1},…,a_{l}}\phi(a_{l+1},\dots,a_{k+l})\quad\text{
sowie}\\
[]_{i_{1},…,i_{l}}^{\bullet_{1},…,\bullet_{l}}\hat{\phi}:(a_{1},…,a_{l+k})&\longmapsto
[]_{i_{1},…,i_{l}}^{a_{1},…,a_{l}}\hat{\phi}(a_{l+1},\dots,a_{k+l})
\end{align*}
beide stetig sind und $\left([]_{i_{1},…,i_{l}}^{\bullet_{1},…,\bullet_{l}}\hat{\phi}\right)\Big|_{\SsV^{k}}=[]_{i_{1},…,i_{l}}^{\bullet_{1},…,\bullet_{l}}\phi$
gilt. Mit der Eindeutigkeit der stetigen Fortsetzung zeigt
dies
$[]_{i_{1},…,i_{l}}^{\bullet_{1},…,\bullet_{l}}\hat{\phi}=\widehat{\left([]_{i_{1},…,i_{l}}^{\bullet_{1},…,\bullet_{l}}\phi\right)}$
und für $\phi\in \DiffOpS{k}{L,\operatorname{\mathrm{cont}}}$ folgt
\begin{equation*}
[]_{i_{1},…,i_{l+1}}^{\bullet_{1},…,\bullet_{l+1}}\hat{\phi}=\widehat{\left([]_{i_{1},…,i_{l+1}}^{\bullet_{1},…,\bullet_{l+1}}\phi\right)}=0,
\end{equation*}
also $\hat{\phi}\in \mathrm{DiffOp}_{k}^{L,
\operatorname{\mathrm{cont}}}(\Hol,\mathcal{M})$.
\end{beweis}
\end{lemma}
\begin{definition}[Differentieller Hochschild-Komplex]
\label{def:DiffKomplexe}
Gegeben eine kommutative Algebra $\mathcal{A}$ und ein symmetrischer $\mathcal{A}-\mathcal{A}$-Bimodul $\mathcal{M}$. Wir betrachten die $\mathbb{K}$-Vektorräume
\begin{equation*}
HC_{\operatorname{\mathrm{diff}}}^{k}(\mathcal{A},\mathcal{M}):=
\begin{cases} \{0\} & k<0\\
\mathcal{M} & k=0\\
\DiffOp{k}{\bullet}& k\geq 1
\end{cases}
\end{equation*}
sowie die durch \eqref{eq:Hochschilddelta} definierten $\mathbb{K}$-lineare Abbildungen:
\begin{equation*}
\delta^{k}_{\operatorname{\mathrm{diff}}}\colon HC_{\operatorname{\mathrm{diff}}}^{k}(\mathcal{A},\mathcal{M})\longrightarrow HC_{\operatorname{\mathrm{diff}}}^{k+1}(\mathcal{A},\mathcal{M}).
\end{equation*}
Hierf"ur beachte man, dass im symmetrischen Falle $*_{L}=*_{R}$ gilt und Proposition \ref{prop:MultidiffOps}~\textit{v.)} dann zeigt, dass die $\delta_{\operatorname{\mathrm{diff}}}^{k}$ in der Tat in die behauptete Menge abbilden.
Sind $\mathcal{A}$ und $\mathcal{M}$ lokalkonvex, so definieren wir
\begin{equation*}
HC_{\mathrm{c,d}}^{k}(\mathcal{A},\mathcal{M}):=
\begin{cases} \{0\} & k<0\\
\mathcal{M} & k=0\\
\DiffOp{k}{\bullet, \operatorname{\mathrm{cont}}}& k\geq 1
\end{cases}
\end{equation*}
sowie die zugeh"origen Kettendifferentiale:
\begin{equation*}
\delta^{k}_{\mathrm{c,d}}: HC_{\mathrm{c,d}}^{k}(\mathcal{A},\mathcal{M})\longrightarrow HC_{\mathrm{c,d}}^{k+1}(\mathcal{A},\mathcal{M}).
\end{equation*}
\end{definition}
Abschließend erhalten wir folgendes Korollar:
\begin{korollar}
\label{kor:DiffKohom}
\begin{enumerate}
\item
Sei $\mathcal{M}$ ein symmetrischer $\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul. Dann induzieren $\xi$ und $\hat{\xi}$ wohldefinierte Kettenabbildungen $\wt{\xi^{k}}$ und $\wt{\hat{\xi}^{k}}$ zwischen $(HC_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M}),\delta_{\operatorname{\mathrm{diff}}})$ und\\ $(KC(\V,\mathcal{M}),\Delta)$.
Des Weiteren ist $\wt{\xi^{k}}$ injektiv und $\wt{\hat{\xi}^{k}}$ surjektiv.
\item
Sei $\mathcal{M}$ ein symmetrischer, lokalkonvexer
$\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul. Dann induzieren $\xi$
und $\hat{\xi}$ wohldefinierte Kettenabbildungen zwischen
$(HC_{\mathrm{c,d}}(\SsV,\mathcal{M}),\delta_{\mathrm{c,d}})$
und\\ $(KC^{\operatorname{\mathrm{cont}}}(\V,\mathcal{M}),\Delta)$. Des Weiteren ist
$\wt{\xi^{k}}$ injektiv und $\wt{\hat{\xi}^{k}}$ surjektiv.
\item
Sei $\mathcal{M}$ ein vollständiger, symmetrischer, hausdorffscher, lokalkonvexer
$\Hol-\Hol$-Bimodul, so sind
$\big(HC_{\mathrm{c,d}}(\Hol,\mathcal{M}),\hat{\delta}_{\mathrm{c,d}}\big)$
und $\big(HC_{\mathrm{c,d}}(\SsV,\mathcal{M}),\delta_{\mathrm{c,d}}\big)$ kettenisomorph
vermöge Einschränkung und stetiger
Fortsetzung. Des Weiteren gilt:
\begin{equation*}
HH^{k}_{\mathrm{c,d}}(\Hol,\mathcal{M})\cong HH^{k}_{\mathrm{c,d}}(\SsV,M).
\end{equation*}
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item[\textit{i.),ii.)}]
Zunächst ist klar, dass sowohl
$\hat{\xi}^{k}\colon HC_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})\longrightarrow
KC(\V,\mathcal{M})$ als auch $\hat{\xi}^{k}\colon HC_{\mathrm{c,d}}(\SsV,\mathcal{M})\longrightarrow KC^{\operatorname{\mathrm{cont}}}(\V,\mathcal{M})$ gilt. Des Weiteren ist das Bild
unter $\xi^{k}$ nach Proposition \ref{prop:wichEizuHKR}~\textit{iii.)}
derivativ in jedem Argument, also
nach Proposition \ref{prop:MultidiffOps}~\textit{vi.)} ein Element in
$\DiffOpS{k}{1}$. Dies zeigt $\xi^{k}\colon KC(\V,\mathcal{M})\longrightarrow
HC_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})$, und da $\xi^{k}$ im lokal
konvexen Fall stetige Elemente auf stetige Elemente abbildet,
gilt gleichermaßen $\xi^{k}\colon KC^{\operatorname{\mathrm{cont}}}(\V,\mathcal{M})\longrightarrow
HC_{\mathrm{c,d}}(\SsV,\mathcal{M})$. Die Injektivität von
$\wt{\xi^{k}}$ sowie die Surjektivität von $\wt{\hat{\xi}^{k}}$ folgen
in beiden Fällen wieder unmittelbar aus Lemma \ref{lemma:Fkettenabb}~\textit{i.)}.
\item[\textit{iii.)}]
Dies folgt mit Lemma \ref{lemma:stetfortDiffOp} analog zu
Satz \ref{satz:HochschildHol}, da auch hier für
\begin{align*}
&\hat{\delta}^{k}_{\mathrm{c,d}}\colon HC^{k}_{\mathrm{c,d}}\big(\Hol,\mathcal{M}\big)\longrightarrow
HC^{k+1}_{\mathrm{c,d}}\big(\Hol,\mathcal{M}\big)\\
&\delta^{k}_{\mathrm{c,d}}\colon HC^{k}_{\mathrm{c,d}}\big(\Ss^{\bullet}(\V),\mathcal{M}\big)\longrightarrow
HC^{k+1}_{\mathrm{c,d}}\big(\Ss^{\bullet}(\V),\mathcal{M}\big)
\end{align*}gilt, dass:
\begin{equation*}
\hat{\delta}_{\mathrm{c,d}}^{k}\left(\hat{\phi}\right)\Big|_{\SsV^{k+1}}= \delta_{\mathrm{c,d}}^{k}\left(\hat{\phi}\big|_{\SsV^{k}}\right).
\end{equation*}
\end{enumerate}
\end{beweis}
\end{korollar}
\begin{bemerkung}
Es ist im Allgemeinen nicht klar, dass die $\wt{\xi^{k}}$
Isomorphismen sind, dass also
\begin{align*}
&HH^{k}_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})\cong
\Hom_{\mathbb{K}}^{a}\big(\V^{k},\mathcal{M}\big)\cong
HH^{k}(\SsV,\mathcal{M})\\
&HH^{k}_{\mathrm{c,d}}(\SsV,\mathcal{M})\cong
\Hom_{\mathbb{K}}^{a,\operatorname{\mathrm{cont}}}\big(\V^{k},\mathcal{M}\big)\cong
HH_{\operatorname{\mathrm{cont}}}^{k}(\SsV,\mathcal{M})
\end{align*}gilt. Dies ist ein Phänomen, dass
bei Unterkomplexen immer auftreten kann.
Hierfür beachte man, dass es wegen $\ker\left(\delta^{k}_{\operatorname{\mathrm{diff}}}\right)\subseteq\ker\left(\delta^{k}\right)$
Elemente $[\nu]\in HH^{k}(\SsV,\mathcal{M})$
geben kann, in denen kein $[\eta_{\operatorname{\mathrm{diff}}}]\in HH^{k}_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})$
enthalten ist. Wegen
\begin{equation*}
HH^{k}_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})\stackrel{\wt{\hat{\xi}^{k}}}{\longrightarrow}\Hom_{\mathbb{K}}^{a}(\V^{k},\mathcal{M})\cong HH^{k}(\SsV,\mathcal{M})
\end{equation*}
und der Surjektivität von
$\wt{\hat{\xi}^{k}}$ ist die bei uns aber nicht der Fall. Des
Weiteren kann es wegen
$\im\left(\delta^{k-1}_{\operatorname{\mathrm{diff}}}\right)\subseteq\im\left(\delta^{k-1}\right)$
passieren, dass $[\eta_{\operatorname{\mathrm{diff}}}],[\mu_{\operatorname{\mathrm{diff}}}]\in
HH^{k}_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})$ existieren, für die sowohl
$[\eta_{\operatorname{\mathrm{diff}}}]\neq [\mu_{\operatorname{\mathrm{diff}}}]$ als auch
$[\eta_{\operatorname{\mathrm{diff}}}],[\mu_{\operatorname{\mathrm{diff}}}] \subseteq [\nu] \in
HH^{k}(\SsV,\mathcal{M})$ gilt. In unserem Fall ist eben dies das Problem, da $\wt{\hat{\xi}^{k}}$ nicht notwendigerweise
injektiv ist.
Abhilfe würde hier die Homotopie $s$ schaffen, wenn
gewährleistet wäre, dass
\begin{equation*}
\zeta_{-1}^{k}s_{k}^{*}\zeta^{k+1}\colon HC^{k+1}_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})\longrightarrow
HC^{k}_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})
\end{equation*}
gilt. In der Tat
erhielten wir dann analog zu \eqref{eq:dgh}, dass
\begin{equation}
\label{eq:ggg}
\id_{HC_{\operatorname{\mathrm{diff}}}^{k}}-\:\xi^{k}\cp\:\hat{\xi}^{k}=\big(\zeta_{-1}^{k}s^{*}_{k}\zeta^{k+1}\big)\delta^{k}+\delta^{k-1}\big(\zeta_{-1}^{k-1}s^{*}_{k-1}\zeta^{k}\big),
\end{equation}
also $\id_{HC_{\operatorname{\mathrm{diff}}}}\sim \xi^{k}\cp\hat{\xi}^{k}\:$ und
$\id_{HH^{k}_{\operatorname{\mathrm{diff}}}}=\wt{\xi^{k}\cp\hat{\xi}^{k}}=\wt{\xi^{k}}\cp\wt{\hat{\xi}^{k}}$,
mithin die Surjektivität von $\wt{\xi^{k}}$ und die
Injektivität von $\wt{\hat{\xi}^{k}}$. In der Tat könnte dann
obiger Fall nicht mehr eintreten, denn für
$[\eta_{\operatorname{\mathrm{diff}}}],[\mu_{\operatorname{\mathrm{diff}}}]\subseteq [\nu]$ wäre jede
Differenz $\phi=\phi_{\eta}-\phi_{\mu}$ von Repräsentanten
$\phi_{\eta}\in[\eta_{\operatorname{\mathrm{diff}}}]$ und
$\phi_{\mu}\in[\mu_{\operatorname{\mathrm{diff}}}]$ ein differentieller
Korand. Unter Berücksichtigung von \eqref{eq:ggg} erhielten
wir
$\phi=\delta^{k-1}\big(\zeta_{-1}^{k-1}s^{*}_{k-1}\zeta^{k}\phi\big)$
mit $\zeta_{-1}^{k-1}s^{*}_{k-1}\zeta^{k}\phi \in
HC^{k-1}_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})$, also
$[\eta_{\operatorname{\mathrm{diff}}}]=[\mu_{\operatorname{\mathrm{diff}}}]$.
Nun gilt, dass $\zeta_{-1}^{k}h^{*}_{k}\zeta^{k+1}$, $\zeta_{-1}^{k}\Omega^{*}_{k}\zeta^{k}$
und $\zeta_{-1}^{k}d^{*}_{k}\zeta^{k}$ die Eigenschaft
besitzen, differentielle Elemente auf differentielle Elemente
abzubilden und dass die $\mathcal{A}^{e}$-Linearisierung
einer derartigen Abbildung weiterhin diese Eigenschaft
besitzt. Jedoch dürfen wir in
$\zeta_{-1}^{k}\Omega_{k}^{*}h^{*}_{k}\zeta^{k+1}$ nicht
einfach die Eins $\zeta^{k}\cp\zeta_{-1}^{k}$ einfügen, da
$h_{k}$ nicht $\mathcal{A}^{e}$-linear ist. Nun könnte
trotzdem
$\zeta_{-1}^{k}\Omega_{k}^{*}h^{*}_{k}\zeta^{k+1}\colon HC^{k+1}_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})\longrightarrow
HC^{k}_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})$ richtig sein. Jedoch ist
\begin{align*}
\left(\zeta_{-1}^{k}\Omega_{k}^{*}h^{*}_{k}\zeta^{k+1}\right)\big(\phi\big)(u_{1},…,u_{k})=&\:\left(\Omega_{k}^{*}h^{*}_{k}\zeta^{k+1}\right)\big(\phi\big)(1\ot
u_{1}\ot…\ot u_{k}\ot 1)
\\=&\:\left(h^{*}_{k}\zeta^{k+1}\right)\big(\phi\big)(\Omega_{k}(1\ot
u_{1}\ot…\ot u_{k}\ot 1))
\\=&\:\left(\zeta^{k+1}\phi\right)(1\ot\Omega_{k}(1\ot
u_{1}\ot…\ot u_{k}\ot 1)),
\end{align*}
also die $\mathcal{A}^{e}$-Linearität von
$\left(\zeta^{k+1}\phi\right)$ in
Kombination mit \eqref{eq:aeTrickt} nur
noch für Elemente $\phi\in \DiffOpS{k}{(0,l_{2},…,l_{k+1})}$
nutzbringend einsetzbar. In diesen Fällen ist dann
$\left(\zeta_{-1}^{k}\Omega_{k}^{*}h^{*}_{k}\zeta^{k+1}\right)\big(\phi\big)\in
\DiffOpS{k}{1}$,
was man mit der $\mathcal{A}^{e}$-Linearität von $F_{k}$,
durch eine ähnliche Rechnung wie in
Proposition \ref{prop:wichEizuHKR}, sieht.\\
Im Falle $\mathcal{A}=C^{\infty}(V)$ mit einer konvexen
Teilmenge $V\subseteq \mathbb{R}^{n}$ kann gezeigt werden
(vgl. \cite[Kapitel 5]{Weissarbeit}), dass
$\zeta_{-1}^{k}s^{*}_{k}\zeta^{k+1}$ tatsächlich die
gewünschte Eigenschaft besitzt, differentielle Elemente
auf differentielle Elemente abzubilden. In diesem Fall ist
dies aber der speziellen Beschaffenheit des
Differentialoperator-Begriffes geschuldet, der wegen der
endlichen Dimension von $\mathbb{R}^{n}$, konsistent zur algebraischen
Definition, durch Verkettung
von partiellen Ableitungen definiert werden kann, siehe
\cite[Def~5.3.2]{Weissarbeit}. Insbesondere gelten dann
Kettenregeln der Form $\pt_{y} f(tx +(1-t)y)=f'(tx
+(1-t)y)(1-t)$, die im Beweis zu
\cite[Prop~5.6.6]{Weissarbeit} von essentieller Bedeutung
sind. Um also die Vorgehensweise aus \cite[Kapitel
5]{Weissarbeit} auf unsere Situation zu übertragen, könnte
man sich im differentiellen Hochschild-Komplex von Anfang
an auf Differentialoperatoren beschränken, die als
endliche Summe in der Form
\begin{equation*}
\phi=\sum_{l=0}^{s}\sum_{|\alpha|=l}\delta_{\alpha_{1}}^{|\alpha_{1}|}…\delta_{\alpha_{k}}^{|\alpha_{k}|}*_{L}m^{\alpha_{1},…,\alpha_{k}}
\end{equation*}mit Derivationen
$\delta_{\alpha_{i}}\in\mathrm{DiffOp}_{1}{1}(\SsV,\SsV)$
geschrieben werden können. Hierbei darf $k$ für jeden Summanden
variieren, und mit $\delta^{|\alpha_{i}|}_{\alpha_{i}}$ ist
die $|\alpha_{i}|$-fache Anwendung von
$\delta_{\alpha_{i}}$ gemeint. Dabei ist die Reihenfolge der Verkettungen wegen
Derivationseigenschaft der $\delta_{\alpha_{i}}$ unwichtig.
Für eine Derivation $\delta$ gilt dann mit
$\delta_{2}(x\ot y):=x\ot \delta(y)$ ebenfalls
\begin{equation*}
\delta_{2}(\hat{i}_{1}(1\ot x\ot
1))=(1-t_{1})\hat{i}_{1}(1\ot \delta(x)\ot 1),
\end{equation*}also die Kettenregel. Um nun jedoch
sicherzustellen, dass $G_{k}^{*}$ und somit $\xi^{k}$ in
diesen Unterkomplex abbildet, wird man sich im allgemeinen
auch auf einen Unterkomplex von $(\K^{*}, \pt^{*})$
beschränken müssen.\\\\
Abseits dieser ganzen Diskussion besteht natürlich
durchaus auch die Möglichkeit, dass $\id_{HC_{\operatorname{\mathrm{diff}}}}\sim
\xi^{k}\cp\hat{\xi}^{k}$ vermöge anderer Homotopieabbildung
$\mathrm{s}\colon HC^{k+1}_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})\longrightarrow
HC^{k}_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M})$ gilt. Dies würde dann
die Surjektivität von $\wt{\xi^{k}}$ und die Injektivität
von $\wt{\hat{\xi}^{k}}$ zeigen.
\end{bemerkung}
\section{Differentielle Bimoduln}
Motiviert durch Korollar \ref{kor:DiffKohom} wollen wir in diesem Abschnitt
der Frage nachgehen, inwiefern obige Aussagen auch f"ur
nicht-symmetrische Bimoduln zu erwarten sind. Seien hierf"ur
$(\mathcal{A},*)$ eine kommutative Algebra und $(\mathcal{M},*_{L})$
ein $\mathcal{A}$-Modul wie im letzten Abschnitt. F"ur $1\leq l\leq s$
seien $\mathbb{K}$-bilineare Abbildungen $D_{l}\colon \mathcal{A}\times \mathcal{M}\longrightarrow \mathcal{M}$ derart gegeben, dass folgende Konsistenzbedingungen erf"ullt sind:
\begin{itemize}
\item[\textbf{a.)}]
$D_{l}(a,b*_{L}m)=b*_{L}D_{l}(a,m)\qquad\qquad\forall\:a,b\in \mathcal{A},\:\forall\: m\in \mathcal{M},\forall\:1\leq l\leq s$
\item[\textbf{b.)}]
F"ur festes $a\in \mathcal{A}$ und $1\leq l\leq s$ bezeichne
$D_{l}^{a}\colon\mathcal{M}\longrightarrow \mathcal{M}$ die
$*_{L}$-lineare Abbildung $D_{l}^{a}\colon m\longmapsto D_{l}(a,m)$. Sei
des Weiteren $D_{l_{1},\dots,l_{p}}^{a_{1},\dots,a_{p}}=D_{l_{1}}^{a_{1}}\cp
\dots \cp D_{l_{p}}^{a_{p}}$, dann soll f"ur alle $a_{i}\in \mathcal{A}$ gelten, dass:
\begin{equation*}
D_{l_{1},\dots,l_{p}}^{a_{1},\dots,a_{p}}= 0,\qquad \text{ falls } \displaystyle\sum_{i=1}^{p}l_{i}> s.
\end{equation*}
\item[\textbf{c.)}]
F"ur $1\leq l\leq s$ gilt:
\begin{align*}
D_{l}(a*b,m)=&\:b*_{L}D_{l}(a,m)+ D_{1}(b,D_{l-1}(a,m))+D_{2}(b,D_{l-2}(a,m))+\dots\\ &\qquad+D_{l-2}(b,D_{2}(a,m))+ D_{l-1}(b,D_{1}(a,m))+a*_{L}D_{l}(b,m).
\end{align*}
\item[\textbf{d.)}]
F"ur fixiertes $m\in \mathcal{M}$ ist $D_{l}(\cdot,m)\in \DiffOp{1}{l}\qquad \forall\:m\in \mathcal{M},\:\forall\: 1\leq l\leq s$.
\end{itemize}
Hiermit erhalten wir folgendes Lemma:
\begin{lemma}
\label{lemma:defBim}
Gegeben eine kommutative Algebra $(\mathcal{A},*)$ und ein $\mathcal{A}$-Modul $(*_{L},\mathcal{M})$. Seien weiter $D_{1},\dots,D_{s}$ Abbildungen, die \textit{i.)} - \textit{iv.)} erf"ullen. Dann wird $\mathcal{M}$ verm"oge
\begin{equation*}
*_{R}=*_{L}+ D_{1}+\dots+D_{s}
\end{equation*}
zu einem $\mathcal{A}-\mathcal{A}$-Bimodul.
\begin{beweis}
Mit \textbf{a.)} folgt unmittelbar, dass $a*_{L}(m *_{R} b)=
(a*_{L}m) *_{R}b$ gilt, und f"ur die Bedingung $m*_{R}(a*b)=(m*_{R}a)*_{R}b$ rechnen wir:
\begin{align*}
(m*_{R}a)*_{R}b=\left[a*_{L}m+D_{1}(a,m)+D_{2}(a,m)+\dots+D_{s}(a,m)\right]*_{R}b.
\end{align*}
Durch Ausmultiplizieren und Anwenden von \textbf{a.)} und \textbf{b.)} ergibt dies:\\\\
{\footnotesize
\begin{array}[t]{cccccccccccc}
&(a*b)*_{L} m &+& a*_{L}D_{1}^{b}(m) &+& a*_{L} D_{2}^{b}(m) &+&\dots &+&a*_{L}D_{s-1}^{b}(m)& +&a*_{L}D_{s}^{b}(m)\\\\
+&b*_{L}D_{1}^{a}(m)&+&D_{1,1}^{b,a}(m)&+&D_{2,1}^{b,a}(m) &+& \dots&+&D_{s-1,1}^{b,a}(m)&+& \cancel{D_{s,1}^{b,a}(a,m)}\\\\
+&b*_{L}D_{2}^{a}(m)&+&D_{1,2}^{b,a}(m)&+&D_{2,2}^{b,a}(m) &+& \dots&+&\cancel{D_{s-1,2}^{b,a}(m)}&+& \cancel{D_{s,2}^{b,a}(m)}\\\\
+&b*_{L}D_{3}^{a}(m)&+&D_{1,3}^{b,a}(m)&+&D_{2,3}^{b,a}(m) &+& \dots&+&\cancel{D_{s-1,3}^{b,a}(m)}&+& \cancel{D_{s,3}^{b,a}(m)}\\\\
&&&&&&\vdots&&&&&\\\\
+&b*_{L}D_{s-1}^{a}(m)&+& D_{1,s-1}^{b,a}(m)&+&\cancel{D_{2,s-1}^{b,a}(m)} &+& \dots&+&\cancel{D_{s-1,s-1}^{b,a}(m)}&+& \cancel{D_{s,s-1}^{b,a}(m)}\\\\
+&b*_{L}D_{s}^{a}(m)&+& \cancel{D_{1,s}^{b,a}(m)}&+&\cancel{D_{2,s}^{b,a}(m)} &+& \dots&+&\cancel{D_{s-1,s}^{b,a}(m)}&+& \cancel{D_{s,s}^{b,a}(m)}.
\end{array}}\\\\\\Durch Zusammenfassen der Diagonalen von links unten nach recht oben folgt mit \textbf{c.)}:
\begin{align*}
(m*_{R}a)*_{R}b=&\:(a*b)*_{L}m+\Big[b*_{L}D_{1}(a,m)+
a*_{L}D_{1}(b,m)\Big]
\\ &+\Big[b*_{L}D_{2}(a,m)+D_{1}(b,D_{1}(a,m))+a*_{L}D_{2}(b,m)\Big]+\dots
\\ &+ \Big[b*_{L}D_{s}(a,m)+ D_{1}(b,D_{s-1}(a,m))+D_{2}(b,D_{s-2}(a,m))+\dots\\ &\qquad+D_{s-2}(b,D_{2}(a,m))+ D_{s-1}(b,D_{1}(a,m))+a*_{L}D_{s}(b,m)\Big]
\\=&\:(a*b)*_{L}m+D_{1}(a*b,m)+\dots+D_{s}(a*b,m)
\\=&\: m*_{R}(a*b).
\end{align*}
Schließlich ist $*_{R}$ bilinear, da $*_{L}$ und alle $D_{l}$
bilinear sind. Des Weiteren folgt f"ur alle $l$ mit $a=b=1$ aus \textbf{c.)},
dass $D_{l}(1,m)=0$, also
$m*_{R}1_{\mathcal{A}}=1_{\mathcal{A}}*_{L}m$ gilt, womit
$\mathcal{M},*_{L},*_{R}$ alle unsere Anforderungen an einen
$\mathcal{A}-\mathcal{A}$-Bimodul erfüllt.
\end{beweis}
\end{lemma}
Wir geben nun die zentrale Definition dieses Kapitels:
\begin{definition}
Gegeben die kommutative Algebra $\Ss^{\bullet}(\V)$ und ein $\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul wie in Lemma \ref{lemma:defBim}.
Wir nennen $\mathcal{M}$ einen differentiellen Bimodul
"uber $\Ss^{\bullet}(\V)$, falls folgende Zusatzbedingung erfüllt ist:
\begin{itemize}
\item[\textbf{e.)}]
F"ur alle $m\in \mathcal{M}$ und alle $\omega\in \SsV$ mit $\deg(\omega)< l$ ist $D_{l}(\omega,m)=0$.
\end{itemize}
Mit \textbf{c.)} ist dies gleichbedeutend mit der Forderung, dass
f"ur alle $m\in \mathcal{M}$ und alle $l\geq 2$
$D_{l}(\mathcal{\operatorname{\mathrm{v}}},m)=0$, falls $\deg(\operatorname{\mathrm{v}})=1$.
\end{definition}
\begin{bemerkung}
Sei $\mathcal{M}$ ein differentieller $\SsV-\SsV$-Bimodul, so ist
mit $\DiffOpS{k}{\bullet}$ im Folgenden immer der
Differentialoperator-Begriff bezüglich der $*_{L}$-Multiplikation gemeint.
\end{bemerkung}
\begin{beispiel}
\begin{enumerate}
\item
Sei $(\mathcal{M},*_{L})$ ein $\Ss^{\bullet}(\V)$-Modul, $s=2$ und $\wt{\mathcal{M}}=\mathcal{M}\times\mathcal{M} \times\mathcal{M}$. Sei weiter $i_{u}$ wie in Bemerkung \ref{bem:DeriBeiStet} und:
\begin{align*}
\mathrm{sh}\colon (m_{1},m_{2},m_{3})&\longmapsto (0,m_{1},m_{2}) \\
*_{L}\colon\big(\omega, (m_{1},m_{2},m_{3})\big)&\longmapsto (\omega *_{L}m_{1},\omega *_{L}m_{2},\omega *_{L}m_{3}).
\end{align*}
Wir setzen $D_{1}(\omega,\wt{m})=\sqrt{2}\cdot i_{u}(\omega)*_{L}
\mathrm{sh}^{1}(\wt{m})$ und
$D_{2}(\omega,\wt{m})=i_{u}^{2}*_{L}\mathrm{sh}^{2}(\wt{m})$. Dann
sind \textbf{a.)} und \textbf{b.)} per Definition erf"ullt, und f"ur \textbf{c.)} rechnet man:
\begin{align*}
D_{2}(v\vee w,\wt{m})=&\:i_{u}(v\vee i_{u}(w)+w\vee i_{u}(v))*_{L}\mathrm{sh}^{2}(\wt{m})\\
=&\:w\vee i_{u}^{2}(v)*_{L}\mathrm{sh}^{2}(\wt{m})+2\cdot i_{u}(v)\vee i_{u}(w)*_{L}\mathrm{sh}^{2}(\wt{m})\\
&+v\vee i_{u}^{2}(w)*_{L}\mathrm{sh}^{2}(\wt{m})\\
=&\: w*_{L} D_{2}(v,\wt{m})+D_{1}(w,D_{1}(v,\wt{m}))+v*_{L} D_{2}(w,\wt{m}).
\end{align*}
Des Weiteren ist
$D_{2}(\mathrm{v},\wt{m})=i_{u}^{2}(\mathrm{v})*_{L}\mathrm{sh}^{2}(\wt{m})=(0,0,0)$
falls $\deg{\mathrm{v}}=1$, und für \textbf{d.)} beachte man, dass $i_{u}\in \mathrm{DiffOp}_{1}^{1}(\Ss^{\bullet}(V),\Ss^{\bullet}(V))$ sowie $i_{u}^{2}\in \mathrm{DiffOp}_{1}^{2}(\Ss^{\bullet}(V),\Ss^{\bullet}(V))$, womit
\begin{align*}
&[]_{2}^{a,b} D_{1}(\cdot, m)=\Big([]_{2}^{a,b}i_{u}\Big) *_{L}\mathrm{sh}^{1}(\wt{m})=0\\
&[]_{3}^{a,b,c} D_{2}(\cdot, m)=\Big([]_{3}^{a,b,c}i_{u}^{2}\Big) *_{L}\mathrm{sh}^{2}(\wt{m})=0.
\end{align*}
\item
Sei $(\mathcal{A},*)=(C^{\infty}(\mathbb{R}^{n},\mathbb{R}),\cdot)$ und
$\mathcal{M}=\mathrm{DiffOp}_{1}^{s}(\mathcal{A},\mathcal{A})$,
versehen mit den Modul-Multiplikationen aus
Proposition \ref{prop:MultidiffOps}~\textit{iii.)}. Dann ist
$\mathcal{M}$ in der Tat ein
$\mathcal{A}-\mathcal{A}$-Bimodul, und unter Verwendung der
Multiindex-Konventionen
\begin{equation*}
|\alpha|=\sum_{i=1}^{n}\alpha_{i}\quad\qquad
\pt^{\alpha}=\frac{\pt^{\alpha_{1}}}{\pt
x_{1}}…\frac{\pt^{\alpha_{n}}}{\pt x_{n}}\quad\qquad\alpha!=\prod_{i=1}^{n}\alpha_{i}!
\end{equation*}
\begin{equation*}
\alpha+\beta=(\alpha_{1}+\beta_{1},…\alpha_{n}+\beta_{n})\quad\qquad \binom{\alpha}{\beta}=\frac{\alpha!}{(\alpha-\beta)!\:\beta!}=\prod_{i=1}^{n}\binom{\alpha_{i}}{\beta_{i}}
\end{equation*} für $\alpha,\beta \in \mathbb{N}^{n}$
erhalten wir aus Bemerkung \ref{bem:DiffOpCRn}, dass wir jedes $m
\in\mathrm{DiffOp}_{1}^{s}(\mathcal{A},\mathcal{A})$ in der
Form $\displaystyle
m=\sum_{l=0}^{s}\sum_{|\alpha|=l}\phi_{\alpha}\pt^{\alpha}$
mit Elementen $\phi_{\alpha}\in
C^{\infty}(\mathbb{R}^{n},\mathbb{R})$
schreiben können. Mit der Derivationseigenschaft der
partiellen Ableitungen folgt
\begin{equation*}
\pt^{\alpha}(f\cdot g)=\sum_{\beta\leq
\alpha}\binom{\alpha}{\beta}\pt^{\beta}f \cdot
\pt^{\alpha-\beta}g\qquad\forall\:f,g\in C^{\infty}(\mathbb{R}^{n},\mathbb{R})
\end{equation*}und wir erhalten für
$m=\phi_{\alpha}\pt^{\alpha}$, dass
\begin{align*}
(m *_{R}f)(g) = \phi_{\alpha}\sum_{\beta\leq\alpha}\binom{\alpha}{\beta}\pt^{\beta}f \cdot
\pt^{\alpha-\beta}g
=\sum_{l=0}^{|\alpha|}\overbrace{\sum_{\substack{|\beta|=l\\\beta\leq
\alpha}}\phi_{\alpha}\pt^{\beta}f
\pt^{\alpha-\beta}}^{D_{l}(f,m)}g
\end{align*}
mit $D_{0}(f,m)=f\cdot\phi_{\alpha}\pt^{\alpha}=f*_{L}m$ gilt. Hierbei sind die
$D_{l}$ ganz allgemein durch lineare Fortsetzung auf
ganz $\mathrm{DiffOp}_{1}^{s}(\mathcal{A},\mathcal{A})$ von
\begin{align*}
D_{l}\colon
\left(f,\phi_{\alpha}\pt^{\alpha}\right)&\longmapsto\sum_{\substack{|\beta|=l\\\beta\leq
\alpha}}\binom{\alpha}{\beta}\phi_{\alpha}\pt^{\beta}f
\pt^{\alpha-\beta}\in \mathrm{DiffOp}_{1}^{|\alpha|-l}(\mathcal{A},\mathcal{A})
\end{align*}mit $D_{l}|_{\mathcal{A}\times\mathrm{DiffOp}_{1}^{m\leq
l}(\mathcal{A},\mathcal{A})}=0$ definiert. Insgesamt zeigt dies \textbf{a.)} und
\textbf{b.)}. Nun ist $D_{l}(\cdot,m)$ linear und wegen
Proposition \ref{prop:MultidiffOps}~\textit{ii.)} folgt
\textbf{d.)} unmittelbar aus:
\begin{equation*}
[]_{l+1}^{f_{1},…,f_{k+1}}D_{l}(\cdot,m)=\sum_{\substack{|\beta|=l\\\beta\leq
\alpha}}\binom{\alpha}{\beta}\phi_{\alpha}\left([]_{l+1}^{f_{1},…,f_{k+1}}\pt^{\beta}\right) \pt^{\alpha-\beta}=0.
\end{equation*}
Für \textbf{c.)} beachten wir, dass $\mathcal{M}$ ein Bimodul ist, also
$m*_{R}(f\cdot g)=(m*_{R}f)*_{R} g$ gilt. Wir betrachten nun
das Schema aus Lemma \ref{lemma:defBim}, welches wir durch
ausmultiplizieren von $(m*_{R}f)*_{R} g$ erhielten. Es ist
dann zu zeigen, dass die $l$-te Diagonale mit $D_{l}(f\cdot
g,m)$ übereinstimmt. Hierfür reicht, es diese
Aussage für Elemente der Form
$m_{k}=\sum_{|\alpha|=k}\phi_{\alpha}\pt^{\alpha}$ mit
$k\leq s$, welche wir
im Folgenden als "`exakt der Ordnung $k$"' bezeichnen wollen, nachzuweisen.
Denn jedes
$m\in\mathrm{DiffOp}_{1}^{s}(\mathcal{A},\mathcal{A})$ kann
offenbar als eindeutige Linearkombination solcher Elemente dargestellt
werden. Für ein derartiges $m_{k}$ ist $D_{l}(f,m_{k})$ exakt
der Ordnung $k-l$ und ebenfalls ist $D_{l_{2}}(g,D_{l_{1}}(f,m_{k}))$ exakt
der Ordnung $k-l_{1}-l_{2}$. Hiermit enthält die $l$-te
Diagonale nur exakte Elemente der Ordnung $l$, was
\textbf{c.)} zeigt. Schränken wir uns auf die
Unteralgebra $\Pol(\mathbb{R}^{n},\mathbb{R})\subseteq
C^{\infty}(\mathbb{R}^{n},\mathbb{R})$ ein, so ist
schließlich auch \textbf{e.)} erfüllt.
\item
In Analogie zu \textit{ii.)} betrachten wir den Unterraum $\mathcal{M}$ aller
Differentialoperatoren
$m\in\mathrm{DiffOp}_{1}^{s}(\SsV,\SsV)$, die als eine endliche
Summe der Form
\begin{equation*}
m=\sum_{l=0}^{s}\sum_{|\alpha|=l}\delta_{\alpha_{1}}^{|\alpha_{1}|}…\delta_{\alpha_{k}}^{|\alpha_{k}|}
\end{equation*}mit Derivationen
$\delta_{\alpha_{i}}\in\mathrm{DiffOp}_{1}{1}(\SsV,\SsV)$
geschrieben werden können. Hierbei ist in der zweiten Summe
$\alpha \in \mathbb{N}^{k}$, wobei $k$ für jeden Summanden
variieren darf. Mit $\delta^{|\alpha_{i}|}_{\alpha_{i}}$ ist
die $|\alpha_{i}|$-fache Anwendung von
$\delta_{\alpha_{i}}$ gemeint, und wegen der
Derivationseigenschaft ist die Reihenfolge Verkettungen
unwichtig. Der Summand für $l=0$ soll dann lediglich aus einem
Element $m_{0}\in \SsV$ bestehen.
Vermöge Proposition \ref{prop:MultidiffOps} wird
$\mathcal{M}$ zu einem $\SsV-\SsV$-Bimodul, und wir erhalten
für
$m=\delta_{\alpha_{1}}^{|\alpha_{1}|}…\delta_{\alpha_{k}}^{|\alpha_{k}|}=:\delta^{\alpha}$
mit $|\alpha|\leq s$ sowie $v,w\in\SsV$, dass
\begin{align*}
(m *_{R}v)(w) = \sum_{\alpha\geq\beta\in \mathbb{N}^{k}}\binom{\alpha}{\beta}\delta^{\beta}(v) \vee
\delta^{\alpha-\beta}(w)
=\sum_{l=0}^{|\alpha|}\overbrace{\sum_{\substack{|\beta|=l\\\alpha\geq\beta\in \mathbb{N}^{k}}}\binom{\alpha}{\beta}\delta^{\beta}(v)\vee
\delta^{\alpha-\beta}}^{D_{l}(v,m)}(w),
\end{align*}also $*_{R}=\displaystyle\sum_{l=1}^{s}D_{l}$ mit Abbildungen
\begin{equation*}
D_{l}\colon\left(v,\delta^{\alpha}\right)\longmapsto \begin{cases}
0\qquad\qquad\qquad\qquad\qquad \text{ falls }l>|\alpha|,\\
\sum_{\substack{|\beta|=l\\\alpha\geq\beta\in \mathbb{N}^{k}}}\binom{\alpha}{\beta}\delta^{\beta}(v)\vee
\delta^{\alpha-\beta}\:\quad\text{ sonst}.
\end{cases}
\end{equation*}gilt. Nun folgt \textbf{a.)} unmittelbar aus
$\delta^{0}(v)=v$ und \textbf{b.)} mit
$\deg(D_{l}(v,m))=|\alpha|-l$. Die Bedingungen \textbf{d.)} und \textbf{c.)}
folgen analog zu \textit{ii.)} und \textbf{e.)} ist
wegen $\delta^{\alpha}(v)=0$ für $\deg(v)<|\alpha|$ ebenfalls klar.
\end{enumerate}
\end{beispiel}
\begin{lemma}
\label{lemma:DiffBimodWichEi}
Für differentielle
$\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimoduln $\mathcal{M}$ erhalten
wir nun folgende Aussagen:
\begin{enumerate}
\item
Sei $m\in \mathcal{M}$ und $u\in \Ss^{\bullet}(\V)$, dann gilt:
\begin{equation*}
\hat{i}(1\ot u\ot 1)*_{e} m= u*_{L}m+ (1-t) D_{1}(u,m)+\dots+ (1-t)^{s} D_{s}(u,m).
\end{equation*}
\item
Sei $\wt{\phi} \in \Hom_{\mathcal{A}^{e}}(\Ss^{\bullet}(\V),\mathcal{M})$ und $\mathcal{D}=D_{l_{1},\dots,l_{k}}^{a_{1},\dots,a_{k}}$ mit $q=s-\displaystyle\sum_{i=1}^{p}l_{i}\geq 0$. Dann ist die Abbildung
\begin{align*}
\tau_{\mathcal{D}}^{\wt{\phi}}\colon(u_{1},\dots,u_{k})\longmapsto \mathcal{D}\int_{0}^{1}dt_{1}\dots\int_{0}^{t_{k-1}}dt_{k}\:\mathrm{p}(t)\:\wt{\phi}\left(\prod_{s=1}^{k}(i_{s}\cp\delta)(1\ot u_{s}\ot 1)\right)
\end{align*}
f"ur jedes $\mathrm{p}(t)\in \mathrm{Pol}(t_{1},\dots,t_{k})$
ein Element in $\DiffOpS{k}{q+1}$. Des Weiteren ist $\xi^{k}(\phi) \in \DiffOpS{k}{s+1}$ f"ur alle $\phi\in \Hom_{\mathbb{K}}^{a}(\V^{k},\mathcal{M})$.
\item
Unter den gegebenen Voraussetzungen sind die Kokettenkomplexe aus Definition \ref{def:DiffKomplexe} ebenfalls wohldefiniert.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Wir zeigen dies per Induktion "uber $\deg(u)$. Sei hierf"ur $\deg(\mathrm{v})=1$, so erhalten wir mit \textbf{e.)}:
\begin{align*}
\hat{i}(1 \ot \mathrm{v} \ot 1)*_{e}m=&\:\left[t \operatorname{\mathrm{v}} \ot 1 +(1-t)1\ot \operatorname{\mathrm{v}}\right]*_{e} m
\\=&\:t \operatorname{\mathrm{v}}*_{L}m+ (1-t)\operatorname{\mathrm{v}}*_{L} m +(1-t) D_{1}(\operatorname{\mathrm{v}},m)
\\=&\: \operatorname{\mathrm{v}}*_{L}m +(1-t) D_{1}(\operatorname{\mathrm{v}},m)+\dots+(1-t)^{s}D_{s}(\operatorname{\mathrm{v}},m).
\end{align*}
Sei nun die Aussage f"ur $\deg(u)=l$ korrekt, dann folgt gleicherma"sen:
{\allowdisplaybreaks\small\begin{align*}
\hat{i}(1\ot &\operatorname{\mathrm{v}}\vee\: u \ot 1)*_{e}m\\
=&\:\hat{i}(1\ot \operatorname{\mathrm{v}} \ot 1)*_{e}\Big(\hat{i}(1\ot u \ot 1)*_{e} m\Big)\\
=&\:\left[t\operatorname{\mathrm{v}}\ot 1+(1-t)1\ot\operatorname{\mathrm{v}}\right]*_{e}\left[u*_{L}m+ (1-t) D_{1}(u,m)+\dots+(1-t)^{l}D_{l}(u,m)\right]
\\=&\:t \operatorname{\mathrm{v}}\vee\: u *_{L} m +(1-t)\: \operatorname{\mathrm{v}}\vee\: u *_{L} m + (1-t)\: u*_{L}D_{1}(v,m) + \cancel{\dots}
\\ &+t(1-t) \operatorname{\mathrm{v}}*_{L}D_{1}(u,m)+(1-t)^{2}\operatorname{\mathrm{v}}*_{L}D_{1}(u,m)+(1-t)^{2}D_{1}(\operatorname{\mathrm{v}},D_{1}(u,m))+ \cancel{\dots}
\\ &+t(1-t)^{2} \operatorname{\mathrm{v}}*_{L}D_{2}(u,m)+(1-t)^{3}\operatorname{\mathrm{v}}*_{L}D_{2}(u,m)+(1-t)^{3}D_{1}(\operatorname{\mathrm{v}},D_{2}(u,m))+ \cancel{\dots}
\\ &+t(1-t)^{3} \operatorname{\mathrm{v}}*_{L}D_{3}(u,m)+(1-t)^{4}\operatorname{\mathrm{v}}*_{L}D_{3}(u,m)+(1-t)^{4}D_{1}(\operatorname{\mathrm{v}},D_{3}(u,m))+ \cancel{\dots}
\\ &+\dots
\\ &+t(1-t)^{l} \operatorname{\mathrm{v}}*_{L}D_{l}(u,m)+(1-t)^{l+1}\operatorname{\mathrm{v}}*_{L}D_{l}(u,m)+(1-t)^{l+1}D_{1}(\operatorname{\mathrm{v}},D_{l}(u,m))\:.
\end{align*}}Fasst man den jeweils letzten Term mit den ersten beiden Termen der n"achsten Reihe zusammen, so folgt mit $(1-t)^{m}=t(1-t)^{m}+ (1-t)^{m+1}$:
{\begin{equation}
\label{eq:klupop}
\begin{split}
\hat{i}(1\ot \operatorname{\mathrm{v}}\vee\: u \ot 1)*_{e}m=&\:\operatorname{\mathrm{v}}\vee\: u *_{L} m
\\ &+(1-t)\:u*_{L}D_{1}(\operatorname{\mathrm{v}},m)+(1-t)\operatorname{\mathrm{v}}*_{L}D_{1}(u,m)
\\ &+(1-t)^{2}D_{1}(\operatorname{\mathrm{v}},D_{1}(u,m))+(1-t)^{2}\operatorname{\mathrm{v}}*_{L}D_{2}(u,m)
\\ &+(1-t)^{3}D_{1}(\operatorname{\mathrm{v}},D_{2}(u,m))+(1-t)^{3}\operatorname{\mathrm{v}}*_{L}D_{3}(u,m)
\\ &+\dots
\\ &+(1-t)^{l}D_{1}(\operatorname{\mathrm{v}},D_{l-1}(u,m))+(1-t)^{l}\operatorname{\mathrm{v}}*_{L}D_{l}(u,m)
\\ &+(1-t)^{l+1}D_{1}(\operatorname{\mathrm{v}},D_{l}(u,m)).
\end{split}
\end{equation}}
Unter Ber"ucksichtigung von \textbf{c.)} und \textbf{e.)} erhalten wir hieraus im Falle $l<s$\::
\begin{align*}
\hat{i}(1\ot \operatorname{\mathrm{v}}\vee \:u \ot 1&)*_{e}m
\\=&\:\operatorname{\mathrm{v}}\vee\: u *_{L} m +(1-t) D_{1}(\operatorname{\mathrm{v}}\vee\: u,m) +\dots + (1-t)^{l+1}D_{l+1}(\operatorname{\mathrm{v}}\vee \:u,m)
\\ &+ \cancel{(1-t)^{l+2}D_{l+2}(\operatorname{\mathrm{v}}\vee \:u,m)}+\dots+\cancel{(1-t)^{s}D_{s}(\operatorname{\mathrm{v}}\vee\: u,m)}
\end{align*}
Im Falle $\deg(u) \geq s$ gilt \eqref{eq:klupop} mit s anstelle von l und wegen \textbf{b.)} verschwindet der letzte Summand. Dies zeigt die Behauptung.
\item
F"ur den Induktionsanfang sei $q=0$. Dann folgt mit
\begin{align*}
\wt{\phi}_{\omega_{j}}:=&\:\wt{\phi}\left(\prod_{s\neq
j}(i_{s}\cp\delta)(1\ot u_{s}\ot 1) \cdot
(i_{j}\cp\delta)(1\ot \omega_{j}\ot 1)\right)
\end{align*} für $u_{1},…,u_{k},\omega\in \SsV$, dass
{\allowdisplaybreaks
\footnotesize
\begin{align*}
\tau_{\mathcal{D}}^{\wt{\phi}}(u_{1}&,\dots, u_{j}\vee u'_{j},\dots,u_{k})
\\ =&\:\mathcal{D}\int_{0}^{1}dt_{1}\dots \int_{0}^{t_{k-1}}dt_{k}\:\mathrm{p}(t)\:\hat{i}_{j}(1\ot u_{j}\ot 1)*_{e} \wt{\phi}\left(\prod_{s\neq j}(i_{s}\cp\delta)(1\ot u_{s}\ot 1) \cdot (i_{j}\cp\delta)(1\ot u'_{j}\ot 1)\right)
\\ & +\mathcal{D}\int_{0}^{1}dt_{1}\dots \int_{0}^{t_{k-1}}dt_{k}\:\mathrm{p}(t)\:\hat{i}_{j}(1\ot u'_{j}\ot 1)*_{e} \wt{\phi}\left(\prod_{s\neq j}(i_{s}\cp\delta)(1\ot u_{s}\ot 1) \cdot (i_{j}\cp\delta)(1\ot u_{j}\ot 1)\right)
\\ =&\:\mathcal{D}\int_{0}^{1}dt_{1}\dots \int_{0}^{t_{k-1}}dt_{k}\:\mathrm{p}(t)\left[u_{j}*_{L}\wt{\phi}_{u'_{j}}+\:(1-t_{j})D_{1}^{u_{j}}\left(\wt{\phi}_{u'_{j}}\right)+\dots+(1-t_{j})^{s}D_{s}^{u_{j}}\left(\wt{\phi}_{u'_{j}}\right) \right]
\\ & +\mathcal{D}\int_{0}^{1}dt_{1}\dots \int_{0}^{t_{k-1}}dt_{k}\:\mathrm{p}(t)\left[u'_{j}*_{L}\wt{\phi}_{u_{j}}+\:(1-t_{j})D_{1}^{u'_{j}}\left(\wt{\phi}_{u_{j}}\right)+\dots+(1-t_{j})^{s}D_{s}^{u'_{j}}\left(\wt{\phi}_{u_{j}}\right) \right]
\\ =&\int_{0}^{1}dt_{1}\dots \int_{0}^{t_{k-1}}dt_{k}\:\mathrm{p}(t)\:\mathcal{D}\left[u_{j}*_{L}\wt{\phi}_{u'_{j}}+\:(1-t_{j})D_{1}^{u_{j}}\left(\wt{\phi}_{u'_{j}}\right)+\dots+(1-t_{j})^{s}D_{s}^{u_{j}}\left(\wt{\phi}_{u'_{j}}\right) \right]
\\ & +\int_{0}^{1}dt_{1}\dots \int_{0}^{t_{k-1}}dt_{k}\:\mathrm{p}(t)\:\mathcal{D}\left[u'_{j}*_{L}\wt{\phi}_{u_{j}}+\:(1-t_{j})D_{1}^{u'_{j}}\left(\wt{\phi}_{u_{j}}\right)+\dots+(1-t_{j})^{s}D_{s}^{u'_{j}}\left(\wt{\phi}_{u_{j}}\right) \right]
\\ =&\int_{0}^{1}dt_{1}\dots \int_{0}^{t_{k-1}}dt_{k}\:\mathrm{p}(t)\:u_{j}*_{L}\mathcal{D}\left(\wt{\phi}_{u'_{j}}\right)
+\int_{0}^{1}dt_{1}\dots \int_{0}^{t_{k-1}}dt_{k}\:\mathrm{p}(t)\:u'_{j}*_{L}\mathcal{D}\left(\wt{\phi}_{u_{j}}\right)
\\ =&\:u_{j}*_{L}\mathcal{D}\int_{0}^{1}dt_{1}\dots \int_{0}^{t_{k-1}}dt_{k}\:\mathrm{p}(t)\:\wt{\phi}_{u'_{j}}
+\:u'_{j}*_{L}\mathcal{D}\int_{0}^{1}dt_{1}\dots \int_{0}^{t_{k-1}}dt_{k}\:\mathrm{p}(t)\:\wt{\phi}_{u_{j}}
\\=&\: u_{j}*_{L} \tau_{\mathcal{D}}^{\wt{\phi}}(u_{1},\dots,u'_{j},\dots,u_{k})+u'_{j}*_{L} \tau_{\mathcal{D}}^{\wt{\phi}}(u_{1},\dots,u_{j},\dots,u_{k}),
\end{align*}}also $\tau_{\mathcal{D}}^{\wt{\phi}}\in
\DiffOpS{k}{1}$ nach
Proposition \ref{prop:MultidiffOps}~\textit{v.)} gilt. Hierbei durften wir $\mathcal{D}$ wegen seiner $\mathbb{K}$-Linearität mit den Integralen vertauschen.
Sei nun $s-\displaystyle\sum_{i=1}^{p}l_{i}=q$ und die Aussage f"ur $q-1$ korrekt. Dann ist:
\begin{align*}
\Big([]_{j}^{a_{q+2}}\tau_{\mathcal{D}}^{\wt{\phi}}\Big)(u_{1},\dots,u_{k})
=&\:a_{q+2}*_{L}\tau_{\mathcal{D}}^{\wt{\phi}}(u_{1},\dots,u_{k}) - \tau_{\mathcal{D}}^{\wt{\phi}}(u_{1},\dots, a_{q+2}\vee u_{j},\dots,u_{k}).
\end{align*}
Der zweite Summand ergibt ausgeschrieben:
{\footnotesize
\begin{align*}
\tau_{\mathcal{D}}^{\wt{\phi}}&(u_{1},\dots, a_{q+2}\vee u_{j},\dots,u_{k})
\\ =&\int_{0}^{1}dt_{1}\dots \int_{0}^{t_{k-1}}dt_{k}\:\mathrm{p}(t)\:\mathcal{D}\left[a_{q+2}*_{L}\wt{\phi}_{u_{j}}+\:(1-t_{j})D_{1}^{a_{q+2}}\left(\wt{\phi}_{u_{j}}\right)+\dots+(1-t_{j})^{s}D_{s}^{a_{q+2}}\left(\wt{\phi}_{u_{j}}\right) \right]
\\ &+\int_{0}^{1}dt_{1}\dots \int_{0}^{t_{k-1}}dt_{k}\:\mathrm{p}(t)\:\mathcal{D}\left[u_{j}*_{L}\wt{\phi}_{a_{q+2}}+\:(1-t_{j})D_{1}^{u_{j}}\left(\wt{\phi}_{a_{q+2}}\right)+\dots+(1-t_{j})^{s}D_{s}^{u_{j}}\left(\wt{\phi}_{a_{q+2}}\right) \right]
\\ =&\:a_{q+2}*_{L}\tau_{\mathcal{D}}^{\wt{\phi}}+ \mathcal{D} D_{1}^{a_{q+2}}\mathbf{\int}\:\mathrm{p}(t)(1-t_{j})\wt{\phi}_{u_{j}}+\dots+\mathcal{D} D_{q}^{a_{q+2}}\mathbf{\int}\:\mathrm{p}(t)(1-t_{j})^{q}\wt{\phi}_{u_{j}}
\\ & + u_{j}*_{L}\tau_{\mathcal{D}}^{\wt{\phi}}+ \mathcal{D} D_{1}^{u_{j}}\mathbf{\int}\:\mathrm{p}(t)(1-t_{j})\wt{\phi}_{a_{q+2}}+\dots+\mathcal{D} D_{q}^{u_{j}}\mathbf{\int}\:\mathrm{p}(t)(1-t_{j})^{q}\wt{\phi}_{a_{q+2}},
\end{align*}}so dass insgesamt:
{\allowdisplaybreaks\small\begin{align*}
\Big([]_{j}^{a_{q+2}}\tau_{\mathcal{D}}^{\wt{\phi}}\Big)&(u_{1},\dots,u_{k})
\\ =&\: -\underbrace{\mathcal{D} D_{1}^{a_{q+2}}\mathbf{\int}\:\overbrace{\mathrm{p}(t)(1-t_{j})}^{\mathrm{p}'(t)}\:\wt{\phi}_{u_{j}}}_{\DiffOpS{k}{q}}-\dots-\underbrace{\mathcal{D} D_{q}^{a_{q+2}}\mathbf{\int}\:\mathrm{p}(t)(1-t_{j})^{q}\wt{\phi}_{u_{j}}}_{\DiffOpS{k}{1}}
\\ & - u_{j}*_{L}\tau_{\mathcal{D}}^{\wt{\phi}}- \mathcal{D} D_{1}^{u_{j}}\mathbf{\int}\:\mathrm{p}(t)(1-t_{j})\:\wt{\phi}_{a_{q+2}}-\dots-\mathcal{D} D_{q}^{u_{j}}\mathbf{\int}\:\mathrm{p}(t)(1-t_{j})^{q}\wt{\phi}_{a_{q+2}}.
\end{align*}}Die Summanden in der ersten Reihe sind nach Induktionsannahme Differentialoperatoren der Ordnung $L=q,\dots,1$, verschwinden also nach Anwendung von $[]_{j}^{a_{1},\dots,a_{q+1}}$.
Das gleiche gilt f"ur die Terme in der zweiten Reihe, denn mit der $*_{L}$-Linearit"at der $D_{l}$ in $\mathcal{M}$ folgt:
\begin{equation*}
a*_{L}\mathcal{D}D_{l}(u,m)-\mathcal{D}D_{l}(a*u,m)=\mathcal{D}\big[a*_{L}D_{l}(u,m)-D_{l}(a*u,m)\big].
\end{equation*}
Insgesamt ist somit
$[]_{j}^{a_{1},\dots,a_{q+2}}\tau_{\mathcal{D}}^{\wt{\phi}}=0$
f"ur alle $1\leq j\leq s$, und
Proposition \ref{prop:MultidiffOps}~\textit{ii.)} zeigt dann, dass
$\tau_{\mathcal{D}}^{\wt{\phi}}\in \DiffOpS{k}{q+1}$ gilt.
Um die letzte Behauptung einzusehen erinnern wir daran, dass
\begin{align*}
\left(\xi^{k}\phi\right)(u_{1},\dots,u_{k})=\big(\wt{\phi}\cp G_{k}\big)(1\ot u_{1}\ot \dots \ot u_{k}\ot 1)
\end{align*}
mit $\wt{\phi}=\left(\left(\Upsilon^{k}\right)^{-1}\cp
\left(\Theta^{k}\right)^{-1}\right)(\phi)$ gilt.
Hieraus wird durch Anwendung von $[]_{j}^{a_{s+2}}$:
\begin{align*}
\left([]_{j}^{a_{s+2}}\xi^{k}\phi\right)&(u_{1},\dots,u_{k})= -\underbrace{D_{1}^{a_{s+2}}\mathbf{\int}\:(1-t_{j})\:\wt{\phi}_{u_{j}}}_{\DiffOpS{k}{s}}-\dots-\underbrace{D_{s}^{a_{s+2}}\mathbf{\int}\:(1-t_{j})^{s}\wt{\phi}_{u_{j}}}_{\DiffOpS{k}{1}}
\\ & \qquad- u_{j}*_{L}\tau_{\mathcal{D}}^{\wt{\phi}}- D_{1}^{u_{j}}\mathbf{\int}\:(1-t_{j})\:\wt{\phi}_{a_{s+2}}-\dots- D_{s}^{u_{j}}\mathbf{\int}\:(1-t_{j})^{s}\wt{\phi}_{a_{s+2}}.
\end{align*
Aus dem bisher Gezeigten folgt nun unmittelbar $[]^{a_{1},\dots,a_{s+2}}_{j}\left(\xi^{k}\phi\right)=0$ f"ur alle $1\leq j\leq k$, also $\xi^{k}(\phi)\in \DiffOpS{k}{s+1}$, wie behauptet.
\item
Sei $\phi\in \DiffOpS{k}{\bullet}$. Dann ist:
\begin{align*}
\big(\delta^{k}\phi\big)(a_{1},…,a_{k+1})=a_{1}*_{L}\phi(a_{2},…,a_{k+1})&+\sum_{j=1}^{k}(-1)^{j}\phi(a_{1},…,a_{i}a_{j+1},…,a_{k+1})\\
&+(-1)^{k+1}\phi(a_{1},…,a_{k})*_{R}a_{k+1}.
\end{align*}
Die differentielle Natur der ersten beiden Summanden hatten wir bereits eingesehen und der letzte ergibt ausgeschrieben:
\begin{align*}
\phi(a_{1},…,a_{k})*_{R}a_{k+1}=a_{k+1}*_{L}\phi(a_{1},…,a_{k})&+D_{1}(a_{k+1},\phi(a_{1},…,a_{k}))+\dots\\ &+D_{s}(a_{k+1},\phi(a_{1},…,a_{k})).
\end{align*}
Die Behauptung folgt nun unmittelbar aus Proposition \ref{prop:MultidiffOps}~\textit{ii.)}, \textbf{d.)} und mit:
\begin{align*}
[]_{j}^{a}D_{l}(a_{k+1},\phi(a_{1},…,a_{k}))&=D_{l}(a_{k+1},[]_{j}^{a}\phi(a_{1},…,a_{k}))\qquad\quad\forall\:j\neq k+1,
\end{align*}da hiermit $\delta^{k}\colon\DiffOpS{k}{\bullet}\longrightarrow \DiffOpS{k}{\bullet}$ gilt. Der lokalkonvexe Fall folgt analog.
\end{enumerate}
\end{beweis}
\end{lemma}
Mit Lemma \ref{lemma:DiffBimodWichEi} erhalten wir abschlie"send folgendes Resultat:
\begin{satz}
\begin{enumerate}
\item
Sei $\mathcal{M}$ ein differentieller $\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul. Dann besitzt jede Kohomologieklasse $[\eta]\in HH^{k}(\SsV,\mathcal{M})$ mindestens einen Repr"asentanten $\phi\in \DiffOpS{k}{s+1}$. Des Weiteren induzieren $\xi$ und $\hat{\xi}$ Kettenabbildungen zwischen $\left(HC_{\operatorname{\mathrm{diff}}}(\SsV,\mathcal{M}),\delta_{\operatorname{\mathrm{diff}}}\right)$ und $(KC(\V,\mathcal{M}),\Delta)$.
Hierbei ist $\wt{\xi^{k}}$ injektiv und $\wt{\hat{\xi}^{k}}$ surjektiv.
\item
Sei $\mathcal{M}$ ein differentieller, lokalkonvexer
$\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$-Bimodul. Dann besitzt
jede Kohomologieklasse $[\eta]\in
HH_{\operatorname{\mathrm{cont}}}^{k}(\SsV,\mathcal{M})$ mindestens einen
differentiellen Repr"asentanten $\phi\in \DiffOpS{k}{s+1,\operatorname{\mathrm{cont}}}$. Des Weiteren induzieren $\xi$
und $\hat{\xi}$ wohldefinierte Kettenabbildungen zwischen
$(HC_{\mathrm{c,d}}(\SsV,\mathcal{M}),\delta_{\mathrm{c,d}})$
und $(KC^{\operatorname{\mathrm{cont}}}(\V,\mathcal{M}),\Delta)$. Hierbei ist
$\wt{\xi^{k}}$ injektiv und $\wt{\hat{\xi}^{k}}$ surjektiv.
\end{enumerate}
\end{satz}
\clearpage
\thispagestyle{empty}
\section{Grundlagen (Homologische Algebra)}
\label{sec:HomologAlgebr}
Ziel dieses Kapitels ist die Bereitstellung der
Homologisch-Algebraischen Begrifflichkeiten und Zusammenhänge, die
dieser Arbeit als Ausgangspunkt dienen sollen. Es enthält haupsächlich
Resultate aus \cite[Kapitel 6]{jacobson:1989a}.
\subsection{Komplexe und Homologien}
\label{subsec:kompundhomolog}
Aufbauend auf \ref{sec:AlgebraischeDefinitionen} beginnen wir mit elementaren Definitionen:
\begin{definition}[(Ketten)Komplex]
\label{def:Komplex}
Gegeben ein Ring $R$,
\begin{enumerate}
\item
Ein $R$-Komplex ist eine Menge $\{C_{i},d_{i}\}_{i\in
\mathbb{Z}}$ von Paaren $(C_{i},d_{i})$, von $R$-Moduln
$C_{i}$ und $R$-Homomorphismen $d_{i}:C_{i}\rightarrow
C_{i-1}$ derart, dass $d_{i-1}\circ d_{i}=0$ für alle $i\in
\mathbb{Z}$.
\item
Ein $R$-Kettenkomplex ist ein $R$-Komplex, für den
$C_{i}=0,\: d_{i}=0\:\forall i<0$.
\item
Ein $R$-Kokettenkomplex ist ein $R$-Komplex, für den
$C_{i}=0,\: d_{i}=0\:\forall\: i>0$. Man setzt dann
$(C^{i},d^{i}):=(C_{-i},d_{-i})$, womit $d^{i}:C^{i}\rightarrow C^{i+1}$.
\item
Gegeben zwei $R$-Komplexe $(C,d)$ und $(C',d')$, so heißt eine
Menge $\alpha=\{\alpha_{i}\}_{i\in \mathbb{Z}}$ von
$R$-Homomorphismen $\alpha_{i}:C_{i}\rightarrow C_{i}'$
Kettenabbildung von $(C,d)$ nach $C',d')$, falls folgendes Diagramm für alle $i\in
\mathbb{Z}$ kommutiert.
$$ \diagram C_{i} \rto^{d_{i}} \dto_{\alpha_{i}} &C_{i-1} \dto^{\alpha_{i-1}} \\
C'_{i} \rto_{d'_{i}} &C'_{i-1}
\enddiagram $$
\end{enumerate}
\end{definition}
\begin{bemerkung}
\begin{enumerate}
\item
Gegeben ein Ring $R$, so bilden die $R$-Komplexe zusammen mit
den Kettenabbildungen die Kategorie $R$-comp (auch hier muss man
eigentlich zwischen Ketten-/ Kokettenkomplexen, sowie Rechts- und
Linksmoduln unterscheiden).
\item
Aus der Additivität von $R$-mod erhalten wir die von $R$-comp vermöge
\begin{equation*}
(\alpha+\beta)_{i}:=\alpha_{i}+\beta_{i},
\end{equation*} für Kettenabbildungen $\alpha,\:\beta:
(C,d)\rightarrow (C',d')$. In der Tat liefert diese Definition mit
\begin{equation*}
(\alpha_{i-1}+\beta_{i-1})d_{i}=\alpha_{i-1}d_{i}+\beta_{i-1}d_{i}=d'_{i}\alpha_{i}+d'_{i}\beta_{i}=d'_{i}(\alpha_{i}+\beta_{i})
\end{equation*} wieder eine Kettenabbildung zwischen besagten Komplexen. Diese Addition ist zudem
kommutativ und es ist klar, dass auch die
Distributivgesetze \eqref{eq:addkatDistlaws1} und
\eqref{eq:addkatDistlaws2} erfüllt sind.
\end{enumerate}
\end{bemerkung}
\begin{definition}[(Ko)Homologiemodul]
\label{def:kohomol}
\begin{enumerate}
\item
Gegeben ein $R$-Kettenkomplex $(C,d)$ und seien $Z_{i}=\ker(d_{i})\subseteq C_{i}$, $B_{i}=\im(d_{i+1})\subseteq
C_{i}$, dann sind sowohl $Z_{i}$ als auch $B_{i}$ Untermoduln von
$C_{i}$.\\ Da $d_{i}\cp d_{i+1}=0$ gilt zudem $B_{i}\subseteq Z_{i}$,
womit $B_{i}$ sogar ein Untermodul von $Z_{i}$ ist.\\
Wir betrachten
den Quotienten $H_{i}=Z_{i}/B_{i}$, womit $\alpha\in
[z_{i}]\subseteq Z_{i}$, wenn
$\alpha=z_{i}+b_{i}$ für ein $b_{i}\in B_{i}$. Die
Elemente in $Z_{i}$ nennt man i-Zykeln oder
geschlossen, die Elemente aus $B_{i}$ i-Ränder oder exakt. Die
Elemente $[\eta_{i}]\in H_{i}$ i-te Homologieklassen und
$H_{i}$ selbst i-ten Homologiemodul. Dabei sind
Moduladdition und Modulmultiplikation durch
$[\eta_{i}]+[\mu_{i}]=[\eta_{i}+\mu_{i}]$ sowie
$r[\eta_{i}]:=[r\eta_{i}]$ auf Repräsentantenniveau
definiert. Die Wohldefiniertheit dieser Operationen folgt
dabei unmittelbar aus der Untermoduleigenschaft von
$B_{i}\subseteq Z_{i}$.
\item
Für einen $R$-Kokettenkomplex definieren wir analog
$Z^{i}=\ker(d_{i})$, $B^{i}=\im(d^{i-1})$ und
$H^{i}=Z^{i}/B^{i}$. Die Element $[\eta_{i}]\in H^{i}$ heißen nun
i-te Kohomologieklassen und $H^{i}$ selbst i-ter
Kohomologiemodul.
\item
Sprechen wir ganz allgemein von einem $R$-Komplex, so benutzen
wir die Nomenklatur aus \textit{i.)}.
\end{enumerate}
\end{definition}
\begin{definition}
\label{def:exakt}
Ein $R$-Komplex $(C,d)$ heißt exakt, falls
$\ker(d_{i})=\im(d_{i+1})$ und für Ketten-/Kokettenkomplexe ist dies nach \thref{def:kohomol} gleichbedeutend mit $H_{i}=0\:\forall\:i\in \mathbb{Z}$.
\end{definition}
Wir wollen nun aufzeigen, inwiefern uns eine Kettenabbildung $\alpha$ zwischen zwei
$R$-Komplexen $(C,d)$ und $(C',d')$ Abbildungen zwischen deren
Homologiemoduln $H_{i},H'_{i}$ und sogar Funktoren von $R$-comp nach
$R$-mod definiert.
\begin{lemma}
\label{lemma:kettenabzu}
Gegeben zwei $R$-Komplexe $(C,d), (C',d')$ und eine
Kettenabbildung $\alpha$ von $(C,d)$ nach $(C',d')$, dann gilt:
\begin{enumerate}
\item
$\alpha_{i}(Z_{i})\subseteq Z'_{i}$,
\item
$\alpha_{i}(B_{i})\subseteq B'_{i}$,
\item
Die Abbildungen
\begin{align*}
\widetilde{\alpha}_{i}:H_{i}&\longrightarrow H'_{i}\\
[z_{i}]&\longmapsto [\alpha_{i}(z_{i})]
\end{align*} sind $R$-Homomorphismen. Ist $\alpha_{i}$ ein
Isomorphismus, so auch $\widetilde{\alpha}_{i}$.
\item
Die Abbildungen
\begin{align*}
\sim^{Ob}_{i}:R\text{-Kompl.}& \longrightarrow R\text{-Moduln}\\
(C,d)&\longmapsto H_{i}(C)
\end{align*} und
\begin{align*}
\sim^{Mor}_{i}:\Hom_{R}(C,C')&\longrightarrow \Hom_{R}(H_{i}(C),H_{i}(C'))\\
\alpha& \longrightarrow \widetilde{\alpha}_{i}
\end{align*} definieren einen additiven, kovarianten Funktor
$\sim_{i}$ von $R$-comp
nach $\Rm$, den i-ten Homologie-Funktor.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Mit \thref{def:Komplex}.\textit{iv.)} erhalten wir
\begin{equation*}
(d'_{i}\cp\alpha_{i})(z_{i})=(\alpha_{i-1}\cp d_{i})(z_{i})=\alpha_{i-1}(0)=0
\qquad\quad \forall\:z_{i}\in Z_{i}
\end{equation*}
\item
\begin{equation*}
\alpha_{i}(b_{i})=(\alpha_{i}\cp
d_{i+1})(b_{i+1})=d'_{i+1} (\alpha_{i+1}\cp
b_{i+1})\in B'_{i}\quad\quad \text{für }b_{i+1}\in
C_{i+1},\:b_{i}\in B_{i}.
\end{equation*}
\item
Die $R$-Linearität ist klar mit der von $\alpha_{i}$ und der
Definition der Modulmultiplikation und Addition in
$H'_{i}$. Für die Wohldefiniertheit sei $z'_{i}\in [z_{i}]$ mit
$z_{i}\in Z$, dann ist zu zeigen, dass
$[\alpha_{i}(z'_{i})]=[\alpha_{i}(z_{i})]$. Dies folgt so:
\begin{align*}
[\alpha_{i}(z'_{i})]=[\alpha_{i}(z_{i}+b_{i})]\glna{\textit{ii.)}}[\alpha_{i}(z_{i})]+[b'_{i}]=[\alpha_{i}(z_{i})].
\end{align*} Sind die $\alpha_{i}$ Isomorphismen, so
folgt aus \thref{def:Komplex}.\textit{v)}
\begin{equation*}
\alpha_{i-1}^{-1}\cp d_{i}'=d_{i}\cp\alpha_{i}^{-1},
\end{equation*} womit auch die $\alpha_{i}^{-1}$
Kettenabbildungen sind. Mit \textit{i.)} und \textit{ii.)} folgt unmittelbar
\begin{align*}
Z_{i}&\stackrel{\alpha_{i}}{\cong}Z'_{i},\\
B_{i}&\stackrel{\alpha_{i}}{\cong}B'_{i}
\end{align*} und für $[z'_{i}]\in H'_{i}$ finden wir somit
$z_{i}\in Z_{i}$ mit
$\widetilde{\alpha_{i}}([z_{i}])=[\alpha_{i}(z_{i})]=[z'_{i}]$,
was die Surjektivität von $\wt{\alpha}_{i}$ zeigt.
Für die Injektivität nehmen wir
an, es wäre
$\widetilde{\alpha_{i}}([z_{i}])=\widetilde{\alpha_{i}}([z'_{i}])$
mit $[z_{i}]\neq[z'_{i}]$, dann folgte
$\alpha_{i}(z_{i})=\alpha_{i}(z'_{i})+b'_{i}$ mit
$b'_{i}=\alpha_{i}(b_{i})$ für ein $b_{i}\in B_{i}$ und somit
$z_{i}-z'_{i}-b_{i}=0$, im Widerspruch
zu $[z_{i}]\neq[z'_{i}]$.
\item
Zunächst ist klar, dass
$\wt{\id_{C_{i}}}=\id_{H_{i}}$. Weiter haben wir zu
zeigen, dass
\begin{equation*}
\sim^{Mor}_{i}(\beta\circ\alpha):=\widetilde{(\beta\circ\alpha)}_{i}\:\glna{!}\:
\widetilde{\beta}_{i}\circ\widetilde{\alpha}_{i}
=:\:\sim^{Mor}_{i}(\beta)\:\circ\sim^{Mor}_{i}(\alpha),
\end{equation*}
für Kettenabbildungen
\begin{align*}
\alpha:(C,d)&\rightarrow (C',d')\\
\beta:(C',d')&\rightarrow (C'',d'').
\end{align*} Sei dafür $[z_{i}]\in H_{i}$, dann folgt
\begin{align*}
\widetilde{(\beta\circ\alpha)}_{i}\big([z_{i}]\big)&=
\big[(\beta_{i}\circ
\alpha_{i})(z_{i})\big]=[\beta_{i}(\alpha_{i}(z_{i}))]\\
&=\widetilde{\beta}_{i}([\alpha_{i}(z_{i})])=\widetilde{\beta}_{i}\big(\widetilde{\alpha}_{i}([z_{i}])\big)=(\widetilde{\beta}_{i}\circ\widetilde{\alpha}_{i})([z_{i}]).
\end{align*} Für die Additivität beachte man, dass sowohl
$R$-comp als auch $R$-mod additive Kategorien sind und es
ist zu zeigen, dass
\begin{equation*}
\sim^{Mor}_{i}(\alpha+\beta)=\:\sim^{Mor}_{i}(\alpha)\:+\sim^{Mor}_{i}(\beta)\qquad\quad\forall\: \alpha,\beta \in \Hom_{R}(C,C').
\end{equation*} Dies folgt sofort mit
\begin{align*}
\widetilde{(\alpha+\beta)_{i}}([\eta_{i}])=[(\alpha_{i}+\beta_{i})(\eta_{i})]=[\alpha_{i}(\eta_{i})]+[\beta_{i}(\eta_{i})]=\widetilde{\alpha}_{i}([\eta_{i}])+\widetilde{\beta}_{i}([\eta_{i}])
\end{align*} und wir sind fertig.
\end{enumerate}
\end{beweis}
\end{lemma}
\subsection{Homotopie}
\label{subsec:homotopie}
Ein Kriterium das festlegt, ob zwei Kettenabbildungen
$\alpha,\beta:(C,d)\rightarrow (C',d')$ auf Homologieniveau die
gleiche Abbildung induzieren, wird durch die sogenannte
Homotopieeigenschaft bereitgestellt:
\begin{definition}[Homotopie]
\label{def:homotopie}
Gegeben zwei Kettenabbildungen $\alpha,\:\beta:(C,d)\rightarrow
(C',d')$ von $R$-Komplexen $(C,d)$ und $(C',d')$, so heißen $\alpha$
und $\beta$ homotop ($\alpha\sim \beta$), falls $R$-Homomorphismen $\{s_{i}\}_{i\in
\mathbb{Z}}$ mit $s_{i}:C_{i}\mapsto C'_{i+1}$ derart existieren, dass
\begin{equation}
\label{eq:homotopiebed}
\alpha_{i}-\beta_{i}=d'_{i+1}s_{i}+s_{i-1}d_{i}\qquad\qquad
\forall i\in \mathbb{Z}.
\end{equation}
$$ \diagram
...\rto^{d_{i+1}} &C_{i}\dto_{\alpha_{i}} \dlto_{s_{i}}\rto^{d_{i}} &C_{i-1} \dto_{\alpha_{i-1}}\dlto_{s_{i-1}}\rto^{d_{i-1}} &C_{i-2} \dto_{\alpha_{i-2}}\dlto_{s_{i-2}}\rto^{d_{i-2}}
&... &\\
...\rto^{d'_{i+1}} &C'_{i} \rto^{d'_{i}} &C'_{i-1} \rto^{d'_{i-1}} &C'_{i-2} \rto^{d'_{i-2}}.
&... &
\enddiagram $$ ($\bold{Warnung}$: die $s_{i}$-Pfeile sind keine
kommutativen Elemente des Diagrammes)
\end{definition}
\begin{bemerkung}
\label{bem:kokettenhomotopiekram}
\begin{enumerate}
\item
Für $R$-Kettenkomplexe gilt dann \eqref{eq:homotopiebed} mit der
Zusatzbedingung $s_{i}=0$, falls $i< 0$.
\item
Für $R$-Kokettenkomplexe $(C^{i},d^{i})$ übersetzt sich
\thref{def:homotopie} zu
\begin{equation}
\label{eq:homotopiebedkokett}
\alpha^{i}-\beta^{i}=d'^{i-1}s^{i}+s^{i+1}d^{i}\quad\quad
\forall i\in \mathbb{Z}
\end{equation} mit $R$-Homomorphismen $s^{i}:C^{i}\rightarrow
C^{i-1}$ und $s^{i}=0$, falls $i\leq0$.
$$ \diagram
C^{0}\dto_{\alpha_{0}} \rto^{d^{0}} &C^{1} \dto_{\alpha_{1}}\rto^{d^{1}}\dlto_{s^{1}} &C^{2} \dto_{\alpha_{2}}\rto^{d^{2}}\dlto_{s^{2}}
&C^{3} \dto_{\alpha_{3}}\dlto_{s^{3}}\rto&...\\
C'^{0} \rto^{d'^{0}} &C'^{1} \rto^{d'^{1}} &C'^{2} \rto^{d'^{2}}
&C'^{3} \rto &...
\enddiagram $$
\end{enumerate}
\end{bemerkung}
\begin{korollar}
Homotopie induziert eine Äquvivalenzrelation.
\begin{beweis}
Die Symmetrie ist klar und ebenso die Reflexivität mit der
Wahl $s_{i}=0$. Für die Transitivität sei $\alpha\sim \beta$ via $s$ und
$\beta\sim \gamma$ via $t$, dann folgt
\begin{align*}
\alpha_{i}-\beta_{i}=d'_{i+1}s_{i}+s_{i-1}d_{i}\\
\beta_{i}-\gamma_{i}=d'_{i+1}t_{i}+t_{i-1}d_{i}&
\end{align*} und somit
\begin{equation*}
\alpha_{i}-\gamma_{i}=d'_{i+1}(s_{i}+t_{i})+(s_{i-1}+t_{i-1})d_{i}.
\end{equation*} Dies zeigt $\alpha\sim \gamma$ via
$u=s+t= \{s_{i}+t_{i}\}_{i \in \mathbb{Z}}$.
\end{beweis}
\end{korollar}
\begin{lemma}
\label{lemma:tildeabbeind}
Ist in der Situation von \thref{def:homotopie} $\alpha\sim \beta$,
so gilt
$\widetilde{\alpha}_{i}=\widetilde{\beta}_{i}$.
\begin{beweis}
\begin{align*}
\widetilde{\alpha}_{i}([z_{i}])=[\alpha_{i}(z_{i})]&=[\beta_{i}(z_{i})+(d'_{i+1}
s_{i})(z_{i})+(s_{i-1}d_{i})(z_{i})]\\
&=[\beta_{i}(z_{i})]+[(d'_{i+1}s_{i})(z_{i})]+\underbrace{[(s_{i-1}d_{i})(z_{i})]}_{[0]}\\
&=[\beta_{i}(z_{i})]+[b'_{i}]\\ &=\widetilde{\beta}_{i}([z_{i}]),
\end{align*} für $z_{i}\in Z_{i},\:B'_{i}\ni b_{i}'= (d'_{i+1}s_{i})(z_{i})$.
\end{beweis}
\end{lemma}
\subsection{Auflösungen}
\label{subsec:aufloesung}
\begin{definition}
\label{def:komplexubermodul}
Gegeben ein $R$-Modul $M$, so bezeichnen wir mit einem Komplex $(C,d,\epsilon)$ über $M$ einen
$R$-Kettenkomplex $(C,d)$ zusammen mit einem $R$-Epimorphismus
$\epsilon:C_{0}\rightarrow M$ derart, dass
$\epsilon\circ d_{1}=0$. Solch ein $\epsilon$ bezeichnet man auch als
Augmentierung von $(C,d)$. Grafisch bedeutet dies
$$ \diagram
...\rto^{d_{i+1}} &C_{i} \rto^{d_{i}} &C_{i-1} \rto^{d_{i-1}}
&... \rto^{d_{2}} &C_{1} \rto^{d_{1}} &C_{0} \rto^{\epsilon}|>>\tip &M,
\enddiagram $$ wobei der letzte Pfeil die
Surjektivität von $\epsilon$ beutet. In obiger Kette von
Homomorphismen ergibt somit die Hintereinanderausführung je
zweier aufeinander folgender die $0-Abbildung$.
\end{definition}
\begin{definition}
Gegeben ein Komplex $(C,d,\epsilon)$ über einem $R$-Modul $M$.
\begin{enumerate}
\item
$(C,d,\epsilon)$ heißt Auflösung, falls die Kette von
Homomorphismen aus \thref{def:komplexubermodul} exakt ist,
also zusätzlich zu der Exaktheit von $(C,d)$ auch
$\im(d_{1})=\ker(\epsilon)$ gilt.
Das bedeutet insbesondere $H_{i}(C)=0$ für $i>0$ und
$H_{0}:=C_{0}/\im(d_{1})=C_{0}/\ker(\epsilon)\cong M$.
\item
$(C,d,\epsilon)$ heißt projektiv, falls alle $C_{i}$
projektive Moduln sind.
\end{enumerate}
\end{definition}
\begin{satz}
Gegeben ein projektiver Komplex $(C,d,\epsilon)$ über einem $R$-Modul
$M$ und eine Auflösung $(C',d',\epsilon')$ eines $R$-Moduls
$M'$. Sei weiterhin $\mu: M\rightarrow M'$ ein $R$-Homomorphismus,
dann existiert eine Kettenabbildung $\alpha$ von $C$ nach $C'$
derart, dass folgendes Diagramm kommutiert
$$ \diagram
...\rto^{d_{i+1}} &C_{i}\dto^{\alpha_{i}} \rto^{d_{i}} &C_{i-1} \dto^{\alpha_{i-1}}\rto^{d_{i-1}} &C_{i-2} \dto^{\alpha_{i-2}}\rto^{d_{i-2}}
&... \rto^{d_{1}} &C_{0} \dto^{\alpha_{0}}\rto^{\epsilon}|>> \tip
&
\dto^{\mu}\\
...\rto^{d'_{i+1}} &C'_{i} \rto^{d'_{i}} &C'_{i-1} \rto^{d'_{i-1}} &C'_{i-2} \rto^{d'_{i-2}}
&... \rto^{d'_{1}} &C'_{0} \rto^{\epsilon'}|>> \tip &M
,
\enddiagram $$
weiterhin sind alle derartigen Kettenabbildungen zueinander homotop.
\begin{beweis}
Es ist $\epsilon'$ ein Epimorphismus nach Vorraussetzung, also
surjektiv. Da $C_{0}$ projektiv, finden wir ein
$\alpha_{0}\in \Hom_{R}(C_{0},C'_{0})$ derart, dass
$$ \diagram C_{0} \dto_{\alpha_{0}} \drto^{\mu\circ\epsilon} & \\
C'_{0} \rto_{\epsilon'}|>>\tip & M'
\enddiagram $$ kommutiert. Dies bildet den Induktionsanfang
und wir müssen dann lediglich für $\alpha_{i-1}$ vorgegeben
die Existenz eines $\alpha_{i}$ nachweisen, welches die
zugehörige Zelle zum kommutieren bringt. Dafür wollen wir
zunächst $C'_{i-1}$ durch $\im(d'_{i})$ ersetzen dürfen, dann
wäre nämlich $d'_{i}$ wieder surjektiv und mit der
Projektivität von $C_{i}$ folgte abermals die Existenz eines solchen
$\alpha_{i}$. Dafür reicht es zu zeigen, dass
$\im(\alpha_{i-1}\:d_{i})\subseteq \im(d'_{i})$. Nun
folgt
\begin{equation*}
d'_{i-1}\alpha_{i-1}\:d_{i}=\alpha_{i-2}\:d_{i-1}d_{i}=0,
\end{equation*} womit $\im(\alpha_{i-1}\cp d_{i})\subseteq
\ker(d'_{i-1})=\im(d'_{i})$ mit der Exaktheit von
$(C',d',\epsilon')$. Das zeigt die erste Aussage des Satzes.\\\\
Für die Zweite seien
$\alpha,\:\beta$ beide so, dass sie obiges Diagramm zum
kommutieren bringen. Wir wollen dann eine Homotopie $s$ derart finden,
dass
\begin{equation}
\label{eq:homotopbed}
\gamma_{i}=\alpha_{i}-\beta_{i}=d'_{i+1}s_{i}+s_{i-1}d_{i}.
\end{equation} Zunächst folgt unmittelbar
\begin{align}
\label{eq:kerepsilonstrich}
\epsilon'\gamma_{0}&=\epsilon'\alpha_{0}-\epsilon'\beta_{0}=\mu\:\epsilon-\mu\:\epsilon=0,\\
\label{eq:vertasuschiGamma}
d'_{i}\gamma_{i}&=\gamma_{i-1}d_{i}\quad i\geq 1.
\end{align}
Wir betrachten den Diagrammausschnitt
$$ \diagram &C_{0} \dto_{\gamma_{0}} &\\
C'_{1}\rto^{d'_{1}} &C'_{0} \rto^{\epsilon'}|>>\tip & M'
\enddiagram $$ und beachten, dass mit
\eqref{eq:kerepsilonstrich} und der Exaktheit von $(C',d',\epsilon')$
\begin{equation*}
\im(\gamma_{0})\subseteq \ker(\epsilon')=\im(d'_{1}),
\end{equation*}
womit vermöge der Projektivität von $C_{0}$
$$ \diagram &C_{0} \dlto_{s_{0}} \dto_{\gamma_{0}} &\\
C'_{1}\rto^{d'_{1}}|>>\tip &\im(d'_{1}).
\enddiagram $$ Es folgt, dass durch $\gamma_{0}=d'_{1}s_{0}$ die
Homotopiebedingung \eqref{eq:homotopbed} für i=0 mit $s_{-1}=0$
erfüllt ist.
Sei nun wieder \eqref{eq:homotopbed} für $0\leq
i\leq n-1$ erfüllt, wir definieren dann die Hilfsabbildung
$\widetilde{\gamma}_{n}:C_{n}\rightarrow C'_{n}$ induktiv durch
$\widetilde{\gamma}_{n}:=\gamma_{n}-s_{n-1}d_{n}$ und erhalten
\begin{align*}
d'_{n}\widetilde{\gamma}_{n}&=d'_{n}(\gamma_{n}-s_{n-1}d_{n})=d'_{n}\gamma_{n}-d'_{n}s_{n-1}d_{n}\glna{\eqref{eq:vertasuschiGamma}}\gamma_{n-1}d_{n}-d'_{n}s_{n-1}d_{n}\\
&=(\gamma_{n-1}-d'_{n}s_{n-1})d_{n}\glna{\eqref{eq:homotopbed}}(\gamma_{n-1}-\gamma_{n-1}+s_{n-2}d_{n-1})d_{n}=0.
\end{align*} Es ist abermals
$\im(\widetilde{\gamma}_{n})\subseteq \ker(d'_{n})=\im(d'_{n+1})$
und mit der Projektivität von $C_{n}$ erhalten wir ein
$s_{n}:C_{n}\rightarrow C'_{n+1}$ derart, dass
$d'_{n+1}s_{n}=\widetilde{\gamma}_{n}=\gamma_{n}-s_{n-1}d_{n}$.
$$ \diagram &C_{n} \dlto_{s_{n}} \dto_{\wt{\gamma}_{n}} &\\
C'_{n+1}\rto^{d'_{n+1}}|>>\tip &\im(d'_{n+1})
\enddiagram $$
Es folgt unmittelbar
$\gamma_{n}=d'_{n+1}s_{n}+s_{n-1}d_{n}$ und somit \eqref{eq:homotopbed}.
\end{beweis}
\end{satz}
\subsection{Induzierte Funktoren}
Zunächst wollen wir die in \ref{subsec:kompundhomolog} und
\ref{subsec:homotopie} behandelten Begrifflichkeiten und Zusammenhänge
von $R$-Moduln auf abelsche Gruppen übertragen ($R$-Moduln waren ja
lediglich derartige Gruppen mit Zusatzstruktur). Aus \thref{def:Komplex}
wird:
\begin{definition}
\label{def:KomplexG}
\begin{enumerate}
\item
Ein Gruppenkomplex ist eine Menge $\{G_{i},d_{i}\}_{i\in
\mathbb{Z}}$ von Paaren $(G_{i},d_{i})$, von abelschen Gruppen
$G_{i}$ und Gruppenhomomorphismen $d_{i}:G_{i}\rightarrow
G_{i-1}$ mit $d_{i-1}\circ d_{i}=0$ für alle $i\in
\mathbb{Z}$.
\item
Ein Gruppen-Kettenkomplex ist ein Gruppenkomplex, für den
$G_{i}=0,\: d_{i}=0$ für alle $i<0$.
\item
Ein Gruppen-Kokettenkomplex ist ein Gruppenkomplex mit
$G_{i}=0,\: d_{i}=0\:\forall\: i>0$. Man setzt
$(G^{i},d^{i}):=(G_{-i},d_{-i})$, womit $d^{i}:G^{i}\rightarrow G^{i+1}$.
\item
Gegeben zwei Gruppenkomplexe $(G,d)$ und $(G',d')$, so heißt
Menge $\alpha=\{\alpha_{i}\}_{i\in \mathbb{Z}}$ von
Gruppenhomomorphismen $\alpha_{i}:G_{i}\rightarrow G_{i}'$
Kettenabbildung von $(G,d)$ nach $(G',d')$, falls folgendes Diagramm für alle $i\in
\mathbb{Z}$ kommutiert
$$ \diagram G_{i} \rto^{d_{i}} \dto_{\alpha_{i}} &G_{i-1} \dto^{\alpha_{i-1}} \\
G'_{i} \rto_{d'_{i}} &G'_{i-1}\:.
\enddiagram $$
\end{enumerate}
\end{definition}
\thref{def:kohomol} wird zu:
\begin{definition}[(Ko)Homologie]
\label{def:kohomolG}
\begin{enumerate}
\item
Gegeben ein Gruppen-Kettenkomplex $(G,d)$. Sei $Z_{i}=\ker(d_{i})\subseteq G_{i}$, $B_{i}=\im(d_{i+1})\subseteq
G_{i}$, dann ist $B_{i}\subseteq Z_{i}$ eine Untergruppe von $Z_{i}$. Wir betrachten wieder den Quotienten $H_{i}=Z_{i}/B_{i}$
und nenne die $[g_{i}]\in H_{i}$ i-te Homologieklassen, sowie
$H_{i}$ selbst i-te Homologiegruppe.
\item
Für einen Guppen-Kokettenkomplex definieren wir analog
$Z^{i}=\ker(d_{i})$, $B^{i}=\im(d^{i-1})$ und
$H^{i}=Z^{i}/B^{i}$. Die Element $[g_{i}]\in H^{i}$ heißen
i-te Kohomologieklassen und $H^{i}$ selbst, i-te
Kohomologiegruppe.
\item
Sprechen wir von einem Gruppen-Komplex, so benutzen wir die
Nomenklatur aus \textit{i.)}.
\end{enumerate}
\end{definition}
\thref{def:exakt} und \thref{lemma:kettenabzu} übertragen sich
sinngemäß, ebenso der gesamte Abschnitt \ref{subsec:homotopie}.
In \thref{lemma:kettenabzu}.\textit{iv.)} haben wir dann additive,
kovariante Funktoren
$\sim_{i}$
\begin{align*}
\sim^{Ob}_{i}:\text{Gruppenkompl.}&\longrightarrow \text{ab. Gr.}\\
(G,d)&\longmapsto H_{i}(G)
\end{align*} und
\begin{align*}
\sim^{Mor}_{i}:\Hom(G,G')&\longrightarrow \Hom(H_{i}(G),H_{i}(G'))\\
\alpha& \longrightarrow \widetilde{\alpha}_{i}.\nonumber
\end{align*}
Für einen $R$-Modul $M$, einen Komplex $(C,d,\epsilon)$ über $M$ und einen kovarianten,
additiven Funktor $\mathcal{F}:\Rm\rightarrow \Ab$, erhalten wir
durch Anwendung dieses Funktors auf $(C,d,\epsilon)$ einen
Gruppen-Kettenkomplex
$(\mathcal{F}C,\mathcal{F}d,\mathcal{F}\epsilon)$
$$ \diagram
...\rto^{\mathcal{F}d_{3}} &\mathcal{F}C_{2}\rto^{\mathcal{F}d_{2}} &\mathcal{F}C_{1} \rto^{\mathcal{F}d_{1}} &\mathcal{F}C_{0} \rto^{\mathcal{F}\epsilon} &\mathcal{F}M.
\enddiagram $$
In der Tat sind mit der Additivität von $\mathcal{F}$ alle Pfeile
Gruppenhomomorphismen, des Weiteren gilt $\mathcal{F}d_{i}\circ
\mathcal{F}d_{i+1}=\mathcal{F}(d_{i}\circ\: d_{i+1})=\mathcal{F}(0)=0$,
wegen der Additivität von $\mathcal{F}$.
Ist $(C,d,\epsilon)$ exakt, so überträgt sich dies nicht
notwendigerweise auf $(\mathcal{F}C,\mathcal{F}d\mathcal{F}\epsilon)$.
Somit besteht die Chance, durch diese Prozedur nicht triviale
Homologie-Gruppen $H_{i}(\mathcal{F}C)$, sowie
$H_{0}(\mathcal{F}C):=\mathcal{F}C_{0}/\mathcal{F}d_{1}(\mathcal{F}C_{1})$
zu erhalten.
\begin{lemma}
\label{lemma:GruppenhomsausprojaufloesundFunktoren}
Gegeben $R$-Moduln $M,M'$, ein projektiver Komplex
$(C,d,\epsilon)$ über $M$,
eine Auflösungen $(C',d',\epsilon')$ von $M'$ und ein
additiver, kovarianter Funktor $\mathcal{F}:\Rm \rightarrow \Ab$. Sei des Weiteren $\mu$
ein Homomorphismus $\mu:M\rightarrow M'$, dann gilt:
\begin{enumerate}
\item Es existieren
von diesen Daten abhängige Gruppen-Homomorphismen\\
$\widetilde{\mathcal{F}\alpha_{i}}:H_{i}(\mathcal{F}C)\rightarrow
H_{i}(\mathcal{F}C')$.
\item
Sind
$(C,d,\epsilon)$ und
$(\overline{C},\overline{d},\overline{\epsilon})$
projektive
Auflösungen des selben $R$-Moduls $M$, so gilt $H_{i}(\mathcal{F}C)\simeq H_{i}(\mathcal{F}\overline{C})$.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Zunächst haben wir eine bis auf Homotopie eindeutig
bestimmte Kettenabbildung
$\alpha:(C,d,\epsilon)\rightarrow (C',d',\epsilon')$ derart, dass
$$ \diagram
...\rto^{d_{3}}
&C_{2} \dto^{\alpha_{2}}\rto^{d_{2}}
&C_{1} \dto^{\alpha_{1}}\rto^{d_{1}} &C_{0}
\dto^{\alpha_{0}}\rto^{\epsilon}|>>\tip &M
\dto^{\mu}\\
... \rto^{d'_{3}}
&C'_{2} \rto^{d'_{2}}\rto^{d'_{2}}
&C'_{1} \rto^{d'_{1}} &C'_{0} \rto^{\epsilon'}|>>\tip &M
\enddiagram $$
kommutiert. Dieses Diagramm übersetzt sich vermöge
$\mathcal{F}$ zu
$$ \diagram
...\rto^{\mathcal{F}d_{3}}
&\mathcal{F}C_{2}
\dto^{\mathcal{F}\alpha_{2}}\rto^{\mathcal{F}d_{2}}
&\mathcal{F}C_{1}
\dto^{\mathcal{F}\alpha_{1}}\rto^{\mathcal{F}d_{1}}
&\mathcal{F}C_{0}
\dto^{\mathcal{F}\alpha_{0}}\rto^{\mathcal{F}\epsilon}
&\mathcal{F}M
\dto^{\mathcal{F}\mu}\\
... \rto^{\mathcal{F}d'_{3}}
&\mathcal{F}C'_{2}\rto^{\mathcal{F}d'_{2}}
&\mathcal{F}C'_{1} \rto^{\mathcal{F}d'_{1}} &\mathcal{F}C'_{0} \rto^{\mathcal{F}\epsilon'} &\mathcal{F}M'
\enddiagram $$
Wegen der Addititivität und Kovarianz von $\mathcal{F}$
erhalten wir für jedes $\beta\sim \alpha$
\begin{equation*}
\mathcal{F}\alpha_{i}-\mathcal{F}\beta_{i}=\mathcal{F}(\alpha_{i}-\beta_{i})=\mathcal{F}(d'_{i+1}s_{i}+s_{i-1}d_{i})=\mathcal{F}d'_{i+1}\mathcal{F}s_{i}+\mathcal{F}s_{i-1}\mathcal{F}d_{i},
\end{equation*} also $\mathcal{F}\alpha\sim
\mathcal{F}\beta$. Mit der Gruppenversion von
\thref{lemma:tildeabbeind} folgt
$\widetilde{\mathcal{F}\alpha_{i}}=\widetilde{\mathcal{F}\beta_{i}}$
und somit die erste Aussage.
\item
Wir haben für $\mu=id_{M}$
$$ \diagram
...\rto^{\mathcal{F}d_{3}}
&\mathcal{F}C_{2}
\dto^{\mathcal{F}\beta_{2}}\rto^{\mathcal{F}d_{2}}
&\mathcal{F}C_{1}
\dto^{\mathcal{F}\beta_{1}}\rto^{\mathcal{F}d_{1}}
&\mathcal{F}C_{0}
\dto^{\mathcal{F}\beta_{0}}\rto^{\mathcal{F}\epsilon}
&\mathcal{F}M \dto^{id_{\mathcal{F}M}}\\
...\rto^{\mathcal{F}\overline{d}_{3}}
&\mathcal{F}\overline{C}_{2} \dto^{\mathcal{F}\widehat{\beta}_{2}}\rto^{\mathcal{F}\overline{d}_{2}}
&\mathcal{F}\overline{C}_{1} \dto^{\mathcal{F}\widehat{\beta}_{1}}\rto^{\mathcal{F}\overline{d}_{1}} &\mathcal{F}\overline{C}_{0} \dto^{\mathcal{F}\widehat{\beta}_{0}}\rto^{\mathcal{F}\overline{\epsilon}} &\mathcal{F}M \dto^{id_{\mathcal{F}M}}\\
... \rto^{\mathcal{F}d_{3}}
&\mathcal{F}C_{2} \rto^{\mathcal{F}d_{2}}
&\mathcal{F}C_{1} \rto^{\mathcal{F}d_{1}} &\mathcal{F}C_{0} \rto^{\mathcal{F}\epsilon} &\mathcal{F}M,
\enddiagram $$ womit
\begin{equation*}
id_{\mathcal{F}C_{i}}=\mathcal{F}(id_{C_{i}})\sim
\mathcal{F}(\widehat{\beta}_{i}\circ\beta_{i}),
\end{equation*}
da $\id_{C}$ ebenfalls eine Kettenabbildung zu $\id_{\mu}$ ist.
Mit \thref{lemma:tildeabbeind},
\thref{lemma:kettenabzu}.\textit{iv.)}, sowie den
Funktoreigenschaften von $\mathcal{F}$ und $\sim_{i}$
erhalten wir
\begin{equation*}
id_{H_{i}(\mathcal{F}C)}=\widetilde{id_{\mathcal{F}C_{i}}}=\widetilde{\mathcal{F}(\widehat{\beta}_{i}\circ\beta_{i})}=\widetilde{\mathcal{F}\widehat{\beta}_{i}\circ
\mathcal{F}\beta_{i}}=\widetilde{\mathcal{F}\widehat{\beta}_{i}}\circ
\widetilde{\mathcal{F}\beta_{i}}.
\end{equation*} Analog folgt
\begin{equation*}
id_{H_{i}(\mathcal{F}\overline{C})}=\widetilde{\mathcal{F}\beta}_{i}\circ\widetilde{\mathcal{F}\widehat{\beta}_{i}},
\end{equation*} womit
\begin{equation*}
\widetilde{\mathcal{F}\beta}_{i}=\widetilde{\mathcal{F}\widehat{\beta}_{i}}^{-1},
\end{equation*} also $H_{i}(\mathcal{F}C)\simeq H_{i}(\mathcal{F}\overline{C})$ vermöge $\eta_{i}=\widetilde{\mathcal{F}\beta_{i}}$.
\end{enumerate}
\end{beweis}
\end{lemma}
\begin{korollar}
\label{kor:homologgruppenisomorphdurchMMstrichundF}
Sind in der Situation von \thref{lemma:GruppenhomsausprojaufloesundFunktoren}
beide Komplexe
$(C,d,\epsilon)$, $(C',d',\epsilon')$ projektive Auflösungen, so
sind die zugehörigen Kohomologie-Gruppen $H^{i}(\mathcal{F}C)$,
$H^{i}(\mathcal{F}C')$, sowie die Homomorphismen aus
\thref{lemma:GruppenhomsausprojaufloesundFunktoren}.\textit{i.)} bereits
bis auf Verkettung mit Isomorphismen eindeutig festgelegt durch $M,M', \mu$ und
$\mathcal{F}$, hängen also nicht von der Wahl der beiden
Auflösungen ab.
\begin{beweis}
Die Isomorphie der Kohomologie-Gruppen folgt sofort aus Teil
\textit{ii.)} obigen Lemmas, da wir zwischen verschiedenen
Auflösungen des selben Moduls via Isomorphismen wechseln können. Mit den Funktoreigenschaften von
$\mathcal{F}$ und $\sim_{i}$ kann dann für Kettenabbildungen
\begin{align*}
\beta:(C,d,\epsilon)&\rightarrow
(\overline{C},\overline{d},\overline{\epsilon})\\
\beta':(C',d',\epsilon')&\rightarrow
(\overline{C}',\overline{d}',\overline{\epsilon}')
\\\alpha:(C,d,\epsilon)&\rightarrow (C',d',\epsilon')
\end{align*} der Kohomologiegruppenhomomorphismus
\begin{align*}
\tau_{i}:H^{i}(\mathcal{F}\overline{C})&\longrightarrow H^{i}(\mathcal{F}\overline{C}')\\
x&\longrightarrow \eta'_{i}\widetilde{\mathcal{F}\alpha_{i}}\eta_{i}^{-1}
\end{align*} geschrieben werden, als
\begin{equation*}
\tau_{i}=\widetilde{\mathcal{F}(\beta'_{i}\alpha_{i}\widehat{\beta}_{i})}.
\end{equation*} Da nun aber $\beta'_{i}\alpha_{i}\widehat{\beta}_{i}$:
$(\overline{C},\overline{d},\overline{\epsilon})\rightarrow
(\overline{C}',\overline{d}',\overline{\epsilon}')$ selbst eine
Kettenabbildung ist,
stimmt $\tau$ auf Homologieniveau mit allen anderen durch
$\mathcal{F}$ gewonnenen Kettenabbildungen überein, dass zeigt die zweite Aussage des Korollars.
\end{beweis}
\end{korollar}
Als nächstes wollen wir ein Kriterium angeben welches uns garantiert,
dass überhaupt projektive Auflösungen eines $R$-Moduls existieren.
\begin{lemma}
\label{lemma:unitarereRingModulnhabenfreieAufloesung}
Gegeben ein unitärer Ring $R$ und ein $R$-Modul $M$, so existiert eine freie
Auflösung $(C,d,\epsilon)$ von $M$. Dabei bedeutet frei, dass die
$C_{i}$ freie $R$-Moduln sind.
\begin{beweis}
Wir wählen eine Indizierung $J$ der Elemente von $M$ und
betrachten die Direkte Summe
$C_{0}=\bigoplus^{|J|}R$. Bezeichne
$j_{m}\in J$ den zu $m\in M$ gehörigen Index, so definieren wir eine surjektive Abbildung
$\epsilon:\bigoplus^{|J|}R \longrightarrow M$ durch
\begin{align*}
0\oplus…\oplus \overbrace{1}^{j_{m}}\oplus…\oplus
0\longrightarrow m \quad\quad m\in M
\end{align*} und setzen diese
$R$-linear auf ganz $C_{0}$ fort. $C_{0}$ ist per
Definition ein freier $R$-Modul (je nachdem ob $M$ ein Links-
oder Rechtsmodul ist,
definiert man die $R$-Multiplikation in $C_{0}$ in
entsprechender Weise) und wir erhalten
$$ \diagram
& \ker(\epsilon) \rto|<\ahook^{i} &C_{0} \rto|>>\ti
^{\epsilon} &M,
\enddiagram $$ wobei $i$ die Injektion von $\ker(\epsilon)$ in
$C_{0}$ bezeichnet. Da $\ker(\epsilon)$ ein Untermodul und somit
selbst ein Modul ist, dürfen wir setzen: $C_{1}=\ker(\epsilon)$, $d_{0}=i$ und
erhalten $\epsilon\cp d_{0}=0$, sowie $\im(i)=\ker(\epsilon)$. Durch
Iteration von Schritt 1 mit $C_{1}$ an Stelle von $M$ erhalten wir eine
freie Auflösung von $M$.
\end{beweis}
\end{lemma}
\begin{korollar}
\label{kor:exprojAufl}
Gegeben ein unitärer Ring $R$ und ein $R$-Modul $M$, so existiert eine
projektive Auflösung $(C,d,\epsilon)$ von $M$.
\begin{beweis}
Das folgt sofort mit
\thref{lemma:unitarereRingModulnhabenfreieAufloesung} und \thref{kor:freiemoduberunitRingensindProjektiv}.
\end{beweis}
\end{korollar}
\begin{definition}[Linksinduzierte Funktoren]
Gegeben ein unitärer Ring $R$ und ein kovarianter Funktor
$\mathcal{F}: \Rm\rightarrow \Ab$, so ist der i-te linksinduzierte
Funktor $L_{i}\mathcal{F}: \Rm\rightarrow \Ab$ definiert durch
\begin{align*}
L_{i}\mathcal{F}^{Ob}: R\text{-Moduln}&\longrightarrow
\text{abelsche Gruppen}\\
M&\longmapsto H_{i}(\mathcal{F}C)
\end{align*}
\begin{align*}
L_{i}\mathcal{F}^{Mor}:\Hom_{R}(M,M') &\longrightarrow \Hom_{G}(H_{i}(\mathcal{F}C),H_{i}(\mathcal{F}C'))\\
\mu&\longmapsto \widetilde{\mathcal{F}\alpha_{i}}.
\end{align*} Dabei stehen $C,C'$ für, nach \thref{kor:exprojAufl} immer
exisitierende projektive Auflösung $(C,d,\epsilon)$ von $M$ und
$(C',d',\epsilon')$ von $M'$ und
$\alpha$ für eine ebenfalls immer existente, durch $\mu$ sowie
$C,C'$ induzierte Kettenabbildung. Die Wohldefiniertheit besagten
Funktors folgt
unmittelbar aus den vorhergehenden
Lemmata und Korollaren.
\end{definition}
\subsubsection{Rechtsinduzierte Funktoren}
Wir wollen nun die für uns eigentlich relevanten rechtsinduzierten
Funktoren definieren. Diese unterscheiden sich von den linksinduzierten lediglich
dadurch, dass man sie durch die Anwendung von
kontravarianten Funktoren auf Kettenkomplexe erhält.\\
Sei dafür $(C,d,\epsilon)$ ein Komplex über einem
$R$-Modul $M$ und $\mathcal{F}$ ein kontravarianter,
additiver Funktor $\mathcal{F}:\Rm\rightarrow \Ab$, dann erhalten wir
durch dessen Anwendung auf $(C,d,\epsilon)$ einen
Gruppen-Kokettenkomplex
$(\mathcal{F}C,\mathcal{F}d,\mathcal{F}\epsilon)$
$$ \diagram
\mathcal{F}M \rto^{\mathcal{F}\epsilon} &\mathcal{F}C_{0}
\rto^{\mathcal{F}d_{1}} &\mathcal{F}C_{1} \rto^{\mathcal{F}d_{2}}
&\mathcal{F}C_{2} \rto^{\mathcal{F}d_{3}} &...
\enddiagram $$
In der Tat sind mit der Additivität von $\mathcal{F}$ wieder alle Pfeile
Gruppenhomomorphismen und es gilt $\mathcal{F}d_{i+1}\circ
\mathcal{F}d_{i}=\mathcal{F}(d_{i}\circ d_{i+1})=\mathcal{F}(0)=0$. Exaktheit von $(C,d,\epsilon)$ überträgt sich wieder
nicht notwendigerweise auf
$(\mathcal{F}C,\mathcal{F}d\mathcal{F}\epsilon)$ und wir erhalten
nicht-triviale Kohomologie-Gruppen $H^{i}(\mathcal{F}C)$, sowie
$H^{0}(\mathcal{F}C):=\ker(\mathcal{F}d_{1})$. Wir wollen nun ein Analogon zu
\thref{lemma:GruppenhomsausprojaufloesundFunktoren} formulieren und
beweisen.
\begin{lemma}
\label{lemma:GruppenKOhomsausprojaufloesundFunktoren}
Gegeben $R$-Moduln $M,M'$, ein projektiver Komplex
$(C,d,\epsilon)$ über $M$,
eine Auflösung $(C',d',\epsilon')$ von $M'$ und ein
additiver, kontravarianter Funktor $\mathcal{F}:\Rm \rightarrow \Ab$. Sei des Weiteren $\mu$
ein Homomorphismus $\mu:M\rightarrow M'$, dann gilt:
\begin{enumerate}
\item Es existieren
nur von diesen Daten abhängige Gruppenhomomorphismen\\
$\widetilde{\mathcal{F}\alpha_{i}}:H^{i}(\mathcal{F}C')\rightarrow
H^{i}(\mathcal{F}C)$.
\item
Sind
$(C,d,\epsilon))$ und
$(\overline{C},\overline{d},\overline{\epsilon})$
projektive
Auflösungen des selben $R$-Moduls $M$, so gilt $H^{i}(\mathcal{F}C)\simeq H^{i}(\mathcal{F}\overline{C})$.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Zunächst haben wir eine bis auf Homotopie eindeutig
bestimmte Kettenabbildung
$\alpha:(C,d,\epsilon)\rightarrow (C',d',\epsilon')$ derart, dass
$$ \diagram
...\rto^{d_{3}}
&C_{2}\dlto_{s_{2}} \dto_{\alpha_{2}}\rto^{d_{2}}
&C_{1}\dlto_{s_{1}} \dto_{\alpha_{1}}\rto^{d_{1}}
&C_{0}\dlto_{s_{0}}
\dto_{\alpha_{0}}\rto^{\epsilon}|>>\tip &
\dto_{\mu}\\
... \rto^{d'_{3}}
&C'_{2} \rto^{d'_{1}}\rto^{d'_{2}}
&C'_{1} \rto^{d'_{1}} &C'_{0} \rto^{\epsilon'}|>>\tip &M
\enddiagram $$ kommutiert (die schrägen Pfeile sind keine
kommutativen Elemente). Dieses Diagramm übersetzt sich vermöge $\mathcal{F}$ zu
$$ \diagram
\mathcal{F}M'
\dto_{\mathcal{F}\mu}\rto^{\mathcal{F}\epsilon'}
&\mathcal{F}C'_{0}
\dto_{\mathcal{F}\alpha_{0}}\rto^{\mathcal{F}d'_{1}}
&\mathcal{F}C'_{1}\dlto_{\mathcal{F}s_{0}}
\dto_{\mathcal{F}\alpha_{1}}\rto^{\mathcal{F}d'_{2}}
&\mathcal{F}C'_{2}\dlto_{\mathcal{F}s_{1}}
\dto_{\mathcal{F}\alpha_{2}}\rto^{\mathcal{F}d'_{3}} &…\dlto_{\mathcal{F}s_{2}}\\
\mathcal{F}M \rto^{\mathcal{F}\epsilon} &\mathcal{F}C_{0}
\rto^{\mathcal{F}d_{1}} &\mathcal{F}C_{1}
\rto^{\mathcal{F}d_{2}} &\mathcal{F}C_{2} \rto^{\mathcal{F}d_{3}} &…
\enddiagram $$ (schräge Pfeile keine kommutativen Elemente)
und wegen der Addititivität und Kontravarianz von $\mathcal{F}$
erhalten wir für jedes $\beta\sim \alpha$
\begin{equation*}
\mathcal{F}\alpha_{i}-\mathcal{F}\beta_{i}=\mathcal{F}(\alpha_{i}-\beta_{i})=\mathcal{F}(d'_{i+1}s_{i}+s_{i-1}d_{i})=\mathcal{F}
d_{i}\cp \mathcal{F}s_{i-1}+\mathcal{F}s_{i}\cp \mathcal{F}d'_{i+1}.
\end{equation*} Mit den Definitionen
$d^{i}=\mathcal{F}d'_{i+1}$;
$d'^{i}=\mathcal{F}d_{i+1}$; $s^{i}=\mathcal{F}s_{i-1}$ erhalten
wir
\begin{equation*}
\mathcal{F}\alpha_{i}-\mathcal{F}\beta_{i}=d'^{i-1}s^{i}+s^{i+1}d^{i},
\end{equation*}
also $\mathcal{F}\alpha\sim
\mathcal{F}\beta$ nach \eqref{eq:homotopiebedkokett}. Mit der Gruppen-Version von
\thref{lemma:tildeabbeind} folgt abermals
$\widetilde{\mathcal{F}\alpha_{i}}=\widetilde{\mathcal{F}\beta_{i}}$
und somit die erste Aussage.
\item
Wir haben für Kettenabbildungen $\beta:(C,d)\rightarrow
(\overline{C},\overline{d})$ und $\widehat{\beta}:(\overline{C},\overline{d})\rightarrow(C,d)$
\begin{equation*}
id_{\mathcal{F}C_{i}}=\mathcal{F}(id_{C_{i}})\sim
\mathcal{F}(\widehat{\beta}_{i}\circ\beta_{i})
\end{equation*}
und mit \thref{lemma:tildeabbeind}, sowie \thref{lemma:kettenabzu}.\textit{iv.)}
\begin{equation*}
id_{H^{i}(\mathcal{F}C)}=\widetilde{id_{\mathcal{F}C_{i}}}=\widetilde{\mathcal{F}(\widehat{\beta}_{i}\circ\beta_{i})}=\widetilde{\mathcal{F}\beta_{i}\circ\mathcal{F}\widehat{\beta}_{i}}
=\widetilde{\mathcal{F}\beta_{i}}\circ\widetilde{\mathcal{F}\widehat{\beta}_{i}}.
\end{equation*} Analog folgt
\begin{equation*}
id_{H_{i}(\mathcal{F}\overline{C})}=\widetilde{\mathcal{F}\widehat{\beta}_{i}}\circ\widetilde{\mathcal{F}\beta}_{i},
\end{equation*} womit
\begin{equation*}
\widetilde{\mathcal{F}\beta}_{i}=\widetilde{\mathcal{F}\widehat{\beta}_{i}}^{-1},
\end{equation*} also $H^{i}(C)\simeq H^{i}(\overline{C})$
vermöge
$\eta^{i}=\widetilde{\mathcal{F}\beta_{i}}:H^{i}(\overline{C})\rightarrow
H^{i}(C)$.
\end{enumerate}
\end{beweis}
\end{lemma}
\begin{korollar}
\label{kor:Extkor}
Sind in der Situation von
\thref{lemma:GruppenKOhomsausprojaufloesundFunktoren} beide Komplexe
$(C,d,\epsilon)$, $(C',d',\epsilon')$ projektive Auflösungen, so
sind die zugehörigen Homologie-Gruppen $H_{i}(\mathcal{F}C)$,
$H_{i}(\mathcal{F}C')$, sowie die Homomorphismen aus
$\thref{lemma:GruppenhomsausprojaufloesundFunktoren}.i.)$ bereits
bis auf Verkettung mit Isomorphismen eindeutig bestimmt, durch $M,M', \mu$ und
$\mathcal{F}$, hängen also nicht von der Wahl der Auflösungen ab.
\begin{beweis}
Die Isomorphie der Homologie-Gruppen folgt sofort aus Teil
$ii.)$ obigen Lemmas. Mit der Funktoreigenschaft von
$\mathcal{F}$ und $\sim_{i}$ kann dann für Kettenabbildungen
\begin{align*}
\beta:(C,d,\epsilon)&\rightarrow
(\overline{C},\overline{d},\overline{\epsilon})\\
\beta':(C',d',\epsilon')&\rightarrow
(\overline{C}',\overline{d}',\overline{\epsilon}')
\\\alpha:(C,d,\epsilon)&\rightarrow (C',d',\epsilon')
\end{align*} der Homologiegruppenhomomorphismus
\begin{align*}
\tau^{i}:H^{i}(\mathcal{F}\overline{C}')&\longrightarrow H^{i}(\mathcal{F}\overline{C})\\
x&\longrightarrow (\eta^{i})^{-1}\widetilde{\mathcal{F}\alpha_{i}}\eta'^{i}
\end{align*} geschrieben werden als
\begin{equation*}
\tau^{i}=\widetilde{\mathcal{F}(\beta'_{i}\alpha_{i}\widehat{\beta}_{i})}.
\end{equation*} Da nun wieder
$\beta'_{i}\alpha_{i}\widehat{\beta}_{i}$: $(\overline{C},\overline{d},\overline{\epsilon})\rightarrow
(\overline{C}',\overline{d}',\overline{\epsilon}')$ eine
Kettenabbildung ist, folgt die zweite Aussage des Korollars.
\end{beweis}
\end{korollar}
\begin{definition}[Rechtsinduzierte Funktoren]
Gegeben ein unitärer Ring $R$ und ein kontravarianter Funktor
$\mathcal{F}: \Rm\rightarrow \Ab$, so ist der $i$-te rechtsinduzierte
Funktor $R^{i}\mathcal{F}: \Rm\rightarrow \Ab$ definiert durch
\begin{align*}
R^{i}\mathcal{F}^{Ob}: R\text{-Moduln}&\longrightarrow
\text{abelsche Gruppen}\\
M&\longmapsto H^{i}(\mathcal{F}C)
\end{align*}
\begin{align*}
R^{i}\mathcal{F}^{Mor}:\Hom_{R}(M,M') &\longrightarrow \Hom_{G}(H_{i}(\mathcal{F}C'),H_{i}(\mathcal{F}C))\\
\mu&\longmapsto \widetilde{\mathcal{F}\alpha_{i}}.
\end{align*} Dessen Wohldefiniertheit folgt aus
\thref{kor:Extkor}, wobei wir beliebige Auflösungen
$(C,d,\epsilon)$ und $(C',d',\epsilon')$ von $M$ und $M'$ wählen dürfen.
\end{definition}
Wir wollen nun ein Beispiel vorstellen, welches eine große Rolle im nächsten
Kapitel spielen wird.
\begin{beispiel}
\label{bsp:ExtBeisp}
Gegeben ein $R$-Modul $N$, dann ist der additive, kontravariante
Funktor $\mathrm{hom}(\cdot,N)$ definiert durch
\begin{equation}
\label{eq:homfktOb}
\begin{split}
\mathrm{hom}(\cdot,N)^{Obj}:R\text{-Moduln}&\longrightarrow
\text{abelsche Gruppen}\\
M&\longmapsto \Hom_{R}(M,N)
\end{split}
\end{equation}
\begin{equation}
\label{eq:homfktrMor}
\begin{split}
\mathrm{hom}(\cdot,N)^{Mor}:\Hom_{R}(M,M')&\longrightarrow \Hom(\Hom_{R}(M',N),\Hom_{R}(M,N))\\
\mu &\longmapsto
\left[\Hom_{R}(M',N)\ni\alpha\mapsto\alpha\circ \mu\in \Hom_{R}(M,N)\right].
\end{split}
\end{equation}
Die Kontravarianz ist klar und die Additivität
sieht man mit der Bilinearität von $\circ$ bezüglich der Addition
in $\Hom_{R}(M',N)$ und $\Hom_{R}(M,N)$.\\\\
Für unitären Ring $R$ und $R$-Modul $N$ definieren wir den Funktor
\begin{align*}
Ext^{i}_{R}(\cdot,N)=R^{i} \mathrm{hom}(.,N).
\end{align*}
Es ist dann
\begin{equation*}
Ext^{i}_{R}(\cdot,N)(M)=H^{i}(\mathrm{hom}(\cdot,N)C)=H^{i}(\Hom_{R}(C,N)),
\end{equation*} für eine projektive Auflösung $(C,d,\epsilon)$ von
$M$, wobei $\Hom_{R}(C,N)$ den durch Anwendung des $\hom$-Funktors auf
$(C,d,\epsilon)$ entstehenden Gruppen-Kokettenkomplex mit Kettengliedern
$\mathcal{F}C_{i}=\mathrm{hom}(\cdot,N)C_{i}=\Hom_{R}(C_{i},N)$ bezeichnet.\\\\
Als Anwendung, berechnen wir $Ext^{0}_{R}(\cdot,N)(M)$, wir haben
$$ \diagram
...\rto^{d_{3}} &C_{2} \rto^{d_{2}} &C_{1} \rto^{d_{1}}
&C_{0}\rto^{\epsilon}|>>\tip
&M \\
\enddiagram $$
$$ \diagram
\mathrm{Hom}_{R}(M,N)\rto^{\epsilon^{*}}
&\mathrm{Hom}_{R}(C_{0},N)\rto^{d_{1}^{*}} &\mathrm{Hom}_{R}(C_{1},N)\rto &…\\
\enddiagram $$ und es folgt
\begin{align*}
Ext^{0}_{R}(\cdot,N)(M)&
=H^{0}(\Hom_{R}(C,N))=\ker(\overbrace{\mathrm{hom}^{Mor}(\cdot,N)d_{1}}^{d_{1}^{*}}) \\
&=\Menge{\alpha\in \Hom_{R}(C_{0},N)}{\alpha\circ d_{1}=0}\\ &=
\Menge{\alpha\in \Hom_{R}(C_{0},N)}{\ker(\alpha)\supseteq
\im(d_{1})=\ker(\epsilon)},
\end{align*} womit
$Ext^{0}_{R}(\cdot,N)(M)\cong\Hom_{R}(C_{0}/\ker(\epsilon),N)$. Da nun
$C_{0}/\ker(\epsilon)\cong \im(\epsilon)=M$, folgt
$Ext^{0}_{R}(\cdot,N)(M)\cong \Hom_{R}(M,N)$.
\end{beispiel}
\newpage
\section{Berechnung von Hochschild-Kohomologien}
Nachdem wir in \ref{sec:AlgebraischeDefinitionen} die verschiedenen
Facetten des Algebra und Algebramodulbegriffes herausgestellt haben,
wollen wir von nun an unter einer $\mathbb{K}$-Algebra $A$ immer einen $\mathbb{K}$-Vektorraum $\mathcal{A}$ mit assoziativer Algebramultiplikation
verstehen. Im ursprünglichen Sinne waren dies gerade die verträglichen
$\mathbb{K}$-Algebren. Des Weiteren wollen wir im Folgenden unter einem Körper
$\mathbb{K}$ immer $\mathbb{R}$ oder $\mathbb{C}$ verstehen
Sprechen
wir von einem $\mathcal{A}-\mathcal{A}$ Bimodul meinen wir
einen $\mathbb{K}$ Vektorraum mit $\mathcal{A}$ verträglicher,
präkompatibler (also insbesondere kompatibler) $\mathcal{A}-\mathcal{A}$
Bimodulstruktur. Dies bedeutet dann insbesondere immer, dass $\mathcal{A}$
unitär. Der Verständlichkeit halber werden wir jedoch die an gegebener
Stelle wichtigen Eigenschaften nochmals explizit erwähnen. Es sei
zudem darauf hingewiesen, dass wir im Folgenden immer wieder
(insbesondere in Angang \ref{sec:TechnBew})
$\mathbb{C}$-lineare Abbildungen von Tensorprodukten,
bzw. $\mathbb{C}$-bilineare Abbildungen vom kartesischen Produkt
zweier Tensorprodukte in Tensorprodukte mit Hilfe von
\thref{kor:WohldefTensorprodabbildungen} durch deren Bilder auf
elementaren Tensoren definieren werden. Bei den ersten Beispielen
dieses Kapitels führen wir dies noch explizit aus, an späterer Stelle
folgt die Wohldefiniertheit besagter Abbildungen analog.
\subsection{Einführung (vgl. \cite[Kapitel
2.1,5]{Weissarbeit})}
Gegeben eine $\mathbb{K}$-Algebra $\mathcal{A}$ und ein
$\mathcal{A}-\mathcal{A}$ Bimodul $\mathcal{M}$, so betrachten für $k\in \mathbb{Z}$ die $\mathbb{K}$-Vektorräume
\begin{equation*}
HC^{k}(\mathcal{A},\mathcal{M}):=
\begin{cases} \{O_{\mathcal{M}}\} & k<0\\
\mathcal{M} & k=0\\
\Hom_{\mathbb{K}}(\underbrace{\mathcal{A}\times…\times
\mathcal{A}}_{k-mal},\mathcal{M})& k\geq 1,
\end{cases}
\end{equation*} die $k$-multilinearen Abbildungen von
$\mathcal{A}\times…\times \mathcal{A}$ nach $\mathcal{M}$.
Vermöge der Links- und Rechtsmodulstruktur auf $\mathcal{M}$
definieren wir $\mathbb{K}$-lineare
Abbildungen
\begin{equation*}
\delta^{k}\in \Hom_{\mathbb{K}}(HC^{k}(\mathcal{A},\mathcal{M}),HC^{k+1}(\mathcal{A},\mathcal{M}))
\end{equation*}
durch
\begin{equation}
\label{eq:Hochschilddelta}
\begin{split}
(\delta^{k}\phi)(a_{1},…,a_{k})=a_{1}\phi(a_{2},…,a_{k+1})&+\sum_{i=1}^{k}(-1)^{k}\phi(a_{1},…,a_{i}a_{i+1},…,a_{k+1})\\
&+(-1)^{k+1}\phi(a_{1},…,a_{k})a_{k+1}.
\end{split}
\end{equation}
Es folgt $\delta^{k}\cp\delta^{k+1}=0$ und wir erhalten einen
Kokettenkomplex $(HC^{\bullet}(\mathcal{A},\mathcal{M}),\delta)$ mit
$HC^{k}(\mathcal{A},\mathcal{M})$ als $\mathbb{K}$-Moduln und
$\delta^{k}$ als $\mathbb{K}$-Homomorphismen.\\\\
Unter besagten
Vorraussetzungen definieren wir die $k$-te
Hochschildkohomologiegruppe in bekannter Weise durch
\begin{equation*}
HH^{k}(\mathcal{A},\mathcal{M})=\ker(\delta^{k})/\im(\delta^{k-1}).
\end{equation*}
Vermöge der Universellen Eigenschaft des Tensorproduktes erhalten wir
einen Isomorphismus
\begin{equation*}
\ot_{*}:
\Hom_{\mathbb{C}}(\mathcal{A}\times…\times
\mathcal{A},\mathcal{M})\longrightarrow \Hom_{\mathbb{C}}(\mathcal{A}\ot…\ot
\mathcal{A},\mathcal{M}),
\end{equation*}dessen Inverses $\ot^{*}=\ot_{*}^{-1}$ einfach der Pullback mit $\ot$ ist.
Die Tensorvariante von \eqref{eq:Hochschilddelta} ist gegeben durch
\begin{equation}
\label{eq:THochschilddelta}
\begin{split}
(\delta^{k}\phi)(a_{1}\ot…\ot a_{k})=a_{1}\phi(a_{2}\ot…\ot
a_{k+1}) &+\sum_{i=1}^{k}(-1)^{k}\phi(a_{1}\ot…\ot
a_{i}a_{i+1}\ot…\ot a_{k+1})\\
&+(-1)^{k+1}\phi(a_{1}\ot …\ot a_{k})a_{k+1}
\end{split}
\end{equation} und wir erhalten
\begin{equation}
\label{eq:TensorglKetteniso}
\delta^{k}_{\ot}\cp \ot^{k}_{*}= \ot^{k+1}_{*}\cp \delta^{k}_{\times},
\end{equation}womit $\ot_{*}$ ein Kettenisomorphismus zwischen diesen
beiden Kokettenkomplexen ist. Dies bedeutet insbesondere die
Isomorphie beider Kohomologiegruppen und wir dürfen uns im Folgenden
darauf beschränken, die einfacher handhabbaren Tensorvariante des
Hochschildkomplexes zu betrachten.\\\\
Wir wollen nun zunächst einsehen, dass dessen
Kohomologiegruppen erhalten werden können, durch
Anwendung eines Ext-Funktors auf die Algebra $\mathcal{A}$, mithin
\begin{equation*}
HH^{k}(\mathcal{A},\mathcal{M})\cong \mathrm{Ext}_{R}^{k}(\cdot,\mathcal{M})(\mathcal{A})=H^{k}(\mathrm{hom}(\cdot,\mathcal{M})\:C)=H^{k}(\Hom_{R}(C,\mathcal{M})).
\end{equation*} Dabei bezeichnet $C$ eine projektive Auflösung
$(C,d,\epsilon)$ von $\mathcal{A}$ und $R$ einen geschickt zu
wählenden Ring. Mit \thref{bsp:ExtBeisp} folgt dann bereits
$HH^{0}(\mathcal{A},\mathcal{M})\cong \Hom_{R}(\mathcal{A},\mathcal{M})$.
\begin{lemma}
\label{lemma:AewirdzuunitRing}
Gegeben eine assoziative $\mathbb{K}$-Algebra $(\mathcal{A},*)$.
\begin{enumerate}
\item
Die Menge
$\mathcal{A}^{e}=\mathcal{A}\otimes \mathcal{A}$ wird vermöge
der Multiplikation
\begin{equation}
\label{eq:AeRingmultdef}
(a\otimes b) *_{e} (\widetilde{a}\otimes
\widetilde{b}):=(a*\widetilde{a})\otimes
(b*^{opp}\widetilde{b})=(a*\widetilde{a})\otimes
(\widetilde{b}*b)
\end{equation} zu einem Ring. Ist $\mathcal{A}$
unitär, so auch $\mathcal{A}^{e}$ vermöge $1_{\mathcal{A}^{e}}=1_{\mathcal{A}}\ot 1_{\mathcal{A}}$.
\item
Jeder $\mathcal{A}-\mathcal{A}$ Bimodul $\mathcal{M}$ wird durch
\begin{equation*}
(a\otimes b)*_{e} m=a(m b)=(a m)b\quad\quad (a\otimes b)\in \mathcal{A}^{e},\: m\in \mathcal{M}
\end{equation*} zu einem $A^{e}$-Links-Modul.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Die Ringeigenschaft ist klar mit der Assoziativität von
$\mathcal{A}$, ebenso die zweite Aussage. Nicht klar ist
die Wohldefiniertheit von $*_{e}$, dafür definieren
wir die Abbildung
\begin{align*}
*: \mathcal{A}\ot \mathcal{A} \ot \mathcal{A} \ot
\mathcal{A}&\longrightarrow \mathcal{A}\ot \mathcal{A}\\
a\ot b \ot \tilde{a}\ot \tilde{b} & \longmapsto
a\tilde{a}\ot b\tilde{b}
\end{align*} durch lineare Fortsetzung vermöge
\thref{kor:WohldefTensorprodabbildungen} und setzen
$*_{e}= *\cp \cong \cp\ot$ mit $\ot: \mathcal{A}^{e}\times
\Ae\rightarrow \Ae\ot \Ae \cong \mathcal{A}\ot
\mathcal{A}\ot \mathcal{A}\ot \mathcal{A}$. Man beachte, dass wir die
Vektorraumeigenschaft von $\mathcal{A}$ benötigt haben.
\item
Für die Wohldefiniertheit beachte man, dass für festes
$m\in \mathcal{M}$ die Abbildung $*_{m}: a\ot b\rightarrow
amb$ die Bedingungen von \thref{kor:WohldefTensorprodabbildungen} erfüllt, mithin linear auf ganz
$\mathcal{A}^{e}$ fortsetzt. Mit \textit{i.)}
folgt
\begin{align*}
[(a\otimes b) *_{e} (\widetilde{a}\otimes\widetilde{b})]*_{e}m=a\widetilde{a}m\widetilde{b}b=(a\otimes b)*_{e} (\widetilde{a}m\widetilde{b})=(a\otimes b)*_{e}[(\widetilde{a}\otimes\widetilde{b})*_{e}m],
\end{align*} was die Behauptung zeigt.
\end{enumerate}
\end{beweis}
\end{lemma}
\begin{definition}[Bar-Komplex]
Gegeben eine assoziative $\mathbb{K}$-Algebra $\mathcal{A}$, so definieren wir
den Bar-Komplex $(\C,d)$ durch $\mathcal{A}^{e}$-Moduln
\begin{align*}
\C_{k}=\mathcal{A}\otimes \underbrace{\mathcal{A}\otimes …
\otimes \mathcal{A}}_{k-mal} \otimes \mathcal{A}
\end{align*} $\qquad\qquad\qquad\qquad \C_{0}=\mathcal{A}\otimes \mathcal{A},\quad\quad
\C_{1}=\mathcal{A}\otimes\mathcal{A}\otimes \mathcal{A},\quad\quad \C_{2}=\mathcal{A}\otimes\mathcal{A}\otimes \mathcal{A}\otimes \mathcal{A}$ \\\\ mit $\mathcal{A}^{e}$-Multiplikation
\begin{equation}
\label{eq:ModMultiplBarkom}
(a\otimes b)(x_{0}\otimes x_{1}\otimes … \otimes x_{k}\otimes
x_{k+1}):=ax_{0}\otimes x_{1}\otimes … \otimes x_{k}\otimes
x_{k+1}b\quad\quad a\otimes b\in \mathcal{A}^{e}
\end{equation} und $\mathcal{A}^{e}$-Homomorphismen
\begin{align*}
d_{k}:\C_{k}&\longrightarrow \C_{k-1}\\
(x_{0}\otimes … \otimes x_{k+1})&\longmapsto
\sum_{j=0}^{k}(-1)^{j}x_{0}\otimes…\otimes
x_{j}x_{j+1}\otimes…\otimes x_{k+1},
\end{align*} für $k\geq 1$ mit $d_{k}\cp d_{k+1}=0$.
Man beachte, dass die $\C_{k}$ alle nur wünschenswerten Eigenschaften
wie Kompatibilität und $\mathcal{A}$ Verträglichkeit besitzen. Die
Wohldefiniertheit der $d_{k}$ folgt unmittelbar mit
\thref{kor:WohldefTensorprodabbildungen} und die der
Modulmultiplikation analog zu \thref{lemma:AewirdzuunitRing}.
\end{definition}
Für unitäres $\mathcal{A}$ ist $\mathcal{A}^{e}$ unitär und wir
hatten gesehen
(\thref{lemma:unitarereRingModulnhabenfreieAufloesung}), dass es dann
rein abstrakt eine projektive (sogar freie) Auflösung von $\mathcal{A}$ geben
muss. Ein essentielles Beispiel liefert folgendes Lemma.
\begin{lemma}[Bar-Auflösung für unitäre $\mathbb{K}$-Algebren]
\label{lemma:unitAlhabeBarAufloesProj}
Gegeben eine unitäre, assoziative $\mathbb{K}$-Algebra $\mathcal{A}$, so wird
der Bar-Komplex vermöge der Abbildung
\begin{align*}
\epsilon:\C_{0}&\longrightarrow \mathcal{A}\\
(a\otimes b)&\longmapsto ab
\end{align*} zu einer projektiven Auflösung $(\C,d,\epsilon)$ von
$\mathcal{A}$.
\begin{beweis}
Zunächst ist $\epsilon$ ein wohldefinierter $\mathcal{A}^{e}$-Homomorphismus
\begin{equation*}
\epsilon((a\otimes b)(x_{0}\otimes
x_{1}))\glna{\eqref{eq:ModMultiplBarkom}}\epsilon\:(ax_{0}\otimes
x_{1}b)=ax_{0}x_{1}b\glna{\eqref{eq:ModMultiplBarkom}}(a\otimes b)\:x_{0}x_{1}=(a\otimes
b)\:\epsilon(x_{0}\otimes x_{1}).
\end{equation*} Des Weiteren ist $\epsilon$ surjektiv, da
$\epsilon\:(1\otimes a)=a\in \mathcal{A}$ und es gilt zudem
\begin{equation*}
(\epsilon\circ d_{1})(x_{0}\otimes x_{1}\otimes
x_{2})=\epsilon(x_{0}x_{1}\otimes x_{2}-x_{0}\otimes x_{1}x_{2})=0.
\end{equation*} Für die Projektivität reicht es, die
$\mathcal{A}^{e}$-Freiheit jedes $\C_{k}$ zu zeigen. Dafür
beachte man, dass
$\C_{0}\cong \mathcal{A}^{e}$, $\C_{1}\cong
\mathcal{A}^{e}\otimes \mathcal{A}$, $…$, $\C_{k}\cong
\mathcal{A}^{e}\bigotimes^{k-2} \mathcal{A}$, $k\geq
2$ und wir erhalten $\C_{k}\cong \mathcal{A}^{e}\otimes V_{k}$
für $\mathbb{K}$-Vektorräumen $V_{k}$. Dann liefert
die $\mathcal{A}^{e}$-lineare Fortsetzung der Abbildung
\begin{align*}
\tau_{k}:\mathcal{A}^{e}\otimes V_{k}&\longrightarrow
(\mathcal{A}^{e})^{dim(V_{k})}\\
a^{e}\otimes\vec{e}_{j}&\longmapsto (0\times…\times
\underbrace{\widehat{a}}_{j} \times…\times0)\quad\quad \forall\:a^{e}\in \mathcal{A}^{e}
\end{align*} mit $\{\vec{e}_{j}\}_{j\in J}$ eine Basis von
$V_{k}$, einen Isomorphismus $\C_{k}\rightarrow(\mathcal{A}^{e})^{dim(V_{k})}$ und es bleibt die Exaktheit
des Komplexes nachzuweisen. Hierfür betrachteten wir die
Kettenabbildungen
\begin{align*}
h_{k}:\C_{k}&\longrightarrow \C_{k+1}\\
x_{0}\otimes…\otimes x_{k+1}&\longmapsto 1\otimes
x_{0}\otimes…\otimes x_{k+1}\qquad k\geq -1,
\end{align*}für welche wir erhalten, dass
\begin{equation}
\label{eq:Homotbar}
\begin{split}
\epsilon\circ h_{-1}&=id_{A},\\
d_{1}\cp h_{0}+h_{-1}\cp\epsilon &=id_{\C_{0}}, \text{ sowie}\\
d_{k+1}\cp h_{k}+h_{k-1}\cp d_{k}&=id_{\C_{k}} \text{ für
}k\geq 1.
\end{split}
\end{equation} Es folgt für $\alpha\in \ker(d_{k})$ sowie
$k\geq 1$
\begin{equation*}
\alpha=(d_{k+1}\cp h_{k})(\alpha)+(h_{k-1}\cp
d_{k})(\alpha)=(d_{k+1}\cp h_{k})(\alpha)\in im(d_{k+1}),
\end{equation*} analog für $\alpha\in \ker(\epsilon)$.
\end{beweis}
\end{lemma}
\begin{proposition}
\label{prop:barauffuerunitalgebraIsomozuHochschildkohomo}
Gegeben eine unitäre, assoziative $\mathbb{K}$-Algebra $\mathcal{A}$ und bezeichne
$(\C,d,\epsilon)$ die Bar-Auflösung, dann gilt
\begin{equation*}
HH^{k}(\mathcal{A},\mathcal{M})\cong \mathrm{Ext}^{k}_{\mathcal{A}^{e}}(\cdot,
\mathcal{M})(\mathcal{A})=H^{k}(\mathrm{hom}(\cdot,\mathcal{M})\C)=H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M})),
\end{equation*} für einen $\mathcal{A}$ verträglichen,
präkompatiblen $\mathcal{A}-\mathcal{A}$ Bimodul $\mathcal{M}$.
\begin{beweis}
Sowohl $(HC^{\bullet}(\mathcal{A},\mathcal{M}),\delta)$, als auch
$(\C^{*},d^{*})=(\mathrm{hom}(\cdot,\mathcal{M})^{Ob}(\C),\mathrm{hom}(\cdot,\mathcal{M})^{Mor}(d))$
sind Komplexe von $\mathbb{K}$-Moduln.\\\\
Mit den Abbildungen
\begin{align*}
\Xi^{k}:\Hom_{\mathcal{A}^{e}}(\C_{k},\mathcal{M})&\longrightarrow
HC^{k}(\mathcal{A},\mathcal{M})\\
\psi &\longmapsto \left(\widetilde{\psi}:(x_{1}\otimes…\otimes x_{k})\mapsto
\psi(1\otimes x_{1}\otimes…\otimes
x_{k}\otimes1)\right),
\end{align*} erhalten wir mit der Präkompatibilität und
$\mathcal{A}$-Verträglichkeit von $\mathcal{M}$
\begin{align*}
\Xi^{k}(\psi)(\lambda\: x_{1}\otimes…\otimes
x_{k})&=\psi\:(1\otimes x_{1}\otimes…\ot \lambda x_{i}\ot…\otimes
x_{k}\otimes1)\\ &=\psi\:(\lambda 1\otimes x_{1}\otimes…\otimes
x_{k}\otimes1)\\ & =\lambda 1\ot 1 *_{e}\psi\:(1\otimes x_{1}\otimes…\otimes
x_{k}\otimes1)\\ &=\lambda\: \psi\:(1\otimes x_{1}\otimes…\otimes
x_{k}\otimes1)\\ &=\lambda\: \Xi^{k}(\psi)(x_{1}\otimes…\otimes
x_{k}).
\end{align*} Damit bilden die $\Xi^{k}$ in der Tat in die behauptete
Menge ab und da
\begin{equation*}
\Xi^{k}(\lambda \psi+\phi)=\lambda\:
\Xi^{k}(\psi)+ \Xi(\phi),
\end{equation*}sind diese zudem
$\mathbb{K}$-Homomorphismen.
Nun ist die Injektivität obiger
Abbildung klar mit der $\mathcal{A}^{e}$-Linearität der
Urbilder. Für die Surjektivität betrachten wir ein\\
$\widetilde{\psi}\in HC^{k}(\mathcal{A},\mathcal{M})$ und definieren
\begin{equation*}
\Hom_{\mathcal{A}^{e}}(\C_{k},\mathcal{M})\ni\psi(x_{0}\otimes
x_{1}\otimes…\otimes x_{k}\otimes
x_{k+1})=x_{0}\widetilde{\psi}(x_{1}\otimes…\otimes
x_{k})x_{k+1},
\end{equation*}
womit $\Xi^{k}(\psi)=\widetilde{\psi}$ und somit die
$\Xi^{k}$ $\mathbb{K}$-Isomorphismen sind. Es folgt
\begin{equation}
\label{eq:isobarHsch}
\Xi^{k+1}d^{*}_{k+1}=\delta^{k}\:\Xi^{k}
\end{equation} und mit der $\mathbb{K}$-Linearität der $ \Xi^{k}$ zeigt \thref{lemma:kettenabzu}.\textit{iii)}, dass die
$\widetilde{\Xi^{k}}$ $\mathbb{K}$-Isomorphismen auf den
Kohomomologie-Gruppen sind.
In der Tat erhalten wir \eqref{eq:isobarHsch} mit
{\allowdisplaybreaks
\begin{align*}
\left[\left(\Xi^{k+1}\cp d_{k+1}\right)(\psi)\right](x_{1}\ot…\ot
x_{k+1})=\:&(d_{k+1}\psi)(1\ot x_{1}\ot…\ot x_{k+1}\ot 1)
\\ =\:& \psi(x_{1}\ot…\ot x_{k+1}\ot 1)
\\ &+\sum_{j=1}^{k}(-1)^{j}
\psi(1\ot x_{1}\ot…\ot x_{j}x_{j+1}\ot…\ot x_{k+1}\ot 1)
\\ &+ (-1)^{k+1}\psi(1\ot…\ot x_{k+1})
\\=\:& x_{1}\psi(1\ot x_{2}\ot…\ot x_{k+1}\ot 1)
\\ &+\sum_{j=1}^{k}(-1)^{j}
\psi(1\ot x_{1}\ot…\ot x_{j}x_{j+1}\ot…\ot x_{k+1}\ot 1)
\\ &+ (-1)^{k+1}\psi(1\ot x_{1}\ot…\ot x_{k}\ot 1)x_{k+1}
\\=\:& x_{1}\left(\Xi^{k}\cp\psi\right)(x_{2}\ot…\ot x_{k+1})
\\ &+ \sum_{j=1}^{k}(-1)^{k}\left(\Xi^{k}\cp\psi\right)(x_{1}\ot…\ot
x_{j}x_{j+1}\ot…\ot x_{k+1})
\\ &+ (-1)^{k+1}\left(\Xi^{k}\cp \psi\right)(x_{1}\ot…\ot x_{k})x_{k+1}
\\ =& \left[\left(\delta^{k}\cp\Xi^{k}\right)(\psi)\right](x_{1}\ot…\ot x_{k+1}).
\end{align*}}
\end{beweis}
\end{proposition}
\newpage
\subsection{Die Hochschild-Kohomologie der Algebra
$\Poly$}
\label{subsec:HschKPol}
Wir wollen als erstes einfaches Beispiel die
Hochschild-Kohomologie der $\mathbb{R}$-Algebra $\Poly$, der Polynome
auf $\mathbb{R}^{n}$ berechnen. Diese ist sicher unitär und assoziativ und wir haben gemäß
\thref{lemma:unitAlhabeBarAufloesProj} und
\thref{prop:barauffuerunitalgebraIsomozuHochschildkohomo} bereits eine
projektive Auflösung, deren Kohomologiegruppen isomorph sind zu den
gesuchten Hochschild-Kohomologien.\\
Als nächstes wollen wir uns eine
weitere projektive Auflösung $(C',d',\epsilon')$ von $\mathcal{A}$ verschaffen und wissen
bereits, dass dann aus abstrakten Gründen
\begin{equation*}
H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}))\cong \mathrm{Ext}^{k}_{\mathcal{A}^{e}}(\cdot,
\mathcal{M})(\mathcal{A})\cong
H^{k}(\Hom_{\mathcal{A}^{e}}(C',\mathcal{M})).
\end{equation*}
Alle in diesem Abschnitt auftretenden Tensorprodukte sind klarerweise
als Tensorprodukte über dem körper $\mathbb{R}$ aufzufassen.
\begin{definition}[Koszul-Komplex]
Wir definieren für $\mathcal{A}=\Poly$ den Koszul-Komplex
$(\K,\partial)$ durch $\mathcal{A}^{e}$-Moduln
\begin{equation*}
\K_{0}=\mathcal{A}^{e}\text{ sowie }
\K_{k}=\mathcal{A}^{e}\otimes \Lambda^{k}(\mathbb{R}^{n*})\quad\forall\:k>0,
\end{equation*} versehen mit der offensichtlichen
$\mathcal{A}^{e}$-Multiplikation im ersten Faktor. Es ist dann klarerweise
$\K_{k}=0$, falls $k>n$ und wir definieren für $0< k\leq n$ die
$\mathcal{A}^{e}$-Homomorphismen
\begin{align*}
\partial_{k}:\K_{k}&\longrightarrow \K_{k-1}\\
\omega&\longmapsto \left[(v,w)(x_{1},…,x_{k-1})\mapsto \omega(v,w)((v-w),x_{1},…,x_{k-1})\right].
\end{align*}Es folgt unmittelbar $\pt_{k}\cp
\pt_{k+1}=0$ und man beachte zudem, dass die $\pt_{k}$ mit Hilfe
der Einsetzabbildung $i_{a}(v,\lambda)(x_{2},…,x_{k})=\lambda(v,x_{2},…,x_{k})$
\begin{align*}
&i_{a}:\mathbb{R}^{n}\times \Lambda^{k}(\mathbb{R}^{n*})\longrightarrow
\Lambda^{k-1}(\mathbb{R}^{n*})\\
&(v, \lambda^{1}\wedge…\wedge \lambda^{k})\longmapsto
\sum_{l=1}^{k}(-1)^{l-1}\lambda^{l}(v)\:\lambda^{1}\wedge…\blacktriangle^{l}…\wedge \lambda^{k}
\end{align*}
auch geschrieben werden können als
\begin{equation*}
\pt_{k}=\sum_{j=1}^{n}\xi^{j}i_{a}(\vec{e}_{j},\cdot)\quad\text{
mit }\quad \mathcal{A}^{e}\ni\xi^{j}=x^{j}\otimes 1-1\otimes
x^{j}.
\end{equation*}
\end{definition}
\begin{lemma}[Koszul-Auflösung für $\Poly$]
Sei $\mathcal{A}=\Poly$, so wird der Koszulkomplex
vermöge der Abbildung
\begin{align*}
\epsilon:\K_{0}&\longrightarrow \mathcal{A}\\
(a\otimes b)&\longmapsto ab
\end{align*} zu einer projektiven Auflösung $(\K,\partial,\epsilon)$
von $\Poly$.
\begin{beweis}
Wir hatten bereits gesehen, dass $\epsilon$ ein surjektiver
$\mathcal{A}^{e}$-Homomorphismus ist. Die
Projektivität der $\K_{k}$ folgt ebenso wie für die
Bar-Auflösung, da die $\Lambda^{k}(\mathbb{R}^{n*})$
ebenfalls Vektorräume $V$ mit Basen sind.
Für die Exaktheit definieren wir die Abbildungen
\begin{align*}
h_{-1}:\mathcal{A}&\longrightarrow
\K_{0}\\
p&\longmapsto [(v,w)\mapsto p(w)],
\end{align*} sowie $h_{k}:\K_{k}\longrightarrow
\K_{k+1}$, $k\geq 0$ durch
\begin{equation}
\label{eq:exakthAbbHvonPol}
\begin{split}
h_{k}(\omega)(v,w)&=\sum_{j=1}^{n}e^{j}\wedge\int_{0}^{1}dt\:
t^{k}\frac{\partial\omega}{\partial v^{j}}(tv+(1-t)w,w)\\ &=\frac{1}{k!}
\sum_{i_{1},…,i_{k},j=1}^{n}\int_{0}^{1}dt\:
t^{k}\frac{\partial\omega_{i_{1},…,i_{k}}}{\partial
v^{j}}(tv+(1-t)w,w)\:e^{j}\wedge e^{i_{1}}\wedge
…\wedge e^{i_{k}}\nonumber,
\end{split}
\end{equation} wobei
\begin{equation*}
\omega=\frac{1}{k!}
\sum_{i_{1},…,i_{k}}^{n}\omega_{i_{1},…,i_{k}}\ot
e^{i_{1}}\wedge…\wedge e^{i_{k}}\quad\text{ und }\quad
\omega_{i_{1},…,i_{k}}\in \mathcal{A}^{e}.
\end{equation*} Zunächst überzeugt man sich, dass besagte
Abbildungen in der Tat nach
\begin{equation*}
\mathrm{Pol}(\mathbb{R}^{n})\ot\mathrm{Pol}(\mathbb{R}^{n})\ot
\Lambda^{k}(\mathbb{R}^{n}) =
\mathrm{Pol}(\mathbb{R}^{n}\times\mathbb{R}^{n})\ot \Lambda^{k}(\mathbb{R}^{n*})
\end{equation*}
abbilden, denn es ist ja jedes
$\frac{\partial\omega_{i_{1},…,i_{k}}}{\partial v^{j}}$ als Ableitung eines
Polynoms nach den ersten Argumenten wieder ein
Polynom in $\mathbb{R}^{n}\times \mathbb{R}^{n}$.
Ebenso haben wir $p(t\vec{x}+(1-t)\vec{y})\in
\mathrm{Pol}(\mathbb{R}\times \mathbb{R}^{n}\times \mathbb{R}^{n})$ für $p\in \mathrm{Pol}(\mathbb{R}^{n}\times \mathbb{R}^{n})$
und die Integration ist nichts weiter als die
$\mathcal{\mathbb{R}}$-lineare Fortsetetzung der Abbildung
\begin{equation*}
\int_{0}^{1}dt\:t^{k}: t^{l}x^{m}\longmapsto
\frac{1}{l+k+1}x^{n}.
\end{equation*}
Behändiges rechnen unter Verwendung der Derivationeigenschaft
\begin{equation*}
i_{a}(v)(\phi\wedge \psi )=i_{a}(v)(\phi)\wedge \psi +(-1)^{deg(\phi)}\phi\wedge i_{a}(v)(\psi)
\end{equation*}
zeigt
\begin{align*}
\epsilon\circ h_{-1}&=id_{\mathcal{A}}, \\
h_{-1}\circ\epsilon+\partial_{1}\circ h_{0}&=id_{\K_{0}}\quad\quad\text{und}\\
h_{k-1}\circ \partial_{k}+\partial_{k+1}\circ h_{k}&=id_{\K_{k}}\quad\quad k\geq1,
\end{align*}
mithin die Exaktheit von
$(\K,\partial,\epsilon)$ (siehe auch \cite[Kapitel 5]{Weissarbeit}).
\end{beweis}
\end{lemma}
Mit der in Kapitel \ref{sec:HomologAlgebr} bereitgestellten Abstrakte folgt:
\begin{satz}
\label{satz:PolsatzHochsch}
Sei $\mathcal{A}=\Poly$ und $\mathcal{M}$
ein $\mathcal{A}$ verträglicher, präkompatibler
$\mathcal{A}-\mathcal{A}$ Bimodul, so gilt
\begin{equation}
\label{eq:HochschPol11}
HH^{k}(\mathcal{A},\mathcal{M})\cong
H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}))\cong
H^{k}(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{M})).
\end{equation}
Ist $\mathcal{M}$ zudem symmetrisch, so folgt
\begin{equation}
\label{eq:HochschPol22}
HH^{k}(\mathcal{A},\mathcal{M})\cong \Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})=\mathcal{M}\ot\Lambda^{k}(\mathbb{R}^{n}).
\end{equation}
\begin{beweis}
\eqref{eq:HochschPol11} ist klar. Für
\eqref{eq:HochschPol22} betrachten wir den Komplex
$(\K^{*},\partial^{*},\epsilon^{*})$,
der durch Anwendung des
$\mathrm{hom}(\cdot,\mathcal{M})$-Funktors auf $(\K,\partial,\epsilon)$
gewonnen wird.\\\\
Sei nun $\phi\in
\K^{*}_{k}=\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})$ und
$\omega=\sum_{i_{1},…,i_{k+1}}\omega_{i_{1},…,i_{k+1}}\ot e^{i_{1}}\wedge…\wedge
e^{i_{k+1}}\in \K_{k+1}$, dann folgt für
$\pt^{*}_{k+1}:\K^{*}_{k}\longrightarrow \K^{*}_{k+1}$
\begin{align*}
(\partial^{*}_{k+1}\phi)(\omega)\glna{\eqref{eq:homfktrMor}}(\phi\circ
\partial_{k+1})(\omega)&=\phi\left(\partial_{k+1}\left(\sum_{i_{1},…,i_{k+1}}\omega_{i_{1},…,i_{k}}\ot
u^{i_{1}}\wedge…\wedge
u^{i_{k+1}}\right)\right)\\ &=
\sum_{i_{1},…,i_{k+1}}\phi\left(\sum_{j=1}^{n}\xi^{j}*_{e}\omega_{i_{1},…,i_{k+1}}\ot
i_{a}\left(\vec{e}_{j},u^{i_{1}}\wedge…\wedge
u^{i_{k+1}}\right)\right)\\
&=\sum_{i_{1},…,i_{k+1}}\sum_{j=1}^{n}\xi^{j}*_{e}\phi\left(\omega_{i_{1},…,i_{k+1}}\ot
i_{a}\left(\vec{e}_{j},u^{i_{1}}\wedge…\wedge
u^{i_{k+1}}\right)\right) \\ &= \sum_{i_{1},…,i_{k+1}}\sum_{j=1}^{n}\:(x^{j}\otimes
1-1\otimes x^{j})*_{e}\phi\left(\omega_{i_{1},…,i_{k+1}}\:i_{a}\left(\vec{e}_{j},u^{i_{1}}\wedge…\wedge
u^{i_{k+1}}\right)\right)\\ &=0,
\end{align*} nach Vorraussetzung.
Damit ist $\im(\partial^{*}_{k+1})=0\:\forall\: k\geq 0$, mithin
$\ker(\partial^{*}_{k+1})= \K^{*}_{k}$ und somit
$H^{k}(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{A})\cong\Hom_{\mathcal{A}^{e}}(\K_{i},\mathcal{A})$.\\\\
Für die letzte Gleichheit erinnern wir, dass
$\Lambda^{k}(\mathbb{R}^{n*})^{*}=\Lambda^{k}(\mathbb{R}^{n})$
und erhalten für $\phi \in \Hom_{A^{e}}(\K_{k},\mathcal{M})$
{\allowdisplaybreaks
\begin{align*}
\phi(a^{e}\ot \lambda)=\:&a^{e}*_{e}\phi\left(1^{e} \ot
\left(\sum_{j_{1},…,j_{k}=1}^{n}\lambda_{j_{1},…,j_{k}}e^{j_{1}}\wedge…\wedge
e^{j_{k}}\right)\right)
\\=\:& a^{e}*_{e}
\left[\sum_{j_{1},…,j_{k}}\lambda_{j_{1},…,\lambda_{k}}\ot
1*_{e}\phi\:(1^{e}\ot e^{j_{1},…,j_{k}})\right]
\\=\:&
a^{e}*_{e}\sum_{j_{1},…,j_{k}=1}^{n}\lambda_{j_{1},…,j_{k}}\:\phi^{j_{1},…,j_{k}}
\\=\:& a^{e}*_{e} \left(\sum_{j_{1},…,j_{k}=1}^{n} \phi^{j_{1},…,j_{k}}\ot \vec{e}_{j_{1}}\wedge…\wedge
\vec{e}_{j_{k}}\right)(1^{e}\ot\lambda)
\\=\:& \left(\sum_{j_{1},…,j_{k}=1}^{n} \phi^{j_{1},…,j_{k}}\ot \vec{e}_{j_{1}}\wedge…\wedge
\vec{e}_{j_{k}}\right)(a^{e}\ot\lambda).
\end{align*}} und mit der Endlichkeit der Summe in der Tat
\begin{equation*}
\left(\sum_{j_{1},…,j_{k}=1}^{n} \phi^{j_{1},…,j_{k}}\ot e_{j_{1}}\wedge…\wedge
e_{j_{k}}\right)\in \mathcal{M}\ot \Lambda^{k}(\mathbb{R}^{n}).
\end{equation*} Dabei haben wir im zweiten Schritt sowohl die
$\mathcal{A}$-Verträglichkeit, als auch die Präkompatibilität von
$\mathcal{M}$ benutzt. Die letzten beiden Schritte folgen mit
der Konvention
\begin{equation}
\label{eq:Multi}
(m\ot \mu)(a^{e}\ot
\lambda):=a^{e}*_{e}[\lambda(\mu)\: m].
\end{equation}
Umgekehrt ist klar, dass jedes Element aus
$\mathcal{M}\ot \Lambda^{k}(\mathbb{R}^{n})$ vermöge \eqref{eq:Multi} ein Element in
$\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})$ definiert.
\end{beweis}
\end{satz}
\begin{bemerkung}
Für einen expliziten Isomorphismus benötigen wir zunächst eine zur
Identität ($\mu=id_{A^{e}}$) gehörige Kettenabbildung
\begin{equation}
\label{eq:Gpol}
G:(\C,d)\rightarrow (\K,\partial)
\end{equation} oder
\begin{equation}
\label{eq:FPol}
F:(\K,\partial)\rightarrow (\C,d).
\end{equation}
Aus dieser
erhalten wir durch anwenden des
$\mathrm{hom}(\cdot,\mathcal{M})$-Funktors, Isomorphismen
\begin{align*}
\widetilde{G_{k}^{*}}:&H^{k}(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{M})\longrightarrow
H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M})\\
\widetilde{F_{k}^{*}}:&H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M})\longrightarrow
H^{k}(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{M}).
\end{align*} Es ist dann unmittelbar klar, dass
$\wt{G_{k}^{*}}^{-1}=\wt{F_{k}^{*}}$ und die Verkettung von $\wt{G_{k}^{*}}$ mit
\begin{equation*}
\wt{\Xi^{k}}:H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M})\rightarrow
HH^{k}(\mathcal{A},\mathcal{M})
\end{equation*}
liefert den gewünschten Isomorphismus nach $HH^{k}(\mathcal{A},\mathcal{M})$. Explizite Kettenabbildungen sind Beispielsweise
gegeben durch (siehe \cite[Kapitel 5/Anhang C]{Weissarbeit})
\begin{align*}
G_{k}&: \bigotimes^{k+2}\Poly \longrightarrow
\mathrm{Pol}(\mathbb{R}^{n} \times \mathbb{R}^{n})\otimes
\Lambda^{k}(\mathbb{R}^{n*})\\ (G_{k}p)(v,w)&=
\sum_{i_{1},…,i_{k}}^{n}e^{i_{1}}\wedge…\wedge
e^{i_{k}}\int_{0}^{1}dt_{1}\int_{0}^{t_{1}}dt_{2}...\int_{0}^{t_{k-1}}dt_{k}\\
&\frac{\partial
p}{\partial q_{1}...\partial q_{k}}(v,t_{1}v+(1-t_{1})w,…,t_{k}v+(1-t_{k})w,w),
\end{align*} mit $\partial_{k+1}\cp G_{k+1}=G_{k}\cp d_{k+1}$ und $G_{k}=0$,
falls $k>n$, sowie $G_{0}=id_{A^{e}}$.
\begin{align*}
F_{k}: \mathrm{Pol}(\mathbb{R}^{n} \times \mathbb{R}^{n})\otimes
\Lambda^{k}(\mathbb{R}^{n*})&\longrightarrow \bigotimes^{k+2}\Poly
\\ \omega & \longmapsto [(v,w)(x_{1},…,x_{k})\mapsto \omega(v,w)(x_{1}-v,…,x_{k}-v)],
\end{align*} mit $d_{k+1}\cp F_{k+1}=F_{k}\cp \partial_{k+1}$ und $
F_{k}=0$, für $k>n$, sowie $F_{0}=id_{A^{e}}$.
\end{bemerkung}
\newpage
\subsection{Die Hochschild-Kohomologie der Algebra
$\Ss^{\bullet}(\mathbb{V})$}
\label{subsec:HochschKohSym}
Als Abstraktion von $\Poly$ betrachten wir nun
die unitäre $\mathbb{C}$-Algebra $\sym$, für beliebigen $\mathbb{C}$-Vektorraum
$\mathbb{V}$. Diese ist die gradierte Menge
\begin{align*}
\bigoplus_{k=0}^{\infty}\mathrm{S}^{k}(\mathbb{V}),
\end{align*} versehen mit der kommutativen, assoziativen
Algebramultiplikation
\begin{align*}
\vee: \symd{l}\times\symd{m} &\longrightarrow \symd{l+m}\\
(\alpha,\beta)&\longmapsto \alpha\vee\beta.
\end{align*} Dabei bezeichnet $\mathrm{S}^{k}(\mathbb{V})$ die Menge aller
symmetrischen k-fachen Tensoren von $\mathbb{V}$ und
$\alpha\vee \beta = S(\alpha\ot \beta)$ mit
\begin{align*}
S: \Tt^{k}(\V)&\longrightarrow \symd{k}\\
\alpha_{1}\ot…\ot \alpha_{k} &\longmapsto \frac{1}{k!}\sum_{\sigma\in
S^{k}}\alpha_{\sigma(1)}\ot…\ot \alpha_{\sigma(k)}.
\end{align*}
Wir benutzen die Konvention
$\alpha_{1}\vee…\vee \alpha_{k}=\frac{1}{k!}\displaystyle\sum_{\sigma\in
S^{k}}\alpha_{\sigma{1}}\ot…\ot \alpha_{\sigma(k)}$ was den Vorteil
hat, dass
\begin{equation}
\label{eq:Symtioll}
(\alpha_{1}\vee…\vee \alpha_{l})\: \boldsymbol{\vee}\:
(\beta_{1}\vee…\vee \beta_{m})= \alpha_{1}\vee…\vee \alpha_{l}\vee
\beta_{1}\vee …\vee\beta_{m}.
\end{equation}
In weiser Voraussicht setzen wir analog $u_{1}\wedge…\wedge u_{k}=\frac{1}{k!}\displaystyle\sum_{\sigma\in
S^{k}} \mathrm{sign}(\sigma)\text{ }u_{\sigma(1)}\ot…\ot u_{\sigma(k)}$, sowie
$u\wedge \overline{u}= A(u\ot \overline{u})$ mit
\begin{align*}
A: \Tt^{k}(\V)&\longrightarrow \Lambda^{k}(\V)\\
u_{1}\ot…\ot u_{k} &\longmapsto \frac{1}{k!}\sum_{\sigma\in
S^{k}}\sign(\sigma)\:u_{\sigma(1)}\ot…\ot u_{\sigma(k)},
\end{align*}
es gilt dann \eqref{eq:Symtioll} Sinngemäß mit $\wedge$ anstelle von $\vee$.\\\\
Das Einselement $\symd{0}$ erhält man vermöge
\begin{equation*}
1_{\mathbb{C}}\otimes \hat{\mathbb{V}} \cong \hat{\mathbb{V}} \cong
\hat{\mathbb{V}}\otimes 1_{\mathbb{C}},
\end{equation*} für beliebigen $\mathbb{C}$-Vektorraum
$\hat{\mathbb{V}}$. Man setzt dann $1_{\Ss^{\bullet}(\V)}=1_{\mathbb{C}}$
und definiert
\begin{equation*}
1\vee \alpha := (\cong\cp
\vee)(1_{\mathbb{C}},\alpha)=\alpha\qquad\qquad \forall\:
\alpha\in \symd{k}
\end{equation*} mit $\cong$ der zu $\hat{\mathbb{V}}=\symd{k}$
gehörige Isomorphismus.\\\\
A priori haben wir die Existenz einer Bar-Auflösung $(\C,d,\epsilon)$ von
$\sym$ und die Isomorphie $HH^{k}(\mathcal{A},\mathcal{M})\cong
H^{k}(\Hom_{R}(\C,\mathcal{M}))$ für präkompatiblen, $\mathcal{A}$ verträglichen Bimodul $\mathcal{M}$. Was wir noch benötigen ist
lediglich eine Koszulauflösung $(\K,\partial,\epsilon)$ für $\Ss^{\bullet}(\V)$.\\\\ Zu diesem Zwecke seien die $\K_{k}$ wie in
\ref{subsec:HschKPol} gegeben durch
$\mathcal{A}^{e}$-Moduln
\begin{equation*}
\K_{0}=\mathcal{A}^{e}\qquad\text{ und }\qquad
\K_{k}=\mathcal{A}^{e}\otimes \Lambda^{k}(\mathbb{V})\:\forall\:k>0
\end{equation*} mit der bekannten $\mathcal{A}^{e}$-Multiplikation im
ersten Faktor.
Die Kettenabbildungen $\partial_{k}$ seien dann wie
folgt definiert:
\begin{definition}[Für die Bezeichnungsweisen siehe Anfang Anhang \ref{sec:TechnBew}]
\begin{align*}
\partial_{k}: \mathcal{A}^{e}\ot
\Lambda^{k}(\mathbb{V})&\longmapsto \mathcal{A}^{e}\ot
\Lambda^{k-1}(\mathbb{V})\\
\omega&\longmapsto
\big[\pt^{k}_{1}-\pt^{k}_{2}\big](\omega)
\end{align*}
mit
\begin{align*}
\pt_{1}^{k}:S^{l}(\mathbb{V})\otimes
S^{m}(\mathbb{V})\otimes\Lambda^{k}(\mathbb{V})&\longrightarrow S^{l+1}(\mathbb{V})\otimes
S^{m}(\mathbb{V})\otimes\Lambda^{k-1}(\mathbb{V})\\
\KE{\alpha}{\beta}{u}&\longmapsto
\sum_{j=1}^{k}(-1)^{j-1}\:\KE{u_{j}\vee
\alpha}{\beta}{u^{j}}\\
\pt_{2}^{k}:S^{l}(\mathbb{V})\otimes
S^{m}(\mathbb{V})\otimes\Lambda^{k}(\mathbb{V})&\longrightarrow S^{l}(\mathbb{V})\otimes
S^{m+1}(\mathbb{V})\otimes\Lambda^{k-1}(\mathbb{V})\\
\KE{\alpha}{\beta}{u}&\longmapsto
\sum_{j=1}^{k}(-1)^{j-1}\:\KE{\alpha}{u_{j}\vee\beta}{u^{j}},
\end{align*} wobei $\partial^{k}\cp \partial^{k+1}=0$ nach
\thref{lemma:EigenschHomotBaukloetze}.\textit{iii.)}.
\end{definition} Vermöge
\begin{align*}
\epsilon:\mathcal{A}\otimes \mathcal{A}&\longrightarrow \mathcal{A}\\
\alpha\otimes \beta&\longmapsto \alpha\vee \beta
\end{align*} wird $(\K,\partial,\epsilon)$ zu einem
projektiven Komplex über $\mathcal{A}$ und es bleibt dessen Exaktheit
nachzuweisen. Hierfür definieren wir die abstrakte Variante von
\eqref{eq:exakthAbbHvonPol} durch:
\begin{definition}
\label{def:KosSymExakthHomothidelta}
Wir definieren
\begin{align*}
h_{-1}: \mathcal{A}&\longrightarrow \K_{0}\\
\alpha &\longmapsto 1\ot\alpha
\end{align*}
und für $k\geq 0$
\begin{align*}
h_{k}: S^{\bullet}(\mathbb{V})\otimes
S^{\bullet}(\mathbb{V})\otimes\Lambda^{k}(\mathbb{V})&\longrightarrow S^{\bullet}(\mathbb{V})\otimes
S^{\bullet}(\mathbb{V})\otimes\Lambda^{k+1}(\mathbb{V})\\
\mu
&\longmapsto\int_{0}^{1}dt\:t^{k}(i_{t}\circ\delta)(\mu)
\end{align*}
vermöge $\mathbb{C}$-linearer Fortsetzung von:
\begin{enumerate}
\item
\begin{align*}
i_{t}\KE{\alpha}{\beta}{u}=&\Big[t\longmapsto \:t^{l}\KE{\alpha}{\beta}{u}\:+\:t^{l-1}(1-t)\sum_{j=1}^{l}\KE{\alpha^{j}}{\alpha_{j}\vee\beta}{u}\:+…\\
&+t^{l-s}(1-t)^{s}\sum_{j_{1},…,j_{s}}^{l}\KE{\alpha^{j_{1},…,j_{s}}}{\alpha_{j_{1},…,j_{s}}\vee\beta}{u}\:+…\\
& + (1-t)^{l}\KE{1}{\alpha\vee \beta}{u}\Big]
\end{align*} mit $\deg(\alpha)=l$ und $i_{t}(1\ot \beta\ot
u)=(1\ot \beta\ot u)$.
\item
\begin{align*}
\delta:S^{l}(\mathbb{V})\otimes S^{m}(\mathbb{V})\otimes
\Lambda^{k}(\mathbb{V})&\longrightarrow S^{l-1}(\mathbb{V})\otimes S^{m}(\mathbb{V})\otimes \Lambda^{k+1}(\mathbb{V})\\
\KE{\alpha}{\beta}{u}&\longmapsto
\sum_{j=1}^{l}\KE{\alpha^{j}}{\beta}{\alpha_{j}\wedge u}
\end{align*} mit $\delta(1\ot\beta\ot u)=0$.
\end{enumerate}
\end{definition} \thref{prop:ExaktheitsbewSym} zeigt
\begin{align*}
\epsilon\circ h_{-1}&=id_{\mathcal{A}}, \\
h_{-1}\circ\epsilon+\partial_{1}\circ h_{0}&=id_{\K_{0}}\quad\quad\text{und}\\
h_{k-1}\circ \partial_{k}+\partial_{k+1}\circ h_{k}&=id_{\K_{k}}\quad\quad k\geq1,
\end{align*} also die Exaktheit von $(\K,\partial,\epsilon)$. Wie in Abschnitt \ref{subsec:HschKPol} folgt
\begin{equation*}
HH^{k}(\mathcal{A},\mathcal{M})\cong
H^{k}(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}))\cong H^{k}(\Hom_{\mathcal{A}^{e}}(\K,\mathcal{M}))
\end{equation*} und falls $\mathcal{M}$ symmetrisch zudem
\begin{equation*}
HH^{k}(\mathcal{A},\mathcal{M})\cong\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M}),
\end{equation*} da für $\phi\in
\K^{*}_{k}=\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})$ und
$\omega=\alpha\ot \beta \ot u \in \K_{k+1}$
\begin{align*}
(\partial^{*}_{k+1}\phi)(\omega)=&\:\phi(\partial_{k+1}(\alpha\ot
\beta \ot u))\\=&\:
\phi\left(\sum_{j=1}^{n}(-1)^{j-1}\left[u_{j}\vee \alpha
\ot \beta \ot u^{j}- \alpha
\ot u_{j}\vee\beta \ot u^{j} \right]\right)
\\=&\sum_{j=1}^{n}(-1)^{j-1} [u_{j}\ot 1 - 1\ot u_{j}]*_{e} \phi\left(\alpha
\ot \beta \ot u^{j}\right)
\\=&\:0.
\end{align*}
Ohne zusätzliche Annahmen über $\mathbb{V}$ und
$\Hom_{\mathcal{A}^{e}}(\K_{k},\mathcal{M})$ erhalten wir jedoch im
Allgemeinen kein Analogon zur letzten Gleichheit in \thref{satz:PolsatzHochsch}.\textit{ii.)}.\\\\
Abstrakte Varianten von \eqref{eq:Gpol} und
\eqref{eq:FPol} sind gegeben durch
\begin{equation}
\label{eq:SymF}
\begin{split}
F:S^{\bullet}(\mathbb{V})\otimes S^{\bullet}(\mathbb{V})\otimes \Lambda^{k}(\mathbb{V})&\longrightarrow \bigotimes^{k+2}S^{\bullet}(\mathbb{V})\\
\KE{\alpha}{\beta}{u}&\longmapsto\sum_{\sigma\in
S_{k}}\mathrm{sign}(\sigma)\: (\alpha\otimes
u_{\sigma(1)}\otimes…\otimes u_{\sigma(k)}\otimes\beta)
\end{split}
\end{equation} und
\begin{equation}
\label{eq:SymG}
\begin{split}
G_{k}:\bigotimes^{k+2}S^{\bullet}(\mathbb{V})&\longrightarrow S^{\bullet}(\mathbb{V})\otimes
S^{\bullet}(\mathbb{V})\otimes \Lambda^{k}(\mathbb{V})\\
\omega&\longmapsto
\int_{0}^{1}dt_{1}\int_{0}^{t_{1}}dt_{2}…\int_{0}^{t_{k-1}}dt_{k}\:(i\circ\delta)(\omega).
\end{split}
\end{equation} Die Definitionen von $i$ und $\delta$ (dies sind nicht
die Abbildungen aus \thref{def:KosSymExakthHomothidelta}), sowie den Beweis
der Kettenabbildungseigenschaften findet man im Anhang
\ref{subsec:KettenabrechSym} (\thref{lemma:Fkettenabb}, \thref{prop:GcirFistidundGKettenabb}). Die $F_{k}^{*}$ und $G_{k}^{*}$ definieren dann wieder auf
Kohomologieniveau zueinander
inverse Abbildungen $\widetilde{F_{k}^{*}}$ und $\widetilde{G_{k}^{*}}$.
\newpage
\section{Topologische Komplexe und stetige Hochschildkohomologien}
Die mathematische Grundlagen dieses Kapitels finden sich im
Anhang \ref{sec:lokalkonvAnal} und wir
werden im Folgenden von den dort gewonnenen Erkenntnissen und bereitgestellten
Stetigkeitskriterien freien Gebrauch machen.
\subsection{Vorbereitendes}
Gegeben eine lokal konvexe Algebra $\mathcal{A}$,
sprich ein lokal konvexer $\mathbb{K}$-Vektorraum $\mathcal{A}$ mit stetiger Algebramultiplikation
sowie ein lokal konvexer $\mathbb{K}$-Vektorraum $\mathcal{M}$ mit
stetiger $\mathcal{A}-\mathcal{A}$ Bimodulstruktur, so
betrachten wir die $\mathbb{K}$-Vektorräume
\begin{equation*}
HC_{cont.}^{k}(\mathcal{A},\mathcal{M}):=
\begin{cases} \{O_{\mathcal{A}}\} & k<0\\
\mathcal{M} & k=0\\
\Hom^{cont.}_{\mathbb{C}}(\underbrace{\mathcal{A}\times…\times
\mathcal{A}}_{k-mal},\mathcal{M})& k\geq 1,
\end{cases}
\end{equation*} die stetigen $k$-multilinearen Abbildungen von
$\mathcal{A}\times…\times \mathcal{A}$ nach $\mathcal{M}$.
Vermöge \eqref{eq:Hochschilddelta} seien $\mathbb{K}$-lineare Abbildungen
\begin{equation*}
\delta^{k}_{c}: HC_{cont.}^{k}(\mathcal{A},\mathcal{M})\longrightarrow HC_{cont.}^{k+1}(\mathcal{A},\mathcal{M})
\end{equation*} definiert und es ist zunächst zu zeigen, dass besagtes
Bild unter $\delta^{k}_{c}$ stetig ist. Für die Stetigkeit des ersten Summanden in
\eqref{eq:Hochschilddelta} rechnen wir
\begin{align*}
p\:\left(a_{1}*_{L}\phi(a_{2},…,a_{k+1})\right)\leq&
\:c\:p_{1}(a_{1})\:p'_{2}(\phi(a_{2},…,a_{k+1}))
\\\leq&\: \hat{c}\:p_{1}(a_{1})\:p_{2}(a_{2})…p_{k+1}(a_{k+1})
\end{align*}mit der Stetigkeit von $*_{L}$. Die Stetigkeit des Letzten
Summanden folgt analog, da $*_{R}$ stetig. Für die mittleren
Summanden rechnen wir mit der Stetigkeit der Algebramultiplikation
\begin{align*}
p(\phi(a_{1},…,a_{i}a_{i+1},…,a_{k+1}))\leq& \:c\:
p_{1}(a_{1})…p'(a_{i}a_{i+1})…p_{k+1}(a_{k+1})
\\\leq & \:\hat{c}\:p_{1}(a_{1})…p_{k+1}(a_{k+1}).
\end{align*}Die Stetigkeit des Bildes unter $\delta_{c}^{k}$ folgt
nun unmittelbar mit der Stetigkeit der Vektoraddition in
$\mathcal{M}$. Mit $\delta^{k}_{c}\cp\delta_{c}^{k+1}=0$ erhalten wir so einen
Kokettensubkomplex $(HC_{cont.}^{\bullet}(\mathcal{A},\mathcal{M}),\delta_{c})$ von
$(HC^{\bullet}(\mathcal{A},\mathcal{M}),\delta)$, für den wir
definieren
\begin{equation*}
HH_{cont.}^{k}(\mathcal{A},\mathcal{M}):=\ker(\delta^{k}_{c})/\im(\delta^{k-1}_{c}).
\end{equation*}
Für die Tensorvariante des Hochschildkomplexes erhalten wir analoge
Aussagen, dabei folgt die Stetigkeit des Bildes unter
$\delta_{c\ot}^{k}$ zum einen durch elementare Rechnung, bzw. mit
\eqref{eq:TensorglKetteniso} und aus dem Fakt, dass $\ot_{*}$ vermöge
der Definition der $\pi$ Topologie stetige auf stetige Elemente abbildet.
Insbesondere bedeutet dies, dass $\ot_{*}$ ein Kettenisomorphismus ist
zwischen diesen beiden Subkomplexen, was zudem die Isomorphie derer
Kohomologiegruppen impliziert. Wir dürfen uns also wieder auf die
Tensorvariante besagten stetigen Hochschildkomplexes beschränken und es soll
nun unter anderem darum gehen
einzusehen, dass
\begin{equation*}
HH_{cont.}^{k}(\mathcal{A},\mathcal{M})\cong H^{k}(\Hom^{cont.}_{A^{e}}(\C_{c},\mathcal{M})),
\end{equation*} wobei $\C_{c}$ den topologische Bar-Komplex bezeichnet.\\
\begin{lemma}
\label{lemma:AezuunittopRing}
Gegeben eine unitäre, assoziative lokal konvexe $\mathbb{K}$-Algebra $(\mathcal{A},*)$,
\begin{enumerate}
\item
dann wird die Menge
$\mathcal{A}^{e}=\mathcal{A}\pite \mathcal{A}$ versehen mit der
distributiven Fortsetzung der Multiplikation
\begin{equation*}
(a\pite b) *_{e} (\tilde{a}\pite
\tilde{b}):=(a*\tilde{a})\pite
(b*^{opp}\tilde{b})=(a*\tilde{a})\pite
(\tilde{b}*b)
\end{equation*} auf ganz
$\mathcal{A}\pite \mathcal{A}$, zu einem unitären topologischem
Ring.
\item
Jeder lokal konvexe $\mathcal{A}-\mathcal{A}$ Bimodul $\mathcal{M}$ wird vermöge
\begin{equation*}
(a\pite b) m:=a(m b)=(a m)b\quad\quad (a\pite b)\in \mathcal{A}^{e},\: m\in \mathcal{M}
\end{equation*} zu einem lokal konvexen $A^{e}$-Linksmodul.
\end{enumerate}
\begin{beweis}
Die Ring und Moduleigenschaften folgen wie in \thref{lemma:AewirdzuunitRing}, die Stetigkeit
der Ringaddition ist die Stetigkeit der Vektorraumaddition in
$(\mathcal{A}^{e},\pi)$ als lkVR und
die Stetigkeit der Ringmultiplikation erhalten wir mit der
Stetigkeit von $*$ wie folgt
\begin{align*}
\pi_{p,q}(z*_{e}\tilde{z})=\:&\pi_{p,q}\left(\sum_{i}a_{i}\pite b_{i}*_{e}\sum_{j}\tilde{a}_{j}\pite \tilde{b}_{j}\right)
\\=\:& \pi_{p,q}\left(\sum_{i,j}a_{i}\tilde{a}_{j}\pite
\tilde{b}_{j}b_{i}\right)
\\\leq& \sum_{i,j}p(a_{i}\tilde{a}_{j})\:q(b_{i}\tilde{b}_{j})
\\\leq&\:c\sum_{i,j}p_{1}(a_{i})\:p_{2}(\tilde{a}_{j})\:q_{1}(b_{i})\:q_{2}(\tilde{b}_{j})
\\=&\:c\left(\sum_{i}p_{1}(a_{i})q_{1}(b_{i})\right)\Bigg(\sum_{j}p_{2}(\tilde{a}_{j})q_{2}(\tilde{b}_{j})\Bigg),
\end{align*} für alle Zerlegungen von $z,\tilde{z}$ und somit
\begin{align*}
\pi_{p,q}(z*_{e}\tilde{z})\leq&\:c \inf\left(\sum_{i}p_{1}(a_{i})\:q_{1}(b_{i})\right)\:\inf\left(\sum_{j}p_{2}(\wt{a}_{j})\:q_{2}(\wt{b}_{j})\right)
\\=&\:c\: \pi_{p_{1}.q_{1}}(z)\:\pi_{p_{2},q_{2}}(\tilde{z}),
\end{align*} was die Stetigkeit zeigt und folglich $i.)$.\\\\
Für $ii.)$ sei $m\in \mathcal{M}$ und $\mathcal{A}^{e}\ni z=
\displaystyle\sum_{i}a_{i}\pite b_{i}$ und wir erhalten
\begin{align*}
p(zm)=&\:p\left(\sum_{i}a_{i}*(m* b_{i})\right)
\leq\sum_{i}p(a_{i}*(m* b_{i}))
\\\leq&\:c\:\sum_{i}p_{1}(m)\:q_{1}(a_{i})\:q_{2}(b_{i})
\leq\: c\: p_{1}(m)\sum_{i}q_{1}(a_{i})\:q_{2}(b_{i}),
\end{align*} für alle Zerlegungen $\sum_{i}a_{i}\pite b_{i}$
von $z$, womit
\begin{equation*}
p(zm)\leq c\: p_{1}(m)\:\pi_{q_{1},q_{2}}(z).
\end{equation*}
\end{beweis}
\end{lemma}
\begin{definition}[topologischer Bar-Komplex]
\label{def:topBarkompl}
Gegeben eine lokal konvexe, assoziative $\mathbb{K}$-Algebra $\mathcal{A}$, so definieren wir
den topologischen Bar-Komplex $(\C_{c},d_{c})$ durch
präkompatible, $\mathcal{A}^{e}$ verträgliche $\mathcal{A}^{e}$-Moduln
\begin{align*}
\C^{c}_{k}=\mathcal{A}\pite \underbrace{\mathcal{A}\pite …
\pite \mathcal{A}}_{k-mal} \pite \mathcal{A}
\end{align*} $\quad\quad\quad \C_{0}=\mathcal{A}\pite \mathcal{A}\quad\quad
\C_{1}=\mathcal{A}\pite\mathcal{A}\pite \mathcal{A}\quad\quad \C_{2}=\mathcal{A}\pite\mathcal{A}\pite \mathcal{A}\pite \mathcal{A}$ \\\\ mit $\mathcal{A}^{e}$-Multiplikation
\begin{equation*}
(a\pite b)(x_{0}\pite x_{1}\pite … \pite x_{k}\pite
x_{k+1}):=ax_{0}\pite x_{1}\pite … \pite x_{k}\pite
x_{k+1}b
\end{equation*} und $\mathcal{A}^{e}$-Homomorphismen
\begin{align*}
d^{c}_{k}:\C^{c}_{k}&\longrightarrow \C^{c}_{k-1}\quad\quad\quad\quad ,k\geq 1\\
(x_{0}\otimes … \otimes x_{k+1})&\longmapsto
\sum_{j=0}^{k}(-1)^{k}x_{0}\otimes…\otimes
x_{j}x_{j+1}\otimes…\otimes x_{k+1}
\end{align*} mit $d^{c}_{k}\cp d^{c}_{k+1}=0$.
\end{definition}
Die relevanten Eigenschaft klärt folgende Proposition.
\begin{proposition}
\label{prop:topBarKomplexprop}
\begin{enumerate}
\item
Die $X^{c}_{k}$ sind topologische Moduln.
\item
Die $d^{c}_{k}$ sind stetig, ebenso die
exaktheitsliefernde Homotopie
\begin{equation*}
h^{c}_{k}:x_{0}\pite…\pite x_{k+1}\mapsto 1\pite
x_{0}\pite…\pite x_{k+1}.
\end{equation*}
\item
Sei $\mathcal{A}$ zudem unitär, so gilt
\begin{equation*}
HH_{c}^{k}(\mathcal{A},\mathcal{M})=H_{c}^{k}(\Hom^{cont.}_{\mathcal{A}^{e}}(\C_{c},\mathcal{M})).
\end{equation*}
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Sei $z\in \mathcal{A}^{e}$ und $x\in X^{c}_{k}$, dann folgt
\begin{align*}
\pi_{p_{0},…,p_{k+1}}(zx)=&\:\pi_{p_{0},…,p_{k+1}}\left(\sum_{i}a_{i}\pite
b_{i}\cdot\sum_{j}x_{0}^{j}\pite…\pite x^{j}_{k+1}\right)
\\ \leq&
\sum_{i,j}p_{0}(a_{i}x_{0}^{j})\:p_{k+1}(x^{j}_{k+1}b_{i})\:\pi_{p_{1},…,p_{k}}(x_{1}^{j}\pite…\pite
x_{k}^{j})
\\\leq&\:
c\sum_{i,j}p'_{0}(a_{i})\:p''_{0}(x_{0}^{j})\:p'_{k+1}(x_{k+1}^{j})\:p''_{k+1}(b_{i})\:\pi_{p_{1},…,p_{k}}(x_{1}^{j}\pite…\pite
x_{k}^{j})
\\=&\:c\sum_{i}p'_{0}(a_{i})\:p''_{k+1}(b_{i}) \cdot\sum_{j}
p''_{0}(x_{0}^{j})\:p_{1}(x_{1}^{j})…p_{k}(x_{k}^{j})\:p'_{k+1}(x_{k+1}^{j}).
\end{align*} Das zeigt
\begin{equation*}
\pi_{p_{0},…,p_{k+1}}(zx)\leq c\: \pi_{p'_{0},p''_{k+1}}(z)\:\pi_{p''_{0},…,p'_{k+1}}(x)
\end{equation*} und somit die Stetigkeit der
Modulmultiplikation. Die Stetigkeit der Addition in den
$\C^{c}_{k}$ ist klar, da diese vermöge $\pite$
topologische Vektorräume sind.
\item
Da $+$ stetig, sind Summen stetiger Funktionen ebenfalls stetig und es reicht
daher, dies für die Abbildungen
$x_{0}\pite…\pite
x_{k+1}\mapsto x_{0}\pite…\pite x_{i}x_{i+1}\pite…\pite
x_{k+1}$ nachzuweisen. Wir erhalten diese Aussage mit der Definition
der $\pi$-Topologie auf altbewährte Weise vermöge
\begin{align*}
\pi_{p_{0,…,p_{k}}}&(x_{0}\pite…\pite x_{i}x_{i+1}\pite…\pite
x_{k+1})\\ &\leq\:c\:
\pi_{p_{0},…,p'_{i}}(x_{0}\pite…\pite
x_{i})\:\pi_{p''_{i},…,p_{k}}(x_{i+1}\pite…\pite
x_{k+1})
\\=&\:c\: p_{0}(x_{0})…p'_{i}(x_{i})\:p''_{i}(x_{i+1})…p_{k}(x_{k+1}).
\end{align*} Die Stetigkeit der $h^{c}_{k}$ folgt ebenfalls ohne
große Mühe.
\item
Wir erinnern zunächst, dass die Assoziativität essentiell
dafür war, dass $\mathcal{A}^{e}$ ein Ring und der
$\mathcal{A}^{e}$-Homomorphismusbegriff wohldefiniert.\\\\
Mit der Stetigkeit der $d_{k}^{c}$ ist sofort
einsichtig, dass in der Tat
\begin{equation*}
d_{k+1}^{c*}:
\Hom^{cont}_{\mathcal{A}^{e}}(\C^{c}_{k},\mathcal{M})\rightarrow \Hom^{cont.}_{\mathcal{A}^{e}}(\C^{c}_{k+1},\mathcal{M})
\end{equation*} und somit $(\C^{*}_{c},d^{*}_{c})$ ein
wohldefinierter Gruppenkokettenkomplex.
Es bleibt dann lediglich nachzuweisen, dass die Isomorphismen
$\Xi^{k}$ (wohldefiniert mit der Unitarität von $\mathcal{A}$) aus
\thref{prop:barauffuerunitalgebraIsomozuHochschildkohomo}
in beide Richtungen stetige auf stetige Homomorphismen
abbilden, womit die Einschränkungen $\Xi^{k}_{c}$ auf
diese ebenfalls Isomorphismen
sind.\\\\
Für die $\Xi^{k}$ folgt das mit der Stetigkeit der
Abbildungen
\begin{equation*}
\tau_{k}: x_{1}\pite…\pite x_{k}\mapsto 1\pite
x_{1}\pite…\pite x_{k}\pite 1,
\end{equation*} da
\begin{align*}
\pi_{p_{0},…,p_{k+1}}(1\pite \omega_{k}\pite 1)=c\: \pi_{p_{1},…,p_{k}}(\omega_{k})
\end{align*} mit $\omega_{k} \in
\bigotimes_{\pi}^{k}\mathcal{A}$ und $c=
p_{0}(1)\:p_{k+1}(1)$.\\\\ Für $(\Xi^{k})^{-1}_{k}$ erhalten
wir die Stetigkeit mit der der Projektionen, der von $\phi$ selbst und der
Stetigkeit der $\mathcal{A}^{e}$-Multiplikation in
$\mathcal{M}$ nach \thref{lemma:AezuunittopRing}.ii.), bzw. mit
\begin{align*}
p(x_{0}\phi(x_{1}\pite…\pite x_{k}) x_{k+1})\leq&\: c\:
p_{0}(x_{0})\:q(\phi(x_{1}\pite…\pite x_{k}))\:p_{k+1}(x_{k+1})
\\\leq& \:\hat{c}\prod_{i=0}^{k+1}p_{i}(x_{i}).
\end{align*} Da nun wieder
\begin{equation*}
\Xi^{k+1}_{c}d^{c*}_{k+1}=\delta^{k}_{c}\Xi^{k}_{c},
\end{equation*} also die $\Xi^{k}_{c}$ Kettenabbildungen
sind, induzieren diese abermals Isomorphismen auf Kohomologieniveau.
\end{enumerate}
\end{beweis}
\end{proposition}
\subsection{Die stetige Hochschildkohomologie der Algebra $S^{\bullet}(\mathbb{V})$}
Wir wollen zunächst $\Ss^{\bullet}(\mathbb{V})$ mit einer lokal konvexen
Topologie derart ausstatten, dass die Algebramultiplikation $\vee$
stetig ist.\\
Sei dafür $(\mathbb{V},P)$ ein lkVR und bezeichne
$\hat{P}$ das filtrierende Halbnormensystem, bestehend aus allen bezüglich $\T_{P}$
stetigen Halbnormen. Mit $p$ enthält dann dieses
ebenfalls alle Halbnormen der Form $c\:p$ für positive Konstanten
$c$.\\
Jedes $\Ss^{l}(\mathbb{V})$ sei nun via $\hat{P}$
$\pi$-topologisiert und wir merken an, dass es dann ausreichend ist, auf
$\Ss^{l}(\mathbb{V})$ das Halbnormensystem $\{p^{l}\}_{p\in
\hat{P}}=\{\pi_{p,…,p}\}_{p\in \hat{P}}$
zu betrachten.\\ In der Tat ist ja $\pi_{p_{1},…,p_{l}}\leq
\pi_{p,…,p}=p^{l}$ für ein $p\geq p_{i}$, $i=1,…,k$ mit der Filtrationseigenschaft von
$\hat{P}$ und es folgt, dass beide Halbnormensysteme die gleiche Topologie
auf $\Ss^{l}(\mathbb{V})$ erzeugen. Die direkte Summe statten wir nun
aus mit dem Halbnormensystem $\Pp$, bestehend aus Halbnormen
\begin{equation*}
\p(\omega)=\sum_{l=0}^{\infty}p^{l}(\omega_{l})\qquad\qquad
\Ss^{\bullet}(\mathbb{V})\ni\omega=\sum_{l}\omega_{l},\text{ mit }\omega_{l}\in \Ss^{l}(\mathbb{V})
\end{equation*} und $p^{0}=\|\|_{\mathbb{C}}$. Es ist dann insbesondere klar, dass die Teilraumtopologien auf den $\Ss^{l}(\mathbb{V})$
gerade mit den $\pi$-Topologien übereinstimmen.\\\\
Es sei an dieser Stelle
explizit darauf hingewiesen, dass die Halbnormen
\begin{align*}
\tilde{\p}=\sum_{l=0}^{\infty}p_{i}^{l}
\end{align*} mit paarweise verschiedenen $p_{i}\in \hat{P}$ in der Tat
eine andere Topologie definieren, da die Summe unendlich ist und wir
im Allgemeinen kein $p\in \hat{P}$ derart finden, dass $p\geq
p_{i}\:\forall\:i\in \mathbb{N}$. Ein Grund für diese Wahl wird in
nächsten Lemma klar werden, einen weiteren liefert \thref{bsp:holomorpheFunkts}.ii.).\\\\
Eine essentielle Eigenschaft besagten Halbnormensystems ist, dass mit $\p \in \Pp$ per Konstruktion ebenfalls die Halbnorm
\begin{align*}
\p_{c}(v)=\sum_{l=0}^{\infty}c^{l}p^{l},
\end{align*} in $\Pp$ enthalten ist, was für spätere Stetigkeitsabschätzungen von hohem Nutzen
sein wird.\\
\begin{lemma}
Vermöge $\vee$ wird $(\Ss^{\bullet}(\mathbb{V}),\Pp)$ zu einer
assoziativen, unitären, lokal konvexen Algebra.
\begin{beweis}
Es ist die Stetigkeit von $\vee$ zu zeigen, der Rest ist klar.\\\\
Zunächst erhalten wir die Stetigkeit von
\begin{align*}
\ot: T^{\bullet}(\mathbb{V})\times T^{\bullet}(\mathbb{V})
&\longrightarrow T^{\bullet}(\mathbb{V})\\
\left(\sum_{l}\alpha_{l},\sum_{m}\beta_{m}\right)&\longmapsto\sum_{l,m}\alpha_{l}\ot\beta_{m}\qquad \alpha_{l}\in T^{l}(\mathbb{V}),\:\beta_{m}\in T^{m}(\mathbb{V}),
\end{align*} da
\begin{align*}
\p\left(\sum_{l,m}\alpha_{l}\ot\beta_{m}\right)=&\:\sum_{k}p^{k}\left(\sum_{l+m=k}\alpha_{l}\ot\beta_{m}\right)
\leq \sum_{l,m}p^{l+m}\left(\alpha_{l}\ot\beta_{m}\right)\\
\leq& \sum_{l,m,i_{l},j_{m}}p^{l+m}\left(\alpha^{i_{l}}_{l}\ot\beta^{j_{m}}_{m}\right)
= \sum_{l,m,i_{l},j_{m}}p^{l}(\alpha^{i_{l}}_{l})\:p^{m}(\beta^{j_{m}}_{m})\\
=& \left(\sum_{l,i_{l}}p^{l}\big(\alpha_{l}^{i_{l}}\big)\right)\left(\sum_{m,j_{m}}p^{m}\left(\beta_{m}^{j_{m}}\right)\right),
\end{align*} für alle Zerlegungen $\alpha^{i_{l}}_{l}$,
$\beta^{j_{m}}_{m}$ von $\alpha_{l}$ und $\beta_{m}$ und somit ebenfalls
\begin{align*}
\p\left(\sum_{l,m}\alpha_{l}\ot\beta_{m}\right)\leq&
\left[\sum_{l}\inf\left(\sum_{i}p^{l}\big(\alpha^{i}_{l}\big)\right)\right]\:\left[\sum_{m}\inf\left(\sum_{j}p^{m}\big(\beta^{j}_{m}\big)\right)\right]
\\=& \p\left(\sum_{l}\alpha_{l}\right)\:\p\left(\sum_{m}\beta_{m}\right).
\end{align*} Damit ist $\ot$ insbesondere stetig in der von
$T^{\bullet}(\mathbb{V})$ auf $\Ss^{\bullet}(\mathbb{V})$ induzierten
Teilraumtopologie, als Abbildung
$\ot\big|_{\Ss^{\bullet}(\V)}\Ss^{\bullet}(\V)\rightarrow
T^{\bullet}(\V)$, da die Ungleichungen natürlich auch für alle
symmetrische Elemente gelten.\\\\ Mit analoger Argumentation ist
\begin{align*}
S:T^{\bullet}(\mathbb{V})&\longrightarrow
\Ss^{\bullet}(\mathbb{V})\subseteq T^{\bullet}(\mathbb{V})\\
\alpha_{1}\ot…\ot \alpha_{l}&\longmapsto \frac{1}{l!}\sum_{\sigma\in S_{l}}\alpha_{\sigma(1)}\ot…\ot\alpha_{\sigma(k)}
\end{align*} stetig auf $\Ss^{\bullet}(\mathbb{V})$ vermöge
\begin{align*}
\p\left(S\left(\sum_{l}\alpha_{l}\right)\right)=\sum_{l}p^{l}(S(\alpha_{l}))\leq \sum_{l}p^{l}(\alpha_{l})=\p\left(\sum_{l}\alpha_{l}\right)
\end{align*} mit $\alpha=\displaystyle\sum_{l}\alpha_{l}$ und
$\alpha_{l}\in T^{l}(\mathbb{V})$ nicht notwendigerweise separabel, da
\begin{equation*}
p^{l}(S(\alpha_{l}))=p^{l}\left(\frac{1}{l!}\sum_{\sigma\in S_{l}}\sigma^{*}\alpha_{l}\right)\leq
\frac{l!}{l!}\:p^{l}(\alpha_{l}).
\end{equation*} Nun ist $\vee=S\cp \ot\cp\kappa\times\kappa$ mit
der Abbildung
\begin{align*}
\kappa:
T^{\bullet}(\mathbb{V})&\longrightarrow
T^{\bullet}(\mathbb{V})\\
\alpha_{l}&\longmapsto
\frac{1}{l!}\:\alpha_{l},
\end{align*} die offensichtlich stetig ist, da für $T^{\bullet}(\mathbb{V})\ni\omega=\displaystyle\sum_{l}\alpha_{l}$
\begin{align*}
\p(\kappa(\omega))=\p\left(\sum_{l}\frac{1}{l!}\:\alpha_{l}\right)=\sum_{l}\frac{1}{l!}\:p^{l}(\alpha_{l})\leq \sum_{l}p^{l}(\alpha_{l})=\p(\omega).
\end{align*} Es folgt unmittelbar die Stetigkeit von $\kappa\times \kappa$
in der Produkttopologie, womit $\vee$ als Verkettung stetiger
Funktionen stetig ist.
\end{beweis}
\end{lemma}
Als Resultat dieses Lemmas erhalten wir mit Hilfe des vorherigen
Abschnittes
\begin{equation*}
HH^{k}_{cont.}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong H^{k}(\Hom_{\mathcal{A}^{e}}^{cont.}(\C_{c},\mathcal{M})),
\end{equation*} mit $\C_{c}$ der zu $\Ss^{\bullet}(\mathbb{V})$
gehörige topologische Bar-Komplex und
$\mathcal{A}^{e}=\Ss^{\bullet}(\mathbb{V})\pite \Ss^{\bullet}(\mathbb{V})$.\\\\
Wir wollen nun den in \ref{subsec:HochschKohSym} betrachteten
Koszul-Komplex in geeigneter Weise derart topologisieren, dass die
Modulhomomorphismen $\partial_{k}$ stetige Abbildungen sind und wir somit in
wohlbegründeter Weise vom topologischen Koszulkomplex
$(\K_{c},\pt_{c})$ und folglich auch vom stetigen Kokettenkomplex
$(\Hom_{\mathcal{A}^{e}}^{cont.}(\C_{c},\mathcal{M}),\pt_{c}^{*})$
sprechen dürfen.\\
\begin{definition}[Topologischer Koszulkomplex]
Gegeben der Koszulkomplex aus \ref{subsec:HochschKohSym}, so
definieren wir die topologischen Räume
\begin{equation*}
\K_{k}^{c}=\Ss^{\bullet}(\mathbb{V})\pite
\Ss^{\bullet}(\mathbb{V})\pite \Lambda^{k}(\mathbb{V})
\end{equation*} und erhalten ein erzeugendes
Halbnormensystem $\Pp_{k}$ vermöge den Halbnormen
\begin{equation*}
\p_{k}=\p\ot\p\ot p^{k}=\p^{2}\ot p^{k}.
\end{equation*} Mit $T^{c}_{k}\supseteq \K_{k}^{c}$ wollen wir den mit selbigem
Halbnormensystem ausgestatteten Raum
\begin{equation*}
T^{\bullet}(\mathbb{V})\pite
T^{\bullet}(\mathbb{V})\pite T^{k}(\mathbb{V})
\end{equation*} bezeichnen, der klarerweise obige Topologie als
Teilraumtopologie auf $\K_{k}^{c}$ induziert.
\end{definition}
\begin{bemerkung}
\label{bem:Teilraumargument}
Für
Stetigkeitsaussagen von Abbildungen $\phi:\K^{c}_{k}\rightarrow
\K^{c}_{k'}$ reicht es aus, die Stetigkeit für Abbildungen
$\tilde{\phi}:T^{c}_{k}\rightarrow T^{c}_{k'}$ nachzuweisen,
die $\phi$ auf $\K^{c}_{k}$ induzieren, für die also
$\tilde{\phi}\big|_{\K^{c}_{k}}=\phi$. Dies sieht man wieder
daran, dass sich die Stetigkeitsabschätzungen für $\phi$
klarerweise auf besagte Unterräume übertragen. Abstrakter in
Teilraumtopologiesprache
folgt dies auch da $\eta\subseteq \K^{c}_{k'}$ offen
bedeutet, dass $\eta=\mu\cap \K^{c}_{k'}$ für offenes
$\mu\subseteq T^{c}_{k'}$ und folglich
\begin{equation*}
\tilde{\phi}^{-1}(\eta)=\phi^{-1}\left(\K^{c}_{k'} \cap
\mu\right)=\phi^{-1}(\mu)\cap \phi^{-1}\left(\K^{c}_{k'}\right)=\phi^{-1}(\mu)\cap \K^{c}_{k}.
\end{equation*}Da nun $\phi^{-1}(\mu)$ offen, folgt die
Behauptung.
In der Tat, angenommen letztere Gleichheit gelte
nicht, so fänden wir $x\in \K^{c}_{k}$ mit
\begin{equation*}
\phi^{-1}(\mu)\cap \K^{c}_{k}\ni x\notin\phi^{-1}(\mu)\cap \phi^{-1}\left(\K^{c}_{k'}\right).
\end{equation*} Da dann aber $x\in\phi^{-1}(\mu)$ und nach Vorraussetzung
$\phi(x)=\tilde{\phi}(x)\in \K^{c}_{k'}$, folgt $x\in
\phi^{-1}(\K^{c}_{k'})$, im Widerspruch zur Wahl von $x$.
\end{bemerkung}Wir erhalten folgende wichtige Proposition.
\begin{proposition}
\label{prop:wichpropKoszStetSym}
\begin{enumerate}
\item
Eine lineare Abbildung $\phi:T^{c}_{k}\rightarrow T^{c}_{k'}$
ist stetig genau dann, wenn
\begin{equation}
\label{eq:stetrelSummenKos}
\p_{k'}(\phi(\alpha_{l}\ot \beta_{m}\ot u))\leq c\: \p_{k}(\alpha_{l}\ot \beta_{m}\ot u)
\end{equation} für separable $\alpha_{l}\in T^{l}(\mathbb{V})$, $\beta_{m}\in
T^{m}(\mathbb{V})$ und $u\in T^{k}(\mathbb{V})$.
\item
\label{item:partkStetig}
Es sind alle $\pt_{k}$ stetig.
\item
\label{item:KettenabbStet}
Die Homotopie $h$ ist stetig, ebenso die
Kettenabbildungen $F$ und $G$.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Ist $\phi$ stetig, so ist besagte Relation per Definition
erfüllt, für die andere Richtung rechnen wir für separables
$\alpha\ot \beta\ot u \in T^{c}_{k}$ mit zudem $u$ separabel
\begin{align*}
\p_{k'}(\phi(\alpha\ot\beta\ot
u))=&\p_{k'}\left(\sum_{l,m,i_{l},j_{m}}\phi\left(\alpha^{i_{l}}_{l}\ot\beta^{j_{m}}_{m}\ot
u\right)\right)
\leq\sum_{l,m,i_{l},j_{m}}\p_{k'}\left(\phi\left(\alpha_{l}^{i_{l}}\ot\beta_{m}^{j_{m}}\ot
u\right)\right)
\\\leq&\sum_{l,m,i_{l},j_{m}}c\:\p_{k}\left(\alpha_{l}^{i_{l}}\ot\beta_{m}^{j_{m}}\ot
u\right)=c\sum_{l,m,i_{l},j_{m}}p^{l}\big(\alpha_{l}^{i_{l}}\big)\:p^{m}\left(\beta_{m}^{j_{m}}\right)\:p^{k}(u)
\end{align*} für alle derartigen Zerlegungen der
$\alpha_{l}$ und $\beta_{m}$, mithin
\begin{align*}
\label{eq:stetmuhmuh}
\p_{k'}(\phi(\alpha\ot\beta \ot u))\leq&\:
c\sum_{l}\inf\left(\sum_{i}
p^{l}\big(\alpha_{l}^{i}\big)\right)\sum_{m}\inf\left(\sum_{i}p^{m}\left(\beta_{m}^{i}\right)\right)\:p^{k}(u)\\=&\:c\:
\p(\alpha)\p(\beta)\:p^{k}(u).
\end{align*} Das zeigt die Stetigkeit von $\phi\cp
\ot^{2}$ und somit die Stetigkeit von $\phi$ in $\pi$.
\item
Wir definieren
\begin{align*}
\pt'^{k*}_{1}:T^{\bullet}(\mathbb{V})\bbot
T^{\bullet}(\mathbb{V})\bbot
T^{k}(\mathbb{V})&\longrightarrow T^{\bullet+1}(\mathbb{V})\bbot
T^{\bullet}(\mathbb{V})\bbot
T^{k-1}(\mathbb{V})\\
\alpha\bbot \beta\bbot u&\longmapsto k\:(u_{1}\ot \alpha\bbot\beta\bbot u^{1})\\
\pt'^{k*}_{2}:T^{\bullet}(\mathbb{V})\bbot
T^{\bullet}(\mathbb{V})\bbot
T^{k}(\mathbb{V})&\longrightarrow T^{\bullet}(\mathbb{V})\bbot
T^{\bullet+1}(\mathbb{V})\bbot
T^{k-1}(\mathbb{V})\\
\alpha\bbot \beta\bbot u&\longmapsto k\:(\alpha\bbot u_{1}\ot\beta\bbot u^{1})
\end{align*} und es
folgt
\begin{align*}
(S\ot S\ot A)\cp
\pt'^{k*}_{1}\big|_{K^{c}_{k}}&=\pt^{k*}_{1}\\
(S\ot S\ot A)\cp
\pt'^{k*}_{2}\big|_{K^{c}_{k}}&=\pt^{k*}_{2},
\end{align*} wobei
\begin{equation*}
A(u_{1}\ot…\ot u_{k})=\frac{1}{k!}\sum_{\sigma\in
S_{k}}\sign(\sigma)\:u_{\sigma(1)}\ot…\ot u_{\sigma(k)}
\end{equation*} und $S\ot S\ot A$ stetig in $T^{c}_{k}$
mit i.), da für
$\alpha_{l}\ot\beta_{m}\ot u\in T^{l}(\mathbb{V})\ot
T^{\bullet}(\mathbb{V})\ot T^{k}(\mathbb{V})$ mit
separablen Faktoren
\begin{align*}
\p_{k}((S\ot S\ot A)(\alpha_{l}\ot \beta_{m} \ot
u))=&\:\p_{k}\left(S(\alpha_{l})\ot
S(\beta_{m})\ot A(u)\right)\\
=&\:p^{l}(S(\alpha_{l}))\:p^{m}(S(\beta_{m}))\:p^{k}(A(u))
\\\leq&\:p^{l}(\alpha_{l})\:p^{m}(\beta_{m})\:p^{k}(u)
\\=&\: \p_{k}(\alpha_{l}\ot \beta_{m} \ot u).
\end{align*}\\\\ Es bleibt die Stetigkeit von
$\pt'^{k*}_{1}$ und $\pt'^{k*}_{2}$ zu zeigen, dafür
rechnen wir
\begin{align*}
\p_{k-1}\left(\pt'^{k*}_{1}\left(\alpha_{l}\bbot\beta_{m}\bbot
u\right)\right)
=&\:k\p_{k-1}\left(u_{1}\ot\alpha_{l}\bbot\beta_{m}\bbot
u^{1}\right)
\\=&\:k\:p^{l+1}(u_{1}\ot
\alpha_{l})\:p^{m}(\beta_{m})\:p^{k-1}(u^{1})
\\=&\:k\:p^{l}(\alpha_{l})\:p^{m}(\beta_{m})\:p^{k}(u)
\\=&\:k\p_{k}(\alpha_{l}\bbot\beta_{m}\bbot u),
\end{align*} für separable $\alpha_{l},\beta_{m}, u$. Mit
$i.)$ folgt die Stetigkeit und mit \thref{bem:Teilraumargument},
die von $\pt^{k*}_{1}$ in $\K^{c}_{k}$, analog für $\pt'^{k*}_{2}$.
\item
Beweise hierfür finden sich in \ref{subsec:Stetigkeitsrechnungen}.
\end{enumerate}
\end{beweis}
\end{proposition}
\begin{satz}
Es ist
\begin{equation*}
HH^{k}_{cont.}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong
H^{k}(\Hom_{\mathcal{A}^{e}}^{cont.}(\C_{c},\mathcal{M}))\cong
H^{k}(\Hom_{\mathcal{A}^{e}}^{cont.}(\K_{c},\mathcal{M})).
\end{equation*} für $\mathcal{M}$ präkompatibel und $\Ss^{\bullet}(\V)$
verträglich. Ist $\mathcal{M}$ symmetrisch so gilt zudem
\begin{equation*}
HH^{k}_{cont.}(\Ss^{\bullet}(\mathbb{V}),\mathcal{M})\cong \Hom_{\mathcal{A}^{e}}^{cont.}(\K_{c},\mathcal{M}).
\end{equation*}
\begin{beweis}
Die erste Isomorphie hatten wir bereits eingesehen. Für die
Zweite beachten wir, dass mit
\thref{prop:wichpropKoszStetSym}.\ref{item:partkStetig})
der Kokettenkomplex
$(\Hom_{\mathcal{A}^{e}}^{cont.}(\C_{c},\mathcal{M}),\pt_{c}^{*})$
in der Tat ein Subkomplex ist von
$(\Hom_{\mathcal{A}^{e}}(\C,\mathcal{M}),\pt^{*})$, die
gleiche Aussage erhielten wir bereits für
$(\Hom_{\mathcal{A}^{e}}^{cont.}(\C_{c},\mathcal{A}),d_{c}^{*})$.
Mit
\thref{prop:wichpropKoszStetSym}.\ref{item:KettenabbStet})
sind $F$ und $G$ stetig, folglich bilden
$F^{*}$ und $G^{*}$ stetige auf stetige Homomorphismen ab und
diese Aussage ist klarerweise immer noch korrekt für die auf den
Kohomologiegruppen induzierten Isomorphismen $\wt{F^{*}}$ und
$\wt{G^{*}}$. $\wt{F^{*}}^{-1}=\wt{G^{*}}$ zeigt dann, dass
$\wt{F^{*}}$ der gesuchte Isomorphismus ist. Die zweite
Aussage ist klar.
\end{beweis}
\end{satz}
\subsection{Die stetige Hochschildkohomologie der Algebra $\Hol$}
Sei im Folgenden $\V$ ein hlkVR, wir beginnen dann mit folgender klärenden Proposition.
\begin{proposition}
\label{prop:HollkAlgebra}
\begin{enumerate}
\item
Gegeben ein hlkVR $\V$, so existiert eine bis auf lineare
Homöomorphie eindeutig bestimmte vollständige, hausdorffsche, lokal konvexe
Algebra $(\Hol,\Pp_{H},*)$, die $(\Ss^{\bullet}(\V),\Pp,\vee)$ im
isometrischen Sinne als Teilraum enthält, diese ist zudem
assoziativ und unitär.
\item
Jeder hausdorffsche, lokal konvexe
$\Ss^{\bullet}(\V)-\Ss^{\bullet}(\V)$ Bimodul vervollständigt
zu einem hausdorffschen, lokal konvexen $\Hol-\Hol$
Bimodul. Ist $\mathcal{M}$ präkompatibel und
$\Ss^{\bullet}(\V)$ verträglich, so ist $\hat{\mathcal{M}}$
präkompatibel und $\Hol$ verträglich.
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Zunächst ist nach \thref{satz:PiTopsatz}.v.) mit $\V$
ebenfalls jedes $\left(\Ss^{k}(\V),p^{k}\right)$ hausdorffsch und es ist
offensichtlich, dass dies dann ebenfalls für
$(\Ss^{\bullet}(V),\Pp)$ der Fall ist.
Es folgt mit \thref{satz:vervollsthlkVR}, dass die bis auf lineare Homöomorphie eindeutig
bestimmte Vervollständigung
$\big(\Hol,\Pp_{H}\big)=\big(\widehat{\Ss^{\bullet}(\V)},\hat{\Pp}\big)$
existiert und ebenfalls hausdorffsch ist.
Weiter erhalten wir,
dass die billineare Abbildung
\begin{align*}
\tilde{\vee}:i(\Ss^{\bullet}(\V))\times i(\Ss^{\bullet}(\V))&\longrightarrow
i(\Ss^{\bullet}(\V))\subseteq\Hol\\
(i(v),i(w))&\longmapsto i(v \vee w)
\end{align*} wohldefiniert und stetig ist, als Verkettung
stetiger Funktionen. Dabei bezeichnet $i$ die zugehörige stetige Isometrie aus
\thref{satz:vervollsthlkVR}. Da
$\overline{i(\Ss^{\bullet}(\V))}=\Hol$, liefert uns
\thref{satz:stetfortsBillphi} eine eindeutig bestimmte, stetige billineare Fortsetzung
\begin{align*}
*: \Hol\times \Hol\longrightarrow \Hol.
\end{align*}
Für die Unitarität betrachten wir das Element $\Hol\ni
\hat{1}= i(1_{\Ss^{\bullet}(\V)})$ und erhalten für ein
Cauchynetz $\net{i(x)}{I}\subseteq i(\Ss^{\bullet}(\V))$ mit
$\net{i(x)}{I}\rightarrow x\in \Hol$ und
$\{i(1_{\Ss^{\bullet}(\V)})\}$ die konstante Folge $i(1_{\Ss^{\bullet}(\V)})$
\begin{align*}
\hat{1}* x = & \lim_{n\times \alpha}
\{i(1_{\Ss^{\bullet}(\V)})\}\:\tilde{\vee}\: i(x)_{\alpha}=\lim_{\alpha}\:
i(x)_{\alpha}=x.
\end{align*}Spätestens hier ist nun auch klar, dass wir
vermöge $i$ die Räume $(\Ss^{\bullet}(\V),\vee)$ und
$i(\Ss^{\bullet}(\V),\tilde{\vee})$ identifizieren dürfen und für die
Assoziativität rechnen wir daher in Kurzschreibweise mit
$x,y,z\in \Hol$
\begin{align*}
x*(y*z)=\lim_{\alpha\times \beta\times
\gamma}x_{\alpha}\vee (y_{\beta}\vee z_{\gamma})=\lim_{\alpha\times \beta\times
\gamma}(x_{\alpha}\vee y_{\beta})\vee z_{\gamma}=(x*y)*z,
\end{align*}da definitionsgemäß $\{y_{\beta}\vee
z_{\gamma}\}_{\beta\times \gamma\in J\times L}\rightarrow y*z$
und $\{x_{\alpha}\vee
y_{\beta}\}_{\alpha\times \beta\in I\times J}\rightarrow x*y$.
\item
Zunächst ist wieder klar, dass für jeden solchen Bimodul
$\mathcal{M}$ eine Vervollständigung $\hat{\mathcal{M}}$
existiert. Da sowohl Links als auch
Rechtsmodulmultiplikation stetige billineare Abbildungen
auf den Dichten Teilräumen $i(\Ss^{\bullet}(\V))\subseteq
\Hol$ und $i'(M)\subseteq \hat{M}$ induzieren, setzen
diese stetig fort. Die $\Hol$ Verträglichkeit folgt wie in i.)
die Unitarität von $\Hol$ und die Präkompatibilität wie die
Assoziativität.
\end{enumerate}
\end{beweis}
\end{proposition}
Punkt ii.) ist hier als Motivation dafür gedacht, dass
überhaupt derartige $\Hol-\Hol$ Bimoduln existieren. Als wichtiges
Resultat aus i.) erhalten wir umgehend.\\
\begin{korollar}
\label{kor:HolBarBimodul}
Gegeben ein hlkVR $\V$ und $\mathcal{A}=\Hol$, so gilt
\begin{align*}
HH_{cont.}^{k}(\Hol,\mathcal{M})\cong
H^{k}(\Hom^{cont.}_{\mathcal{A}^{e}}(\C_{c},\mathcal{M})),
\end{align*}dabei bezeichnet $\C_{c}$ den zu $\Hol$ gehörigen
topologischen Bar-Komplex und $\mathcal{M}$ einen lokal konvexen,
präkompatiblen, $\Hol$ verträglichen Bimodul.
\begin{beweis}
Dies folgt schon wie im letzten Abschnitt aus
\thref{prop:topBarKomplexprop}.iii.), da $(\Hol,*)$ mit
\thref{prop:HollkAlgebra} eine
lokal konvexe, unitäre und assoziative $\mathbb{C}$-Algebra ist.
\end{beweis}
\end{korollar}
Um \thref{prop:HollkAlgebra} mit etwas Leben zu füllen, nun zunächst ein paar Beispiele.\\
\begin{beispiel}
\label{bsp:holomorpheFunkts}
\begin{enumerate}
\item
Wir betrachten den topologischen Vektorraum $\mathbb{C}^{n*}$,
versehen mit der üblichen euklidischen Normtopologie. Mit der
Äquvivalenz aller Normen auf $\mathbb{C}^{n*}$ dürfen wir uns
auf das Halbnormensystem, bestehend aus allen bezüglich der
Maximumsnorm
\begin{equation*}
|x|_{max}=\sum_{i=1}^{n}|x_{i}|\quad\quad
x=\sum_{i=1}^{n}x_{i}\: e^{i}
\end{equation*} stetigen Halbnormen festlegen und verschaffen uns so
ein filtrierendes Halbnormensystem $P$ auf
$\mathbb{C}^{n*}$. Dieses enthält dann insbesondere wieder alle
Normen der Form $c\:||_{max}$ für positive Konstanten $c$. Die
Symmetrische Algebra sei topologisiert vermöge den Halbnormen
\begin{equation*}
\p=\sum_{k=0}^{\infty}p^{k}\quad\quad p\in p
\end{equation*}und wir erhalten
\begin{equation}
\label{eq:pitopendlCn}
|z|^{k}_{max}=\sum_{i_{1},…,i_{k}}|a_{i_{1},…,a_{i_{k}}}|\quad\text{für}\quad
\mathrm{T}^{\bullet}(\mathbb{C}^{n*})\ni
z=\sum_{i_{1},…,i_{k}}^{n}
a_{i_{1},…,i_{k}}e^{i_{1}}\vee…\vee e^{i_{k}}.
\end{equation}
In der Tat, bezeichne $||^{\ot k}$ die durch
\eqref{eq:pitopendlCn} charackterisierte Norm, so folgt zunächst Definitionsgemäß
\begin{equation*}
|z|_{max}^{k}\leq
\sum_{i_{1},…,i_{k}}^{n}|a_{i_{1},…,a_{i_{k}}}e^{i_{1}}||e^{i_{2}}|…|e^{i_{k}}|=|z|^{\ot
k}
\end{equation*} und mit der Normeigenschaft von $||^{\ot k}$
\begin{equation*}
|z|^{\ot k}\leq \sum_{i}|z^{i}|^{\ot k}=\sum_{i}|z^{i}|_{max}^{k},
\end{equation*} für alle elementaren Zerlegungen $z=\displaystyle\sum_{i}
z^{i}$ in separable $z^{i}=x_{1}^{i}\ot…\ot x_{k}^{i}$. Dabei
folgt die zweite Gleichheit aus
\begin{align*}
|x_{1}\ot…\ot x_{k}|^{\ot
k}=&\sum_{i_{1},…,i_{k}}^{n}|x_{1}^{i_{1}}|…|x_{k}^{i_{k}}|
=\:|x_{1}|_{max}…|x_{k}|_{max}=|x_{1}\ot…\ot x_{k}|_{max}^{k},
\end{align*}
für $x_{j}=\displaystyle\sum_{i_{j}}x_{j}^{i_{j}}$. Wir
identifizieren jetzt $\mathrm{Pol(\mathbb{C}^{n})}$ mit
$\Ss^{\bullet}(\mathbb{C}^{n*})$ vermöge
\begin{equation}
\label{eq:isomPolSym}
\begin{split}
\cong:\mathrm{Pol(\mathbb{C}^{n})}&\longrightarrow
\Ss^{\bullet}(\mathbb{C}^{n*})\\
a_{i_{1},…,i_{k}}z_{i_{1}}…z_{i_{k}}&\longmapsto
a_{i_{1},…,i_{k}} e^{i_{1}}\vee…\vee e^{i_{k}}
\end{split}
\end{equation}
und erhalten für $\mathrm{Pol}(\mathbb{C}^{n})\ni
h=
\displaystyle\sum_{l=0}^{k}\sum_{i_{1},…,i_{k}}^{n}a_{i_{1},…,i_{k}}z_{i_{1}}…z_{i_{k}}$,
sowie $p=||_{max}$
\begin{align*}
\p(\cong(h))=&\p\left(\sum_{l=0}^{k}\sum_{i_{1},…,i_{k}}^{n}a_{i_{1},…,i_{k}}e^{i_{1}}…e^{i_{k}}\right)
\\=&\sum_{l=0}^{k}\left|\sum_{i_{1},…,i_{k}}^{n}a_{i_{1},…,i_{k}}e^{i_{1}}…e^{i_{k}}\right|_{max}^{k}
\\=&\sum_{l=0}^{k}\sum_{i_{1},…,i_{k}}^{n}|a_{i_{1},…,i_{k}}|.
\end{align*}
Sei $\left(\begin{array}{c} z_{1} \\ \vdots\\
z_{n} \end{array}\right)\in \mathbb{C}^{n}$,
$|z|=\displaystyle\max_{0\leq i\leq n}|z_{i}|$ und
$p_{z}=|z|\cdot||_{max}$, so folgt gleichfalls
\begin{equation}
\label{eq:abskonvnorm}
\p_{z}(h)=\sum_{l=0}^{k}\sum_{i_{1},…,i_{l}}^{n}|a_{i_{1},…,i_{l}}|\cdot|z|^{l}.
\end{equation}
Dies zeigt insbesondere, dass die Menge aller
bezüglich $||_{max}$ stetigen Halbnormen in der Tat eine
feinere Topologie auf $\Ss^{\bullet}(\mathbb{C}^{n*})$
induziert, als es der Norm $||_{max}$ alleine möglich wäre. In
Formeln bedeutet dies, dass wir im Allgemeinen für $|z|\neq
|z'|$ keine Abschätzung der Form $\p_{z}\leq c\:\p_{z'}$
erhalten können.
Für die Vervollständigung $(\mathrm{Hol}(\mathbb{C}^{n}),\hat{\Pp})$ von
$(\mathrm{Pol}(\mathbb{C}^{n}),\Pp)$ sei nun $\net{h}\subseteq
(\mathrm{Pol}(\mathbb{C}^{n})$ ein Cauchynetz, womit
$\p_{z}(h_{\alpha}-h_{\beta})<\epsilon\quad
\forall\:\alpha,\beta\geq \alpha_{\epsilon}$ und folglich
\begin{equation*}
\p_{z}(h_{\alpha})\leq
\p_{z}(h_{\alpha}-h_{\alpha_{\epsilon}})+
\p_{z}(h_{\alpha_{\epsilon}})<\epsilon +
c=\hat{c}<\infty\quad\quad \forall\:\alpha,\beta\geq \alpha_{\epsilon}.
\end{equation*}
Es folgt $\displaystyle\lim_{\alpha}\p_{z}(h_{\alpha})\leq
\hat{c}<\infty$ und mit \eqref{eq:abskonvnorm} zeigt dies,
dass jedes Element aus $\mathrm{Hol}(\mathbb{C}^{n})$ eine
auf ganz $\mathbb{C}^{n}$ absolut konvergente Potenzreihe
definiert. Umgekehrt gilt für jede solche Potenzreihe
$h=\displaystyle\sum_{l=0}^{\infty}\sum_{i_{1},…,i_{l}}^{n}a_{i_{1},…,i_{l}}z_{i_{1}}…z_{i_{l}}$
\begin{equation*}
\sum_{l=k}^{\infty}\sum_{i_{1},…,i_{l}}^{n}|a_{i_{1},…,i_{l}}|\:|z|^{l}<
\frac{1}{n}\quad\quad \forall \:k\geq k_{n,|z|},
\end{equation*}
wobei wir $z_{1}…z_{n}=|z|> 0$ gesetzt haben. Wir erhalten
dann für die Folge $\{h_{k}\}_{k\in \mathbb{N}}\subseteq
\Ss^{\bullet}(V^{*})$ definiert durch
\begin{equation*}
h_{k}=\sum_{l=0}^{k}\sum_{i_{1},…,i_{l}}^{n}a_{i_{1},…,i_{l}}e^{i_{1}}\vee…\vee
e^{i_{l}},
\end{equation*} vermöge $p\leq |z|\:||_{max}$ für eine
stetige Halbnorm $p$, sowie $k\leq k'$
\begin{align*}
\p(h_{k}-h_{k'})\leq&\:
\sum_{l=k}^{k'}\sum_{i_{1},…,i_{l}}^{n}|a_{i_{1}},…,a_{i_{l}}|\:|z|^{l}
\\\leq&\:\sum_{l=k}^{\infty}\sum_{i_{1},…,i_{l}}^{n}|a_{i_{1}},…,a_{i_{l}}|\:|z|^{l}
\\<&\:\frac{1}{n},
\end{align*}
falls $k,k'\geq k_{n,|z|}$. Das zeigt die Cauchyeigenschaft von
$\{h_{k}\}_{k\in \mathbb{N}}$ und somit $h\in \mathrm{Hol}(\mathbb{C}^{n*})$.
Zusammen folgt, dass $\mathrm{Hol}(\mathbb{C}^{n})$
genau aus den konvergenten Potenzreihen auf $\mathbb{C}^{n}$ besteht und
somit identisch ist mit den ganz holomorphen Funktionen
auf diesem Raum. Die stetige Fortsetzung
der Halbnorm $\p_{z}$ ist dann in
Multiindexschreibweise auch darstellbar in der Form
\begin{equation*}
\hat{\p}_{z}(\phi)=\sum_{k=0}^{\infty}\frac{z^{|\alpha|}}{\alpha!}\left|\frac{\pt^{|\alpha|}\phi}{\pt^{\alpha}}(0)\right|\quad\quad
\forall\:\phi\in \mathrm{Hol}(\mathbb{C}^{n}).
\end{equation*}
\item
Sei $\mathbb{V}=\mathbb{C}^{|\mathbb{N}|}$ oder ein anderer
unendlichdimensionaler $\mathbb{C}$-Vektorraum. Man beachte
zunächst, dass dann nicht mehr $\mathbb{V}\cong
\mathbb{V}^{*}$, denn sei $\{e_{\alpha}\}_{\alpha\in I}$ eine
Basis von $\mathbb{V}$ und $\{e^{\alpha}\}_{\alpha\in I}$ die
durch $e^{\alpha}(\epsilon_{\beta})=\delta_{\alpha,\beta}$
definierten dualen Elemente, so erhalten wir mit
$\phi=\displaystyle\sum_{\alpha\in I}e^{\alpha}$ ein
wohldefiniertes Element aus $\mathbb{V}^{*}$, welches
offensichtlich nicht als endliche Linearkombination der
$e^{\alpha}$ geschrieben werden kann. Dies zeigt, dass
$\mathbb{V}^{*}$ mehr Elemente als $\mathbb{V}$ selbst
enthält und dass die Teilmenge $\{e^{\alpha}\}_{\alpha\in
I}\subseteq \mathbb{V}^{*}$ keine Basis des Dualraumes
ist. Nebenbei bemerkt lässt sich aber jedes Element aus
$\mathbb{V}^{*}$ als eine solche unendliche Summe darstellen,
da Lineare Abbildungen auf den Bildern einer Basis eindeutig
bestimmt ist.
Obige Ausführungen bedeuten insbesondere, dass wir a priori
keine Isomorphie der Form \eqref{eq:isomPolSym} angeben
können und uns im Folgenden im abstrakten $\Ss^{\bullet}(\V^{*})$ Rahmen
bewegen müssen. Sei dafür $\V^{*}$ schwach* topologisiert
vermöge den Halbnormen $p_{v}\in P^{*}$
\begin{equation*}
p_{v}(\phi)=|\phi(v)|\quad\quad \forall\:\phi\in
\V^{*},\: v\in \V
\end{equation*}
und man beachte, dass dann bereits $|c|\:p_{v}=p_{cv}\in
P^{*}$.
Bezeichne wieder $P$ das Halbnormensystem aller bezüglich
dieser Topologie stetigen Halbnormen, so sehen wir vermöge
$||_{max}=\displaystyle\sum_{i=1}^{n}p_{e_{i}}$, dass wir es hier in
der Tat mit einer Verallgemeinerung von i.) zu tun haben.
Für die Vervollständigung sei $\net{h}\subseteq
\Ss^{\bullet}(\V^{*})$ ein Cauchynetz und
$\p_{v}(h_{\alpha})\leq \hat{c}$ für $\alpha\geq
\alpha_{\hat{c}}$. Dies bedeutet
\begin{equation*}
\lim_{\alpha}\p_{v}(h_{\alpha})=\lim_{\alpha}\sum_{l=0}^{k_{\alpha}}p_{v}^{l}\left(\sum_{i_{1},…,i_{l}}u_{\alpha}^{i_{1}}\vee…\vee
u_{\alpha}^{i_{l}}\right)\leq \hat{c}\quad\quad
\forall\:\alpha \geq \alpha_{\hat{c}}
\end{equation*}
und da vermöge $(\Delta_{k}^{*}z)(v)=z\:(\overbrace{v,…,v}^{k})$,
$z\in \mathrm{T}^{k}(\V^{*})$
\begin{equation}
\label{eq:abskonv}
\begin{split}
(\Delta_{k}^{*}z)(v)=&\sum_{i=1}^{n}u^{i}_{1}(v)…u^{i}_{k}(v)\leq \left|\sum_{i=1}^{n}u^{i}_{1}(v)…u^{i}_{k}(v)\right|
\\\leq&
\sum_{i=1}^{n}\left|u^{i}_{1}(v)…u^{i}_{k}(v)\right|=
\sum_{i=1}^{n}p_{v}(u^{i}_{1})…p_{v}(u^{i}_{k}),
\end{split}
\end{equation}
für alle Zerlegungen
$\displaystyle\sum_{i=1}^{n}u^{i}_{1}\ot…\ot u^{i}_{k}$ von
$z$, folgt $(\Delta_{k}^{*}z)(v)\leq p_{v}^{k}(z)$.
Das bedeutet
\begin{align*}
(\Delta^{*}h_{\alpha})(v)=&\:\sum_{l=0}^{k_{\alpha}}\left(\Delta_{l}^{*}\sum_{i_{1},…,i_{l}}u_{\alpha}^{i_{1}}\vee…\vee
u_{\alpha}^{i_{l}}\right)(v)
\\=&\:\sum_{l=0}^{k_{\alpha}}\left(\sum_{i_{1},…,i_{l}}u_{\alpha}^{i_{1}}\vee…\vee
u_{\alpha}^{i_{l}}\right)(v,…,v)
\\\leq&\:\sum_{l=0}^{k_{\alpha}}p_{v}^{l}\left(\sum_{i_{1},…,i_{l}}u_{\alpha}^{i_{1}}\vee…\vee
u_{\alpha}^{i_{l}}\right)
\\\leq&\:\hat{c},
\end{align*}
für alle $\alpha\leq \alpha_{\hat{c}}$ und folglich
$(\Delta^{*}h)(v)=\displaystyle\lim_{\alpha}(\Delta^{*}h_{\alpha})(v)\leq
\hat{c}$, die selbe Aussage erhalten wir mit
\eqref{eq:abskonv} auch für
\begin{align*}
|\Delta^{*}h\:|(v)=\lim_{\alpha}|\Delta^{*}h_{\alpha}|(v)=\lim_{\alpha}\sum_{l=0}^{k_{\alpha}}\left|\Delta_{l}^{*}\sum_{i_{1},…,i_{k}}u_{\alpha}^{i_{1}}\vee…\vee
u_{\alpha}^{i_{l}}\:\right|(v).
\end{align*}
In diesem Sinne definiert jedes Element $h\in
\mathrm{Hol}(\V^{*})$ eine wohldefinierte, komplexwertige
Funktion auf $\V$ und in der Tat dürfen wir diese sogar als
absolut konvergente
Potenzreihen mit abzählbar vielen Summanden in den
$\widehat{\Ss^{k}(V^{*})}=\widehat{\Ss^{k}\left(\widehat{V^{*}}\right)}$
auffassen. Dies folgt, da wir uns $\Ss^{\bullet}(\V^{*})$
Eingebettet vorstellen dürfen in das Kartesische Produkt
\begin{equation*}
\Ss^{0}(\V*)\times \Ss^{1}(\V*)\times \Ss^{3}(\V*)\times… \Ss^{|\mathbb{N}|}(\V^{*})
\end{equation*}
topologisiert vermöge dem Halbnormensystem $\hat{\Pp}$, bestehend aus
\begin{equation*}
\p_{I}=\sum^{endl}_{j}p^{j}\quad\quad p\in P
\end{equation*}
Beispielsweise definiert jede absolut konvergente Potenzreihe
$\displaystyle\sum_{k=0}^{\infty}a_{k}x^{k}$ auf $\mathbb{R}$
für jedes $u\in \V^{*}$ vermöge $h=\displaystyle\sum_{k=0}^{\infty}a_{k}
\overbrace{u\vee…\vee u}^{k}$ ein Element aus
$\mathrm{Hol}(V^{*})$. In der Tat erhalten wir zunächst
\begin{equation*}
\hat{\p}_{v}(h)=\sum_{k=0}^{\infty}|a_{k}|\:|u(v)|^{k}<\infty
\end{equation*} und
\begin{equation*}
\p_{v}(h_{m}-h_{n})=\p_{v}\left(\sum_{k=m}^{n}a_{k}
u\vee…\vee
u\right)=\sum_{k=m}^{n}|a_{k}|\:|u(v)|^{k}<
\epsilon\quad\quad\: m,n
\end{equation*}
Beispiel Exponentialfunktion.
\end{enumerate}
\end{beispiel}
Ziel dieses Abschnittes soll es letztlich wieder sein zu zeigen, dass
\begin{equation*}
HH_{cont.}^{k}(\Hol,\mathcal{M})\cong H^{k}(\Hom^{cont.}_{\mathcal{A}^{e}}(\K_{c},\mathcal{M})).
\end{equation*}
Leider wird dies mit den bisherigen Methoden allein nicht ohne
weiteres möglich sein. Mehr dazu nach der nächsten wichtigen Definition.
\begin{definition}[Vervollständigte Bar-und Koszulkomplexe]
\label{def:vervollstBarKoszulkompl}
\begin{enumerate}
\item
Wir betrachten die Räume
\begin{equation*}
\cB^{c}_{0}=\widehat{\mathcal{A}\pite \mathcal{A}}=\widehat{\mathcal{A}^{e}}\qquad\text{und}\qquad\cB^{c}_{k}=\widehat{\mathcal{A}\pite
\underbrace{\mathcal{A}\pite…\pite
\mathcal{A}}_{k-\text{mal}} \pite \mathcal{A}},
\end{equation*} also die Vervollständigungen der Bar-Moduln
über $\mathcal{A}=\Hol$.
Bezeichne $*_{L}$ die stetige
$\mathcal{A}^{e}$-Linksmodulmultiplikation in den
$\C^{c}_{k}$, so folgt mit der Kompatibilität der $\C^{c}_{k}$ deren $\mathbb{C}$-Bilinearität und wir
erhalten vermöge der kanonischen isometrischen
Einbettungen $\C^{c}_{k}\subseteq\cB^{c}_{k}$ für $k\geq 0$,
wieder stetige $\mathbb{C}$ bilineare Fortsetzungen
\begin{equation*}
\hat{*}_{L}: \hat{\mathcal{A}}^{e}\times
\cB^{c}_{k}\longrightarrow \cB^{c}_{k},
\end{equation*}die die $\cB^{c}_{k}$ zu lokal konvexen,
präkompatiblen, $\hat{\mathcal{A}}^{e}$ verträglichen und
zudem hausdorffschen $\hat{\mathcal{A}^{e}}$-Linksmoduln
machen.
Mit
\begin{equation*}
\hat{d}^{c}_{k}:\cB^{c}_{k}\longrightarrow \C^{c}_{k-1}\subseteq \cB^{c}_{k-1}
\end{equation*}
bezeichnen wir die stetigen $\mathbb{C}$-linearen
Fortsetzungen der stetigen $\mathbb{C}$-linearen
Kettenhomomorphismen
\begin{equation*}
d^{c}_{k}:\C^{c}_{k}\longrightarrow \C^{c}_{k-1}\subseteq \cB^{c}_{k-1},
\end{equation*}
die vermöge
\begin{align*}
\hat{d}^{c}_{k}\:\left(\hat{a}^{e}\:\hat{*}_{L}\:
\hat{x}^{k}\right)=&\hat{d}^{c}_{k}\left(\lim_{\alpha\times\beta}\left[a^{e}_{\alpha}\:*_{L}\:
x^{k}_{\beta}\right]\right)=\lim_{\alpha\times\beta}d^{c}_{k}\left(a^{e}_{\alpha}\:*_{L}\:
x^{k}_{\beta}\right)
\\=&\lim_{\alpha\times\beta}\left[a^{e}_{\alpha}\:*_{L}\:
d^{c}_{k}\big( x^{k}_{\beta}\big)\right]=\hat{a}^{e}\:\hat{*}_{L}\: \hat{d}^{c}_{k}(\hat{x}^{k})
\end{align*}zu $\hat{\mathcal{A}}^{e}$-Homomorphismen
werden. Der so definierte topologischen Kettenkomplex
$(\cB_{c},\hat{d}_{c})$ soll im Folgenden auch als
vervollständigter Barkomplex über der Algebra $\Hol$
bezeichnet werden.
\item
Wir rufen uns den topologischen Koszulkomplex der Algebra
$\Ss^{\bullet}(\V)$ ins Gedächtnis und bezeichnen diesen mit
$(\wt{\K}_{c},\tilde{\pt}_{c})=\Ss^{\bullet}(\V)\pite
\Ss^{\bullet}(\V) \pite
\Lambda^{k}(\V)$, die zugehörige $\Ss^{\bullet}(\V)\pite
\Ss^{\bullet}(\V)$ Modulmultiplikation bezeichnen wir
mit $*_{S}$.
Wir definieren
\begin{align*}
\cK_{k}^{c}=\widehat{\tilde{\K}^{c}_{k}}=\widehat{\Bigg(\Ss^{\bullet}(\V)\pite
\Ss^{\bullet}(\V) \pite
\Lambda^{k}(\V)\Bigg)}=&\:\widehat{\Bigg(\widehat{\Ss^{\bullet}(\V)}\pite
\widehat{\Ss^{\bullet}(\V)}\pite
\widehat{\Lambda^{k}(\V)}\Bigg)}
\\=& \:\widehat{\Bigg(\Hol\pite \Hol \pite
\widehat{\Lambda^{k}(\V)}\Bigg)}
\\=&\:\widehat{\Bigg(\Hol\pite \Hol \pite
\Lambda^{k}(\V)\Bigg)},
\end{align*} sowie
\begin{equation*}
\cK_{0}^{c}=\widehat{\Ss^{\bullet}(\V)\pite
\Ss^{\bullet}(\V)}=\widehat{\Hol\pite \Hol}= \hat{\mathcal{A}}^{e},
\end{equation*}
wobei die Gleichheiten mit
\thref{satz:PiTopsatz}.vi.) folgen.
Analog zu i.) werden diese vermöge $\widehat{\Ss^{\bullet}(\V)\pite
\Ss^{\bullet}(\V)}= \hat{\mathcal{A}}^{e}$ durch die stetige
Fortsetzung $\hat{*}_{S}$ von $*_{S}$ zu
hausdorffschen, lokal konvexen $\hat{\mathcal{A}^{e}}$
Linksmoduln und wir erhalten zudem stetige
$\hat{\mathcal{A}}^{e}$ lineare Fortsetzungen
$\cpt^{k}_{c}$ der $\Ss^{\bullet}(\V)\pite
\Ss^{\bullet}(\V)$ Modulhomomorphismen $\tilde{\pt}^{c}_{k}$.
Den so gewonnenen topologischen Kettenkomplex
$(\cK_{c},\cpt_{c})$ bezeichnen wir als vervollständigten
Koszulkomplex über $\mathcal{A}=\Hol$.
\item
Mit $\K_{k}^{c}$ bezeichnen wir im Folgenden die
hausdorffschen, lokal konvexen $\mathcal{A}^{e}$-Linksmoduln
$\Hol\pite\Hol\pite \Lambda^{k}(\V)$ mit der offensichtlichen
$\hat{\mathcal{A}}^{e}$ Multiplikation $*_{Hol}$ in den ersten beiden
Faktoren. Diese induziert dann ebenfalls eine stetige
$\mathcal{A}^{e}$ Multiplikation $\hat{*}_{Hol}$ auf
$\cK_{k}^{c}$ und da
\begin{equation*}
*_{Hol}\big|_{\Ss^{\bullet}(\V)\pite \Ss^{\bullet}(\V)\times
\Lambda^{k}(\V)}=*_{S}=\hat{*}_{S}\big|_{\Ss^{\bullet}(\V)\pite \Ss^{\bullet}(\V)\times
\Lambda^{k}(\V)},
\end{equation*} stimmen beide auf einer Dichten Teilmenge
von $\cK_{c}^{k}$ überein. Mit der Eindeutigkeit besagter
stetigen Fortsetzung von $*_{S}$ folgt nun beruhigenderweise
\begin{equation}
\label{eq:Modulmultsallegleich}
\hat{*}_{S}=\hat{*}_{Hol}\qquad\text{und somit}\qquad\hat{*}_{S}\big|_{\K_{c}^{k}}=*_{Hol}.
\end{equation}
Mit $(\K_{k}^{c},\pt_{c})$ bezeichnen wir den
Kettenkomplex mit Kettendifferentialen
\begin{equation*}
\pt^{c}_{k}=\hat{\pt}^{c}_{k}\big|_{\K_{k}^{c}}
\end{equation*}für dessen
Wohldefiniertheit wir zeigen müssen, dass die $\pt^{k}_{c}$ in der Tat
Bilder in $\K_{c}^{k-1}$ besitzen. Seien dafür
$\Ss^{\bullet}(\V)\supseteq\net{x}{I}\rightarrow x\in \Hol$,
$\Ss^{\bullet}(\V)\supseteq\{y_{\beta}\}_{\beta\in
J}\rightarrow y\in \Hol$, dann folgt $\{x_{\alpha}\pite
y_{\beta}\pite u\}_{\alpha\times \beta \in I\times J}\rightarrow
x\pite y\pite u \in \K_{k}^{c}$ vermöge zweimaliger Anwendung
der Dreiecksungleichung und der Definition der $\pi$-Topologie
(siehe auch Beweis zu \thref{satz:PiTopsatz}.vi.)) und es folgt
\begin{align*}
\pt_{k}^{c}(x\pite& y \pite u)=\lim_{\alpha\times
\beta}\tilde{\pt}_{k}^{c}\:(x_{\alpha}\pite y_{\beta}\pite u)\\=&\lim_{\alpha\times
\beta}\sum_{j=1}^{k}(-1)^{j-1}(u_{j}\vee x_{\alpha}\pite
y_{\beta}\pite u^{j}) - \lim_{\alpha\times
\beta}\sum_{j=1}^{k}(-1)^{j-1}(x_{\alpha}\pite
u_{j}\vee y_{\beta}\pite u^{j})
\\=& \sum_{j=1}^{k}(-1)^{j-1}(u_{j}* x \pite
y \pite u^{j}) - \sum_{j=1}^{k}(-1)^{j-1}(x\pite
u_{j}* y\pite u^{j})\in \K_{c}^{k-1}
\end{align*}mit $*$ die Algebramultiplikation in $\Hol$.
\end{enumerate}
\end{definition}
Folgende Proposition klärt die Sinnhaftigkeit obiger Definitionen.
\begin{proposition}
\label{prop:SinnhvervollstKomplexe}
\begin{enumerate}
\item
Es sind sowohl $(\C_{c},d_{c})$, $(\cB_{c},\hat{d}_{c})$, als auch
$(\cK_{c},\cpt_{c})$ exakte Kettenkomplexe.
\item
Gegeben ein vollständiger, hausdorffscher, lokal konvexer,
präkompatibler $\Hol$ verträglicher
$\Hol-\Hol$ Bimodul $\mathcal{M}$ (z.B. $\mathcal{M}=\Hol$
oder ein durch Vervollständigung gewonnener Bimodul), so gilt
\begin{equation}
\label{eq:BarKohomIsom}
H^{k}(\Hom^{cont.}_{\mathcal{A}^{e}}(\C_{c},\mathcal{M}))\cong H^{k}(\Hom^{cont.}_{\hat{\mathcal{A}}^{e}}(\cB_{c},\mathcal{M})),
\end{equation}sowie
\begin{equation}
\label{eq:KoszKohomIso}
H^{k}(\Hom^{cont.}_{\mathcal{A}^{e}}(\K_{c},\mathcal{M}))\cong H^{k}(\Hom^{cont.}_{\hat{\mathcal{A}}^{e}}(\cK_{c},\mathcal{M})).
\end{equation}
\end{enumerate}
\begin{beweis}
\begin{enumerate}
\item
Die Exaktheit von $(\C_{c},d_{c})$ ist klar, da die
exaktheitsliefernde und zudem stetige Homotopie $h^{c}$ rein
algebraisch definiert war. Da nun die $h^{c}_{k}$ stetige
$\mathbb{C}$-lineare Abbildungen sind, existieren stetige
lineare Fortsetzungen $\hat{h}^{c}_{k}$ auf
$(\cB_{c},\hat{d}_{c})$. Diese sind eindeutig bestimmt,
da alle beteiligten Räume hausdorffsch sind. Damit bleibt die Homotopieeigenschaft
erhalten, da mit
\begin{equation*}
h^{c}_{k-1}\cp d^{c}_{k}+d^{c}_{k+1}\cp h^{c}_{k}=id_{\C^{k}_{c}}
\end{equation*} und
$id_{\cB^{c}_{k}}=\hat{id}_{\C^{c}_{k}}$ folgt, dass
\begin{align*}
id_{\cB^{c}_{k}}=\widehat{\Big(h^{c}_{k-1}\cp
d^{c}_{k}+d^{c}_{k+1}\cp
h^{c}_{k}\Big)}=\widehat{h^{c}_{k-1}\cp
d^{c}_{k}}+\widehat{d^{c}_{k+1}\cp h^{c}_{k}}
=\hat{h}^{c}_{k-1}\cp \hat{d^{c}_{k}}+\hat{d}^{c}_{k+1}\cp \hat{h}^{c}_{k}.
\end{align*}
Die zweite Gleichheit ist klar, für die
Letzte beachte man, dass
\begin{equation*}
\widehat{f\cp g}\:(x)=\lim_{\alpha}\:(f\cp
g)(x_{\alpha})=\lim_{\alpha}
f(g(x_{\alpha}))=\hat{f}\left(\lim_{\alpha}g(x_{\alpha})\right)=\hat{f}\cp
\hat{g}\:(x),
\end{equation*}für $\C_{c}^{k}\supseteq\net{x}{I}\rightarrow
x\in \cB^{k}_{c}$, da $\C_{c}^{k+1}
\supseteq\{g(x_{\alpha})\}_{\alpha\in I}\rightarrow
\hat{g}(x)\in \cB^{k+1}_{c}$ per Definition.
Für $(\cK_{c},\cpt_{c})$ folgt die
Behauptung genauso, da wir auch die Stetigkeit der zu
$(\tilde{\K_{c}},\pt_{c})$ gehörigen Homotopie
nachgewiesen hatten (\thref{prop:wichpropKoszStetSym}.iii.)).
\item
$\mathcal{M}$ ist vollständig und hausdorffsch und jedes $\phi\in
\Hom^{cont.}_{\mathcal{A}^{e}}(\C_{c}^{k},\mathcal{M})$
insbesondere $\mathbb{C}$ linear mit der Präkompatibilität
und $\Hol$ Verträglichkeit von $\mathcal{M}$, da
\begin{align*}
\phi(\lambda x^{c}_{k})=&\phi(\:\lambda1\pite 1 *_{e}
x_{k})=\lambda1\pite 1 *_{e}\phi(x^{c}_{k})
\\=& \lambda (1\pite 1*_{e} \phi(x^{c}_{k}))= \lambda \phi(x^{c}_{k}).
\end{align*} Es existiert daher eine
eindeutig bestimmtes stetige $\mathbb{C}$ lineare
Fortsetzung $\hat{\phi}$ auf $\cB_{c}$, deren
Einschränkung $\phi$ ist. Die $\hat{\phi}$ Linearität von
$\hat{\mathcal{A}}^{e}$ sieht man analog zu
\thref{def:vervollstBarKoszulkompl}.i).
Umgekehrt liefert jedes $\hat{\phi}\in
\Hom^{cont.}_{\hat{\mathcal{A}}^{e}}(\cB_{c},\mathcal{M})$ via
Einschränkung ein $\phi\in
\Hom^{cont.}_{\mathcal{A}^{e}}(\C_{c},\mathcal{M})$,
dessen eindeutig bestimmte stetige $\mathbb{C}$ und
$\hat{\mathcal{A}}^{e}$ lineare Fortsetzung sie ist.
Das zeigt zunächst
\begin{equation*}
\Hom^{cont.}_{\mathcal{A}^{e}}(\C_{c},\mathcal{M})\cong \Hom^{cont.}_{\hat{\mathcal{A}}^{e}}(\cB_{c},\mathcal{M}).
\end{equation*}
Für die Isomorphie der Kohomologiegruppen beachten wir,
dass
\begin{equation*}
d^{c*}_{k}(\phi)=\phi\cp d^{c}_{k}=0 \Longleftrightarrow \hat{d}^{c*}_{k}(\hat{\phi})=\hat{\phi}\cp \hat{d}^{c}_{k}=0,
\end{equation*}da einerseits $\hat{\phi}\cp
\hat{d}^{c}_{k}\big|_{\C_{c}}=\phi\cp d^{c}_{k}$ und
andererseits mit $\phi\cp d^{c}_{k}=0$ auch\\
$\hat{\phi}\cp \hat{d}^{c}_{k}=\widehat{\phi\cp
d^{c}_{k}}=0$. Mit dem gleichen Argument folgt
\begin{equation*}
d^{c*}_{k}\cp d^{c*}_{k+1}(\phi)=0 \Longleftrightarrow
\hat{d}^{c*}_{k}\cp \hat{d}^{c*}_{k+1}(\hat{\phi})=0.
\end{equation*} Es folgen Injektivität und
Surjektivität von $[\phi]\longleftrightarrow
[\hat{\phi}]$ und somit \eqref{eq:BarKohomIsom}.
Für \eqref{eq:KoszKohomIso} beachten wir, dass wieder
jedes $\phi \in \Hom_{\mathcal{A}^{e}}(\K_{k}^{c},\mathcal{M})$ eine
eindeutige stetige $\mathbb{C}$ und $\hat{\mathcal{A}}^{e}$ lineare
Fortsetzung $\hat{\phi}\in \Hom_{\hat{\mathcal{A}}^{e}}(\cK_{k}^{c},\mathcal{M})$ besitzt. Umgekehrt
induziert jedes $\hat{\phi}\in
\Hom_{\hat{\mathcal{A}}^{e}}(\cK_{k}^{c},\mathcal{M})$ vermöge
Einschränkung ein $\mathbb{C}$ lineare Abbildung
\begin{equation*}
\phi:\K_{c}^{k}\rightarrow \mathcal{M},
\end{equation*}die vermöge
\eqref{eq:Modulmultsallegleich} $\mathcal{A}^{e}$
linear ist. Es folgt
\begin{equation*}
\Hom^{cont.}_{\mathcal{A}^{e}}(\K_{c},\mathcal{M})\cong \Hom^{cont.}_{\hat{\mathcal{A}}^{e}}(\cK_{c},\mathcal{M})
\end{equation*} und mit
\begin{align*}
\pt_{k}^{c*}(\phi)=0&\Longleftrightarrow
\hat{\pt}_{k}^{c*}(\hat{\phi})=0\\
\pt_{k}^{c*}\cp \pt_{k+1}^{c*} (\phi)=0&\Longleftrightarrow \hat{\pt}_{k}^{c*}\cp\hat{\pt}_{k}^{c*}(\hat{\phi})=0
\end{align*} der Rest der Behauptung.
\end{enumerate}
\end{beweis}
\end{proposition}
\begin{bemerkung}
\label{bem:VervollstKomplexeBem}
Man beachte, dass weder $(\cB_{c}, \hat{d}_{c})$ noch
$(\cK_{c},\cpt_{c})$ projektiv sind. $(\C_{c},d_{c})$ ist
projektiv und exakt und $(\K_{c},\pt_{c})$ nur projektiv, da die
Einschränkung $\hat{\tilde{h}}_{c}\big|_{\K_{c}}$ der stetigen Fortsetzung der exaktheitsliefernden
Homotopie $\tilde{h}_{c}$ des topologischen Koszulkomplexes
$(\tilde{\K}_{c},\tilde{\pt}_{c})$ im Gegensatz zur Einschränkung
der Kettendifferentiale
$\cpt_{c}\big|_{\K_{c}}$ im allgemeinen sicher nicht
ausschließlich in die Unterräume $\K_{k}^{c}\subseteq \cK_{k}^{c}$
abbildet, sonder in der Tat in den Vervollständigungen
$\cK_{k}^{c}$ landet. Das Selbe überlegt man sich auch für die
Einschränkungen der stetigen Fortsetzungen $\hat{F}$, $\hat{G}$ der Kettenabbildungen
$F$ und $G$.
Rein algebraisch haben wir somit nichts in der Hand,
dass uns die Isomorphie
\begin{equation*}
H^{k}(\Hom_{\mathcal{A}^{e}}(\K_{c},\mathcal{M}))\cong
H^{k}(\Hom_{\mathcal{A}^{e}}(\C_{c},\mathcal{M}))
\end{equation*}liefern würde,
von der Isomorphie der stetigen Kohomologien ganz zu schweigen.
Der Ausweg
sind nun obige vervollständigte Kettenkomplexe, denn für
diese haben wir wohldefinierte stetige Kettenabbildungen $\hat{F}$
und $\hat{G}$ nach \thref{lemma:Fkettenabb} und
\thref{prop:GcirFistidundGKettenabb}.ii.), da sich besagte
Algebraische Relationen mit der Hausdorffeigenschaft aller
beteiligter Moduln abermals auf die Fortsetzungen übertragen. Für
diese müssen wir nun im Folgenden per Hand nachweisen, dass
die aus ihnen durch Anwendung des $\hom$-Funktors enstehenden
Kettenabbildungen $\hat{F}^{*}$ und $\hat{G}^{*}$ Isomorphismen
auf Kohomologieniveau induzieren. Mit deren Stetigkeit induzierten sie dann
ebenfalls Isomorphismen auf den stetigen Kohomologiegruppen und
wir wären am Ziel angelangt. Dafür beachte
man, dass mit \thref{prop:GcirFistidundGKettenabb}.i)
\begin{equation*}
\hat{G}_{k}\cp \hat{F}_{k}=id_{\cK_{k}^{c}} \Longrightarrow
\hat{F}^{*}_{k}\cp \hat{G}^{*}_{k}=id_{\Hom_{\mathcal{\hat{A}}^{e}}(\cK_{k}^{c},\mathcal{M})},
\end{equation*} also
\begin{equation*}
\widetilde{\hat{F}^{*}}_{k}\cp\widetilde{\hat{G}^{*}}_{k} =id_{H^{k}(\Hom_{\mathcal{\hat{A}}^{e}}(\cK_{k}^{c},\mathcal{M})}
\end{equation*} und mit der Stetigkeit der $\hat{F}_{k}$,
$\hat{G}_{k}$ ebenfalls
\begin{equation*}
\widetilde{\hat{F}^{*}}_{k}\cp\widetilde{\hat{G}^{*}}_{k} =id_{H^{k}(\Hom^{cont.}_{\mathcal{\hat{A}}^{e}}(\cK_{k}^{c},\mathcal{M})}.
\end{equation*}
Es folgt, dass $\widetilde{\hat{F}^{*}}$ surjektiv und
$\widetilde{\hat{G}^{*}}$ injektiv und für die gewünschte Aussage
benötigen wir dann lediglich, dass
\begin{equation*}
\hat{F}\cp \hat{G} \sim id_{\cB_{c}},
\end{equation*} vermöge einer $\hat{\mathcal{A}}^{e}$ linearen
Homotopie $\hat{s}_{k}:\cB_{k}^{c}\rightarrow
\cB_{k+1}^{c}$. Gerade wegen dieser Linearitätseigenschaft
erhalten wir durch Anwendung des
$\hom$-Funktors die Homotopie
\begin{equation*}
\hat{G}^{*}\cp \hat{F}^{*}\sim id_{\Hom_{\hat{\mathcal{A}}^{e}}(\cB_{c},\mathcal{M})}.
\end{equation*} Es folgt
\begin{equation*}
\widetilde{\hat{G}^{*}}_{k}\cp\widetilde{\hat{F}^{*}}_{k} =id_{H^{k}(\Hom^{cont.}_{\mathcal{\hat{A}}^{e}}(\cB_{k}^{c},\mathcal{M})},
\end{equation*}mithin die Injektivität von
$\widetilde{\hat{F}^{*}}_{k}$ und die Surjektivität von $\widetilde{\hat{G}^{*}}_{k}$.
\end{bemerkung}
Im Sinne obiger Bemerkung definieren wir die stetige Abbildung
\begin{equation*}
\theta_{k}=F_{k}\cp
G_{k}
\end{equation*}
und bezeichnen die topologischen Bar-Moduln
über der Algebra $\tilde{\mathcal{A}}=\Ss^{\bullet}(\V)$ mit $\wt{\C}^{c}_{k}$.
\begin{lemma}
Es gilt
\begin{equation}
\label{eq:thetarel}
\theta_{k}\cp d^{c}_{k+1}=d^{c}_{k+1}\cp \theta_{k+1}.
\end{equation}
\begin{beweis}
Es folgt für $G_{k}:\wt{\C}^{c}_{k}\longrightarrow \wt{\K}^{c}_{k}$
und $F_{k}:\wt{\K}^{c}_{k}\longrightarrow \wt{\C}^{c}_{k}$ mit
$G_{k}\cp d^{c}_{k+1}=\pt^{c}_{k+1}\cp G_{k+1}$ und $F_{k}\cp \pt^{c}_{k+1}=d^{c}_{k+1}\cp F_{k+1}$
\begin{align*}
\theta_{k}\cp d^{c}_{k+1}=&F^{c}_{k}\cp\:(G_{k}\cp
d^{c}_{k+1})=F_{k}\cp\:(\pt^{c}_{k+1}\cp
G_{k+1})
\\=&(F_{k}\cp\pt^{c}_{k+1})\cp
G_{k+1}=(d^{c}_{k+1}\cp F_{k+1})\cp G_{k+1}
\\=&\: d^{c}_{k+1}\cp \theta_{k+1}.
\end{align*}
\end{beweis}
\end{lemma}
Um besagte Homotopie $\hat{s}$ zu konstruieren bedienen wir uns bei
\cite[Kapitel 4.6]{Weissarbeit}
geben zunächst eine Methode an, aus $\mathbb{C}$ linearen Abbildungen $\phi:\wt{\C}^{c}_{k}\rightarrow
\wt{\C}^{c}_{k'}$, $\mathcal{\mathcal{A}}^{e}$ lineare
Abbildungen $\overline{\phi}:\wt{\C}^{c}_{k}\rightarrow
\wt{\C}^{c}_{k'}$ zu konstruieren.
\begin{definition}
Gegeben eine $\mathbb{C}$ lineare Abbildung $\phi:\wt{\C}^{c}_{s}\rightarrow \wt{\C}^{c}_{r}$ $\mathbb{C}$, so
definieren wir die zu $\phi$ gehörige $\mathcal{A}^{e}$-lineare
Abbildung
\begin{equation}
\begin{split}
\label{eq:AelinausClin}
\overline{\phi}: \wt{\C}^{c}_{s}&\longrightarrow \wt{\C}^{c}_{r}\\
v\pite \alpha_{s}\pite w&\longmapsto v\pite w *_{e} \phi(1\pite
\alpha_{s}\pite 1).
\end{split}
\end{equation}Diese ist vermöge der Präkompatibilität und $\tilde{\mathcal{A}}$ Verträglichkeit
der $\wt{\C}^{c}_{r}$ ebenfalls $\mathbb{C}$-linear, mithin mit
\thref{kor:WohldefTensorprodabbildungen} durch \eqref{eq:AelinausClin}
wohldefiniert. Betrachten wir die Räume
\begin{equation*}
\wt{\C}'^{c}_{s}=\tilde{\mathcal{A}}\pite \wt{\C}^{c}_{s}\pite \tilde{\mathcal{A}}
\end{equation*}und die $\tilde{\mathcal{A}}^{e}$, sowie $\mathbb{C}$-linearen Abbildungen
\begin{align*}
\phi':\wt{\C}'^{c}_{s}&\longrightarrow \wt{\C}'^{c}_{r}\\
v\pite v'\pite \alpha_{s}\pite w'\pite w&\longmapsto v\pite \phi(v'\pite
\alpha_{s}\pite w')\pite w,
\end{align*}
\begin{align*}
p_{s}:\wt{\C}^{c}_{s}&\longrightarrow \wt{\C}'^{c}_{s}\\
v\pite \alpha_{s}\pite w&\longmapsto v\pite 1\pite
\alpha_{s}\pite 1\pite w,
\end{align*}
\begin{align*}
i_{r}:\wt{\C}'^{c}_{r}&\longrightarrow \wt{\C}^{c}_{r}\\
v\pite v'\pite \alpha_{s}\pite w'\pite w&\longmapsto vv'\pite
\alpha_{s}\pite w'w,
\end{align*}so lässt sich $\overline{\phi}$ offensichtlich auch schreiben
als $\overline{\phi}=i_{r}\cp \phi'\cp p_{s}$, zudem folgt unmittelbar
$i_{t}\cp p_{t}= \id_{\wt{\C}^{c}_{t}}$.
\end{definition}
\begin{proposition}
\label{prop:gedoens}
Gegeben $\mathbb{C}$-lineare Abbildungen
$\psi:\wt{\C}^{c}_{s}\rightarrow \wt{\C}^{c}_{t}$ und
$\phi:\wt{\C}^{c}_{t}\rightarrow \wt{\C}^{c}_{r}$.
\begin{enumerate}
\item
Sei $s=t=r$, so folgt
\begin{equation*}
\overline{\phi+\psi}=\overline{\phi}+\overline{\psi},\quad
\overline{\id}_{\wt{\C}^{c}_{s}}=\id_{\wt{\C}^{c}_{s}},\quad
\overline{0}=0
\end{equation*}
\item
Ist $\phi$ $\mathcal{A}^{e}$-linear, so gilt
$\overline{\phi}=\phi$, mithin
$\overline{d}^{c}_{k}=d^{c}_{k}$ und
$\overline{\theta}^{c}_{k}=\theta^{c}_{k}$.
\item
Ist $i_{r}\cp \phi'=\phi\cp i_{t}$, also insbesondere $\overline{\phi}=\phi$, so folgt $\phi\cp
\overline{\psi}=\overline{\phi\cp \psi}$, im Speziellen ist dies für
alle $d^{c}_{k}$ der Fall.
\item
Ist $\phi$ stetig, so auch $\overline{\phi}$.
\end{enumerate}
\begin{beweis}
i.) und ii.) sind unmittelbar klar. Für iii.) rechnen wir
\begin{align*}
(\phi\cp\psi)'(v\pite v'\pite \alpha_{s}\pite w'\pite
w)=&\:v\pite \big[(\phi\cp\psi)(v'\pite \alpha_{s}\pite w')\big]\pite
w
\\=&\:\phi'(v\pite \psi(v'\pite \alpha_{s}\pite w')\pite
w)
\\=&\:(\phi'\cp \psi')\:(v\pite v'\pite \alpha_{s}\pite w'\pite
w).
\end{align*} und erhalten
\begin{equation*}
\overline{\phi\cp \psi}= i_{r}\cp
\phi'\cp \psi'\cp p_{s}=\phi\cp i_{t}\cp \psi'\cp
p_{s}=\phi\cp \overline{\psi}.
\end{equation*}
Für die $d^{c}_{k}$ rechnen wir
\begin{align*}
(d^{c}_{k}\cp i_{k})(v\pite x_{0}\pite…\pite x_{k+1}\pite
w)=&\sum_{j=0}^{k}(-1)^{k} vx_{0}\pite…\pite
x_{j}x_{j+1}\pite…\pite x_{k+1}w
\\=&\: i_{k-1}\left(v\pite d^{c}_{k}(x_{0}\pite…\pite
x_{k+1})\pite w\right)
\\=&\: (i_{k-1}\cp d'^{c}_{k})(v\pite x_{0}\pite…\pite
x_{k+1}\pite w).
\end{align*}
Sei für iv.) $\phi$ stetig, so folgt mit der Stetigkeit von $*_{e}$
\begin{align*}
p^{k+2}\left(\overline{\phi}\:(v\pite \alpha_{1}\pite…\pite \alpha_{k}\pite
w)\right)=&\:p^{k+2}(v\pite w *_{e} \phi(1\pite\alpha_{1}\pite…\pite \alpha_{k}\pite1))
\\\leq&\:c\:q_{1}^{2}(v\pite w)\:
q_{2}^{k+2}(1\pite\alpha_{1}\pite…\pite \alpha_{k}\pite 1)
\\=&\: \hat{c}\: q_{1}(v)\:q_{1}(w)\:q_{2}(\alpha_{1})…q_{2}(\alpha_{k})
\end{align*} mit $\hat{c}=c\: q_{2}(1)^{2}$, was die
Stetigkeit in $\pi$ zeigt und den Beweis beendet.
\end{beweis}
\end{proposition}
\begin{lemma}
\label{lemma:Homotopiejdfgjkf}
Es ist $\id_{\wt{\C}^{c}_{k}}-\theta_{k}=d^{c}_{k+1} s_{k} +
s_{k-1} d^{c}_{k}$, also $\id_{\wt{\C}^{c}}\sim \theta$ vermöge
der stetigen $\mathcal{A}^{e}$-linearen Homotopie
$s_{k}:\wt{\C}^{c}_{k}\rightarrow \wt{\C}^{c}_{k+1}$, $k\geq 0$
rekursiv definiert durch
\begin{equation*}
s_{k}=\overline{h^{c}_{k}
(\id_{\wt{\C}^{c}_{k}}-\theta^{c}_{k}-s_{k-1}
d^{c}_{k})}\qquad\text{mit}\qquad s_{0}=0,
\end{equation*} dabei bezeichnet $h^{c}$ die stetige,
exaktheitsliefernde Homotopie aus
\thref{prop:topBarKomplexprop}.ii.).
\begin{beweis}
Die Stetigkeit der $s_{k}$ folgt unmittelbar aus der
Stetigkeit der definierenden Abbildungen und
\thref{prop:gedoens}.iv.), die $\mathcal{A}^{e}$-Linearität
ist ebenfalls klar. Für den Induktionsanfang rechnen wir
\begin{align*}
d^{c}_{2}\cp
s_{1}-\cancel{s_{0}\cp d^{1}_{c}}^{0}=&\:d^{c}_{2}\cp
\overline{h^{c}_{1}
\left(\id_{\wt{\C}^{c}_{1}}-\theta_{1}-\cancel{s_{0}\cp
d^{c}_{1}}\right)}\glna{iii.)}
\overline{(d^{c}_{2}\cp h^{c}_{1})
\left(\id_{\wt{\C}^{c}_{1}}-\theta_{1}\right)}
\\\glna{\eqref{eq:Homotbar}}&\:\overline{(\id_{\wt{\C}^{c}_{1}}-h^{c}_{0}\cp d^{c}_{1})\cp
\left(\id_{\wt{\C}^{c}_{1}}-\theta_{1}\right)}\glna{i.),ii.)}\id_{\wt{\C}^{c}_{1}}-\theta_{1}-\overline{h^{c}_{0}\cp
d^{c}_{1}}+\overline{h^{c}_{0}\cp d^{c}_{1}\cp
\theta_{1}}
\\\glna{\eqref{eq:thetarel}}&\id_{\wt{\C}^{c}_{1}}-\theta_{1}
\end{align*}
mit \thref{prop:gedoens}. und da
$\theta_{0}=\id_{\mathcal{A}}$. Für die höheren Grade
rechnen wir
\begin{align*}
d^{c}_{k+1}\cp s_{k}=&\:\overline{(d^{c}_{k+1}\cp
h^{c}_{k})
(\id_{\wt{\C}^{c}_{k}}-\theta_{k}-s_{k-1}\cp
d^{c}_{k})}
\\=&\:\overline{(\id_{\wt{\C}^{c}_{k}}-h^{c}_{k-1}\cp d^{c}_{k})\cp
(\id_{\wt{\C}^{c}_{k}}-\theta_{k}-s_{k-1}
d^{c}_{k})}
\\=&\:\id_{\wt{\C}^{c}_{k}}- \theta^{k}_{c} - \overline{s_{k-1}
d^{c}_{k}} -
\overline{h^{c}_{k-1}(d^{c}_{k}-d^{c}_{k}\cp\theta_{k}-(d^{c}_{k}
s_{k-1})\cp d^{c}_{k})}
\\=&\:\id_{\wt{\C}^{c}_{k}}- \theta^{k}_{c} - s_{k-1}
d^{c}_{k} -
\overline{h^{c}_{k-1}(d^{c}_{k}-\theta_{k-1} d^{c}_{k}-(\id_{\wt{\C}^{c}_{k}}-\theta_{k-1}-s_{k-2}
d^{c}_{k-1})\cp d^{c}_{k})}
\\=&\:\id_{\wt{\C}^{c}_{k}}- \theta^{k}_{c} - s_{k-1}
d^{c}_{k}.
\end{align*}
\end{beweis}
\end{lemma}
\begin{satz}
Es ist
\begin{equation*}
HH^{k}_{cont.}(\Hol),\mathcal{M})\cong
H^{k}(\Hom_{\mathcal{A}^{e}}^{cont.}(\C_{c},\mathcal{M}))\cong
H^{k}(\Hom_{\mathcal{A}^{e}}^{cont.}(\K_{c},\mathcal{M})),
\end{equation*} für $\mathcal{M}$ präkompatibel und $\Hol$
verträglich. Ist $\mathcal{M}$ symmetrisch, so gilt
\begin{equation*}
HH^{k}_{cont.}(\Hol,\mathcal{M})\cong \Hom_{\mathcal{A}^{e}}^{cont.}(\K_{c},\mathcal{M}).
\end{equation*}
\begin{beweis}
Die erste Isomorphie lieferte \thref{kor:HolBarBimodul}. Für
die Zweite betrachten wir die stetige $\mathbb{C}$, sowie
$\hat{\mathcal{A}}^{e}$-lineare Fortsetzungen $\hat{s}_{k}$ der
Homotopieabbildungen $s_{k}$ aus
\thref{lemma:Homotopiejdfgjkf}. Es folgt
\begin{equation*}
\id_{\cB^{c}_{k}}-\hat{\theta}_{k}=\hat{d}^{c}_{k+1} \hat{s}_{k} +
\hat{s}_{k-1} \hat{d}^{c}_{k}\qquad \text{also }\qquad \id_{\cB^{c}}\sim
\hat{F}\cp \hat{G}.
\end{equation*}
Mit \thref{bem:VervollstKomplexeBem} folgt
$H^{k}(\Hom_{\mathcal{A}^{e}}^{cont.}(\cB_{c},\mathcal{M}))\cong
H^{k}(\Hom_{\mathcal{A}^{e}}^{cont.}(\cK_{c},\mathcal{M}))$
und mit \thref{prop:SinnhvervollstKomplexe} schließlich die
zweite Isomorphie, die letzte Behauptung folgt analog zum
$\Ss^{\bullet}$ Fall, da mit
\thref{def:vervollstBarKoszulkompl}.iii.) für $\phi \in \K_{k}^{c*}=
\Hom_{\mathcal{A}^{e}}(\K^{c}_{k},\mathcal{M})$ und\\ $\omega=
\alpha\pite \beta\pite u \in \K_{k+1}^{c}$
\begin{align*}
(\pt_{k}^{c*}\phi)(\omega)=&\:\phi(\pt_{k}^{c}(\alpha\pit\beta\pite
u))
\\=&\:\phi\left(\sum_{j=1}^{n}(-1)^{j-1}[u_{j}*\alpha \pite\beta \pite
u^{j}-\alpha\pite u_{j}*\beta\pite u^{j}]\right)
\\=&\:\sum_{j=1}^{n}(-1)^{j-1}[u_{j}\pite 1 - 1\pite
u_{j}]\:*_{Hol}\phi(\alpha\pite \beta\pite u^{j})
\\=&\:0
\end{align*}
\end{beweis}
\end{satz}
\newpage
|
{
"timestamp": "2010-10-12T02:04:20",
"yymm": "1009",
"arxiv_id": "1009.4365",
"language": "de",
"url": "https://arxiv.org/abs/1009.4365"
}
|
\section{Introduction}
Triangulated categories and derived categories were introduced by
Grothendieck and Verdier \cite{Ver}. Today, they have widely been
used in many branches: algebraic geometry, stable homotopy theory,
representation theory, etc. In the representation theory of
algebras, we will restrict our attention to the equivalences of
derived categories, that is, derived equivalences. Derived
equivalences have been shown to preserve many invariants and provide
new connection. For instance, Hochschild homology and cohomology
\cite{Ri3}, finiteness of finitistic dimension \cite{PX} have been
shown to be invariant under derived equivalences. Moreover, derived
equivalences are related to cluster categories and cluster tilting
objects \cite{BMRR}. As is known, Rickard's Morita theory for
derived categories leaves something to be desired, though, as for
some pairs of rings, or algebras, it is currently difficult,
sometimes even impossible to verify whether there exists a tilting
complex. It is of interest to construct a new derived equivalence
from given one by finding a suitable tilting complex. Rickard
\cite{Ri2,Ri3} used tensor products and trivial extensions to get
new derived equivalences. In the recent years, Hu and Xi have
provided various techniques to construct new derived equivalences.
In \cite{HX1} they established an amazing connection between derived
equivalences and Auslander-Reiten sequences via BB-tilting modules,
and obtained derived equivalences from Auslander-Reiten triangles.
In \cite{HX3} they constructed new derived equivalences between
$\Phi$-Auslander-Yoneda algebras from a given almost $\nu$-stable
equivalence.
In \cite[Corollary 3.13]{HX3} Hu and Xi proved that, if two
representation finite self-injective Artin algebras are derived
equivalent, then their Auslander algebras are derived equivalent. In
this paper, we generalize their result and prove that, if two
Cohen-Macaulay finite Gorenstein Artin algebras are derived
equivalent, then their Cohen-Macaulay Auslander algebras are also
derived equivalent.
This paper is organized as follows. In Section 2, we review some
facts on derived categories and derived equivalences. In Section 3,
we state and prove our main result.
\section{Preliminaries}
In this section, we shall recall some definitions and notations on
derived categories and derived equivalences.
Let $\mathscr{A}$ be an abelian category. For two morphisms $\alpha:
X\ra Y$ and $\beta: Y\ra Z$, their composition is denoted by
$\alpha\beta$. An object $X\in\mathscr{A}$ is called a additive
generator for $\mathscr{A}$ if $\add(X)=\mathscr{A}$, where
$\add(X)$ is the additive subcategory of $\mathscr{A}$ consisting of
all direct summands of finite direct sums of the copies of $X$. A
complex $\cpx{X}=(X^i,d_{X}^i)$ over $\mathscr{A}$ is a sequence of
objects $X^i$ and morphisms $d_{X}^i$ in $\mathscr{A}$ of the form:
$\cdots \ra X^i\stackrel{d^i}\ra X^{i+1}\stackrel{d^{i+1}}\ra
X^{i+1}\ra\cdots$, such that $d^id^{i+1}=0$ for all
$i\in\mathbb{Z}$. If $\cpx{X}=(X^i,d_{X}^i)$ and
$\cpx{Y}=(Y^i,d_{Y}^i)$ are two complexes, then a morphism $\cpx{f}:
\cpx{X}\ra\cpx{Y}$ is a sequence of morphisms $f^i: X^i\ra Y^i$ of
$\mathscr{A}$ such that $d^i_{X}f^{i+1}=f^id^i_{Y}$ for all
$i\in\mathbb{Z}$. The map $\cpx{f}$ is called a chain map between
$\cpx{X}$ and $\cpx{Y}$. The category of complexes over
$\mathscr{A}$ with chain maps is denoted by $\C{\mathscr{A}}$. The
homotopy category of complexes over $\mathscr{A}$ is denoted by
$\K{\mathscr{A}}$ and the derived category of complexes is denoted
by $\D(\mathscr{A})$.
Let $R$ be a commutative Artin ring. And let $A$ be an Artin
$R$-algebra. We denote by $A$-mod the category of finitely generated
left $A$-modules. The full subcategory of $A$-mod consisting of
projective modules is denoted by $_A\mathcal {P}$. Recall that a
homomorphism $f: X\ra Y$ of $A$-modules is called a radical map
provided that for any $A$-module $Z$ and homomorphisms $g: Y\ra Z$ and $h: Z\ra
X$, the composition $hfg$ is not an isomorphism. A complex of
$A$-modules is called a radical complex if its differential maps
are radical maps.
Let $\Kb{A}$ denote the
homotopy category of bounded complexes of $A$-modules. We denote by
$\Db{A}$ by the bounded derived category of $A$-mod.
The fundamental theory on derived equivalences has been established.
Rickard \cite{Ri1} gave a Morita theory for derived categories in
the following theorem.
\begin{Theo}$\rm \cite[Therem 6.4]{Ri1}$
Let $A$ and $B$ be rings. The following conditions are equivalent.
$(i)$ $\Db{A\text{-}\Mod}$ and $\Db{B\text{-}\Mod}$ are equivalent
as triangulated categories.
$(ii)$ $\Kf{\Pmodcat{A}}$ and $\Kf{\Pmodcat{B}}$ are equivalent as
triangulated categories.
$(iii)$ $\Kb{\Pmodcat{A}}$ and $\Kb{\Pmodcat{B}}$ are equivalent as
triangulated categories.
$(iv)$ $\Kb{_{A}\mathcal {P}}$ and $\Kb{_{B}\mathcal {P}}$ are
equivalent as triangulated categories.
$(v)$ $B$ is isomorphic to $\End_{\Db{A}}(\cpx{T})$ for some complex
$\cpx{T}$ in $\Kb{_{A}\mathcal {P}}$ satisfying
\qquad $(1)$ $\Hom_{\Db{A}}(\cpx{T},\cpx{T}[n])=0$
for all $n\neq 0$.
\qquad $(2)$ $\add(\cpx{T})$, the category of direct summands of
finite direct sums of copies of $\cpx{T}$, generates
$\Kb{_{A}\mathcal {P}}$ as a triangulated category.
Here $A$-Proj is the subcategory of $A$-Mod consisting of all
projective $A$-modules.
\end{Theo}
\noindent{\bf Remarks.} (1) The rings $A$ and $B$ are said to be
derived equivalent if $A$ and $B$ satisfy the conditions of the
above theorem. The complex $\cpx{T}$ in Theorem 2.1 is called a
tilting complex for $A$.
(2) By \cite[Corollary 8.3]{Ri1}, two Artin $R$-algebras $A$ and $B$
are said to be derived equivalent if their derived categories
$\Db{A}$ and $\Db{B}$ are equivalent as triangulated categories. By
Theorem 2.1, Artin algebras $A$ and $B$ are derived equivalent if
and only if $B$ is isomorphic to the endomorphism algebra of a
tilting complex $\cpx{T}$. If $\cpx{T}$ is a tilting complex for
$A$, then there is an equivalence $F: \Db{A}\ra\Db{B}$ that sends
$\cpx{T}$ to $B$. On the other hand, for each derived equivalence
$F: \Db{A}\ra\Db{B}$, there is an associated tilting complex
$\cpx{T}$ for $A$ such that $F(\cpx{T})$ is isomorphic to $B$ in
$\Db{B}$.
\section{Derived equivalences for Cohen-Macaulay Auslander Algebras}
In this section, we shall prove the main result of this paper.
First, let us recall the definition of Cohen-Macaulay Auslander
algebras.
\subsection{Cohen-Macaulay Auslander algebras}
Let $A$ be an Artin algebra. Recall that $A$ is of finite
representation type provided that there are only finitely many
indecomposable finitely generated $A$-modules up to isomorphism. If
an $A$-module $X$ satisfies $\Ext_{A}^{i}(X,A)=0$ for $i>0$, then
$X$ is said to be a Cohen-Macaulay $A$-module. Denote by
$_{A}\mathcal {X}$ the category of Cohen-Macaulay $A$-modules. It is
easy to see that if $A$ is a self-injective algebra, then
$_{A}\mathcal {X}=A$-mod. By a $\Hom_{A}(-,X)$-exact sequence
$\cpx{Y}=(Y^{i},d^{i})$, we mean that the sequence $\cpx{Y}$ itself
is exact, and that $\Hom_{A}(\cpx{Y},X)$ remains to be exact. An
$A$-module $X$ is said to be Gorenstein projective if there is a
$\Hom_{A}(-,Q)$-exact sequence
$$\cdots\ra
P^{-1}\stackrel{d^{-1}}\ra P^{0}\stackrel{d^{0}}\ra
P^{1}\stackrel{d^{1}}\ra\cdots
$$ such that $X\simeq Imd^{0}$, where $P^{i}$ (for each $i$) and
$Q$ are projective $A$-modules. Denote by $A$-Gproj the subcategory
of $A$-mod consisting of Gorenstein projective $A$-modules. Note
that Gorenstein projective modules are Cohen-Macaulay $A$-modules.
Following \cite[Example 8.4(2)]{Be} an Artin algebra $A$ is said to
be of Cohen-Macaulay finite type provided that there are only
finitely many indecomposable finitely generated Gorenstein
projective $A$-modules up to isomorphism. It is easy to see that
algebras of finite representation type are of Cohen-Macaulay finite
type. Suppose that $A$ is of Cohen-Macaulay finite type. In other
words, $A$-Gproj has an additive generator $M$, that is,
$\add(M)=A$-Gproj.
\begin{Def} $\rm\cite{Ch}$ Suppose that an Artin algebra $A$ is of
Cohen-Macaulay finite type. Let $M$ be an additive generator in
$A$-Gproj. We call $\Lambda=\End(M)$ a Cohen-Macaulay Auslander
algebra of $A$.
\end{Def}
\noindent{\bf Remark.} For a Cohen-Macaulay finite algebra $A$, its
Cohen-Macaulay Auslander algebra is unique up to Morita
equivalences.
\noindent{\bf Example.} Let $A=k[x]/(x^{2})$ and consider the Artin
algebra
$$
T_{2}(A)=\left(\begin{array}{cc}
A&A\\
0&A
\end{array}\right).
$$
Then $T_{2}(A)$ is a $1$-Gorenstein Artin algebra of Cohen-Macaulay
type \cite{FGR} or \cite{Ha2}. $T_{2}(A)$ has indecomposable
Gorenstein projective modules \cite[p.101]{BR}:
$$
M_1=\left(\begin{array}{cc}
k\\
0
\end{array}\right),
M_2=\left(\begin{array}{cc}
A\\
0
\end{array}\right),
M_3=\left(\begin{array}{cc}
A\\
A
\end{array}\right),
M_4=\left(\begin{array}{cc}
k\\
k
\end{array}\right),
M_5=\left(\begin{array}{cc}
A\\
k
\end{array}\right).
$$
Set $M=\oplus_{1\leq i\leq 5} M_{i}$. Then Cohen-Macaulay Auslander
algebra $\End_{T_{2}(A)}(M)$ of $T_{2}(A)$ is given by the following
quiver and relations $xy=0=v\alpha-yu=\alpha z=\alpha\beta\gamma$
\cite{GZ}. \vspace{-0.5cm}
\begin{center}
\setlength{\unitlength}{1mm}
\begin{picture} (50,20)
\put(5,10){\circle*{1}} \put(40,10){\circle*{1}}
\put(5,-10){\circle*{1}} \put(22.5,0){\circle*{1}}
\put(40,-10){\circle*{1}}
\put(2,9){1} \put(43,9){2} \put(2,-11){3} \put(22,-4){4}
\put(43,-11){5}
\put(7,9){\vector(1,0){31}} \put(5,8){\vector(0,-1){16}}
\put(38,11){\vector(-1,0){31}} \put(38,8){\vector(-2,-1){13}}
\put(7,-8){\vector(2,1){13}} \put(20,1){\vector(-2,1){13}}
\put(25,-1){\vector(2,-1){13}} \put(38,-10){\vector(-1,0){31}}
\put(22.5,6.5){$y$} \put(2,0){$v$} \put(22.5,12){$x$}
\put(32,3){$u$} \put(10,-5){$\alpha$} \put(10,3){$z$}
\put(33,-4){$\beta$} \put(22.5,-13){$\gamma$}
\end{picture}
\end{center}
\bigskip
\bigskip
\subsection{The proof of the main result}
We shall give the proof of the main result of this paper.
Suppose $A$ and $B$ are Artin algebras. Let $F: \Db{A}\lra \Db{B}$
be a derived equivalence and let $\cpx{P}$ be the tilting complex
associated to $F$. Without loss of generality, we assume that
$\cpx{P}$ is a radical complex of the following form
$$
0\ra P^{-n}\ra P^{-n+1} \ra \cdots\ra P^{-1}\ra P^{0}\ra 0.
$$
Then we have the following fact.
\begin{Lem} $\rm\cite[lemma\, 2.1]{HX1}$
Let $F: \Db{A}\lra \Db{B}$ be a derived equivalence between Artin
algebras $A$ and $B$. Then we have a tilting complex
$\bar{P}^{\bullet}$ for $B$ associated to the quasi-inverse of $F$
of the form
$$
0\ra \bar{P}^{0}\ra \bar{P}^{1} \ra \cdots\ra \bar{P}^{n-1}\ra
\bar{P}^{n}\ra 0,
$$ with the differential being radical maps.
\end{Lem}
Suppose that $\cpx{X}$ is a complex of $A$-modules. We define the
following truncations:
$\tau_{\geq 1}(\cpx{X}): \cdots\ra0\ra0\ra X^{1}\ra X^{2}\ra\cdots$,
$\tau_{\leq 0}(\cpx{X}): \cdots\ra X^{-1}\ra X^{0}\ra 0\ra0\cdots$.
Using the properties of Cohen-Macaulay $A$-modules, we can prove the
following lemma.
\begin{Lem}\label{3.3} Let $F: \Db{A}\lra \Db{B}$ be a derived equivalence between
Artin algebras $A$ and $B$, and let $G$ be the quasi-inverse of $F$.
Suppose that $\cpx{P}$ and $\bar{P}^{\bullet}$ are the tilting
complexes associated to $F$ and $G$, respectively. Then
$(i)$ For $X\in _{A}\mathcal {X}$, the complex $F(X)$ is isomorphic
in $\Db{B}$ to a radical complex $\bar{P}^{\bullet}_{X}$ of the form
$$
0\ra \bar{P}_{X}^{0}\ra \bar{P}_{X}^{1} \ra \cdots\ra
\bar{P}_{X}^{n-1}\ra \bar{P}_{X}^{n}\ra 0
$$
with $\bar{P}_{X}^{0}\in _{B}\mathcal {X}$ and $\bar{P}_{X}^{i}$
projective $B$-modules for $1\leq i\leq n$.
$(ii)$ For $Y\in_{B}\mathcal {X}$, the complex $G(Y)$ is isomorphic
in $\Db{A}$ to a radical complex $P^{\bullet}_{Y}$ of the form
$$
0\ra P_{Y}^{-n}\ra P_{Y}^{-n+1} \ra \cdots\ra P_{Y}^{-1}\ra
P_{Y}^{0}\ra 0
$$
with $P_{Y}^{-n}\in_{A}\mathcal {X}$ and $P_{Y}^{i}$ projective
$A$-modules for $-n+1\leq i\leq 0$.
\end{Lem}
\textbf{ Proof.} We just show the first case. The proof of ($ii$) is
similar to that of ($i$).
($i$) For $X\in _{A}\mathcal {X}$, by \cite[Lemma 3.1]{HX2}, we see
that the complex $F(X)$ is isomorphic in $\Db{B}$ to a complex
$\bar{P}^{\bullet}_{X}$ of the form
$$
0\ra \bar{P}_{X}^{0}\ra \bar{P}_{X}^{1} \ra \cdots\ra
\bar{P}_{X}^{n-1}\ra \bar{P}_{X}^{n}\ra 0,
$$
with $\bar{P}_{X}^{i}$ projective $B$-modules for $i>0$. We only
need to show that $\bar{P}_{X}^{0}$ is in $_{B}\mathcal {X}$. It
suffices to prove that $\End^{i}_{B}(\bar{P}_{X}^{0},B)=0$ for
$i\geq 1$. Indeed, there exists a distinguished triangle
$$\bar{P}^{+}_{X}\ra\bar{P}^{\bullet}_{X}\ra\bar{P}^{0}_{X}\ra
\bar{P}^{+}_{X}[1]$$ in $\Kb{B}$, where $\bar{P}^{+}_{X}$ denotes
the complex $\tau_{\geq 1}(\cpx{\bar{P}_{X}})$. For each
$i\in\mathbb{Z}$, applying the functor $\Hom_{\Db{B}}(-,B[i])$ to
the above distinguished triangle, we get an exact sequence
\begin{eqnarray*}
\cdots\ra\Hom_{\Db{B}}(\bar{P}^{+}_{X}[1],B[i])\ra
\Hom_{\Db{B}}(\bar{P}^{0}_{X},B[i])\ra
\Hom_{\Db{B}}(\bar{P}^{\bullet}_{X},B[i])\\\ra
\Hom_{\Db{B}}(\bar{P}^{+}_{X},B[i])\ra\cdots. \end{eqnarray*} On the
other hand, $\Hom_{\Db{B}}(\bar{P}^{+}_{X},B[i])\simeq
\Hom_{\Kb{B}}(\bar{P}^{+}_{X},B[i])=0$ for $i\geq0$. By \cite[lemma
2.1]{PX} and $\End^{i}_{A}(X,A)=0$ for $i\geq 1$, we get
$\Hom_{\Db{B}}(\bar{P}^{\bullet}_{X},B[i])\simeq\Hom_{\Db{A}}(X,P^{\bullet}[i])=0$
for all $i\geq 1$. Consequently, we get
$\Hom_{\Db{B}}(\bar{P}^{0}_{X},B[i])=0$ for all $i\geq 1$ by the
above exact sequence. Therefore,
$$
\End^{i}_{B}(\bar{P}_{X}^{0},B)\simeq
\Hom_{\Db{B}}(\bar{P}^{0}_{X},B[i])=0, \;\;\;\text{for}\;\;\; i\geq
1.$$ This implies that $\bar{P}_{X}^{0}\in_{B}\mathcal {X}$.
$\square$
Now we give a lemma, which is useful in the following argument.
\begin{Lem} Let $A$ be an Artin algebra and $f: X\ra Y$ a
homomorphism of $A$-modules with $X,Y\in_{A}\mathcal {X}$. Suppose
$\cpx{Q}$ is a complex in $\Kb{_{A}\mathcal {P}}$. If $f$ factors
through $\cpx{Q}$ in $\Db{A}$, then $f$ factors through a projective
$A$-module.
\end{Lem}
\textbf{Proof.} There is a distinguished triangle
$$\tau_{\leq
0}(\cpx{Q})\ra\tau_{\geq
1}(\cpx{Q})\stackrel{a}\ra\cpx{Q}\stackrel{b}\ra\tau_{\leq
0}(\cpx{Q})[1]\quad\text{in}\quad \Db{A}.$$ Suppose that $f=gh$,
where $g: X\ra \cpx{Q}$ and $h:\cpx{Q}\ra Y$. Since
$\Hom_{\Db{A}}(\tau_{\geq
1}(\cpx{Q}),Y)\simeq\Hom_{\Kb{A}}(\tau_{\geq 1}(\cpx{Q}),Y)=0$, it
follows that $ah=0$. Then there is a map $x: \tau_{\leq
0}(\cpx{Q})[1]\ra Y$, such that $h=bx$. Thus, we get $f=gbx$. Now,
it is sufficient to show that $f$ factors through $\tau_{\leq
0}(\cpx{Q})$. Consider the following distinguished triangle
$$Q^{0}\stackrel{c}\ra\tau_{\leq 0}(\cpx{Q})\stackrel{d}\ra\tau_{\leq
-1}(\cpx{Q})\ra Q^{0}[1]\quad\text{in}\quad\Db{A}.$$ Note that
$\Ext_{A}^{i}(X,A)=0$ for $i\geq 1$. By \cite[Lemma 2.1]{PX}, we
have $\Hom_{\Db{A}}(X,\tau_{\leq -1}(\cpx{Q}))=0$. Thus, we get
$gbd=0$. Then there is a morphism $u: X\ra Q^{0}$ such that $gb=uc$.
Consequently, $f=ucx$, which implies that $f$ factors through a
projective $A$-module $Q^{0}$. $\square$
Choose an $A$-module $X\in_{A}\mathcal {X}$, by Lemma 3.3, we know
that $F(X)$ is isomorphic to a radical complex of the form
$$
0\ra \bar{P}_{X}^{0}\ra \bar{P}_{X}^{1} \ra \cdots\ra
\bar{P}_{X}^{n-1}\ra \bar{P}_{X}^{n}\ra 0
$$
such that $\bar{P}_{X}^{0}\in _{B}\mathcal {X}$ and
$\bar{P}_{X}^{i}$ are projective $B$-modules for $1\leq i\leq n$. In
the following, we try to define a functor $\underline{F}:
\underline{_{A}\mathcal {X}}\ra \underline{_{B}\mathcal {X}}$.
\begin{Prop}
Let $F:\Db{A}\lra \Db{B}$ be a derived equivalence. Then there is an
additive functor $\underline{F}: \underline{_{A}\mathcal
{X}}\ra\underline{_{B}\mathcal {X}}$ sending $X$ to
$\bar{P}_{X}^{0}$, such that the following diagram
$$\xymatrix{
\underline{_{A}\mathcal {X}}\ar[r]^(.35){{\rm can}}\ar[d]_{\underline{F}} &
\Db{A}/\Kb{_{A}\mathcal {P}}\ar[d]^{{F}}\\
\underline{_{B}\mathcal {X}}\ar[r]^(.35){{\rm can}} & \Db{B}/\Kb{_{B}\mathcal {P}}
}$$
is commutative up to natural isomorphism.
\end{Prop}
\textbf{ Proof.} The idea of the proof is similar to that of
\cite[Proposition 3.4]{HX1}. For convenience, we give the details
here.
For each $f:X\ra Y$ in $_{A}\mathcal {X}$, we denote by
$\underline{f}$ the image of $f$ in $\underline{_{A}\mathcal {X}}$.
By Lemma \ref{3.3}, we have a distinguished triangle
$$\bar{P}^{+}_{X}\stackrel{i_X}\ra
F(X)\stackrel{j_X}\ra\bar{P}^{0}_{X}\stackrel{m_X}\ra\bar{P}^{+}_{X}[1]\quad
\text{in}\quad \Db{B}.$$ Moreover, for each $f:X\ra Y$ in
$_{A}\mathcal {X}$, there is a commutative diagram
$$\xymatrix{
\bar{P}^{+}_{X}\ar^{i_{X}}[r]\ar^{\alpha_{f}}[d]
&F(X)\ar^{j_{X}}[r]\ar^{F(f)}[d]
& \bar{P}^{0}_{X}\ar^{m_{X}}[r]\ar^{\beta_{f}}[d]&\bar{P}^{+}_{X}[1]
\ar^{\alpha_{f[1]}}[d]\\
\bar{P}^{+}_{Y}\ar^{i_{Y}}[r] & F(Y)\ar^{j_{Y}}[r]
& \bar{P}^{0}_{Y}\ar^{m_{Y}}[r]& \bar{P}^{+}_{Y}[1] .}$$
Since $\Hom_{\Db{B}}(\bar{P}^{+}_{X},\bar{P}^{0}_{Y})\simeq
\Hom_{\Kb{B}}(\bar{P}^{+}_{X},\bar{P}^{0}_{Y})=0$, it follows that
$i_{X}F(f)j_{Y}=0$. Then there exists a homomorphism $\alpha_{f}:
\bar{P}^{+}_{X}\ra \bar{P}^{+}_{Y}$. Note that $B$-mod is fully
embedding into $\Db{B}$, hence $\beta_{f}$ is a morphism of
$B$-modules. If there is another morphism $\beta'_{f}$ such that
$j_{X}\beta'_{f}=F(f)j_{Y}$, then $j_{X}(\beta_{f}-\beta'_{f})=0$.
Thus $\beta_{f}-\beta'_{f}$ factors through $\bar{P}^{+}_{X}[1]$,
which implies that $\beta_{f}-\beta'_{f}$ factors through a
projective $B$-module by Lemma 3.4. Therefore, the morphism
$\bar{\beta_{f}}$ is uniquely determined by $f$.
Let $f: X\ra Y$ and $g: Y\ra Z$ be morphisms in $_{A}\mathcal {X}$.
Then we have $F(fg)j_{Z}=j_{X}\beta_{fg}$ and
$F(fg)j_{Z}=j_{X}\beta_{f}\beta_{g}$. By the uniqueness of
$\underline{\beta_{fg}}$, we have
$\underline{\beta_{fg}}=\underline{\beta_{f}}$
$\underline{\beta_{g}}$. Moreover, if $f$ factors through a
projective $A$-module, then $\beta_{f}$ also factors through a
projective $B$-module.
For each $X\in_{A}\mathcal {X}$, we define
$\underline{F}(X):=\bar{P}_{X}^{0}$. Set
$\underline{F}(\underline{f})=\underline{\beta_{f}}$, for each
$\underline{f}\in\Hom_{\underline{\mathcal {X}_{A}}}(X,Y)$. Then $F$
is well-defined and an additive functor.
To complete the proof of the lemma, it remains to show that $j_X:
F(X)\ra \underline{F}(X)$ is a natural isomorphism in
$\Db{B}/\Kb{_{B}\mathcal {P}}$. Since $\bar{P}^{+}_{X}$ is in
$\Kb{_{B}\mathcal {P}}$, then $j_X: F(X)\stackrel{\simeq}\ra
\underline{F}(X)$ in $\Db{B}/\Kb{_{B}\mathcal {P}}$. The following
commutative diagram
$$\xymatrix{
F(X)\ar^{j_{X}}[r]\ar^{F(f)}[d]
& \underline{F}(X)=\bar{P}^{0}_{X}\ar^{\beta_{f}}[d]\\
F(Y)\ar^{j_{Y}}[r]
& \underline{F}(Y)=\bar{P}^{0}_{Y} }$$
shows that $j_X: F(X)\ra \underline{F}(X)$ is a natural isomorphism
in $\Db{B}/\Kb{_{B}\mathcal {P}}$. $\square$
The following lemma is quoted from \cite{HX1} which will be used
frequently.
\begin{Lem} $\rm \cite[Lemma\, 2.2]{HX1}$
Let $R$ be an arbitrary ring, and let $R$-Mod be the category of all
left $A$-modules. Suppose $X^{\bullet}$ is a bounded above complex
and $Y^{\bullet}$ is a bounded below complex over $R$-Mod. Let $m$
be an integer. If $X^{i}$ is projective for all $i>m$ and $Y^{j}=0$
for all $j<m$, then
$\Hom_{\K{R\text{-}\Mod}}(X^{\bullet},Y^{\bullet})\simeq\Hom_{\D(R\text{-}Mod)}(X^{\bullet},Y^{\bullet})$.
\end{Lem}
Let $A$ be an Artin algebra and let $X$ be in $_{A}\mathcal {X}$
which is not a projective $A$-module. Set $\Lambda=\End_{A}(A\oplus
X)$, $N=B\oplus\underline{F}(X)$ and $\Gamma=\End_{B}(N)$. Let
$\bar{T}^{\bullet}$ be the complex
$\bar{P}^{\bullet}\oplus\bar{P}^{\bullet}_{X}$. Then
$\bar{T}^{\bullet}$ is in $\Kb{\add_{B}N}$.
The proof of the following lemma is different from \cite[Lemma
3.6]{HX3}, and in fact extends Hu and Xi's original methods for the
self-injective case.
\begin{Lem} Keep the notations above. We have the following statements.
$(1)$
$\Hom_{\Kb{\add_{B}N}}(\bar{T}^{\bullet},\bar{T}^{\bullet}[i])=0$
for $i\neq0$.
$(2)$ $\add\bar{T}^{\bullet}$ generates $\Kb{\add_{B}N}$ as a
triangulated category.
\end{Lem}
\textbf{Proof.} (1) Decompose the complex $\bar{T}^{\bullet}$ as
$\bar{P}^{\bullet}\oplus\bar{P}^{\bullet}_{X}$. Then we have the
following isomorphisms
\begin{eqnarray*}
\Hom_{\Kb{B}}(\bar{T}^{\bullet},\bar{T}^{\bullet}[i])\simeq
\Hom_{\Kb{B}}(\bar{P}^{\bullet}\oplus
\bar{P}^{\bullet}_{X},(\bar{P}^{\bullet}\oplus\bar{P}^{\bullet}_{X})[i])
\simeq\Hom_{\Kb{B}}(\bar{P}^{\bullet},\bar{P}^{\bullet}[i])\oplus\\
\Hom_{\Kb{B}}(\bar{P}^{\bullet},\bar{P}^{\bullet}_{X}[i])
\oplus\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}[i])\oplus
\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},
\bar{P}^{\bullet}_{X}[i]).\end{eqnarray*}
The proof falls naturally into three parts.
(a) Since $\bar{P}^{\bullet}$ is a tilting complex over $B$, we have
$\Hom_{\Kb{B}}(\bar{P}^{\bullet},\bar{P}^{\bullet}[i])=0$ for all
$i\neq0$. Furthermore,
$$\Hom_{\Kb{B}}(\bar{P}^{\bullet},\bar{P}^{\bullet}_{X}[i])\simeq
\Hom_{\Db{B}}(\bar{P}^{\bullet},\bar{P}^{\bullet}_{X}[i])\simeq
\Hom_{\Db{A}}(A,X[i])=0\quad \text{for all}\quad i\neq0.$$
(b) We claim that
$\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}[i])=0$ for
$i\neq0$.
Indeed, applying the functors $\Hom_{\K{B}}(-,\bar{P}^{\bullet}[i])$
and $\Hom_{\Db{B}}(-,\bar{P}^{\bullet}[i])$ to the distinguished
triangle $
\bar{P}^{+}_{X}\ra\bar{P}^{\bullet}_{X}\ra\bar{P}^{0}_{X}\ra
\bar{P}^{+}_{X}[1]$ in $\Kb{B}$, we obtain the following commutative
diagram
$$\xymatrix{
\Hom_{\Kb{B}}(\bar{P}^{+}_{X}[1],\bar{P}^{\bullet}[i])\ar[r]\ar^{\simeq}[d]&
\Hom_{\Kb{B}}(\bar{P}^{0}_{X},\bar{P}^{\bullet}[i])\ar[r]\ar[d]
&\Hom_{\Kb{A}}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}[i])\ar[r]\ar[d]
& \cdots \\
\Hom_{\Db{B}}(\bar{P}^{+}_{X}[1],\bar{P}^{\bullet}[i])\ar[r]&
\Hom_{\Db{A}}(\bar{P}^{0}_{X},\bar{P}^{\bullet}[i])\ar[r] &
\Hom_{\Db{A}}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}[i])\ar[r]
& \cdots.}$$
Note that
$$
\Hom_{\Kb{B}}(\bar{P}^{0}_{X},\bar{P}^{\bullet})\simeq
\Hom_{\Db{B}}(\bar{P}^{0}_{X},\bar{P}^{\bullet}).
$$
Indeed, since $\cpx{P}$ is a bounded complex and Lemma 3.6, it
suffices to show that for the complex $\bar{P}^{\bullet}$ of length
$2$ of the form $0\ra \bar{P}^{0}\ra \bar{P}^{1}\ra0$, we get
$$
\Hom_{\Kb{B}}(\bar{P}^{0}_{X},\bar{P}^{\bullet})\simeq
\Hom_{\Db{B}}(\bar{P}^{0}_{X},\bar{P}^{\bullet}).
$$
In this case, we have a distinguished triangle
$$\bar{P}^{1}[-1]\ra\bar{P}^{\bullet}\ra\bar{P}^{0} \ra
\bar{P}^{1}\quad \text{in}\quad \Kb{B}.$$ Applying the functors
$\Hom_{\Kb{B}}(\bar{P}^{0}_{X},-)$ and
$\Hom_{\Db{B}}(\bar{P}^{0}_{X},-)$ to the distinguished triangle
$\bar{P}^{1}[-1]\ra\bar{P}^{\bullet}\ra\bar{P}^{0} \ra \bar{P}^{1}$,
we obtain the following commutative diagram
$$
\xymatrix{
\Hom_{\Kb{B}}(\bar{P}^{0}_{X},\bar{P}^{1}[-1])\ar[r]\ar^{\simeq}[d]
&\Hom_{\Kb{B}}(\bar{P}^{0}_{X},\bar{P}^{\bullet})\ar[r]\ar[d]
& \Hom_{\Kb{B}}(\bar{P}^{0}_{X},\bar{P}^{0})\ar[r]\ar^{\simeq}[d]&
\Hom_{\Kb{B}}(\bar{P}^{0}_{X},\bar{P}^{1}) \ar^{\simeq}[d]\\
\Hom_{\Db{B}}(\bar{P}^{0}_{X},\bar{P}^{1}[-1])\ar[r] &
\Hom_{\Db{B}}(\bar{P}^{0}_{X},\bar{P}^{\bullet})\ar[r]
& \Hom_{\Db{B}}(\bar{P}^{0}_{X},\bar{P}^{0})\ar[r]&
\Hom_{\Db{B}}(\bar{P}^{0}_{X},\bar{P}^{1}). }
$$
Since $\Hom_{\Db{B}}(\bar{P}^{0}_{X},\bar{P}^{1}[-1])=0$ and
$\Hom_{\Kb{B}}(\bar{P}^{0}_{X},\bar{P}^{1}[1])=0$, we conclude that
that, for $i\neq0$, we have
$$\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}[i])\simeq
\Hom_{\Db{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}[i])\simeq
\Hom_{\Db{A}}(X,A[i])=0.$$
(c) We claim that $\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},
\bar{P}^{\bullet}_{X}[i])=0$ for $i\neq0$.
Indeed, it follows that $\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},
\bar{P}^{\bullet}_{X}[i])=0$ for $i<0$ by Lemma 3.6. It suffices to
show that
$\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[i])=0$
for $i>0$. Note that there is a distinguished triangle $$
(\star)\quad\quad\bar{P}^{+}_{X}\ra\bar{P}^{\bullet}_{X}\ra\bar{P}^{0}_{X}\ra
\bar{P}^{+}_{X}[1]\quad \text{in}\quad \Kb{B},
\text{\;where\;}\bar{P}^{+}_{X}\text{\;denotes the complex\;}
\tau_{\geq 1}(\cpx{\bar{P}_{X}}) .$$ Applying the functor
$\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},-)$ to ($\star$), we get a long
exact sequence
$$
\cdots\ra\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[i])\ra
\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[i])\ra
\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{0}_{X}[i])\ra\cdots
(\star\star).
$$
From the distinguished triangle
$\bar{P}^{+}_{X}\ra\bar{P}^{\bullet}_{X}\ra\bar{P}^{0}_{X}\ra
\bar{P}^{+}_{X}[1]$, we conclude that $ H^{i}(G(\bar{P}^{+}_{X}))=0$
for $i>1$ and $G(\bar{P}^{+}_{X})$ is a radical complex
$Q^{\bullet}_{X}$ of the form
$$
\cdots\ra Q^{-1}_{X}\ra Q^{0}_{X}\ra Q^{1}_{X}\ra 0.
$$
Applying the functors $\Hom_{\Kb{B}}(-,\bar{P}^{+}_{X}[i])$ and
$\Hom_{\Kb{B}}(-,\bar{P}^{+}_{X}[i])$ to ($\star$) again, we have
the following commutative diagram
$$\xymatrix{
\Hom_{\Kb{B}}(\bar{P}^{+}_{X}[1],\bar{P}_{X}^{+}[i])\ar[r]\ar^{\simeq}[d]&
\Hom_{\Kb{B}}(\bar{P}^{0}_{X},\bar{P}_{X}^{+}[i])\ar[r]\ar^{\simeq}[d]
&\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}_{X}^{+}[i])\ar[r]\ar[d]
& \cdots \\
\Hom_{\Db{B}}(\bar{P}^{+}_{X}[1],\bar{P}_{X}^{+}[i])\ar[r]&
\Hom_{\Db{B}}(\bar{P}^{0}_{X},\bar{P}_{X}^{+}[i])\ar[r] &
\Hom_{\Db{B}}(\bar{P}^{\bullet}_{X},\bar{P}_{X}^{+}[i])\ar[r]
& \cdots.}$$
Therefore,
\begin{eqnarray*}
\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[i])\simeq
\Hom_{\Db{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[i])\simeq
\Hom_{\Db{A}}(G(\bar{P}^{\bullet}_{X}),G(\bar{P}^{+}_{X}[i]))\\\simeq
\Hom_{\Db{A}}(X,G(\bar{P}^{+}_{X})[i]).
\end{eqnarray*}
By \cite[lemma 2.1]{PX}, it follows that
$\Hom_{\Db{A}}(X,G(\bar{P}^{+}_{X})[i])=0$ for all $i>1$.
Consequently,
$\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[i])=0$ for
$i>1$. Since
$\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{0}_{X}[i])=0$ for
$i>0$ by shifting, it follows that
$\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[i])=0$
for $i>1$ by the long exact sequence $(\star\star)$. It remains to
prove that
$\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{\bullet}_{X}[1])=0$.
To get $\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},
\bar{P}^{\bullet}_{X}[1])=0$, it suffices to show that the map
$$
(\maltese)\quad\quad
\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{0}_{X})\ra
\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[1])\quad
\text{is \; surjective}.
$$
From the above argument, we have the following commutative diagram
$$
\xymatrix{ Q^{\bullet}_{X} \ar^{a}[r]\ar_{\simeq}[d] &
X\ar[r]\ar_{\simeq}[d]
& M(a) \ar[r]\ar_{\simeq}[d]& Q^{\bullet}_{X}[1] \ar_{\simeq}[d] \\
G(\bar{P}^{+}_{X}) \ar[r] & G(\bar{P}^{\bullet}_{X})\ar[r]
& G(\bar{P}^{0}_{X})\ar[r]&
G(\bar{P}^{+}_{X})[1]}
$$in $\Db{A}$, where all the vertical maps are isomorphisms, and
the morphism $a$ is chosen in $\Kb{A}$ such that the first square is
commutative. Applying the functor $\Hom_{\Kb{A}}(X,-)$ to the first
horizontal distinguished triangle, we get an exact sequence
$$\Hom_{\Kb{A}}(X,M(a))\ra \Hom_{\Kb{A}}(X,Q^{\bullet}_{X}[1])\ra
0,\quad \text{since}\quad \Hom_{\Kb{A}}(X,X[1])=0.$$ We have the
following formulas.
\begin{eqnarray*}
\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{0}_{X})\stackrel{(\ast)}\simeq
\Hom_{\Db{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{0}_{X})\simeq
\Hom_{\Db{A}}(G(\bar{P}^{\bullet}_{X}),G(\bar{P}^{0}_{X}))\\\simeq
\Hom_{\Db{A}}(X,M(a))
\end{eqnarray*}
and
\begin{eqnarray*}
\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[1])\stackrel{(\ast\ast)}\simeq
\Hom_{\Db{B}}(\bar{P}^{\bullet}_{X},\bar{P}^{+}_{X}[1])\simeq
\Hom_{\Db{A}}(G(\bar{P}^{\bullet}_{X}),G(\bar{P}^{+}_{X})[1])\\\simeq
\Hom_{\Db{A}}(X,Q^{\bullet}_{X}[1]). \end{eqnarray*} The
isomorphisms $(\ast)$ and $(\ast\ast)$ are deduced by Lemma 3.6.
Then we have the following commutative diagram
$$
\xymatrix{
\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}_{X}^{0})\ar[r]\ar^{\simeq}[d]&
\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X},\bar{P}_{X}^{+})\ar^{\simeq}[d] \\
\Hom_{\Db{A}}(X,M(a))\ar[r]&
\Hom_{\Db{A}}(X,Q^{+}[1]) .}
$$
From the above diagram, to show the map ($\maltese$) is surjective,
it is sufficient to show the map
$$
\Hom_{\Db{A}}(X,M(a))\ra \Hom_{\Db{A}}(X,\cpx{Q}_{X}[1])\quad
\text{is \; surjective}.
$$
Applying the functor $\Hom(X,-)$ and $\Hom(X,-)$ to the
distinguished triangle $\cpx{Q}\ra X\ra M(a)\ra \cpx{Q}[1]$, we get
the following commutative diagram
$$\xymatrix{
\Hom_{\Kb{A}}(X,M(a))\ar[r]\ar[d]
& \Hom_{\Kb{A}}(X,Q_{X}^{\bullet}[1])\ar[r]\ar[d]& 0 \ar[d]\\
\Hom_{\Db{A}}(X,M(a))\ar[r]
& \Hom_{\Db{A}}(X,Q_{X}^{\bullet}[1])\ar[r]& \Hom_{\Db{A}}(X,X[1]).}$$
Thus, to get the map
$$
\Hom_{\Db{A}}(X,M(a))\ra \Hom_{\Db{A}}(X,\cpx{Q}_{X}[1])\quad
\text{is\; surjective},
$$it suffices to show the following isomorphisms
$$(i)\quad\Hom_{\Db{A}}(X,Q^{\bullet}_{X}[1])\simeq\Hom_{\Kb{A}}(X,Q^{\bullet}_{X}[1])$$
and
$$
(ii)\quad\Hom_{\Db{A}}(X,M(a))\simeq\Hom_{\Kb{A}}(X,M(a)).
$$
Firstly, we show that
$$
(i)\quad\Hom_{\Db{A}}(X,Q^{\bullet}_{X}[1])\simeq\Hom_{\Kb{A}}(X,Q^{\bullet}_{X}[1]).$$
Indeed, it suffices to show that for the complex $Q^{\bullet}_{X}$
of the form $ 0\ra Q_{X}^{-1}\ra Q_{X}^{0}\ra0$, we get ($i$). There
is a distinguished triangle $$(\clubsuit)\quad\quad Q_{X}^{-1}\ra
Q_{X}^{0}\ra Q_{X}^{\bullet}\ra Q_{X}^{-1}[1]\quad \text{in} \quad
\Kb{A}.$$ Applying the functors $\Hom_{\Kb{A}}(X,-)$,
$\Hom_{\Db{A}}(X,-)$ to ($\clubsuit$), we obtain the following
commutative diagram
$$\xymatrix{
\Hom_{\Kb{A}}(X,Q_{X}^{-1})\ar[r]\ar^{\simeq}[d] &
\Hom_{\Kb{A}}(X,Q_{X}^{0})\ar[r]\ar^{\simeq}[d]
& \Hom_{\Kb{A}}(X,Q_{X}^{\bullet})\ar[r]\ar[d]& \Hom_{\Kb{A}}(X,P^{-1}[1]) \ar^{\simeq}[d]\\
\Hom_{\Db{A}}(X,Q_{X}^{-1})\ar[r] &
\Hom_{\Db{A}}(X,Q_{X}^{0})\ar[r]
& \Hom_{\Db{A}}(X,Q_{X}^{\bullet})\ar[r]& \Hom_{\Db{A}}(X,P^{-1}[1]).}$$ Since
$\End^{i}_{A}(X,A)=0$ for $i\geq 1$, it follows that
$\Hom_{\Db{A}}(X,P^{-1}[1])=0$. Moreover,\\
$\Hom_{\Kb{A}}(X,P^{-1}[1])=0$. We thus get
$\Hom_{\Db{A}}(X,Q^{\bullet}_{X}[1])\simeq\Hom_{\Kb{A}}(X,Q^{\bullet}_{X}[1])$.
Next, we prove that
$$
(ii)\quad\Hom_{\Db{A}}(X,M(a))\simeq\Hom_{\Kb{A}}(X,M(a)).
$$
Indeed, there exists a distinguished triangle
$$
(\spadesuit)\quad\quad M(a)^{0}\ra M(a)\ra M(a)^{-}\ra
M(a)^{0}[1]\quad \text{in}\quad \Kb{A},$$ where $M(a)^{-}$ denotes
the truncated complex $\tau_{\leq-1}(M(a))$. Applying the
homological functors $\Hom_{\Kb{A}}(X,-)$ and $\Hom_{\Db{A}}(X,-)$
to ($\spadesuit$), we obtain the following commutative diagram
$$\xymatrix{
_{\Kb{A}}(X,M(a)^{-}[-1])\ar[r]\ar^{\simeq}[d]&
_{\Kb{A}}(X,M(a)^{0})\ar[r]\ar^{\simeq}[d] &
_{\Kb{A}}(X,M(a))\ar[r]\ar[d]
& _{\Kb{A}}(X,M(a)^{-}) \ar[d]\\
_{\Db{A}}(X,M(a)^{-}[-1])\ar[r]& _{\Db{A}}(X,M(a)^{0})\ar[r] &
_{\Db{A}}(X,M(a))\ar[r]
& _{\Db{A}}(X,M(a)^{-}).}$$
According to ($i$), we have
$\Hom_{\Kb{A}}(X,M(a)^{-})\simeq\Hom_{\Db{A}}(X,M(a)^{-})=0$.
Therefore, $\Hom_{\Db{A}}(X,M(a))\simeq\Hom_{\Kb{A}}(X,M(a))$.
From the above argument, we have shown that
$\Hom_{\Kb{B}}(\bar{P}^{\bullet}_{X}, \bar{P}^{\bullet}_{X}[i])=0$
for $i\neq0$. Since $\Kb{B}$ is a full subcategory of
$\Kb{\add_{B}N}$, it follows that
$\Hom_{\Kb{\add_{B}N}}(\bar{P}^{\bullet}_{X},
\bar{P}^{\bullet}_{X}[i])=0$ for $i\neq0$.
(2) Since $\bar{P}^{\bullet}$ is a tilting complex for $B$, we see
that $\add\bar{P}^{\bullet}$ generates $\Kb{\add_{B}B}$ as
triangulated category. All the terms of $\bar{P}^{+}_{X}$ are in
$\add_{B}B$. From the distinguished triangle
$$\bar{P}^{+}_{X}\ra\bar{P}^{\bullet}_{X}\ra\bar{P}^{0}_{X}\ra
\bar{P}^{+}_{X}[1],$$ it follows that $\bar{P}^{0}_{X}$ is in the
triangulated subcategory generated by
$\add(\bar{P}^{\bullet}\oplus\bar{P}^{\bullet}_{X})$. Therefore,
$\add\bar{T}^{\bullet}$ generates $\Kb{\add_{B}N}$ as a triangulated
category. $\square$
\begin{Prop}
The complex $\Hom(N,\bar{T}^{\bullet})$ is a tiling complex over
$\Gamma$ with the endomorphism
$\End(\Hom(N,\bar{T}^{\bullet}))\simeq\Lambda$. In particular, Artin
algebras $\Lambda$ and $\Gamma$ are derived equivalent associated
with the tilting complex $\Hom(N,\bar{T}^{\bullet})$.
\end{Prop}
\textbf{Proof.} We have an equivalence of categories
$$
\Hom_{B}(N,-): \add_{B} N\stackrel{\simeq}\lra _{\Gamma}\mathcal
{P}.
$$
We thus get an equivalence of triangulated categories induced by
$\Hom_{B}(N,-)$ as follows
$$
\Kb{\add_{B} N}\stackrel{\simeq}\lra \Kb{_{\Gamma}\mathcal {P}}.
$$
Then $\Hom(N,\bar{T}^{\bullet})\in \Kb{_{\Gamma}\mathcal {P}}$. By
the Lemma 3.7, we see that $\add\Hom(N,\bar{T}^{\bullet})$ generates
$\Kb{_{\Gamma}\mathcal {P}}$ as a triangulated category, and $
\End(\Hom(N,\bar{T}^{\bullet}))\simeq\End(\bar{T}^{\bullet})\simeq\Lambda
$. $\square$
We have the following lemma, its proof is due to Happel \cite[Lemma
4.4]{Ha2}.
\begin{Lem} Suppose that $\id_{A}A<\infty$. Then the following statements are
equivalent.
$(i)$ $\pd_{A}(\D(A_{A}))<\infty$.
$(ii)$ For $X\in _{A}\mathcal {X}$, there exists an exact sequence $
0\ra X\ra P\ra X^{'}\ra 0$, with $X^{'}\in_{A}\mathcal {X}$ and $P$
a projective $A$-module.
$(iii)$ If $X\in_{A}\mathcal {X}$ satisfies $\id_{A}X<\infty$, then
$X$ is a projective $A$-module.
\end{Lem}
Recall that an Artin algebra $A$ is called Gorenstein if the regular
module A has finite injective dimension on both sides. If $A$ is a
Gorenstein algebra, then it follows from Lemma $7.2.8$ that
$_{A}\mathcal {X}=A$-Gproj, and that $_{A}\mathcal {X}$ is a
Frobenius category and its category $\underline{_{A}\mathcal {X}}$
is a triangulated category.
\begin{Prop}
Let $A$ and $B$ be Gorenstein Artin algebras. Suppose that $F$
is a derived equivalence between $A$ and $B$.
Then we have the following statements.
$(1)$ There is an equivalence
$\underline{F}: \underline{_{A}\mathcal {X}}\ra\underline{_{B}\mathcal {X}}$.
$(2)$ If $A$ and $B$ are finite dimensional algebras over a field
$k$, then there exist bimodules $_{A}M_{B}$ and $_{B}L_{A}$ such
that the pair of functors
$$
_{A}M_{B}\otimes-: A\text{-}mod\ra B\text{-}mod,\;
_{B}L_{A}\otimes-: B\text{-}mod\ra A\text{-}mod
$$
induces an equivalence of triangulated categories
$\underline{_{A}\mathcal {X}}$ and $\underline{_{B}\mathcal {X}}$.
\end{Prop}
\textbf{ Proof.} We refer to \cite[Theorem 4.6]{Ha2} and \cite
[Theorem 5.4]{Ka} for the proofs of (1) and (2), respectively.
$\square$
Our main result in this chapter is the following theorem.
\begin{Theo}\label{T} Let $A$ and $B$ be Gorenstein Artin algebras of
Cohen-Macaulay finite type. If $A$ and $B$ are derived equivalent,
then the Cohen-Macaulay Auslander algebras $\Lambda$ and $\Gamma$ of
$A$ and $B$ are also derived equivalent.
\end{Theo}
\textbf{ Proof.} In fact, if Artin algebras $A$ and $B$ are derived
equivalent, then $A$ is Gorenstein if and only if $B$ is Gorenstein.
By Proposition 3.10 or \cite[Theorem 8.11]{Be}, if Gorenstein Artin
algebras $A$ and $B$ are derived equivalent, then $A$ is of
Cohen-Macaulay finite type if and only if $B$ is. Let $F:\Db{A}\lra
\Db{B}$ be a derived equivalence. Set $\Lambda=\End(A\oplus X)$ with
$X=\oplus_{0\leq i\leq m} X_{i}$, where each $X_{i}$ is
indecomposable non-projective Gorenstein projective $A$-module. Then
$\Lambda$ is the Cohen-Macaulay Auslander algebra of $A$. By
Proposition 3.10, it follows that $Y_{i}=\underline{F}(X_{i})$ is
the indecomposable non-projective Gorenstein projective $B$-module.
Set $Y=\oplus_{0\leq i\leq m} Y_{i}$. Then $\Gamma=\End(B\oplus Y)$
is the Cohen-Macaulay Auslander algebra of $B$. Let $N$ be the
$B$-module $(B\oplus Y)$ and let $\cpx{\bar{T}}$ be the complex
$F(A\oplus X)$. Thus, we construct a tilting complex
$\Hom(N,\cpx{\bar{T}})$. The result follows from Proposition 3.8.
$\square$
\noindent{\bf Remark.} Let $A$ and $B$ be Gorenstein Artin algebras
of Cohen-Macaulay finite type. According to a result of Liu and Xi
\cite[Theorem 1.1]{LX2}, we see that, if $A$ and $B$ are stably
equivalent of Morita type, then the Cohen-Macaulay Auslander
algebras of $A$ and $B$ are also stably equivalent of Morita type.
As a corollary of Theorem \ref{T}, we re-obtain the following result
of Hu and Xi \cite{HX3} since self-injective Artin algebras of
finite representation type are Gorenstein Artin algebras of
Cohen-Macaulay finite type.
\begin{Koro} $\rm\cite[Corollary\,3.13]{HX3}$
Suppose that $A$ and $B$ are self-injective Artin algebras of finite
representation type. If $A$ and $B$ are derived equivalent, then the
Auslander algebras of $A$ and $B$ are also derived equivalent.
\end{Koro}
\noindent{\bf Acknowledgements.} The author would like to thank his
supervisor Professor Changchang Xi. He is grateful to him for his
guidance, patience and kindness.
|
{
"timestamp": "2011-03-30T02:00:53",
"yymm": "1009",
"arxiv_id": "1009.3794",
"language": "en",
"url": "https://arxiv.org/abs/1009.3794"
}
|
\section{Introduction}
Let $G$ be a finite group. A set $\delta$ of proper subgroups $H_1,\ldots ,H_n$ of $G$ is called
a {\em covering} of $G$ if $G$ is the set-theoretical union of the $H_i$ and moreover, to avoid
redundance, there are no inclusions between the $H_i$:
\[G=\bigcup_{i=1}^{n}H_i,\quad\quad H_i\not \leq H_j\quad \hbox{for } i\neq j.\]
We refer to the $H_i$ as the {\em components} of the covering. From the well known fact that
{\em a group is never the union of two proper subgroups} it follows that {\em a covering of a group needs
at least three components}.
A group $G$ for which a cover exists is called {\em coverable}.
It is clear that each group admitting a partition is coverable, hence we have soon a lot of
available examples.
On the other hand, here, there are no assumption about the intersection of the components, so
that the great part of the arguments used in the partition's theory becomes useless.
This is also one of the reason for the difficulty to develop a general and, at the same time,
expressive description of the coverable groups. Not by accident, this theme appears in the
recent literature always from a particular point of view.
Brandl {\cite{bra81}} considers the coverings $\delta=\{H^{\alpha}: \alpha\in Aut(G)\}$ where
$H\leq G$; Praeger in {\cite{pra88} and {\cite{pra94}} explores the more general coverings
$\delta=\{H^{\alpha}:\alpha\in A\}$ where $Inn(G)\leq A\leq Aut(G)$ to study the coverings of
the Galois group of certain extensions of fields.
We believe that a reasonable idea in exploring the coverable groups is to begin with some
"natural" sets of subgroups which cannot be used to cover a group, trying to add to them some
"natural" new components in order to obtain a covering. In this sense our starting point is a
well known fact: {\em a finite group $G$ is never the set-theoretical union of the
$G$-conjugates of a proper subgroup $H$}. Adding to the
$\,G$-conjugates of $H$ another subgroup $K$ or the set of the
$\,G$-conjugates of a subgroup $K,$ we have the opportunity, in
some cases, to obtain a covering; this happens, for example, in a
Frobenius group if $H$ and $K$ are respectively a complement and
the kernel. In other words we are led to consider two kinds of
collections of subgroups for a group $G$:
\[(\ast)\;\delta=\{H^g,K:g\in G\},\]
\[(\ast\ast)\;\delta=\{H^g,K^g:g\in G\}\]
where $H$ and $K$ are fixed proper subgroups of $G$. Observe that if $\delta$ is of the type
$(\ast)$ or $(\ast\ast)$ and $G$ is the set-theoretical union of the subgroups in $\delta,$ then
no inclusions among the elements of $\delta$ are possible, that is $\delta$ is a covering of $G.$
If a group $G$ is coverable with a set of subgroups of the type $(\ast)$ or $(\ast\ast)$, we
will say respectively that $G$ is
{\em $(\ast)$-coverable} or {\em $(\ast\ast)$-coverable}. In the next section we give a
characterization of the $(\ast)$-coverable groups, from which we deduce, in particular,
that no simple group is $(\ast)$-coverable. But it is easily seen that there exist simple
$(\ast\ast)$-coverable groups, such as $A_5$; the whole family of the alternating groups
looks interesting on this regard and we devote to this subject the section $3$, showing that
$A_n$ is $(\ast\ast)$-coverable if and only if $\;4\leq n\leq 8$. In the last section
we obtain a similar result for the symmetric group proving that $S_n$ is
$(\ast\ast)$-coverable if and only if $\;3\leq n\leq 6$.
All groups in this paper are finite. All unexplained notation is standard ( see
\cite{hup67} ).
This paper has greatly benefited from very helpful discussions with C. Casolo.\\
I would also like to thank S. Dolfi regard to \ref{3.4} and P. Neumann regard to the proof
of \ref{2.10}. A special thank to E. De Tomi for the help in preparing this final version.
\bigskip
\section{$(\ast)$-coverable groups
In this section, we aim to describe the $(\ast)$-coverable groups $G,$ that is the groups
$G$ coverable with a set of subgroups
of the type \[(\ast)\;\delta=\{H^g,K:g\in G\},\] where $H$ and $K$ are fixed proper subgroups
of $G$.
We begin with some elementary facts.
\begin{lem}\label{2.1}
Let $G$ be a group and $\delta=\{H^g,K:g\in G\}$ a covering of $G$.Then:
\begin {enumerate}
\item[(i)] $H^G=G$ and $H_G \leq K$;
\item[(ii)]if $K\lhd G$, then $G=KH$.
\end{enumerate}
\end{lem}
\begin{proof}
(i) Since $G=K \cup \left( \bigcup_{g\in G}H^g \right)$, it follows that $G=K\cup H^G$ with
$K\neq G$ hence $G=H^G$.
Assume now to have $x \in H_G$ with $x\notin K$. If we pick any $\,y\in K$, then we get
$xy\notin K$ hence there exists $g\in G$ such that $xy\in H^g$. But $x\in H^g$ and
therefore $y\in H^g$. This means that $K \subseteq \bigcup_{g\in G}H^g$, hence
$G=\bigcup_{g\in G}H^g$ and $H=G$, a contradiction.
\\
(ii) If $K\lhd G$, then we simply observe that $G=\bigcup_{g\in G}KH^g=\bigcup_{g\in G}(KH)^g$
concluding $G=KH$.
\end{proof}
The next result shows that there is no loss of generality in assuming $K$ normal in $G$.
\begin{lem} \label{2.3}
Let $G$ be a group and $\delta=\{H^g,K:g\in G\}$ a covering of $G$. Then also
$\hat{\delta}=\{K_G,H^g:g\in G\}$ is a covering of $G$.
\end{lem}
\begin{proof}
Let $g\in G-\bigcup_{a\in G}H^a$. Then $\delta$ covering yields $g\in K$ and if $y\in G$, then
we necessarily have $g^{y^{-1}}\in K$, otherwise there would exist an element $x\in G$ with
$g^{y^{-1}}\in H^x$, that is $g\in H^{xy}$, against the assumption on $g$. Therefore for
every $y\in G$ we have $g^{y^{-1}}\in K$, that is $g\in K^y$ and then $g\in K_G$.\\
Clearly $K_G\neq G$ because $K\neq G$ by definition of covering and hence
$\hat{\delta}$ is a covering.
\end{proof}
\begin{co}\label{2.4}
No simple group is $(\ast)$-coverable.
\end{co}
To exhibit some crucial examples of $(\ast)$-coverable groups, we need to recall the definition
of the Frobenius-Wielandt groups and the fundamental theorem related to them.
\begin{de}{\em (Wielandt){\cite{wi58}}}\label{2.5}
{\em A group $G$ is said to be a} Frobenius-Wielandt group {\em provided that it has a subgroup $H$,
with $1\neq H\neq G$, and a proper normal subgroup $N$ of $H$ such that}
\end{de}
\[H \cap H^g\leq N\quad \mbox{if}\;g\in G-H.\]
For brevity, we will refer to them as {\em F-W groups} and we will use the locution
{\em $(G,H,N)$ is a F-W group} to indicate more closely the situation.
\begin{te}{\em (Wielandt)\cite {wi58}}\label{2.6}
If $(G,H,N)$ is a F-W group, then there exists a unique normal subgroup $K$ of $G$ ( called the
$\underline{kernel}$ ) such that:
\[G=HK, \quad H\cap K=N.\]
Moreover $K=G-\bigcup_{g\in G}(H-N)^g$.
\end{te}
\begin{lem}\label{2.7}
Let $(G,H,N)$ be a F-W group with kernel $K$. Then \[\delta=\{H^g,K:g\in G\}\] is a
covering of $G$.
\end{lem}
\begin{proof}
In fact if $x\in G-K$, by \ref{2.6}, we get $x\in \bigcup_{g\in G}(H-N)^g$ hence $x\in H^g$
for some $g\in G$. Moreover \ref{2.6} and \ref{2.5} yield that $H,K\neq G,$ hence
$\delta=\{H^g,K:g\in G\}$ is a covering of $G$.
\end{proof}
It is clear that each Frobenius group $G$ with complement $H$ is a $(G,H,1)$ F-W group.
Furthermore, as a trivial consequence of the definition \ref{2.5}, we get that each group with
a F-W quotient is itself a F-W group. Particularly, we get:
\begin{os}\label{2.8}
If a group has a Frobenius quotient, then it is a F-W group.
\end{os}
This fact allows us to be more concrete in the construction of examples of $(\ast)$-coverable
groups.
\begin{exs}\label{2.9}
$S_3$, $A_4$ and $S_4$ are $(\ast)$-coverable groups.
\end{exs}
\begin{proof}
In fact, by \ref{2.8}, these are all F-W groups and then \ref{2.7} applies.
\end{proof}
Now an application of the counting method leads to the converse of \ref{2.7}.
\begin{lem}\label{2.10}
If the group $G$ has a covering $\delta=\{H^g,K:g\in G\}$, then $(G,H,H\cap K_G)$ is a F-W
group with kernel $K_G$.
\end{lem}
\begin{proof}
Due to \ref{2.3}, we may assume that $K\lhd G$. First of all we show that $H$ is
selfnormalizing in $G$. Let $H_1,\ldots ,H_n$ be the distinct conjugates of $H$ in $G$ and
$m=|N_G(H):H|$. Then we have $|G|=mn|H|<\sum_{i=1}^n|H_i| +|K|$, because
$G=\left(\bigcup_{i=1}^{n} H_i\right)\cup K$.
From $K<G$ we get also $\sum_{i=1}^n|H_i| +|K|\leq n|H|+\frac{|G|}{2}=n|H|+
\frac{mn}{2}|H|$, hence $mn<n\left(1+\frac{m}{2} \right)$, that is $m<2$ and
then $m=1$.
Next we define the subgroups $K_i=H_i\cap K$ and we prove that the subsets $X_i=H_i-K_i$ have
empty intersection for $i\neq j$. By \ref{2.1}(ii), we have $G=KH_i$ hence
$|G|=\frac{|K||H|}{|K_i|}$ that is $|K_i|=\frac{|K|}{n}$ for each $i=1,\ldots,n$. It
follows that $|K|+\sum_{i=1}^n|X_i|=\sum_{i=1}^n|K_i|+\sum_{i=1}^n\left(|H_i|-|K_i|
\right)=n|H|=|G|$.
On the other hand, obviously, we have $G=K\cup \left( \bigcup_{i=1}^{n} X_i\right)$ and then
the previous relation implies $X_i\cap X_j=\emptyset$ for $i\neq j$. Clearly this means that $H_i\cap
H_j \leq K$. Let $g\in G-H=G-N_G(H)$, then $H^g\neq H$ and therefore $H^g\cap H\leq H\cap K$,
that is $(G,H,H\cap K)$ is a F-W group. Finally from the characterization of the kernel of
a F-W group given in \ref{2.6}, it follows that $K$ is actually the kernel of $G$.
\end{proof}
Collecting \ref{2.7} and \ref{2.10}, we can state the main result of this section
\begin{te}\label{2.11}
A group $G$ is $(\ast)$-coverable if and only if $G$ is a F-W group.
\end{te}
\bigskip
\section{The $(\ast\ast)$-coverable alternating groups
Given a set $\Omega$, we denote the symmetric and the alternating groups on
$\Omega$ respectively with {\em $Sym\,\Omega$} and {\em $Alt\,\Omega$}.
When $\Omega=\{1,\ldots,n\}$ we use, more simply, the notations {\em $S_n$} and {\em $A_n$}
and we consider the natural immersions of $S_n$ into $S_{n+1}$ and of $A_n$ into $A_{n+1}$ as
inclusions.
If $\sigma \in S_n$ decomposes into the product of disjoint cycles $\sigma_1,\ldots,\sigma_k$
of lengths $l_1,\ldots,l_k$ we will say that the {\em type} of $\sigma$ is {\em $[l_1;
\ldots;l_k]$}. Sometimes, when not misleading, we will omit the lengths equal to $1$.\\
Obviously it holds:
\begin{os}\label{3.1}
Let $G=S_n$ and $H,\;K<G$.Then $\delta=\{H^g,K^g:g\in G\}$ is a covering of $G$ if and only if
each type of permutation appears at least one time either in $H$ or in $K$.
\end{os}
Though no simple group is $(\ast)$-coverable ( \ref{2.4} ), it looks reasonable to investigate the
simple groups $G$ which admit a $(\ast\ast)$-covering that is which are covered by a set of
subgroups of the type $(\ast\ast)\;\delta=\{H^g,K^g:g\in G\}$ for some $H,\ K<G.$
There is in fact a natural example: $A_5$ is
covered by \{$A_4^g, P^g : g\in A_5$\} where $P\in Syl_5(G)$, just because
$\bigcup_{g\in A_5}A_4^g$ contains all the permutations in $A_5$ with at least a fixed
point and $\bigcup_{g\in A_5}P^g$ all the permutations with no fixed point. On the other
hand this construction is peculiar for $A_5$ and not extendible to the alternating groups
of higher degree. Thus it is not evident for which $n\in \bfn$, $A_n$ is
$(\ast\ast)$-coverable.\\
Observe that if $H,\ K<A_n$ and each type of even permutation lies in $H$ or in $K$,
this does not guarantee any more that $\delta=\{H^g,K^g:g\in A_n\}$ is a covering of $A_n$
( cf. \ref{3.1} ). The problem is of course that certain types of even permutations
decompose into two conjugacy classes in $A_n.$ We need to be more specific about this
question:
\begin{lem}\label{3.4}
If $\sigma \in A_n$, then the $S_n$-conjugacy class $\sigma^{S_n}$ splits into two
$A_n$-conjugacy classes if and only if $\sigma$ is of the type $[l_1;\ldots;l_r]$ with
$l_i\geq 1$ distinct in pairs and odd for $i=1,\ldots,r.$
\end{lem}
\begin{proof} {\em ( probably folklore )} First of all we observe that if $\sigma \in A_n,$ then
$\sigma^{S_n}$ splits into two $A_n$-conjugacy classes if and only if $C_{S_n}(\sigma)\leq A_n.$
Next we note that if $\mu \in S_n$ is a cycle, then
\begin{equation}
C_{S_n}(\mu)= \gen{\mu}\times Sym(\Omega - supp(\mu))
\end{equation}
where, for each $\sigma \in S_n,$ we denote with {\em supp($\sigma$)} ( the support of $\sigma$ )
the set of $i\in \Omega=\{ 1,\ldots,n\}$ such that $i^{\sigma}\neq i.$ Namely it is enough to
show $(1)$ for the $k$-cycle $\mu=(1\,2\ldots k)$ with $2\leq k\leq n$ and obviously
$\gen{\mu}\times Sym(\Omega - supp(\mu))\leq C_{S_n}(\mu).$
Moreover if $\alpha \in C_{S_n}(\mu),$ then we can write $\alpha=
\overline{\alpha}\beta$ where $\overline{\alpha}$ is a product of disjoint cycles whose support
has non empty intersection with $K=\{1,\ldots,k\}$ and $\beta$ is a product of disjoint cycles
whose support has empty intersection with $K.$ Since $\alpha$ stabilizes $K,$ we get that $supp(
\overline{\alpha}) \subseteq K,$ $\beta \in Sym\{k+1,\ldots,n\}$ and $\overline{\alpha}\in
C_{S_n}(\mu).$ Therefore we have $(1\,2\ldots k)^{\overline{\alpha}}=(1^{\overline{\alpha}}\,2^
{\overline{\alpha}}\ldots k^{\overline{\alpha}})=(1\,2\ldots k)$ and then, if we put
$1^{\overline{\alpha}}=i\in K,$ we get $\,j^{\overline{\alpha}}=i+j-1\,(\,\hbox{mod}\ k\,)\,$
for each $1\leq j\leq k.$ But we have also $j^{\mu^{i-1}}=i+j-1\,(\,\hbox{mod}\ k\,)\,$ for each
$1\leq j\leq k,$ that is $\,\overline{\alpha}=\mu^{i-1} \in \gen{\mu}\,$ and so
$\alpha \in \gen{\mu}\times Sym(\Omega - supp(\mu)).$
Now we observe that if $\sigma \in S_n$ and $\sigma=\mu_1\cdots \mu_r$ with $\mu_i$ disjoint
cycles of lengths $l_1,\ldots,l_r$ distinct in pairs, then
\begin{equation}
C_{S_n}(\sigma)= \bigcap_{i=1}^{r}C_{S_n}(\mu_i).
\end{equation}
In fact the inclusion $\bigcap_{i=1}^{r}C_{S_n}(\mu_i)\leq C_{S_n}(\sigma)$ is trivial;
moreover if $\alpha \in C_{S_n}(\sigma),$ then $\,\mu_1^{\alpha}\ldots \mu_r^{\alpha}=
\mu_1\ldots \mu_r\,$ are two decompositions in the product of disjoint cycles and
$\mu_i^{\alpha}$ is
a $\,l_i$-cycle. Since $\mu_i$ is the only $l_i$-cycle among the $\mu_1,\ldots ,\mu_r,$ we
argue that $\mu_i^{\alpha}=\mu_i$ for each $i=1,\ldots,r$ that is
$\alpha \in \bigcap_{i=1}^{r}C_{S_n}(\mu_i).$
Let $\sigma \in A_n$ be the product of the disjoint cycles $\mu_1,\ldots ,\mu_r,$ of lengths
$l_i\geq 1.$\\ If the $l_i$ are odd and distinct in pairs, then by $(2)$ and $(1)$ we obtain
\[C_{S_n}(\sigma)= \bigcap_{i=1}^{r}C_{S_n}(\mu_i)=\bigcap_{i=1}^{r}\left[\gen{\mu_i}\times
Sym(\Omega - supp(\mu_i))\right]=\times_{i=1}^{r}\gen{\mu_i}\leq A_n.\]
Therefore $\sigma^{S_n}$ splits into two $A_n$-conjugacy classes.\\ Now assume that there exist
at least two $\mu_i$ of the same odd length, say $l_1=l_2=l$ odd and let $\mu_1=(i_1\ldots i_l), \
\mu_2=(i_1'\ldots i_l').$ Then we have $(i_1\,i_1')\ldots (i_l\,i_l')\in C_{S_n}(\sigma)-A_n.$
Finally if at least one among the $\mu_i$ has even length, then clearly $\mu_i\in
C_{S_n}(\sigma)-A_n.$ In both cases we have $C_{S_n}(\sigma)\not \leq A_n$ and hence
$\sigma^{S_n}=\sigma^{A_n}.$
\end{proof}
We approach the problem of the determination of the $n\in \bfn$ for which $A_n$ is
$(\ast\ast)$-coverable, beginning with $n\leq 8$. To do this, first of all, we state an
elementary lemma which will be useful also in the sequel and analyse $S_5$ and $S_6.$
\begin{lem}\label{3.3}
Let $G$ be a group covered by $\delta=\{H^g,K^g:g\in G\}$. If $N\unlhd G$ and $G=NH=NK$,
then $N$ admits the covering $\delta_N=\{(H \cap N)^x,(K\cap N)^x\-:x\in N\}$.
\end{lem}
We refer to the covering {\em $\delta_N$} defined in the previous lemma as a {\em $(\ast\ast)$-covering
obtained by intersection}.
\begin{os}\label{3.2}
$S_5,\;S_6$ are $(\ast\ast)$-coverable subgroups.
\end{os}
\begin{proof}
Let $G=S_5,$\[ H_1=G_{\{1,2\}}\] and \[K_1=N_G\gen{(12345)}.\]Then $K_1=\gen{(12345)}\rtimes
\gen{(2354)}<G$
contains permutations of the types $[4],\ [5],\ [2;2]$ and $H_1<G$ contains permutations of
the types $[2],\ [3],\ [2;3]$. Hence, by \ref{3.1}, $\delta=\{H_1^g,K_1^g:g\in G\}$ is a
covering of $G$.
Now let $G=S_6$, \[H_2\hbox { the stabilizer in G of the partition }\{1,2,3\},\ \{4,5,6\}\]
and\[K_2=S_5.\]Then $H_2\cong S_3\ wr\ S_2$ contains the permutation $(142536)$ of type $[6]$
and therefore also a permutation of the type $[2;2;2]$ and one of the type $[3;3]$. Moreover
$H_2$ contains the permutation $(14)(2536)$ of the type $[2;4]$; hence for each type of
fixed-point-free permutation $H_2$ contains at least a representative. On the other hand
$K_2$ contains representatives for each type of permutation with at least a fixed point and
again \ref{3.1} applies.
\end{proof}
\begin{os}\label{3.5}
$A_5,\;A_6,\;A_7$ and $A_8$ are $(\ast\ast)$-coverable groups.
\end{os}
\begin{proof}
We easily obtain a covering of $A_5$ and $A_6$ by intersection from the coverings constructed in
\ref{3.2} for $S_5$ and $S_6$. In fact $A_n \lhd S_n$ and, in
order to apply \ref{3.3}, we only need to observe that each of the subgroups $H_1,H_2,K_1,K_2$
contains an odd permutation. But it is trivially checked that
\[(12)\in H_1,\ (2354)\in K_1,\ (12)\in H_2\hbox{ and }(23)\in K_2.\]
Next let $G=A_7$ and consider \[H=N_G\gen{(1234567)}\] and \[K=[Sym\{1,2\}\times
Sym\{3,4,5,6,7\}]\cap A_7.\] Since
$N_{S_7}\gen{(1234567)}=\gen{(1234567)}\rtimes \gen{\mu}$ with $\mu$ a $6$-cycle, we
get that $H=\gen{(1234567)}\rtimes\gen{\mu^2}<G$ where $\mu^2$ is of type $[1;3;3].$
By \ref{3.4}, the
permutations of type $[1;3;3]$ constitute a single conjugacy class hence
$\bigcup_{g\in G}H^g$ contains all the permutations of this type. Moreover $H$ contains
a $7$-Sylow of $G$ hence $ \bigcup_{g\in G}H^g$ contains every $7$-cycle of $G$.
On the other hand, from $|K|_2=|G|_2$, it follows that
in $\bigcup_{g\in G}K^g$ there is each
$2$-element of $G$. Next observe that $K$ contains at least a permutation of type
$[3],\ [5],\ [2;2;3]$ and therefore, again by \ref{3.4}, each permutation of these
types lies in $\bigcup_{g\in G}K^g$. Since we have examined all the possible types of
permutations in $G$, then it follows that $\{H^g,K^g:g\in G\}$ is a covering of $G=A_7$.
Finally we explore the group $A_8.$ Let $H$ be the group of all the affine transformations
of the vector space $V=GF(2)^3,$ that is of all the transformations $\tau_{_{A,a}}
:V\rightarrow V$ of the form
\[\tau_{_{A,a}}(x)=Ax+a\]
where $A\in GL(3,2)$ and $a\in V.$\\
Obviously, $H\leq Sym\,V\cong S_8.$ We want to show that actually $H<Alt\,V\cong A_8.$ It is
well known that $H=T\rtimes GL(3,2),$ where $T$ is the elementary abelian subgroup of order
$2^3$ consisting of all the translations $\tau_{_{I,a}}\in H.$ Moreover, it is clear that each
$1\neq \tau \in T$ has no fixed points on $V,$ hence $\tau \in Sym\,V$ is a permutation of
the type $[2;2;2;2]$ and $T\leq Alt\,V.$ On the other hand, $GL(3,2)$ is simple and
therefore we necessarily have $GL(3,2) \leq Alt\,V,$ otherwise $GL(3,2)$ would contain a
normal subgroup of index two. Thus we have $H<Alt\,V$ and, because $|H|=2^6\cdot 3\cdot 7,$
it is clear that $H$ contains a $7$-cycle.\\
Next we observe that, putting
\[a=\left(\begin{array}{c}0\\1\\0\end{array}\right)\in V \hbox{ and }
A=\left(\begin{array}{ccc}1&1&0\\0&1&0\\0&0&1\end{array}\right)\in GL(3,2),\]
it is easily checked that $\tau_{_{A,a}}\in H$
is a permutation on $V$ of the type $[4;4].$\\
Moreover, putting
\[b=\left(\begin{array}{c}1\\0\\0\end{array}\right)\in V \hbox{ and }
B=\left(\begin{array}{ccc}1&0&0\\0&0&1\\0&1&1\end{array}\right)\in GL(3,2),\]
we get that $\tau_{_{B,b}}\in H$
is a permutation on $V$ of the type $[2;6].$\\
It follows that $H$ contains permutations of the types:
\[(1)\quad\quad [2;2;2;2],\ [4;4],\ [2;6],\ [7].\]
Therefore, by \ref{3.4} and by the Sylow theorem, we get that $\bigcup_{g\in A_8}H^g$
contains every permutation of these types.\\
Now we denote with the natural numbers $1,\ldots,8$ the elements in $V,$ we identify
$Alt\,V$ with $A_8$ and we put
\[K=\left[Sym\,\{1,2,3\}\times Sym\,\{4,5,6,7,8\}\right]\cap A_8.\]
Thus $K$ contains at least a permutation of the types:
\[[3],\ [5],\ [2;2],\ [2;4],\ [3;3],\ [2;2;3]\]
and also the two non-conjugated permutations $(123)(45678)$ and $(123)^{(12)}(45678)$ of
the type $[3;5].$ Again, by \ref{3.4}, this implies that $\bigcup_{g\in A_8}K^g$ contains
all the permutations of the types
\[(2)\quad \quad[3],\ [5],\ [2;2],\ [2;4],\ [3;3],\ [3;5],\ [2;2;3].\]
Since each type of permutation in $A_8$ belongs either to the list $(1)$ or to the
list $(2),$ we conclude that $\{H^g,K^g:g\in A_8\}$ is a covering of $A_8.$
\end{proof}
Note that if $A_n$ is $(\ast\ast)$-coverable, {\em there is no sort of uniqueness for the
$(\ast\ast)$-coverings} of $A_n$: we have already showed two different
$(\ast\ast)$-coverings for $A_5$ and it is easily seen that we can construct another
$(\ast\ast)$-covering for $A_6$ using
\[H=Alt\{2,3,4,5,6\},\]\[K=\gen{(14)(2356),(15)(24)}\cong S_4.\]
At this point, to continue our investigation on the $(\ast\ast)$-coverable $A_n$, the
leading concept becomes the primitivity. In fact we will appeal
several times to the following classical result.
\begin{te}{\em ( \cite{wipe}, 13.9)}\label{3.6}
Let $G\leq S_n$ be a primitive group and $p$ a prime such that $p\leq n-3.$ If $G$ contains
a $p$-cycle, then $G\geq A_n.$
\end{te}
We recall that if $G\leq S_n$ is $2$-transitive then $G$ is primitive and also that
if $G\leq S_p,$ with $p$ a prime, is transitive then $G$ is primitive.
Moreover, for our purpose, it will be often appropriate to deduce the primitivity of a
permutation group from the next lemma.
\begin{lem}\label{3.7}
Let $G\leq S_n$ be a transitive subgroup and $n_0$ the minimal non trivial divisor of $n.$ If there
exists a prime $p$ such that $p>\frac{n}{n_0}$ and $p\mid\,|G|$, then $G$ is primitive.
\end{lem}
\begin{proof}
Assume that $\Delta$ is a non trivial block of imprimitivity for $G$. By the transitivity of $G$,
$|\Delta|\mid \,n\,$ and therefore $n_0\leq |\Delta|\leq \frac {n}{n_0}.$
Let $\overline{\Omega}=\{\Delta_1=\Delta,\ldots,\Delta_l\}$ be a complete system of blocks
for $G$ and consider the action of $\gen{\sigma}$ on $\overline{\Omega}$, where $\sigma
\in G$ is an element of order $p.$
This action cannot be faithful, otherwise $Sym\,\overline{\Omega}$ would contain an element
of order $p$ that is the product of some disjoint $p$-cycles; consequently
$\,p\leq |\overline{\Omega}|=\frac{n}{|\Delta|}\leq \frac {n}{n_0}$ and we would reach a
contradiction.
This means that the action of $\gen{\sigma}$ on $\overline{\Omega}$ is trivial and if
$(i_1,\ldots,i_p)$ is one of the disjoint $p$-cycles in which $\sigma$ splits then,
renumbering the $\Delta_i$, we can assume that $i_1\in \Delta=\Delta^{\sigma}.$
In particular $\Delta\supseteq\{i_1,\ldots,i_p\}$ and then
$|\Delta|\geq p>\frac {n}{n_0}$ which gives again a contradiction.
\end{proof}
\begin{os} \label{trans}
Let $\{H^g,K^g:g\in A_n\}$ be a covering of $A_n.$ If $n\geq 5,$ then at least one among $H$
and $K$ is transitive. If $n\geq 7,$ then exactly one among $H$ and $K$ is transitive.
\end{os}
\begin{proof}
Let $\{H^g,K^g:g\in A_n\}$ be a covering of $A_n$ and $n\geq 5.$\\
We can assume that $H$ contains a cycle $\sigma$ of maximal length in $A_n.$ If
$n$ is odd, then $H$ is clearly transitive.\\
If $n$ is even $\sigma$ has length $n-1$ and if
$H$ has more than one orbit, then it has exactly two orbits of lengths $1$ and $n-1$.
In this case in $\bigcup_{g\in A_n}H^g$ we get exclusively permutations with at least a fixed
point and therefore $K$ contains a permutation $\lambda$ of the type $[2;n-2]$ and a
permutation $\mu$ of the type $[3;n-3]$. To fix the ideas assume that $\lambda$
interchanges $1$ and $2$. Since $n\geq 6$, it follows that in the decomposition of $\mu$
as a product of disjoint cycles there are no transpositions hence $\mu$ takes $1$ or
$2$ into an element belonging to $\{3,\ldots ,n\}$. But $\lambda$ cyclically permutes
the elements of $\{3,\ldots ,n\}$ and consequently $K$ is transitive on
$\{1,\ldots ,n\}.$
Next let $n\geq 7$ and assume that $H$ and $K$ are both transitive. If $n=7,$ then $H$ and
$K$ are primitive and one of them
must contain a $3$-cycle hence, by \ref{3.6}, it coincides with $A_n$. Then we can
assume $n\geq 8$. By the Bertrand's postulate, there exists
a prime $p$ with $n/2<p\leq n-3$ and either $H$ or $K$ must contain a $p$-cycle
$\sigma$, say $\sigma \in H.$ By \ref{3.7}, we get that $H$ is primitive and, from
\ref{3.6}, it follows that $H=A_n,$ a contradiction.
\end{proof}
We are now in position to prove the goal of this section.
\begin{te}\label{3.9}
$A_n$ is $(\ast\ast)$-coverable if and only if $\,4\leq n\leq 8.$
\end{te}
\begin{proof}
From \ref{2.9} and \ref{3.5} we know that if $4\leq n\leq 8,$ then $A_n$ is a
$(\ast\ast)$-coverable group.
Thanks to \ref{trans} we only need to show that for $n>8,$ $A_n$ admits no covering
$\{H^g,K^g:g\in A_n\}$ with $H,\ K<A_n,\ H$ transitive and $K$ not transitive.
Assume the contrary. First of all we observe that
if $[l_1;\ldots;l_r]$ is the type of a permutation in $A_n$, then at least a
permutation of this type lies in $H$ or in $K.$\\
Let $\gamma_1,\ldots,\gamma_k$, with $k\geq 2$ be the orbits of $K$ on $\Omega=
\{1,\ldots,n\}$ and $\gamma_1$ that of minimal length $a$. Then
\[K\leq\left[Sym\,\gamma_1 \times Sym\,(\gamma_2\cup\ldots\cup\gamma_k)\right]\cap A_n<
A_n.\] Renaming the elements in $\Omega$, we can assume $\gamma_1=\{1,\ldots,a\},\
\gamma_2\cup \ldots\cup\gamma_k=\{a+1,\ldots,n\}$ and, passing from $K$ to a proper
subgroup of $A_n$ containing $K$, we can assume \[K=\left[Sym\,\{1,\ldots,a\}\times Sym\,\
\{a+1,\ldots,n\}\right]\cap A_n\] with orbits $\{1,\ldots,a\}$ and $\{a+1,\ldots,n\}$ of
lengths $a$ and $b$ such that \[1\leq a\leq b\leq n-1,\ a+b=n.\]
Note that $a\leq \left[\frac{n}{2}\right];$ moreover no even permutation of the type
$[l_1;\ldots;l_r]$ with $l_i\geq n-a+1$ for some $i\in \{1,\ldots,r\}$ belongs to $K$,
because $b=n-a$ is the maximal length of an orbit for $K.$ Therefore $H$ contains at
least one permutation for each of these types. Observe that if we can identify a permutation
$\mu \in H$ of the type $[p\,;l_2;\ldots ;l_r]$ with $p$ prime and $p\nmid l_2,\ldots ,l_r,$ then
$\mu^{l_2\cdots l_r}\in H$ is a $p$-cycle. From now on our duty consists essentially in the
chase of these kinds of permutations and to do this we need to divide our argument into six
cases according to whether \[a\geq 6\,\hbox{ or }\, a=5,4,3,2,1.\]
\noindent {\em Case 1: $a\geq 6$}
\noindent
Due to \ref{3.6}, it is sufficient to show that $H$ is primitive and contains a
$3$-cycle.
Let first $n$ be odd. Then $H$ contains permutations of the types: \[[n-2],\ [3;n-4],
\ [2;3;n-5].\] Because $H$ is transitive, $H_1$ contains a $(n-2)$-cycle and a
permutation of the type $[3;n-4]$ which has no fixed point on $\Omega-\{1\}.$ This
implies that $H_1$ is transitive on $\Omega-\{1\}$ and therefore $H$ is $2$-transitive
on $\Omega$ and, in particular, primitive.\\
If $3\nmid n-4,$ then a suitable power of a permutation in $H$ of type $[3;n-4]$ is a $3$-cycle.\\
If $3\mid n-4,$ then $3\nmid n-5$ and we get a $3$-cycle in $H$ as a power of a permutation
of the type $[2;3;n-5].$
Let now $n$ be even. Then $H$ contains permutations of the types:
\[[n-1],\ [3;n-3],\ [3;n-5].\]
$H$ is clearly $2$-transitive and then, as before, primitive.\\
If $3\nmid n,$ then a suitable power of a permutation of the type $[3;n-3]$ gives a $3$-cycle
in $H$.\\
If $3 \mid n,$ then $3 \nmid n-5$ and, we get $3$-cycle considering a power of a permutation
of the type $[3;n-5].$
~\\
\noindent {\em Case 2: $a=5$}
\noindent
It is $b=n-5$ and $5\leq \left[\frac{n}{2}\right]$ leads to $n\geq 10.$ For $n\neq 12,$ it
is sufficient to show that $H$ is primitive and it contains a $3$-cycle or a $7$-cycle; the
case $n=12$ needs a particular argument.
Let $n$ be odd. Then $H$ contains permutations of the types: \[[n-2],\ [3;n-4].\]
Therefore $H_1$, containing a $(n-2)$-cycle and a fixed point free permutation, is
transitive. This means that $H$ is $2$-transitive and in particular primitive.\\
If $3\nmid n-1,$ we get a $3$-cycle in $H$ as a power of a permutation of the type $[3;n-4].$\\
If $3\mid n-1,$ then $n=3k+1$ with $k\geq 4$ even. We examine first the case $k>4,$ that
is $n>13:$ then the integer $\frac{n-3}{2}$ is greater than $5$ and no
permutation of the type $[3;\frac{n-3}{2};\frac{n-3}{2}]$ lies in $K.$ On the other
hand $3\nmid n-3$ hence, if $\mu\in H$ is of that type, we get that
$\mu^{\frac{n-3}{2}} \in H$ is a $3$-cycle.\\ Next let $n=13:$
observe that, because $b=8$, $K$ contains no permutations of type $[2;4;7]$ and then
there exists $\mu \in H$ of the type $[2;4;7].$ Thus $\mu^4\in H$ is a $7$-cycle.
Let $n$ be even. Then $H$ contains permutations of the types: \[[n-1],\ [3;n-3].\]
This forces $H$ to be $2$-transitive and hence primitive.\\
If $3\nmid n,$ then $H$ contains a $3$-cycle. \\
If $3\mid n,$ then $n=6k$ with $k\geq 2.$ We explore first the case $k>2,$ that is
$n\geq 18:$ then the integer $\frac{n-4}{2}$ is greater than $5$ and no permutation
of the type $[3;\frac{n-4}{2};\frac{n-4}{2}]$ is in $K;$ on the other hand $3\nmid
n-4$ and, as usual, $H$ contains a $3$-cycle.\\
Next let $n=12.$ Then
\[K=\left[Sym\,\{1,2,3,4,5\}\times Sym\,\{6,7,8,9,10,11,12\}\right]\cap A_{12}\]
and $H$ contains permutations of the types:
\[[11],\ [9],\ [8;2],\ [8;4],\ [6;2;2;2],\ [3;3;3;3].\]
Then $H_1$ contains a $11$-cycle, $H_{1\,2}$ a permutation of the type $[9]$ and one
of the type $[8;2]$ and $H_{1\,2\,3}$ a permutation of the type $[9].$ This implies that $H$ is
$4$-transitive. Moreover we observe that $H_{1\,2\,3\,4}$ contains a
permutation of the type $[3;3]$ and one of the type $[4;4]$ as a power respectively of an
element of the type $[6;2;2;2]$ and of an element of the type $[8;2].$ Thus
$12\mid |H_{1\,2\,3\,4}|$ and hence
$2^5\cdot 3^4\cdot 5\cdot 11\mid |H|.$ On the other hand, by the well known Bochert's
result on the limitation of the index of a primitive subgroup of the symmetric group
( see \cite{hup67}, II,\,4.6 ), we have $|S_{12}:H|\geq 6!$ and thus
$|H|\leq 2^6\cdot 3^3\cdot 5\cdot 7\cdot 11.$ This gives
$$|H|=2^5\cdot 3^4\cdot 5\cdot 11\cdot k$$
with $1\leq k\leq 4.$\\
Yet we cannot have $k=3,$ otherwise $|H|_3=|A_{12}|_3$ and $H$
would contain a $3$-Sylow of $A_{12},$ hence also a $3$-cycle and, by \ref{3.6}, it would
coincide with $A_{12}.$ It follows that
\[|H|\in \{ 2^5\cdot 3^4\cdot 5\cdot 11,\, 2^6\cdot 3^4\cdot 5\cdot 11,\,
2^7\cdot 3^4\cdot 5\cdot 11\}.\]
Renaming the elements in $\Omega,$ we can assume $\sigma=(1\,2\,3\,4\,5\,6\,7\,8\,9\,\,10\,\,11)\in H$ and
$\gen{\sigma}\in Syl_{11}(H).$ It is easily observed that $N_{A_{12}}\gen{\sigma}=\gen{\sigma}
\rtimes \gen{(2\,5\,6\,10\,4)(3\,9\,\,11\,\,8\,7)}.$ Therefore we can have either
$N_H\gen{\sigma}=\gen{\sigma},$ or $N_H\gen{\sigma}=N_{A_{12}}\gen{\sigma}.$ Yet both these
possibilities are incompatible with $|H:N_H\gen{\sigma}|\equiv 1\,(\,\hbox{mod}\ 11\,).$ Namely, if
$N_H\gen{\sigma}=\gen{\sigma},$ we get $|H:N_H\gen{\sigma}|\in \{2^5\cdot 3^4\cdot 5,\,
2^6\cdot 3^4\cdot 5,\, 2^7\cdot 3^4\cdot 5\}$ while if $N_H\gen{\sigma}=N_{A_{12}}\gen{\sigma}$
we get $|H:N_H\gen{\sigma}|\in \{2^5\cdot 3^4,\, 2^6\cdot 3^4,\, 2^7\cdot 3^4\}$ and calculation
shows that neither of these numbers is congruent $1\ (\,\hbox{mod}\ 11\,).$ Therefore there
is no $H$ as required.
~\\
\noindent {\em Case 3: $a=4$}
\noindent
As usual we examine separately the case $n$ odd and the case $n$ even, remembering that
the maximal length of an orbit for $K$ is $b=n-4.$ For $n\neq 9,$ we will show that $H$
is primitive and it contains a $3$-cycle or a $5$-cycle; the case $n=9$ needs a peculiar
argument.
Let $n$ be odd. Then $H$ contains permutations of the types: \[[n-2],\ [2;n-3]\] and
the usual argument show that $H$ is primitive.\\
Let $n>9,$ then $K$ contains no permutation of the type $[2;3;n-5].$\\
If $3\nmid n-2,$ then a suitable power of an element of $H$ of the type $[2;3;n-5]$ is a
$3$-cycle.\\
If $3\mid n-2,$ then $3 \nmid n-3,\ n-3$ is even and $n=2+3k$ where $k\geq 3$ is odd.
If $k>3,$ we have $n>11$ and hence $\frac{n-3}{2}>4;$ this implies that $K$ does not
contain permutations of the type $[3;\frac{n-3}{2};\frac{n-3}{2}]$ and then $H$ contains
a $3$-cycle.\\
For $k=3,$ that is $n=11,$ we can observe that \[K=\left[Sym\,\{1,2,3,4\}\times
Sym\,\{5,6,7,8,9,10,11\}\right]\cap A_{11}\] contains no permutations of the type
$[3;3;5],$ hence $H$ contains at least a permutation of this type and hence
a $5$-cycle.\\
Finally let $n=9.$ Then
\[K=\left[Sym\,\{1,2,3,4\}\times Sym\,\{5,6,7,8,9\}\right]\cap A_{9}\]
contains no permutations of the types $[9],\ [7],\ [2;6],$ hence $H$ contains at
least a permutation for each of these types and consequently also a permutation of the type
$[3;3].$ It is clear that $H$ is primitive and, since no power of a $9$-cycle is of the type
$[3;3],$ we argue that $|H|_3\geq 3^3.$ But we cannot have $|H|_3> 3^3$ otherwise
$|H|_3=|A_9|_3$ and $H$ would contain a $3$-cycle. Therefore if $P\in Syl_3(H),$ we have
$|P|=27$ and
$P$ contains permutations of the types $[9]$ and $[3;3].$\footnote{We emphasize that $A_9$
really admits a subgroup $H$ with $|H|_3=3^3$ containing permutations of the types $[9],\
[7],\ [2;6].$ In fact you can consider $H=SL(2,8)\rtimes\gen{\alpha}$ where $\alpha$ is
the automorphism of $SL(2,8)$ which extend to the matrices the action of the Frobenius
automorphism of $GF(8)$ of degree $3,\ x\mapsto x^2;$ the natural action of this group on the
$9$ points of the projective line $\mathcal{P}(1,8)$ represents faithfully $H$ as a
subgroup of $A_9$ with the required properties.}\\
Renaming the elements in
$\Omega,$ we may assume $\mu=(123456789)\in P$ and there must exist $\beta\in P$ of the
type $[3;3],$
such that $P=\gen{\mu}\rtimes \gen{\beta}.$ This gives $P\leq N_{A_9}\gen{\mu}.$ But
$N_{A_9}\gen{\mu}={\gen{\mu}}\rtimes \gen{\alpha}$ with $\alpha=(235986)(47)$ and
$\mu^{\alpha}=(135792468)=\mu^2.$ It follows that
\[P=\gen{\mu,\ \beta: \mu^9=\beta^3=1,\ \mu^{\beta}=\mu^4},\] where we have put
$\beta=\alpha^2=(258)(396).$ Therefore $P$ is nilpotent of class $2,$ $P'=\gen{\mu^3}$
and each element in $P$ has a unique representation as $\mu^i\beta^j$ with $0\leq i\leq
8$ and $0\leq j\leq 2.$\\
We have $(\mu^i\beta^j)^3=\mu^{3i}\,\beta^{3j}\,[\beta^j,
\mu^i]^3=\mu^{3i},$ hence the elements of order $9$ in $P$, that is the $9$-cycles in $P,$
are exactly the $\mu^i\beta^j$ with $i\in I=\{1,2,4,5,7,8\}$ and $j\in\{0,1,2\}.$\\
Now we show that these elements are $A_9$-conjugate.\\ First of all we have
\[\mu^{\alpha^k}=\mu^{2^k}\quad \hbox{for}\quad 0\leq k\leq 5\]
and, since mod $9,$ the powers $2^k$ describe the elements of $I,$ the $9$-cycles
$\mu^i$ with $i\in I$ are $A_9$-conjugates.\\ Next consider $\gamma=(258)\in A_9$ and observe
that $\mu^{\gamma}=(153486729)=\mu \beta.$ Then
\[\mu^{\gamma\alpha^k}=(\mu\beta)^{\alpha^k}=\mu^{\alpha^k}\beta\quad\hbox{for}\quad
0\leq k\leq 5,\]
runs over all the $9$-cycles of the type $\mu^i\beta.$\\
Similarly, since $\gamma\in C_{A_9}(\beta),$
\[\mu^{\gamma^2\alpha^k}=(\mu\beta)^{\gamma\alpha^k}=(\mu^{\gamma}\beta)^{\alpha^k}=
(\mu\beta^2)^{\alpha^k}=\mu^{\alpha^k}\beta^2\quad\hbox{for}\quad 0\leq k\leq 5,\]
describes all the $9$-cycles of the type $\mu^i\beta^2.$\\
Since the $3$-Sylow subgroups of $H$ are $H$-conjugates, we deduce that the $9$-cycles
of $H$ are $A_9$-conjugates. Now, by \ref{3.4}, the $9$-cycles of $A_9$ split into two
conjugacy classes $\gamma_1,\ \gamma_2$ and calling $C$ the set of the $9$-cycles in
$H,$ we can assume $C\subseteq \gamma_1.$ But then picking any $\sigma\in \gamma_2,$
we get that $\sigma\notin \bigcup_{g\in {A_{9}}}K^g$ because $K$ contains no $9$-cycle
and, on the other hand, $\sigma\notin \bigcup_{g\in {A_{9}}}H^g$ because the $9$-cycles
in $\bigcup_{g\in {A_{9}}}H^g$ are given by $\bigcup_{g\in {A_{9}}}C^g
\subseteq \gamma_1,$ against the definition of a covering.
Let $n$ be even. Then $H$ contains permutations of the types: \[[n-1],\ [3;n-3]\] and it is
primitive.\\
If $3\nmid n,$ then $H$ contains a $3$-cycle.\\
If $3\mid n,$ then $n=6k$ with $k\geq 2.$ If $k>2,$ that is $n\geq 18,$ then $K$ contains
no permutations of the type $[2;3;5;n-10]$ and moreover $3\nmid n-10;$
thus there exists a $3$-cycle in $H.$\\
If $n=12,$ then \[K=\left[Sym\,\{1,2,3,4\}\times Sym\,\{5,6,7,8,9,10,11,12\}\right]
\cap A_{12}.\]
This implies that $H$ contains a permutation of type $[5;7]$ and hence a $5$-cycle.
~\\
\noindent {\em Case 4: $a=3$}
\noindent
For $n\neq 10,$ it is sufficient to show that $H$ is $2$-transitive and
that it contains a $5$-cycle or, provided that $n>10,$ a $7$-cycle; the case $n=10,$
needs a special argument.
Let $n$ be odd. We consider first the case $n\neq 9,11.$ Then, because $n-5>3,$ $H$ contains permutations of the types:
\[[n-2],\ [4;n-5]\]
and it is primitive.\\
Because $n-5$ is even and $\frac{n-5}{2}>3$ for $n\neq 9,11,$ then
$H$ contains also a permutation of the type $[5;\frac{n-5}{2};\frac{n-5}{2}].$\\
If $5\nmid n,$ then we find a $5$-cycle in $H.$ \\
If $5\mid n,$ we observe that $5\nmid \,\frac{n-5}{2}-1,\,\frac{n-5}{2}
+1;$ moreover $\frac{n-5}{2}-1$ and $\frac{n-5}{2}+1$ have the same parity and therefore
there exists an even permutation of the type $[5;\frac{n-5}{2}-1;\frac{n-5}{2}+1].$ Yet, since
$n\geq 15\,$ implies $\,\frac{n-5}{2}-1 >3,$ no permutation of this type belongs to
$K$ and $H$ contains a $5$-cycle.\\
If $n=9,$ we observe that $K$ does not contain permutations of the type $[2;2;5]$
and we deduce that $H$ contains a $5$-cycle.\\ If $n=11,$ we note that $K$ contains no
permutation of the type $[2;2;7],$ hence $H$ contains a $7$-cycle.
Let $n$ be even. Then $H$ contains a $(n-1)$-cycle and it is primitive
while $K$ contains no permutation of the type $[5;n-5].$ Hence, provided that
$5\nmid n,$ we have a $5$-cycle in $H$. On the other hand, if $5\mid n$ and $n\neq 10,$ then
we observe that $n-6$ is even and
$\frac{n-6}{2}>3;$ hence $K$ contains no permutation of the type $[5;\frac{n-6}{2};
\frac{n-6}{2}].$ Because $5\nmid\ \frac{n-6}{2},$ it turns out that $H$ contains a $5$-cycle.\\
Finally let $n=10.$ Then
\[K=\left[Sym\,\{1,2,3\}\times Sym\,\{4,5,6,7,8,9,10\}\right]\cap A_{10},\]
$H$ contains permutations of the types:
\[[9],\ [4;6]\]
and consequently also permutations of the type $[3;3].$ Since no power of a $9$-cycle
is of the type $[3;3],$ we argue that $|H|_3\geq 3^3.$ On the other hand, $H$ is clearly
primitive hence it contains no $3$-cycle and therefore $|H|_3<|A_{10}|_3.$ Thus,
if $P\in Syl_3H,$ then $|P|=3^3$ and $P$ contains permutations of the types $[9]$ and
$[3;3].$ Now, the same
argument used in the case $a=4,\ n=9,$ enables us to assume $P=\gen{\mu}\rtimes\gen{\beta},$
where $\mu=(123456789)$ and $\beta$ is of the type $[3;3].$ Then we have
$P\leq N_{A_{10}}\gen{\mu}=\gen{\mu}\rtimes\gen{\alpha}$ where $\alpha=(235986)(47)$ and,
setting $\beta=\alpha^2,$ we get
$$P=\gen{\mu,\ \beta: \mu^9=\beta^3=1,\ \mu^{\beta}=\mu^4}$$
At this point observing that, by \ref{3.4}, the $9$-cycles in $A_{10}$ split into two conjugacy
classes, we can repeat word by word the reasoning developed in the case
$a=4,\ n=9$ reaching, as there, a contradiction.
~\\
\noindent
{\em Case 5: $a=2$}
\noindent
It is enough to show that $H$ is primitive and that it contains a $3$-cycle.
Let $n$ be odd. Then $H_1$ contains permutations of the type:
\[[3;n-4],\ [4;n-5].\]
This implies that $H_1$ is transitive on $\Omega-\{1\},$ otherwise it would have two orbits
of lengths $\{3,n-4\}=\{4,n-5\}$ which is impossible because $n\neq8.$
Therefore $H$ is primitive.\\
If $3\nmid n-1,$ then $H$ contains a $3$-cycle.\\
If $3\mid n-1,$ then $3\nmid \frac{n-3}{2}>2$ and $K$ contains no permutation of the type
$[3;\frac{n-3}{2};\frac{n-3}{2}]$ and therefore $H$ contains a $3$-cycle.
Let $n$ be even. Then $H$ contains permutations of the types: \[[n-1],\ [3;n-3]\] and is
primitive.\\
If $3\nmid n$ we obtain a $3$-cycle in $H$ in the usual way.\\
If $3\mid n,$ then $n=6k\geq 12$ and $K$ contains no permutation of the type $[3;4;n-8].$
Since $3\nmid n-8,$ we obtain a $3$-cycle in $H.$
~\\
\noindent
{\em Case 6: $a=1$}
\noindent
In this case \[K=Alt\{2,\ldots,n\}\] and $\bigcup_{g\in {A_n}}K^g$ consists exactly of the
permutations of $A_n$ with at least a fixed point. Therefore if $[l_1;\ldots;l_r]$ where
$l_i\geq 2$ are integers such that $\sum_{i=1}^r l_i=n,$ is the type of an even
permutation, then $H$ contains at least one permutation of this type.
Let $n$ be odd. Observe that for $k$ even $H$ contains permutations $\sigma_k$ of the types:
\[[\underbrace{2;\ldots;2}_k;n-2k],\]
provided that $0\leq k\leq \frac{n-3}{2},$ and for $k$ odd $H$ contains permutations
$\sigma_k$ of the types:
\[[\underbrace{2;\ldots;2}_{k-2};4;n-2k],\]
provided that $3\leq k\leq \frac{n-3}{2}.$\\
Since $\sigma_k^4\in H$ is a $(n-2k)$-cycle, we argue that $H$ contains at least
a $d$-cycle for each $d\in D=\{x\in \bfn: x\hbox{ odd, }3\leq x\leq n,\ x\neq n-2\}.$ But, by
the Bertrand's postulate, there exists a prime $p$ with $\frac{n+1}{2}\leq p \leq n-4$
and obviously $p\in D.$ Then \ref{3.7} implies that $H$ is primitive and, by \ref{3.6}, we
obtain $H=A_n.$
Let $n$ be even. Then $H$ contains permutations $\sigma_k$ of the types:
\[[k;n-k]\]
for each $k\in \bfn$ with $2\leq k\leq \frac{n}{2}.$\\
By \ref{3.7} and \ref{3.6}, it is sufficient to show that $H$ contains a $p$-cycle for
some $p$ prime with $\frac{n}{2}<p\leq n-3.$\\
Now, by the Bertrand's postulate, there exists such a prime $p$ and we can write
$p=n-k_0$ for some $3\leq k_0\leq \frac{n-2}{2}.$ Then, since $k_0<p,$ we get that
$\sigma_{k_0}^{k_0}\in H$ is a $p$-cycle.
\end{proof}
\bigskip
\section{The $(\ast\ast)$-coverable symmetric groups
The main result of the previous section ( \ref{3.9} ) and the opportunity to build coverings
by intersection ( \ref{3.3} ), enable us to
extend immediately our investigation to the $(\ast\ast)$-coverable symmetric groups of
degree greater than $8.$ To complete
our analysis we need essentially to clear what happens for $S_7$ and $S_8:$ we do this
re-echoing the methods used in the last section.
\begin{lem}\label{S_7}
$S_7$ and $S_8$ are not $(\ast\ast)$-coverable groups.
\end{lem}
\begin{proof}
Let $\{H^g,K^g:g\in S_7\}$ be a covering of $S_7$ and, without loss of generality,
let $H$ contain a $7$-cycle. Then, being $7$ a prime, $H$ is primitive and $K$ can not be
primitive either, otherwise one of them contains transpositions and hence, by \ref{3.6}, coincides with
$S_7.$ This implies that $K$ is not transitive and therefore we can assume
\[K=Sym\,\{1,\ldots,a\}\times Sym\,\{a+1,\ldots,7\}\]
with $1\leq a\leq 3$ minimal length of an orbit of $K.$\\
If $a=1,3,$ then $K$ contains no permutations of the type $[2;5]$ and hence there
exists $\mu\in H$ of this type; consequently the transposition $\mu^5$ belongs to $H$
and we can again appeal to \ref{3.6} to conclude that $H=S_7.$\\
If $a=2,$ we have that $K=Sym\,\{1,2\}\times Sym\,\{3,4,5,6,7\}$ contains no permutations of
the types $[6]$ and $[3;4].$ Therefore $H$ contains at least a permutation of the type
$[6]$ and a permutation $\mu$ of the type $[3;4].$ Then $H\neq A_7,$ $\mu^4\in H$ is
a $3$-cycle and \ref{3.6} leads to the contradiction $H=S_7.$
Next let $\{H^g,K^g:g\in S_8\}$ be a covering of $S_8$ with $H$ containing a $7$-cycle.\\
If $H$ is not transitive, then we may assume $H=S_7$ and $K$ must contain all the types of
permutations with no fixed points. In particular $K$ contains at least a permutation of the
types $[8],\ [5;3]$ and $[3;3;2].$ Since $K$ is transitive and $5\mid\,|K|,$ from \ref{3.7},
we argue that $K$ is primitive. But because it contains a transposition, by \ref{3.6}, we
obtain that $K=S_8.$\\
Hence $H$ is transitive and indeed $2$-transitive.\\We observe that $H$ contains neither
$3$-cycles nor $5$-cycles. Namely if we assume the contrary, then by \ref{3.6}, we get $H=A_8;$
yet this implies that $K$ contains all the types of odd permutations and in particular $K$
contains a $8$-cycle and permutations of the types $[3;4]$ and $[6].$ Hence
$K$ is $2$-transitive and, since it contains transpositions, we deduce $K=S_8.$\\
In particular
$H$ contains no permutation of the types $[3;5],\ [3;4]$ and $ [2;5].$ Hence $K$ contains at
least a permutation of these types. Next observe that $K$ cannot be transitive otherwise $K_1$
contains permutations of the types $[3;4],$ $[2;5]$ and $K$ is $2$-transitive. Hence, since at
least a transposition belongs to $K$ we would have $K=S_8.$\\
This means that $K$ admits two orbits of lengths $3$ and $5$ and we may assume that
$$K=Sym\,\{1,2,3\}\times Sym\,\{4,5,6,7,8\}.$$
Then $H$ contains a $6$-cycle and is $3$-transitive. But it is well known that a
$k$-transitive group of degree $\,n\,$ with $\,k>\frac{n}{3}\,$ contains $A_n$
( see \cite{hup67}, p. $154$ ).
Then $H\geq A_8$ and, since $H$ contains also odd permutations, we argue that $H=S_8,$ a
contradiction.
\end{proof}
\begin{te}\label{sim}
$S_n$ is $(\ast\ast)$-coverable if and only if $\,3\leq n\leq 6.$
\end{te}
\begin{proof}
Due to \ref{2.9}, \ref{3.4} and \ref{S_7}, what remains to show is that if $n\geq 9,$
then $S_n$ is not $(\ast\ast)$-coverable. Assume the contrary and let
$\delta=\{H^g,K^g:g\in S_n\}$ be a covering of $S_n$ where $n\geq 9.$ If $H$ and $K$
contain both some odd permutation, then we have $S_n=A_nK=A_nH$ and, by \ref{3.3},
$\delta$ defines by intersection a covering of $A_n$, against \ref{3.9}. Hence exactly
one among $H$ and $K$ is included in $A_n$: we can assume that $H=A_n,\ K\not\leq A_n.$
It follows that $K$ contains all the types of odd permutations and, in particular,
transpositions. Evidently, to reach a contra\-diction, it is sufficient to show that $K$ is
$2$-transitive. If $n$ is even, then $K$ contains a $n$-cycle and is transitive;
moreover $K_1$ containing a permutation of the type $[3;n-4]$ and a permutation of the
type $[n-2]$ is certainly transitive. If $n$ is odd, then $K$ contains a permutation of
the type $[n-1]$ and a permutation of the type $[3;n-3]$ with no fixed point; therefore
$K$ is transitive and, since $K_1$ contains a $(n-1)$-cycle, we get that $K$ is
$2$-transitive.
\end{proof}
\bigskip
|
{
"timestamp": "2010-09-21T02:03:47",
"yymm": "1009",
"arxiv_id": "1009.3866",
"language": "en",
"url": "https://arxiv.org/abs/1009.3866"
}
|
\section{Introduction}\label{18}
In a series of seminal papers of the 1980's, Gel'fand, Graev, Kapranov and Zelevinski{\u\i} introduced {\em $A$-hypergeometric systems} $H_A(\beta)$, a class of maximally over\-determined systems of linear PDEs.
These systems, today also known as \emph{GKZ-systems}, are induced by an integer $d\times n$-matrix $A$ and a parameter vector $\beta\in\mathbb{C}^d$.
$A$-hypergeometric structures are nearly ubiquitous, generalizing most classical differential equations.
Indeed, toric residues, generating functions for intersection numbers
on moduli spaces, and special functions (Gau\ss, Bessel, Airy, etc.)
may all be viewed as solutions to GKZ-systems, and the same is true for varying Hodge structures on families of Calabi--Yau toric hypersurfaces as well as the space of roots of univariate polynomials with undetermined coefficients.
We shall identify $A$ with its set of columns ${\mathbf{a}}_1,\dots,{\mathbf{a}}_n$.
A parameter $\beta$ is \emph{nonresonant} if it is not contained in the locally finite subspace arrangement of \emph{resonant} parameters
\begin{equation}\label{17}
\Res(A):=\bigcup_{\tau}\left(\mathbb{Z} A+\mathbb{C}\tau\right),
\end{equation}
the union being taken over all linear subspaces $\tau\subseteq \mathbb{Q}^n$ that form a boundary component of the rational polyhedral cone $\mathbb{Q}_+ A$.
Assuming that the toric ring $\mathbb{C}[\mathbb{N} A]=\mathbb{C}[{\mathbf{a}}_1,\dots,{\mathbf{a}}_n]$ is Cohen--Macaulay and standard graded (the latter is equivalent to the classical notion of nonconfluence, see \cite{SW08}),
Gel'fand et al.~\cite{GKZ89,GKZ90} proved the following fundamental theorems:
\begin{enumerate}[(I)]
\item\label{1a} $H_A(\beta)$ is holonomic;
\item\label{1b} the rank (dimension of the solution space) of $H_A(\beta)$ equals the degree of $\mathbb{C}[\mathbb{N} A]$ for generic $\beta$;
\item\label{1c} if $\beta$ is nonresonant, the monodromy representation of the solutions of $H_A(\beta)$ in a generic point is irreducible.
\end{enumerate}
More recent research has shown that statements \eqref{1a} and \eqref{1b} hold true irrespective of whether $\mathbb{C}[\mathbb{N} A]$ is Cohen--Macaulay or standard graded, \cite{Ado94,SST00,MMW05}.
In Theorems~\ref{8} and \ref{9}, we prove the same of statement \eqref{1c} while providing a converse inspired by \cite{Beu10}.
The crucial tool for the proof of \eqref{1c} in \cite[Thm.~2.11]{GKZ90} is the Riemann--Hilbert correspondence of Kashiwara and Mebkhout, relating regular holonomic $D$-modules to perverse sheaves.
Confluence (i.e., irregularity) of $M_A(\beta)$
rules out the use of the Riemann--Hilbert correspondence
in the general case.
A powerful way of studying $H_A(\beta)$ is to consider the corresponding $D$-module $M_A(\beta)$ on $\mathbb{C}^n$ as a $0$-th homology of the \emph{Euler--Koszul complex} $K_\bullet(\mathbb{C}[\mathbb{N} A],\beta)$.
This idea can be traced back to \cite{GKZ89} and was developed into a functor in \cite{MMW05}.
Results from \cite{MMW05} show that $K_\bullet(\mathbb{C}[\mathbb{N} A],\beta)$ is a
resolution of $M_A(\beta)$ if and only if $\beta$ is not in the \emph{$A$-exceptional arrangement} $\mathscr{E}_A$ (see Remark~\ref{30}), a well-understood (finite) subspace arrangement of $\mathbb{C}^n$ comprised of the parameters $\beta$ for which the solution space of $H_A(\beta)$ is unusually large.
Surprisingly, the Euler--Koszul technique combined with the $D$-module/represen\-tation-theoretic description of GKZ-systems from \cite{SW09} serves as a replacement for the Riemann--Hilbert correspondence in the proof of \eqref{1c}.
This provides an approach that is simultaneously conceptually simpler
and more widely applicable.
\subsubsection*{Acknowledgments}
We are grateful to the referees for their comments, and for informing us that Mutsumi Saito has an article in press with \emph{Compositio~Math.}\ that also discusses reducibility of GKZ-systems (in much greater detail).
We would also like to thank Alan Adolphson for raising a relevant question.
\section{Hypergeometric system and Euler--Koszul homology}\label{15}
\subsection{Hypergeometric D-module}
Let $A=(a_{i,j})\colon\mathbb{Z}^n\to\mathbb{Z}^d$ be an integer $d\times n$-matrix, which we view both as a map, and as the finite subset $\{{\mathbf{a}}_1,\dots,{\mathbf{a}}_n\}$ of columns.
We assume that the additive group $\mathbb{Z} A$ generated by the columns of $A$ is the free Abelian group $\mathbb{Z}^d$, but we do not assume that $A$ is positive, i.e., we do allow nontrivial units in the semigroup $\mathbb{N} A$ (see Remarks~\ref{41} and \ref{42}).
Let $x_A=x_1,\dots,x_n$ be coordinates on $X:=\mathbb{C}^n$, and let $\partial_A=\partial_1,\dots,\partial_n$ be the corresponding partial derivative operators on $\mathbb{C}[x_A]$.
Then the \emph{Weyl algebra}
\[
D_A=\mathbb{C}\ideal{x_A,\partial_A\mid[x_i,\partial_j]=\delta_{i,j},\,[x_i,x_j]=0=[\partial_i,\partial_j]}
\]
is the ring of algebraic differential operators on $\mathbb{C}^n$.
With ${\mathbf{u}}_+=(\max(0,u_j))_j$ and ${\mathbf{u}}_-={\mathbf{u}}_+-{\mathbf{u}}$, write $\square_{\mathbf{u}}$ for $\partial^{{\mathbf{u}}_+}-\partial^{{\mathbf{u}}_-}$, where here and elsewhere we freely use multi-index notation.
The \emph{toric relations of $A$} are then
\[
\square_A:=\{\square_{\mathbf{u}} \,\mid\, A{\mathbf{u}}=0\}\subseteq R_A:=\mathbb{C}[\partial_A],
\]
and generate the \emph{toric ideal} $I_A=R_A\cdot\square_A$, whose residue ring is the \emph{toric ring}
\[
S_A:=R_A/I_A\cong\mathbb{C}[\mathbb{N} A]=\mathbb{C}[{\mathbf{a}}_1,\dots,{\mathbf{a}}_n].
\]
The \emph{Euler vector fields} $E=E_1,\dots,E_d$ induced by $A$ are defined as
\[
E_i:=\sum_{j=1}^na_{i,j}x_i\partial_j.
\]
Then, for $\beta\in\mathbb{C}^d$, the \emph{$A$-hypergeometric ideal} and \emph{$D$-module} are by \cite{GGZ87,GKZ89} the left $D_A$-ideal and -module
\[
H_A(\beta)=D_A\cdot\{E-\beta\}+D_A\cdot\square_A\quad\text{and}\quad
M_A(\beta)=D_A/H_A(\beta).
\]
The structure of the solutions to $H_A(\beta)$ is tightly interwoven with the combinatorics of the pair $(A,\beta)\in(\mathbb{Z} A)^n\times\mathbb{C} A$ (see, for instance, \cite{ST98,CAD99,MM06,Oku06,Ber08}).
\begin{rmk}\label{41}
Suppose we were to weaken the condition $\mathbb{Z} A=\mathbb{Z}^d$ to ``the rank of $\mathbb{Z} A$ is $d$\,''.
Pick a basis $B$ for $\mathbb{Z} A$, interpreted as elements of $\mathbb{Z}^d$.
In terms of $B$, $A$ takes the form of the $d\times n$ matrix $A'$ (say) which satisfies $A=BA'$ and $\mathbb{Z} A'=\mathbb{Z}^d$.
Choose $\beta\in\mathbb{C} A=\mathbb{C} A'$.
The hypergeometric systems $H_A(\beta)$ and $H_{A'}(B^{-1}\beta)$ are equivalent since $\ker_{\mathbb{Z}^n}(A)=\ker_{\mathbb{Z}^n}(A')$.
\end{rmk}
\subsection{Torus action}\label{33}
Consider the algebraic $d$-torus $T:=\Spec(\mathbb{C}[\mathbb{Z} A])\cong(\mathbb{C}^*)^d$ with coordinate functions $t=t_1,\dots,t_d$.
The columns ${\mathbf{a}}_1,\dots,{\mathbf{a}}_n$ of $A$ can be viewed as characters ${\mathbf{a}}_i(t)=t^{{\mathbf{a}}_i}$ on $T$, and the parameter vector $\beta\in\mathbb{C}^d$ as a character on its Lie algebra via $\beta(t_i\partial_{t_i})=-\beta_i+1$.
These characters define an action of $T$ on $X^*:=\Spec(\mathbb{C}[\mathbb{N}^n])$, interpreted as the cotangent space $T^*_0X$ of $X$ at $0$, by
\[
t\cdot\partial_A=(t^{{\mathbf{a}}_1}\partial_1,\dots,t^{{\mathbf{a}}_n}\partial_n).
\]
The toric ideal $I_A$ is the ideal of the closure of the orbit $T\cdot{\mathbf{1}}_A$ of ${\mathbf{1}}_A=(1,\ldots,1)$ in $X^*$, whose coordinate ring is $S_A$.
The contragredient action of $T$ on the coordinate ring $R_A$ of
$X^*$ is given by
\[
(t\cdot P)(\partial_A)=P(t^{-{\mathbf{a}}_1}\partial_1,\dots,t^{-{\mathbf{a}}_n}\partial_n)
\]
for $P\in R_A$. It yields a \emph{$\mathbb{Z} A$-grading} on $R_A$ on the
coordinate ring $\mathbb{C}[x_A,\partial_A]$ of $T^*X$:
\begin{equation}\label{10}
-\deg(\partial_j)={\mathbf{a}}_j=\deg(x_j).
\end{equation}
In particular, $\deg(\partial^{\mathbf{u}})=A{\mathbf{u}}$, and $E-\beta$ and $\Box_A$
are homogeneous.
The following description of $M_A(\beta)$ was given in \cite{SW09}.
Consider the algebraic $\mathscr{D}_T$-module
\[
\mathscr{M}(\beta):=\mathscr{D}_T/\mathscr{D}_T\cdot\ideal{\partial_tt+\beta},
\]
where $\partial_tt:=\partial_1t_1,\dots,\partial_dt_d$.
It is $\mathscr{O}_T$-isomorphic to $\mathscr{O}_T$ but equipped with a twisted $\mathscr{D}_T$-module structure expressed symbolically as
\[
\mathscr{M}(\beta)=\mathscr{O}_T\cdot t^{-\beta-1}
\]
on which $\mathscr{D}_T$ acts via the product rule.
The orbit inclusion
\[
\phi\colon T\to T\cdot {\mathbf{1}}\hookrightarrow X
\]
gives rise to a (derived) direct image functor $\phi_+\colon\mathscr{D}_T\mods\to \mathscr{D}_X\mods$.
On $X$ one has access to the \emph{Fourier transform}: $\mathscr{F}(x_i)=\partial_i$, $\mathscr{F}(\partial_i)=-x_i$.
By \cite[Prop.~2.1]{SW09}, $\mathscr{F}\circ \phi_+\mathscr{M}(\beta)$ is represented by the Euler--Koszul complex $K_\bullet(S_A[\partial_A^{-1}],\beta)$.
Thus, the latter is quasiisomorphic to $K_\bullet(S_A,\beta)$ if $\beta\not\in\Res(A)$ by \cite[Thm.~3.6]{SW09} and hence \cite[Cor.~3.8]{SW09} yields
\begin{equation}\label{43}
M_A(\beta)=\mathscr{F}\circ\phi_+\mathscr{M}(\beta)\text{ if }\beta\not\in\Res(A).
\end{equation}
\subsection{Euler--Koszul functor}\label{40}
We say that $\beta\in\mathbb{Z} A$ is a \emph{true degree} of the graded $R_A$-module $M$ if $\beta$ is the degree of a nonzero homogeneous element of $M$.
The \emph{quasidegrees} of $M$ are the points $\qdeg(M)$ in the Zariski closure of $\tdeg(M)\subseteq\mathbb{Z} A\subseteq\mathbb{C} A$.
A graded $R_A$-module $M$ is called a \emph{toric module} if it has a finite filtration by graded $R_A$-modules such that each filtration quotient is a finitely generated $S_A$-module.
The toric modules with $\mathbb{Z} A$-homogeneous maps of degree zero form a category that is closed under subquotients and extensions.
For every toric module the quasidegrees form a finite subspace arrangement where each participating subspace is a shift of a complexified face of $\mathbb{Q}_{\ge 0}A$ by a lattice element.
For all $\beta\in\mathbb{C}^d$ and for any toric $R_A$-module $M$ one can define a collection of $d$ commuting $D_A$-linear endomorphisms denoted $E_i-\beta_i$, $1\le i\le d$, on the $D_A$-module $D_A\otimes_{R_A}M$ which operate on a homogeneous element $m\in D_A\otimes_{R_A} M$ by $m\mapsto
(E_i-\beta_i)\circ m$, where
\[
(E_i-\beta_i)\circ m=
(E_i-\beta_i-\deg_i(m))\cdot m.
\]
There is an exact functor $K_\bullet(-,\beta)=K_\bullet(-,E-\beta)$ from the category of graded $R_A$-modules to the category of complexes of graded $D_A$-modules; it sends $M$ to the Koszul complex defined by all morphisms $E_i-\beta_i$.
On toric modules, the functor returns complexes with holonomic homology.
A short exact sequence
\[
0\to M'\to M\to M''\to 0
\]
of graded $R_A$-modules with homogeneous maps of degree zero induces a long exact sequence of \emph{Euler--Koszul homology}
\[
\cdots\to H_i(M'',\beta)\to H_{i-1}(M',\beta)\to
H_{i-1}(M,\beta)\to H_{i-1}(M'',\beta)\to\cdots
\]
where $H_i(-,\beta)=H_i(K_\bullet(-,\beta))$.
If $M=S_A$ then $H_0(M,\beta)=M_A(\beta)$.
We refer to \cite{MMW05,SW09} for more details.
\subsection{Rank (jumps) and monodromy reducibility}
We shall write $D_A(x_A)$ for the ring of $\mathbb{C}$-linear differential
operators on $\mathbb{C}(x_A)$; note that $D_A(x_A)
=\mathbb{C}(x_A)\otimes_{\mathbb{C}[x_A]}D_A$ as left $D_A$-module. We further set
$M(x_A):=\mathbb{C}(x_A)\otimes_{\mathbb{C}[x_A]}M$ for any $D_A$-module $M$.
The \emph{rank} $\rk(M)$ of a $D_A$-module $M$ is the
$\mathbb{C}(x_A)$-dimension of $M(x_A)$.
By Kashiwara's Cauchy--Kovalevskaya Theorem (see \cite[Thm.~1.4.19]{SST00}), it equals the $\mathbb{C}$-dimension of the \emph{solution space} $\Sol(M)=\Hom_{D_A}(M,\mathbb{C}\{x_A-\varepsilon\})$ of $M$ with coefficients in the convergent power series near the generic point $x_A=\varepsilon$ in (the analytic space associated to) $X$.
\begin{rmk}\label{30}
By \cite[Thm.~5.15]{Ado94} and \cite[Thms.~2.9, 7.5]{MMW05},
\[
\rk M_A(\beta)\ge\vol_A(A)
\]
with equality for generic $\beta\in\mathbb{C}^n$.
Here $\vol_A(G)$ denotes, for any $G\subseteq\mathbb{Z} A$, the simplicial
volume of the convex hull of $G$ taken in the lattice $\mathbb{Z} A$.
More precisely, equality is equivalent to $\beta\not\in\mathscr{E}_A$ where
\[
\mathscr{E}_A:=\sum_{j=1}^n{\mathbf{a}}_j -
\bigcup_{i=0}^{d-1}\qdeg\left(\Ext^{n-i}_{R_A}(S_A,R_A)\right)
\]
is the \emph{exceptional arrangement}.
\end{rmk}
\begin{dfn}
We say that a $D_A$-module $M$ has \emph{irreducible monodromy} if $M(x_A)$ is an irreducible $D_A(x_A)$-module (i.e.\ it has no nontrivial $D_A(x_A)$-quotients).
\end{dfn}
By \cite[Thm.~3.15]{Wal07}, monodromy irreducibility of $M(\beta)$ is a property of the equivalence class $\beta\in\mathbb{C} A/\mathbb{Z} A$.
The nomenclature is based on the Riemann--Hilbert correspondence:
$D_A(x_A)$-quotients of $M(x_A)$ correspond to monodromy-invariant subspaces of $\Sol(M)$ in nonsingular points of $M$.
(Analytic continuations of an analytic germ satisfy the same differential equations as the germ itself).
\begin{rmk}\label{42}
Careful reading of~\cite{MMW05} reveals that all fundamental results obtained through Euler--Koszul technology do not require $\mathbb{N} A$ to be a positive semigroup.
As a matter of fact, $\mathscr{E}_A$ was defined in \cite{MMW05} in terms of local cohomology with supports at the origin of $X^*$; the translation between this definition and ours here can only be done if $A$ is pointed.
On the other hand, it is the Ext-based definition that is (implicitly) used in all proofs in loc.~cit.
In consequence, the main theorems in \cite{Wal07} and \cite{SW09} remain true in the absence of positivity since the only ingredients in their proofs that are specific to the hypergeometric situation are those of \cite{MMW05}.
\end{rmk}
\section{Pyramids and resonance centers}
\begin{dfn}
For any subset $F$ of the columns of $A$ we write $\bar F$ for the complement $A\smallsetminus F$.
A \emph{face of $A$} is any subset $F\subseteq A$ subject to the condition that there be a linear functional $\phi_F\colon \mathbb{Z} A\to\mathbb{Z}$
that vanishes on $F$ but is positive on $\bar F$. This includes $F=A$
as possibility. Every face contains all units of $\mathbb{N} A$, and
$A$ is positive if and only if the empty set is a face of $A$.
For a given face $F$, we set
\[
I^F_A:=I_A+R_A\cdot \partial_{\bar F}
\]
and note that $R_A/I^F_A=S_F$ as $R_A$-module.
\end{dfn}
\begin{dfn}
Let $F$ be a face of $A$.
The parameter $\beta\in\mathbb{C}^d$ is \emph{$F$-resonant} if $\beta\in \mathbb{Z} A+\mathbb{C} G$ for a proper subface $G$ of $F$.
If $\beta$ is $G$-resonant for all faces $G$ properly containing $F$, but not for $F$ itself, we call $F$ a \emph{resonance center for $\beta$}.
\end{dfn}
A resonance center is a minimal face $F$ for which $\beta\in \mathbb{Z} A+\mathbb{C} F$.
Every parameter $\beta$ has a resonance center; $A$ is a (and then the only) center of resonance for $\beta$ if and only if $\beta$ is nonresonant in the
usual sense (i.e., $\beta\not\in\Res(A)$, defined in \eqref{17}).
On the other hand, for positive $A$, the empty face is a resonance center for $\beta$ if and only if $\beta\in\mathbb{Z} A$.
\begin{exa}
It is easy to have several resonance centers for $\beta$.
For example, consider $\beta=(\frac12,1)$ on the quadric cone $A=\begin{pmatrix}1&1&1\\0&1&2\end{pmatrix}$; $\beta$ has both extremal rays as resonance centers.
\end{exa}
\begin{dfn}
We say that $A$ is a(n iterated) \emph{pyramid over the face $F$} if $d=\dim_\mathbb{Z}(\mathbb{Z} A)$ equals $|\bar F|+\dim_\mathbb{Z}(\mathbb{Z} F)$.
\end{dfn}
The following equivalences are trivial or follow from \cite[Lem.~3.13]{Wal07}.
\begin{lem}\label{4}
The following are equivalent:
\begin{enumerate}
\item $F$ is a face and $A$ is a pyramid over $F$;
\item ${\mathbf{a}}_j\not\in\mathbb{Q} (A\smallsetminus \{{\mathbf{a}}_j\})$ for any $j\not\in F$;
\item\label{5} $\mathbb{Z} A=\mathbb{Z} {\mathbf{a}}_j\oplus\mathbb{Z}(A\smallsetminus \{{\mathbf{a}}_j\})$ for any $j\not\in F$;
\item $\vol_F(F)=\vol_A(A)$;
\item\label{22} for every $\beta\in\mathbb{C} A$, the coefficients $c_j$ in the sum $\beta=\sum_A c_j{\mathbf{a}}_j$ are uniquely determined by $\beta$ for $j\not\in F$;
\item the generators $\square_A$ of $I_A$ do not involve $\partial_j$ for any $j\not\in F$;
\item $S_F\otimes_{\mathbb{C}}\mathbb{C}[\partial_{\bar F}]=S_A$ as $R_A$-modules.
\end{enumerate}
\end{lem}
\begin{ntn}
Suppose $F$ is any nonempty face of $A$, and let $X_F$, $X^*_F$, $T_F$, $H_\bullet^F$, etc.\ be defined as in Section~\ref{15} with $A$ replaced by $F$ (cf.~Remark~\ref{41} for the case where $\mathbb{Z} A/\mathbb{Z} F$ has torsion).
Write $E^F=E^F_1,\dots,E^F_d$ where $E^F_i:=\sum_{j\in F}a_{i,j}x_j\partial_j$ is the part of $E_i$ supported in $F$.
Then, in particular, $M_F(\beta)=D_F/(D_F\cdot\ideal{E^F-\beta}+D_F\cdot I_F)$ for $\beta\in\mathbb{C} F$.
\end{ntn}
Suppose now that $A$ is a pyramid over the face $F$, and let $\beta\in\mathbb{C} A$.
The splitting in Remark~\ref{4}.\eqref{5} corresponds to a splitting of tori $T_A=T_F\times\prod_{{\mathbf{a}}_j\in\bar F}T_{{\mathbf{a}}_j}$ which in turn gives a splitting of the spaces of Lie algebra characters $\mathbb{C} A=\mathbb{C} F\oplus\bigoplus_{{\mathbf{a}}_j\in\bar F}\mathbb{C}{\mathbf{a}}_j$.
Then $\beta$ decomposes correspondingly as
\[
\beta=\beta^F+\sum_{j\in \bar F}\beta^{\bar F}_j.
\]
Let $\iota_F\colon X^*_F\hookrightarrow X^*_A$ be the inclusion.
By \cite[Lem.~4.8]{MMW05}, for $\beta\in\mathbb{C} F$,
\begin{align}\label{29}
(\mathscr{F}\circ\iota_{F,+}\circ\mathscr{F}^{-1})M_F(\beta)
&=\mathbb{C}[x_{\bar F}]\otimes_\mathbb{C} M_F(\beta)\\
\nonumber&\cong H_0(S_F,\beta)=D_A/(D_A\cdot\ideal{E^F-\beta}+D_A\cdot I_A^F)
\end{align}
as $D_A$-modules.
In the following lemma, \eqref{27} follows from \eqref{26} and
\eqref{29} above.
\begin{lem}\label{6}
If $A$ is a pyramid over $F$ then the following conditions hold:
\begin{enumerate}
\setcounter{enumi}{7}
\item\label{26} the ideal $H_A(\beta)$ contains $x_j\partial_j-\beta^{\bar F}_j$ for $j\not\in F$;
\item\label{27} $M_A(\beta)(x_A)=\mathbb{C}(x_A)\otimes_{\mathbb{C}[x_F]}M_F(\beta)$ for $\beta\in\mathbb{C} F$;
\item\label{24} the solutions of $M_A(\beta)$ are the solutions of $M_F(\beta^F)$, multiplied with the unique solution to the system
\[
\{x_j\partial_j\bullet f=\beta^{\bar F}_j\cdot f\}_{j\in\bar F}.
\]
\end{enumerate}
In particular, $\beta\in\mathscr{E}_A$ if and only if $\beta^F\in\mathscr{E}_F$.
\end{lem}
\begin{prp}
If $\beta\in\mathbb{C} A$ has a resonance center $F$ over which $A$ is a
pyramid, then $F$ is the only resonance center for $\beta$.
\end{prp}
\begin{proof}
Let $G$ be a second resonance center for $\beta$ and suppose $G$ meets the complement of $F$; pick ${\mathbf{a}}_k\in G\cap\bar F$.
Since $\mathbb{Z} {\mathbf{a}}_k$ is a direct summand of $\mathbb{Z} A$, it is also a direct summand
of $\mathbb{Z} G$. It follows that $G\smallsetminus \{{\mathbf{a}}_k\}$ is a face $G'$ of
$A$.
As $F$ and $G$ are resonance centers,
\[
\beta=z_k{\mathbf{a}}_k+\sum_{j\in\bar F\smallsetminus\{k\}}z_j{\mathbf{a}}_j+\sum_{j\in F}c_j {\mathbf{a}}_j,\quad
\beta=c'_k{\mathbf{a}}_k+\sum_{j\in\bar G'\smallsetminus\{k\}} z'_j{\mathbf{a}}_j+\sum_{j\in G'}c'_j{\mathbf{a}}_j
\]
where $z_k,z_j,z'_j\in \mathbb{Z}$ and $c'_k,c_j,c'_j\in\mathbb{C}$.
By Lemma~\ref{4}.\eqref{22}, the coefficients for ${\mathbf{a}}_k$ in these sums are identical, $c'_k=z_k\in\mathbb{Z}$.
It follows that
\[
\beta=\left(z_k{\mathbf{a}}_k+\sum_{j\in\bar G'\smallsetminus\{k\}}z'_j{\mathbf{a}}_j\right)+\sum_{G'}c'_j{\mathbf{a}}_j\in\mathbb{Z} A+\mathbb{C} G'.
\]
This contradicts $G$ being a resonance center. Thus $G\cap\bar
F=\emptyset$ and so $G\subseteq F$.
But then $F$ can only be a resonance center if $F=G$.
\end{proof}
\section{Resonance implies reducibility}
The following result generalizes Theorem~3.4 in \cite{Wal07} and Theorem~1.3 in \cite{Beu10}.
\begin{thm}\label{8}
Let $F$ be a resonance center for $\beta\in\mathbb{C} A$.
If $A$ is not a pyramid over $F$ then $M_A(\beta)$ has reducible monodromy.
\end{thm}
\begin{proof}
By hypothesis, we have $\beta-\gamma\in\mathbb{Z} A$ for some $\gamma\in\mathbb{C} F$.
We first dispose of the case $F=\emptyset$.
In that case, $A$ is positive, $\gamma=0$, $\beta\in\mathbb{Z} A$ and, by \cite[Thm.~3.15]{Wal07}, we may assume $\beta=0$.
Then $\mathbb{C}(x_A)$ is a rank-$1$ quotient of $M_A(\beta)(x_A)$.
But $A$ is not a pyramid over $F$, so
\[
\rk(M_A(\beta))\ge\vol_A(A)>\vol_F(F)=1=\rk(\mathbb{C}(x_A))
\]
by Remark~\ref{30} and Lemma~\ref{4}.
So $\mathbb{C}(x_A)$ is a proper quotient of $M_A(\beta)(x_A)$, and hence $M_A(\beta)$ has reducible monodromy.
We can hence assume that $F$ is not empty, and by \cite[Thm.~3.15]{Wal07}, we need to show the reducibility of $M_A(\gamma)$.
Consider the surjection
\[
M_A(\gamma)=H_0(S_A,\gamma)\twoheadrightarrow H_0(S_F,\gamma)
\]
induced by the surjection $S_A\twoheadrightarrow S_F$.
Therefore, it suffices to show that $0<\rk(H_0(S_F,\gamma))<\vol_A(A)$ by Remark~\ref{30}.
Since $F$ is a resonance center for $\beta$, and hence for $\gamma$ as well, $\gamma$ is a nonresonant parameter for the GKZ-system
\[
M_F(\gamma)=D_F/(D_F\cdot\ideal{E^F-\gamma}+D_A\cdot I_F).
\]
Then, by Remark~\ref{30}, $\rk(M_F(\gamma))=\vol_F(F)>0$ and $\rk(M_A(\gamma))\ge\vol_A(A)$.
As $A$ is not a pyramid over $F$, $\vol_F(F)<\vol_A(A)$ by Lemma~\ref{4}.
Finally, $\rk(M_F(\gamma))=\rk(H_0(S_F,\gamma))$ by \eqref{29}.
Combining the above (in)equalities yields the claim.
\end{proof}
\section{Resonance follows from reducibility}\label{13}
We now generalize Theorem~2.11 in \cite{GKZ90}
\begin{thm}\label{9}
Let $F$ be a resonance center for $\beta$.
If $A$ is a pyramid over $F$ then $M_A(\beta)$ has irreducible monodromy.
\end{thm}
\begin{proof}
First consider the case $F=A$.
Then $\beta\not\in\Res(A)$ and hence
$M_A(\beta)=\mathscr{F}\circ\phi_+(\mathscr{M}_\beta)$ by \eqref{43}
As in the proof of \cite[Prop.~2.1]{SW09}, factor $\phi=\varpi\circ\iota$ into the closed embedding of tori
\begin{equation}\label{19}
\iota:T\hookrightarrow\Spec(\mathbb{C}[\mathbb{Z}^n])=Y^*\cong(\mathbb{C}^*)^n
\end{equation}
induced by $\mathbb{Z} A\subseteq\mathbb{Z}^n$, followed by the open embedding
\begin{equation}\label{20}
\varpi:Y^*=X^*\smallsetminus\Var(\partial_1\cdots\partial_n)\hookrightarrow X^*.
\end{equation}
By Kashiwara equivalence, $\iota$ preserves irreducibility.
The same holds for $\varpi$, because $D$-affinity of both the target and the source of the inclusion map allows to detect submodules on global sections.
But global sections on $Y^*$ and $X^*$ agree because we are looking at an open embedding.
Since $\mathscr{M}(\beta)$ is clearly irreducible, $\phi_+\mathscr{M}(\beta)$ is as well.
As Fourier transforms preserve composition chains, $M_A(\beta)$ is irreducible.
It follows that $M_A(\beta)$ has irreducible monodromy.
Suppose now that $F$ is a proper face.
Choose $\gamma\in\mathbb{C} F$ with $\beta-\gamma\in\mathbb{Z} A$.
Then $M_F(\gamma)$ is irreducible by the first part of the proof, and the claim follows from Lemma~\ref{6}.\eqref{27} and \cite[Thm.~3.15]{Wal07}.
Finally, if $F=\emptyset$ then $A$ is positive and Lemma~\ref{6}.\eqref{26} shows that $M_A(\beta)(x_A)=\mathbb{C}(x_A)$ which has clearly irreducible monodromy.
\end{proof}
\bibliographystyle{amsalpha}
|
{
"timestamp": "2012-07-13T02:02:52",
"yymm": "1009",
"arxiv_id": "1009.3569",
"language": "en",
"url": "https://arxiv.org/abs/1009.3569"
}
|
\section{Introduction}
\label{sec:introduction}
Service-Oriented Computing (SOC) has emerged as a new software
development paradigm that enables implementation of Web accessible
software systems that are composed of distributed services which
interact with each other via the exchange of messages. In order to
facilitate integration of independently developed services that may
reside in different organizations, it is necessary to provide some
analysis and verification techniques to check as automatically as
possible that the new system will behave correctly avoiding erroneous
interactions leading to deadlock states for instance.
Let us show a couple of examples to illustrate the previous arguments,
where services are modelled using Labelled Transition Systems
(presented more formally in Section~\ref{section:models}). Services
{\sf S1} and {\sf S2} in Figure~\ref{fig:unspecified} can end up into
a deadlock because after interacting on {\sf a}, {\sf S2} can decide
to evolve through an internal action $\tau$ (right-hand branch of the
choice) and is deadlocked: {\sf S1} cannot interact on {\sf c} with
{\sf S2} at this point. On the other hand, the execution of {\sf S1'}
and {\sf S2} is free of deadlocks because all emissions on both sides
have a matching reception on the other.
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth,clip]{unspecifiedReceptions.eps}
\caption{Deadlocking execution of services}
\label{fig:unspecified}
\end{figure}
In Figure~\ref{fig:sim}, suppose that {\sf S1} is a client and {\sf
S2} a service. {\sf S1} is satisfied because the service is able to
reply his/her request, {\it i.e.}, can receive {\sf a} and send {\sf
b}. However, if we focus on another version of this client {\sf
S1'}, after submitting {\sf a}, the client expects either {\sf b} or
{\sf c}, but {\sf S2} is not able to provide {\sf c}. This is another
kind of issue that one may need to detect: all the messages (in the
client here) must have a counterpart.
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth,clip]{simulation.eps}
\caption{Unmatching messages}
\label{fig:sim}
\end{figure}
In this paper, we do not want to present the many works and papers
which have proposed analysis and verification for Web services, this
would be too long and uninteresting. Our goal is to focus on specific
issues occuring in this area, and present some automated techniques to
work them out. We will also give some key references for each problem
to enable the reader to go deeper in these issues and solutions
existing for them. Beyond giving a quick overview of service analysis
techniques, we also point out at the end of the paper a few challenges
that are still open, to the best of our knowledge.
The organization of this paper is as follows. First, we present in
Section~\ref{section:models} some formal models that are often used to
represent abstract descriptions of services, {\it e.g.}, Petri nets,
automata-based models, process algebras. In
Section~\ref{section:verification}, we focus on automated verification
techniques, namely equivalence-checking on one hand, and temporal
properties and model-checking on the other.
Section~\ref{section:compatibility} is dedicated to the compatibility
of two (or more) services. This section also comments on some
techniques to quantify the compatibility degree between two services,
and on service adaptation which is a solution to work out existing
mismatches detected using compatibility analysis. In
Section~\ref{section:realizability}, we present a slightly different
kind of analysis which aims at checking the realizability (and
conformance) of choreography specifications. Realizability indicates
whether services can be generated from a given choreography
specification in such a way that the interactions of these services
exactly match the choreography specification. Finally, we draw up some
conclusions in Section~\ref{section:conclusion}.
\section{Models of Services}
\label{section:models}
In this section, we focus on formal models. Bringing formality to the
service development process opens the way to the writing and
verification of properties that the designer expects from his/her
system. This is not the case of semi-formal notations such as UML or
BPMN which are often acknowledged as more readable and user-friendly
than formal methods but lack formal semantics and validation tools.
Services are distributed components which communicate exchanging
messages, therefore they are best described using behavioural
description languages. Several candidates have been used in the
literature:
\begin{itemize}
\item Process algebras (or calculi): CCS, CSP, LOTOS, FSP, etc
\item Automata-based models: state diagrams,
Harel's Statecharts, IO-Automata, LTS, etc
\item Petri nets: coloured Petri nets, workflow nets, open nets, etc
\item Temporal Logic: Lamport's TLA
\item Message Sequence Charts
\end{itemize}
Here are a few
references~\cite{BultanWWW04,SalaunBS06,LTSAWS,vanBreugel2006,AalstMSW09}
where the reader can find more details about these models and their
use in the service development process. According to us, process
algebras are one of the best candidates to specify service models for
four reasons: (i)~the existing calculi present several levels of
abstraction useful to have a more faithful representation of a
service, {\it e.g.}, specifying data (LOTOS) or mobility
($\pi$-calculus), (ii)~they are compositional notations, then adequate
to describe composition of services, (iii)~they provide textual
notation which makes them scalable to tackle real-world systems, and
(iv)~there exist some state-of-the-art verification tool-boxes for
these languages, {\it e.g.}, SPIN, CADP, UPPAAL, or $\mu$-CRL2.
In the rest of this paper, for illustration purposes and for the sake
of readability (process algebras are not perfect, unfortunately), we
assume that services are modelled using {\it Labelled Transition
Systems} (LTSs). An LTS is a tuple $(A, S, I, F, T)$ where: $A$ is
an alphabet which corresponds to the set of labels associated to
transitions, $S$ is a set of states, $I \in S$ is the initial state,
$F \subseteq S$ is a nonempty set of final states, and $T \subseteq S
\times A \times S$ is the transition relation. In our model, a {\it
label} is either a $\tau$ (internal action) or a tuple $(m,d)$ where
$m$ is the message name, and $d$ stands for the communication
direction (either an emission $!$ or a reception $?$). Labels can
take typed parameters or arguments into account as well, and in such a
case the transition system is called {\it symbolic} (STS). Using this
model, a choice can be represented using either a state and at least
two outgoing transitions labelled with observable actions (external
choice) or branches of $\tau$ transitions (internal choice). LTSs and
STSs can be easily derived from higher-level description languages
such as Abstract BPEL, see for
instance~\cite{BultanWWW04,SalaunBS06,CamaraMSCOCP09} where such
abstractions were used for verification, composition or adaptation of
Web services. The operational semantics of STSs is given
in~\cite{DOS-Foclasa09}.
Several communication models can be assumed among services. In
particular, we would like to say a word here about synchronous {\it
vs.} asynchronous communication. Synchronous communication
corresponds to handshake communication whereas asynchronous
communication uses message queues for interaction purposes (similarly
to mailboxes). Most existing works rely on synchronous communication.
Asynchronous communication is as realistic as synchronous
communication, however, results are more complicated to obtain and
even sometimes undecidable~\cite{DZ-ACM83} (see
Section~\ref{section:conclusion} for a more detailed discussion). In
this paper, we assume a binary communication model where two services
synchronize if one can evolve through an emission, the other through a
reception, and both labels share the same message.
\medskip
{\bf Internal behaviours.} Service analysis could be worked out
without taking into account their internal evolution because that
information is not observable from its partners point of view
(black-box assumption). However, keeping an abstract description of
the non-observable behaviours while analysing services helps to find
out possible interoperability issues. Indeed, although one service
can behave as expected by its partner from an external point of view,
interoperability issues may occur because of unexpected internal
behaviours that services can execute. For instance,
Figure~\ref{fig:internal-vs-external} shows two versions of one
service protocol without ({\sf S1}) and with ({\sf S1'}) its internal
behaviour. Assuming a synchronous communication model, {\sf S1} and
{\sf S2} can interoperate on {\sf a} and terminate in final states
({\sf b!} in {\sf S1} has no counterpart in {\sf S2} and cannot be
executed). However, if we consider {\sf S1'}, which is an abstraction
closer to what the service actually does, we see that this protocol
can (choose to) execute a $\tau$ transition at state {\sf s1} and
arrives at state {\sf s3} while {\sf S2} is still in state {\sf u1}.
At this point, both {\sf S1'} and {\sf S2} cannot exchange messages,
and the system deadlocks. This issue would not have been detected with
{\sf S1}.
The reader interested in more details about $\tau$ transitions and
their handling can refer to~\cite{OS-WCSI10}.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.7\linewidth,clip]{internal-external-1.eps}
\caption{{\sf S1} and {\sf S2} interoperate successfully, but {\sf S1'} and {\sf S2} can deadlock}
\label{fig:internal-vs-external}
\end{figure}
\section{Automated Verification}
\label{section:verification}
A major interest of using abstract languages grounded on a clear
semantics is that automated tools can be used to check that a system
matches its requirements and operates safely. Specifically, these
tools can help (i)~checking that two services are in some precise
sense \emph{equivalent} -- one behaviour is typically a very abstract
one expressing the specification of the problem, while the other is
closer to the implementation level; this can also be used for checking
the substitutability (or replaceability) of one service by another;
(ii)~checking that a service (possibly composite) verifies desirable
\emph{properties} -- {\it e.g.}, the property that the system will
never reach some unexpected state. Revealing that the composition of a
number of existing services does not match an abstract specification
of what is desired, or that it violates a property which is absolutely
needed can be helpful to correct a design or to diagnose bugs in an
existing service. Note that in the following of this section, we focus
on verification techniques that are of interest for Web services, and
we do not give an overview of the many papers that have been published
on this topic (most of them do the same using different languages and
tools), see for
instance~\cite{BultanWWW04,SalaunBS06,LTSAWS,vanBreugel2006,CDRM-JLAP-2010}.
\subsection{Verifying Equivalences}
Intuitively, two services are considered to be equivalent
if they are \emph{indistinguishable} from the viewpoint of an external
observer interacting with them. This notion has been formally defined
in the process algebra community, and several notions of equivalence
have been proposed~\cite{MilCC}. Equivalences are strong yet suitable
relations for these checks, because they preserve all observable
actions. However, these notions exhibit some subtleties relevant to
the context of Web services.
A first approach is to consider two services to be equivalent if the
set of \emph{traces} they can produce is the same
(\emph{trace-equivalence}). For instance, the possible executions of
the services shown in Fig.~\ref{figure:traces} part (A), where
messages {\sf a}, {\sf b} and {\sf c} can be respectively understood
as requests for reservation, editing data and cancellation. Both of
these two services will have {\sf a.b} and {\sf a.c} as possible
traces: they will either receive the messages {\sf a} then {\sf b}, or
{\sf a} then {\sf c}.
Nevertheless, it is not fully satisfactory to consider these two
services equivalent since they exhibit the following subtle
difference. After receiving message {\sf a}, the first service will
accept either message {\sf b} or {\sf c}. The second service behaves
differently: on receiving message {\sf a}, it will either choose to
move to a state where it expects message {\sf b}, or to a state where
it expects message {\sf c}. Depending on the choice it makes, it will
not accept one of the messages whereas the first service leaves both
possibilities open. The second service does not guarantee that a
request for reservation ({\sf a}) followed by, {\it e.g.},
cancellation ({\sf c}) will be handled correctly ({\sf c} might not be
possible if the service has chosen the left-hand side branch). The
notion of equivalence called \emph{bisimulation}~\cite{MilCC} is a
refinement of trace equivalence which takes these differences into
account.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.9\linewidth,clip]{bisimul.eps}
\caption{Classical examples of services not observationally equivalent.}
\label{figure:traces}
\end{figure}
Further subtleties arise when one has a partial knowledge of the
service behaviour. This may happen for two reasons: (i)~during the
design stage, where the specification which is being defined is
abstract and incomplete; (ii)~when one finds or reuses an existing
service, and only an interface or a partial description hiding private
details is available. $\tau$ actions must be taken into account when
reasoning on the equivalence of two services, as evidenced by
Fig.~\ref{figure:traces} part (B). Both of the services depicted here
can receive {\sf b} (edition of reservation data) or {\sf c}
(cancellation). But whereas the first one can receive any of the two,
the second one can choose to first execute some unobservable action
which will lead it to a state where it can only receive message {\sf
c}. Once again it cannot be guaranteed that the second service will
accept cancellation requests, and this depends on some decisions it
takes internally.
Weak (or observational) and branching equivalences are the strongest
of the weak equivalences~\cite{ChapterHPA-Intro}, branching
equivalence being the strongest of these two. They preserve
behavioural properties (do not add deadlocks for instance) on
observable actions, and are therefore acknowledged as the most
appropriate notions of process equivalence, in the context of Web
services. They are implemented in tools like CADP~\cite{CADP2006}
which can automatically check that two transition systems denote the
same observational (or branching) behaviour.
Another notion called \emph{strong bisimulation} exists. It is
nevertheless too restrictive in our context because it imposes a
strict matching of the $\tau$ actions. Also note the notion of
\emph{congruence}, an observational equivalence which should be
preferred when one wants services to be equivalent \emph{in any
context}, {\it i.e.}, in all possible systems using them.
\subsection{Verifying Properties}
The properties of interest in concurrent systems typically involve
reasoning on the possible scenarii that the system can go through. An
established formalism for expressing such properties is given by
\emph{temporal logics}\footnote{This name should not give the
impression that these logics introduce a quantitative notion of
time, they are indeed used to express constraints on the possible
executions of a system.} like
CTL$\star$~\cite{Manna-Pnueli:BOOK:1995}. These logics present constructs
allowing to state in a formal way that, for instance, all scenarii
will respect some property at every step, or that some particular
event will eventually happen, and so on.
An introduction to temporal logic goes beyond the aims of this paper, but it
suffices to say that a number of classical properties typically appear as
patterns in many applications. Reusing them diminishes the need to learn all
subtleties of a new formalism. The most noticeable properties are:
\smallskip
$\bullet$ \textbf{Safety properties}, which state that an undesirable situation
will never arise. For instance, the requirements can forbid that the system
reserves a room without having received the credit information from the
bank;
$\bullet$ \textbf{Liveness properties}, which state that some actions will
always be followed by some reactions; a typical example is to check that
every request for a room will be acknowledged.
The techniques used to check whether a system respects temporal logic
properties are referred to as \emph{model checking} methods
\cite{Clarke-Grumberg-Peled:BOOK:2000}. Several tools exist and can be
used to model-check abstract descriptions of services, {\it e.g.},
CADP, or SPIN.
\section{Compatibility and Adaptation}
\label{section:compatibility}
\subsection{Compatibility Notions}
\label{section:compnotions}
Compatibility aims at ensuring that services will be able to interact
properly, that is satisfy a specific criterion on observable actions
and terminate in final states. Typically, compatibility is needed at
design-time as a previous step (discovery) in a service composition
construction in order to avoid erroneous executions at run-time.
Substitutability is a similar issue and aims at replacing one service
by another without introducing flaws. Substitutability can be checked
using equivalence-checking techniques presented in the previous
section. Compatibility checking, if defined in a formal way, can be
automated using state space exploration tools such as CADP or SPIN, or
rewriting-based tools such as Maude.
In the rest of this subsection, we introduce three notions of
compatibility, namely deadlock-freeness,
unidirectional-complementarity and unspecified-receptions, that make
sense in the Web services area. These notions have often been studied
in the
literature~\cite{DZ-ACM83,YS-ACM97,CanalPT01,tes04,DOS-Foclasa09,OS-WCSI10}.
\noindent\textbf{Deadlock-freeness.} This notion says that two service
protocols are compatible if and only if, starting from their initial
states, they can evolve together until reaching final states.
Figure~\ref{fig:dead} presents a simple example to illustrate this
notion. {\sf S1} and {\sf S2} are not compatible because after
interacting on action {\sf a}, both services are stuck. On the other
hand, {\sf S1'} and {\sf S2} are deadlock-free compatible since they
can interact successively on {\sf a} and {\sf c}, and then both
terminate into a final state.
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth,clip]{deadlock.eps}
\caption{Deadlock-freeness compatibility}
\label{fig:dead}
\end{figure}
\medskip
\noindent \textbf{Unidirectional-complementarity.} Two services are
compatible with respect to this notion if and only if there is one
service which is able to receive (send, respectively) all messages
that its partner expects to send (receive, respectively) at all
reachable states. Hence, the ``bigger'' service may send and receive
more messages than the ``smaller'' one. Additionally, both services
must be free of deadlocks.
This notion is different to what is usually called simulation or
preorder relation~\cite{MilCC} because the two protocols under
analysis here aim at being composed, and accordingly present opposite
directions. However, both definitions share the inclusion concept: one
of the two protocols is supposed to accept all the actions that the
other can do.
Figure~\ref{fig:sim} first shows two services {\sf S1} and {\sf S2}
which respect this unidirectional-complementarity compatibility: all
actions possible in {\sf S1} can be captured by {\sf S2}. However,
{\sf S2} does not complement {\sf S1'} because {\sf S2} is not able to
synchronize on action {\sf c} with {\sf S1'}.
\medskip
\noindent \textbf{Unspecified-receptions.} This definition requires
that if one service can send a message at a reachable state, then the
other service must receive that emission. Furthermore, one service is
able to receive messages that cannot be sent by the other service,
{\it i.e.}, there might be additional unmatched receptions. It is
also possible that one protocol holds an emission that will not be
received by its partner as long as the state from which this emission
goes out is unreachable when protocols interact together.
Additionally, both services must be free of deadlocks.
In Figure~\ref{fig:unspecified}, {\sf S1} and {\sf S2} are not
compatible because {\sf S1} cannot receive all actions that {\sf S2}
can send ({\sf c!}). But {\sf S1'} and {\sf S2} are compatible because
all emissions on both sides have a matching reception on the other.
\medskip
The reader interested in the formal definitions for these
compatibility notions can refer to~\cite{tes04,DOS-Foclasa09}.
\subsection{Compatibility Degree}
\label{section:compdegree}
Most of the approaches existing for checking compatibility return a
``True'' or ``False'' result to detect whether services are compatible
or not. Unfortunately, a Boolean answer is not very helpful for many
reasons. First, in real world case studies, there will seldom be a
perfect match, and when service protocols are not compatible, it is
useful to differentiate between services that are slightly
incompatible and those that are totally incompatible. Furthermore, a
Boolean result does not give a detailed measure of which parts of
service protocols are compatible or not. To overcome the
aforementioned limits, a new solution aims at measuring the
compatibility degree (or similarity degree if the idea is to replace
and not to compose services) of service interfaces. This issue has
been addressed by a few recent works, see for
instance~\cite{SokolskyKL06,NejatiEtAl07,Lohmann08,Ait-Bachir-ICSOC08,wu2009computing,OSP10}.
Let us illustrate with a simple example (Fig.~\ref{fig:excd}) the kind
of results one can compute with these compatibility measuring
approaches. Here, we use the compatibility measuring algorithms
presented in~\cite{OSP10}. This approach takes as input two STSs and
computes a compatibility degree for each global state, {\it i.e.,}
each couple of states $(s_i, s_j)$ with $s_i \in S_1$ and $s_j \in
S_2$. All compatibility scores range between 0 and 1, where 1 means a
perfect compatibility. To measure the compatibility of two service
protocols, the protocol compatibility degrees are computed for all
possible global states using a set of static compatibility measures.
This work uses three static compatibility measures, namely state
natures, labels, and exchanged parameters. These measures are used
next to analyse the behavioural part (ordering of labels) of both
protocols. Intuitively, two states are compatible if their backward
and forward neighbouring states are compatible, where the backward and
forward neigbours of state $s'$ in transitions $(s,l,s')$ and
$(s',l',s'')$ are respectively the states $s$ and $s''$. Hence, in
order to measure the compatibility degree of two service protocols, an
iterative approach is considered which propagates the compatibility
degree from one state to all its neighbours. This process is called
compatibility flooding.
Table~\ref{table:matrix} shows the matrix computed for the example
depicted in Figure~\ref{fig:excd} according to the
unidirectional-complementarity notion. Let us comment the
compatibility of states ${\sf c0}$ and ${\sf s0}$. The measure is
quite high because both states are initial and the emission {\sf
search!} at ${\sf c0}$ perfectly matches the reception {\sf search?}
at ${\sf s0}$. However, the compatibility degree is less than 1 due to
the backward propagation of the deadlock from the global state $({\sf
s1},{\sf c3})$ to $({\sf s1},{\sf c1})$, and then from $({\sf
s1},{\sf c1})$ to $({\sf s0},{\sf c0})$.
\begin{figure}
\centering
\includegraphics[width=0.6\linewidth,clip]{simulation2}
\caption{An online store}
\label{fig:excd}
\end{figure}
\begin{table}
\center
\begin{tabular}{c|cccccc|}
& {\sf s0} & {\sf s1} & {\sf s2} & {\sf s3} & {\sf s4}\\
\hline
{\sf c0}& \textbf{0.78}& 0.01& 0.01& 0.01& 0.01\\
{\sf c1}& 0.01 &0.68 &0.01 &0.35 &0.01\\
{\sf c2}& 0.01 &0.01& 0.90 &0.01 &0.67\\
{\sf c3}&0.01 &0.45& 0.76& 0.35 &0.76
\end{tabular}
\caption{The compatibility matrix computed for the example in Figure~\ref{fig:excd}}
\label{table:matrix}
\end{table}
\subsection{Service Adaptation}
\label{section:adaptation}
While searching a service satisfying some specific requirements, one
can find a candidate which exhibits the expected functionality but
whose interface does not exactly fit in the rest of the system. {\it
Software Adaptation}~\cite{BeckerDagstuhl2005} is a very promising
solution to compose in a non-intrusive way black-box components or
(Web) services although they present interface mismatches. Adaptation
techniques aim at automatically generating new components called {\it
adaptors}, and usually rely on an {\it adaptation contract} which is
an abstract description of how mismatches can be worked out. All the
messages pass through the adaptor which acts as an orchestrator, and
makes the involved services work correctly together by compensating
mismatches. The generation of this adaptor is a complicated task,
especially when interfaces take into account a behavioural description
of the service execution flow. Recently, several approaches have been
proposed to generate service adaptors, see for
example~\cite{BrogiICSOC06,BenatallahWWW07,MateescuPS08,Canal-Poizat-Salaun-08,AalstMSW09}.
Figure~\ref{fig:interfaces} gives an example: the first interface
corresponds to an SQL service which can receive ({\sf req?}) and
answer ({\sf result!}) requests, stops ({\sf halt!}), or halts
temporarily for maintenance purposes ({\sf maintenance?} and {\sf
activation?}). The client can submit requests ({\sf request!}), and
receive responses ({\sf request?}).
\begin{figure}[h]
\centerline{\includegraphics[width=0.75\linewidth]{interfaces.eps}}
\caption{An SQL service}
\label{fig:interfaces}
\end{figure}
Several notations exist for writing adaptation contracts. In this
paper, we use {\it vectors}~\cite{MateescuPS08} which specify
interactions between several services. They express correspondences
between messages, like bindings between ports, or connectors in
architectural descriptions. Each label appearing in one vector is
executed by one service and the overall result corresponds to an
interaction between all the involved services. Furthermore,
variables are used as placeholders in message parameters. The same
variable name appearing in different labels (possibly in different
vectors) enables one to relate sent and received arguments of
messages.
As far as our example is concerned, the following vectors constitute a
contract from which the adaptor protocol given in
Figure~\ref{fig:adaptor} is automatically generated by using
techniques and tools presented in~\cite{MateescuPS08}.
This approach respectively generates (i) LOTOS code\footnote{LOTOS is
a value passing process algebra proposed in the late 80s,
see~\cite{BolognesiB87} for more details.} for service interfaces
and the contract, and (ii) the corresponding state space by applying
on-the-fly simplification (deadlock suppression) and reduction
techniques ($\tau$ transition removal).
\smallskip
{\sf V1} = $\langle \mathrm{{\sf s}}\!:\! \mathrm{{\sf req?X}}; \mathrm{{\sf c}}\!:\! \mathrm{{\sf request!X}} \rangle \;\;\;\;\;\;\;\;\;\;$
{\sf V2} = $\langle \mathrm{{\sf s}}\!:\! \mathrm{{\sf result!Y,Z}}; \mathrm{{\sf c}}\!:\! \mathrm{{\sf request?Z}} \rangle \;\;\;\;\;\;\;\;\;\;$
{\sf V3} = $\langle \mathrm{{\sf s}}\!:\! \mathrm{{\sf halt?}} \rangle$
\begin{figure}[h]
\centerline{\includegraphics[width=0.75\linewidth]{adaptor.eps}}
\caption{The adaptor protocol for the SQL example}
\label{fig:adaptor}
\end{figure}
From adaptor protocols, either a central adaptor can be implemented,
or several service wrappers can be generated to distribute the
adaptation. In the former case, the implementation of executable
adaptors from adaptor protocols can be achieved for instance using
techniques presented in~\cite{MateescuPS08} and~\cite{CuboSCPP08} for
BPEL and Windows Workflow Foundation, respectively. In the latter
case, each wrapper constrains its service functionality to make it
respect the adaptation contract~\cite{salaun-SEFM-2008}.
\section{Realizability and Conformance}
\label{section:realizability}
Interactions among a set of services involved in a new system can be
described from a global point of view using {\it choreography}
specification languages. Several formalisms have already been proposed
to specify choreographies: WS-CDL, collaboration diagrams, process
calculi, BPMN, SRML, etc. Given a choreography specification, it would
be desirable if the local implementations, namely {\it peers}, can be
automatically generated via projection. However, generation of peers
that precisely implement the choreography specification is not always
possible: This problem is known as {\it realizability}. A related
problem is known as {\it conformance} where the question is to check
whether a choreography and a set of service implementations (not
obtained by projection from the choreography) produce the same
executions.
A couple of unrealizable collaboration diagrams~\cite{Bultan-Fu-08}
are presented in Figure~\ref{figure:unrealizable}. The first one (left
hand side) is unrealizable because it is impossible for the peer {\tt
C} to know when the peer {\tt A} sends its {\tt request} message
since there is no interaction between {\tt A} and {\tt C}. Hence, the
peers cannot respect the execution order of messages as specified in
the collaboration diagram. The second one is slightly more subtle
because this diagram is realizable for synchronous communication, and
unrealizable for asynchronous communication. Indeed, in case of
synchronous communication, the peer {\tt C} can synchronize
(rendez-vous) with the peer {\tt A} only after the {\tt request}
message is sent, so the message order is respected. This is not the
case for asynchronous communication since {\tt A} cannot block {\tt C}
from sending the {\tt update} message. Hence, {\tt C} has to send the
{\tt update} message to {\tt A} without knowing if {\tt A} has sent
the {\tt request} message or not. Therefore, the correct order
between the two messages cannot be satisfied. We also show in
Figure~\ref{figure:unrealizable} the LTS generated for peer {\tt A} by
projection.
\begin{figure}[ht]
\centerline{\includegraphics[width=0.9\linewidth]{unrealizable.eps}}
\caption{Examples of unrealizable collaboration diagrams}
\label{figure:unrealizable}
\end{figure}
Several works aimed at studying and defining the realizability (and
conformance) problem for choreography, here are a few
references~\cite{KP-FORTE06,Busi-Coordination2006,LiEtAlTASE07,Fu05,Bultan-Fu-08}.
In~\cite{Busi-Coordination2006,LiEtAlTASE07}, the authors define
models for choreography and orchestration, and formalize a conformance
relation between both models. Other
works~\cite{Carbone-ESOP07,ZongyanWWW07} propose well-formedness rules
to enforce the specification to be realizable. A few
works~\cite{ZongyanWWW07,SB-IFM09} also propose to add messages in
order to implement unrealizable choreographies. Fu {\it et
al.}~\cite{FuBS04} proposed three conditions (lossless join,
synchronous compatible, autonomous) that guarantee a realizable
conversation protocol under asynchronous communication. These
conditions have been implemented in the WSAT tool~\cite{FuBS-CAV04}
which takes a conversation protocol as input, and says if it satisfies
the three realizability conditions. \cite{SuBFZ07} discusses some
interesting open issues in this area.
\section{Concluding Remarks}
\label{section:conclusion}
In this paper, we have surveyed some issues in Web services which
require analysis and verification techniques. Using these techniques
seems natural when one wants to ensure that a composition of services
will work correctly or satisfy some high-level requirements. But they
have also other applications in SOC, {\it e.g.}, to check the
compatibility of a service with a possible client (discovery), or to
generate some service adaptors if some interface mismatches prevent
their direct composition. Last, we have showed that when specifying a
system using choreography languages, some analysis are useful to check
that the corresponding distributed implementation will behave as
described in the global specification.
We would like to conclude with a few challenges which are still some
open issues, as far as analysis techniques are concerned, in the Web
services domain. All these challenges assume an asynchronous
communication model (that is based on message queues). A few works
already exist, in~\cite{Fu05} for example the authors define a
synchronizability condition which makes systems under asynchronous
communication verifyable with tools working with synchronous
comunication. Some sufficient conditions have also been proposed to
guarantee the realizability of conversation protocols~\cite{FuBS04}.
Nevertheless, in both works, if these conditions are not satisfied,
nothing can be concluded on the system being analysed.
Some open challenges assuming an asynchronous communication model are
the following: (i)~providing automated techniques to check the
compatibility of two or more services, (ii)~checking the adaptability
of a set of services being given an adaptation contract, and if the
system is adaptable, generating the corresponding adaptor,
(iii)~finding a decidable algorithm for checking the realizability of
a choreography specification language with loops (such as conversation
protocols~\cite{FuBS04}).
\medskip
{\bf Acknowledgements.} The author would like to thank Meriem Ouederni
for her comments on a former version of this paper.
\bibliographystyle{eptcs}
|
{
"timestamp": "2010-09-21T02:02:19",
"yymm": "1009",
"arxiv_id": "1009.3716",
"language": "en",
"url": "https://arxiv.org/abs/1009.3716"
}
|
\section{Introduction}
One of the most basic and widely studied entanglement measures for
bipartite quantum states is the \emph{entanglement of formation}
(EoF)~\cite{eof}, a quantity so named because it was intended to
quantify the resources needed to create (or form) a given bipartite
entangled state. The EoF of any bipartite pure state is quantified by
the entropy of entanglement, which is equal to the von Neumann entropy
of the reduced state of a subsystem. The EoF of a bipartite mixed
state $\rho_{AB}$, is then defined via the convex roof extension, that
is, as the minimum average entanglement of an ensemble of pure states
that represents $\rho_{AB}$:
\begin{equation}
\label{eq:10}
E_F(\rho_{AB}):=\min_{\eE}\sum_ip_iS(\rho^i_A),
\end{equation}
where $\eE=\{p_i,|\psi^i_{AB}\>\}$ is an ensemble of pure biparite
states such that $\sum_ip_i|\psi^i\>\<\psi^i|=\rho_{AB}$, and
$S(\rho^i_A)$ is the von Neumann entropy of the reduced state
$\rho^i_A=\Tr_B|\psi^i\>\<\psi^i|_{AB}$. The popularity of the EoF is
partly due to its formal elegance and the many nice properties it
enjoys \cite{haya,matthias}, and perhaps also due to its connections
with the additivity problem in quantum information theory
\cite{eof-add}.
From the operational point of view, the
EoF is associated with the entanglement
manipulation protocol by which two
distant parties, say Alice and Bob, prepare a given bipartite
quantum state, starting from an initial entangled state which they
share, by using only local operations and classical
communication (LOCC). It turns out that the optimal
(i.e., minimum) rate, at which entanglement has to be consumed in order
for Alice and Bob to create multiple copies of the state with asymptotically
vanishing error, is given by the regularized EoF of the state~\cite{cost}.
Soon after the introduction of the EoF, another quantity, namely
the \emph{entanglement of assistance} (EoA)~\cite{eoa}, was introduced
as its ``dual''. It is defined analogously to EoF but with the
minimisation over ensembles replaced by a maximisation, i.e.,
\begin{equation}
\label{eq:19}
E_A(\rho_{AB}):=\max_\eE \sum_ip_iS(\rho^i_A).
\end{equation}
Unlike the EoF, the EoA is not an entanglement monotone and hence it
can in general increase under local operations and classical
communication~\cite{gilad-spekkens}. However, like the EoF, the EoA
too can be associated with an entanglement manipulation protocol,
namely the one by which Alice and Bob distill entanglement from an
initial mixed bipartite state which they share, when a third party
(say Charlie), who holds the purification of the state, assists them.
Charlie is allowed to do local operations on his share of the
tripartite pure state, and his assistance is in the form of one-way
classical communication to Alice and Bob. This is the sort of
scenario which occurs, for example, in the case of
\emph{environment-assisted quantum error
correction}~\cite{greg,hay-king,assistance,only-andreas,pavia,fb},
in which errors, incurred from sending quantum information through a
noisy environment, are corrected by using classical information
obtained from a measurement on the environment. In this case the
tripartite structure Alice-Bob-Charlie is mirrored by the structure
sender-receiver-environment, and the assistance from Charlie is
replaced by the ability to perform measurements on the environment and
to exploit the resulting information for error correction.
Another area in which the EoA arises naturally, is in the
study of \emph{localizable entanglement} in spin
systems~\cite{local0,local1,local2,local3}. The scenario here is as
follows: a pure state of a system of $n\gg 1$ interacting spins is
given, and the goal is to localize (or ``focus'') as much entanglement
as possible between two arbitrarily chosen spins, by performing a suitable
measurement on the remaining $n-2$ spins. In this case, the
assisting party is actually divided into many subsystems (which are the
$n-2$ spins) and so it is natural to ask what happens when the assisting
measurements are restricted to be \emph{local} in each subsystem. The
amount of entanglement that can be focussed in this case is referred to as
the localizable entanglement, and it is always at most as much as the
EoA. In fact, it is equal to the EoA when the assisting party is
allowed to act \emph{globally} on all the constituent subsystems.
In the literature, one encounters cases in which the EoA is used to
characterize operational tasks of assisted distillation studied in the
generic scenario, where no assumptions are made on the state to be
distilled. This is often referred to as the ``one-shot'' scenario.
However, the definition of the EoA given in eq.~(\ref{eq:19}) has been
shown to have an operational relevance only in the asymptotic regime,
i.e., when asymptotically many copies of the same state are available
for assisted distillation~\cite{assistance}. This points to an
apparent mismatch between the operational task and the quantity used
to characterize it. In order to remedy this problem, one should start
from the operational task itself, and from it, \emph{evaluate} an
expression quantifying the amount of entanglement that can be
distilled under assistance. This leads to a \emph{one-shot} EoA,
which, by its very construction, has a direct operational
interpretation.
In Section~2 we introduce the necessary notation and definitions. In
Section~3 we evaluate the one-shot distillable entanglement of a pure
bipartite state. The one-shot entanglement of assistance is introduced
in Section~4 and evaluated in Section~5. Section~6 deals with the
asymptotic scenario, where some previous results are
recovered. Finally, Section~7 concludes the paper with a summary and
an open question.
\section{Notation and definitions}
\label{prelim}
\subsection{Mathematical preliminaries}
Let ${\cal B}(\sH)$ denote the algebra of linear operators acting on a
finite--dimensional Hilbert space $\sH$ and let $\states(\sH) \subset
{\cal B}(\sH)$ denote the subset of positive operators of unit trace
(states). Further, let $\openone\in {\cal B}(\sH)$ denote the
identity operator. Throughout this paper we restrict our
considerations to finite-dimensional Hilbert spaces, and we take the
logarithm to base $2$. For any given pure state $|\phi\>$, we denote
the projector $|\phi\>\<\phi|$ simply as $\phi$. Moreover, for any
state $\rho$, we define $\Pi_\rho$ to be the projector onto the
support of $\rho$.
For a state $\rho\in\states(\sH)$, the von Neumann entropy is defined
as $S(\rho):=-\tr\rho\log\rho$. Further, for a state $\rho$ and a
positive operator $\sigma$ such that ${\rm{supp }} \rho \subseteq
{\rm{supp }}\sigma$, the quantum relative entropy is defined as
$S(\rho||\sigma) = \tr \rho \log \rho - \rho \log \sigma,$ whereas the
relative R\'enyi entropy of order $\alpha \in (0,1)$ is defined as
\be\label{relren} S_\alpha (\rho || \sigma) := \frac{1}{\alpha - 1}
\log\bigl[ \tr(\rho^\alpha \sigma^{1-\alpha})\bigr]. \ee
For given orthonormal bases $\{|i_A\rangle\}_{i=1}^d$ and
$\{|i_B\rangle\}_{i=1}^d$ in isomorphic Hilbert spaces
$\sH_A\simeq\sH_B$ of dimension $d$, we define a
maximally entangled state (MES) of rank $M \le d$ to be
\begin{equation}\label{MES-M}
|\Psi^M_{AB}\>= \frac{1}{\sqrt{M}} \sum_{i=1}^M |i_A\rangle\otimes |i_B\rangle.
\end{equation}
In order to measure how close two states are, we will use the
fidelity, defined as
\begin{equation}\label{fidelity-aaa}
F(\rho, \sigma):= \tr \sqrt{\sqrt{\rho} \sigma \sqrt{\rho}}
=\N{\sqrt{\rho}\sqrt{\sigma}}_1,
\end{equation}
and the trace distance
\begin{equation}
\N{\rho-\sigma}_1:=\Tr|\rho-\sigma|.
\end{equation}
The trace distance between two states $\rho$ and $\sigma$ is related
to the fidelity $F(\rho, \sigma)$ as follows (see e.~g.~\cite{nielsen}):
\begin{equation}
1-F(\rho,\sigma) \leq \frac{1}{2} \N{\rho -
\sigma}_1 \leq \sqrt{1-F^2(\rho, \sigma)},
\label{fidelity}
\end{equation}
where we use the notation $F^2(\rho, \sigma) = \bigl(F(\rho,\sigma)
\bigr)^2$
The following lemmas will prove useful.
\begin{lemma}[Gentle measurement lemma~\cite{winter99,ogawanagaoka02}]
\label{gmlemma}
For a state $\rho\in\states(\sH)$ and operator $0\le
\Lambda\le\openone$, if $\Tr(\rho\ \Lambda) \ge 1 - \delta$, then
$$\N{\rho - {\sqrt{\Lambda}}\rho{\sqrt{\Lambda}}}_1 \le {2\sqrt{\delta}}.$$
The same holds if $\rho$ is a subnormalized density operator.
\end{lemma}
\begin{lemma}\label{fid3}
For any pure state $|\phi\rangle$ and any given $\eps\ge 0$,
if $0\le P\le \openone$ is an operator such that $\tr (P\phi) \ge 1-\eps$,
then
\begin{equation}
F( \sqrt{P}|\phi\rangle, |\phi\rangle) \ge 1 - \sqrt{\eps}.
\end{equation}
\end{lemma}
\begin{proof}
Since, $\tr (P\phi) \ge 1-\eps$, by Lemma \ref{gmlemma} we have that
$$\|\sqrt{P} \phi \sqrt{P} - \phi \|_1 \le 2 \sqrt{\eps}.$$
The lower bound on the trace distance in \reff{fidelity} then yields
\begin{equation}
F( \sqrt{P}|\phi\rangle, |\phi\rangle) \equiv F(\sqrt{P} \phi \sqrt{P},\phi)\ge 1 - \sqrt{\eps}.
\end{equation}
\end{proof}
\begin{lemma}\label{lemma:accessory}
For any normalized state $\rho$ and any $0\le P\le\openone$, if
$\Tr[P\rho]\ge 1-\eps$, then
\begin{equation}
F(\omega,\rho)\ge 1-2\sqrt{\eps},
\end{equation}
where $\omega:=\frac{\sqrt{P}\rho\sqrt{P}}{\Tr[P\rho]}$.
\end{lemma}
\begin{proof}
The condition $\Tr[P\rho]\ge 1-\eps$ implies that
$\N{\sqrt{P}\rho\sqrt{P}-\rho}_1\le
2\sqrt{\eps}$,~\cite{winter99,ogawanagaoka02}. Let us define
$\tilde\omega:=\sqrt{P}\rho\sqrt{P}$. Due to Lemma~11
in~\cite{distil}, we have that
\begin{equation}
\begin{split}
F(\tilde\omega,\rho):&=\N{\sqrt{\tilde\omega}\sqrt{\rho}}_1\\
&\ge\frac{\Tr[P\rho]+1}2 - \frac 12\N{\tilde\omega-\rho}_1\\
& \ge 1-\frac{\eps}2-\sqrt{\eps}\\
&\ge 1-2\sqrt{\eps}.
\end{split}
\end{equation}
Let $\omega$ be a normalized state defined as
$\omega:=\frac{\tilde\omega}{\Tr(\tilde\omega)}$. Since $F(\omega,\rho)\ge
F(\tilde\omega,\rho)$, we obtain the statement of the lemma.
\end{proof}
In this paper we consider entanglement distillation under LOCC
transformations. In this context, a result by Lo and Popescu
\cite{lopopescu} on entanglement manipulation of bipartite {\em{pure
states}} plays a crucial role. They proved that any LOCC
transformation ($AB \mapsto A'B'$) on a bipartite pure state
$|\phi_{AB}\rangle$, shared between two distant parties Alice and Bob,
is equivalent to a LOCC transformation with only one-way classical
communication, which can be represented as follows: \be\label{form}
\Lambda(\phi_{AB}) = \sum_k (U_k \otimes E_k)\phi_{AB} (U_k \otimes
E_k)^\dagger,\ee where the operators $U_k$ are unitary and the
operators $E_k$ satisfy the relation $\sum_k E_k^\dagger E_k =
\openone_{B}$. Henceforth, we say that an LOCC transformation is of
the {\em{Lo-Popescu form}} if it can be expressed as in
(\ref{form}). Consequently, for a map $\Lambda$ of the Lo-Popescu
form, we have \bea
\Lambda (\openone_A \otimes \sigma_B) &=& \sum_k U_kU_k^\dagger \otimes E_k\sigma_B E_k^\dagger,\nonumber\\
&=& \openone_{A'} \otimes \tau_{B'},
\label{lopop}
\eea
where $\tau_{B'}:= \sum_k E_k\sigma_B E_k^\dagger$.
.
\subsection{Entropies and coherent
information}\label{entropies}
Optimal rates of the entanglement distillation protocols considered
in this paper are expressible in terms
of the following entropic quantities:
\smallskip
For any $\rho,\sigma\ge 0$,
any $0\le P\le\openone$, and any $\alpha\in(0,\infty)\backslash\{1\}$,
we define the following entropic function (introduced in \cite{qcap})
\begin{equation}\label{quasi-ent}
S_\alpha^P(\rho\|\sigma):=\frac{1}{\alpha-1}\log\Tr[\sqrt{P}\rho^\alpha\sqrt{P}\sigma^{1-\alpha}].
\end{equation}
Notice that, for $P=\openone$, the function defined above reduces to
relative R\'enyi entropy of order $\alpha$ given by \reff{relren}.
In this paper, we are in particular interested in the quantity,
\begin{equation}\label{eq:asda}
S_0^P(\rho\|\sigma):=\lim_{\alpha\searrow 0}S_\alpha^P(\rho\|\sigma)=
-\log\Tr[\sqrt{P}\Pi_\rho\sqrt{P}\ \sigma],
\end{equation}
where $\Pi_\rho$ denotes the projector onto the support of $\rho$.
Note that
\be
S_0^{\openone}(\rho\|\sigma) = S_0(\rho\|\sigma) := - \log (\tr \Pi_\rho \sigma),
\ee
which is the relative R\'enyi entropy of order zero. This quantity acts
as a parent quantity for the {\em{zero-coherent information}}, defined as
follows:
\begin{equation}\label{zero-coh}
I^{A\to B}_{0}(\rho_{AB}):=\min_{\sigma_B\in\states(\sH_B)}S_0(\rho_{AB}\|\openone_A\otimes\sigma_B),
\end{equation}
the nomenclature arising from its analogy with the ordinary coherent information $I^{A\to
B}(\rho_{AB})$,
which is expressible in a similar manner, when the zero-relative R\'enyi entropy is replaced by the ordinary relative entropy:
\bea
\label{coh}
I^{A\to
B}(\rho_{AB})&:=& S(\rho_B)-S(\rho_{AB})\label{coh11}\\
&\equiv &
\min_{\sigma_B\in\states(\sH_B)}S(\rho_{AB}\|\openone_A\otimes\sigma_B).
\eea
If $\Psi^\rho_{ABE}$ is a purification of the state $\rho_{AB}$, then
\be\label{neg}
I^{A\to B}(\rho_{AB}) = - I^{A\to E}(\rho_{AE}),
\ee
where $\rho_{AE}= \tr_B \Psi^\rho_{ABE}$.
Note in particular that for a MES of rank $M$, as defined by
\reff{MES-M},
\begin{equation}
I^{A\to B}_{0}(\Psi^M_{AB}) = I^{A\to B}(\Psi^M_{AB})= \log M.
\end{equation}
Another entropic quantity of relevance in this paper is the {\em{min-entropy of a state}}, which is defined for any state $\rho$ as
follows:
\be
S_{\min}(\rho) = - \log \bigl[\lambda_{\max}(\rho)\bigr],
\ee
where $\lambda_{\max}(\rho)$ denotes the maximum eigenvalue of the state
$\rho$.
For one-shot entanglement distillation protocols it is natural to
allow for a finite accuracy, i.e., a non-zero error (say $\eps \ge
0$), in the extraction of singlets from a given state. In this case
the optimal rates of the protocols are given by ``smoothed versions''
of the entropic quantities introduced above. In order to define them
we consider the following sets of positive operators for any
normalized state $\rho$, and any $\eps >0$:
\begin{equation}
\B(\rho;\eps):=\left\{\sigma:\sigma\ge 0,\ \Tr[\sigma]=1,\ F^2(\rho,\sigma)\ge1-\eps^2\right\},\label{ball}
\end{equation}
\begin{equation}
\P(\rho;\eps):=\left\{P:0\le P\le\openone,\ \Tr[P\rho]\ge1-\eps\right\}.\label{P-ball}
\end{equation}
Henceforth we shall
refer to $\B(\rho;\eps)$ as the $\eps-$ball around the state $\rho$.
Further, by restricting the states $\sigma$ in \reff{ball} to be pure
states, we obtain the subset
\begin{equation}\label{b*}
\B_*(\rho;\eps):=\left\{|\vphi\rangle:\,\vphi\in\B(\rho;\eps)\right\}.
\end{equation}
It was proved in \cite{marco} that for a bipartite pure state
$|\phi_{AB}\rangle$, for any $\eps \ge 0$,
\begin{equation}\label{equi}
\left\{\Tr_A[\vphi_{AB}]:\vphi_{AB}\in \B_*(\phi_{AB};\eps)\right\}= \B(\rho^\phi_B; \eps),
\end{equation}
where $\rho^{B}_\phi:= \Tr_A \phi_{AB}$.
The relevant smoothed entropic quantities are then defined as follows:
\begin{definition}
For any given $\eps \ge 0$ the {\em{smoothed}} min-entropy of a state $\rho$
is defined as
\be
S_{\min}^\eps(\rho) := \max_{\bar{\rho} \in \B(\rho;\eps)} S_{\min} ({\bar{\rho}}).
\label{smooth_min}
\ee
\end{definition}
We consider two different smoothed versions of the zero-coherent information, defined as
follows:
\begin{definition} The state-smoothed zero-coherent information is given by
\begin{equation}\label{eq:i}
I^{A\to B}_{0,\eps}(\rho_{AB}):=\max_{\bar\rho_{AB}\in \B(\rho_{AB};\eps)}\min_{\sigma_B\in\states(\sH_B)}S_0(\bar\rho_{AB}\|\openone_A\otimes\sigma_B),
\end{equation}
and the operator-smoothed zero-coherent information is given by
\begin{equation}\label{eq:itilda}
\I_{0,\eps}^{A\to B}(\rho_{AB}):=\max_{P\in \P(\rho_{AB};\eps)}\min_{\sigma_B\in\states(\sH_B)}S_0^P(\rho_{AB}\|\openone_A\otimes\sigma_B).
\end{equation}
\end{definition}
The following technical lemmas involving the operator-smoothed coherent
information are used in proving some of our main results.
\begin{lemma}\label{lemma1}
If for a bipartite state $\rho_{AB}$ and a pure state $|\psi_{AB}\rangle$,
for any given $\eps \ge 0$,
\be
F^2(\rho_{AB}, \psi_{AB})\equiv \tr[\rho_{AB} \psi_{AB}] \ge 1-\eps,
\label{fid2}
\ee
then
\be
\I_{0,\eps}^{A\to B}(\rho_{AB}) \ge I_{0}^{A\to B}(\psi_{AB}).
\ee
\end{lemma}
\begin{proof}
From \reff{fid2} it follows that $\psi_{AB} \in \P(\rho_{AB};\eps)$.
Using this fact, \reff{eq:itilda} and \reff{quasi-ent}, we obtain
\bea
\I_{0,\eps}^{A\to B}(\rho_{AB}) &\ge & \min_{\sigma_B\in\states(\sH_B)}
\Bigl[- \log \tr\bigl({\sqrt{\psi_{AB}}}\Pi_{\rho_{AB}}{\sqrt{\psi_{AB}}}(\openone_A\otimes \sigma_B) \bigr)\Bigr]\nonumber\\
&\ge & \min_{\sigma_B\in\states(\sH_B)}\Bigl[- \log \tr\bigl({\psi_{AB}}(\openone_A\otimes \sigma_B) \bigr)\Bigr]\nonumber\\
&=& I_{0}^{A\to B}(\psi_{AB}).
\eea
where the second inequality follows from the fact that ${\sqrt{\psi_{AB}}}\Pi_{\rho_{AB}}{\sqrt{\psi_{AB}}}\le{\psi_{AB}}$.
\end{proof}
\begin{lemma}\label{lemma2}
For any bipartite pure state $|\phi_{AB}\>$, any LOCC map
$\Lambda:AB\mapsto A'B'$, and any $\eps\ge0$,
\begin{equation}\label{eq:lemma2-stat1}
\I_{0,2\sqrt{\eps}}^{A\to B}(\phi_{AB}) \ge \I_{0,\eps}^{A'\to
B'}(\Lambda(\phi_{AB})).
\end{equation}
\end{lemma}
\begin{proof}
Since the LOCC map $\Lambda$ acts on a pure state, without loss of generality
we can assume it to be of the Lo-Popescu form (\ref{form}).
Defining $\omega_{A'B'} := \Lambda(\phi_{AB})$, we have, starting from \reff{eq:itilda},
\begin{equation}
\begin{split}
\label{eq1.1}
\I_{0,\eps}^{A'\to B'}(\Lambda(\phi_{AB})) &= \max_{P\in
\P(\omega_{A'B'};\eps)}\min_{\sigma_{B'}\in\states(\sH_{B'})}
\left\{- \log \Tr \left[ \sqrt{P} \Pi_{\omega_{A'B'}} \sqrt{P}\ (\openone_{A'}\otimes\sigma_{B'})\right]\right\}\\
&= \min_{\sigma_{B'}\in\states(\sH_{B'})} \left\{- \log \Tr \left[
\sqrt{P_0} \Pi_{\omega_{A'B'}} \sqrt{P_0}\
(\openone_{A'}\otimes\sigma_{B'})\right]\right\}\\
&\le - \log \Tr \left[ \sqrt{P_0} \Pi_{\omega_{A'B'}} \sqrt{P_0}\ (\openone_{A'}\otimes{\tilde{\sigma}}_{B'})\right]\\
&= - \log \Tr \left[ \sqrt{P_0} \Pi_{\omega_{A'B'}} \sqrt{P_0}\ \Lambda(\openone_A\otimes\sigma_{B})\right]\\
&= - \log \Tr \left[ \Lambda^*\left(\sqrt{P_0} \Pi_{\omega_{A'B'}}
\sqrt{P_0}\right)\ (\openone_A\otimes\sigma_{B})\right],
\end{split}
\end{equation}
for any state $\sigma_B \in \states(\sH_B)$. In the above, $P_0$ is the
operator in $\P(\omega_{A'B'};\eps)$ for which the maximum in the first line is
achieved; ${\tilde{\sigma}}_{B'}$ is a state in $\states(\sH_{B'})$ such that
${\tilde{\sigma}}_{B'}=\Lambda(\openone_A\otimes\sigma_{B})$,
and $\Lambda^*$ denotes the dual map of $\Lambda$, defined,
for any operator $X$ and state $\rho$, as $ \tr[X \Lambda(\rho)] =
\tr[\Lambda^* (X) \rho]$.
Let us now define $\tQ_{AB} := \Lambda^*(\sqrt{P_0}
\Pi_{\omega_{A'B'}} \sqrt{P_0})$. Then, continuing from
equation~\reff{eq1.1}, we obtain
\begin{equation}\label{eq:asdfgh}
\begin{split}
\I_{0,\eps}^{A'\to B'}(\Lambda(\phi_{AB})) &\le - \log \Tr \left[ \tQ_{AB}\ (\openone_A\otimes\sigma_{B})\right]\\
&\le - \log \tr \left[ \sqrt{\tQ_{AB}}\ \phi_{AB}\ \sqrt{\tQ_{AB}}\ (\openone_A \otimes \sigma_B)\right],
\end{split}
\end{equation}
for any state $\sigma_{B}$ and any pure state $\phi_{AB}$, since
$\tQ_{AB}\ge\sqrt{\tQ_{AB}}\phi_{AB}\sqrt{\tQ_{AB}}$. Let us now
choose $\sigma_{B}$ to be the state $\tsigma_B$ achieving the minimum in
the second line of~\reff{eq:asdfgh}, i.~e.
\begin{equation}
\begin{split}
\min_{\sigma_B} &\left\{- \log \tr \left[
\sqrt{\tQ_{AB}}\ \phi_{AB}\ \sqrt{\tQ_{AB}}\ (\openone_A \otimes
\sigma_B)\right]\right\}\\
=&- \log \tr \left[ \sqrt{\tQ_{AB}}\ \phi_{AB}\ \sqrt{\tQ_{AB}}\
(\openone_A \otimes \tsigma_B)\right],
\end{split}
\end{equation}
so that
\begin{equation}\label{31}
\I_{0,\eps}^{A'\to B'}(\Lambda(\phi_{AB}))\le \min_{\sigma_B} \left\{- \log \tr \left[
\sqrt{\tQ_{AB}}\ \phi_{AB}\ \sqrt{\tQ_{AB}}\ (\openone_A \otimes
\sigma_B)\right]\right\}
\end{equation}
We next prove that
$\tQ_{AB}\in\P(\phi_{AB};2\sqrt{\eps})$. In fact, since $P_0 \in
\P(\omega_{A'B'};\eps)$, by the Gentle Measurement Lemma,
\begin{equation}
\N{\Lambda(\phi_{AB}) - \sqrt{P_0} \Lambda(\phi_{AB})\sqrt{P_0}}_1
\le 2 \sqrt{\eps}.
\label{gm1}
\end{equation}
We therefore have, by definition of $\tQ_{AB}$,
\bea\label{33}
\tr \left[ \tQ_{AB} \phi_{AB}\right] &=&
\tr \left[ \sqrt{P_0} \Pi_{\Lambda(\phi_{AB})} \sqrt{P_0}\ \Lambda(\phi_{AB})\right]\nonumber\\
&=& \tr \left[ \Pi_{\Lambda(\phi_{AB})} \sqrt{P_0}
\Lambda(\phi_{AB}) \sqrt{P_0}\right]\nonumber\\
&=&\tr \left[ \Pi_{\Lambda(\phi_{AB})}\Lambda(\phi_{AB})\right] \nonumber\\
& & \quad +
\tr \left[ \Pi_{\Lambda(\phi_{AB})}\bigl(\sqrt{P_0}\
\Lambda(\phi_{AB}) \sqrt{P_0} - \Lambda(\phi_{AB})\bigr)\right]\nonumber\\
& \ge & 1 - \|\sqrt{P_0}\Lambda(\phi_{AB}) \sqrt{P_0} - \Lambda(\phi_{AB})\|_1 \nonumber\\
&\ge & 1- 2\sqrt{\eps},
\eea
where the second line follows from the cyclicity of the trace, and the last inequality follows from (\ref{gm1}).
This implies that $\tQ_{AB} \in
\P(\phi_{AB};2\sqrt{\eps})$. Hence, we have from \reff{31}
\begin{equation}
\begin{split}
\I_{0,\eps}^{A'\to B'}(\Lambda(\phi_{AB}))&\le \min_{\sigma_B} \left\{- \log \tr \left[
\sqrt{\tQ_{AB}}\ \phi_{AB}\ \sqrt{\tQ_{AB}}\ (\openone_A \otimes
\sigma_B)\right]\right\} \\
& \le \max_{P \in \P(\phi_{AB};2\sqrt{\eps})}\min_{\sigma_B} - \log
\tr \left[ \sqrt{P}\ \phi_{AB}\ \sqrt{P}\ (\openone_A \otimes \sigma_B) \right]\\
&\equiv\I_{0,2\sqrt{\eps}}^{A\to B}(\phi_{AB}),
\end{split}
\end{equation}
which completes the proof.
\end{proof}
\begin{lemma}
\label{lemma:opsm-stsm}
For any bipartite pure state $|\phi_{AB}\>$ and any $\eps\ge0$,
\begin{equation}\label{eq:lemma2-stat11}
I_{0,\eps}^{A\to B}(\phi_{AB})\ge S_{\min}^{\eps}(\rho^\phi_A),
\end{equation}
where $\rho^\phi_A:=\Tr_B\phi_{AB}$. Further,
\begin{equation}\label{eq:lemma2-stat}
\I_{0,\eps}^{A\to B}(\phi_{AB})\le S_{\min}^{{2\sqrt{\eps}}}(\rho^\phi_A) -\log(1-\eps).
\end{equation}
\end{lemma}
\begin{proof}
We first prove~(\ref{eq:lemma2-stat11}). Starting from~\reff{eq:i}
we have:
\begin{equation}
\begin{split}
I_{0,\eps}^{A\to B}(\phi_{AB}) :&=\max_{\bar\rho_{AB}\in
\B(\phi_{AB};\eps)}\min_{\sigma_B\in\states(\sH_B)}S_0(\bar\rho_{AB}\|\openone_A\otimes\sigma_B)\\
&\ge \max_{\bar\vphi_{AB}\in
\B_*(\phi_{AB};\eps)}\min_{\sigma_B\in\states(\sH_B)}S_0(\bar\vphi_{AB}\|\openone_A\otimes\sigma_B)\\
&=\max_{\bar\vphi_{AB}\in
\B_*(\phi_{AB};\eps)}\min_{\sigma_B\in\states(\sH_B)}\left\{-\log\Tr\left[\bar\vphi_{AB}(\openone_A\otimes\sigma_B)\right]
\right\}\\
&=\max_{\bar\vphi_{AB}\in
\B_*(\phi_{AB};\eps)}\left\{-\log\lambda_{\max}(\rho_B^{\bar\vphi})
\right\}\\
&=\max_{\bar\rho_B\in\B(\rho_B^{\phi};\eps)}S_{\min}(\bar\rho_B)\\
&=S_{\min}^\eps(\rho_B^\phi),
\end{split}
\end{equation}
where in the fifth line we made use of~\reff{equi}.
Next, we prove~\reff{eq:lemma2-stat}. By Lemma \ref{fid3} for any
$P\in\P(\phi;\eps)$, the state $\sqrt{P}|\phi\>$ is a pure state
such that $F^2\left(\sqrt{P}|\phi\>,|\phi\>\right)\ge
1-2{\sqrt{\eps}}$. Let us define the following two sets, for any
given bipartite pure state $\phi_{AB}$ and any $\eps'=
2{\sqrt{\eps}}$:
\begin{equation}\label{set1} \sA_1^{\eps'}(\phi_{AB}) :=
\left\{|\vphi_{AB}\rangle \in \sH_A \otimes \sH_B: |\vphi_{AB}\> =
\frac{\sqrt{P} |\phi_{AB}\>}{\sqrt{\Tr[P\phi_{AB}]}},P\in\P(\phi_{AB};\eps')\right\}.
\end{equation}
Obviously, $\sA_1^{\eps'}(\phi_{AB})\subseteq
\B_*(\phi_{AB};\eps')$, with the set $\B_*(\phi_{AB};\eps')$ being
defined by \reff{b*}. \smallskip Then,
\begin{align}
\I^{A \rightarrow B}_{0, \eps} (\phi_{AB}) =& \max_{P \in
\P(\phi_{AB}; \eps)} \min_{\sigma_B}
\Bigl[- \log \tr\bigl(\sqrt{P} \phi_{AB} \sqrt{P}(\openone \otimes \sigma_{B}) \bigr) \Bigr]\nonumber\\
\le& \max_{|\vphi_{AB}\rangle \in \sA^{\eps'}_1(\phi_{AB})}
\min_{\sigma_B}
\Bigl[- \log \tr\bigl(\vphi_{AB} (\openone \otimes \sigma_{B})
\bigr) \Bigr] -\log(1-\eps)\nonumber\\
\le & \max_{|\vphi_{AB}\rangle \in \B_*(\phi_{AB}; \eps')}
\min_{\sigma_B} \Bigl[- \log \tr\bigl(\vphi_{AB} (\openone \otimes
\sigma_{B}) \bigr) \Bigr]-\log(1-\eps),
\nonumber\\
=& \max_{{\bar{\rho}}_{B} \in \B(\rho^{B}_\phi; \eps')}
\min_{\sigma_B} \Bigl[- \log \tr\bigl({\bar{\rho}}_{B} \sigma_{B}
\bigr) \Bigr]-\log(1-\eps),
\nonumber\\
=& \max_{{\bar{\rho}}_{B} \in \B(\rho_{B}; \eps')}\bigl[- \log \lambda_{\max} ({\bar{\rho}}_{B})\bigr]-\log(1-\eps) \nonumber\\
=& S^{\eps'}_{\min} (\rho_{B}^\phi)-\log(1-\eps)\\
=& S^{2\sqrt{\eps}}_{\min}
(\rho_{A}^\phi) -\log(1-\eps),
\end{align}
where $\rho_{B}^\phi := \tr_A \phi_{AB}$ and $\rho^{A}_\phi := \tr_B
\phi_{AB}$. In the above, the second inequality follows from the fact
that $\sA^{\eps'}_1(\phi_{AB}) \subseteq \B_*(\phi_{AB};\eps')$, the
third inequality follows from the fact that $ \B_*(\phi_{AB}; \eps')=
\B(\rho_{B}^\phi; \eps')$ as stated in \reff{equi}, and the last
identity holds because $\phi_{AB}$ is a pure state.
\end{proof}
\section{Distillable entanglement of a single pure state}
In order to approach the problem of quantifying the one-shot EoA of an
arbitrary bipartite mixed state, we start from the simple but
insightful case in which two distant parties, say Alice and Bob,
initially share a single copy of a {\em{pure state}}
$|\phi_{AB}\rangle$. Their aim is to distill entanglement from this
shared state (i.e., convert the state to a maximally entangled state)
using local operations and classical communication (LOCC) only.
For sake of generality, we consider the situation where, for any given $\eps\ge 0$, the final state of the protocol is $\eps$-close to a maximally entangled state, with respect to a suitable distance measure. More precisely, we require the fidelity \reff{fidelity-aaa}
between the final state of the protocol and a maximally entangled state to be $\ge 1 - \eps$.
\begin{definition}[$\eps$-achievable distillation rates for pure states\footnote{For the more general case of mixed states, see \cite{distil}}]
For any given $\eps\ge0$, a real number $R\ge 0$ is said to be an
\emph{$\eps$-achievable rate} for one-shot entanglement distillation
of a pure state $\phi_{AB}:=|\phi_{AB}\rangle\langle\phi_{AB}|$, if there exists an integer $M\ge 2^R$ and a maximally entangled state
$\Psi^M_{A'B'}$ such that
\begin{equation}
F^2\left(\Lambda(\phi_{AB}),\Psi^M_{A'B'}\right)\ge 1-\eps ,
\end{equation}
for some LOCC operation $\Lambda:AB\mapsto
A'B'$.
\end{definition}
\begin{definition}[One-shot pure-state distillable entanglement] For any given
$\eps\ge0$, the one-shot distillabe entanglement, $E_D(\phi_{AB};\eps)$, of a pure state $\phi_{AB}$ is the maximum of all $\eps$-achievable
entanglement distillation rates for the state $\phi_{AB}$.
\end{definition}
\medskip
Bounds on the one-shot distillable entanglement of a pure state $\phi_{AB}$
are given by the following theorem.
\bigskip
\framebox[0.95\linewidth]{
\begin{minipage}{0.90\linewidth}
\begin{theorem}\label{theo:pure}
For any bipartite pure state $\phi_{AB}$ and any $\eps\ge 0$,
\begin{equation}
\label{eq:20}
S_{\min}^\eps(\rho^\phi_A)-\Delta\le E_D(\phi_{AB};\eps)\le S_{\min}^{\eps'}(\rho^\phi_A)-\log(1-2\sqrt{\eps}),
\end{equation}
where $\rho^\phi_A:=\Tr_B\phi_{AB}$, $\eps'=\sqrt{2\sqrt{\eps}}$, and
$0\le \Delta \le 1$ is a number included to ensure that the lower
bound in \reff{eq:20} is the logarithm of an integer number.
\end{theorem}
\end{minipage}}
\bigskip
\begin{remark}
The above theorem shows that, for any given $\eps\ge0$, the smoothed
min-entropy $S_{\min}^\eps(\rho^\phi_A)$ essentially characterizes
the one-shot distillable entanglement of the bipartite pure state
$|\phi_{AB}\>$.
\end{remark}
\begin{remark}
It is interesting to compare the lower bound of
Theorem~\ref{theo:pure} with the one-shot hashing bound proved in
Lemma~2 of~\cite{distil} for an arbitrary (possibly mixed) state.
For pure states, using Lemma~\ref{lemma:opsm-stsm}, the latter
yields:
\begin{equation}\label{eq:hashing}
E_D(\phi_{AB};\eps)\ge
S_{\min}^{\eps/8}(\rho^\phi_A)+\log\left(\frac 1d+\frac{\eps^2}{4}\right)-\Delta,
\end{equation}
where $d=\dim\sH_A$. It is evident that the bound in
Theorem~\ref{theo:pure} is tighter than~\reff{eq:hashing}, in
particular because there is no explicit logarithmic dependence on
the smoothing parameter $\eps$. From the technical point of view,
this arises because, for the case of pure states, we can directly
employ Nielsen's majorization criterion and hence do not need to
use random coding arguments, which are necessary in the general case.
\end{remark}
The proof of Theorem~\ref{theo:pure} can be divided into the following
two lemmas.
\begin{lemma}
For any bipartite pure state $\phi_{AB}$ and any $\eps\ge 0$,
\begin{equation}
\label{eq:20a}
E_D(\phi_{AB};\eps)\ge S_{\min}^\eps(\rho^\phi_A)-\Delta,
\end{equation}
where $\Delta\ge0$ is the least number such that the left hand side is
equal to the logarithm of a positive integer.
\end{lemma}
\begin{proof}
Let us begin by considering the case $\eps=0$. In this case,
Nielsen's majorization theorem \cite{nielsen-maj} implies that,
using LOCC, it is possible to exactly convert any pure state
$|\phi_{AB}\>$ to a maximally entangled state of rank equal to
$\left\lfloor\frac 1{\lambda_{\max}}\right\rfloor$, where
$\lambda_{\max}$ denotes the maximum eigenvalue of the reduced
density matrix $\rho^\phi_A$. Using the definition
(\ref{smooth_min}) of the min-entropy we then infer that
\begin{equation}
\label{eq:21}
E_D(\phi_{AB};0)\ge \log\left\lfloor 2^{S_{\min}(\rho^\phi_A)}\right\rfloor.
\end{equation}
If we allow a finite accuracy in the conversion, a lower bound to
the distillable entanglement can be given as follows.
For any $|\bar\phi_{AB}\>\in\B_*(\phi_{AB};\eps)$, by Nielsen's
theorem, there exists an LOCC map $\bar\Lambda$ such that
\begin{equation}\label{eq:inter-fid}
F^2\left(\bar\Lambda\left(\bar\phi_{AB}\right),\Psi_{A'B'}^{\bar M}\right)=1,
\end{equation}
where $\log\bar
M:=S_{\min}\left(\rho^{\bar\phi}_A\right)$.
On the other hand, due to the monotonicity of fidelity under the
action of a completely positive trace-preserving map,
\begin{equation}
\begin{split}
1-\eps\le1-\eps^2&\le F^2(\bar\phi_{AB},\phi_{AB})\\
&\le F^2\left(\bar\Lambda\left(\bar\phi_{AB}\right),\bar\Lambda(\phi_{AB})\right)\\
&=F^2\left(\Psi_{A'B'}^{\bar M},\bar\Lambda(\phi_{AB})\right).
\end{split}
\end{equation}
This yields the bound $ E_D(\phi_{AB};\eps)\ge\log\bar M$, for any
$|\bar\phi_{AB}\>\in\B_*(\phi_{AB};\eps)$. In particular, we have that
\begin{equation}
\label{eq:22}
E_D(\phi_{AB};\eps)\ge \max_{\bar\phi_{AB}\in\B_*(\phi_{AB};\eps)}\log\left\lfloor 2^{S_{\min}(\rho^{\bar\phi}_A)}\right\rfloor.
\end{equation}
Since the two sets
$\{\Tr_B[\bar\phi_{AB}]:\bar\phi_{AB}\in\B_*(\phi_{AB})\}$ and
$\B(\rho^\phi_A;\eps)$ coincide~\cite{marco}, we finally arrive at
\begin{equation}
\label{eq:23}
E_D(\phi_{AB};\eps)\ge \log\left\lfloor2^{S_{\min}^\eps(\rho^\phi_A)}\right\rfloor.
\end{equation}
\end{proof}
\begin{lemma}
For any bipartite pure state $\phi_{AB}$ and any $\eps\ge 0$,
\begin{equation}
\label{eq:20aa}
E_D(\phi_{AB};\eps)\le S_{\min}^{\eps'}(\rho^\phi_A),
\end{equation}
for $\eps'=\sqrt{2\sqrt{\eps}}$.
\end{lemma}
\begin{proof}
Let $r$ be the maximum of all achievable rates of entanglement
distillation for the pure state $\phi_{AB}$, i.e. $\log
r=E_D(\phi_{AB};\eps)$. This means that there exists an LOCC
transformation $\Lambda$ that maps $|\phi_{AB}\>$ into a state
$\omega_{A'B'}=\Lambda(\phi_{AB})$ which is $\eps$-close to a
maximally entangled state $|\Psi^r_{A'B'}\>$ of rank $r$, i.e.,
$F^2(\Lambda(\phi_{AB},\Psi^r_{A'B'}) \ge 1 - \eps$. Then,
\begin{equation}
\label{eq:24}
\begin{split}
E_D(\phi_{AB};\eps)&=\log r\\
&=I^{A'\to B'}_0(\Psi^r_{A'B'})\\
&\le \I^{A'\to B'}_{0,\eps}\left(\Lambda(\phi_{AB})\right)\\
&\le \I^{A\to B}_{0,2\sqrt{\eps}}(\phi_{AB})\\
&\le S_{\min}^{\eps'}(\rho^\phi_A)-\log(1-2\sqrt{\eps}),
\end{split}
\end{equation}
for $\eps'=\sqrt{2\sqrt{\eps}}$, where the first, second and third
inequalities follow from Lemma \ref{lemma1}, Lemma \ref{lemma2} and
Lemma \ref{lemma:opsm-stsm}, respectively.
\end{proof}
\section{One-shot entanglement of assistance}
As stated in the introduction, the definition of the EoA
arises naturally when considering the task in which Alice and Bob
distill entanglement from an initial mixed bipartite state $\rho_{AB}$
which they share, when a third party (say Charlie), who holds the
purification of the state, assists them, by doing local operations on
his share and communicating classical bits to Alice and Bob.
In order to express these ideas in a mathematically sound form, we
start by noticing that any strategy that Charlie may employ can be
described as the measurement of a positive operator-valued measure
(POVM) $\{P^i_C\}_i$, followed by the communication, to both Alice and
Bob, of the resulting classical outcome $i$. Since the state shared
between Alice, Bob, and Charlie is pure, say $|\Psi^\rho_{ABC}\>$,
Charlie's POVM's are in one-to-one correspondence with decompositions
of $\rho_{AB}$ into ensembles $\{p_i,\rho^i_{AB}\}_i$, via the
relation $p_i\rho^i_{AB}:=\Tr_C[\Psi^\rho_{ABC}\ (\openone_{AB}\otimes
P^i_C)]$. The fact that Charlie announces which outcome he got, means
that Alice and Bob can apply a different LOCC map for each value of
$i$.
An important point to stress now is that, in general, the distillation
process is allowed to be approximate. This is needed, in particular,
if one later wants to recover, from the one-shot setting, the usual
asymptotic scenario, where errors are required to vanish
asymptotically but are finite otherwise. In the classically-assisted
case we are studying here, since the index $i$ is visible to Alice and
Bob, they can apply a different LOCC map $\Lambda_i$ for each state
$\rho^i_{AB}$. We can hence choose to evaluate the distillation
accuracy according to a worst-case or an average criterion. Here we
choose the average fidelity as a measure of the ``expected''
accuracy. This leads us to define the maximum amount of entanglement
that can be distilled in the assisted case, namely, the \emph{one-shot
entanglement of assistance}, as,
\begin{equation}\label{eq:eoa-def}
\begin{split}
& E_A(\rho_{AB};\eps)\\
&:=\max_{\{P^i_C\}_i}\max_{M\in\mathbb{N}}\left\{
\log M: \max_{\{\Lambda^i_{AB}\}_i}
F^2\left(\sum_ip_i\Lambda^i(\rho^i_{AB}),\Psi^M_{A'B'}\right)\ge 1-\eps
\right\},
\end{split}
\end{equation}
where each $\Lambda^i$ is an LOCC map from $AB$ to $A'B'$.
As proved in Appendix~\ref{app:A}, the maximization over Charlie's
measurement in the above definition can always be restricted, without
loss of generality, to rank-one POVM's. Since rank-one POVM's at
Charlie's side are in one-to-one correspondence with pure state
ensemble decompositions of $\rho_{AB}$, we can equivalently write
\begin{equation}\label{eq:eoa-def2}
\begin{split}
&E_A(\rho_{AB};\eps)\\
=&\max_{{\{p_i,\phi^i_{AB}\}_i}\atop{\sum_ip_i\phi^i_{AB}=\rho_{AB}}}\max_{M\in\mathbb{N}}\left\{
\log M: \max_{\{\Lambda^i_{AB}\}_i}
F^2\left(\sum_ip_i\Lambda^i(\phi^i_{AB}),\Psi^M_{A'B'}\right)\ge 1-\eps
\right\}.
\end{split}
\end{equation}
In order to quantify $E_A(\rho_{AB};\eps)$ then, it is sufficient to
quantify the maximum expected amount of entanglement that can be
distilled, in average, from any given ensemble of pure bipartite
states. This is the aim of the following section.
\section{Distillable entanglement of an ensemble of pure states}
Given an ensemble $\eE=\{p_i,\phi^i_{AB}\}$ of pure states, we
define, for any given $\eps\ge0$ the one-shot distillable entanglement
of $\eE$ as
\begin{equation}\label{eq:ede-def}
E_D(\eE;\eps):=\max_{M\in\mathbb{N}}\left\{\log M:\max_{\{\Lambda^i_{AB}\}_i}
F^2\left(\sum_ip_i\Lambda^i(\phi^i_{AB}),\Psi^M_{A'B'}\right)\ge 1-\eps\right\},
\end{equation}
where each $\Lambda^i$ is an LOCC map from $AB$ to $A'B'$. According
with equation~(\ref{eq:eoa-def2}), the one-shot entanglement of
assistance $E_A$ of a given mixed state $\rho_{AB}$ is given by
\begin{equation}\label{ea}
E_A(\rho_{AB};\eps)=\max_{\eE}E_D(\eE;\eps),
\end{equation}
where the maximum is over all possible pure state ensemble
decompositions $\eE$ of $\rho_{AB}$.
For any given ensemble $\eE=\{p_i, \phi^i_{AB}\}$ of pure states, we
define the quantity
\begin{equation}\label{eff}
F_{\min}(\eE) := \min_i
S_{\min}(\rho^{\phi^i}_A),
\end{equation}
where $\rho^{\phi^i}_A:= \tr_B {\phi^i_{AB}}$. This quantity can be
intuitively interpreted as a conservative estimate of the amount of
entanglement present in the ensemble $\eE$. Further, for any such
ensemble, and any given $\eps \ge 0$, let us define the set
\begin{equation}\label{set11}
\cS_{\lle}(\eE;\eps):=\left\{\bar\eE=\left\{\bar\vphi^i_{AB}\right\}_i:\Tr\bar\vphi^i_{AB}\le 1, {\sum_ip_iF(\bar\vphi^i_{AB},\phi^i_{AB})\ge 1- {{\eps}}}\right\},
\end{equation}
and let $\cS_{\=}(\eE;\eps)$ denote the set obtained from
$\cS_{\lle}(\eE;\eps)$ by restricting the
pure states $\bar\vphi^i_{AB}$ to be normalized.\bigskip
\framebox[0.95\linewidth]{
\begin{minipage}{0.90\linewidth}
\begin{theorem}\label{thm_2}
For any given ensemble $\eE=\{p_i, \phi^i_{AB}\}$ of pure states, and any $\eps\ge 0$,
\begin{equation}\label{stat}
\max_{\bar\eE\in \cS_{=}(\eE ;\eps')}F_{\min}(\bar\eE) - \Delta
\ \le\ E_D(\eE;\eps)\ \le\
\max_{\bar\eE\in \cS_{\lle}(\eE; \eps'')}F_{\min}(\bar\eE),
\end{equation}
where $\eps'= \eps/2$, $\eps'':= \sqrt{2 \sqrt{\eps}}$, and $0\le
\Delta \le 1$ is a number which is included to ensure that the lower
bound in \reff{stat} is the logarithm of an integer number.
\end{theorem}
\end{minipage}
}\bigskip
\noindent The proof of this theorem is divided into the following two lemmas.
\begin{lemma}[Direct part]
For any pure state ensemble $\eE=\{p_i,\phi^i_{AB}\}$ and any
$\eps\ge 0$,
\begin{equation}
E_D(\eE;\eps)\ge\max_{\bar\eE\in \cS_{=}(\eE ;\eps')}F_{\min}(\bar\eE)-\Delta,
\end{equation}
where $\Delta$ is the minimum number in $[0,1]$ such that the right
hand side is equal to the logarithm of an integer number $M\ge 1$.
\end{lemma}
\begin{proof}
From Theorem~\ref{theo:pure}, we know that, given the pure bipartite
state $\phi^i_{AB}$, Alice and Bob can distill $\log
\left\lfloor2^{S_{\min}\left(\rho^{\phi^i}_A\right)}\right\rfloor$
ebits with zero error. Hence, given the ensemble
$\eE=\{p_i,\phi^i_{AB}\}$, Alice and Bob can distill, {\em{without
error}}, at least $\min_i \log
\left\lfloor2^{S_{\min}\left(\rho^{\phi^i}_A\right)} \right\rfloor$
ebits. For any pure state ensemble $\eE$, let us then introduce the
quantity $M(\eE):=\min_i
\left\lfloor2^{S_{\min}\left(\rho^{\phi^i}_A\right)} \right\rfloor$.
If a finite accuracy $\eps> 0$ is allowed, then it is possible to
give a lower bound on the one-shot distillable entanglement
$E_D(\eE;\eps)$ as follows. Let us consider the set of ensembles of
normalized pure states of the form $\bar\eE=\{p_i,
\bar\vphi^i_{AB}\}$, such that
$\sum_ip_iF(\phi^i_{AB},\bar\vphi^i_{AB})\ge 1-\eps$. Then, for any
ensemble $\bar\eE$ in such a set, there exist LOCC maps
$\Lambda^i:AB\rightarrow A'B'$ such that
\begin{equation}\label{eq:exact-ens-dist}
F\left(\sum_ip_i\Lambda^i(\bar\vphi^i_{AB}),\Psi^{M(\bar\eE)}_{AB}\right)=1,
\end{equation}
where $\Psi^{M(\bar\eE)}_{A'B'}$ denotes a maximally entangled state
of rank $M(\bar\eE)$. Equivalently,
$\Lambda^i(\bar\vphi^i_{AB})=\Psi^{M(\bar\eE)}_{AB}$, for all
$i$. Then,
\begin{equation}
\begin{split}
1-\eps&\le \sum_ip_iF(\phi^i_{AB},\bar\vphi^i_{AB})\\
&\le
\sum_ip_iF\left(\Lambda^i(\phi^i_{AB}),\Lambda^i(\bar\vphi^i_{AB})\right)\\
&\le
F\left(\sum_ip_i\Lambda^i(\phi^i_{AB}),\sum_ip_i\Lambda^i(\bar\vphi^i_{AB})\right)\\
&=F\left(\sum_ip_i\Lambda^i(\phi^i_{AB}),\Psi^{M(\bar\eE)}_{AB}\right),
\end{split}
\end{equation}
where the second line follows from the monotonicity of fidelity
under completely positive trace-preserving (CPTP) maps, the third
line follows from the concavity of the fidelity, and the last
identity follows from~(\ref{eq:exact-ens-dist}). Hence, we conclude
that there exist LOCC maps $\Lambda^i$ for which
\begin{equation}
F^2\left(\sum_ip_i\Lambda^i(\phi^i_{AB}),\Psi^{M(\bar\eE)}_{A'B'}\right)\ge 1-2\eps,
\end{equation}
that is,
\begin{equation}
E_D(\eE;{2\eps})\ge\log M(\bar\eE),
\end{equation}
{\em{for any}} $\bar\eE$ in the set introduced above.
By maximizing $M(\bar\eE)$ over
all such ensembles and comparing the result with the
definition in~(\ref{eff}),
we obtain the statement of the lemma.
\end{proof}
\begin{lemma}[Converse part]\label{lemma:eoe-conv}
For any pure state ensemble $\eE=\{p_i,\phi^i_{AB}\}$ and any $\eps\ge 0$,
\begin{equation}
E_D(\eE;\eps)\le \max_{\bar\eE\in \cS_{\lle}(\eE; \eps')}F_{\min}(\bar\eE),
\end{equation}
where $\eps'= \sqrt{2\sqrt{\eps}}$.
\end{lemma}
\begin{proof}
Let $r$ be a positive integer such that $E_D(\eE;\eps)=\log
r$. According to~\reff{eq:ede-def}, this means that there exist LOCC
maps $\Lambda^i:AB\to A'B'$ such that
\begin{equation}\label{eq:av-fid-cond}
\Tr\left[
\sum_ip_i\Lambda^i(\phi^i_{AB})\
\Psi^{r}_{A'B'}
\right]\ge 1-\eps.
\end{equation}
Since the maps $\Lambda^i$ act on pure states, without loss of
generality we can assume them to be of the Lo-Popescu form
\reff{form}.
Further, equation~(\ref{eq:av-fid-cond}) above, in particular, informs us that
\begin{equation}\label{65}
\Psi^r_{A'B'}\in\P\left(\sum_ip_i\Lambda^i(\phi^i_{AB});\eps\right).
\end{equation}
This fact in turns implies that
\begin{equation}\label{eq:tocontt}
\begin{split}
E_D(\eE;\eps)&=\log r\\
&=I^{A'\to B'}_{0}\left(\Psi^r_{A'B'}\right)\\
&\equiv\min_{\sigma_{B'}}\left\{-\log\Tr\left[\Psi^r_{A'B'}\
(\openone_{A'}\otimes\sigma_{B'})\right]\right\}\\
&\le -\log\Tr\left[\Psi^r_{A'B'}\
(\openone_{A'}\otimes\tilde\sigma_{B'})\right]\\
&\le-\log\Tr\left[\left(\Psi^r_{A'B'}\
\Pi_{\sum_ip_i\Lambda^i(\phi^i_{AB})}\
\Psi^r_{A'B'}\right)\
(\openone_{A'}\otimes\tilde\sigma_{B'})\right],
\end{split}
\end{equation}
for any state $\tilde\sigma_{B'}$. To obtain the last inequality,
we simply used
the fact that $\Psi^r_{A'B'}\ge \Psi^r_{A'B'}\Pi \Psi^r_{A'B'}$, for
any $0\le\Pi\le\openone$. We then choose $\tilde\sigma_{B'}$ so that
\begin{equation}\label{68}
\begin{split}
&-\log\Tr\left[\left(\Psi^r_{A'B'}\
\Pi_{\sum_ip_i\Lambda^i(\phi^i_{AB})}\
\Psi^r_{A'B'}\right)\
(\openone_{A'}\otimes\tilde\sigma_{B'})\right]\\
=\min_{\sigma_{B'}}&\left\{-\log\Tr\left[\left(\Psi^r_{A'B'}\
\Pi_{\sum_ip_i\Lambda^i(\phi^i_{AB})}\
\Psi^r_{A'B'}\right)\
(\openone_{A'}\otimes\sigma_{B'})\right]\right\}.
\end{split}
\end{equation}
From \reff{65}, \reff{eq:tocontt} and \reff{68} we infer that
\begin{equation}
E_D(\eE;\eps)\le
\I^{A'\to
B'}_{0,\eps}\left(\sum_ip_i\Lambda_i(\phi^i_{AB})\right).
\end{equation}
Let us now introduce an auxiliary system $Z$ and an orthonormal basis
for it $\{|i_Z\>\}$ that keeps track of the classical outcome $i$
labeling the states in $\eE$. Let us denote by $\pi^i_Z$ the projector
$|i\>\<i|_Z$. By further introducing the states
$\omega_{A'B'}:=\sum_ip_i\Lambda_i(\phi^i_{AB})$ and
$\omega_{A'B'Z}:=\sum_ip_i\Lambda_i(\phi^i_{AB})\otimes\pi^i_Z$, so
that $\omega_{A'B'}=\Tr_Z \omega_{A'B'Z}$, we have
\begin{equation}
\begin{split}
E_D(\eE;\eps) &\le\I^{A'\to B'}_{0,\eps}(\omega_{A'B'})\\
&\equiv\max_{P\in\P(\omega_{A'B'};\eps)}\min_{\sigma_{B'}}\left\{
-\log\Tr\left[ \sqrt{P}\Pi_{\omega_{A'B'}}\sqrt{P}\
(\openone_{A'}\otimes\sigma_{B'})\right]\right\}\\
&=\min_{\sigma_{B'}}\left\{
-\log\Tr\left[ \sqrt{P_0}\Pi_{\omega_{A'B'}}\sqrt{P_0}\
(\openone_{A'}\otimes\sigma_{B'})\right]\right\}\\
&\le -\log\Tr\left[ \sqrt{P_0}\Pi_{\omega_{A'B'}}\sqrt{P_0}\
(\openone_{A'}\otimes\bar\nu_{B'})\right],
\end{split}
\end{equation}
where the operator $P_0$ in the third line is the one achieving the
maximum, and $\bar\nu_{B'}$ in the fourth line is any state in
$\states(\sH_{B'})$. In particular, since
$\Pi_{\omega_{A'B'}}\otimes\openone_Z \ge\Pi_{\omega_{A'B'Z}}$, we have that
\begin{equation}
\begin{split}
E_D(\eE;\eps)&\le -\log\Tr\left[
\sqrt{P_0}\Pi_{\omega_{A'B'}}\sqrt{P_0}\
(\openone_{A'}\otimes\bar\nu_{B'})\right]\\
&= -\log\Tr\left[
\sqrt{P_0\otimes\openone_Z}(\Pi_{\omega_{A'B'}}\otimes\openone_Z)\sqrt{P_0\otimes\openone_Z}\
(\openone_{A'}\otimes\bar\nu_{B'Z})\right]\\
&\le -\log\Tr\left[
\sqrt{P_0\otimes\openone_Z}\Pi_{\omega_{A'B'Z}}\sqrt{P_0\otimes\openone_Z}\
(\openone_{A'}\otimes\bar\nu_{B'Z})\right],
\end{split}
\end{equation}
for any state $\bar\nu_{B'Z}$.
Let us then choose $\bar\nu_{B'Z}$ to be the state such that
\begin{equation}
\begin{split}
&-\log\Tr\left[
\sqrt{P_0\otimes\openone_Z}\Pi_{\omega_{A'B'Z}}\sqrt{P_0\otimes\openone_Z}\
(\openone_{A'}\otimes\bar\nu_{B'Z})\right]\\
=\min_{\nu_{B'Z}}&\left\{-\log\Tr\left[
\sqrt{P_0\otimes\openone_Z}\Pi_{\omega_{A'B'Z}}\sqrt{P_0\otimes\openone_Z}\
(\openone_{A'}\otimes\nu_{B'Z})\right]\right\}.
\end{split}
\end{equation}
Moreover, note that
$(P_0\otimes\openone_Z)\in\P(\omega_{A'B'Z};\eps)$, since
$P_0\in\P(\omega_{A'B'};\eps)$.
In fact, the operator $(P_0 \otimes \openone_Z)$ also belongs
to the following set of quantum-classical (q-c) operators:
\begin{equation}\label{qc}
\begin{split}
&\P_{\textrm{qc}}(\sigma_{ABZ};\eps):=\\
&\left\{\left.P_{ABZ}=\sum_iP^i_{AB}\otimes\pi^i_Z\right|
0\le P^i_{AB} \le \openone_{AB},\, \Tr\bigl(
P_{ABZ}\sigma_{ABZ}\bigr)\ge1-\eps\right\}.
\end{split}
\end{equation}
Hence, we can write
\begin{equation}\label{eq:qwerty}
E_D(\eE;\eps)\le\max_{Q\in\P_{\textrm{qc}}(\omega_{A'B'Z};\eps)}\min_{\nu_{B'Z}}\left\{-\log\Tr\left[
\sqrt{Q}\Pi_{\omega_{A'B'Z}}\sqrt{Q}\
(\openone_{A'}\otimes\nu_{B'Z})\right]\right\}
\end{equation}
Let the Kraus representations of the CPTP maps $\Lambda_i:AB\mapsto
A'B'$ satisfying~\reff{eq:av-fid-cond} be written as
$\Lambda_i(\rho)=\sum_{\mu_i}V_{\mu_i}\rho V_{\mu_i}^\dag$, so that
$\sum_{\mu_i}V_{\mu_i}^\dag V_{\mu_i}=\openone_{AB}$ for all
$i$. Using these, we construct a CPTP map $\M:ABZ\to A'B'Z$ as
\begin{equation}
\label{eq:11}
\M(\rho_{ABZ}):=\sum_i\sum_{\mu_i}\left(V_{\mu_i}\otimes\pi^i_Z\right)\rho_{ABZ}\left(V_{\mu_i}\otimes\pi^i_Z\right)^\dag.
\end{equation}
In terms of the map $\M$ so constructed,
\begin{equation}
\omega_{A'B'Z}=\M\left(\sum_ip_i\phi^i_{AB}\otimes\pi^i_Z\right).
\end{equation}
Defining the quantum-classical (q-c) state
$\sigma_{ABZ}:=\sum_ip_i\phi^i_{AB}\otimes\pi^i_Z$, we have,
continuing from~(\ref{eq:qwerty}),
\begin{align}
E_D(\eE;\eps)&\le
\max_{Q\in\P_{\textrm{qc}}(\M(\sigma_{ABZ});\eps)}\min_{\nu_{B'Z}}\left\{-\log\Tr\left[\sqrt{Q}\Pi_{\M(\sigma_{ABZ})}\sqrt{Q}\
\left(\openone_{A'}\otimes\nu_{B'Z}\right)\right]\right\}\nonumber\\
&\equiv
\min_{\nu_{B'Z}}\left\{-\log\Tr\left[\sqrt{Q_0}\Pi_{\M(\sigma_{ABZ})}\sqrt{Q_0}\
\left(\openone_{A'}\otimes\nu_{B'Z}\right)\right]\right\},
\end{align}
where $Q_0\in\P_{\textrm{qc}}(\M(\sigma_{ABZ});\eps)$ is the q-c
operator achieving the maximum in the second line. This implies that
\begin{equation}
E_D(\eE;\eps)\le -\log\Tr\left[\sqrt{Q_0}\Pi_{\M(\sigma_{ABZ})}\sqrt{Q_0}\
\left(\openone_{A'}\otimes\nu_{B'Z}\right)\right],
\end{equation}
for any state $\nu_{B'Z}$.
Due to the fact that the maps $\Lambda_i$ are in the Lo-Popescu
form \reff{form}, it follows that the map $\M$ (obtained from the
$\Lambda_i$'s) is also in the Lo-Popescu form. The identity \reff{lopop}
then implies that
\begin{equation}\label{eq:quasi-fin}
E_D(\eE;\eps)\le -\log\Tr\left[\sqrt{Q_0}\Pi_{\M(\sigma_{ABZ})}\sqrt{Q_0}\
\ \M(\openone_{A}\otimes\tilde\nu_{BZ})\right],
\end{equation}
for any state $\tilde\nu_{BZ}$. By using the dual map $\M^*$,
\begin{equation}
E_D(\eE;\eps)\le -\log\Tr\left[\M^*\left(\sqrt{Q_0}\Pi_{\M(\sigma_{ABZ})}\sqrt{Q_0}\right)\
\ (\openone_{A}\otimes\tilde\nu_{BZ})\right],
\end{equation}
for any state $\tilde\nu_{BZ}$. By denoting the operator
$\M^*\left(\sqrt{Q_0}\Pi_{\M(\sigma_{ABZ})}\sqrt{Q_0}\right)$ as $\tilde
Q_{ABZ}$, we have, for any state $\tilde\nu_{BZ}$,
\begin{equation}\label{eq:to_cont}
E_D(\eE;\eps)\le -\log\Tr\left[\sqrt{\tilde
Q_{ABZ}}\Pi_{\sigma_{ABZ}}\sqrt{\tilde Q_{ABZ}}\ (\openone_{A}\otimes\tilde\nu_{BZ})\right],
\end{equation}
since $\tilde Q_{ABZ}\ge \sqrt{\tilde
Q_{ABZ}}\Pi_{\sigma_{ABZ}}\sqrt{\tilde Q_{ABZ}}$. Let us also
choose $\tilde\nu_{BZ}$ so that
\begin{equation}
\begin{split}
&-\log\Tr\left[\sqrt{\tilde
Q_{ABZ}}\Pi_{\sigma_{ABZ}}\sqrt{\tilde Q_{ABZ}}
\ (\openone_{A}\otimes\tilde\nu_{BZ})\right]\\
=\min_{\nu_{BZ}}&\left\{-\log\Tr\left[\sqrt{\tilde
Q_{ABZ}}\Pi_{\sigma_{ABZ}}\sqrt{\tilde Q_{ABZ}}
\ (\openone_{A}\otimes\nu_{BZ})\right]\right\}.
\end{split}
\end{equation}
Using the particular form~(\ref{eq:11}) of $\M$, and the facts that
$\sigma_{ABZ}$ is a q-c state and $Q_0 \in
\P_{\textrm{qc}}(\sigma_{ABZ};\eps)$, we can prove that the operator
$\tilde Q_{ABZ} \in \P_{\textrm{qc}}(\sigma_{ABZ};2\sqrt{\eps})$,
using arguments similar to those leading to \reff{33}.
Hence, continuing from equation~\reff{eq:to_cont}, we can write
\begin{align}
E_D(\eE;\eps)&\le \min_{\nu_{BZ}}\left\{-\log\Tr\left[\sqrt{\tilde
Q_{ABZ}}\Pi_{\sigma_{ABZ}}\sqrt{\tilde Q_{ABZ}}
\ (\openone_{A}\otimes\nu_{BZ})\right]\right\}\nonumber\\
&\le \max_{P\in \P_{\textrm{qc}}(\sigma_{ABZ};2\sqrt{\eps})}\min_{\nu_{BZ}} \left\{-\log\Tr\left[\sqrt{P}\Pi_{\sigma_{ABZ}}\sqrt{P}\
\ (\openone_{A}\otimes\nu_{BZ})\right]\right\}.\label{81}
\end{align}
Let $\eps' := 2\sqrt{\eps}$. Then, for any
$P=\sum_iP^i_{AB}\otimes\pi^i_Z$ in
$\P_{\textrm{qc}}(\sigma_{ABZ};\eps')$, let us define
$|\vphi_{AB}^i\rangle := \sqrt{P^i_{AB}}|\phi_{AB}^i\rangle$. As a
consequence of Lemma~\ref{fid3}, we have that
$\sum_ip_iF(\vphi_{AB}^i, \phi^i_{AB})\ge 1-\sqrt{\eps'}$, so that
\begin{align} E_D(\eE;\eps)
&\le \max_{P\in
\P_{\textrm{qc}}(\sigma_{ABZ};{\eps'})}\min_{\nu_{BZ}}
\left\{-\log\Tr\left[\sqrt{P}\Pi_{\sigma_{ABZ}}\sqrt{P}\
\ (\openone_{A}\otimes\nu_{BZ})\right]\right\}\nonumber\\
&\le
\max_{\bar\eE\in \cS_{\lle}(\eE; \sqrt{\eps'})}
\min_{\nu_{BZ}} \left\{-\log\Tr\left[\bigl(\bar\vphi^i_{AB}\otimes
\pi^i_{Z}\bigr)\
\ (\openone_{A}\otimes\nu_{BZ})\right]\right\}\nonumber\\
&=\max_{\bar\eE\in \cS_{\lle}(\eE; \sqrt{\eps'})}
\min_i \min_{\nu_{B}} \left\{-\log\Tr\left[\rho_B^{\bar\vphi^i}\nu_{B})\right]\right\}\nonumber\\
&=
\max_{\bar\eE\in \cS_{\lle}(\eE; \sqrt{\eps'})}
\min_i \bigl[- \log \lambda_{\max}(\rho_B^{\bar\vphi^i})\bigr],
\nonumber\\
&=
\max_{\bar\eE\in \cS_{\lle}(\eE; \sqrt{\eps'})}\min_i
S_{\min}(\rho_A^{\bar\vphi^i}),
\end{align}
where we used the fact that $\lambda_{\max}(\rho_B^{\bar\vphi^i})=
\lambda_{\max}(\rho_A^{\bar\vphi^i})=S_{\min}(\rho_A^{\bar\vphi^i})$,
since $\bar\vphi^i_{AB}$ is a pure state.
\end{proof}
\section{Asymptotic entanglement of assistance}
Consider the situation in which three parties, Alice, Bob and Charlie
jointly possess multiple (say $n$) copies of a tripartite pure state
$|\Psi_{ABC}\rangle$. Alice and Bob, considered in isolation,
therefore possess $n$ copies of the state $\rho_{AB}:= \tr_C
\Psi_{ABC}$, i.e., they share the state $\rho_{AB}^{\otimes n}$. We
refer to this situation as the ``i.i.d. scenario'', in analogy with
the classical case of independent and identically distributed (i.i.d.)
random variables. We define the asymptotic entanglement of assistance
of a state $\rho_{AB}$ as
\begin{equation}\label{eainfty}
E_A^\infty(\rho_{AB}):= \lim_{\eps \rightarrow 0} \lim_{n \rightarrow \infty} \frac{1}{n} E_A( \rho_{AB}^{\otimes n} ; \eps),
\end{equation}
where for any $\eps \ge 0$, $E_A( \rho_{AB}^{\otimes n} ; \eps)$
denotes the one-shot entanglement of assistance of the state
$\rho_{AB}^{\otimes n}$, defined in~\reff{eq:eoa-def} and quantified
in~\reff{ea} and~\reff{stat}.
The same notation $E_A^\infty(\rho_{AB})$ was used in
Ref.~\cite{assistance} to denote the \emph{regularized} EoA, formally
defined as $\lim_{n\to\infty}\frac 1nE_A(\rho_{AB}^{\otimes n})$
from~\reff{eq:19}. The aim of this section is to show that the two
quantities coincide, so that, in fact, there is no notational
inconsistency. At the same time, this provides an alternative proof of
the operational interpretation of the regularized EoA given
in~\cite{assistance}.
The main result of this section is the following theorem:\bigskip
\framebox[0.95\linewidth]{
\begin{minipage}{0.90\linewidth}
\begin{theorem}\label{thm_4}
For any bipartite state $\rho_{AB}$
\begin{equation}\label{asymp}
E_A^\infty(\rho_{AB}):=\lim_{\eps \rightarrow 0} \lim_{n \rightarrow \infty} \frac{1}{n} E_A( \rho_{AB}^{\otimes n} ; \eps)= \lim_{n \rightarrow \infty} \frac{1}{n} E_A(\rho_{AB}^{\otimes n}),
\end{equation}
where for any state $\omega_{AB}$,
\begin{equation}\label{eadef}
E_A(\omega_{AB}):= \max_{\{p_i, |\vphi^i_{AB}\rangle\}\atop{\omega_{AB} =
\sum_i p_i \vphi^i_{AB}}} \sum_i p_i S(\rho^{\vphi^i}_A),
\end{equation}
denotes its entanglement of assistance, with $\rho^{\vphi^i}_A=
\tr_B[\vphi^i_{AB}]$.
\end{theorem}
\end{minipage}
}\bigskip
In order to prove this, we first need to introduce a few more
definitions. Let $\sigma_{ABZ}$ be a quantum-classical (qc) state,
i.e.
\begin{equation}
\sigma_{ABZ}=\sum_ip_i\sigma^i_{AB}\otimes\pi^i_Z,
\end{equation}
for some probabilities $p_i\ge 0$, $\sum_ip_i=1$, some normalized
states $\sigma^i_{AB}\in\states(\sH_A\otimes\sH_B)$, and some
orthogonal rank-one projectors $\pi^i_Z=|i\>\<i|_Z$ (that we fix here
once and for all). As it has been done already in~\reff{qc}, along the
proof of Lemma~\ref{lemma:eoe-conv}, we define the sets
\begin{equation}\label{pqc2}
\P_{\textrm{qc}}(\sigma_{ABZ};\eps):=\left\{P_{ABZ}=\sum_iP^i_{AB}\otimes\pi^i_Z\left|
\begin{split}
&0\le P^i_{AB} \le \openone_{AB},\\
&\Tr[
P\sigma]\ge1-\eps
\end{split}
\right.\right\},
\end{equation}
and
\begin{equation}\label{bqc2}
\begin{split}
&\B_{\textrm{qc}}(\sigma_{ABZ};\eps):=\\
&\left\{\bar\omega_{ABZ}=\sum_ip_i\bar\vphi^i_{AB}\otimes\pi^i_Z\left|
\begin{split}
&\N{\bar\vphi^i_{AB}}_1=\N{\bar\vphi^i_{AB}}_\infty=1,\\
&F(\bar\omega,\sigma)=\sum_ip_iF(\bar\vphi^i,\sigma^i)\ge1-\eps
\end{split}
\right.\right\} .
\end{split}
\end{equation}
The sets defined above are analogous to those introduced
in~\reff{ball} and~\reff{P-ball}, with the difference that the
quantum-classical structure of the argument $\sigma_{ABZ}$ is here
maintained.
For technical reasons that will be apparent in the proofs, we also
need to introduce an additional smoothed zero-coherent information,
besides those in~\reff{eq:i} and~\reff{eq:itilda}, defined as, for any
qc state $\sigma_{ABZ}$ and any $\eps\ge 0$,
\begin{equation}\label{eq2}
I^{A\leadsto BZ}_{0,\eps}(\sigma_{ABZ}):=\max_{\bar\sigma_{ABZ}\in \B_{\textrm{qc}}
(\sigma_{ABZ};\eps)}\min_{\nu_{BZ}\in\states(\sH_B\otimes \sH_Z)}
S_0(\bar\sigma_{ABZ}\|\openone_A\otimes\nu_{BZ}).
\end{equation}
We then proceed by proving the following lemma, which is nothing but a
convenient reformulation of Theorem~\ref{thm_2}:
\begin{lemma}\label{thm_3}
For any bipartite state $\rho_{AB}$ and any $\eps\ge 0$,
\begin{equation}\label{88}
\max_{\eE}I^{A\leadsto BZ}_{0,\eps/2}(\sigma_{ABZ}^\eE) - \Delta \le E_A(\rho_{AB};\eps)\le
\max_{\eE} \I^{A\to BZ}_{0,2\sqrt{\eps}}(\sigma_{ABZ}^\eE),
\end{equation}
where the maxima are taken over all possible pure state ensembles
$\eE=\{p_i, \phi^i_{AB}\}$ such that $\rho_{AB} = \sum_i p_i
\phi^i_{AB}$, and for a given ensemble $\eE=\{p_i, \phi^i_{AB}\}$,
$\sigma_{ABZ}^\eE = \sum_i p_i \phi^i_{AB} \otimes \pi^i_Z.$ In the
above, the real number $0\le \Delta \le 1$ is included to ensure that
the lower bound is equal to the logarithm of a positive integer.
\end{lemma}
For the sake of clarity, we divide the proof of the Lemma above into
two separate lemmas. The first is the following:
\begin{lemma}\label{cor1}
For any given ensemble $\eE=\{p_i, \phi^i_{AB}\}$ of pure states,
and any $\eps\ge 0$,
\begin{equation}
E_D(\eE;\eps)\le \I^{A\to
BZ}_{0,2\sqrt{\eps}}(\sigma_{ABZ}^\eE),
\end{equation}
where $\sigma_{ABZ}^\eE:= \sum_i p_i \phi^i_{AB} \otimes \pi^i_Z$, and
$\I^{A\to BZ}_{0,2\sqrt{\eps}}(\sigma_{ABZ}^\eE)$ is defined
in~\reff{eq:itilda}.
\end{lemma}
\begin{proof}
The equation number~(\ref{81}) in the proof of Theorem~\ref{thm_2},
that is,
\begin{equation}
E_D(\eE;\eps)
\le \max_{P\in \P_{\textrm{qc}}(\sigma_{ABZ};2\sqrt{\eps})}\min_{\nu_{BZ}} \left\{-\log\Tr\left[\sqrt{P}\Pi_{\sigma_{ABZ}}\sqrt{P}\
\ (\openone_{A}\otimes\nu_{BZ})\right]\right\}
\end{equation}
already proves the statement, since
$\P_{\textrm{qc}}(\sigma_{ABZ};2\sqrt{\eps})\subset
\P(\sigma_{ABZ};2\sqrt{\eps})$.
\end{proof}
\begin{lemma}\label{cor2}
For any given ensemble $\eE=\{p_i, \phi^i_{AB}\}$ of pure states,
and any $\eps\ge 0$,
\begin{equation}
E_D(\eE;\eps)\ge I^{A\leadsto
BZ}_{0,\eps/2}(\sigma_{ABZ}^\eE).
\end{equation}
where $\sigma_{ABZ}^\eE:= \sum_i p_i \phi^i_{AB} \otimes \pi^i_Z$ and
$I^{A\leadsto BZ}_{0,\eps/2}(\sigma_{ABZ}^\eE)$ is defined
in~\reff{eq2}.
\end{lemma}
\begin{proof}
The statement is a direct consequence of the lower bound in Theorem
\ref{thm_2}. This can be shown as follows:
\begin{align}
I^{A\leadsto BZ}_{0,\eps/2}(\sigma_{ABZ}^\eE) :&=
\max_{\bar{\sigma}_{ABZ}\in \B_{\textrm{qc}}({\sigma}_{ABZ};\eps/2)} \min_{\nu_{BZ}}
\left\{-\log \tr\left[\Pi_{\bar{\sigma}_{ABZ}}\ (\openone_A \otimes \nu_{BZ})\right] \right\}\nonumber\\
& = \max_{\{\bar\vphi^i_{AB}\}_i:\Tr\bar\vphi^i_{AB}=1\atop{\sum_ip_iF(\bar\vphi^i_{AB},\phi^i_{AB})\ge1-\eps/2}}\min_i \min_{\nu_B}
\left\{ - \log\tr\left[\rho^{\bar\vphi^i}_{B}\ \nu_B\right]\right\}\nonumber\\
&= \max_{\{\bar\vphi^i_{AB}\}_i:\Tr\bar\vphi^i_{AB}=1\atop{\sum_ip_iF(\bar\vphi^i_{AB},\phi^i_{AB})\ge1-\eps/2}}\min_i \left\{ - \log \lambda_{\max}
\left(\rho^{\bar\vphi^i}_B\right)\right\}\nonumber\\
& = \max_{\{\bar\vphi^i_{AB}\}_i:\Tr\bar\vphi^i_{AB}=1\atop{\sum_ip_iF(\bar\vphi^i_{AB},\phi^i_{AB})\ge 1-\eps/2}} \min_i {S_{\min}
(\rho^{\bar\vphi^i}_A)},
\end{align}
since $ \lambda_{\max}(\rho_B^{\bar\vphi^i})= \lambda_{\max}(\rho_A^{\bar\vphi^i})=S_{\min}(\rho_A^{\bar\vphi^i})$, with $\rho_B^{\bar\vphi^i}:= \tr_A({\bar\vphi^i})$ and $\rho_A^{\bar\vphi^i}:= \tr_B({\bar\vphi^i})$, because $\bar\vphi^i_{AB}$ is a pure state. To obtain the identity on the third line, we made use of
the fact that $\Pi_{\bar{\sigma}_{ABZ}}=\sum_i \bar\vphi^i_{AB} \otimes \pi^i_Z$.
\end{proof}
The proof of Theorem \ref{thm_4} can be divided into the following two lemmas.
\begin{lemma}\label{asymp_direct}
For any bipartite state $\rho_{AB}$,
\be
E_A^\infty(\rho_{AB}) \ge \lim_{n \rightarrow \infty} \frac{1}{n} E_A(\rho_{AB}^{\otimes n}),
\ee
\end{lemma}
\begin{proof}
Let $\eE=\{p_i,\phi^i_{AB}\}$ be an ensemble of pure states for
$\rho_{AB}$ and $\eE_n=\{p_i^n, \phi^i_{A_nB_n}\}$ be an ensemble of
pure states for $\rho_{AB}^{\otimes n}$. First of all, note that the
pure states $\phi^i_{A_nB_n}$ need not be factorized. For this
ensemble, define the tripartite state
\begin{equation}\label{sigma}
\sigma_{ABZ}^{\eE_n} = \sum_i p_i^n \phi^i_{A_nB_n} \otimes
\pi^{n,i}_Z \in {\cal{B}}\bigl(\sH_A^{\otimes n}\otimes
\sH_B^{\otimes n}\otimes\sH_Z^{\otimes n}\bigr),
\end{equation}
where $\pi^{n,i}_Z = |i_n\rangle \langle i_n| \in
\states(\sH_Z^{\otimes n})$, with $\{ |i_n\rangle \}_i$ being an
orthonormal basis of $\sH_Z^{\otimes n}$.
From \reff{88} of Lemma \ref{thm_3} we have, for any given $\eps \ge 0$,
\begin{equation}
E_A(\rho_{AB}^{\otimes n}; \eps)\ge \max_{\eE_n}I^{A_n\leadsto B_nZ_n}_{0,\eps/2}
(\sigma_{ABZ}^{\eE_n}) - \Delta_n
\label{lbbd}
\end{equation}
with $0\le \Delta_n \le 1$. We then have:
\begin{align}
E_A^\infty(\rho_{AB}):=& \lim_{\eps \rightarrow 0} \lim_{n \rightarrow \infty} \frac{1}{n} E_A( \rho_{AB}^{\otimes n} ; \eps),\nonumber\\
\ge & \lim_{\eps \rightarrow 0} \lim_{n \rightarrow \infty} \frac{1}{n}
\max_{\eE_n}I^{A_n\leadsto B_nZ_n}_{0,\eps/2}
(\sigma_{ABZ}^{\eE_n})\nonumber\\
\ge & \lim_{\eps \rightarrow 0} \lim_{n \rightarrow \infty} \frac{1}{n}
\max_{\eE}I^{A_n\leadsto B_nZ_n}_{0,\eps/2}\left((\sigma_{ABZ}^{\eE})^{\otimes n}\right)\nonumber\\
=& \max_{\eE}\Bigl[I^{A\rightarrow BZ}(\sigma_{ABZ}^{\eE})\Bigr].
\label{long1}
\end{align}
The proof of~\reff{long1} can be found in Appendix~\ref{app:c}
From the definition of the state $\sigma_{ABZ}^{\eE}$ it follows that
for the ensemble $\eE=\{p_i, \phi^i_{AB}\}$,
\begin{equation}\label{idf}
I^{A\rightarrow BZ}(\sigma_{ABZ}^{\eE}) = \sum_i p_i
S(\rho^{\phi^i}_B),
\end{equation}
where $\rho^{\phi^i}_B = \tr_{AZ}\bigl(\sigma_{ABZ}^{\eE}\bigr)$. From
\reff{long1} and \reff{idf} we hence obtain
\begin{align}
E_A^\infty(\rho_{AB}) &\ge \max_{\eE} \sum_i p_i S(\rho^{\phi^i}_B)\nonumber\\
&= E_A(\rho_{AB}).
\end{align}
The statement of the lemma can then be obtained by the usual blocking
argument.
\end{proof}
\begin{lemma}\label{asymp_converse}
For any bipartite state $\rho_{AB}$,
\begin{equation}
E_A^\infty(\rho_{AB}) \le \lim_{n \rightarrow \infty} \frac{1}{n} E_A(\rho_{AB}^{\otimes n}),
\end{equation}
\end{lemma}
\begin{proof}
From \reff{88} of Lemma \ref{thm_3} we have, for any given $\eps \ge 0$,
\begin{equation}
E_A(\rho_{AB}^{\otimes n};\eps)\le \max_{\eE_n}I^{A_n\to B_nZ_n}_{0,2\sqrt{\eps}}
(\sigma_{ABZ}^{\eE_n}),
\label{upbd}
\end{equation}
where the maximisation is over all possible pure state decompositions of the
satte $\rho_{AB}^{\otimes n}$.
From Lemma 14 of \cite{qcap} we have the following inequality relating the smoothed zero-coherent information to the ordinary coherent information:
\begin{align}
I^{A_n\to B_nZ_n}_{0,2\sqrt{\eps}}(\sigma_{ABZ}^{\eE_n})
\le& \frac{I^{A_n\to B_nZ_n}(\sigma_{ABZ}^{\eE_n})}{1 - \eps^{''}}\nonumber\\
& \quad +
\frac{4\bigl( \eps^{''} \log\bigl(d_A^n d_{BZ}^n\bigr)
+ 1\bigr)}{1- \eps^{''}},
\end{align}
where $\eps' = 2\sqrt{\eps}$, $ \eps^{''}= 2 \sqrt{\eps'}$, $d_A^n= {\rm{dim }}
\sH_A^{\otimes n}$ and $d_{BZ}^n= {\rm{dim }} \bigl(\sH_B^{\otimes n}\otimes
\sH_Z^{\otimes n}\bigr)$. Moreover, analogous to \reff{idf} we have
\be
I^{A_n\rightarrow B_nZ_n}(\sigma_{ABZ}^{\eE_n}) = \sum_i p_i^n S(\rho_{\phi^i}^{B_n}).
\ee
Hence,
\bea
E_A^\infty(\rho_{AB}) &\le & \lim_{n\rightarrow \infty} \frac{1}{n}\max_{\eE_n}
I^{A_n\rightarrow B_nZ_n}(\sigma_{ABZ}^{\eE_n})\nonumber\\
&=& \lim_{n\rightarrow \infty} \frac{1}{n}\max_{\eE_n}\sum_i p_i^n S(\rho_{\phi^i}^{B_n})\nonumber\\
&=& \lim_{n\rightarrow \infty} \frac{1}{n}E_A\bigl(\rho_{AB}^{\otimes n}\bigr)
\eea
\end{proof}
\section{Discussion}
In this paper we evaluated the one-shot entanglement of assistance for
an arbitrary bipartite state $\rho_{AB}$. In doing this, we proved a
result, which is of interest on its own, namely a characterization of
the one-shot distillable entanglement of a bipartite pure state. This
result turned out to be stronger than what one obtains by simply
specializing the one-shot hashing bound, obtained in~\cite{distil}, to
pure states.
Further, we showed how our one-shot result yields the operational
interpretation of the asymptotic entanglement of assistance in the
asymptotic i.i.d. scenario. In this context, an interesting open
question is to find a one-shot analogue of the result
$E_A^\infty(\rho_{AB})=\min\{S(\rho_A),S(\rho_B)\}$ proved
in~\cite{assistance}.
\section*{Acknowledgments}
FB acknowledges support from the Program for Improvement of Research
Environment for Young Researchers from Special Coordination Funds for
Promoting Science and Technology (SCF) commissioned by the Ministry of
Education, Culture, Sports, Science and Technology (MEXT) of Japan.
ND acknowledges support from the European Community's Seventh
Framework Programme (FP7/2007-2013) under grant agreement number
213681. This work was done when FB was visiting the Statistical
Laboratory of the University of Cambridge.
|
{
"timestamp": "2010-09-24T02:00:16",
"yymm": "1009",
"arxiv_id": "1009.4464",
"language": "en",
"url": "https://arxiv.org/abs/1009.4464"
}
|
\section{Introduction}
Images collected with ground based telescopes show temporal variation in image quality, primarily due to fluctuations in the index of refraction of the air along the light path. These index of refraction fluctuations are caused by differences in air temperature arising from turbulent mixing. These turbulent motions happen on smaller scales than the gross motions of the winds. Still it is known that the wind has a part in the strength of these fluctuations. Hufnagel \cite{hufnagel1974} derived a model for the vertical profile of strength of turbulence $C_n^2(z)$ which depended on a wind speed profile centered at a particular altitude. Furthermore, it has been observed that the index of refraction fluctuations are enhanced where there are shears in winds \cite{beland1993}, so that direction as well as speed can be a factor, although this is difficult to quantify \cite{dewan1993}.
Aside from the effects of wind on turbulence strength, the movement of turbulent regions past an observer induces temporal variations in seeing. If wind speeds are sufficiently high that turbulence does not evolve significantly over a measurement period, we apply Taylor's hypothesis and treat the temperature fluctuations as quantities that are advected by the wind \cite{taylor1938}.
The following wind model is often used in the modeling of adaptive optics and other imaging systems (\textit{e.g.} \cite{parenti1994, andrews1998,tyson1998}),
\begin{equation}
\label{greenwood_fit}
v(z) = 8 + 30 \exp\left(-\left[\frac{(z\sin\theta -9400)}{4800}\right]^2\right),
\end{equation}
\noindent where $v(z)$ is the wind speed in m/s, $z$ is the height in m, and $\theta$ is the angle from the zenith. Bufton \cite{bufton1973} is often cited as the source for this model (\textit{e.g.} \cite{parenti1994, andrews1998,tyson1998}), but that paper does not present an explicit wind model.
The basic form of the model was presented by Greenwood \cite{greenwood1977}. Readers often over look the statement in the body of the original paper that the height $z=0$ corresponds to a mean sea level altitude of 3048 m. This is the height of the Maui Space Surveillance System on Haleakala, where the analysis was being used. So the height of the tropopause term, 9400 m, should really be 12448 m when the model is used for sites other than Haleakala. This caveat has been overlooked in a number of references, leading to misleading results.
The model from Greenwood \cite{greenwood1977} was derived by averaging radiosonde data collected from balloons launched from Lihue on the island of Kauai from 1950-1970 and from Hilo on the island of Hawaii from 1950-1974. This was more fully detailed in a technical report, which did not explicitly generalize the equation, but the authors clearly understood what the various terms signified \cite{greenwood1975}.
The wind model is generalized as,
\begin{equation}
\label{gaussian_fit}
v(z)=v_G + v_T\exp\left(-\left[\frac{z\cos\zeta-h_T}{L_T}\right]^2\right)
\end{equation}
\noindent where $v$ is the wind speed, $z$ is the altitude, $v_G$ is the wind speed at the ground or low altitude, $v_T$ is the wind speed at tropopause, $h_T$ is the altitude of the tropopause, $L_T$ is the thickness of the tropopause layer and $\zeta$ is the zenith angle of the observation \cite{hardy1998}.
There are several uses for a wind profile. It can be used in conjunction with a turbulence model to predict the performance of instrumentation. Most commonly this is applied to the servo bandwidth of closed loop systems \cite{greenwood1977}. The minimum bandwidth of the system (to limit servo lag) is related to the Greenwood frequency. Because the Greenwood frequency depends on a product of $C_n^2$ and a power of the wind speed $v^{5/3}$, it is important to get the wind profile and turbulence profiles as a function of altitude. A wind profile can also be used to simulate turbulence in conjunction with phase screens.
Currently, there is an increased emphasis on modeling effects of turbulence on telescopes with diameters of 20-50 meters. Significantly more spatial variation of wavefront optical path difference is captured with such large telescopes in a short exposure than on a smaller telescope. Some of that variation is due to the presence of multiple layers of turbulence, moving at different speeds and in different directions. The patterns may not simply translate across the telescope aperture as implied by the Taylor hypothesis, but are more likely to``boil" with a complicated wind profile. This can have an impact on the design and performance of an adaptive optics system.
In order to better model atmospheric turbulence, we used archival radiosonde data to compute the wind profiles for a number of astronomical observatories. We also show that the wind profile has a strong seasonal variation. We also extend the wind profiles, by including wind direction. Unfortunately, the wind directions do not always lend themselves to an analytic expression, so we only show them in graphs.
Using the same radiosonde data, we examined the variation of the gradient Richardson number \cite{kundu1990}), which is an indicator of the stability of parcels of air. Since it depends upon strength of wind shear, as opposed to wind speed, it also serves to inform the modeler that wind speed alone is not sufficient. As will be seen in our plots, the wind speed can be modeled with a few simple terms, but the form of the Richardson number plots indicates the true nature of atmospheric turbulence: layered, with sharp boundaries (although these are muted by the resolution of our data).
\section{Data Analysis}\label{data_analysis}
Worldwide there are over 900 sites that launch radiosondes on a routine basis. Radiosondes are small instrument packages mounted on weather balloons. They record atmospheric pressure, temperature, potential temperature, mixing ratio, dew point, and relative humidity as a function of height. Data are taken at about 6 second intervals, but databases may only have the data recorded at 60 second intervals. Readings at certain specific standard barometric pressures are made regardless of when they occur. Since the balloons rise at about 5 m/s, the best vertical resolution would be approximately
30 meters, while the usual recorded resolution would be 300 meters and frequently the recorded data are at irregular intervals. Wind speed and wind direction are obtained from either radar observation or a navigation system such as LOng RAnge Navigation (LORAN) and newer systems may use the Global Positioning System. Almost all of these sites launch balloons at 0 UT and 12 UT. The information is used as entries to weather prediction and simulation programs.
We downloaded atmospheric sounding data for 1973 January till 2006 September for a number of stations from the University of Wyoming's Weather Web (http://weather.uwyo.edu/upperair/sounding.html). The stations were chosen based upon their proximity to an astronomical observatory. Not all the sites had data for the entire time period; the dates for the available data are also listed in the table. In addition there are gaps of varying length in each data set, which are most likely due to equipment problems. These data drop outs are not a significant problem due to the large number of data points that are available. The names of stations and their details along with the nearby observatories are listed in Table \ref{station_table}. The name of the station is given, along with its latitude, longitude and altitude, the dates when data were recorded and the names of nearby observatories. In parenthesis are the distance in km between the observatory and the radiosonde launch site and the altitude of the observatory in km.
\begin{table}[htb]
{\bf \caption{\label{station_table}Details of the radiosonde launch sites used}}
\begin{center}
\begin{tabular}{lcccclcc}
\hline
Station & Lat. & Lon. & Alt.& Dates & Nearby & Dist. & Alt.\\
& ($^\circ$) & ($^\circ$) & (m) & & Observatories& (km) & (km)\\
\hline
Antofagasta & -23.43 & -70.43 & 135 & 1973-2006 & Paranal & 133 & 2.3 \\
& & & & & Armazones &132 & 3.0 \\
Flagstaff & 35.23 & -111.82 & 2192 & 1995-2006 & Lowell &14 & 2.2 \\
Hilo & 19.71 & -155.06 & 11 & 1973-2006 & Haleakala &168 & 3.1 \\
& & & & & Mauna Kea &46 & 4.4\\
Oakland & 37.72 & -122.20 & 3 & 1973-2006 & Lick &65 & 1.3 \\
San Diego & 32.84 & -117.11 & 128 & 1990-2006 & Palomar &61 & 1.7 \\
Tenerife & 28.47 & -16.38 & 105 & 2002-2006 & R. Muchachos &150 & 2.3 \\
& & & & & Teide &23 & 2.4 \\
Tucson & 32.11 & -110.93 & 779 & 1973-2006 & Kitt Peak &64 & 2.1 \\
& & & & & Mt. Graham &117 & 3.1\\
\hline
\end{tabular}
\end{center}
\end{table}
For each sounding the pressure, temperature, dew point, relative humidity, wind speed and direction are measured as a function of height. The heights are irregularly measured and not repeated, so the data were interpolated onto a fixed height grid with an increment of 400 m and the mean was computed for various time periods. The height grid is measured from mean sea level, rather than the local surface. The different soundings have data measured to varying maximum altitudes, depending on when the balloons burst. We did not analyze any data from altitudes higher than 30 km, since there were too few data points for a meaningful analysis.
Fig. \ref{speed_variations} shows the yearly variations that occur for January and July in Hilo, Hawaii. Plotted over this are the mean wind speeds for the respective month for the entire 33 year period of data. This shows the variation in monthly mean wind speeds. Some of this difference is caused by large scale weather patterns and some is the natural chaotic behavior of weather. The overall shape of the wind profiles is highly consistent over the 33 years of data.
\begin{figure}[htbp]
\centering\includegraphics[height=4.5cm]{roberts_fig1a.eps}
\includegraphics[height=4.5cm]{roberts_fig1b.eps}
\caption{\label{speed_variations} The solid line in the figure on the left is the model fit to the mean monthly wind speed for Hilo, Hawaii in the month of January. The dotted lines are the mean wind speed for Hilo, Hawaii in each January from 1973-2006. The figure on the right is the same, but for July. These figures illustrate the variation in the monthly mean wind speed.}
\end{figure}
In addition to the sites listed in Table \ref{station_table}, we also downloaded the wind profiles for Lihue HI. As mentioned above this was used in Greenwood's original model. We computed the correlation coefficient between the Hilo and Lihue data sets for each month of the mean wind profiles. The correlation was highest in February (0.999) and lowest in August (0.981). The average correlation was 0.992 with a standard deviation of 0.006. This correlation was high enough that we felt it was valid to only use the Hilo data. This does not say that Hilo and Lihue constantly experience the exact same wind conditions, but that on average the wind profiles are very similar. This also shows that it is valid to use the same wind model for the entire state of Hawaii including the observatories on Haleakala and Mauna Kea.
Surface winds are highly dependent on local geography, such as being in the lee of a mountain, and are also time dependent due to on-shore and off-shore breezes for the island sites and katabatic flows for sites near mountains. They are also dependent on the solar heating of the ground. This is less of a problem for astronomical uses of the wind profiles since mountain top observatories are usually situated above these surface effects. In addition, the wind profiles do not capture the effect that the local orography will have on the wind profile.
We created the mean wind speed and direction for each of the 12 months for each of the sites listed in Table \ref{station_table}. We used a non-linear least squares fit algorithm to fit the data to
\begin{equation}
\label{gaussian_fit_eqn}
v(z) = A_0 + A_1 \exp\left(-\left[\frac{(z-A_2)}{A_3}\right]^2\right).
\end{equation}
\noindent Tables \ref{hilo_coeffs}-\ref{anto_coeffs} list the monthly coefficients and their associated error bars for all sites listed in Table \ref{station_table}. The error bars are computed by the non-linear least squares fit. The errors bars increase as the winds vary from the pure Gaussian model.
We fit the Gaussian to the wind speed data from the ground to a height of 16 km, beyond this the stratospheric winds start to affect the fit. The effect is marginal and often below the error bars, except for the summer data sets.
The Gaussian wind model of \cite{greenwood1977} (Eq. \ref{greenwood_fit}) only fit the troposphere winds and ignored the stratospheric winds. It is possible to fit both of those winds if a Gaussian is summed with a second order polynomial. This has limited utility to astronomical imaging, as the turbulence above 20km is relatively weak, but high altitude airships are being designed to operate between 20 and 30 km \cite{fesen2006} and require a knowledge of the wind speeds and directions in this region.
\begin{table}[htbp]
{\bf \caption{\label{hilo_coeffs}Hawaii Wind Speed Model Coefficients}}
\begin{center}
\begin{tabular}{lcccc}
\hline
Month& A0 & A1 & A2 & A3 \\
\hline
Jan. & 4.1 $\pm$ 0.4 & 25.8 $\pm$ 0.5 & 12007 $\pm$ 69 & 4047 $\pm$ 115\\
Feb. & 4.2 $\pm$ 0.4 & 27.6 $\pm$ 0.4 & 11952 $\pm$ 61 & 3957 $\pm$ 99\\
Mar. & 5.0 $\pm$ 0.3 & 29.6 $\pm$ 0.4 & 12335 $\pm$ 55 & 3405 $\pm$ 79\\
Apr. & 5.0 $\pm$ 0.3 & 29.4 $\pm$ 0.4 & 12776 $\pm$ 51 & 3001 $\pm$ 68\\
May. & 4.8 $\pm$ 0.4 & 22.9 $\pm$ 0.5 & 12909 $\pm$ 88 & 3161 $\pm$ 117\\
Jun. & 4.8 $\pm$ 0.3 & 17.0 $\pm$ 0.5 & 12637 $\pm$ 95 & 2877 $\pm$ 127\\
Jul. & 5.0 $\pm$ 0.3 & 14.0 $\pm$ 0.5 & 12406 $\pm$ 102 & 2605 $\pm$ 133\\
Aug. & 4.6 $\pm$ 0.3 & 12.8 $\pm$ 0.5 & 12527 $\pm$ 99 & 2575 $\pm$ 128\\
Sep. & 4.2 $\pm$ 0.3 & 13.6 $\pm$ 0.4 & 12263 $\pm$ 93 & 2868 $\pm$ 126\\
Oct. & 4.5 $\pm$ 0.3 & 17.2 $\pm$ 0.4 & 12222 $\pm$ 75 & 2845 $\pm$ 101\\
Nov. & 5.1 $\pm$ 0.3 & 17.4 $\pm$ 0.5 & 12313 $\pm$ 92 & 3214 $\pm$ 129\\
Dec. & 5.0 $\pm$ 0.4 & 21.1 $\pm$ 0.5 & 12199 $\pm$ 84 & 3617 $\pm$ 126\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
{\bf \caption{\label{oakland_coeffs}Oakland Wind Speed Model Coefficients}}
\begin{center}
\begin{tabular}{lcccc}
\hline
Month& A0 & A1 & A2 & A3\\
\hline
Jan. & -1.4 $\pm$ 1.8 & 30.5 $\pm$ 1.7 & 10730 $\pm$ 81 & 8329 $\pm$ 318\\
Feb. & -0.7 $\pm$ 1.6 & 30.0 $\pm$ 1.5 & 10695 $\pm$ 77 & 8047 $\pm$ 283\\
Mar. & -0.9 $\pm$ 1.4 & 29.5 $\pm$ 1.3 & 10548 $\pm$ 66 & 7878 $\pm$ 245\\
Apr. & 0.6 $\pm$ 1.0 & 28.3 $\pm$ 1.0 & 10472 $\pm$ 59 & 7195 $\pm$ 188\\
May. & 2.8 $\pm$ 0.7 & 22.8 $\pm$ 0.7 & 10649 $\pm$ 70 & 6367 $\pm$ 170\\
Jun. & 3.1 $\pm$ 0.7 & 19.7 $\pm$ 0.7 & 11164 $\pm$ 98 & 6430 $\pm$ 215\\
Jul. & 4.0 $\pm$ 0.6 & 16.5 $\pm$ 0.6 & 11581 $\pm$ 122 & 5540 $\pm$ 210\\
Aug. & 4.4 $\pm$ 0.5 & 17.3 $\pm$ 0.6 & 11602 $\pm$ 113 & 5187 $\pm$ 184\\
Sep. & 3.2 $\pm$ 0.6 & 18.8 $\pm$ 0.6 & 11400 $\pm$ 95 & 6240 $\pm$ 192\\
Oct. & 1.3 $\pm$ 0.8 & 22.3 $\pm$ 0.8 & 11073 $\pm$ 79 & 7297 $\pm$ 217\\
Nov. & -2.0 $\pm$ 1.6 & 32.8 $\pm$ 1.5 & 10878 $\pm$ 72 & 8251 $\pm$ 263\\
Dec. & -1.8 $\pm$ 2.0 & 31.1 $\pm$ 1.8 & 10611 $\pm$ 81 & 8358 $\pm$ 333\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
{\bf \caption{\label{sandiego_coeffs}San Diego Wind Speed Model Coefficients}}
\begin{center}
\begin{tabular}{lcccc}
\hline
Month& A0 & A1 & A2 & A3\\
\hline
Jan. & -2.9 $\pm$ 1.4 & 33.9 $\pm$ 1.3 & 11470 $\pm$ 76 & 6104 $\pm$ 249\\
Feb. & 1.4 $\pm$ 1.1 & 32.4 $\pm$ 1.0 & 11400 $\pm$ 81 & 5081 $\pm$ 199\\
Mar. & -1.3 $\pm$ 1.2 & 32.7 $\pm$ 1.1 & 11701 $\pm$ 82 & 5793 $\pm$ 226\\
Apr. & -2.4 $\pm$ 1.4 & 32.7 $\pm$ 1.3 & 11656 $\pm$ 88 & 6062 $\pm$ 266\\
May. & 0.2 $\pm$ 1.2 & 25.6 $\pm$ 1.1 & 11577 $\pm$ 113 & 5471 $\pm$ 295\\
Jun. & 3.6 $\pm$ 0.9 & 17.6 $\pm$ 0.9 & 12081 $\pm$ 201 & 4546 $\pm$ 364\\
Jul. & 3.7 $\pm$ 0.6 & 10.6 $\pm$ 0.6 & 11483 $\pm$ 180 & 3909 $\pm$ 317\\
Aug. & 4.4 $\pm$ 0.4 & 10.5 $\pm$ 0.5 & 11821 $\pm$ 161 & 3318 $\pm$ 239\\
Sep. & 4.4 $\pm$ 0.6 & 18.8 $\pm$ 0.6 & 11897 $\pm$ 121 & 3745 $\pm$ 193\\
Oct. & 2.2 $\pm$ 0.8 & 22.7 $\pm$ 0.7 & 11805 $\pm$ 111 & 4843 $\pm$ 230\\
Nov. & -1.6 $\pm$ 1.3 & 30.6 $\pm$ 1.2 & 11511 $\pm$ 88 & 5776 $\pm$ 256\\
Dec. & -1.3 $\pm$ 1.2 & 32.2 $\pm$ 1.1 & 11354 $\pm$ 76 & 5639 $\pm$ 223\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
{\bf \caption{\label{tucson_coeffs}Tucson Wind Speed Model Coefficients}}
\begin{center}
\begin{tabular}{lcccc}
\hline
Month & A0 & A1 & A2 & A3\\
\hline
Jan. & -1.1 $\pm$ 0.7 & 34.1 $\pm$ 0.6 & 11467 $\pm$ 34 & 5549 $\pm$ 112\\
Feb. & 1.6 $\pm$ 0.6 & 34.1 $\pm$ 0.6 & 11620 $\pm$ 40 & 4993 $\pm$ 104\\
Mar. & 1.4 $\pm$ 0.7 & 34.0 $\pm$ 0.6 & 11779 $\pm$ 46 & 5163 $\pm$ 120\\
Apr. & 2.6 $\pm$ 0.8 & 29.1 $\pm$ 0.7 & 11804 $\pm$ 63 & 5028 $\pm$ 155\\
May. & 4.6 $\pm$ 0.7 & 21.2 $\pm$ 0.6 & 11729 $\pm$ 85 & 4465 $\pm$ 181\\
Jun. & 6.5 $\pm$ 0.4 & 14.7 $\pm$ 0.5 & 12107 $\pm$ 109 & 3358 $\pm$ 167\\
Jul. & 5.4 $\pm$ 0.2 & 6.4 $\pm$ 0.3 & 11852 $\pm$ 128 & 2933 $\pm$ 184\\
Aug. & 5.0 $\pm$ 0.2 & 7.8 $\pm$ 0.3 & 11955 $\pm$ 101 & 2779 $\pm$ 141\\
Sep. & 5.7 $\pm$ 0.3 & 16.3 $\pm$ 0.4 & 12076 $\pm$ 80 & 3282 $\pm$ 120\\
Oct. & 4.8 $\pm$ 0.5 & 22.4 $\pm$ 0.5 & 11889 $\pm$ 76 & 4158 $\pm$ 144\\
Nov. & 0.5 $\pm$ 0.8 & 30.1 $\pm$ 0.8 & 11453 $\pm$ 50 & 5297 $\pm$ 152\\
Dec. & 0.0 $\pm$ 0.7 & 32.3 $\pm$ 0.7 & 11462 $\pm$ 41 & 5295 $\pm$ 123\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
{\bf \caption{\label{flagstaff_coeffs}Flagstaff Wind Speed Model Coefficients}}
\begin{center}
\begin{tabular}{lcccc}
\hline
Month& A0 & A1 & A2 & A3\\
\hline
Jan. & -8.1 $\pm$ 4.0 & 36.8 $\pm$ 3.9 & 11486 $\pm$ 63 & 6939 $\pm$ 528\\
Feb. & 1.3 $\pm$ 1.6 & 29.2 $\pm$ 1.5 & 11404 $\pm$ 55 & 5210 $\pm$ 244\\
Mar. & -0.6 $\pm$ 0.9 & 27.0 $\pm$ 0.9 & 11487 $\pm$ 29 & 5702 $\pm$ 155\\
Apr. & 2.9 $\pm$ 0.8 & 24.6 $\pm$ 0.8 & 11280 $\pm$ 34 & 5106 $\pm$ 151\\
May. & 6.3 $\pm$ 0.7 & 17.7 $\pm$ 0.7 & 11569 $\pm$ 68 & 4266 $\pm$ 192\\
Jun. & 8.7 $\pm$ 0.3 & 11.8 $\pm$ 0.4 & 12084 $\pm$ 80 & 2865 $\pm$ 123\\
Jul. & 6.0 $\pm$ 0.2 & 7.7 $\pm$ 0.2 & 11847 $\pm$ 68 & 2606 $\pm$ 101\\
Aug. & 5.7 $\pm$ 0.2 & 10.3 $\pm$ 0.3 & 11967 $\pm$ 61 & 2583 $\pm$ 88\\
Sep. & 8.0 $\pm$ 0.4 & 16.2 $\pm$ 0.5 & 12097 $\pm$ 82 & 3133 $\pm$ 137\\
Oct. & 4.5 $\pm$ 0.8 & 19.8 $\pm$ 0.7 & 11669 $\pm$ 59 & 4587 $\pm$ 183\\
Nov. & -0.2 $\pm$ 1.7 & 29.0 $\pm$ 1.6 & 11443 $\pm$ 54 & 5484 $\pm$ 266\\
Dec. & -2.9 $\pm$ 2.8 & 31.7 $\pm$ 2.7 & 11246 $\pm$ 61 & 5969 $\pm$ 398\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
{\bf \caption{\label{tenerife_coeffs}Tenerife Wind Speed Model Coefficients}}
\begin{center}
\begin{tabular}{lcccc}
\hline
Month& A0 & A1 & A2 & A3\\
\hline
Jan. & 2.5 $\pm$ 0.9 & 20.3 $\pm$ 0.9 & 12371 $\pm$ 159 & 7964 $\pm$ 341\\
Feb. & 4.0 $\pm$ 0.6 & 29.7 $\pm$ 0.6 & 12102 $\pm$ 82 & 6478 $\pm$ 149\\
Mar. & 1.7 $\pm$ 1.1 & 27.5 $\pm$ 1.0 & 12847 $\pm$ 173 & 8041 $\pm$ 325\\
Apr. & 3.0 $\pm$ 0.9 & 30.6 $\pm$ 0.8 & 12332 $\pm$ 120 & 6784 $\pm$ 216\\
May. & 4.6 $\pm$ 0.5 & 24.9 $\pm$ 0.5 & 12294 $\pm$ 92 & 5646 $\pm$ 143\\
Jun. & 4.9 $\pm$ 0.8 & 14.9 $\pm$ 0.8 & 12711 $\pm$ 282 & 5899 $\pm$ 420\\
Jul. & 6.7 $\pm$ 0.4 & 8.1 $\pm$ 0.6 & 12536 $\pm$ 232 & 3760 $\pm$ 304\\
Aug. & 6.3 $\pm$ 0.4 & 8.3 $\pm$ 0.7 & 12137 $\pm$ 226 & 3630 $\pm$ 296\\
Sep. & 6.0 $\pm$ 0.4 & 11.5 $\pm$ 0.6 & 12137 $\pm$ 172 & 4341 $\pm$ 241\\
Oct. & -0.2 $\pm$ 2.4 & 22.9 $\pm$ 2.2 & 11545 $\pm$ 218 & 8465 $\pm$ 661\\
Nov. & 4.5 $\pm$ 0.7 & 16.2 $\pm$ 0.7 & 12313 $\pm$ 191 & 6579 $\pm$ 336\\
Dec. & 5.6 $\pm$ 0.7 & 20.9 $\pm$ 0.7 & 12074 $\pm$ 139 & 6033 $\pm$ 238\\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[htbp]
{\bf \caption{\label{anto_coeffs}Antofagasta Wind Speed Model Coefficients}}
\begin{center}
\begin{tabular}{lcccc}
\hline
Month& A0 & A1 & A2 & A3\\
\hline
Jan. & 3.8 $\pm$ 0.3 & 13.4 $\pm$ 0.3 & 12169 $\pm$ 92 & 3357 $\pm$ 134\\
Feb. & 3.5 $\pm$ 0.3 & 14.1 $\pm$ 0.3 & 12090 $\pm$ 86 & 3598 $\pm$ 131\\
Mar. & 4.0 $\pm$ 0.3 & 17.3 $\pm$ 0.4 & 12186 $\pm$ 91 & 3540 $\pm$ 135\\
Apr. & 4.0 $\pm$ 0.4 & 25.3 $\pm$ 0.5 & 12318 $\pm$ 75 & 3727 $\pm$ 112\\
May. & 2.7 $\pm$ 0.4 & 29.7 $\pm$ 0.4 & 12419 $\pm$ 68 & 4367 $\pm$ 111\\
Jun. & 3.0 $\pm$ 0.5 & 30.5 $\pm$ 0.5 & 12179 $\pm$ 66 & 4298 $\pm$ 110\\
Jul. & 3.1 $\pm$ 0.6 & 32.5 $\pm$ 0.6 & 11797 $\pm$ 65 & 4148 $\pm$ 114\\
Aug. & 2.7 $\pm$ 0.5 & 31.1 $\pm$ 0.5 & 11671 $\pm$ 53 & 4274 $\pm$ 98\\
Sep. & 3.4 $\pm$ 0.5 & 32.1 $\pm$ 0.5 & 11785 $\pm$ 59 & 3948 $\pm$ 98\\
Oct. & 3.7 $\pm$ 0.4 & 29.1 $\pm$ 0.4 & 12052 $\pm$ 55 & 3785 $\pm$ 86\\
Nov. & 4.0 $\pm$ 0.3 & 26.0 $\pm$ 0.4 & 12192 $\pm$ 60 & 3394 $\pm$ 88\\
Dec. & 4.0 $\pm$ 0.3 & 21.1 $\pm$ 0.4 & 12240 $\pm$ 65 & 3131 $\pm$ 91\\
\hline
\end{tabular}
\end{center}
\end{table}
The results are shown in Figs. \ref{hilo_wind}--\ref{anto_wind} respectively. Each figure is composed of 12 graphs corresponding to the 12 months of the year. The first letter of the corresponding month is in the upper right corner. Each graph has the the wind speed plotted with a solid line and the fitted wind speed computed from the coefficients in Tables \ref{hilo_coeffs}--\ref{anto_coeffs} is plotted as a dashed line. The wind direction is plotted as a solid red line. The wind speed is marked along the bottom of the graph, while the wind direction is marked on the top of the graph. The wind directions often show sharp spikes at the edges of jets. Those are an artificial result of the averaging of directions and can be ignored.
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig2.eps}
\caption{\label{hilo_wind} The wind speed and direction for Hilo, Hawaii. Each of the graphs in this figure has the the measured wind speed as a function of height plotted with a solid black line and the wind speed computed from the model coefficients in Table \ref{hilo_coeffs} is shown as a black dashed line. The wind direction is plotted as a solid red line. The wind speed is marked along the bottom of the graph, and the wind direction is marked on the top of the graph. The first letter of the corresponding month is in the upper right corner. }
\end{figure}
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig3.eps}
\caption{ \label{oakland_wind} The wind speed and direction for Oakland, California. The layout of the figure is the same as in Fig. \ref{hilo_wind}, except the fitted wind model comes from Table \ref{oakland_coeffs}. }
\end{figure}
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig4.eps}
\caption{\label{sandiego_wind} The wind speed and direction for San Diego, California. The layout of the figure is the same as in Fig. \ref{hilo_wind}, except the fitted wind model comes from Table \ref{sandiego_coeffs}.}
\end{figure}
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig5.eps}
\caption{\label{tucson_wind} The wind speed and direction for Tucson, Arizona. The
layout of the figure is the same as in Fig. \ref{hilo_wind}, except the fitted wind model comes from Table \ref{tucson_coeffs}. }
\end{figure}
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig6.eps}
\caption{\label{flagstaff_wind} The wind speed and direction for Flagstaff, Arizona.
The layout of the figure is the same as in Fig. \ref{hilo_wind}, except the fitted wind model comes from Table \ref{flagstaff_coeffs}. }
\end{figure}
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig7.eps}
\caption{\label{tenerife_wind} The wind speed and direction for Tenerife, Canary Islands, Spain, which is useful for the observatories located in the Canary Islands. The layout of the figure is the same as in Fig. \ref{hilo_wind}, except the fitted wind model comes from Table \ref{tenerife_coeffs}. }
\end{figure}
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig8.eps}
\caption{\label{anto_wind} The wind speed and direction for Antofagasta, Chile. The layout of the figure is the same as in Fig. \ref{hilo_wind}, except the fitted wind model comes from Table \ref{anto_coeffs}. }
\end{figure}
For both hemispheres, the tropospheric winds increase in the winter months and decrease in the summer months. Stratospheric winds increase in the summer months, the opposite of the tropospheric winds. During the summer months, Tenerife has a low level ($\approx5$ km) jet form. This is most obvious in July and August, but first appears in June and persists to October. In the Greenwood model, $A_0$ is the low altitude wind speed and that is true for our model for Hawaii. It is not always true for other sites, as these values are sometimes negative. The reason for this is that the wind profiles for those sites has a slightly different shape than the Hawaiian wind profile. This can be seen in Figs. \ref{oakland_wind} and\ref{flagstaff_wind}.
Greenwood \cite{greenwood1977} only provides a model for the wind speed, though \cite{greenwood1975} does show the annual mean wind direction for Hilo and Lihue. As shown by Figs. \ref{hilo_wind}-\ref{anto_wind} the wind direction is not an easily modeled function. It can most easily be modeled as a square wave. Since stability of flows does depend upon the amount of shear present, an analysis that computed the wind shear at the upper and lower boundaries would tend to overestimate the instability. Looking at Figs. \ref{hilo_wind}--\ref{anto_wind}, the wind direction does not change instantaneously, but it does change very quickly, often changing direction by $180^\circ$ in a km. Some regions of slowly changing direction, most commonly above 25 km, are not very well modeled by a square wave. A square wave model will suffice for modeling where phase screens at only a few altitudes are used to model the distribution of turbulence.
Note that the reversal of wind direction at altitudes above 20 km is a well known feature of the zonal wind flow. It is due to a higher altitude jet whose center is north of the equator in winter, and south of the equator in summer. There are also northern and southern jets at higher altitudes which move in the same direction as the jets in the lower stratosphere. These are seasonal and impact the wind speed and directions in our data \cite{newell1972}.
\section{Richardson Number}\label{richardson}
The gradient Richardson number is a dimensionless ratio, $R_i$, related to the buoyant production or consumption of turbulence divided by the shear production of turbulence \cite{kundu1990}. It is defined as
\eq
\label{eq_richardson}
R_i = \frac{g\frac{\partial\ln\Theta}{\partial~z}}{\left (\frac{\partial u}{\partial z}\right)^2 + \left(\frac{\partial v}{\partial z}\right)^2},
\en
where $g$ is the acceleration of gravity, $\Theta$ is the potential temperature, $z$ is the altitude, $u$ and $v$ are the horizontal components of the wind vector. Normally a gradient Richardson number with a value less than 0.25 is considered to be turbulent and a value greater than 0.25 to be non-turbulent. It is used to indicate dynamic stability and the formation of turbulence.
In order to see at which altitudes turbulence forms, we used data from the radiosondes to compute the monthly binned Richardson number for each site in Table \ref{station_table} using Eq. \ref{eq_richardson} at each of the fixed grid points used in \S \ref{data_analysis}. The grid heights points are every 400m and some of the finer atmospheric layers may not be captured with our analysis. The derivatives were computed using a three-point Lagrangian interpolation routine.
The computed Richardson numbers were then binned into two bins: $R_i < 0.25$, and $R_i \geq 0.25$.
For a given site, at each altitude the numbers in each bin were normalized by the total number of data points at that altitude. This gives the Richardson number as a function of altitude. These are shown in Figs. \ref{hilo_rich}-\ref{anto_rich}. As expected, the Richardson numbers above 20 km are very high and have little impact on astronomical seeing; as a result we only plotted the values below 20 km. It is important to note that these values are for radiosondes launched from sites at lower altitudes than astronomical observatories. The highest Richardson number values occur at these low altitudes and are not a concern to most astronomical observatories.
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig9.eps}
\caption{\label{hilo_rich}
The binned Richardson number for Hilo, Hawaii The figure shows the fraction of Richardson numbers falling into two bins. Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \ge 0.25$).}
\end{figure}
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig10.eps}
\caption{\label{oakland_rich} The binned Richardson number for Oakland, California. Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \ge 0.25$).}
\end{figure}
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig11.eps}
\caption{\label{sandiego_rich} The binned Richardson number for San Diego, California. Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \ge 0.25$). }
\end{figure}
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig12.eps}
\caption{ \label{tucson_rich} The binned Richardson number for Tucson, Arizona. Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \ge 0.25$). }
\end{figure}
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig13.eps}
\caption{ \label{flagstaff_rich} The binned Richardson number for Flagstaff, Arizona. Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \ge 0.25$). }
\end{figure}
\
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig14.eps}
\caption{\label{tenerife_rich} The binned Richardson number for Tenerife, Canary Islands, Spain. Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \ge 0.25$). }
\end{figure}
\begin{figure}[htbp]
\centering\includegraphics[height=7.5cm]{roberts_fig15.eps}
\caption{\label{anto_rich} The binned Richardson number for Antofagasta, Chile. Yellow signifies turbulent conditions ($R_i < 0.25$) and blue signifies non-turbulent conditions ($R_i \ge 0.25$).}
\end{figure}
Examining the individual months reveals that the figures are not as smooth as the wind speed figures, but instead have many fine layers. The layers are significant, even if they do not seem to explicitly show turbulence (ie their value of $R_i$ is > 0.25). Because the $R_i$ calculation has a height increment of 400m, and because the precise positions and altitudes of jets vary over periods of a day or less, the effective values of the velocity derivatives in Equation \ref{eq_richardson} are lower in our results than they might be at any given time. This averaging leads to relatively large calculated $R_i$ values. It also means that our data will not show many instances of $R_i < 0.25$. But where is does occur it is significant. Moreover, any drop in the plotted $R_i$ number indicates a persistent region of instability, and it is likely that some turbulence is present there for extended periods,
Since low gradient Richardson numbers indicate the presence of atmospheric turbulence, they can be an indicator of seeing. In Fig. \ref{hilo_rich}, the winter months for Hawaii have the highest proportion of Richardson numbers lower than 0.25 (yellow values), and the summer months have the lowest. In other words the winter months have more turbulent conditions than the summer months. Studies of seeing from Mauna Kea show that this seasonal behavior of seeing correlates with this very well \cite{bely1987,seigar2002, skidmore2009}. There is relatively little seasonal variation in for Antofagasta, Chile (Fig. \ref{anto_rich}) and this correlates well with seeing measurements carried at nearby Cerro Armazones \cite{skidmore2009} which show no clear evidence for any seasonal dependency. The trend in Fig. \ref{tenerife_rich} agrees with seeing measurements for La Palma, Canary Islands, showing that seeing is best May-September \cite{munoz1998, wilson1999}. Flagstaff, Arizona (Fig. \ref{flagstaff_rich} has more turbulent values than Oakland, California, which is agreement with site comparison done between Anderson Mesa (outside of Flagstaff) and Lick Observatory (near Oakland) \cite{walters1997}. This shows that Richardson numbers computed from radiosonde data can be used to differentiate between astronomical sites at a macro level which can be of use for narrowing site selection for new telescopes. It is important to remember that the Richardson numbers computed from radiosonde data do not include local effects such as the turbulent surface boundary layer that can have a profound impact on seeing. Small differences in altitudes and location can alter the observed seeing significantly \cite{walters1997}.
While the Richardson number does not indicate the strength of turbulence, knowing that there are persistent turbulent layers may be useful to site planners. For example, knowing roughly where and when one might expect upper layers of turbulence to be present could be useful for measurements, such as laser communications, which are strongly influenced by scintillation. On the other hand, we have seen that some layers closer to the ground have $R_i$<0.25. These must also be persistent to show up over the averaging. Knowing that these layers exist is useful for large telescopes
that plan to compensate for near ground effects.
\section{Other Wind Models}\label{theothers}
There have been a few other studies of wind speed for different sites, though they have been done for only a limited number of sites and with different methodology. Sarazin \& Tokovinin \cite{sarazin2002} suggested that the wind speed at the 200 mbar pressure isobar (\vtwo)~is a good indication of atmospheric turbulence at a given site (\vtwo~is well monitored for meteorology). The advantage of this is that there are models of \vtwo~for the entire planet, while there are only a fixed number of radiosondes launched each day. Several groups have used this idea to characterize and compare several astronomical sites \cite{chueca2004, carrasco2005, garcia-lorenzo2005} including several sites which are included in this study. During the year, the altitude at which \vtwo~occurs varies, this makes it difficult to use the values in Eq. \ref{greenwood_fit}. It is useful for site comparisons, but it supplements rather than replaces the knowledge of the vertical wind profile.
Garc\'{i}a-Lorenzo et al. \cite{garcia-lorenzo2005} studied \vtwo~for Paranal and La Silla in Chile, San Pedro M\'{a}rtir in Mexico, Mauna Kea in Hawaii and La Palma in the Canary Islands. The last two observatories are also characterized in this paper. We show a seasonal variation in $A_1$ in Tables \ref{hilo_coeffs} and \ref{tenerife_coeffs} which has the highest tropospheric wind speed in the spring and the lowest tropospheric winds in the summer. Garc\'{i}a-Lorenzo et al. showed that the \vtwo wind speeds at Mauna Kea and La Palma have this same seasonal behavior.
\section{Conclusions}
We have presented upper-level wind models computed from archival radiosonde data suitable for the major astronomical observatories in the United States, Chile and the Canary Islands. We find that the commonly used Greenwood model is not suitable for sites other than Hawaii. In addition it produced a single wind profile for the entire year, when it is more correct to use different wind models over the course of the year. We find that, as may be expected from geophysical fluid dynamics, that sites with similar latitudes share similar wind profiles. In addition we have computed Richardson number profiles from the same datasets. Those results indicate the presence of turbulence at different altitudes with seasonal variations that seem to agree with variations in seeing. These models and results will be of use to modelers of atmospheric turbulence and instrument developers.
\section*{Acknowledgments}
This research was funded by the Air Force Office of Scientific Research and by the Air Force Research Laboratory's Directed Energy Directorate. A portion of the research in this paper was carried out at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. Radiosonde data were obtained from the Wyoming Weather Web, maintained by the Department of Atmospheric Science of the University of Wyoming.
\end{document}
|
{
"timestamp": "2010-09-21T02:04:06",
"yymm": "1009",
"arxiv_id": "1009.3902",
"language": "en",
"url": "https://arxiv.org/abs/1009.3902"
}
|
\section{Introduction}
Let $P$ be a $K$-algebra and let $\Sigma$ be a monoid of endomorphisms
of $P$. If $I$ is an ideal of $P$ which is invariant under the maps
in $\Sigma$ then it is possible to codify the action of $P$ and $\Sigma$
over $I$ as a single left module structure with respect to the skew monoid
(or semigroup) ring $S = P * \Sigma$. The study of some properties of $I$,
as for instance its finite $\Sigma$-generation, can be reduced hence
to that of general properties of the operators ring $S$ as its
Noetherianity (see \cite{MCR,La}). Ideals which are stable under the action
of monoids of endomorphisms or groups of automorphisms are natural
in many contexts as the representation theory (a classical reference
is \cite{DCEP}), or in the study of PI-algebras \cite{Dr,GZ} where
$P$ is the free associative algebra and $\Sigma$ the complete
monoid of endomorphisms of $P$. Another context of relevant interest
is the study of so-called ``difference ideals'' \cite{Le} which are
ideals invariant under shift operators in applications to combinatorics,
(nonlinear) differential and difference equations. For the viewpoint
of computing in the ring of (differential-difference) operators
an important contribution is \cite{MS}.
To control in an effective way the structure of the left $S$-module $P/I$
one generally needs to compute a $K$-basis of it. If $P$ is a ring
of polynomials in commutive or non-commutative variables and one fixes
a suitable ordering for the monomials of $P$, then a $K$-linear basis
of monomials for $P/I$ can be obtained by using the elements of a
suitable generating set of $I$ as rewriting rules. Such generating set
is usually called a ``Gr\"obner\ basis'' of $I$. Since $I$ is a $\Sigma$-invariant
ideal, it is natural to consider $\Sigma$-bases of $I$ that is sets
$G\subset I$ such that $I$ is the smallest $\Sigma$-ideal of $P$
containing $G$. In other words, $G$ is a generating set of $I$ as left
$S$-module. It follows that one has to harmonize the notion of Gr\"obner\ basis
with that of $\Sigma$-basis and attempts in this direction can be found
for instance in \cite{AH,BD} and also in \cite{DLS,LSL}. If the elements
of $\Sigma$ are automorphisms, the main obstacle in the definition
of a Gr\"obner\ $\Sigma$-basis is that their action on $P$ does not preserve
the monomial ordering. Then, it has been shown in \cite{BD} and before
in \cite{LSL} that an appropriate setting to define Gr\"obner\ $\Sigma$-bases
is that of a commutative polynomial ring $P = K[X]$ in an infinite
number of variables and a monoid $\Sigma$ of monomial monomorphisms
of infinite order which are compatible with the ordering and divisibility
of monomials of $P$.
In this paper we propose a systematic study of the case when $\Sigma$
is generated by a single map $\sigma$. In this case the skew monoid ring
$S$ coincides with the skew polynomial ring $P[s;\sigma]$ which is
an Ore extension where $\sigma$-derivation is zero. The approach
we follow is to consider an abstract map $\sigma$ satisfying compatibility
conditions able to provide a ``natural'' Gr\"obner\ bases theory. Note that
this generalizes in particular the results contained in \cite{We}
where the map $\sigma:x_i\mapsto x_i^e$ with $e > 1$ has been studied.
We choose to consider a single endomorphism essentially because
a major application of our theory is the unification, in the graded case,
of the Gr\"obner\ bases theory for non-commutative polynomials introduced
in \cite{Gr1,Mo,Uf} with the classical commutative theory based on
the notion of S-polynomial (see for instance \cite{GP}).
In Section 6 we show in fact that there exists an algebra embedding
$\iota:{K\langle X \rangle}\to S$ where ${K\langle X \rangle}$ is the free associative algebra generated
by the variables $x_i$ and $S$ is the skew polynomial ring defined by
the algebra $P$ of commutative polynomials in double indexed variables
$x_i(j)$ and the endomorphism $\sigma:P\to P$ such that
$x_i(j)\mapsto x_i(j+1)$, for all $i,j$. This algebra embedding is a
significant improvement of the linear map $\iota':{K\langle X \rangle} \to P$ defined as
$x_{i_1}\cdots x_{i_d}\mapsto x_{i_1}(1)\cdots x_{i_d}(d)$ and
introduced by \cite{Fe,DRS} for the aims of physics and
invariant/representation theory. In fact, the use of the map $\iota$
clarifies the phenomenon found in \cite{LSL} of a bijective correspondence
between all graded two-sided ideals of ${K\langle X \rangle}$ and some class of
$\Sigma$-invariant ideals of $P$. Note that in the same paper,
a competitive new algorithm for non-commutative homogeneous Gr\"obner\ bases
based on this correspondence has been implemented and experimented
in \textsc{Singular} \cite{DGPS}.
In Section 2 one finds a brief account of the equivalence between
the notion of $\Sigma$-invariant $P$-module and that of left $S$-module,
together with the description of some properties of the generating sets
of graded two-sided ideals of $S = \bigoplus_i S_i$ with $S_i = P s^i$.
A Gr\"obner\ basis theory for such ideals is introduced in Section 3 where
we assume $P = K[x_0,x_1,\ldots], \Sigma = \langle \sigma \rangle$ and
$\sigma:P\to P$ be a monomorphism of infinite order sending monomials into
monomials. Additional assuptions for $\sigma$ are that
$\gcd(\sigma(x_i),\sigma(x_j)) = 1$ for $i\neq j$ and the monomial ordering
of $P$ is such that $m\prec n$ implies that $\sigma(m)\prec \sigma(n)$,
for all monomials $m,n$. Such conditions are quite natural in many
contexts as the shift operators defining difference ideals \cite{Le}
or the maps used in \cite{BD}. Note that the algorithms we introduce
for the computation of homogeneous Gr\"obner\ bases in $S$ are based on
the free $P$-module structure of this ring and hence they appear as
a variant of the classical module Buchberger algorithm where the number
of S-polynomials to be considered is reduced owing to the symmetry
defined by $\Sigma$ on the graded ideals of the ring $S$.
In Section 5 we analyze the notion of Gr\"obner\ $\Sigma$-basis for
$\Sigma$-invariant ideals of $P$. When $P$ can be endowed
with a suitable grading compatible with the action of $\Sigma$,
we describe a bijective correspondence between all graded
$\Sigma$-invariant ideals of $P$ and some class of graded two-sided
ideals of $S$. Such correspondence preserves Gr\"obner\ bases and gives
rise to a ``duality'' between homogeneous algorithms in $P$ and in $S$.
Note that for finitely generated ideals all these procedures admit
termination when truncated at some degree. As we said earlier, in Section 6
the algebra embedding $\iota:{K\langle X \rangle}\to S$ is introduced and a bijective
correspondence between the ideals of ${K\langle X \rangle}$ and suitable ideals of $S$
is hence obtained by extension. The Gr\"obner\ bases are preserved by this
correspondence and one obtains an alternative algorithm for computing
non-commutative homogeneous Gr\"obner\ bases of ${K\langle X \rangle}$ in the free $P$-module $S$.
By means of the bijection of the Section 5, we reobtain in Section 7
the ideal correspondence and related algorithms introduced in \cite{LSL}
which provide the computation of non-commutative homogeneous Gr\"obner\ bases
directly in $P$. Therefore, the theory for such bases can be deduced
by the classical Buchberger algorithm for commutative polynomial rings.
In Section 8 we propose the explicit computation of a finite Gr\"obner\ basis
of an ideal of ordinary difference polynomials that can be obtained
as a special case by the algorithms we introduced. Moreover, in this
section we provide some timings obtained by an improvement of the library
\texttt{freegb.lib} of \textsc{Singular} initially developed for
\cite{LSL}. Finally, in Section 9 we propose some conclusions and
suggestions for future developments of the theory of Gr\"obner\ $\Sigma$-bases
and its methods.
The preliminary full-size version of this paper has been accepted for
oral presentation at MEGA 2011 conference in Stockholm.
\section{Modules over skew monoid rings}
Fix $K$ any field and let $P$ be a commutative $K$-algebra. Let now
$\Sigma\subset{\mathrm{End}}_K(P)$ a submonoid of the monoid of $K$-algebra
endomorphisms of $P$. Denote $S = P*\Sigma$ the {\em skew monoid ring
defined by $\Sigma$ over $P$} that is $S$ is the free $P$-module with
(left) basis $\Sigma$ and the multiplication is defined by the identity
$\sigma f = \sigma(f) \sigma$, for all $f\in P,\sigma\in\Sigma$.
If $\Sigma\neq \{id\}$ then $S$ is a non-commutative $K$-algebra where
the ring $P$ and the monoid $\Sigma$ are embedded. Note that if $\Sigma =
\langle \sigma \rangle$ with $\sigma:P\to P$ a map of infinite order
one has that $S\approx P[s;\sigma]$, the skew polynomial ring in the variable
$s$ and coefficients in $P$ defined by the endomorphism $\sigma$.
Moreover, if $P$ is a domain and all maps in $\Sigma$ are injective then
$S$ is also a domain. To simplify notations, we denote $f^\sigma = \sigma(f)$
for any $f\in P,\sigma\in\Sigma$.
\begin{definition}
Let $M$ be a $P$-module. We call $M$ {\em a $\Sigma$-invariant module}
if there is a monoid homomorphism $\rho:\Sigma\to End_K(M)$ such that
$\rho(\sigma)(f x) = f^\sigma\rho(\sigma)(x)$, for all $f\in P,x\in M$
and $\sigma\in\Sigma$. We denote as usual $\sigma\cdot x = \rho(\sigma)(x)$.
If $M,M'$ are $\Sigma$-invariant modules and $\varphi:M\to M'$ is a
$P$-module homomorphism such that $\varphi(\sigma\cdot x) =
\sigma\cdot\varphi(x)$ for all $x\in M,\sigma\in\Sigma$, then the map
$\varphi$ is called a {\em homomorphism of $\Sigma$-invariant modules}.
\end{definition}
\begin{proposition}
\label{invar2leftmod}
The category of $\Sigma$-invariant $P$-modules is equal to the category
of left $S$-modules.
\end{proposition}
\begin{proof}
Let $M$ be a left $S$-module. Then $M$ is a $P$-module since
$P\subset S$. By restriction to $\Sigma\subset S$, one has a monoid
homomorphism $\rho:\Sigma\to{\mathrm{End}}_K(M)$. Moreover we have
$\sigma\cdot(f x) = (\sigma f)\cdot x = (f^\sigma \sigma)\cdot x =
f^\sigma (\sigma\cdot x)$, for all $f\in P, x\in M$ and $\sigma\in\Sigma$.
Let now $M$ be a $\Sigma$-invariant $P$-module. We can define a left
$S$-module structure by putting $(\sum_i f_i \sigma_i)\cdot x =
\sum_i f_i (\sigma_i\cdot x)$ with $f_i\in P,\sigma_i\in\Sigma$ and
$x\in M$. Consider a homomorphism $\varphi:M\to M'$ of $\Sigma$-invariant
modules. Since $\varphi$ is $P$-linear, one has
$\varphi((\sum_i f_i \sigma_i)\cdot x) = \sum_i f_i \varphi(\sigma_i\cdot x) =
\sum_i f_i (\sigma_i\cdot \varphi(x)) = (\sum_i f_i \sigma_i)\cdot \varphi(x)$.
\end{proof}
Let $M$ be a left $S$-module and let $G = \{g_i\}\subset M$ be a generating
set of $M$. Note that $x\in M$ if and only if $x =
\sum_{i,\sigma} f_{i\sigma} \sigma\cdot g_i$ with $f_{i\sigma}\in P$
that is $M$ is generated by $\Sigma\cdot G = \{\sigma\cdot g_i\}_{i,\sigma}$
as $P$-module. We want now to describe homogeneous bases for graded
two-sided ideals of $S$. In fact, the algebra $S$ has a natural grading
over the monoid $\Sigma$ that is $S = \bigoplus_{\sigma\in\Sigma} S_\sigma$
and $S_\sigma S_\tau\subset S_{\sigma\tau}$ where $S_\sigma = P \sigma$.
Note that $S_{id} = P$, all $S_\sigma$ are $P$-submodules of $S$ and
$S_\sigma \tau = S_{\sigma\tau}$.
\begin{proposition}
\label{gen2lgen}
Let $J\subset S$ be a graded (two-sided) ideal and let $G\subset J$ be a set
of homogeneous elements. Then $G$ is a generating set of $J$ if and only if
$G\,\Sigma$ is a left basis of $J$ that is $\Sigma\,G\,\Sigma$
is a basis of $J$ as $P$-module.
\end{proposition}
\begin{proof}
Assume $G = \{g_i \sigma_i\}$ with $g_i\in P,\sigma_i\in\Sigma$, for all $i$.
Let $p_i,q_i\in S$ with $q_i = \sum_\sigma q_{i\sigma} \sigma$ and
$q_{i\sigma}\in P$. It is sufficient to note that $\sum_i p_i g_i \sigma_i q_i =
\sum_{i,\sigma} p_i g_i \sigma_i q_{i\sigma} \sigma =
\sum_{i,\sigma} p_i q_{i\sigma}^{\sigma_i} g_i \sigma_i\sigma$.
\end{proof}
\begin{corollary}
\label{poldiv2ldiv}
Let $f,g\in S$ and let $g$ be a homogeneous element. Then, one has that
$f = p g q$ with $p,q\in S$ if and only if $f$ belongs to the (graded)
left ideal generated by $\{g \sigma\}_{\sigma\in\Sigma}$.
\end{corollary}
\section{Monomial orderings and Gr\"obner\ bases}
Denote ${\mathbb N} = \{0,1,\ldots\}$ the set of non-negative integers and let
$X = \{x_0,x_1,\ldots\}$ be a countable set. From now on,
{\em we make the assumption} that $P = K[X]$ is a commutative polynomial
ring in the variables set $X$. Starting from Section 6 we will assume
in particular that this set has the form $X\times{\mathbb N}$.
Moreover, we fix $\sigma:P\to P$ an algebra homomorphism of infinite
order and define the monoid $\Sigma = \langle \sigma \rangle \approx {\mathbb N}$.
Then, the skew monoid ring $S = P*\Sigma$ is isomorphic to the
skew polynomial ring $P[s;\sigma]$ and we identify $\Sigma = \{\sigma^i\}$
with the monoid $\{s^i\}$ of powers of the variable $s$. Note that $S$
is a free $P$-module of infinite rank. We denote $f^{s^i} = f^{\sigma^i} =
\sigma^i(f)$, for all $f\in P, i\geq 0$. Moreover, a homogeneous element
$f\in S_i = P s^i$ for some $i$ is also called {\em $s$-homogeneous}
and we put $\deg_s(f) = i$. Note finally that in the theory of difference
ideals \cite{Le}, the ring $S$ is called {\em ring of ordinary
difference operators over $P$}.
Denote by ${\mathrm{Mon}}(P)$ the set of all monomials of $P$ (including 1). Clearly,
${\mathrm{Mon}}(P)$ is a multiplicative $K$-basis of $P$ that is $m n\in{\mathrm{Mon}}(P)$
for all $m,n\in{\mathrm{Mon}}(P)$. By definition of $S$, a $K$-basis of such algebra
is given by the elements $m s^i$ where $m\in{\mathrm{Mon}}(P)$ and $i\geq 0$ is an integer.
We call such elements the {\em monomials of $S$} and we denote the set of them
as ${\mathrm{Mon}}(S)$. Clearly ${\mathrm{Mon}}(P)\subset{\mathrm{Mon}}(S)$. Note that ${\mathrm{Mon}}(S)$ is
in fact the ``monomial basis'' of $S$ as a free $P$-module.
In what follows, {\em we assume} also that the endomorphism $\sigma:P\to P$
is injective and {\em monomial} that is it stabilizes the set ${\mathrm{Mon}}(P)$.
In other words, $\{\sigma(x_i)\}$ is a set of algebraically independent
monomials. Since $P$ is a domain, it follows that $S$ is also a domain
and the $K$-basis ${\mathrm{Mon}}(S)$ is multiplicative since $m s^i n s^j =
m n^{s_i} s^{i+j}\neq 0$, for all $m,n\in{\mathrm{Mon}}(P)$ and $i,j\geq 0$.
We want to study now some divisibility relations in ${\mathrm{Mon}}(S)$.
Let $f,g\in S$. We say that {\em $f$ left-divides $g$} if there is
$a\in S$ such that $g = a f$. Clearly, left divisibility is a partial
ordering (up to units). Since $\sigma$ is a monomial injective map,
one has that if $f,g\in{\mathrm{Mon}}(S)$ then also $a\in{\mathrm{Mon}}(S)$.
\begin{proposition}
Let $v = m s^i, w = n s^j\in{\mathrm{Mon}}(S)$ with $m,n\in{\mathrm{Mon}}(P)$.
Then $v$ left-divides $w$ if and only if $i\leq j$ and $m^{s^{j-i}}\mid n$.
\end{proposition}
\begin{proof}
Let $a = p s^k\in{\mathrm{Mon}}(S)$ with $p\in{\mathrm{Mon}}(P)$ such that $n s^j = p s^k m s^i =
p m^{s^k} s^{k+i}$. Then, we have that $j - i = k\geq 0$ and $m^{s^k}\mid n$.
\end{proof}
Note that $S$ has also a free $P$-module structure and so ${\mathrm{Mon}}(S)$
inherits another notion of divisibility. Precisely, let $v,w\in{\mathrm{Mon}}(S)$.
We say that {\em $v$ $P$-divides $w$} if $\deg_s(v) = \deg_s(w)$ and
there is $a\in{\mathrm{Mon}}(P)$ such that $w = a v$. Clearly $P$-divisibility
is a partial ordering and one has the following result.
\begin{proposition}
Let $v,w\in{\mathrm{Mon}}(S)$. Then $v$ left-divides $w$ if and only if
$s^k v$ $P$-divides $w$ for some $k\geq 0$.
\end{proposition}
Note that left divisibility coincides with $P$-divisibility when
the monomials have the same $s$-degree. If there are $v,w,a,b\in{\mathrm{Mon}}(S)$
such that $w = a v b$ we say that {\em $v$ (two-sided) divides $w$}.
It is easy to prove that such divisibility is also a partial ordering.
\begin{proposition}
\label{div2ldiv}
Let $v,w\in{\mathrm{Mon}}(S)$. Then $w$ is a multiple of $v$ if and only if
there is $j\geq 0$ such that $w$ is a left multiple of $v s^j$, that is
$w$ is a $P$-multiple of $s^i v s^j$ for some $i,j\geq 0$.
\end{proposition}
\begin{proof}
Since monomials are $s$-homogeneous elements of $S$, by applying
Corollary \ref{poldiv2ldiv} we obtain that $w$ is a multiple of $v$
if and only if $w$ belongs to the (graded) left ideal generated by
$\{v s^j\}_{j\geq 0}$. Clearly, this happens when $w$ is a left multiple
of $v s^j$ for some $j$.
\end{proof}
We start now considering monomial orderings.
\begin{definition}
Let $\prec$ be a total ordering on the set ${\mathrm{Mon}}(S)$. We call $\prec$
a {\em monomial ordering of $S$} if it satisfies the following conditions
\begin{itemize}
\item[(i)] $\prec$ is a well-ordering, that is every non-empty set
of ${\mathrm{Mon}}(S)$ has a minimal element;
\item[(ii)] $\prec$ is compatible with multiplication, that is
if $v\prec w$ then $p v q\prec p w q$, for all $v,w,p,q\in{\mathrm{Mon}}(S)$.
\end{itemize}
\end{definition}
It follows immediately that $1\preceq w$ for any $w\in{\mathrm{Mon}}(S)$
and if $w = p v q$ with $p\neq 1$ or $q\neq 1$ then $v\prec w$
for all $v,w,p,q\in{\mathrm{Mon}}(S)$. Note that the above conditions agree
with general definitions of orderings on $K$-bases of associative
algebras that provide a Gr\"obner\ basis theory (see for instance \cite{Gr2,Li}).
The same conditions define monomial orderings of the free algebras ${K\langle X \rangle}$
and $K[X]$. Note that such algebras can be endowed with a monomial
ordering even if the set of variables $X$ is countable. This is
provided by the Higman's lemma \cite{Hi} which implies that any
multiplicatively compatible total ordering of the monomials
such that $1 \prec x_0 \prec x_1 \prec \ldots$ is a monomial ordering.
Recall that $f^s$ stands for $\sigma(f)$ for any $f\in P$.
\begin{definition}
Let $\prec$ be a monomial ordering on $P$. We call $\sigma$ {\em compatible
with $\prec$} if $\sigma$ is a strictly increasing map when restricted
to ${\mathrm{Mon}}(P)$, that is $m\prec n$ implies that $m^s\prec n^s$
for all $m,n\in{\mathrm{Mon}}(P)$.
\end{definition}
The following result is based essentially on Remark 3.2 in
\cite{BD}.
\begin{proposition}
\label{noauto}
Assume $\sigma$ be compatible with $\prec$. Then $\sigma$ is not an
automorphism and $m\preceq m^s$, for all $m\in{\mathrm{Mon}}(P)$.
\end{proposition}
\begin{proof}
Since $\sigma\neq id$, there is $m\in{\mathrm{Mon}}(S)$ such that $m\neq m^s$.
If $m\succ m^s$, by compatibility of $\sigma$ one gets an infinite
descending chain $m\succ m^s\succ m^{s^2}\succ\ldots$ which
contradicts the condition that $\prec$ is a well-ordering. We conclude
that $m\prec m^s$. Assume that $\sigma$ has the inverse $\sigma^{-1}$.
By applying $\sigma$, from $m^{s^{-1}}\prec n^{s^{-1}}$ it follows
that $m\prec n$. Since $\sigma^{-1}$ is injective, we have therefore that
$m\prec n$ implies that $m^{s^{-1}}\prec n^{s^{-1}}$. Now,
by compatibility of $\sigma^{-1}$ we obtain $m\prec m^{s^{-1}}$ which
contradicts $m\prec m^s$.
\end{proof}
There are many endomorphisms $\sigma$ with are compatible with
usual monomial orderings on $P$ like lex, degrevlex, etc. For instance,
we have the following maps.
\begin{itemize}
\item $\sigma(x_i) = x_{f(i)}$ for any $i$, where $f:{\mathbb N}\to{\mathbb N}$ is a
strictly increasing map. Such maps have been considered in \cite{BD}.
In particular, one may define the shift operator $\sigma(x_i) = x_{i+1}$
which is used in difference algebra.
\item $\sigma(x_i) = x_i^e$ for any $i$, with $e > 1$.
This map has been considered in \cite{We}.
\end{itemize}
\begin{proposition}
\label{toS2P}
Let $\prec$ be a monomial ordering on $S$. Then $\sigma$ is compatible with
the restriction of $\prec$ to ${\mathrm{Mon}}(P)$. Moreover, for any $m,n\in{\mathrm{Mon}}(P)$
and $i,j\geq 0$ one has that $m s^i\prec n s^j$ implies that
$m\prec n$ or $i < j$.
\end{proposition}
\begin{proof}
Suppose $m\prec n$ with $m,n\in{\mathrm{Mon}}(P)$. Then $s m\prec s n$ that is
$m^s s\prec n^s s$. If $m^s\succeq n^s$ then
$m^s s\succeq n^s s$ which is a contradiction. We conclude
that $m^s\prec n^s$. Now, assume that $m\succeq n$ and $i\geq j$.
We have $m s^i\succeq m s^j\succeq n s^j$.
\end{proof}
Assume now $\sigma$ be compatible with a monomial ordering $\prec$ of $P$.
We define a total ordering on ${\mathrm{Mon}}(S)$ by putting $m s^i\prec' n s^j$
if and only if $i < j$, or $i = j$ and $m\prec n$, for all $m,n\in{\mathrm{Mon}}(P)$
and $i,j\geq 0$. Clearly, the restriction of $\prec'$ to ${\mathrm{Mon}}(P)$ is
$\prec$.
\begin{proposition}
\label{toP2S}
The ordering $\prec'$ is a monomial ordering on $S$ that extends $\prec$.
\end{proposition}
\begin{proof}
Clearly, an infinite descending sequence in ${\mathrm{Mon}}(S)$ implies
an infinite descending sequence in ${\mathrm{Mon}}(P)$ which contradicts
the condition that $\prec$ is a well-ordering. Let $m s^i,n s^j\in{\mathrm{Mon}}(S)$
and suppose $m s^i\prec n s^j$ that is $i < j$, or $i = j$ and $m\prec n$.
Let $q s^k\in{\mathrm{Mon}}(S)$ and consider right multiplications $m s^i q s^k =
m q^{s^i} s^{i+k}$ and $n s^j q s^k = n q^{s^j} s^{j+k}$. If $i < j$ then
$i + k < j + k$. If $i = j$ and $m\prec n$ then $m q^{s^i}\prec
n q^{s^i} = n q^{s^j}$. We conclude in both cases that
$m q^{s^i} s^{i+k} \prec n q^{s^j} s^{j+k}$. For left multiplications
$q s^k m s^i = q m^{s^k} s^{k+i}$ and $q s^k n s^j = q n^{s^k} s^{k+j}$,
note that $m\prec n$ implies that $m^{s^k}\prec n^{s^k}$. Then,
one may argue in a similar way as for right multiplications.
\end{proof}
Clearly, a byproduct of Proposition \ref{toS2P} and Proposition \ref{toP2S}
is that there exist monomial orderings on the skew polynomial ring $S$
if and only if $\sigma$ is compatible with a monomial ordering of $P$.
Note that $\prec'$ is well-known as module ordering when we consider
$S$ as a free $P$-module. Moreover, by Proposition \ref{toS2P} it follows
also that the monomial ordering of $S$ is uniquely defined by the one of $P$
when one compares monomials of the same $s$-degree.
From now on, {\em we assume} $S$ be endowed with a monomial ordering $\prec$.
\begin{definition}
Let $f\in S, f = \sum_i \alpha_i m_i s^i$ with $m_i\in{\mathrm{Mon}}(P),\alpha_i\in K^*$.
Then, we denote ${\mathrm{lm}}(f) = m_k s^k = \max_\prec\{m_i s^i\}$, ${\mathrm{lc}}(f) = \alpha_k$
and ${\mathrm{lt}}(f) = {\mathrm{lc}}(f){\mathrm{lm}}(f)$. If $G\subset S$ we put ${\mathrm{lm}}(G) =
\{{\mathrm{lm}}(f) \mid f\in G,f\neq 0\}$. We denote as ${\mathrm{LM}}(G)$ and ${\mathrm{LM}}_l(G)$
respectively the two-sided ideal and the left ideal of $S$ generated
by ${\mathrm{lm}}(G)$. Moreover, we denote by ${\mathrm{LM}}_P(G)$ the $P$-submodule of $S$
generated by ${\mathrm{lm}}(G)$.
\end{definition}
\begin{proposition}
\label{normbas}
Let $J$ be an ideal (respectively left ideal) of $S$. Then, the set
$\{w + J \mid w\in{\mathrm{Mon}}(S)\setminus{\mathrm{LM}}(J)\}$
(resp.~$\{w + J \mid w\in{\mathrm{Mon}}(S)\setminus{\mathrm{LM}}_l(J)\}$) is a $K$-basis
of the space $S/J$. If $J\subset S$ is a $P$-submodule, in the same way
one defines the $K$-basis $\{w + J \mid w\in{\mathrm{Mon}}(S)\setminus{\mathrm{LM}}_P(J)\}$.
\end{proposition}
\begin{proof}
Let $w\in{\mathrm{Mon}}(S)$. By induction on the monomial ordering of $S$, we can assume
that for any monomial $v\in{\mathrm{Mon}}(S)$ such that $v\prec w$ there is
a polynomial $f\in S$ belonging to the span of $N = {\mathrm{Mon}}(S)\setminus{\mathrm{LM}}(J)$
such that $v - f\in J$. If $w\notin N$ then there is $g\in J$ such that
$w = p {\mathrm{lm}}(g) q$ with $p,q\in{\mathrm{Mon}}(S)$. Therefore $f = w - (1/{\mathrm{lc}}(g))p g q$
is such that ${\mathrm{lm}}(f)\prec w$ and by induction $f - f'\in J$ for some
$f'\in\langle N \rangle_K$. We conclude that $w - f'\in J$. Finally
if $f\in N\cap J$ then necessarily $f = 0$.
Mutatis mutandis one proves the remaining assertions.
\end{proof}
\begin{definition}
Let $J$ be an ideal (respectively left ideal) of $S$ and $G\subset J$.
We call $G$ a {\em Gr\"obner\ basis (resp.~left basis)} of $J$ if
${\mathrm{LM}}(G) = {\mathrm{LM}}(J)$ (resp.~${\mathrm{LM}}_l(G) = {\mathrm{LM}}_l(J)$). As usual, if $J$
is a $P$-submodule of $S$ then $G$ is a {\em Gr\"obner\ $P$-basis} of $J$
when ${\mathrm{LM}}_P(G) = {\mathrm{LM}}_P(J)$.
\end{definition}
\begin{proposition}
Let $J$ be an ideal (respectively left ideal) of $S$ and $G\subset J$.
The following conditions are equivalent:
\begin{itemize}
\item[(i)] $G$ is a Gr\"obner\ basis (resp.~left basis) of $J$;
\item[(ii)] for any $f\in J$, one has a {\em Gr\"obner\ representation of $f$
with respect to $G$} that is $f = \sum_i f_i g_i h_i$
(resp.~$f = \sum_i f_i g_i$) with ${\mathrm{lm}}(f)\succeq{\mathrm{lm}}(f_i){\mathrm{lm}}(g_i){\mathrm{lm}}(h_i)$
(resp.~${\mathrm{lm}}(f)\succeq{\mathrm{lm}}(f_i){\mathrm{lm}}(g_i)$) and $f_i,h_i\in S$, for all $i$.
\end{itemize}
A similar characterization holds for Gr\"obner\ $P$-bases.
\end{proposition}
\begin{proof}
It follows immediately by the reduction process which is implicit
in the proof of Proposition \ref{normbas}.
\end{proof}
\begin{proposition}
\label{gb2lgb}
Let $J$ be a graded ideal of $S$ and $G\subset J$ be a subset of
$s$-homogeneous elements. The following conditions are equivalent:
\begin{itemize}
\item[(i)] $G$ is a Gr\"obner\ basis of $J$;
\item[(ii)] $G\,\Sigma$ is a Gr\"obner\ left basis of $J$;
\item[(iii)] $\Sigma\,G\,\Sigma$ is a Gr\"obner\ $P$-basis of $J$.
\end{itemize}
\end{proposition}
\begin{proof}
Assume $G = \{g_i\}$ is a Gr\"obner\ basis of $J$ and put $d_i = \deg_s(g_i)$.
If $f\in J$ then one has $f = \sum_i f_i g_i h_i$ where $f_i,h_i\in S$
and ${\mathrm{lm}}(f)\succeq{\mathrm{lm}}(f_i){\mathrm{lm}}(g_i){\mathrm{lm}}(h_i)$, for all $i$. Decompose
$h_i = \sum_j h_{i j} s^j$ with $h_{i j}\in P$ for any $i,j$.
Then, we have ${\mathrm{lm}}(f)\succeq{\mathrm{lm}}(f_i){\mathrm{lm}}(g_i){\mathrm{lm}}(h_{i j}) s^j$, for all $i,j$.
Since ${\mathrm{lm}}(g_i)$ has $s$-degree $d_i$, one obtains
${\mathrm{lm}}(f_i){\mathrm{lm}}(g_i){\mathrm{lm}}(h_{i j}) s^j =
{\mathrm{lm}}(f_i){\mathrm{lm}}(h_{i j})^{s^{d_i}}{\mathrm{lm}}(g_i s^j)$.
Moreover, as in Proposition \ref{gen2lgen}, we have
$f = \sum_{i,j} f_i g_i h_{i j} s^j =
\sum_{i,j} f_i h_{i j}^{s^{d_i}} g_i s^j$.
From $\sigma$ compatible with $\prec$ it follows that
${\mathrm{lm}}(h_{i j}^{s^{d_i}}) = {\mathrm{lm}}(h_{i j})^{s^{d_i}}$ and hence $f$ has a left
Gr\"obner\ representation with respect to $G\,\Sigma$, that is this set
is a left Gr\"obner\ basis of $J$. The rest of the proof is straightforward.
\end{proof}
\section{Buchberger algorithm}
After Proposition \ref{gb2lgb}, in order to obtain a homogeneous Gr\"obner\ basis
$G$ of a (two-sided) graded ideal $J\subset S$ one has to start with a
homogeneous generating set $H$ and consider the $P$-basis
$H' = \Sigma\,H\,\Sigma$. Then, one should transform $H'$ into a
homogeneous Gr\"obner\ $P$-basis $G'$ of $J$ and finally reduce $G'$
as $G' = \Sigma\,G\,\Sigma$ with $G\subset J$.
Apart with problems concerning termination of the module Buchberger
algorithm ($P$ is not Noetherian and $S$ is a $P$-module of countable
rank) that we will show solvable for the truncated algorithm up to
some $s$-degree (see Proposition \ref{termin}), it is more desirable
to have a procedure able to compute $G$ without actually considering
all elements of $G'$. To obtain this, we need an additional requirement
for the endomorphism $\sigma$.
Note that, since $\sigma:P\to P$ is a ring homomorphism, such map
is increasing with respect to the divisibility relation in $P$,
that is $f\mid g$ implies that $f^s\mid g^s$ and in this case $(g/f)^s =
g^s/f^s$ with $f,g\in P$.
\begin{proposition}
\label{divcompat}
The following conditions are equivalent:
\begin{itemize}
\item[(a)] $\gcd(x_i^s,x_j^s) = 1$, for all $i\neq j$;
\item[(b)] $\gcd(m^s,n^s) = \gcd(m,n)^s$, for all
$m,n\in{\mathrm{Mon}}(P)$.
\end{itemize}
Moreover, in this case one has $m\mid n$ if and only if $m^s\mid n^s$
and ${\mathrm{lcm}}(m^s,n^s) = {\mathrm{lcm}}(m,n)^s$ with $m,n\in P$.
In other words, $\sigma$ is a lattice homomorphism with respect to the
divisibility relation in ${\mathrm{Mon}}(P)$.
\end{proposition}
\begin{proof}
Assume (a) and let $m,n\in{\mathrm{Mon}}(P)$ such that $\gcd(m,n) = 1$.
If $m = x_{i_1}\cdots x_{i_k}$ and $n = x_{j_1}\cdots x_{j_l}$ then
$m^s = x_{i_1}^s\cdots x_{i_k}^s$ and $n^s = x_{j_1}^s\cdots x_{j_l}^s$
with $\{i_1,\ldots,i_k\}\cap\{j_1,\ldots,j_l\} = \emptyset$.
Since $\gcd(x_i^s,x_j^s) = 1$ for all $i\neq j$, we conclude that
$\gcd(m^s,n^s) = 1$. Assume now $\gcd(m,n) = u$ and hence
$\gcd(m/u,n/u) = 1$. Then $\gcd(m^s/u^s,n^s/u^s) =
\gcd((m/u)^s,(n/u)^s) = 1$ and therefore $\gcd(m^s,n^s) = u^s$
that is (b) holds. Suppose $m^s\mid n^s$ that is $m^s = \gcd(m^s,n^s) =
\gcd(m,n)^s$. Since $\sigma$ is injective we have that $m = \gcd(m,n)$
that is $m\mid n$. Moreover, one obtains ${\mathrm{lcm}}(m,n)^s = (m n / \gcd(m,n))^s =
(m n)^s / \gcd(m,n)^s = m^s n^s / \gcd(m^s,n^s) = {\mathrm{lcm}}(m^s,n^s)$
for all $m,n\in{\mathrm{Mon}}(P)$.
\end{proof}
\begin{definition}
We say that {\em $\sigma$ is compatible with divisibility in ${\mathrm{Mon}}(P)$}
if for all $i\neq j$, one has $\gcd(x_i^s,x_j^s) = 1$ that is the variables
occuring in the monomials $x_i^s,x_j^s$ are disjoint.
\end{definition}
Note that if a monomial endomorphism of $P$ is compatible with divisibility
then it is automatically injective since the monomials $x_i^s$ are
algebraically independent.
Let $\mid$ be the divisibility relation and $\prec$ a monomial ordering
on ${\mathrm{Mon}}(P)$. Throughout the rest of the paper, {\em we make the assumption}
that the monomial endomorphism $\sigma:P \to P$ is compatible both with
$\mid$ and with $\prec$.
We recall now some basic results in the theory of module Gr\"obner\ bases
by applying them to the free $P$-module $S$ whose (left) free basis is
$\Sigma = \{s^i\}_{i\geq 0}$. Consider $f,g\in S\setminus\{0\}$
two elements whose leading monomials have the same $s$-degree (component),
that is ${\mathrm{lm}}(f) = m s^i, {\mathrm{lm}}(g) = n s^i$ with $m,n\in{\mathrm{Mon}}(P)$ and $i\geq 0$.
If we put ${\mathrm{lc}}(f) = \alpha, {\mathrm{lc}}(g) = \beta$ and $l = {\mathrm{lcm}}(m,n)$,
one defines the \mbox{\em S-polynomial} ${\mathrm{spoly}}(f,g) =
(l/\alpha m) f - (l/\beta n) g$. Clearly ${\mathrm{spoly}}(f,g) = - {\mathrm{spoly}}(g,f)$
and ${\mathrm{spoly}}(f,f) = 0$.
\begin{proposition}[Buchberger criterion]
Let $G$ be a generating set of a $P$-submo\-du\-le $J\subset S$. Then
$G$ is a Gr\"obner\ basis of $J$ if and only if for all $f,g\in G\setminus\{0\}$
such that $\deg_s({\mathrm{lm}}(f)) = \deg_s({\mathrm{lm}}(g))$, the S-polynomial ${\mathrm{spoly}}(f,g)$
has a Gr\"obner\ representation with respect to $G$.
\end{proposition}
Usually the above result, see for instance \cite{Ei,GP},
is stated when $P$ is a polynomial ring with a finite number of variables
and $S$ is a $P$-module of finite rank. In fact such assumptions are
not needed since Noetherianity is not used in the proof, but only the
existence of a monomial ordering for the ring $P$ and the free module $S$.
See also the comprehensive Bergman's paper \cite{Be} where the ``Diamond
Lemma'' is proved without any restriction on the finiteness of the variable
set. In the following results we show how the Buchberger criterion, and
hence the corresponding algorithm, can be reduced up to the symmetry
defined by the monoid $\Sigma$ on $S$.
\begin{lemma}
\label{sigmaspoly}
Let $f,g\in S\setminus\{0\}$ and let $i\leq j$ such that
$\deg_s(lm(f)) + i = \deg_s({\mathrm{lm}}(g)) + j$. Then
${\mathrm{spoly}}(s^i f, s^j g) = s^i {\mathrm{spoly}}(f, s^{j-i} g)$ and ${\mathrm{spoly}}(f s^i, g s^j) =
{\mathrm{spoly}}(f, g s^{j-i}) s^i$.
\end{lemma}
\begin{proof}
Let ${\mathrm{lt}}(f) = \alpha m s^k, {\mathrm{lt}}(g) = \beta n s^l$ with
$\alpha,\beta\in K^*$ and $m,n\in{\mathrm{Mon}}(P)$. Then ${\mathrm{lt}}(s^i f) =
\alpha m^{s^i} s^{i+k}, {\mathrm{lt}}(s^j g) = \beta n^{s^j} s^{j+l}$
and ${\mathrm{lt}}(s^{j-i} g) = \beta n^{s^{j-i}} s^{j-i+l}$.
By compatibility of $\sigma$ with divisibility in ${\mathrm{Mon}}(P)$,
if $q = {\mathrm{lcm}}(m,n^{s^{j-i}})$ then ${\mathrm{lcm}}(m^{s^i}, n^{s^j}) = q^{s^i}$.
Therefore $h = {\mathrm{spoly}}(f, s^{j-i} g) =
(q/\alpha m) f - (q/\beta n^{s^{j-i}}) s^{j-i} g$ and hence we have
$s^i h = (q^{s^i}/\alpha m^{s^i}) s^i f - (q^{s^i}/\beta n^{s^j}) s^j g =
{\mathrm{spoly}}(s^i f, s^j g)$.
Note now that ${\mathrm{lt}}(f s^i) = \alpha m s^{i+k}, {\mathrm{lt}}(g s^j) = \beta n s^{j+l}$
and ${\mathrm{lt}}(g s^{j-i}) = \beta n s^{j-i+l}$. If $q = {\mathrm{lcm}}(m,n)$ and
$h = {\mathrm{spoly}}(f, g s^{j-i}) = (q/\alpha m) f - (q/\beta n) g s^{j-i}$ we have
simply that $h s^i = (q/\alpha m) f s^i - (q/\beta n) g s^j =
{\mathrm{spoly}}(f s^i, g s^j)$.
\end{proof}
\begin{proposition}[Two-sided $\Sigma$-criterion]
\label{sigmacrit}
Let $G$ be an $s$-homogeneous basis of a graded two-sided ideal $J\subset S$.
Then $G$ is a Gr\"obner\ basis of $J$ if and only if for all $f,g\in G\setminus\{0\}$
and for any $i,j\geq 0$, the S-polynomials ${\mathrm{spoly}}(f, s^i g s^j)$
$(\deg_s(f) = \deg_s(g) + i + j)$ and ${\mathrm{spoly}}(f s^i, s^j g)$ $(\deg_s(f) + i =
\deg_s(g) + j)$ have a Gr\"obner\ representation with respect to
$\Sigma\,G\,\Sigma$.
\end{proposition}
\begin{proof}
By Proposition \ref{gb2lgb} we have to prove that $G' = \Sigma\,G\,\Sigma$
is a Gr\"obner\ basis of $J$ as $P$-module, that is $G'$ is $P$-basis of $J$ and
the S-polynomials $h = {\mathrm{spoly}}(s^i f s^k, s^j g s^l)$ have a Gr\"obner\ representation
with respect to $G'$ for all $f,g\in G\setminus\{0\}$ and for any $i,j,k,l\geq 0$
such that $\deg_s(f) + i + k = \deg_s(g) + j + l$. Since $G$ is a homogeneous
basis of $J$ as two-sided ideal, from Proposition \ref{gen2lgen} it follows that
$G'$ is a generating set of $J'$ as $P$-module.
Consider now all possibilities $i\leq j$ or $i\geq j$ and $k\leq l$ or $k\geq l$
and apply Lemma \ref{sigmaspoly}.
If $i\leq j,k\leq l$ one has $h = s^i {\mathrm{spoly}}(f, s^{j-i} g s^{l-k}) s^k$,
if $i\leq j,k\geq l$ then $h = s^i {\mathrm{spoly}}(f s^{l-k}, s^{j-i} g) s^l$, and
so on. Then, assume that a S-polynomial $h = {\mathrm{spoly}}(f,g)$, with
$f,g\in G'\setminus\{0\}$, has a Gr\"obner\ representation with respect to $G'$
as $P$-basis of $J$, that is $h = \sum_i f_i g_i$ with $f_i\in P, g_i\in G'$
and ${\mathrm{lm}}(h)\geq{\mathrm{lm}}(f_i){\mathrm{lm}}(g_i)$, for all $i$. We have to prove that
$s^k h s^l$ has also a Gr\"obner\ representation with respect to $G'$
for any $k,l\geq 0$. One has that $s^k h s^l = \sum_i f_i^{s^k} s^k g_i s^l$
and ${\mathrm{lm}}(s^k h s^l) = s^k {\mathrm{lm}}(h) s^l\geq s^k {\mathrm{lm}}(f_i){\mathrm{lm}}(g_i) s^l =
{\mathrm{lm}}(f_i)^{s^k} s^k {\mathrm{lm}}(g_i) s^l = {\mathrm{lm}}(f_i^{s^k}) {\mathrm{lm}}(s^k g_i s^l)$.
Since $s^k g_i s^l\in G' = \Sigma\,G\,\Sigma$, one obtains that
$s^k h s^l$ has a Gr\"obner\ representation with respect to $G'$.
\end{proof}
A criterion similar to Proposition \ref{sigmacrit} holds clearly for
Gr\"obner\ left bases of left ideals of $S$ where no restrictions about
the $s$-homogeneity of bases and ideals are needed.
\begin{proposition}[Left $\Sigma$-criterion]
\label{leftsigmacrit}
Let $G$ be a basis of a left ideal $J\subset S$. Then $G$ is a Gr\"obner\ basis
of $J$ if and only if for all elements $f,g\in G\setminus\{0\}$ such that
$i = \deg_s({\mathrm{lm}}(f)) - \deg_s({\mathrm{lm}}(g))\geq 0$, the S-polynomial
${\mathrm{spoly}}(f, s^i g)$ has a Gr\"obner\ representation with respect to $\Sigma\,G$.
\end{proposition}
A standard procedure in the (module) Buchberger algorithm is the following.
\suppressfloats[b]
\begin{algorithm}\caption{{\textsc{Reduce}}}
\begin{algorithmic}[0]
\State \text{Input:} $f\in S$ and $G\subset S$.
\State \text{Output:} $h\in S$ such that $f - h\in\langle G\rangle_P$
and $h = 0$ or ${\mathrm{lm}}(h)\notin{\mathrm{LM}}_P(G)$.
\State $h:= f$;
\While{ $h\neq 0$ and ${\mathrm{lm}}(h)\in{\mathrm{LM}}_P(G)$ }
\State choose $g\in G,g\neq 0$ such that ${\mathrm{lm}}(g)$ $P$-divides ${\mathrm{lm}}(h)$;
\State $h:= h - ({\mathrm{lt}}(h)/{\mathrm{lt}}(g)) g$;
\EndWhile;
\State \Return $h$.
\end{algorithmic}
\end{algorithm}
Note that if ${\mathrm{lt}}(g) = \alpha m s^i, {\mathrm{lt}}(h) = \beta n s^i$ with
$\alpha,\beta\in K^*$ and $m,n\in{\mathrm{Mon}}(P)$, by definition
${\mathrm{lt}}(h)/{\mathrm{lt}}(g) = (\alpha m)/(\beta n)$. Moreover, the termination
of ${\textsc{Reduce}}$ is provided since $\prec$ is a well-ordering on ${\mathrm{Mon}}(S)$.
In particular, even if $G$ is an infinite set, there are only a finite
number of elements $g\in G,g\neq 0$ such that ${\mathrm{lm}}(g)$ $P$-divides ${\mathrm{lm}}(h)$
and hence ${\mathrm{lm}}(g)\preceq{\mathrm{lm}}(h)$.
It is well-known that if ${\textsc{Reduce}}(f,G) = 0$ then $f$ has a Gr\"obner\ representation
with respect to $G$. Moreover, if ${\textsc{Reduce}}(f,G) = h\neq 0$ then clearly
we have ${\textsc{Reduce}}(f,G\cup\{h\}) = 0$. Therefore, from Proposition \ref{sigmacrit}
it follows immediately the correctness of the following algorithm.
\suppressfloats[b]
\begin{algorithm}\caption{{\textsc{SkewGBasis}}}
\begin{algorithmic}[0]
\State \text{Input:} $H$, an $s$-homogeneous basis of a graded two-sided
ideal $J\subset S$.
\State \text{Output:} $G$, an $s$-homogeneous Gr\"obner\ basis of $J$.
\State $G:= H$;
\State $B:= \{(f,g) \mid f,g\in G\}$;
\While{$B\neq\emptyset$}
\State choose $(f,g)\in B$;
\State $B:= B\setminus \{(f,g)\}$;
\ForAll{$i,j\geq 0$ such that $i + j = \deg_s(f) - \deg_s(g)$}
\State $h:= {\textsc{Reduce}}({\mathrm{spoly}}(f,s^i g s^j), \Sigma\,G\,\Sigma)$;
\If{$h\neq 0$}
\State $B:= B\cup\{(h,h),(h,k),(k,h)\mid k\in G\}$;
\State $G:= G\cup\{h\}$;
\EndIf;
\EndFor;
\ForAll{$i,j\geq 0$ such that $j - i = \deg_s(f) - \deg_s(g)$}
\State $h:= {\textsc{Reduce}}({\mathrm{spoly}}(f s^i, s^j g), \Sigma\,G\,\Sigma)$;
\If{$h\neq 0$}
\State $B:= B\cup\{(h,h),(h,k),(k,h)\mid k\in G\}$;
\State $G:= G\cup\{h\}$;
\EndIf;
\EndFor;
\EndWhile;
\State \Return $G$.
\end{algorithmic}
\end{algorithm}
\newpage
Clearly, all well-known criteria (product criterion, chain criterion, etc)
can be applied to {\textsc{SkewGBasis}}\ to shorten the number of S-polynomials
to be considered. In fact, this algorithm can be understood as the usual
(module) Buchberger procedure applied to the $P$-basis $\Sigma\,H\,\Sigma$,
where an additional criterion to avoid ``useless pairs'' is provided
by Proposition \ref{sigmacrit}. Note that owing to Proposition
\ref{leftsigmacrit}, one has also a similar procedure for computing a
Gr\"obner\ left basis of any left ideal of $S$. Since the set $\Sigma\,H\,\Sigma$
if infinite even if the basis $H$ is eventually finite ($S$ is a
non-Noetherian ring) one has that {\textsc{SkewGBasis}}\ does not admit general
termination. In particular, the cycle ``{\bf for all} $i,j\geq 0$
such that $j - i = \deg_s(f) - \deg_s(g)$ {\bf do}'' never stops unless
one bounds the $s$-degree $\deg_s(f) + i = \deg_s(g) + j$.
As for other non-Noetherian structures like the free associative
algebra that in fact can be embedded in $S$ (see Section 6),
the termination of homogeneous Gr\"obner\ bases computations can be obtained
only under truncation.
\begin{proposition}
\label{termin}
Let $J\subset S$ be a graded two-sided ideal and fix $d\geq 0$.
Assume that $J$ has a $s$-homogeneous basis $H$ such that $H_d =
\{f\in H\mid \deg_s(f)\leq d\}$ is a finite set. Then, there exists
an $s$-homogeneous Gr\"obner\ basis $G$ of $J$ such that $G_d$ is also finite.
In other words, if we consider a selection strategy for the S-polynomials
based on their $s$-degree, we obtain that the $d$-truncated version
of the algorithm {\textsc{SkewGBasis}}\ terminates in a finite number of steps.
\end{proposition}
\begin{proof}
Denote $H'_d = \{ s^i f s^j\mid f\in H_d, i,j\geq 0, i + j + \deg_s(f)\leq d \}$.
Since $H_d$ is finite one has that $H'_d$ is also finite. Then, consider $X_d$
the finite set of variables of $P$ occurring in the elements of $H'_d$
and define $P^{(d)} = K[X_d]$ and $S^{(d)} = \bigoplus_{i\leq d} P^{(d)} s^i$.
In fact, the $d$-truncated algorithm {\textsc{SkewGBasis}}\ computes a subset
of a Gr\"obner\ basis of the $P^{(d)}$-submodule $J^{(d)}\subset S^{(d)}$
generated by $H'_d$. By Noetherianity of the ring $P^{(d)}$ and the free
$P^{(d)}$-module $S^{(d)}$ which has finite rank, we clearly obtain termination.
\end{proof}
Note that the above result implies algorithmic solution of the membership
problem for graded ideals of $S$ which are finitely generated up to
any degree.
\section{$\Sigma$-invariant ideals of $P$}
In this section we define Gr\"obner\ bases of $\Sigma$-invariant
ideals $I\subset P$ which generates $I$ up to the action of $\Sigma$.
Moreover, if $P$ can be endowed with a suitable grading, we show how
such bases can be computed in the algebra $S$ for a class of graded
$\Sigma$-invariant ideals. As usual, we fix a monomial endomorphism
$\sigma:P\to P$ which is compatible both with the divisibility and
a monomial ordering on ${\mathrm{Mon}}(P)$ and we extend this to an ordering
on ${\mathrm{Mon}}(S)$. From Section 2 we know that $\Sigma$-invariant ideals
of $P$ are just left $S$-submodules of $P$. Since we make use of
identification $\Sigma = \{s^i\}$, for all $f\in P\subset S$ and
for any $i\geq 0$ one has that $s^i\cdot f = f^{s^i} =
\sigma^i(f)$ and $s^i f = (s^i\cdot f) s^i$.
\begin{definition}
Let $I\subset P$ be a $\Sigma$-invariant ideal and $G\subset I$.
We say that $G$ is a {\em $\Sigma$-basis} of $I$ if $G$ is a basis of $I$
as left $S$-module. In other words, $\Sigma\cdot G$ is a basis of $I$
as $P$-ideal.
\end{definition}
\begin{proposition}
Let $G\subset P$. Then ${\mathrm{lm}}(\Sigma\cdot G) = \Sigma\cdot {\mathrm{lm}}(G)$.
In particular, if $I$ is a $\Sigma$-invariant $P$-ideal then ${\mathrm{LM}}_P(I)$
is also $\Sigma$-invariant.
\end{proposition}
\begin{proof}
Since $\sigma$ is compatible with the monomial ordering of $P$, it is sufficient
to note that ${\mathrm{lm}}(s^i\cdot f) = s^i\cdot{\mathrm{lm}}(f)$ for any $f\in P$ and $i\geq 0$.
\end{proof}
\begin{definition}
Let $I\subset P$ be a $\Sigma$-invariant ideal and $G\subset I$.
We call $G$ a {\em Gr\"obner\ $\Sigma$-basis} of $I$ if ${\mathrm{lm}}(G)$ is a basis
of ${\mathrm{LM}}_P(I)$ as left $S$-module. In other words, $\Sigma\cdot G$
is a Gr\"obner\ basis of $I$ as $P$-ideal.
\end{definition}
The computation of Gr\"obner\ $\Sigma$-bases of $\Sigma$-invariant $P$-ideals
is relevant, for instance, in applications to difference algebra
(cf.~\cite{Le}, Chapter 3). Such computations appear also
in other contexts, see for instance \cite{DLS} and \cite{BD}. Note that
in the latter paper Gr\"obner\ $\Sigma$-bases are named ``equivariant
Gr\"obner\ bases''.
In analogy with Proposition \ref{sigmacrit} and Proposition
\ref{leftsigmacrit}, we present here a $\Sigma$-criterion that allows
to reduce the number of S-polynomials to be checked to provide that
a $\Sigma$-basis is of Gr\"obner\ type.
\begin{proposition}[$\Sigma$-criterion in $P$]
\label{Psigmacrit}
Let $G$ be a $\Sigma$-basis of a $\Sigma$-invariant ideal $I\subset P$.
Then $G$ is a Gr\"obner\ $\Sigma$-basis of $I$ if and only if for all
$f,g\in G\setminus\{0\}$ and for any $i\geq 0$, the S-polynomial
${\mathrm{spoly}}(f, s^i\cdot g)$ has a Gr\"obner\ representation with respect to
$\Sigma\cdot G$.
\end{proposition}
\begin{proof}
Consider any pair of elements $s^i\cdot f, s^j\cdot g\in \Sigma\cdot G$
($f,g\in G$) and let $i\leq j$. By compatibility of $\sigma$ with
divisibility in ${\mathrm{Mon}}(P)$ (cf.~Lemma \ref{sigmaspoly}), one has that
${\mathrm{spoly}}(s^i\cdot f, s^j\cdot g) = s^i\cdot {\mathrm{spoly}}(f, s^k\cdot g)$
with $k = j - i$. Assume that ${\mathrm{spoly}}(f, s^k\cdot g) = h =
\sum_l f_l (s^l\cdot g_l)$ ($f_l\in P, g_l\in G$) is a Gr\"obner\
representation with respect to $\Sigma\cdot G$. Since the endomorphism
$\sigma$ is compatible with the monomial ordering of $P$, we have also
the Gr\"obner\ representation ${\mathrm{spoly}}(s^i\cdot f, s^j\cdot g) =
s^i\cdot h = \sum_l (s^i\cdot f_l) (s^{i+l}\cdot g_l)$.
\end{proof}
Note that some version of this criterion can be found in \cite{BD},
Theorem 2.5, where it is called ``equivariant Buchberger criterion''.
Before than this, the same ideas have been used in \cite{LSL}
for the Proposition 3.11. From this criterion it follows immediately
the correctness of the following algorithm.
\suppressfloats[b]
\begin{algorithm}\caption{SigmaGBasis}
\begin{algorithmic}[0]
\State \text{Input:} $H$, a $\Sigma$-basis of a $\Sigma$-invariant ideal
$I\subset P$.
\State \text{Output:} $G$, a Gr\"obner\ $\Sigma$-basis of $I$.
\State $G:= H$;
\State $B:= \{(f,g) \mid f,g\in G\}$;
\While{$B\neq\emptyset$}
\State choose $(f,g)\in B$;
\State $B:= B\setminus \{(f,g)\}$;
\ForAll{$i\geq 0$}
\State $h:= {\textsc{Reduce}}({\mathrm{spoly}}(f,s^i\cdot g), \Sigma\cdot G)$;
\If{$h\neq 0$}
\State $B:= B\cup\{(h,h),(h,k),(k,h) \mid k\in G\}$;
\State $G:= G\cup\{h\}$;
\EndIf;
\EndFor;
\EndWhile;
\State \Return $G$.
\end{algorithmic}
\end{algorithm}
\newpage
As for the algorithm {\textsc{SkewGBasis}}, all criteria to avoid useless pairs
can be added to {\textsc{SigmaGBasis}}. Note that termination of this algorithm
is not provided in general (note the infinite cycle ``{\bf for all}
$i\geq 0$ {\bf do}'') and this is, in fact, one of the main problems
in applications to differential/difference algebra. Nevertheless,
in what follows we describe some class of $\Sigma$-invariant ideals
of $P$ where a truncated version of the algorithm {\textsc{SigmaGBasis}}\ stops
in a finite number of steps. Such ideals are in bijective correspondence
with a class of graded (two-sided) ideals of $S$ which have truncated
termination of {\textsc{SkewGBasis}}\ provided by Proposition \ref{termin}.
Consider now the $P$-module homomorphism $\pi:S\to P$ such that
$s^i\mapsto 1$, for all $i$. Clearly $\pi$ is a left $S$-module
epimorphism whose kernel is the left ideal of $S$ generated by $s - 1$.
\begin{definition}
Let $J$ be a graded ideal of $S$ and put $J^P = \pi(J)$. Clearly
$J^P$ is a $\Sigma$-invariant ideal of $P$.
\end{definition}
\begin{proposition}
\label{genS2P}
Let $J\subset S$ be a graded ideal. If $G$ is a homogeneous basis of $J$
then $G^P = \pi(G)$ is a $\Sigma$-basis of $J^P$.
\end{proposition}
\begin{proof}
Since the map $\pi$ is a left $S$-module homomorphism,
it is sufficient to note that $G\,\Sigma$ is a left basis of $J$ and
$\pi(G\,\Sigma) = \pi(G) = G^P$.
\end{proof}
We introduce now a grading on the algebra $P$ which is compatible
with action of $\Sigma$. We start extending the structure $({\mathbb N},\max,+)$
in the following way.
\begin{definition}
Let $-\infty$ be an element disjoint by ${\mathbb N}$ and put ${\hat{\N}} =
\{-\infty\}\cup{\mathbb N}$. Then, we define a commutative idempotent monoid
$({\hat{\N}},\max)$ with identity $-\infty$ that extends $({\mathbb N},\max)$ (with
identity 0) by imposing that $\max(-\infty,x) = x$ for any $x\in{\hat{\N}}$.
Moreover, we define a commutative monoid $({\hat{\N}},+)$ with identity $-\infty$
extending the monoid $({\mathbb N},+)$ by putting $-\infty + x = -\infty$,
for all $x\in{\hat{\N}}$. Since $+$ clearly distributes over $\max$, one has
that $({\hat{\N}},\max,+)$ is a commutative idempotent semiring, also known
as commutative dioid or max-plus algebra \cite{GM}.
\end{definition}
Note that if $\sigma^{-\infty}\in{\mathrm{End}}_K(P)$ is the map such that
$x_i\mapsto 0$ for any $x_i\in X$, then ${\hat{\Sigma}} =
\{\sigma^{-\infty}\}\cup\Sigma$ is a commutative monoid isomorphic to
$({\hat{\N}},+)$. Denote now $M = {\mathrm{Mon}}(P)$ the set of monomials of the polynomial
ring $P$.
\begin{definition}
\label{weight}
A mapping ${\mathrm{w}}:M\to{\hat{\N}}$ such that for all $m,n\in M$ and $x_i\in X$ one has
\begin{itemize}
\item[(i)] ${\mathrm{w}}(1) = -\infty$;
\item[(ii)] ${\mathrm{w}}(m n) = \max({\mathrm{w}}(m),{\mathrm{w}}(n))$;
\item[(iii)] ${\mathrm{w}}(s\cdot x_i) = 1 + {\mathrm{w}}(x_i)$.
\end{itemize}
is called a {\em weight function of $P$ endowed with $\sigma$}.
\end{definition}
Note that if $m = x_{i_1}\cdots x_{i_d}\neq 1$ with
${\mathrm{w}}(x_{i_1})\leq\ldots\leq{\mathrm{w}}(x_{i_d})$ then ${\mathrm{w}}(m) = {\mathrm{w}}(x_{i_d})$.
Moreover, the condition (iii) implies that ${\mathrm{w}}(s^i\cdot m) = i + {\mathrm{w}}(m)$
for all $i\in{\hat{\N}}, m\in M$ and hence $s^i\cdot m = m$ if and only if
$m = 1$ or $i = 0$. We put $M_i = \{m\in M\mid{\mathrm{w}}(m) = i\}$ for all
$i\in{\hat{\N}}$ and define $P_i\subset P$ the subspace spanned by $M_i$.
We have that $P_{-\infty} = K$. Clearly $P = \bigoplus_{i\in{\hat{\N}}} P_i$
is a grading of the algebra $P$ defined by the monoid $({\hat{\N}},\max)$
by means of the function ${\mathrm{w}}$. Then, an element $f\in P_i$ is said
{\em ${\mathrm{w}}$-homogeneous of weight $i$}.
In what follows, {\em we assume} that $P$ is endowed with a weight function.
In fact, such functions are easily to define. Consider for instance
the polynomial ring $P = K[X\times{\mathbb N}]$ and denote $x_i(j)$ each variable
$(x_i,j)\in X\times{\mathbb N}$. Let $\sigma:P\to P$ be the algebra monomorphism
of infinite order such that $\sigma(x_i(j)) = x_i(j+1)$, for all $i,j$.
Clearly $\sigma$ is a monomial map compatible with divisibility in ${\mathrm{Mon}}(P)$
and many usual monomial orderings on $P$, like lex, degrevlex, etc, are
compatible with $\sigma$. For the algebra $P$ endowed with the map $\sigma$
we can clearly define the weight function ${\mathrm{w}}(x_i(j)) = j$. In Section 6
we show how to embed the free associative algebra ${K\langle X \rangle}$ into the skew polynomial
ring defined by $P$ and the monoid $\Sigma = \langle \sigma \rangle$.
Moreover, if we put $x_i(j) = \sigma^j(u_i)$ where $x_i(0) = u_i = u_i(t)$ is
a set of (algebraically independent) univariate functions and $\sigma$ is
the shift operator $u_i(t)\mapsto u_i(t+h)$ then $P = K[X\times{\mathbb N}]$ is
by definition the {\em ring of ordinary difference polynomials} with constant
coefficients in the field $K$ (see \cite{Le}). Such algebra is used
to study systems of (ordinary) difference equations for applications
in combinatorics or discretization of systems of differential
equations.
\begin{definition}
Let $I$ be an ideal of $P$. We call $I$ {\em ${\mathrm{w}}$-graded} if $I = \sum_i I_i$
with $I_i = I\cap P_i$ for any $i\in{\hat{\N}}$.
\end{definition}
Define now the skew monoid ring ${\hat{S}} = P * {\hat{\Sigma}}$ extending $S = P * \Sigma$
and let ${\hat{\pi}}:{\hat{S}}\to P$ the left ${\hat{S}}$-module epimorphism extending $\pi$
that is $s^i\mapsto 1$, for all $i\in{\hat{\N}}$. The existence of a weight function
for $P$ implies that one has also a mapping $\xi:P\to{\hat{S}}$ such that
${\hat{\pi}} \xi = id$.
\begin{proposition}
Define $\xi:P\to {\hat{S}}$ the homogeneous injective map such that
$f\mapsto \sum_{i\in{\hat{\N}}} f_i s^i$, for all $f = \sum_{i\in{\hat{\N}}} f_i\in P$.
Then $\xi$ is a ${\hat{\Sigma}}$-equivariant map.
\end{proposition}
\begin{proof}
For all $i,j\in{\hat{\N}}$ and $f\in P_j$ one has that $s^i\cdot f\in P_{i+j}$
and therefore $\xi(s^i\cdot f) = (s^i\cdot f) s^{i+j} = s^i f s^j = s^i \xi(f)$.
\end{proof}
Let $I\subset P$ be a ${\mathrm{w}}$-graded $\Sigma$-invariant ideal and consider
$\xi(I)\subset{\hat{S}}$. Note that if $I\neq P$ then $I_{-\infty} = 0$ and
the set $\xi(I)$ is in fact contained in $S$. Then, to get rid of the symbol
$-\infty$ we restrict ourselves to ideals not containing constants.
\begin{definition}
Let $I\subsetneq P$ be a ${\mathrm{w}}$-graded $\Sigma$-invariant ideal of $P$.
Denote by $I^S$ the graded (two-sided) ideal of $S$ generated by
$\xi(I)\subset S$. In other words, if we put $G = \xi(\bigcup_{i\geq 0} I_i) =
\{f s^i \mid f\in I_i, i\geq 0\}$ then $I^S$ is the left ideal
generated by $G\,\Sigma = \{f s^j \mid f\in I_i, j\geq i\geq 0\}$
or equivalently $I^S$ has the basis $G\,\Sigma = \Sigma\,G\,\Sigma$ as
$P$-submodule of $S$. We call $I^S$ the {\em skew analogue} of $I$.
\end{definition}
\begin{proposition}
\label{S2Pcor}
Let $I\subsetneq P$ be a ${\mathrm{w}}$-graded $\Sigma$-invariant ideal.
Then $I^{SP} = I$, that is there is a {\em bijective correspondence}
between all ${\mathrm{w}}$-graded $\Sigma$-invariant ideals different from $P$ and
their skew analogues in $S$.
\end{proposition}
\begin{proof}
Put $J = I^{SP} = \pi(I^S)$. For any $f\in I_i$ and $j\geq i$
we have clearly $\pi(f s^j) = f$. Since the elements $f s^j$ are
a left basis of $I^S$, the ideal $I$ is $\Sigma$-invariant and $\pi$
is a left $S$-module homomorphism, we have that $J\subset I$.
Moreover, the elements $f\in I_i$ are a basis of $I = \sum_i I_i$
and one has also that $I\subset J$.
\end{proof}
The next propositions need the following lemmas.
\begin{lemma}
\label{divbound}
If $s^k\cdot m$ divides $n$, with $m,n\in M$, then ${\mathrm{w}}(n) - k\geq {\mathrm{w}}(m)$.
\end{lemma}
\begin{proof}
Since $n = q (s^k\cdot m)$ with $q\in M$, we have ${\mathrm{w}}(n)\geq
{\mathrm{w}}(s^k\cdot m) = k + {\mathrm{w}}(m)$.
\end{proof}
\begin{lemma}
\label{lcmbound}
Let $m,n\in M$ and put $l = {\mathrm{lcm}}(m,n)$. Then, one has that ${\mathrm{w}}(l) =
\max({\mathrm{w}}(m),{\mathrm{w}}(n))$.
\end{lemma}
\begin{proof}
By property (ii) of Definition \ref{weight}, it is sufficient to note that
$\max$ is an idempotent operation and hence the weight of a monomial
depends only on the variables occurring in the support.
\end{proof}
\begin{proposition}
Let $I\subsetneq P$ be a ${\mathrm{w}}$-graded $\Sigma$-invariant ideal,
then $I^S$ is a graded ideal of $S$. Let $G = \bigcup_i G_i$
be a {\em ${\mathrm{w}}$-homogeneous} $\Sigma$-basis of $I$ that is
$G_i\subset I_i$. Then $G^S = \xi(G) = \{f s^i \mid f\in G_i,i\geq 0\}$
is an $s$-homogeneous basis of $I^S$.
\end{proposition}
\begin{proof}
Consider the elements $f s^j$ with $f\in I_i,j\geq i$ which form
a left basis of $I^S$. Since $G$ is a $\Sigma$-basis, one has
$f = \sum_k f_k (s^k\cdot g_k)$ with $f_k\in P, g_k\in G_{i_k}$.
From ${\mathrm{w}}(f) = i$, by Lemma \ref{divbound} we obtain that
$i - k\geq i_k$. We have therefore that $f s^j =
\sum_k f_k (s^k\cdot g_k) s^k s^{j-k} =
\sum_k f_k s^k (g_k s^{j-k})$ with $j - k\geq i - k\geq i_k$
and hence $g_k s^{j-k}\in G^S\,\Sigma$, for all $k$.
\end{proof}
Note now that by Proposition \ref{toS2P} we have that $m s^i\prec n s^i$
if and only if $m\prec n$, for all $m,n\in M$ and for any $i\geq 0$.
In other words, if $f s^i$ ($f\in P$) is an $s$-homogeneous element
of $S$ then ${\mathrm{lm}}(f s^i) = {\mathrm{lm}}(f) s^i$.
\begin{lemma}
\label{gbgenIS}
Let $I\subsetneq P$ be a ${\mathrm{w}}$-graded $\Sigma$-invariant ideal.
If $G = \bigcup_i I_i$, by definition $I^S$ is the graded ideal
of $S$ generated by $G^S = \xi(G)$. Then $G^S$ is an $s$-homogeneous
Gr\"obner\ basis of $I^S$.
\end{lemma}
\begin{proof}
Let $f s^i, g s^j\in G^S\,\Sigma$ that is the ${\mathrm{w}}$-homogeneous elements
$f,g\in G$ are such that $i\geq {\mathrm{w}}(f), j\geq {\mathrm{w}}(g)$. Assume $i\geq j$
and put $k = i - j$. By Proposition \ref{leftsigmacrit} we have to check for
Gr\"obner\ representations of the S-polynomial ${\mathrm{spoly}}(f s^i, s^k g s^j)$
with respect to $\Sigma\,G^S\,\Sigma$. Since $G$ is clearly a Gr\"obner\
$\Sigma$-basis of $I$, one has that the S-polynomial ${\mathrm{spoly}}(f, s^k\cdot g)$
has a Gr\"obner\ representation with respect to $\Sigma\cdot G$, say $h =
{\mathrm{spoly}}(f, s^k\cdot g) = \sum_l f_l (s^l\cdot g_l)$ with $f_l\in P,g_l\in G$.
Note that ${\mathrm{spoly}}(f s^i, s^k g s^j) = h s^i = \sum_l f_l (s^l\cdot g_l) s^i$.
We have to prove now that $i\geq l + {\mathrm{w}}(g_l)$ for any $l$, because
in this case one has the Gr\"obner\ representation $h s^i =
\sum_l f_l s^l (g_l s^{i-l})$. In fact, by Lemma \ref{divbound}
and Lemma \ref{lcmbound} we have that $\max({\mathrm{w}}(f),{\mathrm{w}}(g)) =
{\mathrm{w}}(h)\geq l + {\mathrm{w}}(g_l)$. Then, from $i\geq {\mathrm{w}}(f)$ and $i\geq j\geq {\mathrm{w}}(g)$
one obtains the claim.
\end{proof}
\begin{proposition}
Let $G\subset \bigcup_{i\geq 0} P_i$. Then ${\mathrm{lm}}(G)^S = {\mathrm{lm}}(G^S)$. Moreover,
if $I\subsetneq P$ is a ${\mathrm{w}}$-graded $\Sigma$-invariant ideal then
${\mathrm{LM}}_P(I)^S = {\mathrm{LM}}(I^S)$.
\end{proposition}
\begin{proof}
If $f\in P_i$ is a ${\mathrm{w}}$-homogeneous element then ${\mathrm{w}}({\mathrm{lm}}(f)) = {\mathrm{w}}(f) = i$ and
${\mathrm{lm}}(f) s^i = {\mathrm{lm}}(f s^i)$. We obtain that ${\mathrm{lm}}(G)^S = {\mathrm{lm}}(G^S)$. Consider now
$G = \bigcup_i I_i$. By definition $I^S$ is the ideal of $S$ generated
by $G^S$. Moreover, since $I = \sum_i I_i$ one has that ${\mathrm{lm}}(G) = {\mathrm{lm}}(I)$
and hence ${\mathrm{LM}}_P(I)^S$ is the ideal generated by ${\mathrm{lm}}(G)^S = {\mathrm{lm}}(G^S)$.
Finally, by Lemma \ref{gbgenIS} one has that ${\mathrm{LM}}(I^S)$ is the ideal
of $S$ generated by ${\mathrm{lm}}(G^S)$.
\end{proof}
\begin{proposition}
\label{gbP2S}
Let $I\subsetneq P$ be a ${\mathrm{w}}$-graded $\Sigma$-invariant ideal.
Let $G = \bigcup_i G_i$ be a ${\mathrm{w}}$-homogeneous Gr\"obner\ $\Sigma$-basis of $I$.
Then, $G^S = \xi(G)$ is an $s$-homogeneous Gr\"obner\ basis of $I^S$.
\end{proposition}
\begin{proof}
By hypothesis ${\mathrm{lm}}(G)$ is a $\Sigma$-basis of ${\mathrm{LM}}_P(I)$.
Then ${\mathrm{lm}}(G^S) = {\mathrm{lm}}(G)^S$ is a basis of ${\mathrm{LM}}_P(I)^S = {\mathrm{LM}}(I^S)$
that is $G^S$ is a Gr\"obner\ basis of $I^S$.
\end{proof}
\begin{proposition}
\label{gbS2P}
Let $I\subsetneq P$ be a ${\mathrm{w}}$-graded $\Sigma$-invariant ideal.
If $G$ is an $s$-homogeneous Gr\"obner\ basis of $I^S$ then $G^P = \pi(G)$
is a Gr\"obner\ $\Sigma$-basis of $I$.
\end{proposition}
\begin{proof}
Let $f\in I_l$ for some $l\geq 0$ and consider the element
$f s^l\in I^S$. Since $G$ is an $s$-homogeneous Gr\"obner\ basis of $I^S$,
there is $g s^k\in G$ ($g\in P,k\geq 0$) such that
${\mathrm{lm}}(f s^l) = q s^i {\mathrm{lm}}(g s^k) s^j$ that is ${\mathrm{lm}}(f) s^l = q s^i {\mathrm{lm}}(g) s^{k+j} =
q (s^i\cdot{\mathrm{lm}}(g)) s^l$ with $q\in M$ and $i + j + k = l$. It follows that
${\mathrm{lm}}(f) = q (s^i\cdot{\mathrm{lm}}(g)) = q {\mathrm{lm}}(s^i\cdot g)$ with $g = \pi(g s^k)\in G^P$
and we conclude that $G^P$ is a Gr\"obner\ $\Sigma$-basis of $I$.
\end{proof}
Note that Proposition \ref{gbP2S} and Proposition \ref{gbS2P} explain that
there is a complete equivalence between Gr\"obner\ bases computations for
${\mathrm{w}}$-graded $\Sigma$-invariant ideals $I\subsetneq P$ and their skew
analogues $I^S$ which are graded two-sided ideals of $S$. In particular,
Gr\"obner\ $\Sigma$-bases of $I$ can be computed by the algorithm {\textsc{SkewGBasis}}\
when applied to $I^S$. Precisely, if $H = \bigcup_i H_i$
is a ${\mathrm{w}}$-homogeneous $\Sigma$-basis of $I$ and $G = {\textsc{SkewGBasis}}(H^S)$ then
$G^P = \pi(G)$ is a ${\mathrm{w}}$-homogeneous Gr\"obner\ $\Sigma$-basis of $I$.
We may call this procedure \SigmaGBasis2.
The following result provides algorithmic solution of the membership
problem for a class of $\Sigma$-invariant ideals. Note that such kind
of results are quite rare, for instance, in the theory of difference
ideals.
\begin{proposition}
\label{Ptermin}
Let $I\subset P$ be a ${\mathrm{w}}$-graded $\Sigma$-invariant ideal and fix $d\geq 0$.
Assume that $I$ has a ${\mathrm{w}}$-homogeneous basis $H$ such that $H_d =
\{f\in H\mid {\mathrm{w}}(f)\leq d\}$ is a finite set. Then, there is
a ${\mathrm{w}}$-homogeneous Gr\"obner\ $\Sigma$-basis $G$ of $I$ such that $G_d$
is also a finite set. In other words, if we consider for the algorithm
{\textsc{SigmaGBasis}}\ a selection strategy of the S-polynomials based on their
weights, we obtain that the $d$-truncated version of {\textsc{SigmaGBasis}}\ stops
in a finite number of steps.
\end{proposition}
\begin{proof}
First of all, note that the algorithm {\textsc{SigmaGBasis}}\ essentially
computes a subset $G$ of a Gr\"obner\ basis $\Sigma\cdot G$ obtained
by applying the Buchberger algorithm to the basis $\Sigma\cdot H$
of $I$. Moreover, by property (iii) of Definition \ref{weight} and
Lemma \ref{lcmbound} the elements of $\Sigma\cdot H$ and hence of
$\Sigma\cdot G$ are all ${\mathrm{w}}$-homogeneous. Denote $H'_d =
\{ s^i\cdot f\mid i\geq 0, f\in H_d, i + {\mathrm{w}}(f)\leq d\}$. Since $H'_d$
is also a finite set, consider $X_d$ the finite set of variables
of $P$ occurring in the elements of $H'_d$ and define $P^{(d)} = K[X_d]$.
In fact, the $d$-truncated algorithm {\textsc{SigmaGBasis}}\ computes a subset
of a Gr\"obner\ basis of the ideal $I^{(d)}$ of $P^{(d)}$ generated
by $H'_d$. By Noetherianity of the ring $P^{(d)}$, we clearly obtain
termination.
\end{proof}
Note that the above result can be obtained also by Proposition \ref{termin}.
In fact, if $I\subsetneq P$ is finitely $\Sigma$-generated up to weight $d$
then $I^S$ is a graded ideal of $S$ which is finitely generated up to
$s$-degree $d$. Precisely, if $H = \bigcup_i H_i$ is a ${\mathrm{w}}$-homogeneous
$\Sigma$-basis of $I$ and the set $\bigcup_{i\leq d} H_i$ is finite
for all $d$, then $\{ f s^i \mid f\in H_i, i\leq d \}$ is also a finite
set that generates $I^S$ up to degree $d$.
\section{The skew letterplace embedding}
Denote ${\mathbb N}^* = \{1,2,\ldots\}$ the set of positive integers and
let $X = \{x_1,x_2,\ldots\}$ be a finite or countable set of variables.
We denote by $x_i(j)$ each element $(x_i,j)$ of the product set
$X\times{\mathbb N}^*$ and define $P = K[X\times{\mathbb N}^*]$ the polynomial ring in the
commuting variables $x_i(j)$. Consider the algebra monomorphism
of infinite order $\sigma:P \to P$ such that $x_i(j)\mapsto x_i(j+1)$
for all $i,j$. Note that $\sigma$ is a monomial map that is compatible
with divisibility in ${\mathrm{Mon}}(P)$. Then, put $S = P[s;\sigma]$ the skew polynomial
ring in the variable $s$ defined by $P$ and $\sigma$. Finally, let
$F = {K\langle X \rangle}$ denote the free associative algebra generated by $X$.
We consider $F$ as a graded algebra with respect to the total degree.
Recall that $S = \bigoplus_{i\in{\mathbb N}} S_i$ is also a graded algebra
with $S_i = P s^i$.
\begin{definition}
Let $A\subset S$ be a $K$-subalgebra. If $A$ is spanned by a submonoid
$M\subset{\mathrm{Mon}}(S)$ then we call $A$ {\em a monomial subalgebra of $S$}
and we denote ${\mathrm{Mon}}(A) = M$. In this case, a monomial ordering of $S$ can
be restricted to $A$.
\end{definition}
For instance, $P$ is a monomial subalgebra of $S$. We have now a result
about the possibility to embed the free associative algebra $F$
into the skew polynomial ring $S$.
\begin{proposition}
The graded algebra homomorphism $\iota:F\to S, x_i\mapsto x_i(1)s$
is injective. Then, the free associative algebra $F$ is isomorphic
to $R = {\mathrm{Im\,}}\iota$, a graded monomial subalgebra of $S$.
\end{proposition}
\begin{proof}
It is sufficient to note that by the commutation rule of the variable $s$
and the definition of the endomorphism $\sigma$, any word
$x_{i_1}\cdots x_{i_d}\in{\mathrm{Mon}}(F)$ maps into
$x_{i_1}(1)\cdots x_{i_d}(d)s^d\in{\mathrm{Mon}}(S)$.
\end{proof}
We call $S$ the {\em skew letterplace algebra} and the algebra monomorphism
$\iota$ the {\em skew letterplace embedding}. In Section 7 we will give
motivation for such names.
Fix now a monomial ordering $\prec$ on the algebra $S$ that is $\sigma$ is
compatible with the restriction of $\prec$ to ${\mathrm{Mon}}(P)$. It is easy to
show that many usual monomial orderings on $P$ (lex, degrevlex, etc)
satisfy such condition. Recall that the existence of monomial orderings
for $P$ is provided by the Higman's lemma which implies the following result
(see for instance \cite{AH}, Corollary 2.3 and remarks at beginning
of page 5175).
\begin{proposition}
Let $\prec$ be a total ordering on the set ${\mathrm{Mon}}(P)$ such that for all
$m,n,t\in {\mathrm{Mon}}(P)$ one has $1\preceq m$ and if $m\prec n$ then $t m \prec t n$.
Then $\prec$ is also a well-ordering of ${\mathrm{Mon}}(P)$ that is a monomial
ordering of $P$ if and only if the restriction of $\prec$ to the variables
set $X\times{\mathbb N}^*$ is a well-ordering.
\end{proposition}
Clearly, it is easy to assign well-orderings to the set $X\times{\mathbb N}^*$
which is in bijective correspondence to ${\mathbb N}^2$. Note that the algebra $P$
has also a multigrading which is defined as follows.
If $m = x_{i_1}(j_1)\cdots x_{i_d}(j_d)\in{\mathrm{Mon}}(P)$, then we denote
$\partial(m) = \mu = (\mu_k)_{k\in{\mathbb N}^*}$ where $\mu_k =
\#\{\alpha \mid j_\alpha = k\}$. If $P_\mu\subset P$ is the subspace
spanned by all monomials of multidegree $\mu$ then $P = \bigoplus_\mu P_\mu$
is clearly a multigrading of the algebra $P$. If $\mu = (\mu_k)$ is
a multidegree, we denote $i\cdot\mu = (\mu_{k-i})_{k\in{\mathbb N}^*}$ where
we put $\mu_{k-i} = 0$ when $k-i < 1$. By definition of the map $\sigma$,
if we denote $S_{\mu,i} = P_\mu s^i$ one obtains that
$S = \bigoplus_{\mu,i} S_{\mu,i}$ and $S_{\mu,i} S_{\nu,j}\subset
S_{\mu + (i\cdot\nu), i + j}$.
The elements of each subspace $S_{\mu,i}\subset S$ are said
{\em multi-homogeneous}. An ideal $J\subset S$ is called {\em multigraded}
if $J = \sum_{\mu,i} J_{\mu,i}$ with $J_{\mu,i} = J\cap S_{\mu,i}$.
In other words, the ideal $J$ is generated by multi-homogeneous elements.
For any integer $i\geq 0$ we denote by $1^i$ the multidegree $\mu =
(\mu_k)_{k\in{\mathbb N}^*}$ such that $\mu_k = 1$ if $k\leq i$ and $\mu_k = 0$
otherwise. Clearly, a homogeneous element $f s^i\in S$ ($f\in P$)
belongs to the graded subalgebra $R$ if and only if $f$ is multi-homogeneous
and $\partial(f) = 1^i$. In other words, $R_i = R\cap S_i = S_{1^i,i} =
P_{1^i} s^i$.
\begin{lemma}
\label{Rgood}
Let $f s^l\in S$ with $f\in P$ a multi-homogeneous element and consider
$f_{ij} s^i, g_j s^j, h_{jk} s^k\in S$ where $f_{ij}, g_j, h_{jk}\in P$
are multi-homogeneous elements such that $f s^l =
\sum_{i+j+k=l} f_{ij} s^i g_j s^j h_{jk} s^k$. Then, from $f s^l\in R$
it follows that $f_{ij} s^i, g_j s^j, h_{jk} s^k\in R$, for all $i,j,k$.
\end{lemma}
\begin{proof}
Clearly we have $f = \sum_{i+j+k=l} f_{ij} g_j^{s^i} h_{jk}^{s^{i+j}}$.
Denote $\mu = \partial(f_{ij}), \nu = \partial(g_j^{s^i})$ and
$\rho = \partial(h_{jk}^{s^{i+j}})$ and put $\alpha = \min\{k\mid\nu_k > 0\}$
and $\beta = \min\{k\mid\rho_k > 0\}$. By definition of the map $\sigma$,
one has that $\alpha\geq i+1$ and $\beta\geq i+j+1$. If we assume $f s^l\in R$
that is $1^l = \partial(f) = \mu + \nu + \rho$, then necessarily
$\mu = 1^i, \nu = i\cdot 1^j$ and $\rho = (i+j)\cdot 1^k$ and hence
$\partial(f_{ij}) = 1^i, \partial(g_j) = 1^j, \partial(h_{jk}) = 1^k$.
\end{proof}
\begin{proposition}
\label{gengood}
Let $I$ be a graded (two-sided) ideal of $R\subset S$ and let $J$ be
the {\em extension of $I$ to $S$} that is $J$ is the (multigraded)
ideal generated by $I$ in $S$. If $G$ is a multi-homogeneous
basis of $J$ then $G\cap R$ is a (homogeneous) basis of $I$.
In particular, the {\em contraction} $J\cap R$ is equal to $I$,
that is there is a {\em bijective correspondence} between all graded
ideals of $R$ and their extensions to $S$.
\end{proposition}
\begin{proof}
Consider $f s^l\in I\subset R$ ($f\in P$) a homogeneous element and
let $G = \{g_j s^j\}$ with $g_j\in P$, $g_j$ multi-homogeneous.
Since $f$ is multi-homogeneous and $G$ is a basis of $J\supset I$,
one has $f s^l = \sum_{i+j+k=l} f_{ij} s^i g_j s^j h_{jk} s^k$
with $f_{ij},h_{jk}\in P$, $f_{ij},h_{jk}$ multi-homogeneous.
From Lemma \ref{Rgood} it follows immediately that all
elements $f_{ij} s^i, g_j s^j, h_{jk} s^k\in R$ that is $G\cap R$
is a basis of $I$.
\end{proof}
\begin{proposition}
\label{gbgood}
Let $I\subset R$ be a graded ideal and let $J\subset S$ be its
extension. If $G\subset J$ is a multi-homogeneous Gr\"obner\ basis of $J$ then
$G\cap R$ is a homogeneous Gr\"obner\ basis of $I$.
\end{proposition}
\begin{proof}
If $f s^l = \sum_{i+j+k=l} f_{ij} s^i g_j s^j h_{jk} s^k$
is a Gr\"obner\ representation in $S$ of a homogeneous element $f s^l\in I\subset J$
with respect to $G = \{g_j s^j\}$, then it is sufficient to use the same
argument of Proposition \ref{gengood} to obtain that $f s^l$ has a Gr\"obner\
representation in $R$ with respect to $G\cap R$.
\end{proof}
We obtain finally an algorithm to compute Gr\"obner\ bases of graded two-sided
ideals of the subring $R\subset S$ which is isomorphic to the free associative
algebra $F$ by the map $\iota$. Note that the considered monomial orderings
on $F$ are obtained as the restriction of monomial orderings on $S$ to the
monomial subalgebra $R$.
By applying Proposition \ref{gbgood}, the computation of homogeneous
Gr\"obner\ bases in $R$ is obtained as a slight modification of the algorithm
{\textsc{SkewGBasis}}\ for the ideals of $S$. It is interesting to note that
the latter procedure is in turn a variant of the Buchberger algorithm
for modules over commutative polynomial rings. Thus, we may say that
these computations in associative algebras are reduced to analogue ones
over commutative rings via the notion of skew polynomial ring (see also
Section 7). This reverses somehow the trivial fact that commutative algebras
are just a subclass of the associative ones.
\suppressfloats[b]
\begin{algorithm}\caption{\FreeGBasis2}
\begin{algorithmic}[0]
\State \text{Input:} $H$, a homogeneous basis of a graded two-sided
ideal $I\subset R$.
\State \text{Output:} $G$, a homogeneous Gr\"obner\ basis of $I$.
\State $G:= H$;
\State $B:= \{(f,g) \mid f,g\in G\}$;
\While{$B\neq\emptyset$}
\State choose $(f,g)\in B$;
\State $B:= B\setminus \{(f,g)\}$;
\ForAll{$i,j\geq 0, i + j = \deg_s(f) - \deg_s(g)$ and
${\mathrm{spoly}}(f,s^i g s^j)\in R$}
\State $h:= {\textsc{Reduce}}({\mathrm{spoly}}(f,s^i g s^j), \Sigma\,G\,\Sigma)$;
\If{$h\neq 0$}
\State $B:= B\cup\{(h,h),(h,k),(k,h)\mid k\in G\}$;
\State $G:= G\cup\{h\}$;
\EndIf;
\EndFor;
\ForAll{$i,j\geq 0, j - i = \deg_s(f) - \deg_s(g)$ and
${\mathrm{spoly}}(f s^i, s^j g)\in R$}
\State $h:= {\textsc{Reduce}}({\mathrm{spoly}}(f s^i, s^j g), \Sigma\,G\,\Sigma)$;
\If{$h\neq 0$}
\State $B:= B\cup\{(h,h),(h,k),(k,h)\mid k\in G\}$;
\State $G:= G\cup\{h\}$;
\EndIf;
\EndFor;
\EndWhile;
\State \Return $G$.
\end{algorithmic}
\end{algorithm}
\newpage
Note explicitely that conditions ${\mathrm{spoly}}(f,s^i g s^j), {\mathrm{spoly}}(f s^i, s^j g)\in R$
are equivalent to ask that such multi-homogeneous elements of $S$ have
multidegrees of type $(1^d,d)$, for some $d\geq 0$.
\begin{proposition}
The algorithm \FreeGBasis2\ is correct.
\end{proposition}
\begin{proof}
Since $G$ is multi-homogeneous implies that $\Sigma\,G\,\Sigma$ is also
multi-homogeneous, the procedure {\textsc{Reduce}}\ clearly preserves multi-homogeneity.
Moreover, any element $f\in G$ ($f\notin H$) is obtained by reduction
of a S-polynomial, say $h$. Owing to Proposition \ref{gbgood} we are
interested only in the elements $f\in R$ and this holds if and only if
$h\in R$.
\end{proof}
Assume now that the graded ideal $I\subset R$ has a finite number of generators
up to some degree $d > 0$. Note that the $d$-truncated algorithm \FreeGBasis2\
has termination provided by termination of {\textsc{SkewGBasis}}\ as stated in
Proposition \ref{termin}. This generalizes a well-known result about
algorithmic solution of the word problem (membership problem)
for finitely presented graded associative algebras.
\section{Letterplace in $P$}
As in Section 5, consider the $P$-linear map $\pi:S\to P$ such that
$s^i \mapsto 1$, for all $i$. Note now that $\iota' = \pi\iota:F\to P$
is an injective $K$-linear map such that
$x_{i_1}\cdots x_{i_d}\in{\mathrm{Mon}}(F)\mapsto x_{i_1}(1)\cdots x_{i_d}(d)\in{\mathrm{Mon}}(P)$.
Recall that $F = \bigoplus_i F_i$ is a graded algebra with respect to
total degree. Moreover, consider the weight map ${\mathrm{w}}:{\mathrm{Mon}}(P)\to{\hat{\N}}$ such that
${\mathrm{w}}(x_i(j)) = j$ for all $i,j\geq 1$ and the corresponding grading
$P = \bigoplus_{i\in{\hat{\N}}} P_i$ defined by the monoid $({\hat{\N}},\max)$.
Then, we have that $\iota'$ is a homogeneous map (note $K = F_0 = P_{-\infty}$
and $P_0 = 0$) and $\iota = \xi \iota'$ which is an algebra homomorphism.
\begin{definition}
Let $I\subsetneq F$ be a graded (two-sided) ideal. Denote by $I'\subsetneq P$
the ${\mathrm{w}}$-graded $\Sigma$-invariant ideal $\Sigma$-generated by $\iota'(I)$.
In other words, if $G = \{\iota'(f) \mid f\in I_i, i > 0\}$ then
$I'$ is the ideal of $P$ generated by $\Sigma\cdot G$. We call $I'$
the {\em letterplace analogue} of $I$.
\end{definition}
\begin{proposition}
Let $I\subsetneq F$ be a graded ideal and $I'\subsetneq P$ its letterplace
analogue. Denote by $J = I'^S$ the skew analogue of $I'$ and call $J$
the {\em skew letterplace analogue} of $I$. We have that $J$ is the extension
to $S$ of the ideal $\iota(I)\subset R$. Then, there is a {\em bijective
correspondence} between all graded ideals of $F$ and their (skew) letterplace
analogues.
\end{proposition}
\begin{proof}
Let $J'$ be the extension of $\iota(I)$ to $S$. By definition $J'$ is the ideal
generated by the elements $\iota(f) = \iota'(f) s^i$, for all $f\in I_i$.
Since $I'$ is $\Sigma$-generated by the ${\mathrm{w}}$-homogeneous elements $\iota'(f)$
of weight $i$, we conclude that $J = I'^S = J'$. Moreover, the bijective
correspondence between graded two-sided ideals of $F$ and their letterplace
analogues in $P$ is obtained by composing the bijections contained
in Proposition \ref{S2Pcor} and Proposition \ref{gengood}.
\end{proof}
The bijection between graded ideals of $F$ and their letterplace analogues
has been introduced in \cite{LSL} and called ``letterplace correspondence''.
The motivation of such name is essentially historical since the linear
map $\iota'$ was first considered in \cite{Fe,DRS}. Note that in these
articles the endomorphism $\sigma$ and the algebra embedding $\iota$
were not introduced. The polynomial ring $P$ was named there the ``letterplace
algebra'' because in the monomial $\iota'(x_{i_1}\cdots x_{i_d}) =
x_{i_1}(1)\cdots x_{i_d}(d)$ the indices $1,\ldots,d$ play the role
of the ``places'' where the ``letters'' $x_{i_1},\ldots,x_{i_d}$ occur
in the word $x_{i_1}\cdots x_{i_d}\in{\mathrm{Mon}}(F)$.
Fix now a monomial ordering $\prec$ on the algebra $S$ that is $\sigma$
is compatible with the restriction of $\prec$ to ${\mathrm{Mon}}(P)$. By restricting
$\prec$ to $R$ one obtains a monomial ordering on $F$. Denote by $V$
the image of the map $\iota'$ that is $V = \bigoplus_i V_i$ is a graded
subspace of $P$ where $V_i = P_{1^i}\subset P_i$. Note that $V$ is a left
$R$-module isomorphic to $R\approx F$. In fact, $V = \pi(R)$ and
the restriction $\pi:R\to V$ has the restriction $\xi:V\to R$ as its inverse.
In \cite{LSL} one has the following result which is now a direct
consequence of Proposition \ref{gbP2S} and Proposition \ref{gbgood}.
\begin{proposition}
Let $I\subsetneq F$ be a graded ideal and denote by $J\subsetneq P$ its
letterplace analogue. Then $J$ is a multigraded (hence ${\mathrm{w}}$-graded)
$\Sigma$-invariant ideal of $P$. If $G$ is a multi-homogeneous
(hence ${\mathrm{w}}$-homogeneous) Gr\"obner\ $\Sigma$-basis of $J$ then $\iota'^{-1}(G\cap V)$
is a homogeneous Gr\"obner\ basis of $I$.
\end{proposition}
From this result and algorithm {\textsc{SigmaGBasis}}\ one obtains the correctness
of the following procedure which also has been introduced in \cite{LSL}.
\suppressfloats[b]
\begin{algorithm}\caption{{\textsc{FreeGBasis}}}
\begin{algorithmic}[0]
\State \text{Input:} $H$, a homogeneous basis of a graded two-sided
ideal $I\subsetneq F$.
\State \text{Output:} $\iota'^{-1}(G)$, a homogeneous Gr\"obner\ basis of $I$.
\State $G:= \iota'(H)$;
\State $B:= \{(f,g) \mid f,g\in G\}$;
\While{$B\neq\emptyset$}
\State choose $(f,g)\in B$;
\State $B:= B\setminus \{(f,g)\}$;
\ForAll{$i\geq 0$ such that ${\mathrm{spoly}}(f,s^i\cdot g)\in V$}
\State $h:= {\textsc{Reduce}}({\mathrm{spoly}}(f,s^i\cdot g), \Sigma\cdot G)$;
\If{$h\neq 0$}
\State $B:= B\cup\{(h,h),(h,k),(k,h),\mid k\in G\}$;
\State $G:= G\cup\{h\}$;
\EndIf;
\EndFor;
\EndWhile;
\State \Return $\iota'^{-1}(G)$.
\end{algorithmic}
\end{algorithm}
\newpage
Assume finally that the graded ideal $I\subsetneq F$ has a finite number
of generators up to some degree $d > 0$. Note that the $d$-truncated algorithm
{\textsc{FreeGBasis}}\ has now termination provided by Proposition \ref{Ptermin}.
\section{Examples and timings}
In this section we propose an explicit computation and some timings
in order to provide some concrete experience with the algorithms
we introduced.
Let $X = \{x\}$ and consider the ring of ordinary difference polynomials
$P = K[X\times{\mathbb N}]$ that is $P$ is the polynomial ring in the variables $x(j)$
which are the shifts of a single univariate function $x = x(0)$.
Moreover, let $P$ be endowed with the lexicographic monomial ordering
where $x(0) < x(1) < \ldots$. Denote by $J$ the difference ideal generated by
the single difference polynomial $g_1 = x(2)x(0) - x(1)$. This ideal
has been considered in \cite{GL} as an example of an ordinary difference
equation with periodic solutions. A variant of this equation has been also
considered in \cite{Ge}.
By the algorithm ${\textsc{SigmaGBasis}}$ one can compute that $J$ has a finite
Gr\"obner\ difference basis with elements
\[
\begin{array}{l}
g_1 = x(2)x(0) - x(1),
g_2 = x(4)x(1) - x(3)x(0),
g_3 = x(3)^2x(0) - x(3), \\
g_4 = x(4)x(3)x(0) - x(4),
g_5 = x(5) - x(4)x(0).
\end{array}
\]
Let us see how the algorithm {\textsc{SigmaGBasis}}\ is able to obtain this.
If $\Sigma = \langle \sigma \rangle$ where $\sigma:P\to P$ is the
shift endomorphism $x(i)\mapsto x(i+1)$, then the ideal $J$ is
by definition $\Sigma$-generated by $\{g_1\}$ that is it is generated
by $\Sigma\cdot\{g_1\} = \{x(2)x(0) - x(1),x(3)x(1) - x(2),
x(4)x(2) - x(3),\ldots\}$. Up to the product criterion, to compute
a Gr\"obner\ basis of $J$ one should consider all the S-polynomials
${\mathrm{spoly}}(\sigma^i\cdot g_1,\sigma^{i+2}\cdot g_1)$ for any $i\geq 0$.
We are interested in fact in computing a Gr\"obner\ $\Sigma$-basis of $J$ and
hence we can apply the $\Sigma$-criterion that kills all these S-polynomials
except for ${\mathrm{spoly}}(g_1,\sigma^2\cdot g_1)$. The reduction of this element
with respect to $\Sigma\cdot\{g_1\}$ leads to $g_2 = x(4)x(1) - x(3)x(0)$.
Now the current $\Sigma$-basis of $J$ is $\{g_1,g_2\}$. The S-polynomials that
survive to product and $\Sigma$-criterion are now
\[
\begin{array}{l}
{\mathrm{spoly}}(g_2,\sigma\cdot g_1),
{\mathrm{spoly}}(g_2,\sigma^2\cdot g_1),
{\mathrm{spoly}}(g_2,\sigma^4\cdot g_1),
{\mathrm{spoly}}(g_1,\sigma\cdot g_2), \\
{\mathrm{spoly}}(g_2,\sigma^3\cdot g_2).
\end{array}
\]
Then ${\mathrm{spoly}}(g_2,\sigma^2\cdot g_1)\to 0$ and ${\mathrm{spoly}}(g_2,\sigma\cdot g_1)$
reduces to $g_3$ with respect to $\Sigma\cdot\{g_1,g_2\}$. The list of new
S-polynomials arising from $g_3$ that pass product and $\Sigma$-criterion is
\[
\begin{array}{l}
{\mathrm{spoly}}(g_3,g_1),
{\mathrm{spoly}}(g_3,\sigma\cdot g_1),
{\mathrm{spoly}}(g_3,\sigma^3\cdot g_1),
{\mathrm{spoly}}(g_3,\sigma^2\cdot g_2), \\
{\mathrm{spoly}}(g_2,\sigma\cdot g_3),
{\mathrm{spoly}}(g_1,\sigma^2\cdot g_3),
{\mathrm{spoly}}(g_3,\sigma^3\cdot g_3),
{\mathrm{spoly}}(g_2,\sigma^4\cdot g_3).
\end{array}
\]
We have now that ${\mathrm{spoly}}(g_3,\sigma\cdot g_1)\to 0,
{\mathrm{spoly}}(g_3,g_1)\to 0$ and ${\mathrm{spoly}}(g_2,\sigma\cdot g_3)\to g_4$.
Up to all criteria, including chain criterion, the list of S-polynomials
has to be updated with the following ones
\[
\begin{array}{l}
{\mathrm{spoly}}(g_4,\sigma\cdot g_1),
{\mathrm{spoly}}(g_4,\sigma^2\cdot g_1),
{\mathrm{spoly}}(g_4,\sigma^3\cdot g_1),
{\mathrm{spoly}}(g_4,\sigma^4\cdot g_1), \\
{\mathrm{spoly}}(g_4,\sigma^3\cdot g_2),
{\mathrm{spoly}}(g_2,\sigma\cdot g_4),
{\mathrm{spoly}}(g_1,\sigma^2\cdot g_4),
{\mathrm{spoly}}(g_3,\sigma^3\cdot g_4), \\
{\mathrm{spoly}}(g_4,\sigma^3\cdot g_4),
{\mathrm{spoly}}(g_2,\sigma^4\cdot g_4),
{\mathrm{spoly}}(g_4,\sigma^4\cdot g_4).
\end{array}
\]
Now, one has the following reductions: ${\mathrm{spoly}}(g_4,\sigma\cdot g_1)\to 0,
{\mathrm{spoly}}(g_4,\sigma^2\cdot g_1)\to 0$ and ${\mathrm{spoly}}(g_1,\sigma\cdot g_2)\to
f = x(5)x(1) - x(3)x(0)^2$. We will show that the element $f$ of the
Gr\"obner\ $\Sigma$-basis of $J$ is in fact redundant because $g_5$ is also
in this basis. The new S-polynomials arising from $f$ are
\[
\begin{array}{l}
{\mathrm{spoly}}(f,\sigma\cdot g_1),
{\mathrm{spoly}}(f,\sigma^5\cdot g_1),
{\mathrm{spoly}}(f,g_2),
{\mathrm{spoly}}(f,\sigma\cdot g_2),
{\mathrm{spoly}}(f,\sigma^4\cdot g_2), \\
{\mathrm{spoly}}(g_1,\sigma\cdot f),
{\mathrm{spoly}}(g_3,\sigma^2\cdot f),
{\mathrm{spoly}}(g_4,\sigma^2\cdot f),
{\mathrm{spoly}}(g_2,\sigma^3\cdot f), \\
{\mathrm{spoly}}(g_4,\sigma^3\cdot f),
{\mathrm{spoly}}(f,\sigma^4\cdot f).
\end{array}
\]
Then, we start again with reductions:
${\mathrm{spoly}}(f,\sigma\cdot g_2)\to 0,{\mathrm{spoly}}(f,\sigma\cdot g_1)\to 0$,
${\mathrm{spoly}}(g_3,\sigma^3\cdot g_1)\to 0,{\mathrm{spoly}}(g_2,\sigma\cdot g_4)\to g_5$
and therefore $f$ is redundant. The last S-polynomials to be added
are
\[
\begin{array}{l}
{\mathrm{spoly}}(g_5,\sigma^3\cdot g_1),
{\mathrm{spoly}}(g_5,\sigma^5\cdot g_1),
{\mathrm{spoly}}(g_5,\sigma\cdot g_2),
{\mathrm{spoly}}(g_5,\sigma^4\cdot g_2),\\
{\mathrm{spoly}}(g_5,f),
{\mathrm{spoly}}(g_5,\sigma^4\cdot f).
\end{array}
\]
If $G = \{g_1,g_2,g_3,g_4,g_5\}$ then we have that all remaining S-polynomials
to be considered reduce to zero with respect to $\Sigma\cdot G$, that is $G$
is a Gr\"obner\ $\Sigma$-basis (difference basis) of the $\Sigma$-ideal
(difference ideal) $J$.
We present now some timings obtained with an implementation of the algorithm
{\textsc{FreeGBasis}}. This implementation, which is still under development,
is an improvement of the one we presented in \cite{LSL}. We decided not
to start implementing also \FreeGBasis2\ until {\textsc{FreeGBasis}}\ will evolute
to some final form. We propose here new comparisons with the system
\textsc{Magma} that contains one of the most effective implementations
of the classical algorithm \cite{Mo,Gr1,Uf} for computing non-commutative
Gr\"obner\ bases. Note that this implementation takes also advantage by the use of
Faug{\`e}re's F4 approach. The tests were performed on a PC with four Intel
Core i7 CPU 940 2.93GHz processors with 12 GB RAM running Ubuntu Linux.
We used \textsc{Singular} 3-1-3 with \texttt{freegb.lib} release 14203
and \textsc{Magma} version 2.17-8. We measured the time for real execution
of the process (thus differently to the way we did comparisons in \cite{LSL})
in "min:sec" format. The number of generators in the input and in the output
are given as well.
\medskip
\begin{center}
\begin{tabular}[h]{|l|c|c|c|c|}
\hline
Example & {\sc Magma} & {\sc Singular} & $\#$In & $\#$Out \\
\hline
\texttt{G3-5-6-2d12} & 0:10 & 1:15 & 11 & 5885 \\
\texttt{G2-3-13-4d10} & 0:05 & 0:01 & 10 & 275 \\
\texttt{G3-8-13d8} & 0:05 & 0:04 & 18 & 1490 \\
\texttt{serf-g2d8} & 0:05 & 0:01 & 17 & 6 \\
\texttt{cliff5d9} & 0:08 & 0:12 & 41 & 168 \\
\texttt{C41d6} & 0:05 & 0:10 & 6 & 50 \\
\texttt{C41Xd5} & 0:08 & 0:04 & 6 & 44 \\
\texttt{C41Yd5} & 0:05 & 0:03 & 6 & 44 \\
\texttt{C41Zd6} & 0:08 & 0:10 & 6 & 44 \\
\texttt{C41Wd6} & 0:05 & 0:01 & 6 & 35 \\
\hline
\end{tabular}
\end{center}
\medskip
This table shows essentially that the letterplace approach to the computation
of non-commutative Gr\"obner\ bases is comparable with the classical algorithms
and hence it is feasible. From the viewpoint of implementations we record
that \textsc{Magma} achieved significant improvements with respect to comparisons
included in \cite{LSL} and this stimulate us to further optimize our code.
In fact, there is an ongoing work to enhance \texttt{freegb.lib} in
\textsc{Singular}. We will make more extensive comparisons in future articles
that will be concentrated on technical aspects of implementing the
letterplace algorithms.
Here is a brief description of the examples we considered for testing.
In all the examples the last integer indicates the total degree that bounds
the computations. The examples \texttt{G3-5-6-2, G2-3-13-4} refer to the class
of presented groups $G(l,m,n;q) = \langle r,s\mid r^l, s^m, (rs)^n,
[r,s]^q \rangle$, where $[r,s]$ denotes the commutator.
As for the example \texttt{G3-8-13}, this is one from the class of groups
$G(m,n,p) = \langle a,b,c\mid a^m, b^n, c^p, (ab)^2, (bc)^2, (ca)^2,
(abc)^2\rangle$. All these groups has been considered by Coxeter \cite{Co}
for the problem of determining their finiteness. For our computations
we considered a homogenization of the ideal of the free associative algebra
defining the group algebra of such groups. The example \texttt{serf-g2}
are modified full Serre relations built from the Cartan matrix $G_2$.
The following non-commutative polynomials are explicitly the generators
we considered for homogenization.
\[
\begin{footnotesize}
\begin{array}{l}
f_1 f_2 f_2 - 2 f_2 f_1 f_2 + f_2 f_2 f_1,
e_1 e_2 e_2 - 2 e_2 e_1 e_2 + e_2 e_2 e_1,\\
f_1 f_1 f_1 f_1 f_2 - 4 f_1 f_1 f_1 f_2 f_1 + 6 f_1 f_1 f_2 f_1 f_1 -
4 f_1 f_2 f_1 f_1 f_1 + f_2 f_1 f_1 f_1 f_1, \\
e_1 e_1 e_1 e_1 e_2 - 4 e_1 e_1 e_1 e_2 e_1 + 6 e_1 e_1 e_2 e_1 e_1 -
4 e_1 e_2 e_1 e_1 e_1 + e_2 e_1 e_1 e_1 e_1,\\
f_2 e_1 - e_1 f_2,
f_1 e_2 - e_2 f_1,
f_1 e_1 - e_1 f_1 + h_1,
f_2 e_2 - e_2 f_2 + h_2,\\
h_1 h_2 - h_2 h_1,
h_1 e_1 - e_1 h_1 - 2 e_1,
f_1 h_1 - h_1 f_1 - 2 f_1,
h_1 e_2 - e_2 h_1 + e_2,\\
f_2 h_1 - h_1 f_2 + f_2,
h_2 e_1 - e_1 h_2 + 3 e_1,
f_1 h_2 - h_2 f_1 + 3 f_1,
h_2 e_2 - e_2 h_2 - 2 e_2,\\
f_2 h_2 - h_2 f_2 - 2 f_2.
\end{array}
\end{footnotesize}
\]
Let $F_5 = K\langle x_1,x_2,x_3,x_4,x_5 \rangle$ and define
$\Gamma\subset {\mathrm{End}}_K(F_5)$ the submonoid of all algebra endomorphisms
sending variables into variables. The example \texttt{cliff5} is the ideal
$I\subset F_5$ which is $\Gamma$-generated by the polynomials
$[x_1\circ x_2,x_3] = (x_1x_2 + x_2x_1) x_3 - x_3 (x_1x_2 + x_2x_1)$ and
$s_5 = \sum_{\pi\in{\mathbb S}_5} {\rm sgn}(\pi)
x_{\pi(1)}x_{\pi(2)}x_{\pi(3)}x_{\pi(4)}x_{\pi(5)}$. The quotient ring $F_5/I$
is the generic Clifford algebra in 5 variables of a 4-dimensional vector space.
Finally, the family \texttt{C41} of examples originates from random linear
substitutions into the ideal of 6 generators, defining the non-cancellative
monoid $C(4,1)$ (see \cite{JO}) and includes also variations of those.
For instance, \texttt{C41W} is given by
\[
\begin{array}{l}
x_4 x_4-25 x_4 x_2-x_1 x_4-6 x_1 x_3-9 x_1 x_2+x_1 x_1, \\
x_4 x_3+13 x_4 x_2+12 x_4 x_1-9 x_3 x_4+4 x_3 x_2+41 x_3 x_1-7 x_1 x_4-x_1 x_2, \\
x_3 x_3-9 x_3 x_2+2 x_1 x_4+x_1 x_1,
17 x_4 x_2-5 x_2 x_2-41 x_1 x_4, \\
x_2 x_2-13 x_2 x_1-4 x_1 x_3+2 x_1 x_2-x_1 x_1,
x_2 x_1+4 x_1 x_2-3 x_1 x_1.
\end{array}
\]
while \texttt{C41} is given by
\[
\begin{footnotesize}
\begin{array}{l}
189 x_4 x_4+63 x_4 x_3-66 x_4 x_2-161 x_4 x_1-103 x_3 x_4+19 x_3 x_3+262 x_3 x_2+467 x_3 x_1 -\\
360 x_2 x_4-144 x_2 x_3+24 x_2 x_2+136 x_2 x_1+175 x_1 x_4+35 x_1 x_3-160 x_1 x_2-315 x_1 x_1,\\
27 x_4 x_4+409 x_4 x_3+82 x_4 x_2-42 x_4 x_1-57 x_3 x_4-403 x_3 x_3+26 x_3 x_2-42 x_3 x_1 -\\
50 x_2 x_4-434 x_2 x_3-12 x_2 x_2-14 x_2 x_1+45 x_1 x_4+435 x_1 x_3+30 x_1 x_2,\\
232 x_4 x_4-29 x_4 x_3+77 x_4 x_2+332 x_4 x_1-147 x_3 x_4+175 x_3 x_3+60 x_3 x_2-269 x_3 x_1 -\\
107 x_2 x_4+184 x_2 x_3+83 x_2 x_2-217 x_2 x_1+28 x_1 x_4-217 x_1 x_3-139 x_1 x_2+120 x_1 x_1,\\
52 x_4 x_4+233 x_4 x_3-129 x_4 x_2+135 x_4 x_1-248 x_3 x_4-205 x_3 x_3+171 x_3 x_2+138 x_3 x_1 +\\
100 x_2 x_4-58 x_2 x_3-177 x_2 x_1+84 x_1 x_4+39 x_1 x_3-43 x_1 x_2-73 x_1 x_1,\\
-225 x_4 x_4-150 x_4 x_3-179 x_4 x_2-262 x_4 x_1+91 x_3 x_4-94 x_3 x_3+225 x_3 x_2+74 x_3 x_1 +\\
214 x_2 x_4+224 x_2 x_3+90 x_2 x_2+266 x_2 x_1-175 x_1 x_4-50 x_1 x_3-205 x_1 x_2-190 x_1 x_1,\\
289 x_4 x_4-170 x_4 x_3-289 x_4 x_2-153 x_4 x_1-186 x_3 x_4+95 x_3 x_3+177 x_3 x_2+106 x_3 x_1 -\\
231 x_2 x_4+35 x_2 x_3+168 x_2 x_2+175 x_2 x_1+241 x_1 x_4+60 x_1 x_3-115 x_1 x_2-233 x_1 x_1.
\end{array}
\end{footnotesize}
\]
\section{Conclusions and future directions}
From the previous sections we can conclude that, owing to the notion
of Gr\"obner\ $\Sigma$-basis and the skew letterplace embedding $\iota$,
the theory of non-commutative Gr\"obner\ bases developed for the free associative
algebra $F = {K\langle X \rangle}$ using the concepts of overlappings, tips or obstructions
\cite{Gr1,Mo,Uf} can be deduced from, unified to the classical Buchberger theory
for commutative polynomial rings based on S-polynomials, at least
in the graded case. From a practical point of view, one obtains
the alternative algorithms {\textsc{FreeGBasis}}\ and \FreeGBasis2\ which are
implementable in any computer algebra system providing
commutative Gr\"obner\ bases. The feasibility of such methods has been already
shown in \cite{LSL} and confirmed by the new timings we have collected
in Section 8.
Moreover, the general theory developed in this paper can be applied
to any context where a monoid of endomorphisms $\Sigma$ acts on
the polynomial algebra $P = K[X]$ in a way which is compatible with
Gr\"obner\ bases theory. We propose not only an abstract definition of what
this may mean contributing to a current research trend (see for instance
\cite{DLS,AH,BD}), but also a method to transfer the related algorithms
from $P$ to the skew monoid ring $S = P *\Sigma$ when a suitable grading
is given for $P$. This theory applies in particular to the shift operators
and hence a stimulating field of applications are the rings of difference
polynomials. The simple calculation proposed in Section 8 gives some
feeling of this. In particular, we aim to extend the Gr\"obner\ $\Sigma$-bases theory
to any finitely generated free commutative monoid $\Sigma =
\langle \sigma_1,\ldots,\sigma_r \rangle$ in order to cover partial difference
ideals and to extend the letterplace method for $F$ to the non-graded
case by means of suitable (de)homogenization techniques. An effective
implementation of all proposed algorithms will be clearly important
to understand the actual performance of the methods.
\section*{Acknowledgments}
First of all, we would like to thank Vladimir Gerdt for introducing us
to the theory of difference polynomial rings and thus leading us to note
that the letterplace algebra $K[X\times{\mathbb N}]$ endowed with the endomorphism
$x_i(j)\mapsto x_i(j+1)$ is just an instance of them (ordinary case).
We are grateful to Albert Heinle, Benjamin Schnitzler and Grischa Studzinski
for adopting the {\sc SymbolicData} project \cite{SD} for handling ideals
in free algebras. The timings in this article were obtained with the help
of the updated {\sc SymbolicData} system. We like also to thank
the anonymous reviewers for their valuable comments and suggestions.
|
{
"timestamp": "2012-05-24T02:00:22",
"yymm": "1009",
"arxiv_id": "1009.4152",
"language": "en",
"url": "https://arxiv.org/abs/1009.4152"
}
|
\section{Introduction}
Let $\mathbb{N}$ be the set of nonnegative integers. Given an infinite set
$A\subseteq \mathbb{N}$, the symbol $A^{[\infty]}$ (resp. $A^{[<\infty]}$)
represents the collection of the infinite (resp. finite) subsets
of A. Let $A^{[n]}$ denote the set of all the subsets of A with n
elements. If $a\in \mathbb{N}^{[<\infty]}$ is an \textbf{initial segment}
of $A\in \mathbb{N}^{[\infty]}$ then we write $a\sqsubset A$. Also, let
$A/a : = \{n\in A : max(a) < n\}$, and write $A/n$ to mean
$A/\{n\}$. For $a\in \mathbb{N}^{[<\infty]}$ and $A\in\mathbb{N}^{[\infty]}$ let
\[
[a,A] : = \{B\in\mathbb{N}^{[\infty]} : a\sqsubset B\subseteq A\}.
\]
The family $Exp(\mathbb{N}^{[\infty]}) : = \{[a,A] : (a,A)\in
\mathbb{N}^{[<\infty]}\times \mathbb{N}^{[\infty]}\}$ is a basis for
\textbf{Ellentuck's topology}, also known as \textbf{exponential
topology}. In \cite{ell}, Ellentuck gave a characterization of
Ramseyness in terms of the Baire property relative to this
topology (see Theorem \ref{Ellentuck} below).
\medskip
Let $(P,\leq)$ be a poset, a subset $D\subseteq P$ is
\textbf{dense} in $P$ if for every $p \in P$, there is $q \in D$
with $q\leq p$. $D\subseteq P$ is \textbf{open} if $p\in D$ and
$q\leq p$ imply $q\in D$. $P$ is $\sigma$-\textbf{distributive} if
the intersection of countably many dense open subsets of $P$ is
dense. $P$ is $\sigma$-\textbf{closed} if every decreasing
sequence of elements of $P$ has a lower bound.
\medskip
\begin{defn}
A family $\mathcal{H}\subset\wp (\mathbb{N})$ is a \textbf{coideal} if it satisfies:
\begin{itemize}
\item[{(i)}] $A\subseteq B$ and $A\in\mathcal{H}$ implies $B\in\mathcal{H}$, and
\item[{(ii)}] $A\cup B \in \mathcal{H}$ implies $A\in\mathcal{H}$ or $B\in\mathcal{H}$.
\end{itemize}
\end{defn}
The complement $\mathcal{I} = \wp(\mathbb{N})\setminus\mathcal{H}$ is the
\textbf{dual ideal} of $\mathcal{H}$. In this case, as usual, we
write $\mathcal{H} = \mathcal{I}^+$. We will suppose that coideals
differ from $\wp (\mathbb{N})$. Also, we say that a nonempty family
$\mathcal{F}\subseteq\mathcal{H}$ is
$\mathcal{H}$-\textbf{disjoint} if for every $A,B\in\mathcal{F}$,
$A\cap B\not\in\mathcal{H}$. We say that $\mathcal{F}$ is a
\textbf{maximal} $\mathcal{H}$-\textbf{disjoint family} if it is
$\mathcal{H}$-disjoint and it is not properly contained in any
other $\mathcal{H}$-disjoint family as a subset.
\bigskip
A subset $\mathcal X$ of $\mathbb{N}^{[\infty]}$ is \textbf{Ramsey} if for
every $[a,A]\neq\emptyset$ with $A\in\mathbb{N}^{[\infty]}$ there exists
$B\in [a,A]$ such that $[a,B]\subseteq\mathcal{X}$ or
$[a,B]\cap\mathcal{X} = \emptyset$. Some authors have used the
term ``completely Ramsey'' to express this property, reserving the
term ``Ramsey'' for a weaker property. Galvin and Prikry
\cite{gapr} showed that all Borel subsets of $\mathbb{N}^{[\infty]}$ are
Ramsey, and Silver \cite{sil} extended this to all analytic sets.
Mathias in \cite{math} showed that if the existence of an
inaccessible cardinal is consistent with $ZFC$ then it is
consistent, with $ZF+DC$, that every subset of $\mathbb{N}^{[\infty]}$ is
Ramsey. Mathias introduced the concept of a selective coideal (or
a happy family), which has turned out to be of wide interest.
Ellentuck \cite{ell} characterized the Ramsey sets as those having
the Baire property with respect to the exponential topology of
$\mathbb{N}^{[\infty]}$.
A game theoretical characterization of Ramseyness was given by
Kastanas in \cite{kas}, using games in the style of Banach-Mazur
with respect to Ellentuck's topology.
\bigskip
In this work we will deal with a game-theoretic characterization
of the following property:
\begin{defn}
Let $\mathcal{H}\subset\mathbb{N}^{[\infty]}$ be a coideal.
$\mathcal{X}\subseteq\mathbb{N}^{[\infty]}$ is
$\mathcal{H}$-\textbf{Ramsey} if for every $[a,A]\neq\emptyset$
with $A\in\mathcal{H}$ there exists $B\in [a,A]\cap\mathcal{H}$
such that $[a,B]\subseteq\mathcal{X}$ or $[a,B]\cap\mathcal{X} =
\emptyset$. $\mathcal{X}$ is $\mathcal{H}$-\textbf{Ramsey null} if
for every $[a,A]\neq\emptyset$ with $A\in\mathcal{H}$ there exists
$B\in [a,A]\cap\mathcal{H}$ such that $[a,B]\cap\mathcal{X} =
\emptyset$.
\end{defn}
$\mathcal{H}$-Ramseyness is also called {\bf local Ramsey property}.
\medskip
Mathias considered sets that are $\mathcal H$-Ramsey with respect
to a selective coideal $\mathcal H$, and generalized Silver's
result to this context. Matet \cite{mate} used games to
characterize sets which are Ramsey with respect to a selective
coideal $\mathcal H$. These games coincide with the games of
Kastanas if $\mathcal H$ is $\mathbb{N}^{[\infty]}$ and with the games of
Louveau \cite{lou} if $\mathcal H$ is a Ramsey ultrafilter.
Given a coideal $\mathcal{H}\subset\mathbb{N}^{[\infty]}$, let
\[
Exp(\mathcal{H}) : = \{[a,A] :\; (a,A)\in \mathbb{N}^{[<\infty]}\times
\mathcal{H}\}.
\]
In general, this is not a basis for a topology on
$\mathbb{N}^{[<\infty]}$, but the following abstract version of the Baire
property and related concepts will be useful:
\begin{defn}
Let $\mathcal{H}\subset\mathbb{N}^{[\infty]}$ be a coideal.
$\mathcal{X}\subseteq\mathbb{N}^{[\infty]}$ has the abstract
$Exp(\mathcal{H})$-\textbf{Baire property} if for every
$[a,A]\neq\emptyset$ with $A\in\mathcal{H}$ there exists
$[b,B]\subseteq [a,A]$ with $B\in\mathcal{H}$ such that
$[b,B]\subseteq\mathcal{X}$ or $[b,B]\cap\mathcal{X} = \emptyset$.
$\mathcal{X}$ is $Exp(\mathcal{H})$-\textbf{nowhere dense} if for
every $[a,A]\neq\emptyset$ with $A\in\mathcal{H}$ there exists
$[b,B]\subseteq [a,A]$ with $B\in\mathcal{H}$ such that
$[b,B]\cap\mathcal{X} = \emptyset$. $\mathcal{X}$ is
$Exp(\mathcal{H})$-\textbf{meager} if it is the union of countably
many $Exp(\mathcal{H})$-nowhere dense sets.
\end{defn}
Given a decreasing sequence $A_0\supseteq A_1\supseteq
A_2\supseteq \cdots$ of infinite subsets of $\mathbb{N}$, a set $B$ is a
\textbf{diagonalization} of the sequence (or $B$
\textbf{diagonalizes} the sequence) if and only if $B/n\subseteq
A_n$ for each $n\in B$. A coideal $\mathcal{H}$ is
\textbf{selective} if and only if every decreasing sequence in
$\mathcal{H}$ has a diagonalization in $\mathcal{H}$.
\bigskip
A coideal $\mathcal H$ has the {\bf $Q^+$-property}, if for every
$A\in\mathcal{H}$ and every partition $(F_n)_n$ of $A$ into finite
sets, there is $S\in \mathcal{H}$ such that $S\subseteq A$ and
$|S\cap F_n|\leq 1$ for every $n\in\mathbb{N}$.
\begin{Prop}\cite{math}
A coideal $\mathcal{H}$ is selective if and only if the poset $(\mathcal{H}, \subseteq^*)$ is $\sigma$-closed
and $\mathcal{H}$ has the $Q^+$-property.
\end{Prop}
Given a coideal $\mathcal{H}$ and a sequence $\{D_n\}_{n\in\mathbb{N}}$ of
dense open sets in $(\mathcal{H},\subseteq)$, a set $B$ is a
\textbf{diagonalization} of $\{D_n\}_{n\in\mathbb{N}}$ if and only if
$B/n\in D_n$ for every $n\in B$. A coideal $\mathcal{H}$ is
\textbf{semiselective} if for every sequence $\{D_n\}_{n\in\mathbb{N}}$ of
dense open subsets of $\mathcal{H}$, the family of its
diagonalizations is dense in $(\mathcal{H},\subseteq)$.
\begin{Prop}\cite{far}
A coideal $\mathcal{H}$ is semiselective if and only if the poset
$(\mathcal{H}, \subseteq^*)$ is $\sigma$-distributive and
$\mathcal{H}$ has the $Q^+$-property.
\end{Prop}
Since $\sigma$-closedness implies $\sigma$-distributivity, then
semiselectivity follows from selectivity, but the converse does
not hold (see \cite{far} for an example).
In section \ref{topchar} we list results of Ellentuck, Mathias
and Farah that characterize topologically the Ramsey property and
the local Ramsey property. In section \ref{main} we define a
family of games, and present the main result, which states that a
coideal $\mathcal H$ is semiselective if and only if the $\mathcal
H$-Ramsey sets are exactly those for which the associated games
are determined. This generalizes results of Kastanas \cite{kas}
and Matet \cite{mate}. The proof is given in section
\ref{mainproof}. In section \ref{frechet} we relate
semiselectivity of coideals with the Fr\'echet-Urysohn property,
and show that in Solovay's model every semiselective coideal has
the Fr\'echet-Urysohn property.
\thanks{We thank A. Blass and J. Bagaria for helping us to correct some deficiencies in previous versions of the article.}
\section{Topological characterization of Ramseyness.}\label{topchar}
The following are the main results concerning the characterization
of the Ramsey property and the local Ramsey property in
topological terms.
\begin{thm}\label{Ellentuck}
[Ellentuck] Let $\mathcal{X}\subseteq \mathbb{N}^{[\infty]}$ be given.
\begin{itemize}
\item[(i)] $\mathcal{X}$ is Ramsey if and only if
$\mathcal{X}$ has the Baire property, with respect to Ellentuck's
topology.
\item[(ii)] $\mathcal{X}$ is Ramsey null if and only if
$\mathcal{X}$ is meager, with respect to Ellentuck's topology.
\end{itemize}
\end{thm}
\begin{thm}\label{Mathias}
[Mathias] Let $\mathcal{X}\subseteq \mathbb{N}^{[\infty]}$ and a selective coideal $\mathcal{H}$ be given.
\begin{itemize}
\item[(i)] $\mathcal{X}$ is $\mathcal{H}$-Ramsey if and only if
$\mathcal{X}$ has the abstract $Exp(\mathcal{H})$-Baire property.
\item[(ii)] $\mathcal{X}$ is $\mathcal{H}$-Ramsey null if and only
if $\mathcal{X}$ is $Exp(\mathcal{H})$-meager.
\end{itemize}
\end{thm}
\begin{thm}\label{Farah} [Farah, Todorcevic] Let $\mathcal{H}$ be a coideal. The following are
equivalent:
\begin{itemize}
\item[(i)] $\mathcal{H}$ is semiselective.
\item[(ii)] The $\mathcal{H}$-Ramsey subsets of $\mathbb{N}^{[\infty]}$
are exactly those sets having the abstract
$Exp(\mathcal{H})$-Baire property, and the following three families of subsets of
$\mathbb{N}^{[\infty]}$ coincide and are $\sigma$-ideals:
\begin{enumerate}
\item[(a)] $\mathcal{H}$-Ramsey null sets,
\item[(b)] $Exp(\mathcal{H})$-nowhere dense, and
\item[(c)] $Exp(\mathcal{H})$-meager sets.
\end{enumerate}
\end{itemize}
\end{thm}
In the next section we state results by Kastanas \cite{kas} and
Matet \cite{mate} (Theorems \ref{Kastanas} and \ref{Matet} below)
which are the game-theoretic counterparts of theorems
\ref{Ellentuck} and \ref{Mathias}, respectively; and we also
present our main result (Theorem \ref{Main} below), which is the
game-theoretic counterpart of Theorem \ref{Farah}.
\section{Characterizing Ramseyness with games.}\label{main}
The following is a relativized version of a game due to Kastanas \cite{kas},
employed to obtain a characterization of the family
of completely Ramsey sets (i.e. $\mathcal{H}$-Ramsey for
$\mathcal{H}=\mathbb{N}^{[\infty]}$). The same game was used by Matet in
\cite{mate} to obtain the analog result when $\mathcal{H}$ is
selective.
\bigskip
Let $\mathcal{H}\subseteq \mathbb{N}^{[\infty]}$ be a fixed coideal. For
each $\mathcal{X} \subseteq \mathbb{N}^{[\infty]}$, $A\in \mathcal{H}$ and
$a\in \mathbb{N}^{[<\infty]}$ we define a two-player game
$G_{\mathcal{H}}(a,A,\mathcal{X})$ as follows: player I chooses an
element $A_{0}\in \mathcal{H}\upharpoonright A$; II answers by
playing $n_{0}\in A_{0}$ such that $a\subseteq n_{0}$, and
$B_{0}\in \mathcal{H}\cap (A_{0}/n_{0})^{[\infty]}$; then I
chooses $A_{1}\in \mathcal{H}\cap B_{0}^{[\infty]}$; II answers by
playing $n_{1}\in A_{1}$ and $B_{1}\in \mathcal{H}\cap
(A_{1}/n_{1})^{[\infty]}$; and so on. Player I wins if and only if
$a\cup \{n_{j}:\; j\in \mathbb{N}\}\in \mathcal{X}$.
\bigskip
$
\begin{array}{lcccccccccc}
I & A_0 & & A_1 & & \cdots & A_k & & \cdots\\
\\
II & & n_0, B_0 & &n_1, B_1&\cdots & & n_k, B_k&\cdots
\end{array}
$
\bigskip
A \textbf{strategy} for a player is a rule that tells him (or her) what to
play based on the previous moves. A strategy is a \textbf{winning
strategy for player I} if player I wins the game whenever she (or he)
follows the strategy, no matter what player II plays. Analogously,
it can be defined what is a winning strategy for player II. The
precise definitions of strategy for two players games can be found
in \cite{kec,mos}.
\bigskip
Let $s = \{s_0, \dots, s_k\}$ be a nonempty finite subset of $\mathbb{N}$,
written in its increasing order, and $\overrightarrow{B} = \{B_0,
\dots, B_k\}$ be a sequence of elements of $\mathcal{H}$. We say
that the pair $(s,\overrightarrow{B})$ is a \textbf{legal position
for player II} if $(s_0, B_0), \dots, (s_k, B_k)$ is a sequence of
possible consecutive moves of II in the game
$G_{\mathcal{H}}(a,A,\mathcal{X})$, respecting the rules. In this
case, if $\sigma$ is a winning strategy for player I in the game,
we say that $\sigma(s,\overrightarrow{B})$ is a \textbf{realizable
move of player I} according to $\sigma$. Notice that if $r\in
B_k/s_k$ and $C\in\mathcal{H}\upharpoonright B_k/s_k$ then $(s_0,
B_0), \dots, (s_k, B_k), (r,C)$ is also a sequence of possible
consecutive moves of II in the game. We will sometimes use the
notation $(s,\overrightarrow{B}, r, C)$, and say that
$(s,\overrightarrow{B}, r, C)$ is a legal position for player II
and $\sigma(s,\overrightarrow{B}, r, C)$ is a realizable move of
player I according to $\sigma$.
\bigskip
We say that the game $G_{\mathcal{H}}(a,A,\mathcal{X})$ is
\textbf{determined} if one of the players has a winning strategy.
\bigskip
\begin{thm} \label{Kastanas}
[Kastanas] $\mathcal{X}$ is Ramsey if and only if for every
$A\in \mathbb{N}^{[\infty]}$ and $a\in \mathbb{N}^{[<\infty]}$ the game
$G_{\mathbb{N}^{[\infty]}}(a,A,\mathcal{X})$ is determined.
\end{thm}
\begin{thm}\label{Matet}
[Matet] Let $\mathcal{H}$ be a selective coideal.
$\mathcal{X}$ is $\mathcal{H}$-Ramsey if and only if for every
$A\in \mathcal{H}$ and $a\in \mathbb{N}^{[<\infty]}$ the game
$G_{\mathcal{H}}(a,A,\mathcal{X})$ is determined.
\end{thm}
Now we state our main result:
\begin{thm}\label{Main}
Let $\mathcal{H}$ be a coideal. The following are
equivalent:
\begin{enumerate}
\item{} $\mathcal{H}$ is semiselective.
\item{} $\forall\mathcal{X}\subseteq\mathbb{N}^{[\infty]}$,
$\mathcal{X}$ is $\mathcal{H}$-Ramsey if and only if for every
$A\in \mathcal{H}$ and $a\in \mathbb{N}^{[<\infty]}$ the game
$G_{\mathcal{H}}(a,A,\mathcal{X})$ is determined.
\end{enumerate}
\end{thm}
So Theorem \ref{Main} is a game-theoretic counterpart to Theorem \ref{Farah} in the previous
section, in the sense that it gives us a game-theoretic
characterization of semiselectivity. Obviously, it also gives us a
characterization of $\mathcal{H}$-Ramseyness, for semiselective
$\mathcal{H}$, which generalizes the main results of Kastanas in
\cite{kas} and Matet in \cite{mate} (Theorems \ref{Kastanas} and
\ref{Matet} above).
It is known that every analytic set is $\mathcal{H}$-Ramsey for
$\mathcal H$ semiselective (see Theorem 2.2 in \cite{far} or Lemma
7.18 in \cite{todo}). We extend this result to the projective
hierarchy. Please see \cite{kec} or \cite{mos} for the definitions
of \textit{projective set} and of \textit{projective determinacy}.
\begin{Coro}
\label{projective sets} Assume projective determinacy for games
over the reals. Let $\mathcal H$ be a semiselective projective
coideal. Then, every projective set is $\mathcal H$-Ramsey.
\end{Coro}
\proof Let $\mathcal{X}$ be a projective subset of
$\mathbb{N}^{[\infty]}$. Fix $A\in \mathcal{H}$, $a\in \mathbb{N}^{[<\infty]}$.
By the projective determinacy over the reals, the game
$G_{\mathcal{H}}(a,A,\mathcal{X})$ is determined. Then, Theorem
\ref{Main} implies that $\mathcal{X}$ is $\mathcal{H}$-Ramsey.
\endproof
\section{Proof of the main result}\label{mainproof}
Throughout the rest of this section, fix a semiselective coideal
$\mathcal{H}$. Before proving Theorem \ref{Main}, in Propositions
\ref{playerI_semiselective} and \ref{playerII_semiselective} below
we will deal with winning strategies of players in a game
$G_{\mathcal{H}}(a,A,\mathcal{X})$.
\begin{Prop}\label{playerI_semiselective}
For every $\mathcal{X}\subseteq\mathbb{N}^{[\infty]}$, $A\in \mathcal{H}$ and $a\in
\mathbb{N}^{[<\infty]}$, I has a winning strategy in
$G_{\mathcal{H}}(a,A,\mathcal{X})$ if and only if there exists $E\in
\mathcal{H}\upharpoonright A$ such that $[a,E]\subseteq
\mathcal{X}$.
\end{Prop}
\begin{proof}
Suppose $\sigma$ is a winning strategy for I. We will suppose that
$a = \emptyset$ and $A = \mathbb{N}$ without loss of generality.
\medskip
Let $A_{0} = \sigma(\emptyset)$ be the first move of I using
$\sigma$. We will define a tree $T$ of finite subsets of $A_0$;
and for each $s\in T$ we will also define a family $M_s\subseteq A_0^{[\infty]}$ and a family $N_s\subseteq (A_0^{[\infty]})^{\lvert s\rvert}$, where $\lvert s\rvert$ is the length of $s$. Put $\{p\}\in T$ for each $p\in A_0$ and let
$$
M_{\{ p\}} \subseteq \{\sigma(p,B) :\; B\in
\mathcal{H}\upharpoonright A_0\}
$$
be a maximal $\mathcal{H}$-disjoint family, and set
$$
N_{\{ p\} } = \{\{B\} :\; \sigma(p,B)\in M_{\{ p\}}\}.
$$
\medskip
Suppose we have defined $T\cap A_0^{[n]}$ and we have chosen a
maximal $\mathcal{H}$-disjoint family $M_s$ of realizable moves of
player I of the form $\sigma(s,\overrightarrow{B})$ for every
$s\in T\cap A_0^{[n]}$ . Let
\[
N_s = \{\overrightarrow{B} :\; \sigma(s,\overrightarrow{B})\in
M_s\}.
\]
Given $s\in T\cap A_0^{[n]}$, $\overrightarrow{B}\in N_s$ and
$r\in \sigma(s,\overrightarrow{B})/s$, we put $s\cup\{r\}\in T$.
Then choose a maximal $\mathcal{H}$-disjoint family
$$
M_{s\cup\{r\}} \subseteq \{\sigma(s,\overrightarrow{B},r,C):\;
\overrightarrow{B}\in N_s,\;
C\in\mathcal{H}\upharpoonright\sigma(s,\overrightarrow{B})/r\}.
$$
Put
$$
N_{s\cup\{r\}} = \{(\overrightarrow{B},C) :
\;\sigma(s,\overrightarrow{B},r,C)\in M_{s\cup\{r\}}\}.
$$
\medskip
Now, for every $s\in T$, let $$\mathcal{U}_s = \{E\in\mathcal{H}
: (\exists F\in M_s)\ E\subseteq F\}\ \ \mbox{and}$$
$$\mathcal{V}_s = \{E\in\mathcal{H} : (\forall F\in
M_{s\setminus\{max(s)\}})\ max(s)\in F\ \rightarrow F\cap
E\not\in\mathcal{H}\}.$$
\bigskip
\begin{clm}\label{dense_moves}
For every $s\in T$, $\mathcal{U}_s\cup\mathcal{V}_s$ is dense open
in $(\mathcal{H}\upharpoonright A_0,\subseteq)$.
\end{clm}
\begin{proof}
Fix $s\in T$ and $A\in\mathcal{H}\upharpoonright A_0$. If
$(\forall F\in M_{s\setminus\{max(s)\}})\ max(s)\in F\ \rightarrow
F\cap A\not\in\mathcal{H}$ holds, then $A\in\mathcal{V}_s$.
Otherwise, fix $F\in M_{s\setminus\{max(s)\}}$ such that
$max(s)\in F$ and $F\cap A \in\mathcal{H}$. Let
$\overrightarrow{B}\in N_{s\setminus\{max(s)\}}$ be such that
$\sigma(s\setminus\{max(s)\}, \overrightarrow{B}) = F$. Notice that
since $max(s)\in F$ then $$(s\setminus\{max(s)\},
\overrightarrow{B}, max(s), F\cap A/max(s))$$ is a legal position
for player II. Then, using the maximality of $M_s$, choose
$\hat{F}\in M_s$ such that $$E : = \sigma(s\setminus\{max(s),
\overrightarrow{B}, max(s), F\cap A/max(s))\cap\hat{F}$$ is in
$\mathcal{H}$. So $E\in\mathcal{U}_s$ and $E\subseteq A$. This
completes the proof of claim \ref{dense_moves}.
\end{proof}
\begin{clm}\label{Compatible_Moves_dense}
There exists $E\in\mathcal{H}\upharpoonright A_0$ such that for
every $s\in T$ with $s\subset E$, $E/s \in \mathcal{U}_s$.
\end{clm}
\begin{proof}
For each $n\in \mathbb{N}$ , let
$$
\begin{array}{lcl}
\mathcal{D}_n & = & \bigcap_{max(s) =
n}\mathcal{U}_s\cup\mathcal{V}_s.
\\
\\
\mathcal{U}_n & = & \bigcap_{max(s) = n}\mathcal{U}_s\ \mbox{,}
\end{array}
$$
(if there is no $s\in T$ with $max(s)=n$, then we put
$\mathcal{D}_n = \mathcal{U}_n =\mathcal{H}\upharpoonright
A_0$). By Claim \ref{dense_moves}, every
$\mathcal{D}_n$ is dense open in $(\mathcal{H}\upharpoonright
A_0,\subseteq)$. Using semiselectivity, choose a diagonalization
$\hat{E}\in\mathcal{H}\upharpoonright A_0$ of the sequence
$(\mathcal{D}_n)_n$. Let
$$E_0 : = \{n\in \hat{E} : \hat{E}/n\in\mathcal{U}_n\}\ \ \mbox{and}\ \ E_1 := \hat{E}\setminus E_0.$$
\medskip
Let us prove that $E_1\not\in\mathcal{H}$:
\medskip
Suppose $E_1\in\mathcal{H}$. By the definitions, $(\forall n\in
E_1)\ \hat{E}/n\not\in \mathcal{U}_n$. Let $n_0 = min(E_1)$ and
fix $s_0\subset \hat{E}$ such that $max(s_0) = n_0$ and
satisfying, in particular, the following: $$(\forall F\in
M_{s_0\setminus\{n_0\}})\ n_0\in F\ \rightarrow\ F\cap
E_1/n_0\not\in\mathcal{H}.$$ Notice that $|s_0|>1$, by the
construction of the $M_s$'s.
Now, let $m = max(s_0\setminus\{n_0\})$. Then $m\in E_0$ and
therefore
$\hat{E}/m\in\mathcal{U}_m\subseteq\mathcal{U}_{s_0\setminus\{n_0\}}$.
So there exists $F\in M_{s_0\setminus\{n_0\}}$ such that
$\hat{E}/m \subseteq F$. Since $m<n_0$ then $n_0\in F$. But $F\cap
E_1/n_0 = E_1/n_0\in\mathcal{H}$. A contradiction.
\medskip
Hence, $E_1\not\in\mathcal{H}$ and therefore $E_0\in\mathcal{H}$. Then $E : = E_0$ is as required.
\end{proof}
\begin{clm}\label{Compatible_Moves}
Let $E$ be as in Claim \ref{Compatible_Moves_dense} and $s\cup \{r\}\in T$ with $s\subset E$ and $r\in E/s$. If
$E/s\subseteq\sigma(s,\overrightarrow{B})$ for some
$\overrightarrow{B}\in N_s$, then there exists
$C\in\mathcal{H}\upharpoonright\sigma(s,\overrightarrow{B})/r$
such that $E/r\subseteq \sigma(s,\overrightarrow{B},r,C)$ and
$(\overrightarrow{B},C)\in N_{s\cup\{r\}}$.
\end{clm}
\begin{proof}
Fix $s$ and $r$ as in the hypothesis. Suppose $E/s\subseteq
\sigma(s,\overrightarrow{B})$ for some $\overrightarrow{B}\in
N_s$. Since $E/r\in\mathcal{U}_{s\cup\{r\}}$, there exists
$(\overrightarrow{D}, C)\in N_{s\cup\{r\}}$ such that
$E/r\subseteq \sigma(s,\overrightarrow{D},r,C)$. Notice that
$E/r\subseteq\sigma(s,\overrightarrow{B})\cap\sigma(s,\overrightarrow{D})$.
Since $M_{s}$ is $\mathcal{H}$-disjoint, then
$\sigma(s,\overrightarrow{D})$ is neccesarily equal to
$\sigma(s,\overrightarrow{B})$ and therefore
$\sigma(s,\overrightarrow{B},r,C) =
\sigma(s,\overrightarrow{D},r,C)$. Hence $(\overrightarrow{B},
C)\in N_{s\cup\{r\}}$ and $E/r\subseteq
\sigma(s,\overrightarrow{B},r,C)$.
\end{proof}
\begin{clm}\label{ganadora}
Let $E$ be as in Claim \ref{Compatible_Moves_dense}. Then $[\emptyset,E]\subseteq\mathcal{X}$.
\end{clm}
\begin{proof}
Let $\{k_i\}_{i\geq 0}\subseteq E$ be given. Since $E/k_0
\in\mathcal{U}_{\{k_0\}}$, there exists $B_0\in
N_{\{k_0\}}$ such that $E/k_0\subseteq\sigma(k_0,B_0)$. Thus, by the choice of $E$ and
applying Claim \ref{Compatible_Moves} iteratively, we prove that $\{k_i\}_{i\geq
0}$ is generated in a run of the game in which player I has used his winning strategy
$\sigma$. Therefore $\{k_i\}_{i\geq 0}\in\mathcal{X}$.
\end{proof}
The converse is trivial. This completes the proof of Proposition \ref{playerI_semiselective}.
\end{proof}
\bigskip
Now we turn to the case when player II has a winning strategy. The proof of the following is similar to
the proof of Proposition 4.3 in \cite{mate}. First we show a
result we will need in the sequel, it should be compared with
lemma 4.2 in \cite{mate}.
\begin{lem}\label{lemmaPlayerII}
Let $B\in \mathcal{H}$, $f:\mathcal{H}\upharpoonright B \rightarrow \mathbb{N}$, and
$g: \mathcal{H}\upharpoonright B \rightarrow
\mathcal{H}\upharpoonright B$ be given such that $f(A)\in A$ and
$g(A)\subseteq A/f(A)$. Then there is $E_{f,g}\in
\mathcal{H}\upharpoonright B$ with the property that for each
$p\in E_{f,g}$ there exists $A\in \mathcal{H}\upharpoonright B$
such that $f(A)=p$ and $E_{f,g}/p\subseteq g(A)$.
\end{lem}
\begin{proof}
For each $n\in \{f(A) : A\in\mathcal{H}\upharpoonright B\}$, let
\medskip
$$
U_n = \{E\in \mathcal{H}\upharpoonright B :\; (\exists A\in
\mathcal{H}\upharpoonright B)\ (f(A) = n \wedge E\subseteq g(A))\}$$
and
$$
V_n = \{E\in \mathcal{H}\upharpoonright B :\; (\forall A\in
\mathcal{H}\upharpoonright B)\ (f(A) = n\ \rightarrow\ \mid
g(A)\setminus E \mid = \infty)\}.
$$
The set $D_n = U_n\cup V_n$ is dense open in
$\mathcal{H}\upharpoonright B$. Choose $E\in
\mathcal{H}\upharpoonright B$ such that for each $n\in E$, $E/n\in
D_n$. Let
$$
E_{0}=\{n\in E :\; E/n\in U_n\} \ \mbox{and} \ E_{1}=\{n\in E
:\; E/n\in V_n\}.
$$
Now, suppose $E_1\in \mathcal{H}$. Then, for each $n\in E_1$,
$E_1/n\in V_n$. Let $n_1= f(E_1)$. So $n_1\in E_1$ by
the definition of $f$. But, by the definition of $g$,
$g(E_1)\subseteq E_1/n_1$ and so $E_1/n_1\not\in
V_{n_1}$; a contradiction. Therefore, $E_{1}\not\in \mathcal{H}$.
Hence $E_{0}\in \mathcal{H}$, since $\mathcal{H}$ is a coideal.
The set $E_{f,g} : = E_{0}$ is as required.
\end{proof}
\begin{Prop}\label{playerII_semiselective}
For every $\mathcal{X} \subseteq \mathbb{N}^{[\infty]}$, $A\in\mathcal{H}$ and $a\in
\mathbb{N}^{[<\infty]}$, II has a winning strategy in
$G_{\mathcal{H}}(a,A,\mathcal{X})$ if and only if $\forall
A'\in\mathcal{H}\upharpoonright A$ there exists
$E\in\mathcal{H}\upharpoonright A'$ such that $[a,E]\cap\mathcal{X}
= \emptyset$.
\end{Prop}
\begin{proof}
Let $\tau$ be a winning strategy for II in $G_{\mathcal{H}}(a,A,\mathcal{X})$
and let $A'\in\mathcal{H}\upharpoonright A$ be given. We are going to define
a winning strategy $\sigma$ for I, in $G_{\mathcal{H}}(a,A',\mathbb{N}^{[\infty]}\setminus
\mathcal{X})$, in such a way that we will get the required result by
means of Proposition \ref{playerI_semiselective}. So, in a play of the game
$G_{\mathcal{H}}(a,A',\mathbb{N}^{[\infty]}\setminus \mathcal{X})$, with II's
successive moves being $(n_j,B_j)$, $j\in\mathbb{N}$, define
$A_j\in \mathcal{H}$ and $E_{f_j,g_j}$ as in Lemma \ref{lemmaPlayerII}, for $f_j$ and $g_j$ such that
\begin{itemize}
\item[{(1)}] For all $\hat{A}\in \mathcal{H}\upharpoonright A'$,
\[
(f_0(\hat{A}),g_0(\hat{A}))= \tau(\hat{A});
\]
\item[{(2)}] For all $\hat{A}\in \mathcal{H}\upharpoonright
B_j\cap g_j(A_j)$,
\[
(f_{j+1}(\hat{A}),g_{j+1}(\hat{A}))=
\tau(A_{0},\cdots,A_j,\hat{A});
\]
\item[{(3)}] $A_{0}\subseteq A'$, and $A_{j+1}\subseteq B_j\cap
g_j(A_j)$;
\item[{(4)}] $n_j=f_j(A_j)$ and $E_{f_j,g_j}/n_j\subseteq
g_j(A_j)$.
\end{itemize}
Now, let $\sigma (\emptyset)$=$E_{f_0,g_0}$ and $\sigma
((n_0,B_0),\cdots ,(n_j,B_j))$=$E_{f_{j+1},g_{j+1}}$.
\bigskip
Conversely, let $A_0$ be the first move of I in the game. Then
there exists $E\in \mathcal{H}\upharpoonright A_0$ such that
$[a,E]\cap \mathcal{X} = \emptyset$. We define a winning strategy
for player II by letting her (or him) play $(\min E,E\setminus
\{\min E \})$ at the first turn, and arbitrarily from there on.
\end{proof}
\bigskip
We are ready now for the following:
\bigskip
\begin{proof}[Proof of Theorem \ref{Main}]
If $\mathcal{H}$ is semiselective, then part 2 of Theorem
\ref{Main} follows from Propositions \ref{playerI_semiselective}
and \ref{playerII_semiselective}.
\medskip
Conversely, suppose part 2 holds and let $(\mathcal{D}_n)_n$ be a
sequence of dense open sets in $(\mathcal{H},\subseteq)$. For
every $a\in\mathbb{N}^{[<\infty]}$, let
$$
\mathcal{X}_a = \{B\in [a,\mathbb{N}] : B/a\ \mbox{diagonalizes some decreasing}\ (A_n)_n\ \mbox{such that}\ (\forall n)\ A_n\in \mathcal{D}_n\}
$$
and define $$\mathcal{X} = \bigcup_{a\in\mathbb{N}^{[<\infty]}} \mathcal{X}_a.$$
Fix $A\in \mathcal{H}$ and $a\in \mathbb{N}^{[<\infty]}$ with
$[a,A]\neq\emptyset$, and define a winning strategy $\sigma$ for
player I in $G_{\mathcal{H}}(a,A,\mathcal{X})$, as follows: let
$\sigma(\emptyset)$ be any element of $\mathcal{D}_0$ such that
$\sigma(\emptyset)\subseteq A$. At stage $k$, if II's successive
moves in the game are $(n_j,B_j)$, $j\leq k$, let
$\sigma((n_0,B_0), \dots, (n_k,B_k))$ be any element of
$\mathcal{D}_{k+1}$ such that $\sigma((n_0,B_0), \dots,
(n_k,B_k))\subseteq B_k$. Notice that $a\cup\{n_0, n_1, n_2,
\dots\}\in\mathcal{X}_a$.
So the game $G_{\mathcal{H}}(a,A,\mathcal{X})$ is determined for
every $A\in \mathcal{H}$ and $a\in \mathbb{N}^{[<\infty]}$ with
$[a,A]\neq\emptyset$. Then, by our assumptions, $\mathcal{X}$ is
$\mathcal{H}$-Ramsey. So given $A\in\mathcal{H}$, there exists
$B\in\mathcal{H}\upharpoonright A$ such that
$B^{[\infty]}\subseteq\mathcal{X}$ or $B^{[\infty]}\cap\mathcal{X}
= \emptyset$. The second alternative does not hold, so
$\mathcal{X}\cap\mathcal{H}$ is dense in $(\mathcal{H},
\subseteq)$. Hence, $\mathcal{H}$ is semiselective.
\end{proof}
\section{ The Fr\'echet-Urysohn property and semiselectivity.}\label{frechet}
We say that an coideal $\mathcal{H}\subseteq\mathbb{N}^{[\infty]}$ has the
\textit{Fr\'echet-Urysohn property} if
\[
(\forall A\in\mathcal{H})\ (\exists B\in A^{[\infty]})\
(B^{[\infty]}\subseteq\mathcal{H}).
\]
The following characterization of the Fr\'echet-Urysohn property
is taken from \cite{si98,Todor96}. It provides a method to
construct ideals with that property. Given $\mathcal{A}\subseteq
\mathbb{N}^{[\infty]}$, define the \textit{orthogonal of} $\mathcal{A}$ as
$\mathcal{A}^\bot : = \{A\in \mathbb{N}^{[\infty]} : (\forall
B\in\mathcal{A})\ (|A\cap B|<\infty)\}$. Notice that
$\mathcal{A}^\bot$ is an ideal.
\begin{Prop}\label{PropFrechet} A coideal
$\mathcal{H}\subseteq \mathbb{N}^{[\infty]}$ has the Fr\'echet-Urysohn
property if and only if $\mathcal{H} = (\mathcal{A}^\bot)^+$ for
some $\mathcal{A}\subseteq \mathbb{N}^{[\infty]}$.
\end{Prop}
\begin{Prop}
\label{H-Ramsey-Frechet} Let $\mathcal{H}$ be a coideal. The
following are equivalent:
\begin{itemize}
\item[(i)] $\mathcal{H}$ is $\mathcal{H}$-Ramsey.
\item[(ii)] $\mathcal{H}$ has the Fr\'echet-Urysohn property.
\end{itemize}
\end{Prop}
\proof Suppose $\mathcal H$ has the Fr\'echet-Urysohn property.
Let $a$ be a finite set and $A\in \mathcal{H}$. Let $B\subseteq A$
be such that $B^{[\infty]}\subseteq\mathcal{H}$. Then $[a,B]\subseteq
\mathcal{H}$ and thus $\mathcal{H}$ is $\mathcal{H}$-Ramsey.
Conversely, suppose $\mathcal{H}$ is $\mathcal{H}$-Ramsey. Let
$A\in\mathcal{H}$. Since $\mathcal{H}$ is $\mathcal{H}$-Ramsey,
there is $B\subseteq A$ in $\mathcal{H}$ such that $[\emptyset,
B]\subseteq \mathcal{H}$ or $[\emptyset, B]\cap
\mathcal{H}=\emptyset$. Since the second alternative does not
hold, then $B^{[\infty]}\subseteq\mathcal{H}$ and thus
$\mathcal{H}$ is Fr\'echet-Urysohn.
\endproof
The following result is probably known but we include its proof
for the sake of completeness.
\begin{Prop}
\label{Frechet implica semiselective} Every coideal $\mathcal H$
with the Fr\'echet-Urysohn property is semiselective.
\end{Prop}
\proof Let $\mathcal H$ be a Fr\'echet-Urysohn coideal. Suppose
$\{D_n\}_{n\in\mathbb{N}}$ is a sequence of dense open sets in
$(\mathcal{H},\subseteq)$ and $B\in \mathcal H$. Since $B\in
\mathcal H$ and $\mathcal H$ has the Fr\'echet-Urysohn property,
then there is $A\subseteq B$ in $\mathcal H$ such that
$A^{[\infty]}\subseteq \mathcal H$. Let $D\subseteq A$ be any
diagonalization of $\{D_n\}_{n\in\mathbb{N}}$. Then $D\in \mathcal H$.
This shows that the collection of diagonalizations is dense in
$(\mathcal{H}, \subseteq)$. Thus $\mathcal{H}$ is semiselective.
\endproof
The converse of the previous result is not true in general, since
a non principal ultrafilter cannot have the Fr\'echet-Urysohn
property. However, as every analytic set is $\mathcal{H}$-Ramsey
\cite{far}, from \ref{H-Ramsey-Frechet} we immediately get the
following.
\begin{Prop} Every analytic semiselective coideal
is Fr\'echet-Urysohn.
\end{Prop}
The previous result can be extended from suitable axioms.
\begin{thm}
Assume projective determinacy over the reals. Then, every
projective semiselective coideal is Fr\'echet-Urysohn.
\end{thm}
\begin{proof}
It follows from corollary \ref{projective sets} and proposition
\ref{H-Ramsey-Frechet}.
\end{proof}
Farah \cite{far} shows that if there is a supercompact cardinal,
then every semiselective coideal in $L(\mathbb R)$ has the
Fr\'echet-Urysohn property.
As we show below, it is also an easy consequence of results of
\cite{far} and \cite{math} that in Solovay's model every
semiselective coideal has the Fr\'echet-Urysohn property.
Recall that the Mathias forcing notion $\mathbb M$ is the collection of all the sets of the form
\[
[a,A] : = \{B\in\mathbb{N}^{[\infty]} : a\sqsubset B\subseteq A\},
\]
ordered by
$[a,A]\leq [b,B]$ if and only if $[a,A]\subseteq [b,B]$.
If $\mathcal H$ is a coideal, then $\mathbb M_{\mathcal H}$, the Mathias partial
order with respect to $\mathcal H$ is the collection of all the
$[a,A]$ as above but with $A\in \mathcal H$, ordered in the same
way.
A coideal $\mathcal H$ has the \textit{Mathias property} if it satisfies
that if $x$ is $\mathbb M_{\mathcal H}$-generic over a model $M$, then every $y\in
x^{[\infty]}$ is $\mathbb M_{\mathcal H}$-generic over $M$. And $\mathcal H$ has the
\textit{Prikry property} if for every $[a,A]\in\mathbb M_{\mathcal H}$ and every formula
$\varphi$ of the forcing language of $\mathbb M_{\mathcal H}$, there is $B\in\mathcal H \upharpoonright A$ such that $[a,B]$ decides $\varphi$.
\begin{thm}{(\cite{far}, Theorem 4.1)}
For a coideal $\mathcal H$ the following are equivalent.
\begin{enumerate}
\item $\mathcal H$ is semiselective.
\item $\mathbb M_{\mathcal H}$ has the Prikry property.
\item $\mathbb M_{\mathcal H}$ has the Mathias property.
\end{enumerate}
\end{thm}
Suppose $M$ is a model of $ZFC$ and there is a inaccessible cardinal
$\lambda$ in $M$. The Levy partial order $Col(\omega , <\lambda)$ produces a generic
extension $M[G]$ of $M$ where $\lambda$ becomes $\aleph_1$.
Solovay's model is obtained by taking the submodel of $M[G]$ formed
by all the sets hereditarily definable in $M[G]$ from a sequence of
ordinals (see \cite{math}, or \cite{je}).
In \cite{math}, Mathias shows that if $V=L$, $\lambda$ is a Mahlo
cardinal, and $V[G]$ is a generic extension obtained by forcing with
$Col(\omega , <\lambda)$, then every set of real numbers defined in the generic
extension from a sequence of ordinals is $\mathcal H-Ramsey$ for
$\mathcal H$ a selective coideal. This result can be extended to
semiselective coideals under suitable large cardinal hypothesis.
\begin{thm}\label{wchramsey}
Suppose $\lambda$ is a weakly compact cardinal. Let $V[G]$ be a
generic extension by $Col(\omega , <\lambda)$. Then, if $\mathcal H$ is a
semiselective coideal in $V[G]$, every set of real numbers in
$L(\mathbb R)$ of $V[G]$ is $\mathcal H$-Ramsey.
\end{thm}
\begin{proof}
Let $\mathcal H$ be a semiselective coideal in $V[G]$. Let $\mathcal
A$ be a set of reals in $L(\mathbb R)^{V[G]}$; in particular,
$\mathcal A$ is defined in $V[G]$ by a formula $\varphi$ from a
sequence of ordinals. Let $[a,A]$ be a condition of the Mathias
forcing $\mathbb M_{\mathcal H}$ with respect to the semiselective
coideal $\mathcal H$. Let finally $\dot{\mathcal H}$ be a name for
$\mathcal H$. Notice that $\dot{\mathcal H}\subseteq V_\lambda$.
Since $V[G]$ satisfies that $\mathcal H$ is semiselective, the
following statement holds in $V[G]$: For every sequence $D=(D_n:
n\in \omega)$ of open dense subsets of $\mathcal H$ and for every
$x\in \mathcal H$ there is $y\in \mathcal H$, $y\subseteq x$, such
that $y$ diagonalizes the sequence $D$.
Therefore, there is $p\in G$ such that, in $V$, the following
statement holds:
\[\begin{aligned}
\notag \forall \dot D \forall \tau ( p\Vdash_{Col(\omega , <\lambda)} (\dot D \mbox{
is a name for a sequence of dense open subsets of } \dot{\mathcal H}
\\\notag \mbox{ and } \tau\in \dot{\mathcal H}) \longrightarrow
(\exists x (x \in \dot{\mathcal H}, x\subseteq \tau ,
x \mbox{ diagonalizes } \dot D))) .
\end{aligned}\]
Notice that every real in $V[G]$ has a name in $V_\lambda$, and
names for subsets of $\mathcal H$ or countable sequences of subsets
of $\mathcal H$ are contained in $V_\lambda$. Also, the forcing
$Col(\omega , <\lambda)$ is a subset of $V_\lambda$. Therefore the same statement is
valid in the structure $(V_\lambda, \in, \dot{\mathcal H}, Col(\omega , <\lambda))$.
This statement is $\Pi^1_1$ over this structure, and since $\lambda$
is $\Pi^1_1$-indescribable, there is $\kappa<\lambda$ such that in
$(V_\kappa, \in ,\dot{\mathcal H}\cap V_\kappa, Col(\omega , <\lambda)\cap V_\kappa)$
it holds
\[\begin{aligned}\notag
\forall \dot D \forall \tau (p\Vdash_{Col(\omega, <\kappa)} (\dot D
\mbox{ is a name for a sequence of dense open subsets of }
\dot{\mathcal H}\cap V_\kappa \\\notag\mbox{ and } \tau\in
\dot{\mathcal H}\cap V_\kappa) \longrightarrow (\exists x
(x\in \dot{\mathcal H}\cap V_\kappa, x\subseteq \tau ,
x \mbox{ diagonalizes } \dot D))).
\end{aligned}\]
We can get $\kappa$ inaccessible, since there is a $\Pi^1_1$ formula expressing that $\lambda$ is
inaccessible. Also, $\kappa$ is such that $p$ and the names for the
real parameters in the definition of $\mathcal A$ and for $A$ belong
to $V_\kappa$.
If we let $G_\kappa = G\cap Col(\omega, <\kappa)$, then
$G_\kappa\subseteq Col(\omega, <\kappa)$, and is generic over $V$.
Also, $p\in G_\kappa$. $\dot{\mathcal H}\cap V_\kappa$ is a
$Col(\omega, <\kappa)$-name in $V$ which is interpreted by
$G_\kappa$ as $\mathcal H\cap V[G_\kappa]$, thus $\mathcal H\cap
V[G_\kappa]\in V[G_\kappa]$. And since every subset (or sequence of
subsets) of $\mathcal H\cap V[G_\kappa]$ which belongs to
$V[G_\kappa]$ has a name contained in $V_\kappa$, we have that, in
$V[G_\kappa]$, $\mathcal H\cap V_\kappa$ is semiselective, and in
consequence it has both the Prikry and the Mathias properties.
Now the proof can be finished as in \cite{math}. Let $\dot r$ be the
canonical name of a $\mathbb M_{\mathcal H \cap V[G_\kappa]}$
generic real, and consider the formula $ \varphi( \dot r)$ in the
forcing language of $V[G_\kappa]$. By the Prikry property of
$\mathcal H \cap V[G_\kappa]$, there is $A'\subseteq A$, $A'\in
\mathcal H \cap V[G_\kappa]$, such that $[a,A']$ decides $\varphi(
\dot r)$. Since $2^{2^\omega}$ computed in $V[G_\kappa] $ is
countable in $V[G]$, there is (in $V[G]$) a $\mathbb M_{\mathcal H
\cap V[G_\kappa]}$-generic real $x$ over $V[G_\kappa]$ such that
$x\in [a,A']$. To see that there is such a generic real in $\mathcal
H$ we argue as in 5.5 of \cite{math} using the semiselectivity of
$\mathcal H$ and the fact that $\mathcal H\cap V[G_\kappa]$ is
countable in $V[G]$ to obtain an element of $\mathcal H$ which is
generic.
By the Mathias property
of $\mathcal H \cap V[G_\kappa]$, every $y\in [a,x\setminus a]$ is
also $\mathbb M_{\mathcal H \cap V[G_\kappa]}$-generic over
$V[G_\kappa]$, and also $y\in [a,A']$. Thus $\varphi(x)$ if and only
if $[a,A']\Vdash\varphi(\dot r)$, if and only if
$\varphi (y)$.
Therefore, $[a,x\setminus a]$ is contained in $\mathcal A$ or is
disjoint from $\mathcal A$.
\end{proof}
As in \cite{math}, we obtain the following.
\begin{Coro}
If $ZFC$ is consistent with the existence of a weakly compact cardinal, then so is $ZF+DC$ and ``every set of reals is $\mathcal H$-Ramsey for every semiselective coideal $\mathcal H$''.
\end{Coro}
\begin{Coro}
Suppose there is a weakly compact cardinal. Then, there is a model of $ZF+DC$
in which every semiselective coideal $\mathcal H$ has
the Fr\'echet-Urysohn property.
\end{Coro}
\begin{proof}
By theorem \ref{wchramsey},
in $L(\mathbb R)$ of the Levy collapse of a weakly compact cardinal, every set of reals is $\mathcal H$-Ramsey for every semiselective coideal . Thus, by proposition
\ref{H-Ramsey-Frechet}, every semiselective coideal $\mathcal H$ in this model has the Fr\'echet-Urysohn property.
\end{proof}
|
{
"timestamp": "2010-11-19T02:02:26",
"yymm": "1009",
"arxiv_id": "1009.3683",
"language": "en",
"url": "https://arxiv.org/abs/1009.3683"
}
|
\section{Introduction}
According to the IMCR research group of BCBS \cite{fi}, {\it ``liquidity conditions interact with market risk and credit risk through the horizon over which assets can be liquidated''}. To face the impact of market liquidity risk, risk managers agree in adopting a longer holding period to calculate the market VaR, for instance ten business days instead of one; recently, BCBS has prudentially stretched such liquidity horizon to three months \cite{gc}. However, even the IMCR group pointed out that {\it ``the liquidity of traded products can vary substantially over time and in unpredictable ways''}, and moreover, {\it ``IMCR studies suggest that banks' exposures to market risk and credit risk vary with liquidity conditions in the market''}. The former statement suggests a stochastic description of the time horizon over which a portfolio can be liquidated, and the latter highlights a dependence issue.
\\\newline
We can start by saying that probably the holding period of a risky portfolio is neither ten business days nor three months; it could, for instance, be 10 business days with probability 99\% and three months with probability 1\%. This is a very simple assumption but it may have already interesting consequences. Indeed, given the FSA requirement to justify liquidity horizon assumptions for the Incremental Risk Charge modelling, a simple example with the two-points liquidity horizon distribution that we develop below could be interpreted as a mixture of the distribution under normal conditions and of the distribution under stressed and rare conditions.
To make the general idea more precise, it is necessary to distinguish between the two processes:
\begin{itemize}
\item the daily P\&L of the risky portfolio;
\item the P\&L of disinvesting and reinvesting in the risky portfolio.
\end{itemize}
In the following we will assume no transaction costs, in order to fully represent the liquidity risk through the holding period variability. Therefore, even if the cumulative P\&L is the same for the two processes above on the long term, the latter has more variability than the former, due to variable liquidity conditions in the market. If we introduce a third process, describing the dynamics of such liquidity conditions, for instance
\begin{itemize}
\item the process of time horizons over which the risky portfolio can be fully bought or liquidated
\end{itemize}
then the P\&L is better defined by the returns calculated over such stochastic time horizons instead of a daily basis. We will use the ``stochastic holding period'' (SHP) acronym for that process, which belongs to the class of {\bf positive processes} largely used in mathematical finance.
We define the liquidity-adjusted VaR or Expexted Shortfall (ES) of a risky portfolio as the VaR or ES of portfolio returns calculated over the horizon defined by the SHP process, which is the `operational time' along which the portfolio manager must operate, in contrast to the `calendar time' over which the risk manager usually measures VaR.
\\ \newline
Earlier literature on extending risk measures to liquidity includes several studies. Jarrow and Subramanian (1997), Bangia et al. (1999), Angelidis and Benos (2005), Jarrow and Protter (2005), Stange and Kaserer (2008), Earnst, Stange and Kaserer (2009), among few others, propose different methods of extending risk measures to account for liquidity risk. Bangia et al. (1999) classify market liquidity risk in two categories: (a) the exogenous illiquidity which depends on general market conditions, is common to all market players and is unaffected by the actions of any one participant and (b) the endogenous illiquidity that is specific to one's position in the market, varies across different market players and is mainly related to the impact of the trade size on the bid-ask spread. Bangia et al. (1999) and Earnst et al. (2009) only consider the exogenous illiquidity risk and propose a liquidity adjusted VaR measure built using the distribution of the bid-ask spreads. The other mentioned studies model and account for endogenous risk in the calculation of liquidity adjusted risk measures.
In the context of the coherent risk measures literature, the general axioms a liquidity measure should satisfy are discussed in \cite{acerbi}.
None of the above works however focuses specifically on our setup with random holding period, which represents a simple but powerful idea to include liquidity in traditional risk measures such as Value at Risk or Expected Shortfall. When analyzing multiple positions, holding periods can be taken to be strongly dependent, in line with the first classification (a) of Bangia et al (1999) above, or independent, so as to fit the second category (b). We will discuss whether adding dependent holding periods to different positions can actually add dependence to the position returns.
\\ \newline
The paper is organized as follows. In order to illustrate the SHP model, first in a univariate case (Section 2) and then in a bivariate one (Section 3), it is considerably easier to focus on examples on (log)normal processes. A brief colloquial hint at positive processes is presented in Section 2, to deepen the intuition of the impact on risk measures of introducing a SHP process. Across Section 3 and Section 4, where we try to address the issue of calibration, we outline a possible multivariate model which could be adopted, in line of principle, in a top-down approach to risk integration in order to include the liquidity risk and its dependence on other risks.
Finally, we point out that this paper is meant as a proposal to open a research effort in stochastic holding period models for risk measures. This paper contains several suggestions on future developments, depending on an increased availability of market data. The core ideas on the SHP framework, however, are presented in this opening paper.
\section{The univariate case}
Let us suppose we have to calculate VaR of a market portfolio whose value at time $t$ is $V_t$. We call $X_t = \ln V_t$, so that the log-return on the portfolio value at time $t$ over a period $h$ is
\[ X_{t + h} - X_t = \ln(V_{t + h}/V_t) \approx \frac{V_{t+h}-V_t}{V_t} . \]
In order to include liquidity risk, the risk manager decides that a realistic, simplified statistics of the holding period in the future will be
\begin{table}[H]
\caption{Simplified discrete SHP}
\centering
\begin{tabular}{c c}
\hline\hline
Holding Period & Probability \\[0.5ex]
\hline
10 & 0.99 \\
75 & 0.01 \\[1ex]
\hline
\end{tabular}
\label{shp_bernoulli}
\end{table}
To estimate liquidity-adjusted VaR say at time 0, the risk manager will perform a number of simulations of $V_{0+H_0}-V_{0}$ with $H_0$ randomly chosen by the statistics above, and finally will calculate the desired risk measure from the resulting distribution. If the log-return $X_{T}-X_{0}$ is normally distributed with zero mean and variance $T$ for deterministic $T$ (e.g. a Brownian motion, i.e. a Random walk), then the risk manager could simplify the simulation using $X_{0+H_0}-X_{0}|_{H_0}\stackrel{d}\backsim\sqrt{H_0}\left(X_{1}-X_{0}\right)$ where $|_{H_0}$ denotes ``conditional on $H_0$".
With this practical exercise in mind, let us generalize this example to a generic $t$.
\subsection{A brief review on the stochastic holding period framework}
A process for the risk horizon at time t, i.e. $t \mapsto H_t$, is a positive stochastic process modeling the risk horizon over time. We have that the risk measure at time $t$ will be taken on the change in value of the portfolio over this random horizon. If $X_t$ is the log-value of the portfolio at time $t$, we have that the risk measure at time $t$ is to be taken on the log-return
\[ X_{t+H_t} - X_t .\]
For example, if one uses a $99\%$ Value at Risk (VaR) measure, this will be the 1st percentile of
$X_{t+H_t} - X_t$. The request that $H_t$ be just positive means that the horizon at future times can both increase and decrease, meaning that liquidity can vary in both directions.
There is a large number of choices for positive processes: one can take lognormal processes with or without mean reversion, mean reverting square root processes, squared gaussian processes, all with or without jumps. This allows one to model the holding period dynamics as mean reverting or not, continuous or with jumps, and with thinner or fatter tails. Other examples are possible, such as Variance Gamma or mixture processes, or Levy processes. See for example \cite{brigotoolkit1} and \cite{brigotoolkit2}.
\subsection{Semi-analytic Solutions and Simulations}
Going back to the previous example, let us suppose that
\begin{assumption}\label{ass:dist} The increments $X_{t+1y}-X_t$ are logarithmic returns of an equity index, normally distributed with annual mean and standard deviation respectively $\mu_{1y} = -1.5\%$ and $\sigma_{1y} =30\%$.
\end{assumption}
We suppose an exposure of 100 in domestic currency.
Before running the simulation, we recall some basic notation and formulas.
The portfolio log-returns under random holding period at time $0$ can be written as
\[
P[X_{H_0} - X_0 < x] = \int_0^\infty
P[X_{h} - X_0 < x] d F_{H,t}(h)
\]
i.e. as a mixture of Gaussian returns, weighted by the holding period distribution. Here $F_{H,t}$ denotes the cumulative distribution function of the holding period at time $t$, i.e. of $H_t$.
\begin{remark}{\bf (Mixtures for heavy-tailed and skewed distributions).}
Mixtures of distributions have been used for a long time in statistics and may lead to heavy tails, allowing for modeling of skewed distributions and of extreme events. Given the fact that mixtures lead, in the distributions space, to linear (convex) combinations of possibly simple and well understood distributions, they are tractable and easy to interpret. The literature on mixtures is enormous and it is impossible to do justice to all this literature here. We just hint at the fact that
static mixtures of distributions had been postulated in the past to fit option prices for a given maturity, see for example \cite{ritchey}, where a mixture of normal
densities for the density of the asset log-returns under the pricing measure is assumed, and subsequently \cite{melick}, \cite{bhupinder},
and \cite{guo}. In the last decade \cite{brigobachelier}, \cite{brigorisk} and \cite{alexander} have extended the mixture distributions to fully dynamic arbitrage-free stochastic processes for asset prices.
\end{remark}
Going back to our notation, $\VaR_{t,h,c}$ and $\ES_{t,h,c}$ are the value at risk and expected shortfall, respectively, for an horizon $h$ at confidence level $c$ at time $t$, namely
\[ \Pb\{X_{t+h} - X_t > - \VaR_{t,h,c} \} = c, \ \ \
\ES_{t,h,c} = - \Ex[ X_{t+h} - X_t | X_{t+h} - X_t \le - \VaR_{t,h,c}]. \]
In the gaussian log-returns case where
\begin{equation}\label{eq:normaldX} X_{t+h} - X_t \ \ \mbox{is normally distributed with mean}\ \ \mu_{t,h}
\ \ \mbox{and standard deviation} \ \ \sigma_{t,h}
\end{equation}
we get
\[ \VaR_{t,h,c} = -\mu_{t,h} + \Phi^{-1}(c) \sigma_{t,h}, \ \
\ES_{t,h,c} = -\mu_{t,h} + \sigma_{t,h} p( \Phi^{-1}(c) )/(1-c) \]
where $p$ is the standard normal probability density function and $\Phi$ is the standard normal cumulative distribution function.
In the following we will calculate VaR and Expected Shortfall referred to a confidence level of $99.96\%$, calculated over the fixed time horizons of 10 and 75 business days, and under SHP process with statistics given by Table \ref{shp_bernoulli}, using Monte Carlo simulations. Each year has 250 (working) days.
\begin{table}[H]
\caption{SHP distributions and Market Risk}
\centering
\begin{tabular}{c c c c c}
\hline\hline
Holding Period & VaR 99.96\% & (analytic) & ES 99.96\% & (analytic) \\[0.5ex]
\hline
constant 10 b.d. & 20.1 & (20.18) & 21.7 & (21.74)\\
constant 75 b.d. & 55.7 & (55.54) & 60.0 & (59.81) \\
SHP (Bernoulli 10/75, $p_{10}$=0.99) & 29.6 & (29.23) & 36.1 & (35.47) \\[1ex]
\hline
\end{tabular}\label{fmr_bernoulli}
\end{table}
More generally, we may derive the VaR and ES formulas for the case where $H_t$ is distributed according to a general distribution
\[ \Pb( H_t \le x ) = F_{H,t}(x), \ \ x \ge 0 \]
and
\[ \Pb( X_{t+h} - X_t \le x ) = F_{X,t,h}(x) . \]
We define VaR and ES under a random horizon $H_t$ at time $t$ and for a confidence level $c$ as
\[ \Pb\{X_{t+H_t} - X_t > - \VaR_{H,t,c} \} = c, \ \ \
\ES_{H,t,c} = - \Ex[ X_{t+H_t} - X_t | X_{t+H_t} - X_t \le - \VaR_{H,t,c}]. \]
Using the tower property of conditional expectation it is immediate to prove that in such a case
$\VaR_{H,t,c}$ obeys the following equation:
\[ \int_0^{\infty} (1- F_{X,t,h}(-\VaR_{H,t,c}))d F_{H,t}(h) = c \]
whereas $\ES_{H,t,c}$ is given by
\[ \ES_{H,t,c} = - \frac{1}{1-c} \int_0^{\infty} \Ex[ X_{t+h} - X_t | X_{t+h} - X_t \le - \VaR_{H,t,c}]
\mbox{Prob}( X_{t+h} - X_t \le - \VaR_{H,t,c}) d F_{H,t}(h) \]
For the specific Gaussian case~(\ref{eq:normaldX}) we have
\[ \int_0^{\infty} \Phi\left(\frac{\mu_{t,h} + \VaR_{H,t,c}}{\sigma_{t,h}} \right) d F_{H,t}(h) = c \]
\[ \ES_{H,t,c} =\frac{1}{1-c} \int_0^\infty \left[-\mu_{t,h} \Phi\left(\frac{-\mu_{t,h} - \VaR_{H,t,c}}{\sigma_{t,h}} \right) + \sigma_{t,h} p\left(\frac{-\mu_{t,h} - \VaR_{H,t,c}}{\sigma_{t,h}} \right)\right] d F_{H,t}(h)
\]
Notice that in general one can try and obtain the quantile $\VaR_{H,t,c}$ for the random horizon case by using a root search, and subsequently compute also the expected shortfall.
Careful numerical integration is needed to apply these formulas for general distributions of $H_t$. The case of Table \ref{fmr_bernoulli} is somewhat trivial, since in the case where $H_0$ is as in Table \ref{shp_bernoulli} integrals reduce to summations of two terms.
We note also that the maximum difference, both in relative and absolute terms, between ES and VaR is reached by the model under random holding period $H_0$. Under this model the change in portfolio value shows heavier tails than under a single deterministic holding period. In order to explore the impact of SHP's distribution tails on the liquidity-adjusted risk, in the following we will simulate SHP models with $H_0$ distributed as an Exponential, an Inverse Gamma distribution\footnote{obtained by rescaling a distribution IG$\left(\frac{\nu}{2},\frac{\nu}{2}\right)$ with $\nu=3$. Before rescaling, setting $\alpha=\nu/2$, the inverse gamma density is
$f(x) = (1/\Gamma(\alpha)) (\alpha)^{\alpha} x^{-\alpha - 1} e^{-\alpha/x}$, $x > 0$, $\alpha >0$, with expected value $\alpha/(\alpha - 1)$. We rescale this distribution by $k = 8.66/(\alpha/(\alpha - 1))$ and take for $H_0$ the random variable with density $f(x/k)/k$} and a Generalized Pareto distribution\footnote{with scale parameter $k =9$ and shape parameter $\alpha = 2.0651$, with cumulative distribution function $F(x)=1-\left(\frac{k}{k+x} \right )^\alpha$, $x \ge 0$, this distribution has moments up to order $\alpha$. So the smaller $\alpha$, the fatter the tails. The mean is, if $\alpha > 1$, $\Ex[H_{0}] = k/(\alpha-1)$} having parameters calibrated in order to obtain a sample with the same 99\%-quantile of 75 business days:
\begin{table}[H]
\caption{SHP statistics and resulting Market Risk}
\centering
\begin{tabular}{c c c c|c c c c c }
\hline\hline
Distribution & Mean & Median & 99\%-q & VaR 99.96\% & VaR 99.96\% & ES 99.96\% & ES 99.96\% & ES/VaR-1\\[0.5ex]
& & & & simulation & root search & simulation & root search & \\[0.5ex]
\hline
Exponential & 16.3 & 11.3 & 75.0 & 39.0 & 39.2 & 44.7 & 44.7 & 14 \%\\
Pareto & 8.45 & 3.7 & 75 & 41.9 & 41.9 & 57.1 & 56.9 & 36\% \\[1ex]
Inverse Gamma & 8.6 & 3.7 & 75.0 & 46.0 & 46.7 & 73.5 & 73.0 & 55 \% \\[1ex]
\hline
\end{tabular}
\label{fmr_various}
\end{table}
The SHP process changes the statistical nature of the P\&L process: the heavier the tails of the SHP distribution, the heavier the tails of P\&L distribution. Notice that our Pareto distribution has tails going to 0 at infinity with exponent around $3$, as one can see immediately by differentiation of the cumulative distribution function, whereas our inverse gamma has tails going to 0 at infinity with exponent about 2.5. In this example we have that the tails of the inverse gamma are heavier, and indeed for that distribution VaR and ES are larger and differ from each other more. This can change of course if we take different parameters in the two distributions.
\section{Dependence modelling: a bivariate case}
Within multivariate modelling, using a common SHP for many normally distributed risks leads to dynamical versions of the so-called {\it normal mixtures} and {\it normal mean-variance mixtures} \cite{qrm}.
Let the log-returns (recall $X^i_t = \ln V^i_t$, with $V^i_t$ the value at time $t$ of the i-th asset)
\[ X^1_{t+h} - X^1_t, \ldots, X^m_{t+h} - X^m_t \]
be normals with means $\mu^1_{t,h}, \ldots, \mu^m_{t,h}$ and covariance matrix $Q_{t,h}$.
Then
\[
P[X^1_{t+H_t} - X^1_t < x_1, \ \ X^m_{t+H_t} - X^m_t< x_m] = \int_0^\infty
P[X^1_{t+h} - X^1_t < x_1, \ \ X^m_{t+h} - X^m_t< x_m] d F_{H,t}(h)
\]
is distributed as a mixture of multivariate normals,
and a portfolio $V_t$ of the assets $1,2,...,m$ whose log-returns $X_{t+h}-X_t$ $(X_t = \ln V_t)$ are a linear weighted combination
$w_1,...,w_m$ of the single asset log-returns $X^i_{t+h}-X^i_t$ would be distributed as
\[ P [ X_{t+H_t} - X_t < z ] =
\int_0^\infty
P[w_1(X^1_{t+h} - X^1_t) + \ldots + w_m(X^m_{t+h} - X^m_t) < z] d F_{H,t}(h)
\]
In particular, in analogy with the unidimensional case, the mixture
may potentially generate skewed and fat-tailed distributions, but when working with more than one
asset this has the further implication that VaR is not guaranteed to
be subadditive on the portfolio. Then the risk manager who wants to
take into account SHP in such a setting should adopt a coherent
measure like Expected Shortfall.
A natural question at this stage is whether the adoption of a common SHP can add dependence to returns that are jointly Gaussian under deterministic calendar time, perhaps to the point of making extreme scenarios on the joint values of the random variables possible.
Before answering this question, one needs to distinguish extreme behaviour in the single variables and in their joint action in a multivariate setting.
Extreme behaviour on the single variables is modeled for example by heavy tails in the marginal distributions of the single variables.
Extreme behaviour in the dependence structure of say two random variables is achieved when the two random variables tend to take extreme values in the same direction together. This is called tail dependence, and one can have both upper tail dependence and lower tail dependence. More precisely, but still loosely speaking, tail dependence expresses the limiting proportion according to which the first variable exceeds a certain level given that the second variable has already exceeded that level. Tail dependence is technically defined through a limit, so that it is an asymptotic notion of dependence. For a formal definition we refer, for example, to \cite{qrm}. ``Finite" dependence, as opposed to tail, between two random variables is best expressed by rank correlation measures such as Kendall's tau or Spearman's rho.
We discuss tail dependence first.
In case the returns of the portfolio assets are jointly Gaussian
with correlations smaller than one, the adoption of a common random
holding period for all assets does not add tail dependence,
\textit{unless the commonly adopted random holding period has a
distribution with power tails}. Hence if we want to rely on one of
the random holding period distributions in our examples above to
introduce upper and lower tail dependence in a multivariate
distribution for the assets returns, we need to adopt a common
random holding period for all assets that is Pareto or Inverse Gamma
distributed. Exponentials, Lognormals or discrete Bernoulli
distributions would not work. This can be seen to follow for
example from properties of the normal variance-mixture model, see
for example \cite{qrm}, page 212 and also Section 7.3.3.
A more specific theorem that fits our setup is Theorem 5.3.8 in
\cite{prestele}. We can write it as follows with our notation.
\begin{proposition}\label{prop:power} {\bf A common random holding period with less than power tails does not add tail dependence to jointly Gaussian returns.}
Assume the log-returns to be $W^i_t = \ln V^i_t$, with $V^i_t$ the value at time $t$ of the i-th asset, $i=1,2$, where
\[ W^1_{t+h} - W^1_t, W^2_{t+h} - W^2_t \]
are two correlated Brownian motions, i.e. normals with zero means, variances $h$ and instantaneous correlation less than 1 in absolute value:
\[ d \langle W^1, W^2 \rangle_t = d W^1_t \ d W^2_t = \rho_{1,2} dt, \ \ |\rho_{1,2}| < 1. \]
Then adding a common non-negative random holding period $H_0$ independent of $W$'s leads to tail dependence in the returns
\[W^1_{H_0},W^2_{H_0} \]
if and only if $\sqrt{H_0}$ is regularly varing at $\infty$ with index $\alpha > 0$.
\end{proposition}
Theorem 5.3.8 in \cite{prestele} also reports an expression for the tail dependence coefficients as functions of $\alpha$ and of the survival function of the student $t$ distribution with $\alpha + 1$ degrees of freedom.
Summarizing, if we work with power tails, the heavier are the tails of the common
holding period process $H$, the more one may expect {\it tail dependence} to emerge
for the multivariate distribution: by adopting a common SHP for all
risks, dependence could potentially appear in the whole
dynamics, in agreement with the fact that liquidity risk is a
systemic risk.
We now turn to finite dependence, as opposed to tail dependence. First we note the well known elementary but important fact that one can have two random variables with very high dependence but without tail dependence. Or one can have two random variables with tail dependence but small finite dependence. For example, if we take two jointly Gaussian Random variables with correlation 0.999999, they are clearly quite dependent on each other but they will not have tail dependence, even if a rank correlation measure such as Kendall's $\tau$ would be $0.999$, still very close to $1$, characteristic of the co-monotonic case. This is a case with zero tail dependence but very high finite dependence.
On the other hand, take a bivariate student $t$ distribution with few degrees of freedom and correlation parameter $\rho = 0.1$. In this case the two random variables have positive tail dependence and it is known that Kendall's tau for the two random variables is
\[ \tau = \frac{2}{\pi} \arcsin(\rho) \approx 0.1 \]
which is the same tau one would get for two standard jointly Gaussian random variables with correlation $\rho$. This tau is quite low, showing that one can have positive tail dependence while having very small finite dependence.
The above examples point out that one has to be careful in distinguishing large finite dependence and tail dependence.
A further point of interest in the above examples comes from the fact that the multivariate student $t$ distribution can be obtained by the multivariate Gaussian distribution when adopting a random holding period given by an inverse gamma distribution (power tails). We deduce the important fact that in this case {\emph{a common random holding period with power tails adds positive tail dependence but not finite dependence}}.
In fact, one can prove a more general result easily by resorting to the tower property of conditional expectation and from the definition of tau based on independent copies of the bivariate random vector whose dependence is being measured. One has the following ''no go" theorem for increasing Kendall's tau of jointly Gaussian returns through common random holding periods, regardless of the tails power.
\begin{proposition}{\bf A common random holding period does not alter Kendall's tau for jointly Gaussian returns.}
Assumptions as in Proposition \ref{prop:power} above.
Then adding a common non-negative random holding period $H_0$ independent of $W$'s leads to the same Kendall's tau for
\[W^1_{H_0},W^2_{H_0} \]
as for the two returns
\[W^1_{t},W^2_{t} \]
for a given deterministic time horizon $t$.
\end{proposition}
Summing up, this result points out that adding further
finite dependence through common SHP's, at least as measured by Kendall's tau, can be impossible if we start
from Gaussian returns. A different popular rank correlation measure, Spearman's rho, does not
coincide for the bivariate $t$ and Gaussian cases though, so that it is not excluded that
dependence could be added in principle though dependent holding periods, at least if we measured dependence with
Spearman's $\rho$. This is under investigation.
More generally, at least from a theoretical point of view, it could be
interesting to model other kinds of dependence than the one stemming
purely from a common holding period (with power tails). One could have
two different holding periods that are themselves dependent on each
other in a less simplistic way, rather than being just identical.
In this case it would be interesting to study the tail dependence implications and also finite dependence as measured by Spearman's rho.
We will investigate this aspect in further research, but increasing dependence may
require, besides the adoption of power tail laws for the random
holding periods, abandoning the Gaussian distribution for the basic
assets under deterministic calendar time.
A further aspect worth investigating is the possibility to calculate semi-closed form risk contributions to VaR and ES under SHP along the lines suggested in \cite{tasche06}, and to investigate the Euler principle as in \cite{tasche08} and \cite{tasche09}.
\section{Calibration over liquidity data}
We are aware that multivariate SHP modelling is a purely theoretical
exercise and that we just hinted at possible initial developments
above. Nonetheless, a lot of financial data is being collected by
regulators, providers and rating agencies, together with a
consistent effort on theoretical and statistical studies. This will
possibly result in available synthetic indices of liquidity risk
grouped by region, market, instrument type, etc. For instance, Fitch
already calculates market liquidity indices on CDS markets
worldwide, on the basis of a scoring proprietary model.
\paragraph{Dependences between liquidity, credit and market risk}
It could be an interesting exercise to calibrate the dependence
structure (e.g. copula function) between a liquidity index (like the
Fitch's one), a credit index (like iTRAXX) and a market index (for
instance Eurostoxx50) in order to measure the possible (non linear)
dependence between the three. The risk manager of a bank could use
the resulting dependence structure within the context of risk
integration, in order to simulate a joint dynamics as a first step,
to estimate later on the whole liquidity-adjusted VaR/ES by assuming
co-monotonicity between the variations of the liquidity index and of
the SHP processes.
\paragraph{Marginal distributions of SHPs} A lot of information on SHP
`extreme' statistics of a OTC derivatives portfolio could be
collected from the statistics, across Lehman's counterparties, of
the time lags between the Lehman's Default Event Date and the trade
dates of any replacement transaction. The data could give
information on the marginal distribution of the SHP of a portfolio,
in a stressed scenario, by assuming a statistical equivalence
between data collected `through the space' (across Lehman's
counterparties) and `through the time' under i.i.d.
hypothesis\footnote{a similar approach is adopted in \cite{or}
within the context of operational risk modelling}. The risk manager
of a bank could examine a more specific and non-distressed dataset
by collecting information on the ordinary operations of the business
units.
\section{Conclusions}
Within the context of risk integration, in order to include
liquidity risk in the whole portfolio risk measures, a stochastic
holding period (SHP) model can be useful, being versatile, easy to
simulate, and easy to understand in its inputs and outputs. In a
single-portfolio framework, as a consequence of introducing a SHP
model, the statistical distribution of P\&L moves to possibly heavier tailed
and skewed mixture distributions\footnote{even if the originary return process was
Brownian, implying normal returns, the resulting P\&L process under
SHP is not: paraphrasing Geman {\it et al.} \cite{apb}, it is not
the time for a process to be Brownian!}. In a multivariate setting,
the dependence among the SHP processes to which marginal P\&L are
subordinated, may lead to dependence on the latter under drastic choices of the SHP distribution,
and in general to
heavier tails on the total P\&L distribution. At present, lack of
synthetic and consensually representative data forces to a
qualitative top-down approach, but it is straightforward to assume
that this limit will be overcome in the near future.
|
{
"timestamp": "2010-10-21T02:03:26",
"yymm": "1009",
"arxiv_id": "1009.3760",
"language": "en",
"url": "https://arxiv.org/abs/1009.3760"
}
|
\section{Introduction}
The problem of defining a Hilbert space operator for the quantum phase \cite{BarVac07} has been an outstanding one, a question that has plagued
physicists from the earliest days of quantum theory. The quantum phase is not only relevant to the radiation field, but also for
other bosonic systems, particularly Bose-Einstein condensates \cite{BarBurVac_BEC}. There has been significant progress in formulating a quantum phase
operator over the years \cite{BarVac07, nieto} , with the hermitian Pegg-Barnett phase formalism \cite{PegBar86, PegBar89} proving to be
very popular. The Pegg-Barnett formalism relies on states of well defined phase, given by,
\begin{equation} \label{phasestate}
\ket{\theta} = \lim_{s \rightarrow \infty} (s+1)^{-1/2} \sum_{n=0}^{s}\exp (i n \theta) \ket{n},
\end{equation}
which are defined in an $s-$dimensional Hilbert space and the limit $s \rightarrow \infty$ is understood to be taken
for physical expectation values; the associated phase operator is the projector,
\begin{equation}
\rho_\text{PB}(\theta) = \ket{\theta}\bra{\theta}.
\end{equation}
Recently, there has been some debate over whether the Pegg-Barnett phase is the actual phase observed in experiments \cite{SkagBerg04}.
An alternative approach to defining a plausible quantum phase is, as the radial integral of one of the various
phase space quasi-probability distributions \cite{GarKni93, VargMoy04}. In this paper, we will focus on the phase probability
distribution defined as the radial integral over the Wigner distribution, viz.,
\begin{equation}\label{W_integral}
P_{\psi}^\text{W}(\theta) = \int_0^\infty dr\, r\, W_\psi (r \cos \theta, r \sin \theta),
\end{equation}
where $W_\psi (x,p)$ is the Wigner transform of the state $\ket{\psi}$ and we refer to the above phase distribution as the Wigner phase
distribution. One of the main advantages of studying a phase probability defined in such a manner is the it becomes easier to understand
the appropriate classical limits via the Wigner-Weyl-Moyal correspondence. Moreover, Schleich \citep{Sch01} has suggested that
the phase states $\ket{\theta}$, defined in \eqref{phasestate}, can be represented as a wedge subtending an angle $\theta$ at the origin in phase space, suggesting
some relation between the Pegg-Barnett phase distribution and the Wigner phase distribution.
Our aim in this paper is to construct a hermitian operator for the Wigner phase distribution, to study its properties
and to elucidate its important Fock state and coherent state representations. Finally, we prove a weak-equivalence between the
Pegg-Barnett phase distribution and the phase distribution arising out of our Wigner phase operator. The final section of the
paper attempts to factorize the operator into a projector form.
\section{Wigner phase operator}
The phase sensitivity of various phase space distribution functions is the reason why they are a very attractive means of defining the quantum phase. In
particular, Schleich \citep{Sch01} has shown how to interpret this phase sensitivity in terms of phase space interference, which
is manifested as the ``overlap of areas'' in phase space. It is this same concept that prompted Knight \cite{GarKni92} to investigate the Wigner
phase \eqref{W_integral}.
We postulate that there exists a Wigner distribution associated with some phase state, such that this Wigner function is maximally sensitive to phase. This
means that this phase Wigner distribution, $W_\text{w}(r \cos \theta, r \sin \theta)$, is sharply localized around the phase angle $\theta$ i.e., if a state $\ket{\psi}$ has the Wigner distribution $W_\psi (x,p)$,
then $W_\text{w} (x,p)$ is characterized by the property,
\begin{equation}\label{Ww_delta_property}
\begin{split}
\int_{0}^{2\pi} W_\psi (r \cos \theta', r \sin \theta') W_\text{w} (r \cos \theta' - \theta, r \sin \theta' - \theta) d\theta' =
W_\psi (r \cos \theta, r \sin \theta).
\end{split}
\end{equation}
The Wigner phase operator that satisfies the above criteria, is given by the normal-ordered series,
\begin{equation}\label{rho_w}
\begin{split}
\rho_\text{w} (\theta) = \frac{1}{2\pi}\, \sum_{m,n=0}^{\infty} \sum_{l=0}^n \frac{(-1)^m\; 2^{m+n/2}\; e^{i(n-2l)\theta}}{m!\; (n-l)!\; l!} \times\,
\Gamma\left( \frac{n}{2}+1 \right) \left( a^\dagger \right)^{m+n-l} a^{m+l},
\end{split}
\end{equation}
so that for a state represented by the density operator $\rho_\psi$, the phase probability distribution is given by,
\begin{equation*}
P_\psi^\text{w} (\theta) = \text{Tr}\left[ \rho_\psi \rho_\text{w} (\theta) \right].
\end{equation*}
The above expression \eqref{rho_w} is derived by using the property \eqref{Ww_delta_property} on the Wigner function of the coherent state
$\ket{\alpha} = D(\alpha)\ket{0}$ and then using the correspondence \citep{CahGlau69} between normal ordered operator expressions and the coherent
state expectations of the operator.
\subsection{Representations of the operator}
The form of the Wigner phase operator, in its normal-ordered form as given in \eqref{rho_w}, is not tenable for easy manipulation and is in general quite complicated.
Any application of this operator formalism for actual computation would necessitate an understanding of the associated representations in various basis. Of course,
the two most important bases used frequently in the study of the radiation field is the Fock basis and the coherent state basis.
It can be shown, after some algebraic gymnastics, that the number state representation of $\rho_\text{w}(\theta)$ is given by,
\begin{eqnarray}\label{rhow_sr}
\bra{s} \rho_\text{w}(\theta) \ket{r} &=& \sqrt{\frac{s!\, r!}{4\pi^2}} \times
\begin{cases} \label{rhow_sr1}
\sum_{n=0}^s \frac{(-1)^{s-n} (\sqrt{2}e^{-i\theta})^{r-s+n} (\sqrt{2}e^{i\theta})^{n} }{n!(s-n)!(r-s+n)!}
\Gamma \left( \frac{r-s}{2}+ n+ 1 \right) & \text{if $r \geq s$} \\
\sum_{n=0}^r \frac{(-1)^{r-n} (\sqrt{2}e^{-i\theta})^{n} (\sqrt{2}e^{i\theta})^{s-r+n} }{n!(r-n)!(s-r+n)!}
\Gamma \left( \frac{s-r}{2}+ n+ 1 \right) & \text{if $s \geq r$}
\end{cases} \\
&=& \sqrt{\frac{s!\,r!}{4\pi}} \times
\begin{cases} \label{rhow_sr2}
\frac{(-1)^s}{s!}\, \left(\frac{e^{-i\theta}}{\sqrt{2}}\right)^{r-s}\,
\frac{{}_2 F_1 \left( -s, 1+ \frac{r-s}{2} ; 1+r-s ; 2 \right)}{\Gamma \left( \frac{1}{2}+\frac{r-s}{2} \right)} & \text{if $r \geq s$} \\
\frac{(-1)^r}{r!}\, \left(\frac{e^{i\theta}}{\sqrt{2}}\right)^{s-r}\,
\frac{{}_2 F_1 \left( 1+ \frac{s-r}{2}, -r ; 1+s-r ; 2 \right)}{\Gamma \left( \frac{1}{2}+\frac{s-r}{2} \right)} & \text{if $s \geq r$}.
\end{cases}
\end{eqnarray}
This expression clearly shows the stark departure of the Wigner phase operator from the Pegg-Barnett phase operator; nevertheless, for the important case $s=r$, both
the formalisms give a uniform phase probability distribution, $P^\text{w}(\theta) = P^\text{PB}(\theta) = \frac{1}{2\pi}$. Moreover, it is easy to see from \eqref{rhow_sr1}
that $\rho_\text{w}(\theta)$ is a hermitian phase operator.
Given two coherent states $\ket{\alpha}$ and $\ket{\beta}$, the Wigner phase operator can be represented in the Bargmann basis as,
\begin{eqnarray}\label{rhow_alpha_beta}
\bra{\beta}\rho_\text{w}(\theta)\ket{\alpha} = \frac{\langle \beta \ket{\alpha} e^{-2\alpha \beta^*}}{2\pi}\, \left[ 1+ \sqrt{\pi}\,z e^{z^2}
\left( 1+ \text{erf}(z) \right) \right],
\end{eqnarray}
where $z=\frac{1}{\sqrt{2}}( \alpha e^{-i\theta} + \beta^* e^{i\theta} )$ and $\text{erf}(z)$ stands for the error-function of the argument $z$. For the case
$\beta= \alpha$, the above expression leads to the phase probability distribution of the coherent state $\ket{\alpha}$, given by,
\begin{equation}\label{rhow_alpha_alpha}
P^\text{w}_{\alpha}(\theta) = \frac{e^{-2\vert \alpha \vert^2}}{\pi} \left[\frac{1}{2}+\sqrt{\frac{\pi}{2}}\, a e^{2a^2} \left( 1+ \text{erf}(a\sqrt{2})
\right) \right],
\end{equation}
where, $a = \vert \alpha \vert \cos (\theta - \text{arg}\, \alpha)$. This is the same expression that was derived by Knight \citep{GarKni92, GarKni93} using the radial
integral \eqref{W_integral}. From the properties of the error-function, it is trivial to note that the above Wigner phase distribution for the coherent state is a
positive quantity, thus enabling it to be interpreted as a true probability distribution.
An important piece of geometrical insight can be obtained from a slight manipulation on the above expression for the Wigner phase distribution of the coherent state. It
is possible to prove, starting from \eqref{rhow_alpha_alpha}, that,
\begin{equation*}
\int_0^{2\pi} d\theta\; e^{im\theta}\; \bra{\alpha} \rho_\text{w}(\theta)\ket{\alpha} =
e^{im\; \text{arg}\; \alpha}\; e^{-2\vert \alpha \vert^2}\; \left(\frac{\vert \alpha \vert}{\sqrt{2}}\right)^m
\frac{\sqrt{\pi}}{\Gamma \left( \frac{m+1}{2} \right)}\; {}_1 F_1 \left( \frac{m}{2}+1; m+1; 2\vert \alpha \vert^2 \right),
\end{equation*}
for any integer $m$. Now taking the limit $\vert \alpha \vert \rightarrow \infty$ on either side, we find,
\begin{equation*}
\lim_{\vert \alpha \vert \rightarrow \infty} \int_0^{2\pi} d\theta\; e^{im\theta}\; \bra{\alpha} \rho_\text{w}(\theta)\ket{\alpha} = e^{im\; \text{arg}\; \alpha},
\end{equation*}
which implies that,
\begin{equation}\label{lim_alpha_infty}
\lim_{\vert \alpha \vert \rightarrow \infty}\; \bra{\alpha} \rho_\text{w}(\theta)\ket{\alpha} = \delta (\theta - \text{arg}\; \alpha).
\end{equation}
Here, $\delta(\theta - \theta')$ is the angular delta function characterized by the property $\int_0^{2\pi} f(\theta') \delta(\theta - \theta') d\theta' = f(\theta)$.
The way to interpret \eqref{lim_alpha_infty} is to first note that the Wigner function of the state $D(\alpha) \ket{\psi}$ is the shape-preserved displacement of the
Wigner function of $\ket{\psi}$, with the displacement affected in the $\text{arg}\; \alpha$ direction by a distance proportional to $\vert \alpha \vert$. It is also
well known that the phase distribution of the coherent state $\ket{\alpha}$ is peaked about $\text{arg}\; \alpha$. Now, \eqref{lim_alpha_infty} is a mathematical
statement of the fact that as the ``hump'' shaped coherent state Wigner function is displaced further away from the origin, the ``hump'' completely falls within the
angular wedge that subtends an angle $\theta$ at the origin, resulting in a sharply defined phase. This geometric interpretation carried by the Wigner phase operator is
inherited from the properties of the radially integrated Wigner phase distribution, the latter of which has been noted previously \citep{Sch01}. From this geometric
idea associated with the Wigner phase operator, it becomes evident that the number state, whose Wigner function is centered at the origin, should have a uniform phase
probability distribution, as has been mathematically proved above. Further, any state whose Wigner function can be localized within an angular wedge is expected to have
a correspondingly localized phase probability distribution.
\subsection{Properties of the operator}
We have already shown in the previous section, using the Fock state representation \eqref{rhow_sr}, that $\rho_\text{w}^\dagger (\theta) = \rho_\text{w}(\theta)$ i.e.,
that the Wigner phase operator is hermitian. In this section, we will enumerate some more of the important properties of the Wigner phase operator.
\subsubsection{Completeness}
Together with hermiticity, completeness is an essential property of an infinite dimensional operator which ensures that its orthonormal eigenstates span
the associated Hilbert space. Before we show that $\rho_\text{w}(\theta)$ is complete, we make use of a Fock decomposition of the operator \eqref{rho_w}, by inserting a set of complete Fock states between the raising and lowering operator powers. Performing the subsequent algebraic simplifications, we get the expression,
\begin{equation*}\label{rhow_fock_decomp}\begin{split}
\rho_\text{w}(\theta) = \frac{1}{2\pi} \sum_{n,k=0}^{\infty} \sum_{l=0}^n \Gamma\left(\frac{n}{2}+1\right) \frac{(-1)^k}{k!}
\frac{\left(e^{-i\theta}\sqrt{2}\right)^{n-l}}{(n-l)!} \frac{\left(e^{i \theta}\sqrt{2}\right)^l}{l!} \times \\
\sqrt{(k+l)!(n+k-l)!}\; \ket{k+l} \bra{n+k-l}.
\end{split}\end{equation*}
Using the above form of the operator, we can perform a straightforward integration of either sides to obtain the completeness result,
\begin{equation}\label{rhow_complete}
\int_0^{2\pi} \rho_\text{w}(\theta)\, d\theta = 1.
\end{equation}
From the geometric interpretation associated with the Wigner phase operator expounded in the previous section, it becomes clear that the above completeness property is
intimately related to the requirement that the Wigner function be normalized.
\subsubsection{Relation with Pegg-Barnett phase operator}
Since the state $\ket{\theta}$ defined in \eqref{phasestate} are states of well-defined phased, it is of interest to investigate the phase of these states as predicted by
our operator formalism. The Wigner function of these phase states has been worked out previously \citep{Herz93}. Here, we attempt to evaluate the Wigner phase distribution
$P^\text{w}_{\phi}(\theta)$ associated with the phase state $\ket{\phi}$. Before we do this, we give a Fock state decomposition of $\rho_\text{w}(\theta)$ which is very
conducive to subsequent developments, viz.,
\begin{equation}\label{rhow_outer_prod1}
\rho_\text{w}(\theta) = \frac{1}{2\pi} \sum_{n,k=0}^\infty \Gamma \left( \frac{n+k}{2}+1 \right) \frac{(\sqrt{2}e^{i\theta})^k}{k!} (a^\dagger)^k
\left[ \sum_{m=0}^\infty (-1)^m \ket{m}\bra{m} \right] a^n \frac{(\sqrt{2}e^{-i\theta})^n}{n!}.
\end{equation}
Using the above expression to evaluate $P^\text{w}_{\phi}(\theta)$ we get the following elegant result,
\begin{equation}\label{Pw_PB}
P^\text{w}_{\phi}(\theta) = \text{Tr}\left[ \rho_\text{w}(\theta -\phi) \rho_\text{PB}(0) \right],
\end{equation}
which implies that whenever four angles satisfy the relation, $\theta - \phi = \theta' - \phi'$, we have the following relation,
\begin{equation}\label{weak_equiv}
\text{Tr}\left[ \rho_\text{w}(\theta) \rho_\text{PB}(\phi) \right] = \text{Tr}\left[ \rho_\text{w}(\theta') \rho_\text{PB}(\phi') \right].
\end{equation}
We term the above relation, the weak-equivalence between the Pegg-Barnett and the Wigner phase operator formalisms. \eqref{Pw_PB} is a statement of the fact that when the
phase of a state is well-defined, then its phase, as given by its expectation over $\rho_\text{w}(\theta)$, gives the phase only upto a relative reference phase; this is of
course a very important property for any phase operator. Similarly, another way of looking at the weak-equivalence \eqref{weak_equiv} is to note that either side of the
equation is actually the Hilbert-Schmidt inner product of the respective operator pairs. Thus, in some sense, the equation claims that the ``overlap'' between the
Wigner and Pegg-Barnett phase operators is invariant under the transformation $\theta - \phi = \theta' - \phi'$.
\subsection{Operator in projector form}
A very important aspect of any hermitian operator is its projection property i.e., to identify the space onto which a measurement of the associated physical property
would finally leave the system in. Whenever an operator has a diagonal representation in terms of some orthogonal basis, it is a projector when it is
idempotent i.e., when the operator squares to itself. The Wigner phase operator is definitely not idempotent, as can be easily deduced from \eqref{rhow_outer_prod1}.
We use the integral representation of the gamma function, viz., $\Gamma(x) = 2 \int_0^\infty u^{2x-1}e^{-u^2} du$ in \eqref{rhow_outer_prod1}, to express it in the form,
\begin{equation}\label{rhow_outer_prod2}
\rho_\text{w}(\theta) = \frac{1}{\pi} \int_0^\infty du\; u e^{-u^2} \sum_{m=0}^{\infty} (-1)^m \ket{\sqrt{2}u e^{i\theta},m}\bra{\sqrt{2}u e^{i\theta},m},
\end{equation}
where the state $\ket{\sqrt{2}u e^{i\theta},m}$ is a particular case of the general state,
\begin{equation}\label{ket_z_m}
\ket{z,m} = \sum_{n=0}^\infty \frac{z^n}{n!} (a^\dagger)^n \ket{m}.
\end{equation}
Together with the above definition, \eqref{rhow_outer_prod2} provides a diagonal representation of the Wigner phase operator, in the sense that the integral and sum
contains only terms that couple projectors of the form $\ket{z,m}\bra{z',m'}$ for $z=z', m=m'$. Note that the state $\ket{z,m}$ is not normalized as given in
\eqref{ket_z_m}; to normalize it, it has to be divided by a factor of $\sqrt{I_0 (2 \vert z \vert)}$, where $I_\nu$ is the Bessel function of the second kind of order
$\nu$. We also have the relation for the un-normalized states,
\begin{equation*}
\bra{z',m'}z,m \rangle = \frac{z^m ({z'}^*)^{m+m'}}{(m+m')!}\sqrt{\frac{m!}{m'!}}\, {}_1 F_1 \left( m+1; m+m'+1; {z'}^* \right),
\end{equation*}
thus showing that these states do not furnish an orthogonal set. Owing to the completeness relation \eqref{rhow_complete}, we have the resolution of unity,
\begin{equation*}
\int dz\; dz^*\; e^{-\vert z \vert^2} \sum_{m=0}^\infty (-1)^m \ket{z,m}\bra{z,m} = 1,
\end{equation*}
so that these states do furnish a complete bases set, although not orthogonal, very much like the Glauber coherent states.
\section{Conclusions}
The hermitian Wigner phase operator that we have constructed is shown to give the expected uniform phase distribution for a number state, while for a coherent state, it
gives a distribution which essentially captures the same phase information as given by the Pegg-Barnett formalism. Subsequently, it is shown that our operator is
complete and we also give a diagonal representation of the same in terms of a set of complete states which seem to be naturally suited for describing phase in the quantum
phase space. Finally, we also prove that the Wigner phase operator satisfies a weak-equivalence relation with the Pegg-Barnett operator, which is the reason why the radially integrated Wigner function captures essentially the same phase information as the latter.
The concept of defining a phase via the Wigner function gives a more general way to understand phase and it maybe the case that one can ascribe a consistent meaning to the
phase of non-bosonic systems and the authors are pursuing work along these directions.
\begin{acknowledgments}
One of the authors (TS) would like to thank Prof. P. C. Deshmukh at the Indian Institute of Technology Madras for a valuable suggestion
on the nature of infinite dimensional operators, made during the course of the work. The other author would like to thank Prof. N. Mukunda at the Indian Institute of
Science for helpful discussions regarding the notion of a Wigner distribution and the associated generalizations.
\end{acknowledgments}
|
{
"timestamp": "2010-09-22T02:01:32",
"yymm": "1009",
"arxiv_id": "1009.4030",
"language": "en",
"url": "https://arxiv.org/abs/1009.4030"
}
|
\section{Introduction}
\label{sec:intro}
Over the past decades, numerical simulations \citep[{\it e.g. }][]{Efstathiou}, and large redshift surveys \citep[{\it e.g. }][]{lapparent86} have highlighted the large-scale structure (hereafter LSS) of our Universe, a cosmic web formed by voids, sheets, elongated filaments and clusters at their nodes \citep{pogo96}. Characterizing quantitatively these striking features of the observed and modeled universe has proven to be both useful \citep{SDSSskel,gay} but challenging. It is useful because these features reflect the underlying dynamics of structure formation, and are therefore sensitive to the content of the universe \citep{pogo}. It is challenging because observations and simulations provide limited and noisy data sets. Recently Soubie (2010, hereafter paper I) presented an algorithm able to estimate the underlying critical sets (walls, filaments, voids) from a given noisy discrete sample of the underlying field. Typically, this situation arises in astrophysics when the aim is to recover the topology or the geometry of the underlying density field while only a catalogue of galaxies is available. For instance, in the context of understanding the history of our Milky Way, it is of interest to identify the filaments of the local group. Yet typically in this context, only a limited number of galaxies at somewhat poorly estimated positions are observed. For redshift catalogues involving hundreds of thousands of galaxies, one would also wish to reconstruct the main features of the cosmic web as best as the non-uniform sampling allows. From a theoretical point of view, it might for instance be of interest to compute the cosmic evolution of the filamentary network, as its history constrains the dark energy content of the universe. From an observational point of view, it could also help solving the missing baryon problem \citep{1998ApJ...503..518F} because most of such baryons has been considered to be located along the filamentary structure in the form of diffuse hot gas called Warm/Hot Intergalactic Medium (WHIM; \cite{1999ApJ...514....1C}, \cite{aracil}). Identifying the filament from galaxy distributions clearly provides good candidates for searching for the WHIM with UV absorptions \citep[e.g.]{2000ApJ...534L...1T,2010ApJ...710..613D}, X-ray absorptions \citep[e.g.][]{2002ApJ...564..604F,2006PASJ...58..657K,2009ApJ...695.1351B, 2010ApJ...714.1715F} and X-ray emission lines by future surveys \citep[e.g.][]{2003PASJ...55..879Y,2006SPIE.6266E..12O}. It is therefore of prime importance to provide a tool which deals consistently with such possibly sparse discrete samples. Quite a few such options have been presented recently (\cite{skel2D,hahn,SDSSskel,rsex,skel,spine, aragonvoid,jaime,neyrinck,platen, stoica05,stoica07,stoica10}, Sousbie (2010)), relying on different strategies on how to deal with these constraints (see Sousbie (2010)).
In paper I, the emphasis was on a formal presentation of the underlying mathematical theory and its extension to the discrete regime. As the corresponding algorithm is fairly
intricate, a certain level of formal jargon was required to describe it unambiguously.
Hence paper I focused on the language of mathematics.
Conversely, let us first now rephrase here
the corresponding framework in the more intuitive language of astrophysical data processing, as our aim is to appeal to both the community of computational geometry and that of astrophysics.
What should be the expected characteristics of the ideal structure finder?
Optimally, one would like to implement an algorithm which recovers the important and robust features of the underlying field even when little information is available, so that the procedure manages to reasonably identify the most striking features of the cosmic field. {\sl Topolology} (in fact discrete topology) therefore provides the natural context in which the optimal algorithm should be implemented.
Indeed, topology de facto characterizes the ``rubber" geometry of the underlying field, {\it i.e. } its most intrinsic and robust features.
More specifically, as argued in Sousbie (2010),
ideally such an optimal tool should produce an ensemble of critical sets (lines, surfaces and volumes) consistent with those defined within the context of Morse theory for sufficiently smooth fields \citep{milnor,jost}.
Morse theory basically provides a rigorous framework in which to formally define such sets for ``regular" density fields (roughly speaking regular means not degenerate so that these sets are well defined).
For instance, the critical lines defined by this theory connect peaks and
maxima via special (extremal) flow lines of the gradient\footnote{Indeed, Morse theory formalizes the process of partitionning space according to the gradient flow of the density into so called ascending and descending manifolds. In other words, it tags space according to where one would end up going ``uphill" or ``downhill". In doing so it identifies special lines or surfaces, where something unusual happens.}. These lines should trace quite well the filaments of the LSS. Similarly, the walls of the LSS should have
a natural equivalent feature as the ``critical" surfaces of Morse theory (the so called 2-manifolds).
Here our purpose is to proceed within the context of its discrete counterpart \citep{forman}.
The corresponding discrete construction should be as consistent as possible with all the topological features of the underlying smooth field (it should globally preserve, at the level of the noise, the salient features of the field, such as the number of connected components, the number of tunnels or holes defined by its iso-contours, etc.; conversely\footnote{this well known duality between the topology of the level sets and the characteristics of the critical points clearly has a discrete analogue through the creation/destruction of discrete cycles, see paper I} the significant discrete critical sets should have the correct ``combinatorial" properties: {\it e.g. } critical lines should only connect at critical points, saddle points connect exactly two peaks together, etc...). The level of complexity of the corresponding network should also reflect the inhomogeneities of the underlying survey: {\it i.e. } it should adapt its level of description to the sampling, hence yielding a parameter-free multiscale description of the cosmic web. In fact, it should also provide a simple diagnostic in order to estimate the robustness of the various components of the network ({\it i.e. } the degree of reproduced details should be tunable to the purpose of the investigation). Finally it should clearly address the shortcomings of watershed-based methods described in paper I
(namely the occurrence of spurious boundary lines induced by resampling in 3D or more).
Paper I presented precisely such a tool, while building up on an extension of Morse theory to discrete unstructured meshes.
Two main challenges were addressed: (i) defining the counterpart of the (topologically consistent) critical sets {\sl on} the mesh; (ii) defining a procedure to simplify the corresponding mesh
at the level of the local shot noise. \\ The first step is achieved by considering simultaneously all the discrete components of the triangulated mesh (its vertices, edges, faces and tetrahedra),
and reassigning a density to all these components in a manner which is heuristically consistent with the sampled density at the vertices; this relabeling procedure also ensures that the discrete flow (which follows from the corresponding discrete gradient) is sufficiently well-behaved to provide such topological consistency (specifically, it ensures the existence of
discrete counterparts of regular critical points). Loosely speaking, amongst the discrete analogues of gradient flows, the algorithm identifies the critical subsets as special sets which cannot be paired to their neighbours through these discrete gradients.
Note that the required level of compliancy to achieve this construction has the virtue of not only producing the discrete set of critical segments, but also the triangulated walls and voids at no extra computational cost.
\\ The second step follows from the concept of topological persistence \citep{edel00,edel}, which assigns a
density ratio to pairs of critical points which are found to be connected together by such discrete integral paths; { these pairs are identified by the destruction/creation of critical points as one describes the level sets}.
If this ratio is below a given threshold, the corresponding critical line/surface is found to be (topologically) insignificant, it is removed from the set and the remaining critical mesh is simplified so as to recover some topological consistency. In other words,
if the shot noise induces the occurrence of the discrete counterparts of, {\it e.g. } spurious loops, disconnected blobs, or tunnels which are found to be insignificant according the the aforementioned criterium, they will be removed. The idea of topological persistence is central in producing a natural (topologically motivated)
self-consistent criterion for infering the significance level of the identified structures. {In particular, it warrants that the removal of pairs of critical points consistently extracts the corresponding topological feature (loop, tunnel, component).}
Importantly, let us emphasize that within this framework, the mathematical theories that we use are intrinsically discrete and readily apply to the measured raw data (modulo the consistent labelling of the elements of the Delaunay tessellation relative to the DTFE densities computed at the sampling points). This warrants that all the well-known and extensively studied mathematical properties of Morse theory are ensured by construction {\sl at the mesh level}, and that the corresponding cosmological structures therefore correspond to well-defined mathematical objects with known mathematical properties.
In the language of computational geometry (see Appendix \ref{sec_terminology} for the relevant definitions), a simplicial complex (the tessellation) is computed from a discrete distribution (galaxy catalogue, N-body simulation, ...) using a Delaunay tessellation. A density $\rho$ is assigned to each galaxy using DTFE (roughly speaking, the density at a vertex is proportional to the inverse volume of its dual Voronoi cell, see \citet{DTFE}). A discrete Morse function (a re-labelling of all elements of the tessellation) is then defined by attributing a properly chosen value to each simplex in the complex ({\it i.e. } the segments, facets and tetrahedron of the tessellation). From this discrete function, we then compute the \hyperref[defDG]{discrete gradient} and deduce the corresponding \hyperref[defDMC]{Discrete Morse-Smale complex} (DMC hereafter, \citet{forman}). The \hyperref[defDMC]{DMC} (the set of critical points connected by \hyperref[defarc]{arcs}, quads, \hyperref[defcrystal]{Crystals} etc..) is used as the link between the topological and geometrical properties of the density field. Its critical points together with their ascending and descending \hyperref[defmanifold]{manifolds} (the ``critical" sets) are identified to the peaks, filaments, walls and voids of the density field. The \hyperref[defDMC]{DMC} is then filtered using persistence theory. For that purpose, we consider the \hyperref[deffiltration]{Filtration} (the discrete counter part of the density-sorted level-sets) of the tessellation according to the values of the discrete Morse function and use it to compute persistence pairs of critical points (pairs of critical points which are linked by their appearance and disappearance as one scans the \hyperref[deffiltration]{Filtration}). The \hyperref[defDMC]{DMC} is simplified by canceling the pairs that are likely to be generated by noise. This is achieved by computing the probability distribution function of the persistence ratio ({\it i.e. } the ratio of the densities {\sl at} the connected pair) of all types of pairs in scale-invariant Gaussian random fields and canceling the pairs with a persistence ratio whose probability is lower than a certain level.
Paper I presented a couple of examples of such a construction in 2D. Let us now illustrate the virtue of the method in the context of the 3D cosmic Web.
We start\footnote{Note that our goal here is not to present an exhaustive review of the geometrical properties of the cosmic web, which is clearly out of the scope of this paper.} by showing that the geometry of the cosmic web is accurately reproduced, while
illustrating the quality of the cosmological structures identification, both in an N-Body simulation
(section~\ref{sec:LSSsimu}) and directly on an unprocessed version of the SDSS DR7 galaxy catalogue (section~\ref{sec_SDSS}).
In particular we show how DisPerSE\,\, allows us to identify various configurations of the filamentary structure of galaxies,
and identify a previously missed X-ray ``optically faint" halo at the intersection of a set of SDSS filaments using the SUZAKU satellite.
We then discuss in section~\ref{subsec_sigthres} the problem of estimating the right value for the persistence level in cosmological simulations,
and illustrate how the measured topological properties of the LSS distributions are affected by varying this threshold.
In particular we show how this criterion compares with the simple friend-of-friend algorithm when attempting to identify
halos in simulations. Section~\ref{sec_conclusion} wraps up and discusses prospects.
\section{Geometry of LSS: simulation}
\label{sec:LSSsimu}
\begin{figure*}
\begin{minipage}[c][\textheight]{\linewidth}
\begin{minipage}[c]{0.49\linewidth}
\centering\includegraphics[height=0.3\textheight]{img/250bis_simu}\\
\centering\includegraphics[height=0.3\textheight]{img/250bis_simu_skl4}\\
\centering\includegraphics[height=0.3\textheight]{img/250bis_simu_skl5}\\
\end{minipage}
\hfill
\begin{minipage}[c]{0.49\linewidth}
\centering\includegraphics[height=0.3\textheight]{img/250bis_skl3}\\
\centering\includegraphics[height=0.3\textheight]{img/250bis_skl4}\\
\centering\includegraphics[height=0.3\textheight]{img/250bis_skl5}\\
\end{minipage}
\caption{The filamentary distribution above a persistence level of $\nsig{3}$, $\nsig{4}$ and $\nsig{5}$ (from top to bottom) in a $250\times250\times20{\,h^{-1}\,{\rm Mpc}}$ slice of a $512^3$ particles and $250{\,h^{-3}\,{\rm Mpc}^{3}}$ large cosmological simulation. The computation was achieved on a $128^3$ particles sub-sample, and the filaments are colored according to the logarithm of the density. The density field was represented using the $512^3$ particles of the N-body simulation. \label{fig_simu250} }
\end{minipage}
\end{figure*}
\begin{figure*}
\begin{minipage}[c][\textheight]{\linewidth}
\centering\subfigure[Simulated dark matter distibution]{\includegraphics[width=0.49\linewidth]{img/F3_simu250_large}\label{fig_simu250_voidA}}
\hfill\subfigure[A void (bottom right) embedded in the filamentary structure]{\includegraphics[width=0.49\linewidth]{img/F3_simu250_large_simu_void_skel_crit}\label{fig_simu250_voidB}}\\
\centering\subfigure[Zoom on the void of panel \ref{fig_simu250_voidB}.]{\includegraphics[width=0.49\linewidth]{img/F3_simu250Z_void_simu}\label{fig_simu250_voidC}}
\hfill\subfigure[The relationship between the detected void, filaments and critical points]{\includegraphics[width=0.49\linewidth]{img/F3_simu250Z_skel_crit_void}\label{fig_simu250_voidD}}\\
\caption{The \hyperref[defarc]{arcs} of the \hyperref[defDMC]{Discrete Morse-Smale complex} ({\it i.e. } the filaments) and an ascending $3$-\hyperref[defmanifold]{manifold} ({\it i.e. } a void) at a significance level of $\nsig{5}$ in the same distribution as that of figure \ref{fig_simu250} (a $250\times250\times20{\,h^{-1}\,{\rm Mpc}}$ slice of a $512^3$ particles and $250{\,h^{-3}\,{\rm Mpc}^{3}}$ large cosmological simulation). The density distribution is represented using all available particles within the simulation (panel \ref{fig_simu250_voidA}) while the DMC was computed using $128^3$ particles sub-sample. The $2$ lower panels (\ref{fig_simu250_voidC} and \ref{fig_simu250_voidD}) show a zoom on the upper panels at the location of the randomly selected void (see panel \ref{fig_simu250_voidB}). On these figures, the maxima, $1$-saddle points, $2$-saddle points and minima are represented as red, yellow, green and blue square respectively and the \hyperref[defarc]{arcs} as well as the manifold are shaded according to the log of the density. Note on panel \ref{fig_simu250_voidD} how the maxima, saddle-points and path of the filaments corresponds to the crests of the 2D density field measured on the surface of the void. This particularly emphasize the coherence of the detection of objects of different nature. \label{fig_simu250_void}}
\end{minipage}
\end{figure*}
Although we have shown in paper I that the method introduced in that paper seems to be able to measure the topological properties of the cosmic web efficiently even in the presence of significant noise, we only illustrated in the 2D case that it could also recover correctly the geometry of the filamentary structure (see paper I). Demonstrating that a given algorithm is able to correctly identify the location of filaments is a difficult task, as it requires the previous knowledge of the location of those filaments. The only solution therefore seems to build an artificial distribution from a previously defined filamentary structure. This method was adopted in \citet{spine}, where the authors use a Voronoi Kinematic Model \citep{weygaert02}. The principle of the Voronoi Kinematic Model is to identify the voids, walls, filaments and clusters to the cells, faces, edges and vertice of the Voronoi tesselation. In practice, randomly distributed particles are moved away from the nuclei of the Voronoi cells following a universal expansion rate, and their displacement being constrained to the faces, edges and finally vertice as they reach them. This results in a distribution of particles where each is said to be a void, wall, filament or cluster particle depending on weather they belong to a cell, face, edge or vertice of a Voronoi cell when the simulation is stopped.\\
We argue that using such a model to quantify the quality of the Morse-Smale complex identification is not as relevant as one would think, mainly because it is too idealized topologically speaking. In fact, it is a built-in property of the Voronoi Kinematic Models that all the cosmological structures overlap neatly: maxima ({\it i.e. } voronoi vertice) are located at the intersection of filaments ({\it i.e. } Voronoi segments) that always intersect with a suitable angle, those filaments are themselves by definition located at the intersection of at least three voids ({\it i.e. } voronoi cells), and each pair of neighboring void have exactly one Voronoi face in common, neatly defining the walls. As was shown in paper I, density functions extracted from actual data sets are in fact quite different, as they do not comply so easily to Morse conditions, in particular when measured from cosmological simulations or observational galaxy catalogues. In that case, and as clearly shown in paper I (see appendix 1), filaments may (and actually often do) merge before reaching a maximum, two apparently neighboring voids (down to the resolution limits) do not necessarily share a common face, and filaments are not necessarily at the intersection of at least three voids (once again, down to the resolution limit). The nature of the Voronoi Kinematic Model is therefore such that it avoids all the difficulty in identifying the Morse-Smale complex of realistic data sets. It might be possible to build more sophisticated Voronoi Models, that would for instance mimic the structure mergers that occur along the course of the evolution of large scale matter distribution in the Universe, but this is clear out of the scope of this paper. For the lack of a simple better way, we therefore use here what is probably to date the most efficient way to detect structures: the human eye and brain.\\
\begin{figure*}
\begin{minipage}[c][\textheight]{\linewidth}
\centering\subfigure[Dark matter distribution in a $50\times50\times20{\,h^{-1}\,{\rm Mpc}}$ sub-box]{\includegraphics[width=0.49\linewidth]{img/simu_manifolds_jpg}\label{fig_simu_manifolds_picA}}
\hfill \subfigure[An ascending $2$-manifold ({\it i.e. } a wall)]{\includegraphics[width=0.49\linewidth]{img/simu_wall_jpg}\label{fig_simu_manifolds_picB}}\\
\centering\subfigure[An ascending $3$-manifold ({\it i.e. } a void)]{\includegraphics[width=0.49\linewidth]{img/simu_voidg_jpg}\label{fig_simu_manifolds_picC}}
\hfill \subfigure[Superposition of an ascending $3$-manifold and an ascending $2$-manifold on its surface.]{\includegraphics[width=0.49\linewidth]{img/simu_wall_void_jpg}\label{fig_simu_manifolds_picD}}\\
\caption{An ascending $2$-manifold ({\it i.e. } blue 2D wall) and an ascending $3$-manifold ({\it i.e. } green 3D void) identified in a $512^3$ particles $100{\,h^{-1}\,{\rm Mpc}}$ $\Lambda{\rm CDM}$ dark matter simulation. The manifolds where computed from a $64^3$ particles sub-sample.\label{fig_simu_manifolds_pic}}
\end{minipage}
\end{figure*}
\subsection{Visual inspection}
The evolution of the geometry of the measured filaments with significance threshold is illustrated on figure \ref{fig_simu250}. The \hyperref[defDMC]{DMC} represented on this figure was computed at significance levels of $3$, $4$ and $\nsig{5}$ (from top to bottom) within $128^3$ particles sub sample of a $512^3$ particles, $250{\,h^{-3}\,{\rm Mpc}^{3}}$ $\Lambda{\rm CDM}$ dark matter only N-body simulation. Note that the dark matter distribution within the $20{\,h^{-1}\,{\rm Mpc}}$ slice is represented in the top left corner to facilitate the visualization of its filamentary structure. Despite the projection effects that create visual artifacts ({\it i.e. } spurious filament looking structures resulting from the projection of dark matter clumps at different depths) and the fact that filaments may enter or leave the slice, therefore seemingly appearing and disappearing for no apparent reasons, it seems fair to recognize that the agreement between the observed and measured filaments is excellent. These good performances are mainly the result of our use of the scale adaptive Delaunay tessellation and the fact that our implementation does not require any pre-treatement of the density field, unlike usual grid based methods which enforce a maximal resolution and resort to some kind of density smoothing technique that affect the geometrical properties of the distribution. As a result, the resolution of the filaments is optimal with respect to the initial sampling whatever the selected significance level: the higher \hyperref[defpers]{persistence} and larger scale filamentary network is, by construction, a subset of its less persistent and lower scale counterpart. Because \hyperref[defpers]{persistence} based topological simplification is used, increasing the \hyperref[defpers]{persistence} threshold actually results in less significant filaments disappearing (when simplifying a $1$-saddle point/$2$-saddle point \hyperref[defpers]{persistence} pair) or merging into each other (when simplifying a $1$-saddle point/maximum \hyperref[defpers]{persistence} pair) to form larger scale more persistent ones, but conserving the exact same resolution in any case. This can easily be observed by comparing the filamentary networks on the right column of figure \ref{fig_simu250}.\\
Another remarkable advantage of constructing cosmological structures identification on Morse theory is the extraordinary built-in coherence of the results, whatever the type of structure, as shown on figures \ref{fig_simu250_void} and \ref{fig_simu_manifolds_pic}. For instance, the intricate pattern of a randomly selected void ({\it i.e. } an ascending $3$ \hyperref[defmanifold]{manifold}) embedded within the filamentary network ({\it i.e. } ascending $1$ \hyperref[defmanifold]{manifolds}) of the cosmic web in the same simulation as previously is shown on figure \ref{fig_simu250_void}. The location of the void within the slice is displayed on panel \ref{fig_simu250_voidB}, each colored square standing for a critical point (see legend). On the zoomed frame (\ref{fig_simu250_voidC} and \ref{fig_simu250_voidD}), the surface of the void has been shaded according to the logarithm of the density, showing how the \hyperref[defDMC]{DMC} correctly traces the filamentary structure at the interface of the ascending $3$-\hyperref[defmanifold]{manifolds}, as expected in Morse theory\footnote{The slight shift in position between the surface of the void and the filament is due to the fact that we smoothed the filaments $4$ times (see paper I)}. Similarly, the neat relationship between a detected void and a wall structure on its surface ({\it i.e. } an ascending $2$ \hyperref[defmanifold]{manifold}) in a $100{\,h^{-1}\,{\rm Mpc}}$ large N-body simulation is displayed on figure \ref{fig_simu_manifolds_pic}.\\
\begin{figure}
\centering
\includegraphics[width=\linewidth]{img/halo_persistence_jpg}
\caption{Distribution of the \hyperref[defpers]{persistence} pairs of the highest density particles within each dark matter halo of mass $M>74 \times 10^{10}\;M_\odot$ (red) and $M>590 \times 10^{10}\;M_\odot$ (green) in a $128^3$ particle sub-sample of a $100{\,h^{-1}\,{\rm Mpc}}$ large $\Lambda{\rm CDM}$ dark matter simulation. The \hyperref[defpers]{persistence} diagram of maxima/$1$-saddle-points pairs with \hyperref[defpers]{persistence} larger than $\nsig{3}$ is shown in the background. The horizontal dashed and dotted lines correspond to overdensity levels of $4\times 10^3$ and $3.2\times 10^4$ respectively and the oblique lines to \hyperref[defpers]{persistence} levels of $\sim\nsig{4}$ and $\sim\nsig{5}$ respectively.\label{fig_halo_per_diag}}
\end{figure}
\begin{figure}
\centering\subfigure[Dark matter distribution in a $50\times50\times20{\,h^{-1}\,{\rm Mpc}}$ sub-box and haloes with mass $M>73.8\;10^{10}\;M_\odot$]{\includegraphics[height=0.26\textheight]{img/simu_fof20}\label{fig_simu_halos_picA}}\\
\subfigure[The filaments at $\nsig{4}$ on a $128^3$ sub-sample]{\includegraphics[height=0.26\textheight]{img/simu_fof20_skl}\label{fig_simu_halos_picB}}\\
\subfigure[The main filaments of the dark matter haloes]{\includegraphics[height=0.26\textheight]{img/simu_fof20_skelselect}\label{fig_simu_halos_picC}}\\
\caption{Distribution of the main filaments of FOF haloes with mass $M>73.8\;10^{10}\;M_\odot$ in a $20{\,h^{-1}\,{\rm Mpc}}$ thick slice of a $512^3$ particles $100{\,h^{-3}\,{\rm Mpc}^{3}}$ $\Lambda{\rm CDM}$ dark matter simulation. The filaments were computed from a $128^3$ particles sub-sample. Note that many filaments are linked to halos outside the slice, giving the false impression to end for no reason.\label{fig_simu_halos_pic} }
\end{figure}
Let us finally address a straightforward question: to what extend does DisPerSE\,\, manage to grasp the main features of the cosmic web with relatively sparse samples?
Figure~\ref{fig_simuD_resolution} illustrates this query while comparing the filaments computed from two sub samples of variing resolution of from a $250{\,h^{-3}\,{\rm Mpc}^{3}}$ large cosmological simulation with $512^3$ particles (namely $64^3$ sub-sample and $128^3$ sub-sample respectively). From this figure, it seems that indeed, the features which are identified in the
sparser sample are real, since they are also found in the more densely sampled catalogue. There seems to be some encouraging level of convergence between the two sets of critical lines.
\subsection{Persistent peak identification}
\begin{figure}
\begin{minipage}[c][\textheight]{\linewidth}
\centering\includegraphics[height=0.28\textheight]{img/simu250D_10_skl128.ps}\\
\centering\includegraphics[height=0.28\textheight]{img/simu250D_10_skl64.ps}\\
\centering\includegraphics[height=0.28\textheight]{img/simu250D_10_skl64_skl128.ps}\\
\caption{The filamentary distribution above a \hyperref[defpers]{persistence} level of $\nsig{4}$ in a $250\times250\times20{\,h^{-1}\,{\rm Mpc}}$ slice of a $512^3$ particles and $250{\,h^{-3}\,{\rm Mpc}^{3}}$ large cosmological simulation. The red segments on the {\sl top} and {\sl central } figures correspond to the segments of the Delaunay tesselation of a $128^3$ and $64^3$ sub-sample, on which the corresponding filaments have been computed. On the {\sl bottom} figure, the thick white filaments correspond to the $64^3$ sub-sample while the blue thin filaments where computed on the $128^3$ sub-sample. This figure clearly demonstrates that DisPerSE\,\, is able to grasp the main features of the cosmic web with relatively sparse sample.
\label{fig_simuD_resolution} }
\end{minipage}
\end{figure}
From visual inspection, it therefore seems relatively clear that the technique developed in this paper is able to correctly decompose the cosmic web into simpler objects of astrophysical interest. However, the approach is based on one fundamental assumption, which is that the ascending and descending \hyperref[defmanifold]{manifolds} of Morse theory, each associated to a specific type of critical point, are representative of the voids, filaments, walls and haloes. While the astrophysical nature of a filament or a wall is not defined very precisely, but is rather understood intuitively, this is not the case of a dark matter halo for instance, which is supposed to be a gravitationally bound structure and the fact that the persistent maxima of the density field correctly identify the gravitationally bound structures is not established. We check this assumption by comparing the distribution of dark matter haloes identified by a simple friend of friend technique (FOF hereafter, see \citet{FOFinfo} for instance) in a $100{\,h^{-3}\,{\rm Mpc}^{3}}$, $512^3$ $\Lambda{\rm CDM}$ dark matter simulation to the \hyperref[defpers]{persistence} diagram in the same simulation, as illustrated on figure \ref{fig_halo_per_diag}.\\
On this figure, the probability distribution function (PDF) of the \hyperref[defpers]{persistence} pairs\footnote{As in section \ref{subsec_sigthres}, each pair is represented by a point with coordinates the density of each of its critical point, see that section for more explanations.} of type $2$ ({\it i.e. } the maxima/$1$-saddle points pairs) measured in a $128^3$ particles sub-sample is displayed in the density/density plane, the horizontal axis corresponding to the density of the $1$-saddle point, and the vertical one to that of the maximum. The green line therefore represents the $0$-\hyperref[defpers]{persistence} limit, while the oblique white dashed and dotted lines delimits the $\nsig{4}$ and $\nsig{5}$ threshold respectively. In order to compare this distribution to that of the astrophysical dark matter haloes, each of them is also represented as a disk with coordinates that of the \hyperref[defpers]{persistence} pair of its most dense particle (the densest particle within a halo is necessarily a local maximum). Each halo was identified using a standard linking length parameter of one fifth of the average inter particular distance, and the red disks represent the haloes with mass $M>73.8\times10^{10}\;M_\odot$ ({\it i.e. } with more than $1,280$ particles in the initial simulation, or $20$ in a $128^3$ sub sample), while the green ones represent the haloes with mass $M>590.4\times 10^{10}\;M_\odot$ ({\it i.e. } with more than $10,240$ particles in the initial simulation, or $1,280$ in a $128^3$ sub sample). It is a very striking result how well the population of dark matter halos is localized in the \hyperref[defpers]{persistence} diagram. While lighter ones (red disks) mostly correspond to maxima with \hyperref[defpers]{persistence} ratio higher than $\nsig{4}$ and overdensity $\rho/\rho_0 > 4\times10^3$, the heavier ones lie in the zone with \hyperref[defpers]{persistence} higher than $\nsig{5}$ and overdensity $\rho/\rho_0 > 3.2\times10^4$.\\
These results mean that \hyperref[defpers]{persistence} selection associated to a global overdensity threshold is naturally ({\it i.e. } without any specific qualibration) a very good halo finder, which is quite encouraging, and validates our initial assumption on the relationship between the persistent topological features and the astrophysical components of the cosmic web. This is illustrated on figure \ref{fig_simu_halos_pic} where each dark matter halo with mass $M>73.8\times10^{10}\;M_\odot$ ({\it i.e. } the red disks of figure \ref{fig_halo_per_diag}) is colored in blue. Once again, it is clear on the central frame that all haloes along large filaments are correctly linked by the \hyperref[defDMC]{DMC}. We also remark that the \hyperref[defDMC]{DMC} and \hyperref[defpers]{persistence} pairs contain unexploited information of the topology as our algorithm explicitly identify the $k$-cycles as sequences of critical points associated to \hyperref[defpers]{persistence} pairs (see Sousbie (2010)). For instance, each \hyperref[defpers]{persistence} pair associated to a halo correspond to a $0$-cycle that define a principal filament, as shown on the bottom frame, where only the filaments corresponding to \hyperref[defpers]{persistence} pairs whose maximum is a dark matter halo are represented. Moreover, using the information contained in the persistence pairs, one basically obtains a hierarchical structure finder that is able to also identify substructures not only within clusters, but also within filaments, walls and voids.
\section{Our universe: the SDSS catalogue}
\label{sec_SDSS}
Let us now illustrate a few prospective measurements of the filamentary structure of the actual galaxy distribution in the Universe. The ultimate goal of such measurements is to allow a complete and precise characterization of the properties of the filamentary structure of the galaxy distribution by measuring their topological properties, such as the Betti number and Euler characteristics, and modeling the geometrical characteristics of the voids, walls and filaments ({\it i.e. } their total length, number, the number of filaments per galaxy clusters, ...).
Such a task is rather challenging, as it requires the construction of realistic mock observations from N-body simulations to asses the influence of observational biases and distortions; it also requires a lot of care in the handling of the observational data themselves (for instance by taking into account the complex survey geometry, among other difficulties). In this paper, we focus on convincing the reader that the method we introduced paper I is particularly suited to such a task by showing how easily and efficiently it can be applied to a real galaxy catalogue. We postpone the full investigation to a future paper.
\subsection{The cosmic web in the SDSS}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{img/SDSS_radec}
\caption{Angular distribution of the $515,458$ galaxies corresponding to a sub-sample of the SDSS DR7 galaxy catalogue that we use in our tests (see main text for selection criterion). The $66,608$ red galaxies are those detected as being on the boundary of the distribution using the method described in the main text. Note that some regions were not fully scanned and exhibit series of thin empty parallel stripes, but we simply ignore that fact when computing the boundaries.\label{fig_SDSS_radec}}
\end{figure}
For that purpose, we use data from the $7^{\rm th}$ data release (DR7) of the SDSS \citep{Abazajian09}, and in particular the large-scale structure subsample called {\em dr72bright0} sample of the New York University Value Added catalogue \citep{blanton05}, which is made of a spectroscopic sample of galaxies with u,g,r,i,z- band (K-corrected) absolute magnitudes, r-band apparent magnitude $m_r$, redshifts, and information on the mask of the survey. In that sample, the spectroscopic galaxies are originially selected under the conditions that $10.0 \leq m_r \leq 17.6$ and $0.001 \leq z \leq 0.5$, but we further cut the sample for the purpose of our tests, restraining it to the galaxies with $z \leq 0.26$ and right ascension $100^{\circ}\leq RA \leq 280^{\circ}$, which removes the three thin stripes in the southern Galactic hemisphere. The resulting angular distribution, containing $515,458$ galaxies among the $567,759$ in the original sample is displayed on figure \ref{fig_SDSS_radec}.\\
\begin{figure}
\centering
\includegraphics[width=\linewidth]{img/SDSS_selection}
\caption{A slice within the delaunay tesselation of the distribution used to compute the \hyperref[defDMC]{DMC} of the SDSS. The plain white contour delimits the SDSS distribution (inside) and the randomly added low density particles that fill the void regions of the bounding box (outside). Any galaxy outside the white dashed contours is considered as being on the boundary.\label{fig_SDSS_select}}
\end{figure}
In order to compute the \hyperref[defDMC]{DMC} of the observed galaxy distribution, we will use the mirror type boundary conditions as introduced in paper I. This type of boundary conditions normally apply to distributions enclosed within parallelepiped boxes, which is not the case here. In the simple case of a box-like geometry, the particles within a given distance of the faces are mirrored, and any particle outside the initial box or whose DTFE density may be affected by the content of the exterior of the box is tagged as a boundary particle. As the geometry of the SDSS catalogue is complex, we simply enclose it within a slightly larger box, fill the empty regions with a low density random distribution of particles, and then mirror the boundaries. The mirrored particles and the random ones are tagged and we then identify the boundaries of the galaxies distribution and tag as well those galaxies whose DTFE density may depend on the distribution outside the observational region. Although the catalogue does contain precise information about the mask of the survey, we prefer to use a simple though automatic method to identify the boundaries of the galaxy distribution. This method simply samples the angular galaxy distribution in the RA/DEC plane over a regular grid of $1\times 1^{\circ}$ pixels, and identifies the galaxies on the boundary of the catalogue as those that belong to a pixel with at least one completely empty neighbor. Note that such a method presents the advantage of being generic, as it does not presume any previous knowledge of the mask, and could therefore be applied directly to other galaxy catalogues. The resulting boundary galaxies are represented in red on figure \ref{fig_SDSS_radec}. We finally also tag those galaxies with redshift $z \leq 0.02$ and $z \geq 0.2$ as boundary and proceed with the computation of the \hyperref[defDMC]{DMC}, as in the regular mirror type boundary condition case. A slice of the Delaunay tesselation of the final distribution is displayed on figure \ref{fig_SDSS_select}.\\
\begin{figure}
\begin{minipage}[c][\textheight]{\linewidth}
\includegraphics[width=\linewidth]{img/delaunay_SDSS_fil3}\\
\includegraphics[width=\linewidth]{img/delaunay_SDSS_fil4}\\
\includegraphics[width=\linewidth]{img/delaunay_SDSS_fil5}\\
\caption{ From top to bottom, the filamentary structure in a $\sim40{\,h^{-1}\,{\rm Mpc}}$ thick slice of the SDSS Dr7 galaxy catalogue at a significance level of $3$, $4$ and $\nsig{5}$ respectively. The distribution is represented by the non bounding subset (see main text) of the Delaunay tesselation used to compute the \hyperref[defDMC]{DMC}, shaded according to the logarithm of the density. The depth of a filament can be judged by how dimmed its shade is. Note that filaments that seem to stop for no apparent reason actually enter or leave the slice.\label{fig_delaunaySDSS}}
\end{minipage}
\end{figure}
\begin{figure*}
\begin{minipage}[c][\textheight]{\linewidth}
\centering\subfigure[A portion of SDSS DR7]{\includegraphics[width=0.32\linewidth]{img/F_SDSS_A}\label{fig_SDSS_picA}}
\hfill \subfigure[The filamentary structure]{\includegraphics[width=0.32\linewidth]{img/F_SDSS_B}\label{fig_SDSS_picB}}
\hfill\subfigure[Three voids]{\includegraphics[width=0.32\linewidth]{img/F_SDSS_C}\label{fig_SDSS_picC}}\\
\centering \subfigure[A zoom on the voids and the filamentary structure]{\includegraphics[height=0.4\textheight]{img/F_SDSS_D}\label{fig_SDSS_picD}}\\
\caption{ The detected filamentary structure at a significance level of $\nsig{5}$ and three voids within a portion of SDSS DR7. Note that only the upper half of the distribution shown on figure \ref{fig_SDSS_radec} is displayed here for clarity reasons. The color of the filaments corresponds the the logarithm of the density field. The filaments of the SDSS extracted with DisPerSE\,\, is readily available online at the URL {\em\tt http://www.iap.fr/users/sousbie/}.\label{fig_SDSS_pic} }
\end{minipage}
\end{figure*}
\begin{figure*}
\begin{minipage}[c][\textheight]{\linewidth}
\begin{minipage}[c]{0.49\linewidth}
\centering\includegraphics[height=0.3\textheight]{img/F2_slice_SDSS}\\
\centering\includegraphics[height=0.3\textheight]{img/F2_slice_skel}\\
\centering\includegraphics[height=0.3\textheight]{img/F2_slice_SDSS_skel}\\
\end{minipage}
\hfill
\begin{minipage}[c]{0.49\linewidth}
\centering\includegraphics[height=0.3\textheight]{img/F2_zoom_SDSS_skel}\\
\centering\includegraphics[height=0.3\textheight]{img/F2_zoom_SDSS_skel_voidfil_density}\\
\centering\includegraphics[height=0.3\textheight]{img/F2_zoom_SDSS_skel_voidwire}\\
\end{minipage}
\caption{The filamentary structure (left) and a void (right) detected at a significance level of $\nsig{5}$ in SDSS DR7. In order to emphasize the filamentary structure, only a $\sim 60{\,h^{-1}\,{\rm Mpc}}$ thick flat slice of the distribution is displayed on each frame. The void surface is shaded according to the log of the density field (central right frame), while the color of each \hyperref[defarc]{arc} of the \hyperref[defDMC]{DMC} tracing the filamentary structure depends on the index of the maximum to which it is connected. Note that the foremost part of the voids on the central and bottom right picture protrudes from the slice, while the filaments are trimmed to its surface.
Given its shape, this void is in fact a good example of why we should identify filaments via a DMC rather than using a Watershed technique, as it displays two strong ``thin wings"
which would lead to the incorrect detection of spurious sets of boundaries.
\label{fig_SDSS_slice}}
\end{minipage}
\end{figure*}
The resulting \hyperref[defDMC]{DMC} covers the $440,950$ galaxies in black on figure \ref{fig_SDSS_radec} and obeying the additional condition $0.02\leq z \leq 0.2$ (or equivalently $85\leq d \leq 860{\,h^{-1}\,{\rm Mpc}}$) and it is displayed on figures \ref{fig_delaunaySDSS}, \ref{fig_SDSS_pic}, and \ref{fig_SDSS_slice}. Figure \ref{fig_delaunaySDSS} illustrates the influence of the significance level on the measured filamentary network. On this figure, the filaments ({\it i.e. } the ascending $1$-\hyperref[defmanifold]{manifolds} or \hyperref[defarc]{arcs}) within a $\sim 40{\,h^{-1}\,{\rm Mpc}}$ slice of the Delaunay tessellations are shown at significance levels of $\nsig{3}$, $\nsig{4}$ and $\nsig{5}$ (from top to bottom); it is quite striking how well more or less significant filaments are accurately identified depending on the value of the \hyperref[defpers]{persistence} ratio threshold. Note how already at a level of $\nsig{3}$ the influence of sampling noise has disappeared and increasing this threshold results in the selection of apparently denser, bigger and longer filaments. As the distant faint galaxies and the nearby bright ones cannot be observed easily, the selection function strongly depends on the distance, and so does the sampling. It reflects in the shade of the Delaunay tessellation, which depends on the logarithm of the density. From a theoretical point of view, the fact that the absolute value of the density is multiplied by the selection function should not affect the detection of the filaments as long as the value of the selection function does not vary much over the typical scale of a filament (or in other word, as long as the topology of the distribution remains unchanged). The measured \hyperref[defpers]{persistence} ratio of \hyperref[defpers]{persistence} pairs may be slightly affected though, when the two critical points in the pair are located at different distances, but this does not seem to have much importance in the present case. A more significant effect results from the scale adaptive nature of DTFE. Because the quality of the sampling decreases with distance, comparatively larger scale filaments are identified as the distance increases and to be able to identify comparable filaments independently of the distance from the observer, one would therefore probably have to resort on volume limited samples.\\
The filamentary structure at $\nsig{5}$ significance level is also shown over larger scales on figure \ref{fig_SDSS_pic} and within a $60{\,h^{-1}\,{\rm Mpc}}$ slice where each galaxy is represented by a point on figure \ref{fig_SDSS_slice}. Three voids ({\it i.e. } ascending $3$-\hyperref[defmanifold]{manifolds}) have been randomly selected within the distribution of figure \ref{fig_SDSS_pic} and are displayed on the bottom frame \ref{fig_SDSS_picD}, showing the intricate relationship between the voids and the filamentary structure that crawls at their surface. As previously observed in simulations, it can be seen on the central right frame of figure \ref{fig_SDSS_slice} that those 3D filaments also trace the 2D filamentary structure at the surface of the voids as expected from Morse theory. Note that it is only because they have been smoothed over four segments to look more appealing and to avoid rendering problems that the filaments do not lie precisely on the surface of the voids. It is in fact a build-in feature of the \hyperref[defDMC]{DMC} and in particular of our implementation that all the different types of identified cosmological structures do form a coherent picture, whatever the properties of the initial discrete sample. This allows for interesting features, such as making possible the count of the number of filaments that belong to a common maxima by intersecting the ascending $1$-\hyperref[defmanifold]{manifolds} with the descending $3$-\hyperref[defmanifold]{manifolds}. This is shown on figure \ref{fig_SDSS_slice} where the color of the filamentary structure corresponds to the index of the maximum it belongs to and individual filaments could be identified the same way, as the two \hyperref[defarc]{arcs} of the \hyperref[defDMC]{DMC} originating from a given saddle point.
\begin{figure*}
\begin{minipage}[c][]{0.49\linewidth}
\centering\includegraphics[width=\linewidth]{img/observation}
\end{minipage}
\begin{minipage}[c][]{0.49\linewidth}
\centering\includegraphics[width=\linewidth]{img/observation_3D}
\end{minipage}
\caption{Left: An X-ray halo observed around an elliptical galaxy in the center of a group at redshift $z=0.083$ and located at the confluence of several filaments. The color map indicates the X-ray combined image of CCD chips (XIS 0,1, and 3), while the white dots stand for the SDSS spectroscopic-identified galaxies within $0.080<z<0.086$. The filamentary structure in the surrounding region is shown by the colored solid curves, extracted from the filaments catalogue shown on figure \ref{fig_SDSS_picB}. Note that the colors (cyan, red and yellow) correspond to that of the filaments represented on the 3D view on the right frame. Right: a 3D view of the configuration of the filaments around the observed region. The vertical axis corresponds to the line of sight (the observer being upward), and the box roughly encompasses the galaxies in the SDSS catalogue with coordinates $233^\circ<{\rm DEC}<243^\circ$, $22^\circ<{\rm RA}<32^\circ$ and $0.075<z<0.092$. The delaunay tesselation of the galaxies, shaded according to the local density, is displayed to help visualizing the filamentary structure. The observational target is identified by a red square and is located at the intersection of the red cyan and yellow filaments, the last two being aligned with the line of sight to a very good approximation. A movie is available for download at {\em\tt http://www.iap.fr/users/sousbie/}. \label{fig_observation} }
\end{figure*}
\subsection{An ``optically faint'' cluster at a filamentary junction}
Because some dark matter haloes are sparsely populated and also as a result of selection effects, classical methods such as FOF are unable to detect them from the observed galaxy distribution. Such ``optically faint'' groups and clusters may nevertheless present a strong astrophysical interest: as they are by nature different from the ``regular'' haloes, one could for instance expect that they have different formation history that needs to be understood. As they are faint though, their properties are poorly assessed, but massive dark matter haloes such as galaxy clusters or galaxy groups are believed to form at intersections of two or several filaments, which can be identified in the SDSS using DisPerSE\,. We demonstrate that this is possible by enlightening the relationship between an X-ray halo and its surrounding filamentary network as identified in the SDSS catalogue (see figure \ref{fig_SDSS_picB}).\\
Because of the particular configuration of the filaments in the region, we submitted an observation proposal to the X-ray satellite SUZAKU \citep{2007PASJ...59S...1M}, which was accepted. We present here the results of this observation, but reserve its analysis to a future article (Kawahara {\it et al} 2010, {\it in prep.}). The observational target was selected for being located at the confluent of galaxy filaments, and because one of those filaments is both straight and aligned with the line of sight as shown on figure \ref{fig_observation} (see the yellow filament on the right frame). While no X-ray signal could be found within the ROSAT All Sky Survey (RASS), X-ray signals emitted by diffuse thermal gas were clearly observed by the high sensitivity detectors of SUZAKU, unveiling the presence of a dark matter halo as shown by the X-ray image reproduced in the central part of the left panel of figure \ref{fig_observation}. It is remarkable that there are no corresponding candidates in the $78,800$ groups catalogue identified by \citep{2010A&A...514A.102T} using a modified friend-of-friend (FOF) algorithm. In fact, because the optically observable member galaxies are not strongly clustered and their number is limited ($N \sim 10$), regular methods have high chances to miss them. It is also very difficult to locate and identify particular filamentary configurations by eye directly from the galaxy distribution using projections or even a real time 3D visualization. Using DisPerSE\,, we showed that it is possible to easily identify such targets, which demonstrate the complimentary of our approach with respect to one based on a traditional halo finders.
\section{Significance of topology of LSS}
\label{subsec_sigthres}
As noted in paper I, it is not an option to use the raw \hyperref[defDMC]{Discrete Morse-Smale complex} as a tool to assess the properties of the cosmic web. Hence we showed there how to simulate a topological simplification of the DTFE density field so that the critical simplexes that were most probably accidentally generated by Poisson noise could practically be removed from the \hyperref[defDMC]{DMC}. This simplification is based on the \hyperref[defpers]{persistence} ratio of critical points pairs ({\it i.e. } \hyperref[defpers]{persistence} pairs), and one must therefore decide a significance level $s=\nsig{n}$ such that all \hyperref[defpers]{persistence} pairs with lower significance ({\it i.e. } or equivalently a higher probability to be generated by Poisson noise) can be removed. We showed in paper I that, at least in the 2D case, such a method allows for what seems to be a very efficient and natural simplification of the \hyperref[defDMC]{DMC}. We did not discuss however how to decide the value of this particular threshold. This is particularly important though, and especially in the context of the cosmic web, as our ultimate goal is to assess physical properties of astrophysical objects identified as features of the \hyperref[defDMC]{DMC} ({\it i.e. } the haloes, filaments, walls and voids of matter distribution on cosmological scales in the Universe). Imagine for instance one is interested in statistically measuring the average number of filaments that branch on dark matter halos. If the threshold is too low, the measure will be equivalent to that in a Gaussian random field because of Poisson noise (see lower left frame of figure 13 of paper I), and if it is too high, then the risk is to systematically ignore weaker filaments (see central right panel of figure 13 of paper I).\\
\begin{figure}
\centering\includegraphics[width=\linewidth]{img/persistence_diag_comp}
\caption{\hyperref[defpers]{persistence} diagrams ({\it i.e. } the probability distribution function (PDF) of \hyperref[defpers]{persistence} pairs) in a cosmological simulation and for Gaussian random noise. Each pair $P_i=\ppair{p_i}{q_{i+1}}$ of critical points of order $i$ and $i+1$ is considered as a point with coordinates $[\rho\left(p_i\right),\rho\left(q_{i+1}\right)]/\rho_0$. The PDF were computed from a $250{\,h^{-1}\,{\rm Mpc}}$ large $\Lambda$CDM dark matter simulation down sampled to $128^3$ particles, $S^{128}$ (left column), the same distribution with $128^3$ additional randomly located particles, $S_N^{2\times 128}$ (central column), and a random distribution of particles within the same volume, $S_R^{128}$ (right column). From top to bottom, each line correspond to a different type of pair: $P_0$ (minima/$2$-saddle points), $P_1$ ($2$-saddle points/$1$-saddle points) and $P_2$ ($1$-saddle points/maxima) respectively. The green, purple dashed and pink dashed lines correspond to $\nsig{0}$, $\nsig{3}$ and $\nsig{4}$ \hyperref[defpers]{persistence} levels respectively.\label{fig_per_diag}}
\end{figure}
\begin{figure}
\centering\includegraphics[width=\linewidth]{img/pair_count_simu_comp}
\caption{Number of \hyperref[defpers]{persistence} pairs of type $k$ as a function of the significance threshold $S_k\left(r\right)$ (in units of $\sigma$) in a $250{\,h^{-1}\,{\rm Mpc}}$ large $\Lambda$CDM dark matter simulation down sampled to $128^3$ particles, $S^{128}$ (filled curves), the same distribution with $128^3$ additional randomly located particles, $S_N^{2\times 128}$ (dash-dotted curve) and a random distribution of particles within the same volume, $S_R^{128}$(dotted curves). The blue, green and red color correspond to \hyperref[defpers]{persistence} pairs of type $0$, $1$ and $2$ respectively (see figure \ref{fig_per_diag} for the corresponding \hyperref[defpers]{persistence} diagrams).\label{fig_pair_count_simu}}
\end{figure}
\subsection{Persistence diagrams}
Figure \ref{fig_per_diag} shows the probability distribution function of \hyperref[defpers]{persistence} diagrams (see \citet{edel00}, \citet{cohen07}) computed from the Delaunay tesselation of a $250{\,h^{-1}\,{\rm Mpc}}$ large, $512^3$ particles $\Lambda$CDM dark matter simulation subsampled to $128^3$ particles (left column, $S^{128}$ hereafter), the same distribution with an identical number of particle added at random locations (central column, $S_N^{2\times 128}$ hereafter), and a completely random distribution of particles within the same volume (right column, $S_R^{128}$ hereafter). Simply speaking, plotting a \hyperref[defpers]{persistence} diagram of a density distribution $\rho$ basically consists in representing each \hyperref[defpers]{persistence} pair $P_i=\ppair{p_i}{q_{i+1}}$, where $p_i$ and $q_{i+1}$ are critical points of order $i$ and $i+1$ respectively, as a point of coordinates $[{\rho_\downarrow},{\rho_\uparrow}]=[\rho\left(p_i\right),\rho\left(q_{i+1}\right)]/\rho_0$ where $\rho_0$ designates the average density in the distribution\footnote{In the following, the term density will generally refer to the normalized density $\rho/\rho_0$ so that different distributions can be fairly compared}. Recall that \hyperref[defpers]{persistence} pairs are pairs of critical simplexes that correspond to the act of creation and destruction of a topological feature in the \hyperref[deffiltration]{Filtration} of the Delaunay tesselation.
On figure \ref{fig_per_diag}, the pairs of type $P_0$, $P_1$ and $P_2$ are represented on the top, central and bottom rows respectively. On those diagrams, the pairs with null \hyperref[defpers]{persistence} lie on the green line of equation ${\rho_\uparrow}={\rho_\downarrow}$ and the farther away from this line a point is, the higher the \hyperref[defpers]{persistence} of its corresponding \hyperref[defpers]{persistence} pair. The purple and pink dashed line stand for $\nsig{3}$ and $\nsig{4}$ \hyperref[defpers]{persistence} respectively. As expected, most \hyperref[defpers]{persistence} pairs in the random distribution $S_R^{128}$ have a \hyperref[defpers]{persistence} ratio below $\nsig{3}$ (right column). Fortunately, the PDF of the \hyperref[defpers]{persistence} pairs in $S^{128}$ is sufficiently different from that in $S_R^{128}$ so that a reasonable fraction of them lie above the $\nsig{3}$ and even $\nsig{4}$ threshold (see left column). By canceling all those pairs that lie below the $\nsig{3}$ or $\nsig{4}$ line, it should therefore seem reasonable to assume that only those topological properties that were imprinted by the physical processes at work in the simulation would be conserved. A good measure of the actual influence of Poisson noise on the distribution of the \hyperref[defpers]{persistence} pairs in the underlying distribution can be gained from the examination of the central column. The distribution $S_N^{2\times 128}$ was created by adding a large number of randomly located particles to $S^{128}$, resulting also in the creation of a very large number of spurious critical points. One can see on the central column that as a result, the \hyperref[defpers]{persistence} diagram tends to concentrate at lower \hyperref[defpers]{persistence} ratio ({\it i.e. } closer from the green line). This means that as expected, those spurious critical points mainly create low \hyperref[defpers]{persistence} ratio pairs which can therefore be removed.\\
This observation is supported by figure \ref{fig_pair_count_simu}, where the actual number of \hyperref[defpers]{persistence} pairs in the three distributions are displayed as a function of the cutting threshold. Whereas the number of critical pairs of all sorts and with significance higher than $\nsig{0}$ is higher in $S_N^{2\times 128}$ (dash-dotted curves) than in $S^{128}$ (plain curves), this number decreases comparatively faster with the increase of the \hyperref[defpers]{persistence} selection threshold. For low \hyperref[defpers]{persistence} thresholds ({\it i.e. } up to $\sim \nsig{2}$), the number of \hyperref[defpers]{persistence} pairs in $S_N^{2\times 128}$ actually decreases as fast as that in the random distribution $S_R^{128}$ (dotted curves). In the case of pairs of type $P_1$ and $P_2$ ($2$-saddle points/$1$-saddle points pairs, green curves, and $1$-saddle point/maxima pairs, red curves, respectively), this tendency actually changes between $\nsig{2}$ and $\nsig{3}$ and the cancellation rates in $S_N^{2\times 128}$ and $S^{128}$ become relatively similar above $\nsig{3}$. This strongly suggests that most of the spurious \hyperref[defpers]{persistence} pairs in $S_N^{2\times 128}$ do in fact have a \hyperref[defpers]{persistence} ratio lower than $\nsig{3}$ and that above that threshold, the remaining \hyperref[defpers]{persistence} pairs have a distribution similar to that in the original N-body simulation $S^{128}$. The \hyperref[defpers]{persistence} pairs of type 0 in $S_N^{2\times 128}$ ( minima/$2$-saddle point pairs, blue filled curves) exhibit a slightly different behavior though, as their number seems to vary more or less accordingly with the \hyperref[defpers]{persistence} threshold in $S_N^{2\times 128}$ and $S_R^{128}$ (blue dotted curve). This number nevertheless always remain higher in $S_N^{2\times 128}$ and there are proportionally more high \hyperref[defpers]{persistence} pairs in $S_N^{2\times 128}$ than in $S_R^{128}$. This suggests that the number of minima resulting from the physical processes at stake in voids formation is relatively low compared to that due to Poisson noise, the reason for this being that the cosmological voids' minima have an intrinsically lower density because of the nature of voids. While Poisson noise creates spurious minima over a wide range of densities, the voids' minima only span the lower densities and therefore stretch over comparatively larger scales due to DTFE properties (resolution being inversely proportional to the density). The addition of random particles in $S_N^{2\times 128}$ particularly affects the wider regions around minima, therefore increasing their density and lowering the \hyperref[defpers]{persistence} ratio of the corresponding \hyperref[defpers]{persistence} pairs, hence the lack of high significance pairs of type $0$ at $S\left(r\right)>\nsig{5}$ (see blue curves) in $S_N^{2\times 128}$ compared to $S^{128}$. Note however that this does not mean that the physically created \hyperref[defpers]{persistence} pairs are destroyed by Poisson noise in $S_N^{2\times 128}$, but only that they are shifted to lower \hyperref[defpers]{persistence}, and that the \hyperref[defpers]{persistence} threshold should not be chosen too high if ones wants to retrieve the full \hyperref[defDMC]{DMC} (which is not the case if one is only interested in the filaments).\\
\begin{figure*}
\begin{centering}
\subfigure[Critical points PDF]{\includegraphics[width=0.49\linewidth]{img/simu_crit_count_comp}\label{fig_simu_crit}}\hfill
\subfigure[Betti numbers and Euler characteristic]{\includegraphics[width=0.49\linewidth]{img/simu_betti_comp}\label{fig_simu_betti}}
\end{centering}
\caption{Evolution of the topological properties in a $512^3$ particles $250 {\,h^{-1}\,{\rm Mpc}}$ dark matter simulation down-sampled to $128^3$ particles, $S^{128}$, for increasing \hyperref[defpers]{persistence} levels (left columns on each figure), and in the same distribution with $128^3$ additional randomly located particles, $S_N^{2\times 128}$ (right columns on each figure). On each frame, the \hyperref[defpers]{persistence} selection level ranges from $\nsig{0}$ for the outer colored curve up to $\nsig{6}$ for the inner curve. {\em Left:} The probability distribution function (PDF) of critical points of type $0$ (top) up to $3$ (bottom) as a function of their overdensity $\rho/\rho_0$. The black curve is the PDF of the vertice in the tessellation while the dashed curve stands for the (volume weighted) PDF of the overdensity $\rho/\rho_0$. The blue and red vertical dotted lines emphasize the critical level $r_v=\rho_v/\rho_0=0.2$ (resp. $r_p=\rho_p/\rho_0=125$) below (resp. above) which a void (resp. a peak) may be considered physically significant. {\em Right:} from top to bottom, the betti numbers, $\beta_2$, $\beta_1$, $\beta_0$, and Euler characteristic $\chi$ of the excursion set with over density greater than $\rho/\rho_0$. \label{fig_simu_curves}}
\end{figure*}
Two complementary measures of the evolution of the topological properties in $S^{128}$ and $S_N^{2\times 128}$ with the \hyperref[defpers]{persistence} threshold are presented on figure \ref{fig_simu_curves}: the PDF of the critical points on figure \ref{fig_simu_crit} and the betti numbers and Euler characteristics on figure \ref{fig_simu_betti}.
\subsection{Critical points}
Let us consider figure \ref{fig_simu_crit} first. On that figure, the PDF of the density at vertice ({\it i.e. } the particles in the studied distribution) is shown by the dark black bold curve, and it is striking how the PDF of the critical points tend to follow it, especially at low \hyperref[defpers]{persistence} (outer curves): the more the $k$-simplexes at a given density level, the higher the number of detected critical points of order $k$. This is an expected result when Poisson noise dominates as it affects indifferently any scales, but it is not desirable though as the filamentary structure of the cosmic web is an intrinsic property which should not depend on the properties of a particular sampling technique. One would in fact rather expect the PDF of the critical points to follow the PDF of the volume weighted density, or equivalently as we use DTFE, of the number of vertice at a given density in the tesselation\footnote{in the case of DTFE, the density of a sample particle is defined as the inverse volume of its dual Voronoi cell, and the volume it occupies is also the volume of this cell, which implies that the PDF of the volume weighted density and that of the number of sample particles are identical.}. The black bold dashed curve traces the {\em volume weighted} PDF of the density at vertice. It is clear on figure \ref{fig_simu_crit} that in the case of the minima, $1$-saddle points and $2$-saddle points PDFs, the bias toward higher better sampled densities due to DTFE is progressively wiped out with increasing \hyperref[defpers]{persistence} ratio threshold, and almost disappears above a significance level threshold of $\sim\nsig{3}$ (see blue, green and purple curves). The PDF of the maxima though (red curves) exhibits an opposite tendency, as their PDF concentrates at higher and higher densities with increasing \hyperref[defpers]{persistence} ratio thresholds. This actually reflects the nature of the distribution of the dark matter over large scales in the universe. In fact, most maxima are expected to be found within gravitationally bound structures undergoing non-linear regime ({\it i.e. } dark matter haloes), which therefore exhibit densities several order of magnitudes higher than the average density and with very steep gradients (note that this fact also prevents them from being affected by Poisson noise too much). Those regions, although numerous, represent only a very small fraction of the total volume, as reflected by the discrepancy between the PDF of the maxima at high \hyperref[defpers]{persistence} ratio and the volume weighted PDF of the density. To confirm these hypothesis, we traced on figures \ref{fig_simu_curves} and \ref{fig_simu_betti} the blue and red vertical dotted lines which mark the characteristic average under-density of a void in a Einstein-de Sitter model, $\rho/\rho_0\leq 0.2$ (see \citet{void_blumenthal92}, \citet{void_sheth04} or \citet{neyrinck}) and the typical critical overdensity above which gravitationally bound structure are identified using friend of friend algorithm, $\rho/\rho_0\geq 125$ \citep{FOFinfo} respectively. While this is not clear at low \hyperref[defpers]{persistence} thresholds because of Poisson noise, all maxima (resp. minima) belonging to \hyperref[defpers]{persistence} pairs with \hyperref[defpers]{persistence} ratio greater than $\sim\nsig{3}$ have densities above (resp. below) those critical thresholds while the two types of saddle points lie within those limits. This means that the detected persistent maxima and minima correspond to physically meaningful objects, which strongly supports the pertinence of using \hyperref[defpers]{persistence} based cancellation of a Morse-Smale complex to identify the characteristics components of the cosmic web such as cosmic voids and filaments.\\
\subsection{Discrete topological invariants}
The Betti numbers and Euler characteristics represented on figure \ref{fig_simu_betti} are slightly more involved topological analysis tools than the PDF of critical points (see paper I for a more formal definition of the Betti numbers and a simple example of their computation). The $k^{\rm th}$ Betti number $\beta_k$ counts the number of \hyperref[defkcycle]{$k$-cycles} in excursion sets as a function of the density threshold of the excursion. Within the context of the 3D cosmological matter distribution, there are $3$ Betti numbers, that count the number of holes or $2$-cycles ($\beta_2$), the number of tunnels or $1$-cycles ($\beta_1$) and the number of distinct components or $0$-cycles ($\beta_0$) enclosed in the set of points with density threshold larger than the aforementioned density threshold. As this threshold decreases, new \hyperref[defkcycle]{$k$-cycles} may be created or destroyed, therefore increasing or decreasing the value of the corresponding Betti numbers. The value of the Betti numbers as a function of the density threshold reflects the global topology of the field ({\it i.e. } the way it connects as function of density threshold) and it is therefore very instructive to compare the Betti numbers of two distributions to appreciate how similar or distinct they may be from a topological point of view. For that reason, we plotted on figure \ref{fig_simu_betti}, from top to bottom, the value of $\beta_2$, $\beta_1$, $\beta_0$ and the Euler characteristic $\chi$ (a topological invariant, computed as the alternate sum of the Betti numbers) as measured in $S^{128}$ and $S_N^{2\times 128}$ (left and right column respectively). As noted in Sousbie (2010), the notions of \hyperref[defpers]{persistence} pairs and Betti numbers are intimately related: the Betti numbers were readily computed from the \hyperref[defpers]{persistence} pairs, the positive critical point of order $k+1$ increasing $\beta_k$ when it enters the excursion and the negative critical point of order $k$ decreasing $\beta_k$. Distribution $S_N^{2\times 128}$ was obtained by adding an equal number of randomly distributed particles to the particles in the N-body simulations $S^{128}$, and the Betti numbers of the two distributions should therefore give some insight on how topology is affected by Poisson noise. Note that the presence of Poisson noise in $S_N^{2\times 128}$ affects the PDF of the sampled density by slightly downscaling it (numerous random particles land in large scale void regions, increasing their densities, while few of them affect the high density regions, therefore lower their density contrast, see black plain curves on figure \ref{fig_simu_curves}). When comparing Betti numbers in the two distributions, one would rather want to know weather the same structures ({\it i.e. } void, tunnel, component) exist in both distributions though, even if it exists at slightly different densities. It is therefore more important to compare the general shape and amplitude of the Betti number in both distributions than their value at a precise density threshold. Inspecting figure \ref{fig_simu_betti}, it is clear that random particles mainly affect the topological properties of the field around the average density $\rho_0$, each Betti number differing of about one order of magnitude in $S^{128}$ (left) and $S_N^{2\times 128}$ (right) at a level around $\rho/\rho_0=1$. The situation largely improves after the cancellation of the lower \hyperref[defpers]{persistence} pairs though and it is striking how the shape and amplitude of the Betti numbers at a level of \hyperref[defpers]{persistence} ratio of $3\sim\nsig{4}$ become similar. Note also that $\beta_0$ is the Betti number that is the least affected by Poisson noise, and for \hyperref[defpers]{persistence} higher than $\nsig{3}$, the values are almost identical in $S^{128}$ and $S_N^{2\times 128}$. This means that individual components in the \hyperref[deffiltration]{Filtration} are created and merge in a very similar way independently of the presence of Poisson noise, which does not affect the filamentary structure of $S^{128}$. It is therefore reasonnable to trust the filaments detected at \hyperref[defpers]{persistence} levels higher than $\sim\nsig{3}$ as being true topological properties of the underlying distribution. One should nonetheless remain cautious with the identification of voids and wall. In fact, although the topology of the $1$-cycles and $2$-cycles seems to be correctly recovered in $S_N^{2\times 128}$ at a significance level of $3\sim\nsig{4}$, this is not the case anymore at higher levels and one should be careful not to set the threshold too high. In fact, the cosmological voids and walls are more affected by Poisson noise as they usually live at densities around $\rho/\rho_0=1$ where the influence of Poisson noise is maximal and the corresponding \hyperref[defpers]{persistence} pairs have statistically lower \hyperref[defpers]{persistence} ratios than that associated to filaments.\\
\section{Conclusion}
\label{sec_conclusion}
We implemented DisPerSE\,\, (Soubie 2010) on realistic 3D dark matter cosmological simulations and
observed redshift catalogues from the SDSS DR7.
We showed that DisPerSE\,\, traces very well the observed filaments, walls, and voids seen both in simulations and observations.
In either setting, filaments are shown to connect onto halos, outskirt walls, which circumvent voids, as is topologically required by Morse theory.
Indeed, DisPerSE\,\, warrants that all the well-known and extensively studied mathematical properties of Morse theory are ensured by construction at the mesh level.
As illustrated in sections~\ref{sec_SDSS}, DisPerSE\,\, assumes nothing about the geometry of the survey or its homogeneity, and yields a natural (topologically motivated)
self-consistent criterion for selecting the significance level of the identified structures.
We demonstrated that the extraction is possible even for very sparsely sampled point processes, as a function of the \hyperref[defpers]{persistence} ratio
(a measure of the significance of topological connections between critical points), which allows us to account consistently for the shot noise of real surveys.
The corresponding recovered cosmic web is also ``persistent" in as much as it is robust because it relies on intrinsic topological features of the underlying density field.
Hence we can now trace precisely the locus of filaments, walls and voids from such samples and assess the confidence of the post-processed sets as a function of this threshold,
which can be expressed relative to the expected amplitude of shot noise.
DisPerSE\,\, also seems to be robust, in as much that more sparsely samples recover filamentary structures which are consistent with those of the better sampled catalogues.
In a cosmic framework, this criterion was shown to level with FoF structure finder for the identifications of peaks, while DisPerSE\,\, also identifies the connected filaments and quantitatively produces on the fly the full set of Beti numbers (number of holes, tunnels, connected components etc...) {\sl directly from the particles}, as a function of the \hyperref[defpers]{persistence} threshold, as these follow from the \hyperref[defpers]{persistence} pairs.
We investigated the evolution of the critical points, the Beti numbers and the Euler characteristic has a function of the \hyperref[defpers]{persistence} ratio: its illustrates the biases
involved in filtering low \hyperref[defpers]{persistence} ratios.
For dark matter simulations, this criterion was shown to be sufficient even if one particle out of two is noise, when the \hyperref[defpers]{persistence} ratio is set to 3-$\sigma$ or more.
We applied this procedure to the localization of a specific filamentary configuration and observed an ``optically faint'' cluster at a galaxy filaments junction, identified in the SDSS catalogue. An X-ray counterpart could indeed be observed (Kawahara et al. in prep) by the X-ray satellite SUZAKU. The filaments of the SDSS extracted with DisPerSE\,\, are available online at the URL {\em \tt http://www.iap.fr/users/sousbie/SDSS-skeleton.html} as a set of segments with extremities in RA, DEC, redshift. All these results are very encouraging for future investigations using DisPerSE\,, for searching galaxy clusters, galaxy groups, and missing baryons of the universe in particular, and for the study of LSS in general.
\subsection*{Acknowledgements}
{
\sl The authors thank T.~Nishimichi for his help dealing with SDSS galaxy catalogue, H.~Yoshitake for his help with X-ray analysis and TS thanks Y.~Suto for his constant help and support.
This work was made possible through an extensive usage of the Yorick programming language by D.~Munro (available at {\em\tt http://yorick.sourceforge.net/}) and also CGAL, the Computational Geometry Algorithms Library, ({\em\tt http://www.cgal.org}), to compute the Delaunay tessellations.
\sl The filaments of the SDSS extracted with DisPerSE\,\, is readily available online at the URL {\em \tt http://www.iap.fr/users/sousbie/SDSS-skeleton.html}.
\sl Funding for the SDSS has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is {\em\tt http://www.sdss.org/}.
\sl TS gratefully acknowledges support from JSPS (Japan Society for the Promotion of Science) Postdoctoral Fellowhip for Foreign Researchers award P08324.
HK is supported by a JSPS (Japan Society for Promotion of Science) Grant-in-Aid for science fellows.
CP acknowledges supports from a Leverhulme visiting professorship at the Astrophysics department of the University of Oxford.
This work is also supported by Grant-in-Aid for Scientific research from JSPS and from the Japanese Ministry of Education, Culture, Sports, Science and Technology (No. 22$\cdot$5467).
}
\bibliographystyle{mn2e}
|
{
"timestamp": "2010-09-22T02:01:16",
"yymm": "1009",
"arxiv_id": "1009.4014",
"language": "en",
"url": "https://arxiv.org/abs/1009.4014"
}
|
\section{Introduction}
\label{sec:intro}
Charged particle identification is critical for a heavy flavour physics experiment.
It is required both for separating signal decays from more copious backgrounds,
such as $B^0_s\rightarrow D^-_s K^+$ from $B^0_s\rightarrow D^-_s\pi^+$,
and for tagging the initial flavour of neutral mesons which may undergo mixing.
One way to accomplish this is with a Ring Imaging Cherenkov (RICH) system,
in which mass hypotheses are tested by measuring the angle ($\theta_C$) and number
of Cherenkov photons emitted by a charged particle passing through a radiator.
For a given detector geometry, the separation power between $K$ and $\pi$ hypotheses
depends on the refractive index $n$ of the radiator and the momentum $p$ of the
particle.
Between the pion and kaon thresholds, pions are positively identified
but kaons are identified only by the absence of photons.
In principle this can provide good separation, but
it is vulnerable to background from fake tracks (e.g. due to
mistakes in pattern recognition in the tracking system), which cannot
create Cherenkov photons and so would be identified as kaons. Likewise,
protons would not be distinguished from kaons. Therefore, it is desirable
to work above the kaon threshold---and in an environment with high
occupancy in the tracking system or a large number of protons, it is
essential.
At LHCb~\cite{bib:DetPaper}, particle identification is needed for the momentum
range from 2\,GeV$/c$ to 100\,GeV$/c$. This is currently accomplished by a
RICH system~\cite{bib:TDR:RICH} consisting of
two detectors with three radiators.
Two of these are gaseous: C$_4$F$_{10}$ and CF$_4$, with
refractive indices of 1.0014 and 1.0005, respectively, for visible light at STP.
These correspond to kaon Cherenkov thresholds of 9.3 and 15.6\,GeV$/c$, and
together provide better than $3\sigma$ kaon-pion separation up to 100\,GeV$/c$.
Positive kaon identification from 2--10\,GeV$/c$ requires a third radiator
with a kaon Cherenkov threshold around 2\,GeV$/c$, corresponding to
$n \approx 1.03$. This excludes conventional materials: the phase transition
from gas to liquid or solid is associated with a large jump in $(n-1)$
from $\mathcal{O}(10^{-3})$ or smaller to $\mathcal{O}(1)$.
The low density of aerogel gives it a suitable refractive index:
LHCb's third radiator is aerogel with $n=1.029$ in air and $n=1.037$
in C$_4$F$_{10}$. However, Rayleigh scattering limits the useful light
yield: approximately 7 detected photoelectrons are expected at LHCb
for a saturated track in aerogel. In a high-background scenario, such
as the proposed LHCb upgrade
from $2 \times \, 10^{32} \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1}$
to $2 \times \, 10^{33} \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1}$,
kaon-pion discrimination from aerogel would be severely
compromised~\cite{bib:young-min}.
Another method of charged particle identification is by time of
flight measurement. The principle is simple: if a particle of
mass $m$ is detected at position $x$ and time $t$ after leaving
the origin, then
\begin{equation}
t = \frac{x}{c} \sqrt{1 + \left(\frac{m}{p}\right)^2}
\approx \frac{x}{c} \left[ 1 + \frac{1}{2}\left( \frac{m}{p} \right)^2 \right]
\end{equation}
and so the difference in expected time between kaons and pions would be
\begin{equation}
t_K - t_\pi \approx \frac{x}{c} \frac{1}{2p^2} \left[ m_K^2 - m_{\pi}^2 \right]
.
\end{equation}
Therefore, to obtain $3\sigma$ separation between the two hypotheses
the time resolution $\sigma_t$ must satisfy
\begin{equation}
\sigma_t < \frac{1}{3} \frac{x}{c} \frac{1}{2p^2} \left[ m_K^2 - m_{\pi}^2 \right]
.
\end{equation}
If this approach were to be used for kaon identification in the range
2--10\,GeV$/c$ and the path length were $x=10$\,m, then a per-track time
resolution $\sigma_t < 12.5$\,ps would be required. A very fast
response would be needed---Cherenkov radiation would be suitable for this
purpose. If the expected light yield were 50 detected photoelectrons,
a per-photon time resolution under 90\,ps would be required.
A third method is to measure the time of propagation $\tau_{\gamma}$ over a path
length $d_{\gamma}$ of Cherenkov photons emitted by a charged particle.
In a dispersive medium, photons propagate at the group velocity $c/n_g$,
and so
\begin{equation}
\label{eq:ng}
n_g = c \, \frac{\tau_{\gamma}}{d_{\gamma}}
.
\end{equation}
The relationship between $n$ and $n_g$ is a non-linear function
that depends on the medium but obeys:
\begin{equation}
n_g = n - \lambda \frac{ \mathrm{d}n }{ \mathrm{d}\lambda}
.
\end{equation}
After computing $n$ from $n_g$, the mass of the particle can be extracted:
\begin{eqnarray}
\beta & = & \frac{1}{n \cos \theta_C} \\
\Rightarrow m & = & p \sqrt{ n^2 \cos^2 \theta_c - 1 }
.
\end{eqnarray}
The methods outlined above may be combined in an elegant way.
Consider a large, thin plane of an optically dense medium such as
quartz. Charged particles passing through the plane emit Cherenkov
photons, which propagate to the edges via total internal reflection
as in the BABAR DIRC~\cite{bib:babarNIM}.
The arrival time of a photon at the edge of the plane is the
sum of the time of flight of the track to the emission point, $t$,
and the time of propagation of the photon in the medium, $\tau_{\gamma}$. Then
\begin{equation}
\label{eq:timeSum}
t + \tau_{\gamma}
= \frac{x}{c} \sqrt{1 + \left(\frac{m}{p}\right)^2}
+ \frac{d_{\gamma} n_g}{c}
\end{equation}
where $n_g$ may be determined from
\begin{equation}
\label{eq:timeSum_n}
n = \frac{1}{\beta \cos \theta_C} = \frac{\sqrt{m^2 + p^2}}{p \cos \theta_C}
.
\end{equation}
Both $\mathrm{d}t/\mathrm{d}m$ and $\mathrm{d}\tau_{\gamma}/\mathrm{d}m$
have the same sign so the two effects combine constructively.
For a given track, the first term of Eq.~\ref{eq:timeSum} is fixed but the second
depends on the direction and path length of the photon,
which must be reconstructed for each photon individually---for the
planar geometry outlined above, this can be done by measuring two
independent angles for each photon.
This is the principle behind the design of the TORCH,
and of the Belle upgrade TOP~\cite{bib:belleTop99}
and PANDA endcap DIRC~\cite{bib:pandaDirc} which inspired it.
\section{TORCH design}
\label{sec:design}
The TORCH detector consists of a
plane of quartz covering the full LHCb angular acceptance and placed
12\,m downstream of the interaction point, plus focusing elements
and photodetectors along each edge. The quartz plate has dimensions
7440\,mm\,$\times$\,6120\,mm, with a hole of 26\,mm\,$\times$\,26\,mm
at the centre for the beampipe, as illustrated in Fig.~\ref{fig:layout}.
The focusing element uses a cylindrical mirror to focus the light
such that the distance along the photodetector plane is approximately
proportional to the angle $\theta_z$, as illustrated in
Fig.~\ref{fig:focus}. This angle is key: the overall resolution
is dominated by the pixelization uncertainty on $\theta_z$.
From simulation studies, we find that a 128-channel segmentation and
a dynamic range of 400\,mrad corresponds to an uncertainty of
0.96\,mrad on $\theta_z$ and per-photon uncertainty on the time of
propagation of $\mathcal{O}(70\,\mathrm{ps})$. The granularity
requirement in the other dimension is much looser, since the azimuthal
angle is typically measured with a lever arm of a few metres:
$\mathcal{O}(1\,\mathrm{cm})$ is sufficient match the 1\,mrad resolution.
\begin{figure}[htb]
\begin{center}
\epsfig{file=Layout.eps,width=0.9\columnwidth}
\end{center}
\vspace*{-2mm}
\caption{
Schematic layout of the TORCH detector.
For clarity, focusing elements have only been shown on the upper
and lower edges.
Photodetectors extend along the full length of each edge.
}
\label{fig:layout}
\end{figure}
\begin{figure}[htb]
\begin{center}
\epsfig{file=Focus.eps,width=0.57\columnwidth}~
\includegraphics*[width=0.41\columnwidth,viewport=325 0 505 240,clip]{Recon.eps}
\end{center}
\vspace*{-2mm}
\caption{
Cross section of the focusing element (left), and definition
of the angle $\theta_z$ (right). The focusing of photons is
shown for five illustrative angles between 450 and 850\,mrad,
emerging at different points across the edge of the plate.
}
\label{fig:focus}
\end{figure}
The key property required of the photodetector is high speed: its
intrinsic time spread should be significantly better than 90\,ps.
The photon detection technology which currently gives the best
time resolution is the use of micro-channel plate photomultipliers
(MCP-PMT). They have been used in R\&D for a related concept under
development by the Belle collaboration for their upgrade~\cite{bib:BelleTOP},
with single-photon time resolutions of 35\,ps achieved.
An example of a commercially-available detector of this type
which comes close to satisfying the requirements of TORCH is the
Photonis$^{\rm TM}$ XP85022 MCP-PMT. This is available in a
1024-channel version, with an array of $32\times 32$ pixels in
a $53\times 53\,$mm$^2$ active area, and overall dimension of 59\,mm.
For the sake of developing a concrete design for TORCH we have
assumed the use of this photodetector, laid side-by-side with a pitch of
60\,mm along the edges of the plane. The anode structure of an MCP-PMT
consists simply of conductive pads to read out the charge,
and the pixellization can therefore in principle be adjusted.
For the simulation of TORCH we have assumed an array of $8\times 128$ pixels
of dimension $6.6 \, \mathrm{mm} \times 0.4 \, \mathrm{mm}$.
\section{Reconstruction and pattern recognition}
Following the approach outlined in Sec.~\ref{sec:intro} for the
design from Sec.~\ref{sec:design}, the reconstruction is
relatively straightforward: we measure the arrival time and position
of a photon on the photodetector plane, infer $\theta_z$, and calculate
its trajectory assuming it was emitted by the track at the midpoint in $z$
of its path through the quartz block. Given measurements of the
track path and momentum from a tracking system and knowledge of the optical
properties of quartz, this is sufficient information to compute all quantities
in Eq.~\ref{eq:timeSum} and~\ref{eq:timeSum_n} and extract the mass
from $t + \tau_{\gamma}$. There are two major practical challenges, though:
first, association of photons to tracks (pattern recognition); and
second, determining the time when the track left the origin.
Pattern recognition is critical: in the environment of the upgraded LHCb
luminosity, there may be $\mathcal{O}(100)$ fully reconstructed tracks
plus a large number of secondaries passing through the TORCH in an event.
However, most photon-track pairs can be rejected as unphysical. We take
advantage of the limited range of photon wavelengths to which the
photodetector is sensitive and reject candidates outside this range.
This is mathematically equivalent to a restriction on $\theta_C$:
for our geometry, this means that relevant photons lie roughly along arcs
on the photodetector plane.
Additionally, we can ignore photon-track pairs with unphysical
timing. Instead of attempting to measure the mass of the particle directly,
we assume $e, \mu, \pi, K, p$ hypotheses and test for consistency
with each in turn. Background photons whose timing is not consistent with
any of these masses can be ignored.
In Sec.~\ref{sec:intro} it was assumed that the track left the origin
at $t=0$---or equivalently, that the relative timing between the
photodetector and the initial collision is known. However, we cannot
rely on an external clock for 10\,ps-level timing---and even if we could,
the luminous region at LHCb extends a few~cm in $z$, which would add a
smearing of order a few tens of ps. We must therefore measure the track
start time ourselves. One approach would be to install a second TORCH-like
detector close to the interaction point, at the price of additional cost,
material, and time resolution. A more efficient solution is to take
advantage of the high pion multiplicity at a hadron
collider and use other tracks from the same event primary vertex to fix the
relative timing. The same reconstruction procedure described above is
used except that the last step is inverted: for each track from the
event primary vertex, the pion mass is assumed as the input and the time
elapsed $(t + \tau_{\gamma})$ is calculated from Eq.~\ref{eq:timeSum}. Subtracting
this from the measured photon detection time gives a measurement of the
track start time. For a primary vertex with $N_{\pi}$ fully reconstructed pion
tracks, the start time resolution is smaller than the per-track time
resolution by roughly $\surd N_{\pi}$.
\section{Performance}
To test the particle identification performance, we take events from the
full LHCb Monte Carlo simulation, determine the set of charged particles that
would pass through the TORCH,
simulate the emission, propagation, and detection of Cherenkov radiation in
the TORCH for these particles with a stand-alone program, then apply the
reconstruction and pattern-recognition procedure outlined above to the output.
No truth information is used for the reconstruction and pattern-recognition.
However, a number of simplifying assumptions were made:
perfect measurements of track position and momentum,
perfect extrapolation of tracks through the magnetic field of the detector,
negligible electronics noise and cross-talk;
negligible spill-over between events; and
no multiple scattering, delta ray emission, or inelastic collisions in the TORCH.
Background from certain sources such as backscatter from the calorimeters was not
available to the simulation and was also neglected.
We characterize the performance in terms of particle identification efficiencies
for those kaon and pion tracks which would be useful for physics analysis.
Specifically, we require tracks to be well-measured (with track segments found
both upstream and downstream of the magnet), well-matched to an event primary
vertex, and associated to a kaon or pion in the Monte Carlo truth information.
Tracks are identified either as kaons or as pions according to which of the
two hypotheses is found to be most consistent by the pattern recognition.
The efficiency is shown as a function of track momentum in
Fig.~\ref{fig:effic-reqPV} for events simulated at a luminosity of
$L = 2 \times \, 10^{32} \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1}$.
The efficiency for correct identification is $> 95\%$ up to
10\,GeV$/c$, dropping at higher momentum as the time difference between
kaon and pion hypotheses diminishes.
\begin{figure}[htb]
\begin{center}
\includegraphics*[width=0.75\columnwidth,viewport=5 2 521 356,clip]{effic-kaon-reqPV.eps}
\includegraphics*[width=0.75\columnwidth,viewport=5 2 521 356,clip]{effic-pion-reqPV.eps}
\end{center}
\caption{
Identification efficiency of the subset of well-measured charged tracks
which are well-matched to a primary vertex.
The plots show the efficiency for a kaon (upper) or pion (lower) track to be
identified correctly (black) or incorrectly (red/dot), considering only the
kaon and pion hypotheses.
}
\label{fig:effic-reqPV}
\end{figure}
\section{Conclusions}
The TORCH concept is intended to provide charged particle identification in
the momentum range 2--10\,GeV$/c$ in a high-rate environment.
It has been shown that the pattern-recognition problem is solvable
and that target performance can be reached under simplified conditions.
The challenge is now to test whether this performance can be maintained
under realistic conditions and at
$L = 2 \times \, 10^{33} \, \mathrm{cm}^{-2} \, \mathrm{s}^{-1}$.
This will require prototype tests and more detailed simulation.
A revised, modular design with a smaller quartz plane is also
under investigation.
\section{Acknowledgements}
The TORCH concept has evolved from ideas that have been demonstrated in the
BABAR DIRC, and related developments that are under study by the PANDA and Belle
collaborations. It is a pleasure to thank Bjoern Seitz for fruitful discussion,
and our colleagues in the LHCb RICH group for their interest.
|
{
"timestamp": "2010-09-21T02:03:07",
"yymm": "1009",
"arxiv_id": "1009.3793",
"language": "en",
"url": "https://arxiv.org/abs/1009.3793"
}
|
\section{Main results}
\label{sec:intro}
Let $(X,\mathcal{X})$ be a measurable space, and let $\mathcal{F}$ be a
family of measurable functions on $(X,\mathcal{X})$. Given a probability
measure $\mu$ on $(X,\mathcal{X})$, the family $\mathcal{F}$ is
said to be a \emph{$\mu$-Glivenko-Cantelli class} (cf.\ \cite{Tal87}
or \cite[section 6.6]{Dud99}) if
$$
\sup_{f\in\mathcal{F}}\left|
\frac{1}{n}\sum_{k=1}^nf(X_k)-\mu(f)
\right|\xrightarrow{n\to\infty}0\quad\mbox{a.s.},
$$
where $(X_k)_{k\ge 1}$ is the i.i.d.\ sequence of $X$-valued random
variables with distribution $\mu$, defined on its canonical product
probability space.\footnote{
The supremum in the definition of the $\mu$-Glivenko-Cantelli
property need not be measurable in general when the class
$\mathcal{F}$ is uncountable. However, measurability will turn
out to hold in the setting of our main results as a consequence
of the proofs. See section \ref{sec:meas} below for
further discussion.
}
The class $\mathcal{F}$ is said to be a \emph{universal Glivenko-Cantelli
class} if it is $\mu$-Glivenko-Cantelli for every probability measure
$\mu$ on $(X,\mathcal{X})$. The goal of this paper is to characterize
the universal Glivenko-Cantelli property in the case that $\mathcal{F}$ is
separable and $(X,\mathcal{X})$ is a standard measurable space (these
regularity assumptions will be detailed below). Somewhat surprisingly, we
find that universal Glivenko-Cantelli classes are in fact uniformity
classes for convergence of (random) probability measures under the
assumptions of this paper, so that their applicability extends
substantially beyond the setting of laws of large numbers for i.i.d.\
sequences that is inherent in their definition.
The following probability-free independence properties for families
of functions will play a fundamental role in this paper.
These notions date back to Marczewski \cite{Mar48} (for sets)
and Rosenthal \cite{Ros74} (for functions, see also \cite{BFT78}).
\begin{defn}
\label{defn:marcz}
A family $\mathcal{F}$ of functions on a set $X$ is said to be
\emph{Boolean independent at levels $(\alpha,\beta)$} if for
every finite subfamily $\{f_1,\ldots,f_n\}\subseteq\mathcal{F}$
$$
\bigcap_{j\in F} \{f_j<\alpha\}\cap\bigcap_{j\not\in F}
\{f_j>\beta\}
\ne\varnothing\quad\mbox{for every }F\subseteq\{1,\ldots,n\}.
$$
A sequence $(f_i)_{i\in\mathbb{N}}$ is said to be \emph{Boolean
$\sigma$-independent at levels $(\alpha,\beta)$} if
$$
\bigcap_{j\in F} \{f_j<\alpha\}\cap\bigcap_{j\not\in F}
\{f_j>\beta\}
\ne\varnothing\quad\mbox{for every }F\subseteq\mathbb{N}.
$$
A family (sequence) of functions is called Boolean
($\sigma$-)independent if it is Boolean ($\sigma$-)independent at levels
$(\alpha,\beta)$ for some $\alpha<\beta$.
\end{defn}
We also recall the well-known notions of bracketing and covering numbers.
\begin{defn}
\label{defn:bracketing}
Let $\mathcal{F}$ be a class of functions on a measurable space
$(X,\mathcal{X})$. Given $\varepsilon>0$ and a probability measure
$\mu$ on $(X,\mathcal{X})$, a pair of measurable functions $f^+,f^-$
such that $f^-\le f^+$ pointwise and $\mu(f^+-f^-)\le\varepsilon$ defines
an \emph{$\varepsilon$-bracket in $L^1(\mu)$} $[f^-,f^+]:=
\{f:f^-\le f\le f^+\mbox{ pointwise}\}$. Denote by
$N_{[]}(\mathcal{F},\varepsilon,\mu)$ the cardinality of the smallest
collection of $\varepsilon$-brackets in $L^1(\mu)$ covering
$\mathcal{F}$, and by $N(\mathcal{F},\varepsilon,\mu)$ the cardinality
of the smallest covering of $\mathcal{F}$ by $\varepsilon$-balls in
$L^1(\mu)$.
\end{defn}
A measurable space $(X,\mathcal{X})$ is said to be \emph{standard} if it
is Borel-isomorphic to a Polish space. A class of functions $\mathcal{F}$
on a set $X$ will be said to be \emph{separable} if it contains a
countable dense subset for the topology of pointwise convergence in
$\mathbb{R}^X$.\footnote{
This notion of separability is not commonly considered in empirical
process theory. A sequential counterpart is more familiar: $\mathcal{F}$
is called pointwise measurable if it contains a countable subset
$\mathcal{F}_0$ such that every $f\in\mathcal{F}$ is the pointwise limit
of a sequence in $\mathcal{F}$ (cf.\ \cite[Example 2.3.4]{VW96}). In
general, separability is much weaker than pointwise measurability.
However, a deep result of Bourgain, Fremlin and Talagrand
\cite[Theorem 4D(viii)$\Rightarrow$(vi)]{BFT78} implies that a separable
uniformly bounded family of measurable functions on a standard space
is necessarily pointwise measurable if it contains no Boolean
$\sigma$-independent sequence. Thus universal Glivenko-Cantelli classes
satisfying the assumptions of Theorem \ref{thm:main} below are always
pointwise measurable, though this is far from obvious a priori. This fact
will not be needed in our proofs.
}
We can now formulate our main result.
\begin{thm}
\label{thm:main}
Let $\mathcal{F}$ be a separable uniformly bounded family of measurable
functions on a standard measurable space $(X,\mathcal{X})$. The
following are equivalent:
\begin{enumerate}
\item $\mathcal{F}$ is a universal Glivenko-Cantelli class.
\item $N_{[]}(\mathcal{F},\varepsilon,\mu)<\infty$ for every
$\varepsilon>0$ and every probability measure $\mu$.
\item $N(\mathcal{F},\varepsilon,\mu)<\infty$ for every
$\varepsilon>0$ and every probability measure $\mu$.
\item $\mathcal{F}$ contains no Boolean $\sigma$-independent sequence.
\end{enumerate}
\end{thm}
A notable aspect of this result is that the four equivalent conditions
of Theorem \ref{thm:main} are quite different in nature: roughly speaking,
the first condition is probabilistic, the second and third are geometric
and the fourth is combinatorial.
The implication $1\Rightarrow 2$ in Theorem \ref{thm:main} is the most
important result of this paper. A consequence of this implication is that
universal Glivenko-Cantelli classes can be characterized as uniformity
classes in a much more general setting.
\begin{cor}
\label{cor:uniformity}
Under the assumptions of Theorem \ref{thm:main}, the following are
equivalent to the equivalent conditions 1--4 of Theorem \ref{thm:main}:
\begin{enumerate}
\setcounter{enumi}{4}
\item For any probability measure $\mu$ on $(X,\mathcal{X})$ and net of
probability measures
$(\mu_\tau)_{\tau\in I}$ such that $\mu_\tau\to\mu$ setwise, we have
$\sup_{f\in\mathcal{F}}|\mu_\tau(f)-\mu(f)|\to 0$.
\item For any probability measure $\mu$ on $(X,\mathcal{X})$ and sequence
of random probability measures (kernels) $(\mu_n)_{n\in\mathbb{N}}$ such
that $\mu_n(A)\to\mu(A)$ a.s.\ for every $A\in\mathcal{X}$,
we have $\sup_{f\in\mathcal{F}}|\mu_n(f)-\mu(f)|\to 0$ a.s.
\item For any countably generated reverse filtration
$(\mathcal{G}_{-n})_{n\in\mathbb{N}}$ and $X$-valued random variable $Z$,
$\sup_{f\in\mathcal{F}}
|\mathbf{P}_{\mathcal{G}_{-n}}(f(Z))-\mathbf{P}_{\mathcal{G}_{-\infty}}
(f(Z))|\to 0$ a.s.
\item For any strictly stationary sequence $(Z_n)_{n\in\mathbb{N}}$ of
$X$-valued random variables,
$\sup_{f\in\mathcal{F}}|\frac{1}{n}\sum_{k=1}^nf(Z_k)-
\mathbf{P}_{\mathcal{I}}(f(Z_0))|\to 0$ a.s.\ ($\mathcal{I}$ is the
invariant $\sigma$-field).
\end{enumerate}
Here $\mathbf{P}_{\mathcal{G}}$ denotes any version
of the regular conditional probability
$\mathbf{P}[\,\cdot\,|\mathcal{G}]$.
\end{cor}
The characterization provided by Theorem \ref{thm:main} and Corollary
\ref{cor:uniformity} is proved under three regularity assumptions: that
$\mathcal{F}$ is uniformly bounded and separable, and that
$(X,\mathcal{X})$ is standard. It is not difficult to show that any
universal Glivenko-Cantelli class is uniformly bounded up to additive
constants (see, for example, \cite[Proposition 4]{DGZ91}), so that the
assumption that $\mathcal{F}$ is uniformly bounded is not a restriction.
We will presently argue, however, that without the remaining two
assumptions a characterization along the lines of this paper cannot be
expected to hold in general.
In the case that $\mathcal{F}$ is not separable, there are easy
counterexamples to Theorem \ref{thm:main}. For example, consider the class
$\mathcal{F}$ consisting of all indicator functions of finite subsets of
$X$. It is clear that this class is not $\mu$-Glivenko-Cantelli for any
nonatomic measure $\mu$, yet condition 3 of Theorem \ref{thm:main} holds.
Conversely, \cite[section 1.2]{AN10} gives a simple example of a universal
Glivenko-Cantelli class (in fact, a Vapnik-Chervonenkis class that is
image admissible Suslin, cf.\ \cite[Corollary 6.1.10]{Dud99}) for which
condition 8 of Corollary \ref{cor:uniformity}, and therefore condition 2
of Theorem \ref{thm:main}, are violated.
In the case that $(X,\mathcal{X})$ is not standard, an easy counterexample
to Theorem \ref{thm:main} is obtained by choosing $X=[0,1]$ and
$\mathcal{X}=2^X$. Assuming the continuum hypothesis, nonatomic
probability measures on $(X,\mathcal{X})$ do not exist \cite[Theorem
C.1]{Dud02}, so that any uniformly bounded family of functions is
trivially universal Glivenko-Cantelli. But we can clearly choose a
uniformly bounded Boolean $\sigma$-independent sequence $\mathcal{F}$ of
functions on $X$, in contradiction to Theorem \ref{thm:main}. This
example is arguably pathological, but various examples given by Dudley,
Gin{\'e} and Zinn \cite{DGZ91} show that such phenomena can appear even in
Polish spaces if we admit universally measurable functions. Therefore,
in the absence of some regularity assumption on $(X,\mathcal{X})$, the
universal Glivenko-Cantelli property can be surprisingly broad. In
Appendix \ref{sec:counter}, we show that it is consistent with the usual
axioms of set theory that the implications in Theorem \ref{thm:main} whose
proof relies on the assumption that $(X,\mathcal{X})$ is standard may fail
in a general measurable space. I do not know whether it is possible to
obtain examples of this type that do not depend on additional
set-theoretic axioms.
For the case where $(X,\mathcal{X})$ is a general measurable space we will
prove the following quantitative result, which is of independent interest.
\begin{defn}
\label{defn:shatter}
Let $\gamma>0$. A family $\mathcal{F}$ of functions on a set $X$ is said
to \emph{$\gamma$-shatter} a subset $X_0\subseteq X$ if there exist
levels $\alpha<\beta$ with $\beta-\alpha\ge\gamma$ such that, for every
finite subset $\{x_1,\ldots,x_n\}\subseteq X_0$, the following
holds:
$$
\forall\,F\subseteq\{1,\ldots,n\},~~
\exists\,f\in\mathcal{F}~~
\mbox{so that}~~
f(x_j)<\alpha\mbox{ for }j\in F,~~
f(x_j)>\beta\mbox{ for }j\not\in F.
$$
The \emph{$\gamma$-dimension} of $\mathcal{F}$ is the maximal
cardinality of $\gamma$-shattered finite subsets of $X$.
\end{defn}
\begin{thm}
\label{thm:scales}
Let $\mathcal{F}$ be a separable uniformly bounded family of
measurable functions on a measurable space $(X,\mathcal{X})$,
and let $\gamma>0$. Consider:
\begin{enumerate}
\renewcommand{\theenumi}{\alph{enumi}}
\item $\mathcal{F}$ has finite $\gamma$-dimension.
\item No sequence in $\mathcal{F}$ is Boolean independent
at levels $(\alpha,\beta)$ with $\beta-\alpha\ge\gamma$.
\item $N_{[]}(\mathcal{F},\varepsilon,\mu)<\infty$ for every
$\varepsilon>\gamma$ and every probability measure $\mu$.
\end{enumerate}
Then the implications $a\Rightarrow b\Rightarrow c$ hold.
\end{thm}
The notion of $\gamma$-dimension appears in Alon et al.\ \cite{ABCH97}
(called $V_{\gamma/2}$-dimension there). The implication $a\Rightarrow c$
of Theorem \ref{thm:scales} contains the recent results of Adams and Nobel
\cite{AN10a,AN10b,AN10}. Let us note that condition $b$ is strictly
weaker than condition $a$: for example, the class
$\mathcal{F}=\{\mathbf{1}_C:C\mbox{ is a finite subset of }\mathbb{N}\}$
has infinite $\gamma$-dimension for $\gamma<1$, but does not contain a
Boolean independent sequence. Similarly, condition $c$ is strictly weaker
than condition $b$: if
$X=\{x\in\{0,1\}^{\mathbb{N}}:\lim_{n\to\infty}x_n=0\}$ and
$\mathcal{F}=\{\mathbf{1}_{\{x\in X:x_j=1\}}:j\in\mathbb{N}\}$, then
$\mathcal{F}$ contains a Boolean independent sequence, but all the
bracketing numbers are finite as $X$ is countable (note that
$\mathcal{F}$ does not contain a Boolean $\sigma$-independent
sequence, so there is no contradiction with Theorem \ref{thm:main}).
Condition $b$ is dual
(in the sense of Assouad \cite{Ass83}) to the nonexistence of a
$\gamma$-shattered sequence in $X$. A connection between the latter and
the universal Glivenko-Cantelli property for families of indicators is
considered by Dudley, Gin{\'e} and Zinn \cite{DGZ91}.
An interesting question arising from Theorem \ref{thm:scales} is as
follows. If $\mathcal{F}$ is uniformly bounded and has finite
$\gamma$-dimension for all $\gamma>0$, then
$\sup_{\mu}N(\mathcal{F},\gamma,\mu)<\infty$ for all $\gamma>0$, that is,
the covering numbers of $\mathcal{F}$ are bounded uniformly with respect
to the underlying probability measure (see \cite{MV03} for a quantitative
statement). If $\mathcal{F}$ is a family of indicators, we have in
fact the polynomial bound
$\sup_{\mu}N(\mathcal{F},\varepsilon,\mu)\lesssim \varepsilon^{-d}$
\cite[Theorem 4.6.1]{Dud99}. In view of Theorem \ref{thm:scales}, one
might ask whether one can similarly obtain uniform or quantitative bounds
on the bracketing numbers of $\mathcal{F}$. Unfortunately, this is not
the case: $N_{[]}(\mathcal{F},\varepsilon,\mu)$ can blow up arbitrarily
quickly as $\varepsilon\downarrow 0$.
The following result is based on a combinatorial
construction of Alon, Haussler, and Welzl \cite{AHW87}.
\begin{prop}
\label{prop:vc}
There exists a countable class $\mathcal{C}$ of subsets of $\mathbb{N}$,
whose Vapnik-Chervonenkis dimension is two (that is, the
$\gamma$-dimension of $\{\mathbf{1}_C:C\in\mathcal{C}\}$ is two for all
$0<\gamma<1$) such that the following holds:
for any function $n(\varepsilon)\uparrow\infty$ as
$\varepsilon\downarrow 0$, there is a probability measure $\mu$ on
$\mathbb{N}$ such that $N_{[]}(\mathcal{C},\varepsilon,\mu)\ge
n(\varepsilon)$ for all $0<\varepsilon<1/3$.
In particular, $\sup_{\mu}N_{[]}(\mathcal{C},\varepsilon,\mu)=\infty$ for
all $0<\varepsilon<1/3$.
\end{prop}
Probabilistically, this result has the following consequence. In contrast
to the universal Glivenko-Cantelli property, it is known that both the
uniform Glivenko-Cantelli property and the universal Donsker property are
equivalent to finiteness of the Vapnik-Chervonenkis dimension for image
admissible Suslin classes of sets (see \cite{Dud99}, p.\ 225 and p.\ 215,
respectively). These results are proved using symmetrization arguments. In
view of Theorem \ref{thm:scales}, one might expect that it is possible to
provide an alternative proof of these results for separable classes using
bracketing methods (as in \cite[Chapter 7]{Dud99}). However, this would
require either uniform or quantitative control of the bracketing numbers,
both of which are ruled out by Proposition \ref{prop:vc}.
The original motivation of the author was an attempt to characterize
uniformity classes for reverse martingales that appear in filtering
theory. In a recent paper, Adams and Nobel \cite{AN10} showed that
Vapnik-Chervonenkis classes of sets are uniformity classes for the
convergence of empirical measures of stationary ergodic sequences; their
proof could be extended to more general random measures. A simplified
argument, which makes the connection with bracketing, appeared
subsequently in \cite{AN10b}. While attempting to understand the results
of \cite{AN10}, the author realized that the techniques used in the proof
are closely related to a set of techniques developed by Bourgain, Fremlin
and Talagrand \cite{BFT78,Tal84} to study pointwise compact sets of
measurable functions. The proof of Theorem \ref{thm:main} is based on
this elegant theory, which does not appear to be well known in the
probability literature (however, the proofs of our main results, Theorem
\ref{thm:main}, Corollary \ref{cor:uniformity}, and Theorem
\ref{thm:scales}, are intended to be essentially self-contained).
A key innovation in this paper is the construction in section
\ref{sec:scales} of a ``weakly dense'' set which allows to prove the
implication $4\Rightarrow 2$ in Theorem \ref{thm:main} (and $b\Rightarrow
c$ in Theorem \ref{thm:scales}). This result is the essential step that
closes the circle of implications in Theorem \ref{thm:main} and Corollary
\ref{cor:uniformity}. Many of the remaining implications are essentially
known, albeit in more restrictive settings and/or using significantly more
complicated proofs: these results are unified here in what appears to be
(in view the simplicity of the proofs and the counterexamples above and in
Appendix \ref{sec:counter}) their natural setting. In a topological
setting (continuous functions on a compact space), the equivalence of
$1,3,4$ in Theorem \ref{thm:main} can be deduced by combining
\cite[Theorem 14-1-7]{Tal84} with Talagrand's characterization of the
$\mu$-Glivenko-Cantelli property \cite[Theorem 11-1-1]{Tal84},
\cite{Tal87} (note that in this setting the distinction between Boolean
independent and $\sigma$-independent sequences is irrelevant). The
equivalence between $3,4$ in Theorem \ref{thm:main} is also obtained in
\cite[Theorem 4D]{BFT78} by a much more complicated method. The
implication $5\Rightarrow 2$ follows from the characterization of
uniformity classes for setwise convergence of Stute \cite{Stu76} and
Tops{\o}e \cite{Top77}. The implications $2\Rightarrow 1,5$--$8$ follow
from the classical Blum-DeHardt argument, up to measurability problems
that are resolved here. Finally, the implication $a\Rightarrow c$ (but
not $b\Rightarrow c$) of Theorem \ref{thm:scales} is shown in \cite{AN10b}
for the special case of Vapnik-Chervonenkis classes of sets.
The remainder of this paper is organized as follows. We first prove
Theorem \ref{thm:scales} in section \ref{sec:scales}. The proofs of
Theorem \ref{thm:main}, Corollary \ref{cor:uniformity}, and Proposition
\ref{prop:vc} are subsequently given in sections \ref{sec:main},
\ref{sec:uniformity}, and \ref{sec:vc}, respectively. Finally, Appendix
\ref{app:boole} and Appendix \ref{app:decomp} develop some properties of
Boolean $\sigma$-independent sequences and decomposition theorems that are
used in the proofs of our main results, while Appendix \ref{sec:counter}
is devoted to the aforementioned counterexamples to Theorem
\ref{thm:main} in nonstandard spaces.
\section{Proof of Theorem \ref{thm:scales}}
\label{sec:scales}
In this section, we fix a measurable space $(X,\mathcal{X})$ and a
separable uniformly bounded family of measurable functions $\mathcal{F}$.
Let $\mathcal{F}_0\subseteq\mathcal{F}$ be a countable family that is
dense in $\mathcal{F}$ in the pointwise convergence topology.
\begin{defn}
Denote by $\Pi(X,\mathcal{X})$ the collection of all finite measurable
partitions of $X$. For $\pi,\pi'\in\Pi(X,\mathcal{X})$, we write
$\pi\preceq\pi'$ if $\pi$ is finer than $\pi'$. For any pair of sets
$A,B\in\mathcal{X}$, finite partition $\pi\in\Pi(X,\mathcal{X})$, and
probability measure $\mu$ on $(X,\mathcal{X})$, define the $\mu$-essential
$\pi$-boundary of $(A,B)$ as
$$
\partial_\pi^\mu(A,B) = \bigcup\{P\in\pi:
\mu(P\cap A)>0\mbox{ and }\mu(P\cap B)>0\}.
$$
\end{defn}
We begin by proving an approximation result.
\begin{lem}
\label{lem:approx}
Let $\mu$ be a probability measure on $(X,\mathcal{X})$ and
let $\gamma>0$. If
$$
\inf_{\pi\in\Pi(X,\mathcal{X})}\sup_{f\in\mathcal{F}_0}
\mu\big(\partial_\pi^\mu(\{f<\alpha\},\{f>\beta\})\big)=0
\quad\mbox{for all}\quad\beta-\alpha\ge\gamma,
$$
then $N_{[]}(\mathcal{F},\varepsilon,\mu)<\infty$ for every
$\varepsilon>\gamma$.
\end{lem}
\begin{proof}
There is clearly no loss of generality in assuming that every
$f\in\mathcal{F}$ takes values in $\mbox{}[0,1]\mbox{}$
and that $\gamma<1$.
Fix $k\ge 1$, and let $\delta:=\gamma/k$. Choose
$\pi\in\Pi(X,\mathcal{X})$ so that
$$
\sup_{f\in\mathcal{F}_0}
\mu\left(\Xi(f)\right)
<\delta,\qquad
\Xi(f):=
\bigcup_{1\le j\le \lfloor\delta^{-1}\rfloor}
\partial_\pi^\mu(\{f<j\delta\},
\{f>j\delta+\gamma\}).
$$
For each $f\in\mathcal{F}_0$, define the functions $f^+$ and $f^-$ as
follows:
\begin{align*}
f^+ &= \delta\,\lceil\delta^{-1}\rceil\,
\mathbf{1}_{\Xi(f)} + \sum_{P\in\pi:P\not\subseteq\Xi(f)}
\delta\,\lceil
\delta^{-1}\esssup_{P}f\rceil\,\mathbf{1}_P,
\\
f^- &= \sum_{P\in\pi:P\not\subseteq\Xi(f)}
\delta\,\lfloor
\delta^{-1}\essinf_{P}f\rfloor\,\mathbf{1}_P.
\end{align*}
Here $\esssup_Pf$ ($\essinf_Pf$) denotes the essential supremum (infimum)
of $f$ on the set $P$ with respect to $\mu$. By construction,
$f^-\le f\le f^+$ outside a $\mu$-null set and $\mu(f^+-f^-) <
\gamma+3\delta$. Moreover, as $f^+,f^-$ are
constant on each $P\in\pi$ and take values in the finite set
$\{j\delta:0\le j\le \lceil\delta^{-1}\rceil\}$, there is only a finite
number of such functions. As $\mathcal{F}_0$ is countable, we can
eliminate the null set to obtain a finite number of
$(\gamma+3\delta)$-brackets in $L^1(\mu)$ covering $\mathcal{F}_0$. But
$\mathcal{F}_0$ is pointwise dense in $\mathcal{F}$, so
$N_{[]}(\mathcal{F},\gamma+3\delta,\mu)<\infty$, and we may choose
$\delta=\gamma/k$ arbitrarily small.
\end{proof}
To proceed, we need the notion of a ``weakly dense'' set, which is the
measure-theoretic counterpart of the corresponding topological notion
defined in \cite{BFT78}.
\begin{defn}
\label{defn:wkdens}
Given a measurable set $A\in\mathcal{X}$ and a probability measure
$\mu$ on $(X,\mathcal{X})$, the family of functions $\mathcal{F}$ is said
to be \emph{$\mu$-weakly dense over $A$ at levels $(\alpha,\beta)$} if
$\mu(A)>0$ and for any finite collection of measurable sets
$B_1,\ldots,B_p\in\mathcal{X}$ such that $\mu(A\cap B_i)>0$ for
all $1\le i\le p$, there exists $f\in\mathcal{F}$ such that
$\mu(A\cap B_i\cap\{f<\alpha\})>0$ and $\mu(A\cap B_i\cap\{f>\beta\})>0$
for all $1\le i\le p$.
\end{defn}
The key idea of this section, which lies at the heart of the results in
this paper, is that we can construct such a set if the bracketing numbers
fail to be finite. The proof is straightforward but requires some
elementary topological notions: the reader unfamiliar with nets is
referred to the classic text \cite{Kel55}, while weak compactness of the
unit ball in $L^2$ follows from Alaoglu's theorem \cite[Theorem
V.3.1]{Con85}.
\begin{prop}
\label{prop:wkdens}
Suppose there exists a probability measure $\mu$ on $(X,\mathcal{X})$
such that $N_{[]}(\mathcal{F},\varepsilon,\mu)=\infty$ for some
$\varepsilon>\gamma$. Then there exist $\alpha<\beta$ with
$\beta-\alpha\ge\gamma$ and a measurable set $A\in\mathcal{X}$ such that
$\mathcal{F}_0$ is $\mu$-weakly dense over $A$ at levels $(\alpha,\beta)$.
\end{prop}
\begin{proof}
By Lemma \ref{lem:approx}, there exist $\alpha<\beta$ with
$\beta-\alpha\ge\gamma$ such that
$$
\inf_{\pi\in\Pi(X,\mathcal{X})}\sup_{f\in\mathcal{F}_0}
\mu\big(\partial_\pi^\mu(\{f<\alpha\},\{f>\beta\})\big)>0.
$$
Choose for every $\pi\in\Pi(X,\mathcal{X})$ a function
$f_\pi\in\mathcal{F}_0$ such that
$$
\mu\big(\partial_\pi^\mu(\{f_\pi<\alpha\},\{f_\pi>\beta\})\big)
\ge \frac{1}{2}
\sup_{f\in\mathcal{F}_0}
\mu\big(\partial_\pi^\mu(\{f<\alpha\},\{f>\beta\})\big).
$$
Define $A_\pi:=\partial_\pi^\mu(\{f_\pi<\alpha\},\{f_\pi>\beta\})$. Then
$(\mathbf{1}_{A_\pi})_{\pi\in\Pi(X,\mathcal{X})}$ is a net of random
variables in the unit ball of $L^2(\mu)$. By weak compactness, there is
for some directed set $T$ a subnet
$(\mathbf{1}_{A_{\pi(\tau)}})_{\tau\in T}$ that
converges weakly in $L^2(\mu)$ to a random variable $H$. We claim that
$\mathcal{F}_0$ is $\mu$-weakly dense over $A:=\{H>0\}$ at levels
$(\alpha,\beta)$.
To prove the claim, let us first note that as $\inf_\pi\mu(A_\pi)>0$,
clearly $\mu(A)>0$. Now fix $B_1,\ldots,B_p\in\mathcal{X}$ such that
$\mu(A\cap B_i)>0$ for all $i$. This trivially implies
that $\mu(H\mathbf{1}_{A\cap B_i})>0$ for all $i$, so we can
choose $\tau_0\in T$ such that
$$
\mu(A_{\pi(\tau)}\cap A\cap B_i)>0\quad
\forall\,1\le i\le p,~\tau\preceq\tau_0.
$$
Let $\pi_0$ be the partition generated by $A,B_1,\ldots,B_p$, and choose
$\tau^*\in T$ such that $\tau^*\preceq\tau_0$ and
$\pi^*:=\pi(\tau^*)\preceq\pi_0$. As $A\cap B_i$ is a union of atoms of
$\pi^*$ by construction, $\mu(A_{\pi^*}\cap A\cap B_i)>0$ must imply
that $A\cap B_i$ contains an atom $P\in\pi^*$ such that
$\mu(P\cap\{f_{\pi^*}<\alpha\})>0$ and
$\mu(P\cap\{f_{\pi^*}>\beta\})>0$. Therefore
$$
\mu(A\cap B_i\cap\{f_{\pi^*}<\alpha\})>0\quad
\mbox{and}\quad
\mu(A\cap B_i\cap\{f_{\pi^*}>\beta\})>0
\quad\forall\,i.
$$
Thus $\mathcal{F}_0$ is $\mu$-weakly dense over $A$ at levels
$(\alpha,\beta)$ as claimed.
\end{proof}
We can now complete the proof of Theorem \ref{thm:scales}.
\begin{proof}[Theorem \ref{thm:scales}]
~
$a\Rightarrow b$: Lemma \ref{lem:assouad} in Appendix \ref{app:boole}
shows that if $\mathcal{F}$ contains a subset of cardinality $2^n$
that is Boolean independent at levels $(\alpha,\beta)$ with
$\beta-\alpha\ge\gamma$, then $\mathcal{F}$ $\gamma$-shatters a subset of
$X$ of cardinality $n$. Therefore, if condition $b$ fails, there exist
$\gamma$-shattered finite subsets of $X$ of arbitrarily large cardinality,
in contradiction with condition $a$.
$b\Rightarrow c$: Suppose that condition $c$ fails. By Proposition
\ref{prop:wkdens}, there exist a probability measure $\mu$, levels
$\alpha<\beta$ with $\beta-\alpha\ge\gamma$, and a set
$A\in\mathcal{X}$ so that $\mathcal{F}_0$ is $\mu$-weakly dense
over $A$ at levels $(\alpha,\beta)$. We now iteratively apply
Definition \ref{defn:wkdens} to construct a Boolean independent sequence.
Indeed, applying first the definition with
$p=1$ and $B_1=X$, we choose $f_1\in\mathcal{F}_0$ so that
$\mu(A\cap\{f_1<\alpha\})>0$ and $\mu(A\cap\{f_1>\beta\})>0$.
Then applying the definition with $p=2$ and $B_1=\{f_1<\alpha\}$,
$B_2=\{f_1>\beta\}$, we choose $f_2\in\mathcal{F}_0$ so that
$\mu(A\cap\{f_1<\alpha\}\cap\{f_2<\alpha\})>0$,
$\mu(A\cap\{f_1<\alpha\}\cap\{f_2>\beta\})>0$,
$\mu(A\cap\{f_1>\beta\}\cap\{f_2<\alpha\})>0$, and
$\mu(A\cap\{f_1>\beta\}\cap\{f_2>\beta\})>0$. Repeating this procedure
yields the desired sequence $(f_i)_{i\in\mathbb{N}}$.
\end{proof}
\section{Proof of Theorem \ref{thm:main}}
\label{sec:main}
Throughout this section, we fix a standard measurable space
$(X,\mathcal{X})$ and a separable uniformly bounded family of measurable
functions $\mathcal{F}$. We will prove Theorem \ref{thm:main} by
proving the implications $1\Rightarrow 4\Rightarrow 2\Rightarrow 1$
and $2\Rightarrow 3\Rightarrow 4$.
\subsection{$1\Rightarrow 4$}
Suppose there exists a sequence
$(f_i)_{i\in\mathbb{N}}\subseteq\mathcal{F}$ that is Boolean
$\sigma$-independent at levels $(\alpha,\beta)$ for some $\alpha<\beta$.
Clearly we must have
$$
\kappa_-<\alpha<\beta<\kappa_+,\qquad
\kappa_-:= \inf_{f\in\mathcal{F}}\inf_{x\in X}f(x),\quad
\kappa_+:= \sup_{f\in\mathcal{F}}\sup_{x\in X}f(x).
$$
Let $p=(\kappa_+-\beta+\varepsilon)/(\kappa_+-\alpha)$, where
we choose $\varepsilon>0$ such that $p<1$. Applying Theorem
\ref{thm:marczewski} in Appendix \ref{app:boole} to the sets
$A_i=\{f_i<\alpha\}$ and $B_i=\{f_i>\beta\}$, there exists a
probability measure $\mu$ on $(X,\mathcal{X})$ such that
$(\{f_i<\alpha\})_{i\in\mathbb{N}}$ is an i.i.d.\ sequence of
sets with $\mu(\{f_i<\alpha\})=\mu(X\backslash\{f_i>\beta\})=p$ for every
$i\in\mathbb{N}$.
We now claim that $\mathcal{F}$ is not $\mu$-Glivenko-Cantelli, which
yields the desired contradiction. To this end, note that we can trivially
estimate for any $f\in\mathcal{F}$
$$
\beta\,\mathbf{1}_{f>\beta}+\kappa_-\,\mathbf{1}_{f\le\beta}
\le
f \le \alpha\,\mathbf{1}_{f<\alpha}+\kappa_+\,\mathbf{1}_{f\ge\alpha}.
$$
We therefore have
\begin{align*}
\sup_{f\in\mathcal{F}}\left|
\frac{1}{n}\sum_{k=1}^nf(X_k)-\mu(f)
\right| &\mbox{}\ge
\sup_{j\in\mathbb{N}}
\frac{1}{n}\sum_{k=1}^n\{f_j(X_k)-\mu(f_j)\} \\
&\mbox{}\ge
(\kappa_--\beta)
\inf_{j\in\mathbb{N}}
\frac{1}{n}\sum_{k=1}^n
\mathbf{1}_{f_j\le\beta}(X_k)
+\varepsilon.
\end{align*}
But if $(X_k)_{k\ge 1}$ are i.i.d.\ with distribution $\mu$ then, by
construction, the family of random variables
$\{\mathbf{1}_{f_j\le\beta}(X_k):j,k\in\mathbb{N}\}$ is i.i.d.\ with
$\mathbf{P}[\mathbf{1}_{f_j\le\beta}(X_k)=0]>0$, so
$$
\inf_{j\in\mathbb{N}}
\frac{1}{n}\sum_{k=1}^n
\mathbf{1}_{f_j\le\beta}(X_k) = 0 \quad\mbox{a.s.}\quad
\mbox{for all }n\in\mathbb{N}.
$$
Thus $\mathcal{F}$ is not a $\mu$-Glivenko-Cantelli class. This
completes the proof.
\subsection{$4\Rightarrow 2$}
Suppose there exists a probability measure $\mu$ and $\varepsilon>0$ such
that $N_{[]}(\mathcal{F},\varepsilon,\mu)=\infty$. By Proposition
\ref{prop:wkdens}, there exist levels
$\alpha<\beta$ and a set $A\in\mathcal{X}$ such that
$\mathcal{F}$ is $\mu$-weakly dense over $A$ at levels $(\alpha,\beta)$.
We will presently construct a Boolean $\sigma$-independent sequence,
which yields the desired contradiction. The idea is to repeat the proof
of Theorem \ref{thm:scales}, but now exploiting the fact that
$(X,\mathcal{X})$ is standard to ensure that the
infinite intersections in the definition of Boolean $\sigma$-independence
are nonempty.
As $(X,\mathcal{X})$ is standard, we may assume without loss of
generality that $X$ is Polish and that $\mathcal{X}$ is the Borel
$\sigma$-field. Thus $\mu$ is inner regular. We now apply Definition
\ref{defn:wkdens} as follows. First, setting $p=1$ and $B_1=X$, choose
$f_1\in\mathcal{F}$ such that
$$
\mu(A\cap\{f_1<\alpha\})>0,\qquad\mu(A\cap\{f_1>\beta\})>0.
$$
As $\mu$ is inner regular, we may choose compact sets
$F_1\subseteq\{f_1<\alpha\}$ and $G_1\subseteq\{f_1>\beta\}$ such that
$\mu(A\cap F_1)>0$ and $\mu(A\cap F_2)>0$. Applying the definition with
$p=2$, $B_1=F_1$, and $B_2=G_1$, we can choose $f_2\in\mathcal{F}$ such
that
\begin{align*}
&\mu(A\cap F_1\cap\{f_2<\alpha\})>0,
&\mu(A\cap F_1\cap\{f_2>\beta\})>0, \\
&\mu(A\cap G_1\cap\{f_2<\alpha\})>0,
&\mu(A\cap G_1\cap\{f_2>\beta\})>0.
\end{align*}
Using again inner regularity, we can now choose compact sets
$F_2\subseteq\{f_2<\alpha\}$ and $G_2\subseteq\{f_2>\beta\}$ such that
$\mu(A\cap F_1\cap F_2)>0$, $\mu(A\cap F_1\cap G_2)>0$, $\mu(A\cap G_1\cap
F_2)>0$, and $\mu(A\cap G_1\cap G_2)>0$. Iterating the above steps, we
construct a sequence of functions
$(f_i)_{i\in\mathbb{N}}\subseteq\mathcal{F}$
and compact sets $(F_i)_{i\in\mathbb{N}}$, $(G_i)_{i\in\mathbb{N}}$ such
that $F_i\subseteq\{f_i<\alpha\}$, $G_i\subseteq\{f_i>\beta\}$ for
every $i\in\mathbb{N}$, and for any $n\in\mathbb{N}$
$$
\mu\left(\bigcap_{j\in Q}F_j\cap\bigcap_{j\in
\{1,\ldots,n\}\backslash Q}G_j
\right)>0\quad\mbox{for every }Q\subseteq\{1,\ldots,n\}.
$$
Now suppose that the sequence $(f_i)_{i\in\mathbb{N}}$ is not Boolean
$\sigma$-independent. Then
$$
\bigcap_{j\in R} \{f_j<\alpha\}\cap\bigcap_{j\not\in R}
\{f_j>\beta\} = \varnothing
$$
for some $R\subseteq\mathbb{N}$. Thus we certainly have
$$
\bigcap_{j\in R} F_j\cap\bigcap_{j\not\in R}
G_j = \varnothing.
$$
Choose arbitrary $\ell\in R$ (if $R$ is the empty set, replace $F_\ell$ by
$G_1$ throughout the following argument). Then clearly $\{X\backslash
F_j:j\in R\}\cup\{X\backslash G_j:j\not\in R\}$ is an open cover of
$F_\ell$. Therefore, there exist finite subsets $Q_1\subseteq R$,
$Q_2\subseteq\mathbb{N}\backslash R$ such that $\{X\backslash
F_j:j\in Q_1\}\cup\{X\backslash G_j:j\in Q_2\}$ covers $F_\ell$.
But then
$$
F_\ell\cap\bigcap_{j\in Q_1} F_j\cap\bigcap_{j\in Q_2}
G_j = \varnothing,
$$
a contradiction. Thus $(f_i)_{i\in\mathbb{N}}$ is Boolean
$\sigma$-independent at levels $(\alpha,\beta)$.
\subsection{$2\Rightarrow 1$}
\label{sec:blum}
This is the usual Blum-DeHardt argument, included here for
completeness. Fix a probability measure $\mu$ and $\varepsilon>0$,
and suppose that $N_{[]}(\mathcal{F},\varepsilon,\mu)<\infty$.
Choose $\varepsilon$-brackets $[f_1,g_1],\ldots,[f_N,g_N]$ in $L^1(\mu)$
covering $\mathcal{F}$. Then
\begin{multline*}
\sup_{f\in\mathcal{F}}|\mu_n(f)-\mu(f)| =
\sup_{f\in\mathcal{F}}\{\mu_n(f)-\mu(f)\} \vee
\sup_{f\in\mathcal{F}}\{\mu(f)-\mu_n(f)\} \\
\le
\max_{i=1,\ldots,N}\{\mu_n(g_i)-\mu(f_i)\} \vee
\max_{i=1,\ldots,N}\{\mu(g_i)-\mu_n(f_i)\} ,
\end{multline*}
where we define the empirical measure $\mu_n:=\frac{1}{n}\sum_{k=1}^n
\delta_{X_k}$ for an i.i.d.\ sequence $(X_k)_{k\in\mathbb{N}}$ with
distribution $\mu$. The right hand side in the above expression is
measurable and converges a.s.\ to a constant not exceeding $\varepsilon$
by the law of large numbers. As $\varepsilon>0$ and $\mu$ were arbitrary,
$\mathcal{F}$ is universal Glivenko-Cantelli.
\subsection{$2\Rightarrow 3\Rightarrow 4$}
As $N(\mathcal{F},\varepsilon,\mu)\le
N_{[]}(\mathcal{F},2\varepsilon,\mu)$, the implication $2\Rightarrow 3$ is
trivial. It therefore remains to prove the implication $3\Rightarrow 4$.
To this end, suppose that there exists a sequence
$(f_i)_{i\in\mathbb{N}}\subseteq\mathcal{F}$ that is Boolean
$\sigma$-independent at levels $(\alpha,\beta)$ for some $\alpha<\beta$.
Construct the probability measure $\mu$ as in the proof of the implication
$1\Rightarrow 4$. We claim that $N(\mathcal{F},\varepsilon,\mu)=\infty$
for $\varepsilon>0$ sufficiently small, which yields the desired
contradiction.
To prove the claim, it suffices to note that for any $i\ne j$
\begin{align*}
\mu(|f_i-f_j|) &\ge
\mu(|f_i-f_j|\mathbf{1}_{f_j<\alpha}\mathbf{1}_{f_i>\beta}) \\
&\ge (\beta-\alpha)\,
\mu(\{f_j<\alpha\}\cap\{f_i>\beta\}) =
(\beta-\alpha)p(1-p)>0
\end{align*}
by the construction of $\mu$. Therefore $\mathcal{F}$ contains an
infinite set of $(\beta-\alpha)p(1-p)$-separated points in $L^1(\mu)$,
so $N(\mathcal{F},(\beta-\alpha)p(1-p)/2,\mu)=\infty$.
\subsection{A remark about a.s.\ convergence and measurability}
\label{sec:meas}
When the class $\mathcal{F}$ is only assumed to be separable, the quantity
$$
\Gamma_n(\mathcal{F},\mu):=
\sup_{f\in\mathcal{F}}\left|\frac{1}{n}
\sum_{k=1}^nf(X_k)-\mu(f)\right|
$$
may well be nonmeasurable. For nonmeasurable functions, there are
inequivalent notions of convergence that coincide with a.s.\ convergence
in the measurable case. In this paper, following Talagrand \cite{Tal87},
we defined $\mu$-Glivenko-Cantelli classes as those for which the quantity
$\Gamma_n(\mathcal{F},\mu)$ converges to zero a.s., that is, pointwise
outside a set of probability zero. A different definition, given by
Dudley \cite[section 3.3]{Dud99}, is to require that
$\Gamma_n(\mathcal{F},\mu)$ converges to zero almost uniformly, that is,
it is dominated by a sequence of measurable random variables converging to
zero a.s.
For nonmeasurable functions, almost uniform convergence is in general much
stronger than a.s.\ convergence. Nonetheless, in the fundamental paper
characterizing the $\mu$-Glivenko-Cantelli property, Talagrand showed
\cite[Theorem 22]{Tal87} that for $\mu$-Glivenko-Cantelli classes a.s.\
convergence already implies almost uniform convergence. Thus this is
certainly the case for universal Glivenko-Cantelli classes. In the
setting of Theorem \ref{thm:main}, the latter can also be seen directly:
indeed, the proof of the implication $1\Rightarrow 4$ requires only a.s.\
convergence, while the Blum-DeHardt argument $2\Rightarrow 1$
automatically yields the stronger notion of almost uniform convergence.
However, let us note that in Corollary \ref{cor:countdens} below we will
prove an even stronger property: for separable uniformly bounded classes
$\mathcal{F}$ with finite bracketing numbers, the quantity
$\sup_{f\in\mathcal{F}}|\nu(f)-\rho(f)|$ is Borel-measurable for arbitrary
random probability measures $\nu,\rho$. Thus $\Gamma_n(\mathcal{F},\mu)$
is automatically measurable for universal Glivenko-Cantelli classes
satisfying the assumptions of Theorem \ref{thm:main}, though this is far
from obvious a priori. Similarly, if any of the equivalent conditions of
Theorem \ref{thm:main} or Corollary \ref{cor:uniformity} holds, then all
the suprema in Corollary \ref{cor:uniformity} are measurable. It follows
that a.s.\ and almost uniform convergence coincide trivially in our main
results.
\section{Proof of Corollary \ref{cor:uniformity}}
\label{sec:uniformity}
Throughout this section, we fix a standard measurable space
$(X,\mathcal{X})$ and a separable uniformly bounded family of measurable
functions $\mathcal{F}$. We will prove Corollary \ref{cor:uniformity} by
proving the implications $2\Leftrightarrow 5$ and $2\Rightarrow
\{6,7,8\}\Rightarrow 1$. The implication $5\Rightarrow 2$ is related to a
result of Tops{\o}e \cite{Top77}, though we give here a direct proof
inspired by Stute \cite{Stu76}. The remaining implications are
straightforward modulo measurability issues.
\subsection{$2\Leftrightarrow 5$}
The implication $2\Rightarrow 5$ follows from the Blum-DeHardt argument as
in section \ref{sec:blum}. Conversely,
suppose that condition 2 does not hold, so that
$N_{[]}(\mathcal{F},\varepsilon,\mu)=\infty$
for some $\varepsilon>0$ and probability measure $\mu$. Then by
Lemma \ref{lem:approx}, there exist $\delta>0$ and $\alpha<\beta$
such that we can choose for every $\pi\in\Pi(X,\mathcal{X})$ a function
$f_\pi\in\mathcal{F}$ with
$$
\mu(D_\pi)\ge\delta,\qquad\quad
D_\pi:=\partial_\pi^\mu(\{f_\pi<\alpha\},\{f_\pi>\beta\}).
$$
We now define for every $\pi\in\Pi(X,\mathcal{X})$ two probability
measures $\mu_\pi^+,\mu_\pi^-$ as follows. For every $P\in\pi$ such that
$P\subseteq D_\pi$, choose two points $x_P^+\in P\cap\{f_\pi>\beta\}$
and $x_P^-\in P\cap\{f_\pi<\alpha\}$ arbitrarily, and define for
every $A\in\mathcal{X}$
$$
\mu_\pi^\pm(A) =
\mu(A\backslash D_\pi) +
\sum_{P\in\pi:P\subseteq D_\pi}
\mu(P)\,\mathbf{1}_A(x_P^\pm).
$$
Then $(\mu_\pi^\pm)_{\pi\in\Pi(X,\mathcal{X})}$ is a net of
probability measures that converges to $\mu$ setwise: indeed,
for every $A\in\mathcal{X}$, we have
$\mu_\pi^\pm(A)=\mu(A)$ whenever $\pi\preceq\pi_A$ with
$\pi_A=\{A,X\backslash A\}$. On the other hand, by construction we
have
$$
\sup_{f\in\mathcal{F}}|\mu_\pi^+(f)-\mu_\pi^-(f)|
\ge |\mu_\pi^+(f_\pi)-\mu_\pi^-(f_\pi)| \ge
(\beta-\alpha)\mu(D_\pi)\ge(\beta-\alpha)\delta
$$
for every $\pi\in\Pi(X,\mathcal{X})$. Therefore either
$(\mu_\pi^+)_{\pi\in\Pi(X,\mathcal{X})}$ or
$(\mu_\pi^-)_{\pi\in\Pi(X,\mathcal{X})}$ does not converge
to $\mu$ uniformly over $\mathcal{F}$, in contradiction to condition 5.
\subsection{$2\Rightarrow\{6,7,8\}$}
The implication $2\Rightarrow 6$ follows immediately from the Blum-DeHardt
argument as in section \ref{sec:blum}. The complication for the
implications $2\Rightarrow\{7,8\}$ is that the limiting measure is a
random measure (unlike $2\Rightarrow 6$ where the limiting measure is
nonrandom). Intuitively one can simply condition on
$\mathcal{G}_{-\infty}$ or $\mathcal{I}$, respectively, so that the
problem reduces to the implication $2\Rightarrow 6$ under the conditional
measure. The main work in the proof consists of resolving the
measurability issues that arise in this approach.
Let $\mathcal{F}_0\subseteq\mathcal{F}$ be a countable family that is
dense in $\mathcal{F}$ in the topology of pointwise convergence.
We first show that $\mathcal{F}_0$ is also $L^1(\mu)$-dense in
$\mathcal{F}$ for any $\mu$: this is not obvious, as the dominated
convergence theorem does not hold for nets.
\begin{lem}
\label{lem:ptmsc}
If $N_{[]}(\mathcal{F},\varepsilon,\mu)<\infty$ for all
$\varepsilon>0$, then $\mathcal{F}_0$ is $L^1(\mu)$-dense in
$\mathcal{F}$.
\end{lem}
\begin{proof}
Fix $\varepsilon>0$, and choose
$\varepsilon$-brackets $[f_1,g_1],\ldots,[f_N,g_N]$ in $L^1(\mu)$
covering $\mathcal{F}$. As topological closure and finite unions commute,
for every $f\in\mathcal{F}$ there exists $1\le i\le N$ such that $f$
is in the pointwise closure of $[f_i,g_i]\cap\mathcal{F}_0$. But then
clearly $f\in[f_i,g_i]$, and choosing any $g\in
[f_i,g_i]\cap\mathcal{F}_0$ we have
$\mu(|f-g|)\le\mu(g_i-f_i)\le\varepsilon$. As $\varepsilon>0$ is
arbitrary, the proof is complete.
\end{proof}
We can now reduce the suprema in conditions $7$ and $8$ to countable
suprema.
\begin{cor}
\label{cor:countdens}
Suppose that $N_{[]}(\mathcal{F},\varepsilon,\mu)<\infty$ for every
$\varepsilon>0$ and probability measure $\mu$. Then for any pair
of probability measures $\mu,\nu$ we have
$$
\sup_{f\in\mathcal{F}}|\mu(f)-\nu(f)| =
\sup_{f\in\mathcal{F}_0}|\mu(f)-\nu(f)|.
$$
In particular, this holds when $\mu$ and $\nu$ are random measures.
\end{cor}
\begin{proof}
Fix (nonrandom) probability measures $\mu,\nu$, and define
$\rho=\{\mu+\nu\}/2$. Then $\mathcal{F}_0$ is $L^1(\rho)$-dense
in $\mathcal{F}$ by Lemma \ref{lem:ptmsc}. In particular, for every
$f\in\mathcal{F}$ and $\varepsilon>0$, we can choose $g\in\mathcal{F}_0$
such that $\mu(|f-g|)+\nu(|f-g|)\le \varepsilon$. Now let
$(f_n)_{n\in\mathbb{N}}\subseteq\mathcal{F}$ be a sequence such that
$
\sup_{f\in\mathcal{F}}|\mu(f)-\nu(f)|=
\lim_{n\to\infty}|\mu(f_n)-\nu(f_n)|.
$
For each $f_n$, choose $g_n\in\mathcal{F}_0$ such that
$\mu(|f_n-g_n|)+\nu(|f_n-g_n|)\le n^{-1}$. Then
$$
\sup_{f\in\mathcal{F}}|\mu(f)-\nu(f)|=
\lim_{n\to\infty}|\mu(g_n)-\nu(g_n)|\le
\sup_{f\in\mathcal{F}_0}|\mu(f)-\nu(f)|,
$$
which clearly yields the result (as $\mathcal{F}_0\subseteq\mathcal{F}$).
In the case of random probability measures, we simply apply the nonrandom
result pointwise.
\end{proof}
To prove $2\Rightarrow 8$ we use the ergodic decomposition (cf.\
Appendix \ref{app:decomp}). Consider a stationary sequence
$(Z_n)_{n\in\mathbb{N}}$ of $X$-valued random
variables on a probability space $(\Omega,\mathcal{G},\mathbf{P})$. Using
Corollary \ref{cor:countdens} and the ergodic theorem, it suffices to
prove that
$$
\mathbf{P}\left[
\limsup_{n\to\infty}
\sup_{f\in\mathcal{F}_0}\left|
\frac{1}{n}\sum_{k=1}^nf(Z_k)-
\limsup_{N\to\infty}
\frac{1}{N}\sum_{k=1}^N f(Z_k)
\right|=0
\right]=1.
$$
The event inside the probability is an
$\mathcal{X}^{\otimes\mathbb{N}}$-measurable function of
$(Z_n)_{n\in\mathbb{N}}$. Therefore, by Theorem
\ref{thm:ergdecomp} in Appendix \ref{app:decomp}, it suffices to prove
the result for the case that $(Z_n)_{n\in\mathbb{N}}$ is
stationary and ergodic. But in the ergodic case
$\frac{1}{N}\sum_{k=1}^N f(Z_k)\to\mathbf{E}(f(Z_0))$ a.s.,
so that the result follows from the Blum-DeHardt argument.
To prove the implication $2\Rightarrow 7$, we aim to repeat the proof of
$2\Rightarrow 8$ with a suitable tail decomposition (cf.\ Theorem
\ref{thm:taildecomp} in Appendix \ref{app:decomp}). On an
underlying probability space $(\Omega,\mathcal{G},\mathbf{P})$, let
$(\mathcal{G}_{-n})_{n\in\mathbb{N}}$ be a reverse filtration such that
$\mathcal{G}_{-n}\subseteq\mathcal{G}$ is countably generated for each
$n\in\mathbb{N}$, and consider a random variable $Z$ taking values in the
standard space $(X,\mathcal{X})$.
Using Corollary \ref{cor:countdens} and the reverse
martingale convergence theorem, it evidently suffices to prove that
$$
\mathbf{P}\left[
\limsup_{n\to\infty}
\sup_{f\in\mathcal{F}_0}\left|
\mathbf{E}(f(Z)|\mathcal{G}_{-n})-
\limsup_{N\to\infty}
\mathbf{E}(f(Z)|\mathcal{G}_{-N})
\right|=0
\right]=1.
$$
If $(\Omega,\mathcal{G})$ is standard, then by Theorem
\ref{thm:taildecomp} it suffices to
prove the result for the case that the tail $\sigma$-field
$\mathcal{G}_{-\infty}=\bigcap_n\mathcal{G}_{-n}$ is trivial.
But in that case
$\mathbf{E}(f(Z)|\mathcal{G}_{-n})\to\mathbf{E}(f(Z))$ a.s.,
so that the result follows from the Blum-DeHardt argument.
It therefore remains to show that there is no loss of generality in
assuming that $(\Omega,\mathcal{G})$ is standard. To this end, choose for
every $n\ge 1$ a countable generating class
$(H_{n,j})_{j\in\mathbb{N}}\subseteq\mathcal{G}_{-n}$, and define the
$\{0,1\}^\mathbb{N}$-valued random variable
$Z_{-n}=(\mathbf{1}_{H_{n,j}})_{j\in\mathbb{N}}$. Then, by construction,
$\mathcal{G}_{-n}=\sigma\{Z_{-k}:k\ge n\}$. If we define $Z_0=Z$, then it
is clear that the implication $2\Rightarrow 7$ depends only on the law of
$(Z_{-n})_{n\ge 0}$. There is therefore no loss of generality in assuming
that $(\Omega,\mathcal{G})$ is the canonical space of the process
$(Z_{-n})_{n\ge 0}$, which is clearly standard as $\{0,1\}^{\mathbb{N}}$
is Polish.
\subsection{$\{6,7,8\}\Rightarrow 1$}
These implications follow from the fact that each of the conditions
$\{6,7,8\}$ contains condition $1$ as a special case. For the implication
$6\Rightarrow 1$, it suffices to choose $\mu_n$ to be the empirical
measure of an i.i.d.\ sequence with distribution $\mu$. Similarly, the
implication $8\Rightarrow 1$ follows from the fact that an i.i.d.\
sequence is stationary and ergodic. Finally, the implication
$7\Rightarrow 1$ follows from the following well known construction.
Let $(X_k)_{k\in\mathbb{N}}$ be an i.i.d.\ sequence of $X$-valued
random variables with distribution $\mu$, let $Z=X_1$, and let
$\mathcal{G}_{-n}=\sigma\{\sum_{k=1}^n\mathbf{1}_A(X_k):A\in\mathcal{X}\}$.
As $(X,\mathcal{X})$ is standard, $\mathcal{X}$ and hence
$\mathcal{G}_{-n}$ are countably generated. Moreover, we have
$$
\mathbf{E}(f(Z)|\mathcal{G}_{-n})=
\mathbf{E}(f(X_\ell)|\mathcal{G}_{-n})=
\frac{1}{n}\sum_{k=1}^n\mathbf{E}(f(X_k)|\mathcal{G}_{-n})=
\frac{1}{n}\sum_{k=1}^nf(X_k)
$$
for any bounded measurable function $f$ and $1\le\ell\le n$,
as the right hand side is $\mathcal{G}_{-n}$-measurable and every element
of $\mathcal{G}_{-n}$ is symmetric under permutations of
$\{X_1,\ldots,X_n\}$. Therefore, $\frac{1}{n}\sum_{k=1}^n\delta_{X_k}$ is
a version of the regular conditional probability
$\mathbf{P}(Z\in\,\cdot\,|\mathcal{G}_{-n})$ for every $n\ge 1$. By the
law of large numbers and the martingale convergence theorem, it follows
that $\mu$ is a version of the regular conditional probability
$\mathbf{P}(Z\in\,\cdot\,|\mathcal{G}_{-\infty})$. The implication
$7\Rightarrow 1$ is now immediate.
\section{Proof of Proposition \ref{prop:vc}}
\label{sec:vc}
The construction of the class $\mathcal{C}$ in Proposition \ref{prop:vc}
is based on a combinatorial construction due to Alon, Haussler, and Welzl
\cite[Theorem A(2)]{AHW87}. We begin by recalling the essential results
in that paper, and then proceed to the proof of Proposition \ref{prop:vc}.
\subsection{Construction}
Let $q\ge 2$ be a prime number, and denote by $\mathbb{F}_q$ the finite
field $\mathbb{Z}/q\mathbb{Z}$ of order $q$. In the following, we
consider the three-dimensional vector space $\mathbb{F}_q^3$ over the
finite field $\mathbb{F}_q$. Denote by $V_q$ the family of all
one-dimensional subspaces of $\mathbb{F}_q^3$, and denote by $E_q$ the
family of all two-dimensional subspaces of $\mathbb{F}_q^3$. Each element
of $E_q$ is identified with a subset of $V_q$ by inclusion, that is,
a two-dimensional subspace $C\in E_q$ is identified with the set of
one-dimensional subspaces $x\in V_q$ contained in it. An elementary
counting argument, cf.\ \cite[section 9.3]{Cam94}, yields the
following properties:
\begin{enumerate}
\item $\card V_q=\card E_q=q^2+q+1$.
\item Every set $C\in E_q$ contains exactly $q+1$ points in $V_q$.
\item Every point $x\in V_q$ belongs to exactly $q+1$ sets in $E_q$.
\item For every $x,x'\in V_q$, $x\ne x'$ there is a unique set
$C\in E_q$ with $x,x'\in C$.
\end{enumerate}
A pair $(V_q,E_q)$ with these properties is called a \emph{finite
projective plane} of order $q$. For our purposes, the key property
of finite projective planes is the following result due to
Alon, Haussler, and Welzl, whose proof is given in \cite[p.\ 336]{AHW87}
(the proof is based on a combinatorial lemma proved in
\cite[Theorem 2.1(2)]{Alon85}).
\begin{prop}
\label{prop:alon}
Let $q\ge 2$ be prime, define $m=q^2+q+1$, and let $\varepsilon>0$.
Then for any partition $\pi$ of $V_q$ such that
$(\card\pi)^2\le m^{1/2}(1-\varepsilon)$, we have
$$
\max_{C\in E_q}
\frac{\card\partial_\pi C}{m}>\varepsilon.
$$
Here we defined the $\pi$-boundary
$\partial_\pi C:=\bigcup\{P\in\pi:P\cap C\ne\varnothing\mbox{ and }
P\not\subseteq C\}$.
\end{prop}
We now proceed to construct the class $\mathcal{C}$ in Proposition
\ref{prop:vc}. Let $q_j\uparrow\infty$ be an increasing sequence of
primes ($q_j\ge 2$), and define $m_j=q_j^2+q_j+1$. We now partition
$\mathbb{N}$ into consecutive blocks of length $m_j$, as follows:
$$
\mathbb{N} = \bigcup_{j=1}^\infty N_j,\qquad
N_j=\left\{
\sum_{i=1}^{j-1}m_i+1,\ldots,\sum_{i=1}^jm_i
\right\}\simeq V_{q_j}.
$$
Define $\mathcal{C}$ as the disjoint union of copies of $E_{q_j}$
defined on the blocks $N_j$: that is, choose for every $j$ a
bijection $\iota_j:V_{q_j}\to N_j$, and define
$$
\mathcal{C}=\bigcup_{j=1}^\infty\mathcal{C}_j,\qquad
\mathcal{C}_j = \{ B\subseteq N_j :
\iota_j^{-1}(B)\in E_{q_j}\}.
$$
We claim that the countable class $\mathcal{C}$ of subsets of $\mathbb{N}$
has $\gamma$-dimension two.
\begin{lem}
$\mathcal{C}$ has Vapnik-Chervonenkis dimension two.
\end{lem}
\begin{proof}
Choose any three distinct points $n_1,n_2,n_3\in\mathbb{N}$. If two of
these points are in distinct intervals $N_j$, then no set in $\mathcal{C}$
contains both points. On the other hand, suppose that all three points
are in the same interval $N_j$. Then by the properties of the finite
projective plane, either there is no set in $\mathcal{C}$ that contains
all three points, or there is no set that contains two of the points but
not the third (as each pair of points must lie in a unique set in
$\mathcal{C}$). Thus we have shown that no family of three points
$\{n_1,n_2,n_3\}$ is $\gamma$-shattered for $0<\gamma<1$. On the other
hand, it is easily seen that the properties of the finite projective
plane imply that any pair of points $\{n_1,n_2\}$ belonging to the same
interval $N_j$ is $\gamma$-shattered for $0<\gamma<1$.
\end{proof}
\subsection{Proof of Proposition \ref{prop:vc}}
The following crude lemma yields lower bounds on the bracketing numbers.
\begin{lem}
\label{lem:crude}
Let $\mu$ be a probability measure on $\mathbb{N}$. Then
$$
\inf_{\card\pi\le 3^N}\sup_{C\in\mathcal{C}}
\mu(\partial_\pi C)>\varepsilon
\qquad\mbox{implies}\qquad
N_{[]}(\mathcal{C},\varepsilon,\mu)>N,
$$
where the infimum ranges over all partitions of $\mathbb{N}$ with
$\card\pi\le 3^N$.
\end{lem}
\begin{proof}
Suppose $N_{[]}(\mathcal{C},\varepsilon,\mu)\le N$. Then there
are $k\le N$ pairs $\{C_i^+,C_i^-\}_{i\le k}$ of subsets of
$\mathbb{N}$ such that $\mu(C_i^+\backslash C_i^-)\le\varepsilon$
for all $1\le i\le k$, and for every $C\in\mathcal{C}$, there exists
$1\le i\le k$ such that $C_i^-\subseteq X\subseteq C_i^+$.
Let $\pi$ be the partition generated by $\{C_i^+,C_i^-:1\le i\le k\}$.
Then $\card\pi\le 3^N$, as $\pi$ is the common refinement of at most $N$
partitions $\{C_i^-,C_i^+\backslash C_i^-,\mathbb{N}\backslash C_i^+\}$
of size three.
Now choose any $C\in\mathcal{C}$, and choose $1\le i\le k$ such that
$C_i^-\subseteq C\subseteq C_i^+$. As $C_i^-$ and $\mathbb{N}\backslash
C_i^+$ are unions of atoms of $\pi$ by construction, and as
$C_i^-\subseteq C$ and $(\mathbb{N}\backslash C_i^+)\cap C=\varnothing$,
we evidently have $\partial_\pi C \subseteq C_i^+\backslash C_i^-$. Thus
$\mu(\partial_\pi C)\le \varepsilon$. As this holds for any
$C\in\mathcal{C}$, we complete the proof by contradiction.
\end{proof}
Denote by $\mu_j$ the uniform distribution on $N_j$.
Let $(p_j)_{j\in\mathbb{N}}$ be a sequence of nonnegative numbers
$p_j\ge 0$ so that $\sum_jp_j=1$, and define the probability measure
$$
\mu = \sum_{j=1}^\infty p_j\mu_j.
$$
We first obtain a lower bound on $N_{[]}(\mathcal{C},\varepsilon,\mu)$.
Subsequently, we will be able to choose the sequence
$(p_j)_{j\in\mathbb{N}}$ such that this bound grows arbitrarily quickly.
To obtain a lower bound, let us suppose that
$N_{[]}(\mathcal{C},\varepsilon,\mu)\le N$. Then applying Lemma
\ref{lem:crude}, there exists a partition $\pi$ of $\mathbb{N}$
with $\card\pi\le 3^N$ such that
$$
\sup_{j\in\mathbb{N}}p_j\min_{\card\pi'\le 3^N}
\max_{C\in E_{q_j}}\frac{\card\partial_{\pi'}C}{m_j}
\le
\sup_{j\in\mathbb{N}}p_j\max_{C\in\mathcal{C}_j}\mu_j(\partial_\pi C)
\le\sup_{C\in\mathcal{C}}\mu(\partial_\pi C)
\le\varepsilon.
$$
By Proposition \ref{prop:alon},
$$
\min_{\card\pi'\le 3^N}
\max_{C\in E_{q_j}}\frac{\card\partial_{\pi'}C}{m_j}
\le\frac{\varepsilon}{p_j}\quad\mbox{implies}\quad
m_j^{1/4}\sqrt{1-\frac{\varepsilon}{p_j}\wedge 1}<3^{N}.
$$
Therefore, $N_{[]}(\mathcal{C},\varepsilon,\mu)\le N$ implies that
$$
N>\frac{1}{4}\log_3 m_j
+ \frac{1}{2}\log_3\left(1-\frac{\varepsilon}{p_j}\wedge 1\right)
$$
for every $j\in\mathbb{N}$. It follows that
$$
N_{[]}(\mathcal{C},\varepsilon,\mu)\ge\sup_{j\in\mathbb{N}}
\left\lfloor
\frac{1}{4}\log_3 m_j
+ \frac{1}{2}\log_3\left(1-\frac{\varepsilon}{p_j}\wedge 1\right)
\right\rfloor.
$$
This bound holds for any choice of $(p_j)_{j\in\mathbb{N}}$.
Fix $n(\varepsilon)\uparrow\infty$ as $\varepsilon\downarrow 0$.
We now choose $(p_j)_{j\in\mathbb{N}}$ such that
$N_{[]}(\mathcal{C},\varepsilon,\mu)\ge n(\varepsilon)$.
First, as $m_j\uparrow\infty$, we can choose a subsequence
$j(k)\uparrow\infty$ such that
$$
m_{j(\lfloor \log_2(2/3\varepsilon)\rfloor)} \ge
3^{4n(\varepsilon)+6}\qquad\mbox{for all }0<\varepsilon<1/3.
$$
Now define $(p_j)_{j\in\mathbb{N}}$ as follows:
$$
p_{j(k)}=2^{-k}\quad\mbox{for }k\in\mathbb{N},\qquad
p_j=0\quad\mbox{for }j\not\in\{j(k):k\in\mathbb{N}\}.
$$
Then we clearly have, setting $J(\varepsilon)
=j(\lfloor \log_2(2/3\varepsilon)\rfloor)$,
$$
N_{[]}(\mathcal{C},\varepsilon,\mu)\ge
\left\lfloor
\frac{1}{4}\log_3 m_{J(\varepsilon)}
+ \frac{1}{2}\log_3\left(1-\frac{\varepsilon}{p_{J(\varepsilon)}
}\wedge 1\right)
\right\rfloor \ge
\left\lfloor
n(\varepsilon)+1
\right\rfloor \ge n(\varepsilon)
$$
for all $0<\varepsilon<1/3$. This completes the proof.
\begin{appendices}
\section{Boolean and stochastic independence}
\label{app:boole}
An essential property of a Boolean $\sigma$-independent sequence of sets
is that there must exist a probability measure under which these sets are
i.i.d. This idea dates back to Marczewski \cite{Mar48}, who showed that
such a probability measure exists on the $\sigma$-field generated by these
sets. For our purposes, we will need the resulting probability measure to
be defined on the larger $\sigma$-field $\mathcal{X}$ of the underlying
standard measurable space $(X,\mathcal{X})$. One could apply an extension
theorem for measures on standard measurable spaces (for example, \cite[p.\
194]{Var63}) to deduce the existence of such a measure from Marczewski's
result. However, a direct proof is easily given.
\begin{thm}
\label{thm:marczewski}
Let $(X,\mathcal{X})$ be a standard measurable space. Let
$(A_i,B_i)_{i\in\mathbb{N}}$ be a sequence of pairs of sets
$A_i,B_i\in\mathcal{X}$ such that $A_i\cap B_i=\varnothing$ for
every $i\in\mathbb{N}$ and
$$
\bigcap_{j\in F}A_j\cap\bigcap_{j\not\in F}B_j
\ne\varnothing\quad\mbox{for every }F\subseteq\mathbb{N}.
$$
Let $p\in[0,1]$. Then there exists a probability measure $\mu$ on
$(X,\mathcal{X})$ such that $\mu(A_i)=\mu(X\backslash B_i)=p$
for every $i\in\mathbb{N}$, and such that $(A_i)_{i\in\mathbb{N}}$
are independent under $\mu$.
\end{thm}
\begin{proof}
Let $\mathcal{B}^*$ be the universal completion of the
the Borel $\sigma$-field of $\{0,1\}^{\mathbb{N}}$, and let
$C_j=\{\omega\in\{0,1\}^{\mathbb{N}}:\omega_j=1\}$ for
$j\in\mathbb{N}$. Moreover, let $\nu$ be the probability measure on
$\mathcal{B}^*$ under which $(C_j)_{j\in\mathbb{N}}$ are independent and
$\nu(C_j)=p$ for every $j\in\mathbb{N}$.
Define for every $\omega\in\{0,1\}^{\mathbb{N}}$ the set
$$
H(\omega) =
\bigcap_{j:\omega_j=1}A_j\cap\bigcap_{j:\omega_j=0}B_j.
$$
It suffices to show that there is a measurable map
$\iota:(\{0,1\}^{\mathbb{N}},\mathcal{B}^*) \to(X,\mathcal{X})$ such that
$\iota(\omega)\in H(\omega)$ for every $\omega\in\{0,1\}^{\mathbb{N}}$.
Indeed, as $\iota^{-1}(A_j)=C_j$ and
$\iota^{-1}(B_j)=\{0,1\}^{\mathbb{N}}\backslash C_j$ for every
$j\in\mathbb{N}$, the measure $\mu(\cdot)=\nu(\iota^{-1}(\cdot))$ has the
desired properties.
It remains to prove the existence of $\iota$. To this end, note that
the set
$$
\Gamma=\{(\omega,x):x\in H(\omega)\} =
\bigcap_{j\in\mathbb{N}}\left\{
C_j\times A_j\cup
\left(\{0,1\}^{\mathbb{N}}\backslash C_j\right)\times B_j
\right\}
$$
is measurable
$\Gamma\in\mathcal{B}(\{0,1\}^{\mathbb{N}})\otimes\mathcal{X}$,
where $\mathcal{B}(\{0,1\}^{\mathbb{N}})$ denotes the Borel
$\sigma$-field of $\{0,1\}^{\mathbb{N}}$. As $H(\omega)$ is nonempty
for every $\omega\in\{0,1\}^{\mathbb{N}}$ by assumption, the
existence of $\iota$ now follows by the measurable section theorem
\cite[Theorem 8.5.3]{Cohn80}.
\end{proof}
\begin{rem}
In the above proof, the assumption that $(X,\mathcal{X})$ is standard
is required to apply the measurable section theorem. When
$(X,\mathcal{X})$ is an arbitrary measurable space, we could of course
invoke the axiom of choice to obtain a map $\iota:\{0,1\}^{\mathbb{N}}
\to X$ such that $\iota(\omega)\in H(\omega)$ for every
$\omega\in\{0,1\}^{\mathbb{N}}$, but such a map need not be measurable
in general.
On the other hand, as $\iota^{-1}(A_j)=C_j$ and
$\iota^{-1}(B_j)=\{0,1\}^{\mathbb{N}}\backslash C_j$, it follows
that $\iota$ is necessarily Borel-measurable if we choose
$\mathcal{X}=\sigma\{A_j,B_j:j\in\mathbb{N}\}$. Thus we recover
a result along the lines of Marczewski by using the same proof.
\end{rem}
The proof of Theorem \ref{thm:scales} uses the following connection
between Boolean independence and $\gamma$-shattering which is a trivial
modification of a result of Assouad \cite{Ass83} (cf.\ \cite[Theorem
4.6.2]{Dud99}). We give the proof for completeness.
\begin{lem}
\label{lem:assouad}
Let $\{f_1,\ldots,f_{2^n}\}$ be a finite family of functions
on a set $X$ that is Boolean independent at levels $(\alpha,\beta)$
with $\beta-\alpha\ge\gamma$. Then the family $\{f_1,\ldots,f_{2^n}\}$
$\gamma$-shatters some finite subset $\{x_1,\ldots,x_n\}\subseteq X$.
\end{lem}
\begin{proof}
Define $\ell(F)=1+\sum_{j\in F}2^{j-1}$ for $F\subseteq\{1,\ldots,n\}$,
so that $\ell(F)$ assigns to every $F\subseteq\{1,\ldots,n\}$ a unique
integer between $1$ and $2^n$. Choose some point
$$
x_j\in \bigcap_{F\ni j}\{f_{\ell(F)}<\alpha\}\cap
\bigcap_{F\not\ni j}\{f_{\ell(F)}>\beta\}
$$
for every $j=1,\ldots,n$.
Then for any $F\subseteq\{1,\ldots,n\}$, we have
$f_{\ell(F)}(x_j)<\alpha$ if $j\in F$ and
$f_{\ell(F)}(x_j)>\beta$ if $j\not\in F$.
Therefore $\{x_1,\ldots,x_n\}$ is $\gamma$-shattered.
\end{proof}
\section{Decomposition theorems}
\label{app:decomp}
Part of the proof of Corollary \ref{cor:uniformity} relies on the
decomposition of stochastic processes with respect to
the invariant and tail $\sigma$-fields. These theorems will be
given presently.
The first theorem is the well-known ergodic decomposition. As this result
is classical, we state it here without proof (see \cite[Theorem
6.6]{Var01} or \cite[Theorem 10.26]{Kal02}, for example, for elementary
proofs). In the following, for any standard space $(Y,\mathcal{Y})$, we
denote by $\mathcal{P}(Y,\mathcal{Y})$ the space of probability measures
on $(Y,\mathcal{Y})$. The space $\mathcal{P}(Y,\mathcal{Y})$ is endowed
with the $\sigma$-field generated by the evaluation mappings
$\pi_B:\mu\mapsto\mu(B)$, $B\in\mathcal{Y}$. Recall that if
$(X,\mathcal{X})$ is standard, then so is
$(X^{\mathbb{N}},\mathcal{X}^{\otimes\mathbb{N}})$.
\begin{thm}
\label{thm:ergdecomp}
Let $(X,\mathcal{X})$ be a standard space, and denote by
$(Z_n)_{n\in\mathbb{N}}$ the canonical process on the space
$(X^{\mathbb{N}},\mathcal{X}^{\otimes\mathbb{N}})$.
Let $\mu\in\mathcal{P}(X^{\mathbb{N}},\mathcal{X}^{\otimes\mathbb{N}})$
be a stationary probability measure. Then there exists a probability
measure $\rho$ on
$\mathcal{P}(X^{\mathbb{N}},\mathcal{X}^{\otimes\mathbb{N}})$ such that
$$
\mu(A) = \int \nu(A)\,\rho(d\nu)
\quad\mbox{for every }
A\in\mathcal{X}^{\otimes\mathbb{N}},
$$
and such that there exists a measurable subset $B$ of\/
$\mathcal{P}(X^{\mathbb{N}},\mathcal{X}^{\otimes\mathbb{N}})$
with $\rho(B)=1$ and with the property that
every $\nu\in B$ is stationary and ergodic.
\end{thm}
The second theorem is similar in spirit to Theorem \ref{thm:ergdecomp},
where we now decompose with respect to the tail $\sigma$-field rather than
with respect to the invariant $\sigma$-field. This result is closely
related to the decomposition theorem for Gibbs measures (see, for example,
\cite{Dyn78}). For completeness, we provide a self-contained proof.
\begin{thm}
\label{thm:taildecomp}
Let $(\Omega,\mathcal{G},\mu)$ be a standard probability space. Let
$(\mathcal{G}_{-n})_{n\in\mathbb{N}}$ be a reverse filtration
with each $\mathcal{G}_{-n}\subseteq\mathcal{G}$ countably
generated. Fix for every $n\in\mathbb{N}$ a version $\mu_{-n}$ of the
regular conditional probability $\mu(\,\cdot\,|\mathcal{G}_{-n})$. Then
there exists a probability measure $\rho$ on
$\mathcal{P}(\Omega,\mathcal{G})$ such that
$$
\mu(A) = \int \nu(A)\,\rho(d\nu)
\quad\mbox{for every }
A\in\mathcal{G},
$$
and such that there is a measurable subset $B$ of\/
$\mathcal{P}(\Omega,\mathcal{G})$ with $\rho(B)=1$ and
\begin{enumerate}
\item
The tail $\sigma$-field $\mathcal{G}_{-\infty}=\bigcap_n\mathcal{G}_{-n}$
is $\nu$-trivial for every $\nu\in B$.
\item $\nu(A|\mathcal{G}_{-n})=\mu_{-n}(A)$ $\nu$-a.s.\ for every
$\nu\in B$, $A\in\mathcal{G}$, and $n\in\mathbb{N}$.
\end{enumerate}
\end{thm}
\begin{proof}
Let $\mu_{-\infty}$ be a version of the regular conditional probability
$\mu(\,\cdot\,|\mathcal{G}_{-\infty})$, whose existence
is guaranteed as $(\Omega,\mathcal{G})$ is standard. We consider
$\mu_{-\infty}:\Omega\to\mathcal{P}(\Omega,\mathcal{G})$ as a
$\mathcal{G}_{-\infty}$-measurable random probability measure
$\omega\mapsto\mu_{-\infty}^\omega$ in the usual manner (e.g.,
\cite[Lemma 1.40]{Kal02}). Let
$\rho\in\mathcal{P}(\mathcal{P}(\Omega,\mathcal{G}))$ be the
law under $\mu$ of the random measure $\mu_{-\infty}$. It follows
directly from the definition of regular conditional probability that
$$
\mu(A) = \int \mu_{-\infty}^\omega(A)\,\mu(d\omega) =
\int \nu(A)\,\rho(d\nu)
\quad\mbox{for every }
A\in\mathcal{G}.
$$
It remains to obtain a set $B$ with the two properties in the
statement of the theorem.
We begin with the second property. Note that
\begin{multline*}
\int |\nu(\mathbf{1}_C\mu_{-n}(A))-\nu(A\cap C)|\,\rho(d\nu)=\\
\int |\mu(\mathbf{1}_C\mu(A|\mathcal{G}_{-n})|\mathcal{G}_{-\infty})
-\mu(A\cap C|\mathcal{G}_{-\infty})|\,d\mu = 0
\end{multline*}
for every $n\in\mathbb{N}$, $A\in\mathcal{G}$,
and $C\in\mathcal{G}_{-n}$. Let $\mathcal{G}_{-n}^0$ be a countable
generating algebra for $\mathcal{G}_{-n}$ and let $\mathcal{G}^0$
be a countable generating algebra for $\mathcal{G}$. Evidently
$$
\int \mathbf{1}_C(\omega)\,\mu_{-n}^\omega(A)\,\nu(d\omega) =
\nu(A\cap C)
\quad\mbox{for every }n\in\mathbb{N},~
A\in\mathcal{G}^0,~C\in\mathcal{G}_{-n}^0
$$
for all $\nu$ in a measurable subset $B_0$ of
$\mathcal{P}(\Omega,\mathcal{G})$ with
$\rho(B_0)=1$. But the monotone class theorem allows to extend this
identity to all $A\in\mathcal{G}$ and
$C\in\mathcal{G}_{-n}$. Thus we have
$\nu(A|\mathcal{G}_{-n})=\mu_{-n}(A)$ $\nu$-a.s.\ for every
$\nu\in B_0$, $A\in\mathcal{G}$, and $n\in\mathbb{N}$.
We now proceed to the first property. For any
$A\in\mathcal{G}$, we have
\begin{multline*}
\int \nu(\nu(A|\mathcal{G}_{-\infty})=\nu(A))\,\rho(d\nu) =
\int
\nu\bigg(\limsup_{n\to\infty}\mu_{-n}(A)=\nu(A)\bigg)\,\rho(d\nu) =
\\
\mu\bigg(\limsup_{n\to\infty}\mu_{-n}(A)=
\mu(A|\mathcal{G}_{-\infty})\bigg)=1,
\end{multline*}
where we have used the martingale convergence theorem and the previously
established fact that $\nu(\mu_{-n}(A)=\nu(A|\mathcal{G}_{-n})
~\mbox{for all }n\in\mathbb{N})=1$ for $\rho$-a.e.\ $\nu$.
Therefore, it follows that $\nu(A|\mathcal{G}_{-\infty})=\nu(A)$
$\nu$-a.s.\ for all $A\in\mathcal{G}^0$
for every $\nu$ in a measurable
subset $B_1$ of $\mathcal{P}(\Omega,\mathcal{G})$
with $\rho(B_1)=1$. By the monotone class theorem
$\nu(A|\mathcal{G}_{-\infty})=\nu(A)$ $\nu$-a.s.\ for every $\nu\in B_1$
and $A\in \mathcal{G}$. But then evidently
$\mathcal{G}_{-\infty}$ is $\nu$-trivial for every $\nu\in B_1$. Choosing
$B=B_0\cap B_1$ completes the proof.
\end{proof}
\section{Counterexamples in nonstandard spaces}
\label{sec:counter}
The assumption that $(X,\mathcal{X})$ is standard is used in the proof of
Theorem \ref{thm:main} to establish the implications $1,3\Rightarrow 4$
and $4\Rightarrow 2$. The goal of this appendix is to show that these
implications may indeed fail when $(X,\mathcal{X})$ is not standard. To
this end we provide two counterexamples, based on the following simple
observation.
\begin{lem}
\label{lem:aleph}
There exists a Boolean $\sigma$-independent sequence of functions on
a set $X$ if and only if $\card X\ge 2^{\aleph_0}$.
\end{lem}
\begin{proof}
Suppose there exists a Boolean $\sigma$-independent sequence
$(f_j)_{j\in\mathbb{N}}$ of functions $f_j:X\to\mathbb{R}$.
Then there exist $\alpha<\beta$ such that for every
$F\subseteq\mathbb{N}$, the set
$$
\bigcap_{j\in F}\{f_j<\alpha\}\cap\bigcap_{j\not\in F}
\{f_j>\beta\}
$$
contains at least one point. As these sets are disjoint for distinct
$F\subseteq\mathbb{N}$, and there are $2^{\aleph_0}$ subsets of
$\mathbb{N}$, it follows that $\card X\ge 2^{\aleph_0}$. Conversely, if
$\card X\ge 2^{\aleph_0}$, there exists an injective map
$\iota:\{0,1\}^{\mathbb{N}}\to X$. Define the sets
$C_j=\{\iota(\omega):\omega\in\{0,1\}^{\mathbb{N}},~
\omega_j=1\}\subset X$. Then the sequence
$(\mathbf{1}_{C_j})_{j\in\mathbb{N}}$ is Boolean $\sigma$-independent.
\end{proof}
Both examples below are consistent with the usual axioms of set theory
(that is, the set theory ZFC) but depend on additional set-theoretic
axioms. I do not know whether it is possible to obtain counterexamples in
the absence of additional axioms.
\subsection{An example where $1,3\not\Rightarrow 4$}
Let $X$ be an uncountable Polish space, and let $\mathcal{X}$ be the
universal completion of its Borel $\sigma$-field. Then $(X,\mathcal{X})$
is certainly not a standard measurable space. It is known, see
Sierpi{\'n}ski and Szpilrajn \cite{SS36}, that there exists a set
$A\in\mathcal{X}$ with $\card A=\aleph_1$ that is \emph{universally null},
that is, $\mu(A)=0$ for every nonatomic probability measure $\mu$ on
$\mathcal{X}$. As every subset $C\subseteq A$ is in the $\mu$-completion
of the Borel $\sigma$-field of $X$ for every probability measure
$\mu$, it follows that $C\in\mathcal{X}$ for every $C\subseteq A$.
As is noted by Dudley, Gin{\'e} and Zinn \cite[p.\ 494]{DGZ91}, the family
of indicators $\mathcal{F}_A=\{\mathbf{1}_C:C\subseteq A\}$ is a universal
Glivenko-Cantelli class. Moreover, as $A$ is a $\mu$-null set for every
nonatomic probability measure, it is evident that
$N(\mathcal{F}_A,\varepsilon,\mu)=N(\mathcal{F}_A,\varepsilon,\mu_{\rm
at})<\infty$ for every $\varepsilon>0$ and
probability measure $\mu$, where $\mu_{\rm at}$ denotes the atomic part
of $\mu$. But assuming the continuum hypothesis, we have
$\card A=2^{\aleph_0}$ and therefore $\mathcal{F}_A$ contains a Boolean
$\sigma$-independent sequence $\mathcal{F}$ by Lemma \ref{lem:aleph}.
Clearly $\mathcal{F}$ is a separable uniformly bounded family of
measurable functions on $(X,\mathcal{X})$ for which the implications
$1,3\Rightarrow 4$ of Theorem \ref{thm:main} fail.
\begin{rem}
The existence of a universally null set does not require the
continuum hypothesis: Sierpi{\'n}ski and Szpilrajn \cite{SS36}
construct such a set in ZFC (the construction follows directly
from Hausdorff \cite{Hau36}, see also \cite[Theorem
1.2]{Lav76}). Nonetheless, the present counterexample
does depend on the continuum hypothesis and may fail in its absence.
Indeed, there exist models of the set theory ZFC in which every
universally null set has cardinality strictly less than $2^{\aleph_0}$,
see Laver \cite[p.\ 152]{Lav76}, Miller \cite[pp.\ 577--578]{Mil83}, or
Ciesielski and Pawlikowski \cite[p.\ xii and Theorem 1.1.4]{CP04}. In
such a model, $\mathcal{F}_A$ cannot contain a Boolean
$\sigma$-independent sequence by Lemma \ref{lem:aleph}.
\end{rem}
\subsection{An example where $4\not\Rightarrow 2$}
The present counterexample follows from the following
result that is proved below.
\begin{prop}
\label{prop:cardindep}
It is consistent with the set theory ZFC that there exists a probability
space $(X,\mathcal{X},\mu)$ with $\card X < 2^{\aleph_0}$ such that
there is a sequence of sets $(C_j)_{j\in\mathbb{N}}\subset\mathcal{X}$
that are independent under $\mu$ with $\mu(C_j)=1/2$ for every
$j\in\mathbb{N}$.
\end{prop}
This result easily yields the desired example. Let $(X,\mathcal{X},\mu)$
and $(C_j)_{j\in\mathbb{N}}$ be as in Proposition \ref{prop:cardindep},
and define the class $\mathcal{F}=\{\mathbf{1}_{C_j}:j\in\mathbb{N}\}$.
The proof of the implication $3\Rightarrow 4$ of Theorem \ref{thm:main}
shows that $N_{[]}(\mathcal{F},\varepsilon,\mu)\ge
N(\mathcal{F},\varepsilon,\mu)=\infty$ for $\varepsilon>0$ sufficiently
small. On the other hand, $\mathcal{F}$ cannot contain a Boolean
$\sigma$-independent sequence by Lemma \ref{lem:aleph}. Thus $\mathcal{F}$
is a separable uniformly bounded family of measurable functions on
$(X,\mathcal{X})$ for which the implication $4\Rightarrow 2$ of Theorem
\ref{thm:main} fails.
\begin{rem}
It is clear that the present counterexample must depend on a model of set
theory in which the continuum hypothesis fails. Indeed, the set $X$ in
Proposition \ref{prop:cardindep} must be uncountable as it supports a
(stochastically) independent sequence. Therefore, if we assume the
continuum hypothesis, then necessarily $\card X \ge 2^{\aleph_0}$ and
we cannot guarantee the nonexistence of a Boolean $\sigma$-independent
sequence.
\end{rem}
Denote by $\lambda$ the Lebesgue measure on $[0,1]$, and denote by
$\lambda^*$ the Lebesgue outer measure. The proof of Proposition
\ref{prop:cardindep} is based on the following remarkable fact: there
exist models of the set theory ZFC in which there is a subset
$X\subset[0,1]$ with $\card X<2^{\aleph_0}$ such that $\lambda^*(X)>0$;
see Martin and Solovay \cite[section 4.1]{MS70},
Kunen \cite[Theorem 3.19]{Kun84}, or Judah and Shelah \cite{JS90}.
The existence of such a set $X$ will be assumed in the proof of
Proposition \ref{prop:cardindep}. Note that the set $X$ cannot be
Lebesgue measurable (if $X$ were measurable it must contain a Borel set of
positive measure, which has cardinality $2^{\aleph_0}$ by the Borel
isomorphism theorem).
\begin{proof}[Proposition \ref{prop:cardindep}]
Assume a model of the set theory ZFC in which there exists a set
$X\subset[0,1]$ with $\card X<2^{\aleph_0}$ such that $\lambda^*(X)>0$.
Let $\mathcal{X}$ be the trace of the Borel $\sigma$-field of $[0,1]$
on $X$, that is, $\mathcal{X}=\{A\cap X:A\in\mathcal{B}([0,1])\}$.
Choose a measurable cover $\tilde X$ of $X$, and note that
$A\cap\tilde X$ is a measurable cover of $A\cap X$ whenever
$A\in\mathcal{B}([0,1])$. We may therefore unambiguously define
$\mu(A\cap X)=\lambda(A\cap \tilde X)/\lambda(\tilde X)$ for
$A\in\mathcal{B}([0,1])$, and it is easily verified that $\mu$ is a
probability measure on $(X,\mathcal{X})$ whose definition does not depend
on the choice of $\tilde X$.
We now claim the following: for every set $C\in\mathcal{X}$ with
$\mu(C)>0$, there exists a set $C'\in\mathcal{X}$, $C'\subset C$
with $\mu(C')=\mu(C)/2$. Indeed, let $C=A\cap X$ for some
$A\in\mathcal{B}([0,1])$. As the function $\phi:t\mapsto
\lambda(A\cap \tilde X\cap [0,t])$ is continuous and
$\phi(0)=0$, $\phi(1)=\lambda(A\cap \tilde X)$, there exists
by the intermediate value theorem
$0<s<1$ such that $\phi(s)=\lambda(A\cap \tilde X)/2$.
Therefore $C'=C\cap[0,s]$ yields the desired set.
Now inductively define for every $n\ge 1$ and
$\omega\in\{0,1\}^n$ a set $A_\omega\in\mathcal{X}$ as follows.
For $n=1$, choose a set $A_0\in\mathcal{X}$ such that
$\mu(A_0)=1/2$, and define $A_1=X\backslash A_0$.
For $n>1$, choose for every $\omega\in\{0,1\}^{n-1}$ a set
$A_{\omega 0}\in\mathcal{X}$ such that $A_{\omega 0}\subset A_{\omega}$
with $\mu(A_{\omega 0})=\mu(A_\omega)/2$, and define
$A_{\omega 1}=A_{\omega}\backslash A_{\omega 0}$. Finally, define
for every $n\ge 1$
$$
C_n=\bigcup_{\omega\in\{0,1\}^n:\omega_n=0}A_\omega.
$$
Then $\mu(C_n)=1/2$ for every $n\ge 1$, and $\mu(C_{i_1}\cap
\cdots\cap C_{i_k})=2^{-k}$ for every $k\ge 1$ and
$1\le i_1<i_2<\cdots<i_k$. This evidently completes the proof.
\end{proof}
\end{appendices}
\subsection*{Acknowledgment}
The author would like to thank Terry Adams and Andrew Nobel for making
available an early version of \cite{AN10b} and for interesting
discussions on the topic of this paper.
|
{
"timestamp": "2012-01-25T02:03:24",
"yymm": "1009",
"arxiv_id": "1009.4434",
"language": "en",
"url": "https://arxiv.org/abs/1009.4434"
}
|
\section{Introduction}\label{sc_intro}
The Magellanic Clouds have provided astronomers with a wide variety of information,
from the small to the large hierarchy of objects in the Universe.
They are the first step in the cosmic-distance ladder,
as well as a proxy for the low-metal
gas-rich galaxies assembled in the early universe, they have zones of recent massive star
formation, young 1-3 Gyr globular clusters and a variety of old pulsating stars and planetary nebulae.
They contain substantial amounts of gas and dust in a violent interstellar medium,
harbor the closest supernova in recent years, and are being used as a testbed
for dark matter searches.
The Magellanic Clouds are also the prime example of a galaxy-galaxy interaction,
based on several lines of evidence, close enough to be studied in detail:
An apparently starless Magellanic Stream trails the Clouds \citep{1974ApJ...190..291M,1982Obs...102..174P};
a bridge of gas and stars connects the Clouds \citep{1963AuJPh..16..570H,1985Natur.318..160I};
a gaseous Leading Arm precedes the clouds \citep{1998Natur.394..752P};
an HI envelope surrounds the whole system and
a collection of High Velocity Clouds seems to be ``raining'' over the Galaxy \citep{2004A&A...423..895O}.
Despite such convenient observational circumstances, on the theoretical side
no dynamical model has been able to reproduce all these phenomena simultaneously.
It is widely accepted that the complexity and intricacies of the Magellanic Clouds'
external and internal features, have been largely determined by the orbit they have
followed in the past few Gigayears. Due to their large distance, about 50 and 60 kpc to
the LMC and the SMC, respectively, only radial velocities
were precise enough to provide some assessment of their spatial
velocity. In fact, line-of-sight measurements of the Clouds began
about a hundred years ago \citep{1915PNAS....1..183W},
while proper motion measurements of a useful accuracy were possible only in the 1990's.
The first proper motion results
\citep{1989BAAS...21.1107J,1991IAUS..148..491T,1993AGAb....8..155B,
1993BAAS...25..783L,1994MNRAS.266..412K,1994AJ....107.1333J,
1996BAAS...28..932I,1997NewA....2...77K,1997AGAb...13...77K,
1997ESASP.402..615K,1999IAUS..190..475A} based on
plate and/or CCD data, were compatible with a picture in which
the Magellanic Clouds were bound to each other and to the
Milky Way. Such scenario relied heavily on the fact that the
Galactic gravitational potential used (isothermal sphere)
yields such results by default, and proper motion errors were not
small enough to refine the tangential velocities.
In the past ten years though, investigations yielded quite a variety of
results \citep{2000AJ....120..845A,2001AAS...199.5205D,2002AJ....123.1971P,
2005A&A...437..339M,2005AAS...20711307K,2006ApJ...638..772K,
2006AJ....131.1461P,2006RMxAC..25...43P,2006RMxAC..26...78P,2006RMxAC..26..183M,
2006ApJ...652.1213K,2008AJ....135.1024P,2009IAUS..256...93K,2009AJ....137.4339C}.
Some \citep{2000AJ....120..845A,2005A&A...437..339M} were found to have
unknown or important systematic errors.
More interestingly, HST-based results \citep{2005AAS...20711307K,2006ApJ...638..772K,
2006ApJ...652.1213K,2008AJ....135.1024P,2009IAUS..256...93K} coupled to more
modern and cosmologically inspired dark matter Halo models, suggest
that the Clouds are not bound to the Galaxy, opposite to the long-held paradigm.
Twenty years have passed since the very first proper motion measurements of the Clouds,
and it is only now with the Yale-San Juan Southern Proper Motion
(SPM) program - briefly explained in Section \ref{sc_spm} - that for the
first time a wide-field astrometric
proper motion survey of the Magellanic System is finally completed.
All the 1st-epoch (early 1970's) and part of the 2nd-epoch (early 1990's)
SPM material used in this work are photographic plates.
Their processing is briefly summarised in Section \ref{sc_plate}, but a more
detailed explanation can be found in Girard et al. (2010).
A substantial part of our 2nd-epoch data comes from SPM CCD observations.
A short explanation of the data acquisition,
quality control and processing is explained in Section
\ref{sc_ccd}. To supplement our 2nd-epoch plate data with CCD-quality
positions, we have included mean positions at Julian epoch 2000.0
from the UCAC2 catalog \citep{2004AJ....127.3043Z}.
Section \ref{sc_ucac2} explains how these data were selected
and included in this investigation.
In contrast to other SPM reductions, as explained in Section \ref{sc_obtpm},
relative proper motions measured in CCD-size fields of view were combined
into a single common extended and accurately defined global system.
Although our zero point accuracy, i.e. how well our reference frame is linked
to the International Celestial Reference System (ICRS), is ultimately
limited by Hipparcos accuracy, our very precise relative proper motions
over the whole field of view, enabled us to measure the proper motion of the SMC
with respect to the LMC, at a precision comparable to the quoted errors of
space-based proper motions.
It is this capability that we exploit to obtain new measurements of the proper motion
of the SMC based on previously published LMC proper motions. Section \ref{sc_pmmcs} contains
the main results of this paper regarding the proper motion of the Clouds, absolute and relative.
Section \ref{sc_implications} has a discussion of the implications of our results
on the current understanding of the dynamics of the Magellanic System.
Finally, Section \ref{sc_end} states the conclusions of this paper, and
future plans already in consideration to improve the current results.
\section{The SPM program}\label{sc_spm}
This investigation is part of the SPM program, a joint venture of Yale University
and Universidad Nacional de San Juan in Argentina.
The SPM program was initiated in the
early 1960s by D. Brower and J. Schilt as a joint enterprise of
the Yale and Columbia Universities \citep{1974IAUS...61..201W}.
The goal of the SPM program is to provide absolute proper motions, positions,
and $BV$ photometry for the Southern sky to a limiting magnitude of $V\sim 18$.
The SPM program makes use of the Yale Southern Observatory's double astrograph
at Cesco Observatory in the foothills of the Andes mountains in El Leoncito,
Argentina. This telescope consists of two 51-cm refractors,
designed for photography in the blue and yellow bands, respectively.
The first-epoch survey, taken between 1965 and 1974 was made on
glass photographic plates, exposed simultaneously in blue-yellow pairs
and always centered on the meridian. The plates' field of view (FOV)
extends over an area of $6.3^o \times 6.3^o$.
The sky south of $\delta=-17^o$ was observed in the first-epoch period.
Second-epoch SPM plate observations were begun in 1988. By the mid
1990's, with only a third of the second-epoch survey completed,
Kodak discontinued the production of the photographic plates.
In 2000, with funding from the NSF, a CCD camera system was installed on the
double astrograph to replace
the photographic plate holders. A PixelVision 4K$\times$4K CCD camera
($0.94^o\times 0.94^o$ FOV) was placed in the yellow
lens focal plane, and an Apogee Ap-8 1K$\times$1K ($0.37^o\times 0.37^o$ FOV)
was fitted in the blue focal plane. In 2004, the Apogee 1K camera was
upgraded to an Apogee Alta E42 2K$\times$2K ($0.42^o\times 0.42^o$ FOV)
with funds from the Argentine CONICET.
In order to achieve a limiting magnitude similar to the first-epoch
plate material, the CCD survey consists of 2-minute exposures
in both cameras.
In the past decade, catalogs of the SPM program covering various parts of the sky
have been published, as the second-epoch material became
available for its astrometric reduction. Catalogs SPM1 \citep{1998AJ....116.2556P},
SPM2 \citep{1999DDA....31.1004V}, and SPM3 \citep{2004AJ....127.3060G}
are based on photographic plates only, in both first and
second epochs. The plates were scanned either with the Yale PDS (using
input lists of selected objects) or the USNO Precision Measuring Machine (PMM)
for the whole plate. Also, two different
centering algorithms have been used to measure the image centers in the scans,
the Yale 2D-Gaussian fit from \cite{1983AJ.....88.1683L}, or the USNO circular
fit from \cite{2003AJ....125..984M}.
Since 2004, regular CCD observations have been carried out to finish the
second-epoch survey of the SPM program. By December 2008,
the survey was effectively completed for the sky south of $\delta=-20^o$.
Subsequently, the SPM4 catalog, based on all available plate
and CCD data, was completed in late 2009 and is currently available,
(Girard et al.~2010).
SPM4 includes $\sim$100 million objects south of $\delta=-20^o$,
brighter than $V\sim 18$. Many of the data-processing procedures,
software and protocols developed in this investigation were also used in the
construction of the SPM4 catalog.
\section{The Plate Data}\label{sc_plate}
\subsection{Observations}\label{ss_plateobs}
The SPM survey fields are on $5^o$ centers in declination and a maximum
of $5^o$ separation in right ascension, providing at least a full degree of
overlap between adjacent plates.
Each 17-inch $\times$ 17-inch plate covers an area of $6.3^o \times 6.3^o$
(55.1 "/mm plate scale) and consists of a
2-hour and a 2-minute offset exposure.
All observations were made with a wire grating over the objectives,
producing measurable diffraction images out to third order.
The grating constant is 3.8 magnitudes, thus, along with the offset
short exposure, effectively increasing the dynamic range of each plate
allowing measurement of external galaxies and bright Hipparcos stars.
Fields were observed simultaneously in blue and yellow passbands, and there
are some fields with repeated blue and/or yellow plates from the same epoch.
See \cite{2004AJ....127.3060G} for a more complete description of the SPM
plate material.
Table \ref{tab_plateslist} lists the SPM plates used in this investigation.
During the course of this research, it was found that some plates yielded unusually
deviant results. A visual examination of the suspect plates revealed that the stellar
images suffered from significant defects, possibly caused by poor guiding,
polar misalignment, or poor focus.
These plates, 0750B, 0751B, 1371B, 1357Y and 1373Y,
were therefore discarded for the research presented here. Coverage
in these areas was not affected, since only one of the two plates at a given epoch
per field affected was discarded.
Figure \ref{fig_plates} shows the distribution on the sky of the
plates used in this work. Twenty two (22) SPM regions were studied, of which
seven (7) have 2nd-epoch plates. For the areas with 2nd-epoch plate data, no 2nd-epoch
CCD observations were made.
\subsection{Astrometric Reduction and Photometric Calibration}\label{plate_red}
The SPM plates used in this investigation were scanned with the Precision Measuring Machine
(PMM) of the USNO Flagstaff Station. For more details about the PMM setup and operations see
\cite{2003AJ....125..984M}. In a collaboration between the USNO-Washington
and the Yale Astrometry Group, the StarScan reduction pipeline \citep{2008PASP..120..644Z} was modified
to analyze the PMM pixel data of the SPM plates to produce a list of detections,
image centers and photometric indices.
The astrometric and photometric reductions then proceeded as follows:
1) cross-identification of detections to an input catalog, including
2) recognition and identification of central and higher grating orders;
3) photometric calibration to obtain $BV$;
4) correction for Atmospheric Refraction;
5) correction for Magnitude Equation, which also combines grating-order systems;
6) transformation of short-exposure positions into the long-exposure system; and
7) astrometric solution into Tycho-2 to obtain $(\alpha,\delta)$.
The input catalog referenced in step 1) is a compilation of the following
external catalogs:
Hipparcos \citep{1997ESASP1200.....P},
Tycho-2 \citep{2000A&A...355L..27H},
UCAC2 \citep{2004AJ....127.3043Z},
2MASS point-source and extended-source \citep{2006AJ....131.1163S},
LEDA galaxies with DENIS measurements \citep{2005A&A...430..751P},
and the QSO catalog of \cite{2006A&A...455..773V}.
In order for an object to be included in our study, it must appear in one
or more of these listed catalogs.
A thorough explanation of all these procedures can be found in Girard et al. (2010).
After the above processing, one has positions $(x,y)$ properly calibrated into
a common system within each plate, with computed positional errors and
astronomical coordinates $(\alpha,\delta)$ on the ICRS, as realized by Tycho-2.
\subsection{Evaluation of the plate data}\label{ss_evaplate}
Well measured stars on the plates in all of the orders have positional errors
between 0.9 $\mu$m and 1.6 $\mu$m (50 mas and 90 mas).
These errors, which only assess measurement uncertainties, are consistent
with the precision expected for a good centering procedure, based on
previous experience with the SPM plates.
A single final position $(x,y)$ per star per plate is obtained, from the positional-error-weighted-average
of the available measurements. As expected, the error of the final position and
magnitude will depend on the number of grating orders contributing to their calculation.
If average measurements from different plates were later averaged to obtain
a final number, then other errors, random and systematic, would come into play,
and the error budget of this final result should include the additional sources
of uncertainty.
As with earlier SPM catalogs, an
approximate estimate of the relative completeness between the plate data and 2MASS,
can be made. Figure \ref{fig_completeness} shows the $V$ magnitude distribution of all stars
detected on a yellow plate, compared to the $V_{JK}$ magnitude distribution of all
2MASS point source stars in the same field. $V_{JK}=J+2.79(J-K)$ is an approximate empirical
transformation from 2MASS JHK to V determined by \cite{2004AJ....127.3060G}
and found to be valid for a relatively wide range of spectral types.
It can be seen that the SPM plates have a completeness similar to that
of 2MASS up to $V=17.5$ and a falloff after that.
In general, compared to previous SPM processing, these plate data have
significantly fewer false detections, and a better correction of systematics
in the detected positions associated with the scanning process.
\subsection{SPM 2nd-epoch plate data}\label{ss_platelet}
As explained in subsection \ref{ss_plateobs}, seven SPM fields have 2nd-epoch
plates. For this reason, no 2nd-epoch CCD observations were made at these locations.
In order to facilitate the managing of files and software, the 2nd-epoch plates
were divided in CCD-size fields, to emulate the overlap scheme of the
2nd-epoch CCD frames (See Section \ref{ccd_obs}). This way, all the 2nd-epoch material, regardless
of its type, would have a uniform format and structure, for programming purposes.
From the about 90 CCD pointings that usually cover one SPM field, half were
used to divide the blue plate and the other half to divide the yellow
plate, in such way that these yellow and blue fields overlap in a
similar way as do the real CCD frames.
\section{The CCD data}\label{sc_ccd}
\subsection{CCD cameras}\label{ccd_cam}
The CCD system consists of four cameras, one each for the blue- and yellow-optimized
lenses and two focus-sensor cameras, again, one for each lens.
The CCD camera for the Yellow telescope is a PixelVision camera with a Cryotiger
Chiller cooled (-85$^o$C) 4K $\times$ 4K Loral chip with 15 $\mu m$ pixels, which translates
into a pixel size of 0.''83 and a total area of $0^o.94 \times 0^o.94$ degrees.
The pixel size is well matched to the YSO site where the seeing conditions usually
yield an image FWHM of 2-3'', corresponding to 3-4 pixels per FWHM.
This is optimal sampling for the derivation of astrometric
image centers, based on SPM experience with digital image centering.
The unthinned and front-illuminated Loral chip is fitted with a fixed
Custom Scientific V-band filter.
The Blue telescope was first fitted with an Apogee AP-8 camera that utilized a
ther\-mo\-elec\-tri\-ca\-lly-cooled (-40$^o$C) and back-illuminated 1K $\times$ 1K Site chip
with 24 $\mu m$ pixels, which translated into a pixel size of 1.''32 and a total
area of 22.'57 $\times$ 22.'57. In May 2005, an upgrade was
made by replacing this camera with a new Apogee Alta E42 back-illuminated 2K $\times$ 2K chip,
with 13.5 $\mu m$ pixels, which correspond to a pixel size of 0.''74 and
a total area of 24.'8 $\times$ 24.'8.
Centered on the same field as the larger PV yellow camera, the purpose of
the blue CCD camera is to provide B-band CCD photometry for the stars that
fall into its FOV.
Data from the yellow PV camera were used for both astrometry and photometry,
while the blue Apogee and Alta cameras observations were used only
for the photometric calibration of the blue plates.
\subsection{Observations}\label{ccd_obs}
As a norm for the SPM, CCD observations are done always within $1^h30^m$
of the meridian, in 2-minute exposures,
with the wire grating placed so that the diffraction pattern is at about $45^o$
from the E-W line. Normally, an E-W orientation is ideal to avoid differential
color refraction effects within the diffraction pattern,
but in this case, a diagonal orientation prevents the saturated central-order
image of a bright star from spoiling its grating images by either row or
column bleeding. The CCD pointings conform to
a two-fold overlap coverage scheme for the PV camera, as shown in Figure \ref{fig_plates}.
About 90 PV CCD frames cover one single SPM field.
Only targets that did not have 2nd-epoch plate data were
observed with the CCD, except for a few special targets.
A total of 1310 CCD pointings were observed for this investigation. Each pointing
was planned to be observed only once, except for 90 of them
extending over a $6^o \times 6^o$ field around $(\alpha,\delta)=(3^h44^m33^s,-71^o40'18")$,
within the area delimited by the bold black line in Figure \ref{fig_plates}.
This area corresponds to the non-SPM VMC field,
for the Variable Stars in the Magellanic Clouds
study done from 1965 to 1968 by A. J. Wesselink,
and comprises seventy 60-minute exposure blue plates without the wire objective
grating, reaching a limiting magnitude of about 18.
Numerous repeated CCD observations were performed on this area
to provide suitable 2nd-epoch observations for this material.
The VMC plates were not used for this work because they require their own special
reduction, different from the one used in normal SPM plates, but the repeated CCD
observations were indeed used for this investigation. The VMC CCD targets have
at least 13 good observations each, with some of them having up to 20 good observations.
The criteria used to qualify an observed frame as acceptable are: $FWHM \leq$ 3."5 , limiting magnitude
$V_{lim}\geq 17$ and standard error $\leq$ 120 mas from an astrometric solution into the UCAC2.
If a frame fails any of these limits then it was taken again until it passed all of them.
Nonetheless, all frames regardless of quality are saved and processed,
and only later in the astrometric reduction are discarded, if they prove
to be too bad for any use. In total, 5422 CCD frames were processed
for this investigation.
\subsection{Astrometric Reduction and Photometric Calibration}\label{ccd_ast}
Data from all the CCD cameras went through the usual processing to calibrate
the flux detected by the electronics, for the zero charge of the chip
(bias), accumulated signal from the electronics dark current (dark) and
different response to light from each pixel (flat). Details of these
procedures can be found in Girard et al. (2010).
Image detection on the processed CCD frames is done using SExtractor
Version 2.4.4 \citep{1996A&AS..117..393B}, from which a preliminary centroid
and an aperture instrumental magnitude are read.
SExtractor centroids are then used as input positions
to compute more precise centers, based on the Yale 2D gaussian centering
algorithm \citep{1983AJ.....88.1683L}. Typically more than 90\%
of SExtractor detections are centered.
A significant reduction in the rms of the positions
for repeated observations of the PV frames was found when using the
Yale-based centers as compared to the SExtractor-based centroids.
The process of transforming the Yale-based centers $(x,y)$ and
SExtractor-based instrumental magnitudes $m_{inst}$ into calibrated
$(\alpha,\delta)$ and $BV$, for the CCD frames, follows a procedure similar
to that of the plates (Section \ref{plate_red}), with the
exception that the external catalog used for the astrometric reduction
of the CCDs is UCAC2 (Girard et al. 2010).
As a result of these procedures, all the detected positions $(x,y)$ are properly
calibrated within each CCD frame, with computed positional errors and equatorial
coordinates $(\alpha,\delta)$ in the ICRS, as realized by UCAC2.
\subsection{Evaluation of the CCD data}\label{ss_evaccd}
The single-image centering precision for well measured stars ($V\leq 15$) in the CCD frames
is 0.5 $\mu$m (25 $mas$), worsening for the faintest
stars where it reaches about 2 $\mu$m (100 $mas$). A single final position $(x,y)$ and magnitude
per star per CCD is obtained, from the positional-error-weighted-average
of the available measurements. The final positional and photometric errors
will depend on the various image orders contributing to the final value.
At this point, each star in each CCD frame has a
master ID identification, a position $(x,y)$ with errors, a calibrated
$B$ or $V$ magnitude and a UCAC2-based $(\alpha,\delta)$.
Because stars were identified in the CCD frames in the same way as
in the plates, similar charateristics regarding completeness
were expected when compared to 2MASS (See Figure \ref{fig_completeness}).
The CCD data show in general a completeness magnitude of about $V=18$.
\section{UCAC2 CCD-positions as supplement for the 2nd-epoch plate data}\label{sc_ucac2}
As seen in Figure \ref{fig_plates}, some fields on and adjacent to the Magellanic
Clouds have only photographic plates as 2nd-epoch material.
Given the lower quality of plate images compared to CCD
observations, the measured proper motions in these
plate-only fields, will have significantly larger errors than those
coming from the combination of plate and CCD data.
In an attempt to counter this and achieve a more homogeneous quality in the final measurements,
we decided to supplement the 2nd-epoch plate measures with epoch 2000 positions
from the UCAC2 Catalog \citep{2004AJ....127.3043Z}.
These are mostly based on CCD observations\footnote{Strictly speaking,
the supplemental UCAC2 epoch-2000 positions we use include up to a few years'
worth of UCAC2 proper motions. For the areas of sky in this study,
the UCAC2 CCD observations were made around 1998. Thus the positions
we employ as supplements are practically those of the UCAC CCD program.}
with the USNO 8 inch (0.2 m) Twin Astrograph from Cerro Tololo International Observatory in Chile.
The UCAC2 data were collected in such way that it mimics our 2nd-epoch fields.
The precision of the positions are 15 to 70 mas, depending on magnitude,
with claimed estimated systematic errors of 10 mas or below.
UCAC2 provides only crude magnitudes in a single nonstandard bandpass
$R_{UCAC2}$ between $V$ and $R$, and its limiting magnitude is about $R\approx 16$.
A significant number of faint stars in our 2nd-epoch SPM plates do not have a counterpart
in the UCAC2 catalog. This means we are only sampling stars about $V<16$ for these
SPM fields when using UCAC2 positions. UCAC2 completeness compared to 2MASS and SPM
data, can be seen in Figure \ref{fig_completeness}.
\section{Obtaining the proper motions}\label{sc_obtpm}
Given that a substantial part of the 2nd-epoch SPM material used in this work
comes from CCD frames, previous SPM procedures for obtaining proper motions
(used on 1st- and 2nd-epoch plates), could not be straightforwardly applied.
Since the CCD's FOV is about 40 times smaller than the plate's FOV, the number of reference
stars available to measure relative proper motions in each field is proportionally
smaller. A simple cut in magnitude, as in past SPM
reductions, would result in too few reference objects per CCD frame;
in particular, the extragalactic objects needed to transform relative proper motions
into absolute ones.
However, if we select reference stars belonging to some specific population of the Galaxy,
it is reasonable to assume that they have a mean absolute motion along
an extended area on the sky\footnote{Stars
behave like a collisionless system, all moving under the influence of
the same general background gravitational field, therefore we can expect
them to have a global smooth distribution in their velocities, with some scatter around
a mean value at any given location.} that can be parametrized as a smooth function
of $(\alpha,\delta)$. Moreover, within a CCD FOV, their mean motion has a
very small gradient, if any. It then becomes a matter of precisely quantifying,
over the whole field of view, the measured mean relative proper motion of all known
extragalactic objects, which is simply the reflex of the mean absolute proper motion
of the reference stars. Applying such a function to the
measured relative proper motions converts them to absolute.
The layout of the CCD fields, where substantial overlap exists (20\% to 50\%),
contains a wealth of information that can provide linkage
of the reference system across the observed region of the Clouds
that is limited only by measurement errors. A key point of this investigation
is therefore to find a procedure to utilize this large overlap to produce a
precise global reference system.
\subsection{Relative proper motions in the small CCD FOVs}\label{sc_relpm}
After all the plate and CCD data processing described above,
the following data are ready for the measurement of proper motions:
\begin{itemize}
\item 1st-epoch positions on the plates, which have been corrected for systematics,
as much as Tycho-2 precision and number of stars available allows.
\item 2nd-epoch positions on SPM CCD frames, SPM plates, or from UCAC2 .
CCD positions are mostly free of systematic errors. Plate positions have been corrected
for systematics, as much as Tycho-2 stars precision and number of stars allows.
\end{itemize}
Henceforth, we will refer to each of the 2nd-epoch CCD-size frames as a {\it brick},
regardless of the source of its data (SPM plate, SPM CCD or UCAC2).
For each 2nd-epoch brick, the corresponding 1st-epoch plate area was reprojected
onto a tangential plane centered on the brick. Then a quadratic solution was computed to
transform the reprojected 1st-epoch plate's $(x,y)$ into the 2nd-epoch brick's $(x,y)$,
forcing the chosen reference stars to have zero mean proper motion.
The quadratic terms in the solution were meant to model systematic errors,
either from the plate (uncorrected Optical Field Angle Distortion) or from Tycho-2 proper motion
systematics\footnote{Such systematics
were indeed found later on in some tests. The Tycho-2 Catalog, although an astrometric
catalog based on late epoch space-based data, has early epoch positions from
ground-based data that are known to suffer from significant magnitude
equation.} unavoidably propagated backwards in time into the computed 1st-epoch $(x,y)$.
These solutions yield measured relative proper motions in each brick.
The vast majority of the solutions only needed linear terms
and the typical standard error of the solutions varied from about 5 to 8 mas yr$^{-1}$,
which is dominated by the intrinsic proper motion dispersion of the reference stars.
The reference stars were chosen with the following criteria:
$1<V-J<1.5$, $0.25<J-K_s<0.65$ and $13<K_s<15$, which according to
\cite{2000ApJ...542..804N} isolates mostly G-M dwarfs in the Galaxy disk,
located between 0.4 to 1.6 kpc from us, with an estimated mean distance
of $\approx$ 650 pc. Their distribution in $V$ magnitude
ranges mostly from $V=15$ to $V=16.5$.
In general there are between 200 and 500 reference stars per field,
depending on the galactic latitude (our fields extend from $b\approx-50^o$ to
$b\approx-20^o$). The intrinsic proper
motion dispersion of the Galactic disk reference stars was seen to increase
with $|b|$, reflecting the changing kinematics of the Galaxy along it.
Despite the photometric cuts to select the reference stars,
contamination by LMC and SMC stars could not be avoided in the densest parts of the Clouds.
This forced us for the time being to restrict our investigation to those fields in
which we trust the relative proper motions, as being measured with respect to
bona fide Galactic foreground stars. These areas were defined as shown
in Figure \ref{fig_refmap}. Although a substantial number of Magellanic Clouds
stars were lost in this selection, on the other hand, confusion due to image crowding
at these locations render these fields useless anyway, due to the
risk of misidentifications.
From the initial 13880 bricks available, 12180 are in the non-contaminated
areas. After rejecting bricks from the discarded plates (Section \ref{ss_plateobs}),
10900 are left to build the catalog of proper motions, mostly outside the Magellanic Clouds.
In order to increase the number of LMC and SMC stars measured at the end,
contaminated fields that overlap with this non-contaminated catalog were later on directly
tied into it, and common stars had their proper motions averaged.
\subsection{Combining the proper motions}\label{sc_overlap}
Once relative proper motions with respect to the Galactic foreground stars
had been measured in the non-contaminated area, different approaches were tried to combine them
into a single global well-defined reference frame. Reference stars
were chosen hoping that their mean motion along the sky could be
described by a smooth function. This goal was indeed attained as confirmed
by the fact that the measured relative proper motion of all known extragalactic objects
were very precisely fit by a quadratic polynomial in $(\alpha,\delta)$.
At this point, applying this polynomial to the relative proper motions
to convert them to absolute, and then averaging all measurements, is
a way to combine all the information available per star.
But this would yield a catalog with a rather noisy
zero point as one moves along the sky, mostly due
to the real intrinsic proper motion dispersion of the reference stars and the fact
that two overlapping frames may not have the same reference stars.
Given two frames with about 50\% overlap between them, means
that both have 50\% of the reference stars in common, while
the other 50\% are different. Given two samples with $N$ data points
each, both with the same dispersion $\sigma$ and both
having 50\% of their points in common, it can be easily shown that
their individual mean values typically differ by $\sqrt{\frac{2}{N}}\sigma$.
In other words, we can expect that two overlapping frames
typically differ in their relative proper motion systems by $\approx 0.6$
mas yr$^{-1}$, for $N=220$ and $\sigma=6.5$ mas yr$^{-1}$.
This is too large for the level of precision that we want to obtain.
A variant of the so-called Block Adjustment solutions
\citep{1960AN....285..233E,1979AJ.....84.1775J,1981RMxAA...6..115S,
1988ApJ...334..465E,1988AJ.....96..409T} was considered to link
every frame's relative system into a global one.
Each frame, containing a sample of the whole population of reference
stars, realizes a local system that could deviate from the global system,
due to statistical, measurement and/or systematic errors. A linear function
per frame was considered sufficient to describe the difference between the frame
and the global system. Therefore three pa\-ra\-me\-ters per frame, per proper-motion component
($\mu_\alpha\cos\delta$ and $\mu_\delta$ are solved separately) need to be determined.
The coefficients of the quadratic polynomial that globally describes the mean motion
of the reference stars are determined as well. To
simultaneously solve several thousands frames,
we would need to invert an approximately $40,000\times40,000$ matrix
to get the parameters values and their errors.
For this project, such a scheme was deemed impractical at this time.
Nonetheless, ideas about using this approach over smaller areas first,
like the SPM fields, and then performing another block adjustment solution
to join these regions, are being adopted for future work.
A more practical approach was considered to bring each individual frame's
reference system {\it closer} to the global one.
A single frame's reference system is statistically more deviant
from the global system than is the system defined
by all stars, since the proper-motion reference stars are only a subset of all stars.
Using all stars in common between overlapping frames, we can adjust
each frame's proper motions to agree with the average of the surrounding fields.
This adjustment also helps to correct residual distortions, as they
are statistically smoothed out in the average frame.
To avoid frames drifting away from the global system as they are being
aligned to one another, all reference stars in the
field are explicitly assigned a relative proper motion of zero.
Once the adjustments are applied, new averages can be computed, and the
whole process is iterated until the adjustments converge to zero.
Before making these adjustments, we first checked to see if systematic
differences existed as a function of the magnitude.
A non-negligible number of frames exhibited systematic trends with magnitude when compared to
the average frame (See Figure \ref{fig_intoave}). In general, a linear function of the coordinates in the
field would take care of the geometrical distortion, but the magnitude equation required a smoothed-localized
median, which can trace the general trend better than any parametrized fit.
Therefore, each frame is first corrected for its differential magnitude
equation with respect to the average frame, and then we proceed to
correct for the distortion, following the iterative procedure
explained above. Each frame has typically about 1500 to 2000 stars, with some
of them having up to 5000 stars, to compute the adjustments.
With these improved relative proper motions, the quadratic
polynomial that describes the mean reflex proper motion of the extragalactic
objects was computed and used to transform the relative proper motions
into absolute ones. Polynomials for $\mu_\alpha\cos\delta$ and
$\mu_\delta$ as functions of $\alpha$ and $\delta$ were calculated separately,
using proper-motion-error-weighted least squares.
A total of 5351 external galaxies were used across the 450 sq-degree area and
the formal errors of the polynomials, computed at the center of LMC
and SMC, based on the full covariance matrix, amount to 0.03 mas yr$^{-1}$
and 0.06 mas yr$^{-1}$ respectively.
At this stage, a final absolute proper motion and its corresponding error are obtained
per star, from the error-weighted average of all the individual absolute proper motions
obtained for it. A total of 1337050 objects in the non-contaminated fields had
final proper motions, including a good number of LMC and SMC stars located in the
outskirts of these galaxies. For such reason, it is named the {\it outside catalog}.
In order to increase the number of LMC and SMC stars with measured absolute
proper motions, contaminated fields that overlapped with the outside catalog just
obtained, were directly tied into it by computing a linear solution
to correct their relative proper motions and put them into its absolute system.
A total of 678 additional fields were added with this procedure, eventually increasing
the number of LMC and SMC stars by about 30\% and 50\%, respectively.
\subsection{Zero point global correction of the Absolute Proper Motions}\label{sc_zerop}
Since we are now theoretically on the absolute reference frame defined by the external galaxies,
our catalog should be -within measurement errors- in the same reference frame system
of other known catalogs of absolute proper motion. When we checked
our measures $(\mu_\alpha\cos\delta,\mu_\delta)$ of 1356 Hipparcos stars
against the Hipparcos Catalogue, we found a significant
difference between the two, that amounts to
\begin{eqnarray}
\mu_{\alpha\cos\delta,Hipparcos}-\mu_{\alpha\cos\delta,This\; work} &=& -0.49 \pm 0.07 \;\;\mbox{mas yr}^{-1} \label{hipa_err}\\
\mu_{\delta,Hipparcos}-\mu_{\delta,This\; work} &=& -1.21 \pm 0.07 \;\;\mbox{mas yr}^{-1} \label{hipd_err}
\end{eqnarray}
The source of this systematic difference could be indeed in any (or both) of
the two catalogs. Although Hipparcos is the most accurate optical
astrometric catalog published so far, it has significant correlations between the astrometric
parameters (position, proper motion and parallax) of different stars,
when they are less than about 5 degrees apart on the sky,
and also between the astrometric parameters for a given
star, due to the special measurement principle of Hipparcos \citep{1997ESASP1200.....P}.
In fact, a new reduction by \cite{2007ASSL..350.....V} was performed
to correct some systematic correlations in the data. However no significant difference
was found between the old reference frame (used in this investigation)
and the new one.
On the other hand, our plate measurements of the galaxies are not error-free.
Tests were run to compare the final absolute proper motions if only galaxies
with 2nd-epoch CCD data, or all of them, were used to compute the quadratic
polynomials to transform relative proper motions into absolute.
The polynomial for $\mu_\alpha\cos\delta$ exhibited noticeable differences
around the SMC. This was not completely unexpected given that at that location,
four SPM fields have plate-only data, but in any case it points to the
fact that galaxies in the 2nd-epoch plates may introduce problems.
Hence, galaxies in the 1st-epoch plates may do the same.
As determined by \cite{1998AJ....115..855G}, the SPM plate material exhibits
magnitude equation for galaxies and for stars that differ in functional form.
However, the two could be brought into approximate agreement by adding an offset
of -0.7 to the magnitudes of galaxy images before calculating the magnitude-equation
correction as determined from stellar images. In the present study, we have
applied the same -0.7 mag offset to galaxy images for the purpose of magnitude-equation
correction only.
A comparison of the $V$ magnitudes for the galaxies in our work,
showed that plate photometry returned signficantly fainter magnitudes than in
CCD calibrations. Not suprisingly, this may have affected the magnitude equation
correction in the plates, since it was the calibrated plate photographic magnitude
that was used for such purpose. More surprising though, was to find that
Eq. \ref{hipa_err} showed a linear trend versus $\alpha$ with a slope
of about 15 $\mu$as yr$^{-1}$ per degree (a bit smaller when using only
galaxies with CCD data).
The LMC and SMC center of mass positions are separated by about $20^o$,
for which the trend above indicates a zero point shift of 0.3 mas yr$^{-1}$,
that could in principle be related to Hipparcos' systematics.
On the other hand, systematics in Eq. \ref{hipd_err}
versus $\delta$ were also seen, that look to be related to the plates location
and layout.
The investigation of the SPM plates magnitude equation done
by \cite{1998AJ....115..855G} found that magnitude equation terms\footnote{A polynomial
in $(X,Y,m)$ is used to describe the magnitude equation correction.} varied
more or less uniformly from 1st to 2nd-epoch plates, so we can expect
more or less uniform offsets with Hipparcos proper motions, if any residual uncorrected
magnitude equation were still present in the data.
Consequently, Eqs. \ref{hipa_err} and \ref{hipd_err} were applied to all the absolute proper motions, putting
our catalog on the system of the ICRS via Hipparcos. The inaccuracy of its zero point is dominated mostly by
Hipparcos' systematic error of 0.25 mas yr$^{-1}$, since the quadratic
polynomial, as defined by the external galaxies, was in general very accurate,
being within 0.1 mas yr$^{-1}$ of error for most of the 450 square-degree area studied.
\subsection{Final Catalogue of Proper Motions - Evaluation of Errors}
The final catalog of absolute proper motions at this point has
1,448,438 objects, with the following data listed: $\alpha, \delta, V$;
$V-J, J-K_s, H-K_s$ when available; absolute proper motions $\mu_\alpha\cos{\delta}, \mu_\delta$
and their formal errors $\epsilon_{\mu_\alpha\cos{\delta}},\epsilon_{\mu_\delta}$
in mas yr$^{-1}$, number of data points used, number of data points rejected
(outliers were rejected based on their normalized errors), a flag to
indicate Hipparcos, Tycho-2, 2MASS extended sources, confirmed LEDA Galaxies
and QSOs, a flag to indicate if the object is or is not a reference star,
and the 2mass ID when available.
The overall distribution of the stars in the catalog can be seen in Figure \ref{fig_catmap}.
In the final catalog of proper motions, stars with $V<12$ have
formal proper motion errors of about 0.5 mas yr$^{-1}$, and well measured stars with $12<V<15.5$, have
values that range from 0.5 to 1.3 mas yr$^{-1}$. These only reflect measurement errors,
as they are based on the positional errors and the epoch difference.
When proper motions for the same star measured in different
bricks are averaged together, statistical and
systematic deviations between the bricks must be added to get the true
error. The iterative method applied in Section \ref{sc_overlap} was designed
to reduce those deviations, but cannot make them zero.
A better way to determine the real proper motion uncertainties, is to compute
the standard deviation of the final error-weighted average proper motions.
Figure \ref{fig_caterrors} shows these scatter-based proper motion errors,
for a random sample of about 5\% the size of the whole catalog.
These errors can still be an underestimate of the real proper
motion uncertainty, because in the error-weighted average the data points
are not independent, two overlapping bricks reduced into the same plate
can produce (positively) correlated proper motions.
In this case, the true uncertainty is larger than the measured one.
The most reliable assessment of the proper motion errors
is obtained from a comparison with external catalogs.
The scatter observed in the differences between our proper motions and those
from Hipparcos, is the combined result of both catalogs' proper motion errors.
Given that Hipparcos errors are about 1 mas yr$^{-1}$,
the measured dispersion indicates that our real proper motion uncertainties are about 2.3 mas yr$^{-1}$, for
stars brighter than $V=10$. This coincides well with the scatter-based errors
in Figure \ref{fig_caterrors}, at the bright end.
The scatter in the proper motion of the LMC and SMC samples can
be used to estimate the proper motion uncertainties, at the mean magnitude
of the Clouds, since their intrinsic internal velocity dispersion\footnote{30
km s$^{-1}$ (0.13 mas yr$^{-1}$ at 50 kpc distance) or less in each \citep{1997CAS....29.....W}.}
makes a very minor contribution to the observed scatter. Results indicate that
our real proper motion uncertainties are about 3.8 mas yr$^{-1}$
for stars around $V=16.4$, entirely consistent with the scatter-based
uncertainties in Figure \ref{fig_caterrors}.
At the faint end, the dispersion in the proper motion of external galaxies
varies substantially, depending on whether they have CCD or plate 2nd-epoch data.
In the first case, the dispersion is 11 mas yr$^{-1}$
at a mean magnitude of $V=17.3$. In the second case, the dispersion is 21 mas yr$^{-1}$
at a mean magnitude of $V=18.5$. Since the plate photometry for the galaxies produced
systematically fainter magnitudes than the CCD photometry, the difference in
these two sets reflects also the difference in precision between (better) CCD and
(worse) plate measurements, and not just the increase of errors with magnitude.
At these magnitudes, our scatter-based proper motion
uncertainties are about 50\% below these values.
In general, the scatter-based proper motion errors are a rather good indicator
of the real uncertainties in the proper motions for stars brighter than
about V=16.5. More importantly,
all the above external estimates: 2.5, 3.8 and 11 mas yr$^{-1}$
at $V\approx 10, 16.4$ and 17.3, respectively,
are smaller in size than what was achieved in SPM3, clearly
showing the increased precision due to having 2nd-epoch CCD data.
\section{Proper Motion of the Magellanic Clouds}\label{sc_pmmcs}
\subsection{Selection of LMC and SMC dominated samples}\label{sc_pmmcs_sel}
A photometric selection was made to choose bona fide red giant LMC and
SMC stars, based on the analysis of the 2MASS LMC
infrared color magnitude diagram (CMD) of \cite{2000ApJ...542..804N} (their sample ``J'').
The photometric cuts applied, as seen in Figure \ref{fig_cmdclouds}, are:
\[
\begin{array}{lcccc}
\mbox{LMC :} & 1.1\leq J-K_s \leq 1.3 & \mbox{ and } & 9.5\leq K_s\leq 12 \\
\mbox{SMC :} & 1.0\leq J-K_s \leq 1.2 & \mbox{ and } & 10\leq K_s\leq 12.5
\end{array}
\]
Only stars with CCD 2nd-epoch data were selected, as stars whose proper motions
were based on plate data only showed a significanly higher dispersion and some visible
systematics. We were also forced to discard an area of high stellar density,
at $71^o\leq\alpha\leq 76^o$ and $-68^o\leq\delta\leq -71^o$,
with CCD data close to the LMC center, which consistently showed deviant results
probably caused by misidentifications.
3822 LMC and 964 SMC stars were selected, as seen in Figure \ref{fig_posclouds},
to measure the mean absolute proper motion of the Clouds. Bluer sequences of the CMDs
in Figure \ref{fig_cmdclouds} containing LMC/SMC stars
have signficant contribution from Milky Way stars and therefore were not considered.
The redder sequences of LMC/SMC
AGB stars have very faint $V\approx 17.5$ magnitudes,
consequently the proper motion errors are too large to be useful.
Also, given the magnitude-related problems in the plates,
it is desirable to have the smallest possible difference in brightness
between the reference stars and the Clouds' stars, and indeed, the
chosen samples overlap sufficiently in magnitude (See Figure \ref{fig_histmag}).
\subsection{Absolute proper motion of the LMC and SMC}\label{sc_pmmcs_lmc}
Probability plots \citep{hamaker1978} of the chosen samples yielded the
mean and dispersion values for the LMC and SMC proper motion listed in Table \ref{tab_motion}.
The errors quoted include: the formal error of the mean value
($\sigma/\sqrt{N_{stars}}$), the error of the quadratic
polynomial at the LMC and SMC centers,
transformation to Hipparcos errors (Eqs. \ref{hipa_err} and \ref{hipd_err}),
and the estimated Hipparcos systematic error (0.25 mas yr$^{-1}$).
As explained before, the error budget is dominated by Hipparcos systematics.
Table \ref{tab_compare} and Figures \ref{fig_lmc_compare} and \ref{fig_smc_compare}
summarize how our results compare with recent measurements of the proper motion of
the Magellanic Clouds. The error bars hinder a more precise conclusion
about the individual tangential velocities of the Clouds based on our data.
Nonetheless, the methodology used to measure the stellar proper motions
in our catalog permits us to make a rather precise measurement of the
proper motion of the SMC with respect to the LMC, as explained in more detail
in Section \ref{ss_cloudsrelpm}.
\subsubsection{Center-of-mass proper motion}\label{sc_rotdis}
The large extent of the Clouds and their non-negligible depth means that
all previous investigations, which measured proper motions on scattered small fields,
had to convert their measured values into a {\it center-of-mass proper motion}.
That is because a given space velocity at a fixed distance projects
differently on radial velocity and proper motion at different locations in the sky,
following the same principle of the Moving Cluster method and the Solar
Motion\footnote{One of the first attempts to estimate the
proper motion of the Magellanic Clouds was done by measuring gradients
in the radial velocity along them \citep{1977A&A....57..265F,1988ApJ...327..651M}}.
Besides, proper motion obviously scales with distance.
Additionally, internal rotation must also be taken into account for the LMC.
In the case of the SMC, such correction is deeemed unnecessary because
its stellar component is mostly supported by velocity dispersion \citep{2006AJ....131.2514H}.
The LMC's rotation curve is obtained from radial velocities of Carbon stars
\citep{2002AJ....124.2639V,2007ApJ...656L..61O} and yield widely accepted values
of $V_{rot,LMC}=50-60$ km s$^{-1}$. Nonetheless,
\cite{2008AJ....135.1024P} estimates its own $V_{rot,LMC}=120$ km s$^{-1}$,
based on the gradient of their measured proper motions along the radius in the LMC disk,
and use such value for the rotation correction.
Given such discrepant values, \cite{2009AJ....137.4339C} actually obtains
two final results for the LMC, \cite{2009AJ....137.4339C}-(1) refers to their final proper motion
when using $V_{rot,LMC}=50$ km s$^{-1}$, while
\cite{2009AJ....137.4339C}-(2) does so for $V_{rot,LMC}=120$ km s$^{-1}$.
For the latter, it must be noted that $\mu_\alpha\cos\delta$ deviates
noticeably from the other determinations in Table \ref{tab_compare}.
Altogether, the typical correction for perspective
effect for the LMC from both methods is about $\pm$ 0.2 mas yr$^{-1}$ and may rise
to about $\pm$ 0.5 mas yr$^{-1}$ for locations farther than $5^o$ from the LMC
center, running more or less in opposite directions at opposite locations on the Cloud.
Rotation effects for the LMC are usually less than 0.1 mas yr$^{-1}$.
Perspective effects for SMC are much smaller. For previous studies, these
corrections are necessary and in some cases yield quite different values
for the same fields, but in our work, given the spatial extent and symmetry of the data, the net effect
on the mean motion of the Clouds is very close to zero, and no correction is done.
\subsection{Relative proper motion of the SMC with respect to the LMC}\label{ss_cloudsrelpm}
As explained in Section \ref{sc_overlap}, we measured the mean motion of our
reference stars precisely all over our field of view,
in particular at the location of LMC and SMC
within 0.03 and 0.06 mas yr$^{-1}$,
respectively. Combined with the relative proper motion
of LMC and SMC stars with respect to these reference stars, we can
indeed measure the proper motion of the SMC with respect to that of
the LMC, with a higher precision, limited by the
error just quoted plus the formal error of the mean coming from
the number of stars and their measured scatter.
From Table \ref{tab_motion} and taking the errors quoted above into account,
it is straightforward to obtain the relative proper motion of the SMC
with respect to the LMC
\begin{eqnarray}
\Delta\mu_{\alpha\cos\delta}(SMC-LMC) & = & -0.91 \pm 0.16\;\; \mbox{mas yr}^{-1} \label{rel_mua}\\
\Delta\mu_{\delta}(SMC-LMC) & = & -1.49 \pm 0.15\;\; \mbox{mas yr}^{-1} \label{rel_mud}
\end{eqnarray}
These values cannot be transformed directly into a measurement
of the relative velocity between the Clouds.
Being at different locations in the sky means that the
planes of their tangential velocities are different as well, and the
necessary rotation and projections to measure the SMC velocity on the
LMC re\-fe\-ren\-ce frame does not allow us to obtain the relative space velocity
as merely a function of the relative proper motion between the Clouds.
But, we can use these values to obtain new independent measurements of the SMC's
absolute proper motion, based on existing measurements of the LMC's absolute proper motion
plus our precise relative proper motion from above.
Moreover, since all authors that directly measured the proper motion of the SMC
had previously measured the LMC's proper motion as well, we can
verify if their original SMC results are consistent with our relative measure.
Figure \ref{fig_smc_improv} shows the absolute proper-motion determination for the SMC
from Table \ref{tab_smcimprov} (made by combining our relative LMC-SMC motion with
absolute LMC motions from the literature), compared to the direct determinations
of the absolute proper motion of the SMC from Table \ref{tab_compare}.
Except for \cite{2008AJ....135.1024P} and \cite{2009AJ....137.4339C}-(2),
all published measurements of the SMC proper motions are consistent with our new ones.
These two works are the only ones that use $V_{rot,LMC}=120$ km s$^{-1}$.
It is worth noting that field L11 in the LMC from \cite{2008AJ....135.1024P} is also
in \cite{2009AJ....137.4339C}, but their measured proper motions are significantly
different, beyond the quoted errors. All this makes us suspect that the rather
small quoted proper motion errors in \cite{2008AJ....135.1024P} underestimate
their real uncertainties.
In summary, although our proper motions for the LMC and the SMC separately
are in agreement - within error bars - with
\cite{2006ApJ...638..772K,2006ApJ...652.1213K}, \cite{2008AJ....135.1024P} and
\cite{2009AJ....137.4339C}, our relative
proper motion of SMC with respect to LMC is consistent only
with \cite{2006ApJ...652.1213K,2006ApJ...638..772K} and \cite{2009AJ....137.4339C}-(1).
\subsection{The space motion of the clouds}\label{sc_pmmcs_spa}
The individual as well as relative space velocities of the Clouds,
as derived from our LMC and SMC proper motions,
are given in Table \ref{tab_vel_param}. The escape
velocity at the distance of the LMC is estimated to be
300-350 km s$^{-1}$, depending on the Galactic potential model
used (either a simple isothermal sphere or a more elaborate ``cosmologically inspired''
Navarro-Frank-White (NFW) dark matter profile \citep{1996MNRAS.278..191G,2002ApJ...573..597K,2008ApJ...684.1143X}.
Taken at face value, the galactocentric velocities in Table \ref{tab_vel_param}
indicate that the LMC is traveling at a speed that is very close to the
escape velocity, while the SMC is still below the escape velocity of the Galaxy.
Unfortunately, the uncertainties hinder a more definitive conclusion regarding their binding status.
From our proper motion measures, we determine that the SMC is moving at 89 $\pm$ 54 km s$^{-1}$
with respect to LMC. Our error bars do not
allow us to determine whether or not the Clouds are bound to each other, given that the escape
velocity from the LMC at the SMC location is about 90 km s$^{-1}$ (assuming a simple point-mass geometry,
a mass for the LMC of $2\times 10^{10} M_\odot$ and 23 kpc for the distance between the Clouds).
Yet, we can use our new SMC proper motions to obtain additional
determinations of the relative velocity between the Clouds, using the more precise
LMC proper motions available in the literature, and the other needed input parameters
listed in Table \ref{tab_vel_param}. Then we obtain more estimates of $||(U,V,W)_{SMC-LMC}||$,
which are listed in Table \ref{tab_relvel}.
For comparison, this table also lists the relative velocity originally quoted
by the references used.
It is important to be aware of how sensitive $||(U,V,W)_{SMC-LMC}||$ is for different values of
the LMC proper motion (See Figure \ref{fig_grid_dv}). It is
clear that even the smallest proper motion errors
quoted translate into a substantial relative velocity error between the Clouds.
On top of that, the measured values are close enough to the escape velocity
of the SMC with respect to the LMC, such that any conclusion regarding the binarity of
the Clouds is still far from decided. The same situation applies to the individual
space velocities of the Clouds, the binding status of the
Clouds to the Milky Way is also extremely sensitive to the individual proper motions.
Thus, although our individual absolute proper motions for the Clouds cannot
tell us much about their orbits, our measured relative proper motion between
LMC and SMC allowed us to identify the best self-consistent measurements of the proper motion
of both clouds, from those authors that had measured both galaxies with
claimed very good accuracies. Our measured space velocity has also helped
to check which results were more consistent with that value,
since it is quite sensitive to the proper motions used.
\section{Implications of our results to the dynamics of the Magellanic System}\label{sc_implications}
The most prevalent scenario for the orbit of the Magellanic Clouds, before the
HST proper motion results, favored them as a binary system in a bound
orbit around the Milky Way. Dynamical simulations were performed
under the presumption that the Clouds were bound to each other,
due to the existence of the common HI envelope surrounding them \citep{1963AuJPh..16..570H}.
Crude timing estimates suggested that for its creation and survival, a long time of
shared orbits was needed. The extent of the Magellanic Stream implied that the Clouds
had undergone multiple orbits gravitationally bound to the Milky Way and to each other.
The orbits of the Clouds are naturally a key ingredient in such dynamical modeling.
Once an orbit is chosen, then a full simulation of the Clouds themselves is run.
The main goal has been to reproduce as much as possible the position and
radial velocity of the Magellanic Stream,
though significant validation is obtained if the Leading Arm,
the Magellanic Bridge, and the distorted structure of the Clouds,
are also replicated. Another ``condition'' imposed is that the
Clouds share the same orbital plane with the Magellanic Stream,
since the latter runs more or less on a great circle on the sky.
Depending on the particular questions under study, Cloud model investigators
use different approaches. Either one or both Clouds are considered,
using collisionless (stars) and/or collisional (gas) particles.
The Cloud(s) is(are) made of one or many particles,
which can be massless test particles under a given potential, or
a conglomerate of self-gravitating particles or even
``sticky'' self-gravitating particles (to model hydrodynamic
processes).
Overall, two competing scenarios have been systematically studied
and have had some degree of success in the modeling of the mentioned
structures: the tidal model and the ram-pressure model. In the first scenario,
the Galaxy extracted a tidal plume from the LMC and/or the SMC, which gave
origin to the Magellanic Stream, in a previous close encounter of the three bodies
about 1.5 Gyr ago. A most recent encounter 200 Myr ago created the
Magellanic Bridge. The second scenario proposes that the Stream and the Leading
Arm consist of material that has been ram-pressure-stripped from the LMC
(and SMC), during its last passage through the extended ionized Halo of the Galaxy,
about 500 Myrs ago.
A complementary scenario to extract substantial amounts of gas and no stars from the Clouds,
the blowout model, has been proposed by \cite{2008ApJ...679..432N}. The main hypothesis
is that star formation outflow in the leading edge of the LMC has been blowing out or puffing up the gas over
the past 2 Gyr, making it easier for ram pressure and tidal forces to strip it off.
The first dynamical simulation of the Magellanic System
\citep{1980PASJ...32..581M} was done knowing only the radial
velocity of the Clouds, and very little about the mass of the Milky Way
at the Clouds' distances. Some dynamically permissible parameters\footnote{
By specifying the orbital inclination of the LMC with respect to the Galactic
Plane and its perigalactic distance, the equations of energy and angular momentum
conservation may be solved to yield the three components of the LMC space
velocity.} were assumed in order to
have all the input parameters necessary to compute an orbit.
Their models required the inclusion of a massive halo (an idea that was just starting to
being accepted then) in order to reproduce the highly negative radial velocities observed
in the tip of the Magellanic Stream. Conversely, assuming that the LMC was in a bound orbit
provided estimates consistent with a massive Galactic halo.
In general, all the simulations for the Magellanic Clouds
\citep{1982MNRAS.198..707L,1994MNRAS.266..567G,1994MNRAS.270..209M,1994A&A...291..743H,
1995ApJ...439..652L,1996MNRAS.278..191G,2003MNRAS.339.1135Y,2005MNRAS.363..509M,
2005MNRAS.356..680B,2006MNRAS.371..108C,2007ApJ...668..949B}
are based on the backwards integration of the equations of motion, first
applied by \cite{1980PASJ...32..581M}. Most papers consider an isothermal
sphere with a given constant rotational velocity at a large galactocentric
distance, while three of the most recent calculations use NFW dark matter halo profiles.
Not surprisingly, the mass (profile and amount) of the Galaxy is an important
source of uncertainty in the orbital models. The masses of the Clouds
are as well another source of error, their current distorted state makes
any dynamical estimate of the mass a difficult task.
Given the large mass of the Clouds, dynamical friction\footnote{The ``retardation'' of a moving object
when it passes through a region with non-vanishing mass density,
caused by its gravitational interaction with the particles of that
continuous mass \citep{1943ApJ....97..255C}.}
can significantly reduce their perigalacticon distance as they move through the
Galactic Halo. N-body simulations of self-gravitating particles
naturally consider it by default, while in other cases, it is accounted for by
using an analytical expression given by \cite{1987gady.book.....B}.
Recent studies \citep{2003ApJ...582..196H,2005A&A...431..861J}
have found though, that the latter tends to circularize orbits to
excess when compared to equivalent N-body simulations, thus
some simulations have scaled down its effects.
The important point to consider here is that a high orbital eccentricity,
or equivalently a high transverse motion with respect to the Galaxy,
is needed to lead to the formation of the high-velocity Magellanic Stream.
Therefore, the initial conditions of the
Clouds' orbits must be such that even after the effects of dynamical friction, orbital
eccentricity is high enough for the Stream to be formed.
Interestingly enough, \cite{1995ApJ...439..652L} had pointed out that
a hyperbolic encounter of the Clouds with the Galaxy, in which
they are passing by for the first (and only) time, could lead to the
tidal stripping of gas segments that would later infall and trail
rapidly behind. Such a model was discarded then, on the basis that the
Clouds were presumed bound to the Galaxy.
Another source of dynamical friction, the LMC dark matter halo,
is considered as well in the calculations of the Magellanic orbits
\citep{2005MNRAS.356..680B}. Its effects are quite important on the binarity of the galaxy pair,
exerting a significant frictional drag on the SMC when it penetrates
the LMC halo. If they get close enough, they would merge quickly, and since
this has not occurred yet, it implies either that this force is
negligible or that the Clouds became bound to each other relatively
recently \citep{1994MNRAS.266..567G}. \cite{2005MNRAS.356..680B}
also found that under these conditions, the Clouds cannot keep their
binary status for more than $\approx$5 Gyr in the past.
This opens the possibility of the Clouds being coupled only recently.
Curiously, the very first dynamical model
of the Magellanic System \citep{1980PASJ...32..581M} had to make an
extensive search for the binary state that could produce the Magellanic
Stream. The choice of tangential velocities for SMC had to be
so specific, that they could - theoretically -
predict it within $\pm$ 5 km s$^{-1}$. This prompted
them and others to make the first inferred estimates of the proper motion of the
Clouds. This early model though, did not include a dark matter halo for the LMC.
A similar situation was faced later on by \cite{1994A&A...291..743H},
who explain that if the perturbing forces acting on the LMC and/or
SMC have to be small enough to leave the binary system intact while
simultaneously producing long streams similar in shape to the Stream,
then the evolution time of these streams has to be rather long and very special,
and properly chosen initial conditions for the test particles
in the simulations have to be adopted.
One point in which all orbital models agree, is that
the Clouds had a recent encounter, sometime between 200 to 500 Myr ago.
In fact, several models also found that the binarity
of the Clouds, regardless of how long it has been in place,
was most probably broken at this last collision, which happened
very close to their perigalacticton distance. In other words,
the Milky Way's powerful gravitational tide has disrupted
the pair.
In general, searching for the appropriate binary-bound orbits for the
Clouds that can later be used to reproduce the Magellanic System,
has been a difficult fine tuning task. The challenge is now even
harder, since the most recent HST measurements of the LMC and
SMC proper motion \citep{2006ApJ...638..772K,2006ApJ...652.1213K,2008AJ....135.1024P}
seem to indicate that the Clouds are traveling too fast to have ever been
bound to the Milky Way.
Numerical simulations of the LMC orbit by \cite{2007ApJ...668..949B},
based on these new numbers, suggest that the Large Cloud is ``plunging'' in a highly eccentric
parabolic orbit, on its first passage about the Milky Way.
At such speed, dynamical friction is negligible,
but the choice of the Galactic potential (isothermal sphere
vs. NFW) introduces dramatic changes in the orbital history of the LMC.
In an isothermal sphere, the LMC has indeed a bound orbit, although
with an increased period and apogalacticon distance compared to
previous models. In an NFW profile, the ``best case scenario''
of a bound orbit has a period of about a Hubble time, and reaches
an apogalacticon distance of $\approx$ 550 kpc.
Despite such 3D differences, their projected orbits on the sky
were in good agreement, so their predicted location compared to
the Magellanic Stream's great circle, could not be used to distinguish
Milky Way mass profile models. More importantly though, in both
cases the projected orbit did not trace the Magellanic Stream,
deviating from it by about 10$^o$ on the sky.
Adding the SMC into the calculations did not reduce the disagreement,
nor using the weighted average of all pre-HST proper motions.
\cite{2007ApJ...668..949B} argues then that the usual criteria to validate
a dynamical model of the Magellanic System, its ability to match an
orbit with the Stream, is no longer acceptable.
The conclusion of \cite{2007ApJ...668..949B}, that the Clouds are on their
{\it first} passage relies heavily on the large value for $\mu_\alpha\cos\delta$ that was measured
with HST for the LMC. \cite{1995ApJ...439..652L} and others have in fact argued
that it is hard to explain how a bound LMC could have mantained
a high angular momentum perpendicular to the Galaxy's rotation axis for so long.
\cite{1989MNRAS.240..195R}, \cite{1992ApJ...386..101S} and \cite{1994AJ....107.2055B} have suggested
that an early interaction with M31 could be the source of such a high
tangential velocity. In any case, such a condition is easier to understand
with an LMC not bound to the Milky Way. In addition, the Clouds are
the only gas-rich dwarf galaxies at small galactrocentric distances
\citep{2006AJ....132.1571V}, different from the rest in the
Local Group. All these reasons are used by \cite{2007ApJ...668..949B} to support the case
of an unbound LMC.
Notably, all LMC proper motion results after \cite{2006ApJ...638..772K}, including
this work, have produced lower values of $\mu_\alpha\cos\delta$. Therefore,
a bound orbit with a long period and a large apogalacticon
distance is still a scenario compatible with the most recent results. As for
the binarity of the Clouds, a recent period of joint orbits could be enough
to explain the common features between the Clouds, that is the HI envelope,
the Bridge and even the Stream, and can account for the different star formation
history and chemical evolution in each Cloud, that point to
a separate origin and place of birth.
To conclude, the search for a realistic orbit of the Magellanic Clouds
is far from over. The space velocities obtained in this investigation are supportive
of a scenario in which the Magellanic Clouds are possibly currently unbound from each other,
with the LMC traveling at a velocity that is high enough to make
it nearly unbound to the Galaxy. But having nowadays a much more precise measurement
of the proper motion of the Clouds, has not facilitated our understanding
of their dynamics and has instead opened new questions and placed
all the constructed models in doubt.
\section{Conclusions and Future Work}\label{sc_end}
A catalog of absolute proper motions containing 1,448,438 objects has been
obtained from SPM material, supplemented with UCAC2 data. The catalog covers
an estimated area of 450 square degrees except for the inner regions of LMC, SMC
and 47 Tucanae, where the high stellar density made it impossible to obtain accurate
cross-identification of the stars.
Samples of 3822 LMC stars and 964 stars were selected from the catalog to measure
the mean proper motion of the Magellanic Clouds. The results obtained are:
\[
(\mu_\alpha\cos\delta,\mu_\delta)_{LMC}=(+1.89,+0.39)\pm (0.27,0.27)\;\;\mbox{mas yr}^{-1}
\]
\[
(\mu_\alpha\cos\delta,\mu_\delta)_{SMC}=(+0.98,-1.01)\pm (0.30,0.29)\;\;\mbox{mas yr}^{-1}
\]
Our much more precise relative proper motions with respect to the photometrically
selected Galactic Disk dwarf stars, enabled us to obtain the proper motion of the SMC with
respect to the LMC, with significantly smaller uncertainties:
\[
(\mu_{\alpha\cos\delta},\mu_\delta)_{SMC-LMC} = (-0.91,-1.49) \pm (0.16,0.15)\;\;\mbox{mas yr}^{-1}
\]
This was used to obtain new independent and more precise proper motions
for the SMC, based on the more accurate LMC proper motions of other authors.
It was also used to confirm if their separate measurements of the SMC
proper motion were consistent or not with our results.
After a comparison in an absolute and relative sense with previous proper motion results,
followed by a discussion of the orbital models of the Magellanic Clouds based on those results,
we conclude that our proper motions are compatible with the LMC and SMC being born and formed
as separate entities, which later joined in a temporary binary state for
the past few Gigayears, being recently disrupted by the Milky Way in their
most recent perigalaticon passage about 200 Myr ago. The Clouds orbits are
{\it marginally} bound to the Milky Way, possibly following a very elongated
but still periodic orbit around the Galaxy.
The search for a realistic orbit of the Magellanic Clouds
is far from over. Having (formally) very accurate and precise space-based
proper motions for the Clouds, has not facilitated our understanding
of their dynamics but has, instead, opened new questions and has placed
all dynamic scenarios of the Magellanic System in doubt. As of today,
it is still unclear if the Magellanic Stream and the Leading Arm are caused
mostly by a tidal interaction or are the result of the ram-pressure of
the Galactic Halo on the gas of the Clouds.
Given the inherent difficulties in
measuring an accurate proper motion for the Magellanic Clouds,
the obvious dangers that systematic errors pose in those
measurements and the fact that the dynamical models of the Magellanic System
are extremely sensitive to small variations in the proper motions
of the Clouds, we believe that we are not yet in the position
of considering them known parameters in the orbital calculation.
But we are getting closer.
\subsection{Future work}
From the very begining, this investigation of the proper motion of
the Magellanic Clouds was known to be restricted by the conditions
and characteristics of the SPM material.
We have proven that even under those constraints, these data are able
to produce independent significant results on the proper motion of nearby
dwarf galaxies. Thus, there is certainly room for improvement.
Second epoch CCD data over the whole field are needed,
especially in the inner areas of the Clouds. Since the definition of an adequate uncontaminated relative
reference frame in those areas is difficult, it is also necessary to devise
an adequate proper motion reduction method, to precisely link these
fields into the general global relative reference frame. A scaled-down
version of the block adjustment method could be put in place here, in which
plate-size fields of view are assembled first, and then another solution is
run to paste those into the larger global system.
Since the dominant limiting factor in terms of precision is the plate measurement
errors, additional improvement can be achieved by re-scanning the first epoch plates
used in this work with the Yale PDS, which yields 2 times
more precise positions than the currently measured ones. As
PDS scanning of a full plate is very time-consuming, a subset of
properly pre-selected objects should be measured.
Although additional HST follow up observations are already planned to improve
the space-based proper motions, and other research teams are still working on
additional CCD ground-based measurements, these studies are still limited since
they must correct their observed proper motions into a center-of-mass value.
Therefore, it is still worth trying to improve our results, since they offer a
wide field extended coverage of the Magellanic Clouds.
Another future work being considered, is to search in the Intercloud region
for coherent structures in the relative proper motion space,
to identify stars whose motion is directed towards the LMC. This will
require to find what possibly is a small number of stars spread over
an extended area that share systematic proper motions.
\acknowledgements
Vieira would like to thank Dr. Burt Jones for being the
external reader of her PhD thesis, on which this manuscript is based.
We would like to thank the many former and present members of the Cesco Observatory,
Universidad Nacional de San Juan and Yale Southern Observatory, who have
contributed to the SPM program over the years.
Vieira would also like to thank the National Science Foundation
(grants AST04-07292, AST04-07293, AST09-08996) and the Yale
Astronomy Dept for their financial support during her graduate career.
She and the other authors are grateful to the NSF and to Yale University
for support of the SPM and this research in particular throughout the many
years required to complete the SPM.
This publication makes use of data products from the Two Micron All Sky Survey, which
is a joint project of the University of Massachusetts and the Infrared Processing and
Analysis Center/California Institute of Technology, funded by the National Aeronautics
and Space Administration and the National Science Foundation.
\bibliographystyle{aa}
|
{
"timestamp": "2010-10-04T20:13:16",
"yymm": "1009",
"arxiv_id": "1009.4218",
"language": "en",
"url": "https://arxiv.org/abs/1009.4218"
}
|
\subsection*{Acknowledgements}
The first author was supported in part by NSF Grant No. DMS-0805206 and in part by EPSRC Grant No. EP/D073626/2, and is currently supported by ARC Grant No. DP110100440. The second author is supported in part by NSF Grant No. DMS-0905891.
We thank Martin Bridson and Pierre-Emmanuel Caprace for helpful conversations.
\section{Right-angled buildings}\label{s:rabs}
In this section we recall the basic definitions and some examples for right-angled buildings. We mostly follow Davis \cite{D}, in particular Section 12.2 and Example 18.1.10. See also \cite[Sections 1.2--1.4]{KT}.
Let $(W,S)$ be a right-angled Coxeter system. That is, $$W = \langle S \mid (st)^{m_{st}} = 1\rangle$$ where $m_{ss} = 1$ for all $s \in S$, and $m_{st} \in \{ 2,\infty \}$ for all $s, t \in S$ with $s \neq t$.
We will discuss the following examples:
\begin{itemize}
\item $W_1 = \langle s, t \mid s^2 = t^2 = 1\rangle \cong D_\infty$, the infinite dihedral group;
\item $W_2 = \langle r, s, t \mid r^2 = s^2 = t^2 = (rs)^2 = 1 \rangle \cong (C_2 \times C_2)* C_2$, where $C_2$ is the cyclic group of order $2$; and
\item The Coxeter group $W_3$ generated by the set of reflections $S$ in the sides of a right-angled hyperbolic $p$--gon, $p \geq 5$. That is, \[ W_3 = \langle s_1, \ldots, s_p \mid s_i^2 = (s_i s_{i+1})^2 = 1 \rangle \] with cyclic indexing.
\end{itemize}
Fix $(q_s)_{s \in S}$ a family of integers with $q_s \geq 2$. Given any family of groups $(H_s)_{s \in S}$ with $|H_s| = q_s$, let $H$ be the quotient of the free product of the $(H_s)_{s \in S}$ by the normal subgroup generated by the commutators $\left\{ [h_s,h_t] : h_s \in H_s, h_t \in H_t, m_{st} = 2\right\}$.
Now let $X$ be the piecewise Euclidean CAT(0) geometric realization of the chamber system $\Phi=\Phi\left(H,\{1\},(H_s)_{s \in S}\right)$. Then $X$ is a locally finite, regular right-angled building, with chamber set $\Ch(X)$ in bijection with the elements of the group $H$.
Let $\delta_W\co\Ch(X) \times \Ch(X) \to W$ be the $W$--valued distance function and let $l_S\co W \to \mathbb{N}$ be word length with respect to the generating set $S$. Denote by $d_W\co \Ch(X) \times \Ch(X) \to \mathbb{N}$ the \emph{gallery distance} $l_S \circ \delta_W$. That is, for two chambers $\phi$ and $\phi'$ of $X$, $d_W(\phi,\phi')$ is the length of a minimal gallery from $\phi$ to $\phi'$.
Suppose that $\phi$ and $\phi'$ are $s$--adjacent chambers, for some $s \in S$. That is, $\delta_W(\phi,\phi') = s$. The intersection $\phi \cap \phi'$ is called an \emph{$s$--panel}. By definition, since $X$ is regular, each $s$--panel is contained in $q_s$ distinct chambers. For distinct $s, t \in S$, the $s$--panel and $t$--panel of any chamber $\phi$ of $X$ have nonempty intersection if and only if $m_{st} = 2$. Each $s$--panel of $X$ is reduced to a vertex if and only if $m_{st} = \infty$ for all $t \in S - \{ s \}$.
For the examples $W_1$, $W_2$, and $W_3$ above, respectively:
\begin{itemize}
\item The building $X_1$ is a tree with each chamber an edge, each $s$--panel a vertex of valence $q_s$, and each $t$--panel a vertex of valence $q_t$. That is, $X_1$ is the $(q_s,q_t)$--biregular tree. The apartments of $X_1$ are bi-infinite rays in this tree.
\item The building $X_2$ has chambers and apartments as shown in \fullref{f:RAB_example2} below. The $r$-- and $s$--panels are $1$--dimensional and the $t$--panels are vertices.
\begin{figure}[ht]
\begin{center}
\scalebox{0.5}{\includegraphics{RAB_example2}}
\caption{A chamber (on the left) and part of an apartment (on the right) for the building $X_2$.}
\label{f:RAB_example2}
\end{center}
\end{figure}
\item The building $X_3$ has chambers $p$--gons and $s$--panels the edges of these $p$--gons. If $q_s = q \geq 2$ for all $s \in S$, then each $s$--panel is contained in $q$ chambers, and $X_3$, equipped with the obvious piecewise hyperbolic metric, is Bourdon's building $I_{p,q}$.
\end{itemize}
\section{Tree-walls}\label{s:tree-walls}
We now generalize the notion of tree-wall due to Bourdon \cite{B}. We will use basic facts about buildings, found in, for example, Davis \cite{D}. Our main results concerning tree-walls are \fullref{c:tree wall trichotomy} below, which describes three possibilities for tree-walls, and \fullref{p:tree-wall} below, which generalizes the separation property 2.4.A(ii) of \cite{B}.
Let $X$ be as in \fullref{s:rabs} above and let $s \in S$. As in \cite[Section 2.4.A]{B}, we define two $s$--panels of $X$ to be \emph{equivalent} if they are contained in a common wall of type $s$ in some apartment of $X$. A \emph{tree-wall of type $s$} is then an equivalence class under this relation.
We note that in order for walls and thus tree-walls to have a well-defined type, it is necessary only that all finite $m_{st}$, for $s \neq t$, be even. Tree-walls could thus be defined for buildings of type any even Coxeter system, and they would have properties similar to those below. We will however only explicitly consider the right-angled case.
Let ${\mathcal T}$ be a tree-wall of $X$, of type $s$. We define a chamber $\phi$ of $X$ to be \emph{epicormic at ${\mathcal T}$} if the $s$--panel of $\phi$ is contained in ${\mathcal T}$, and we say that a gallery $\alpha = (\phi_0, \ldots, \phi_n)$ \emph{crosses ${\mathcal T}$} if, for some $0 \leq i < n$, the chambers $\phi_i$ and $\phi_{i + 1}$ are epicormic at ${\mathcal T}$.
By the definition of tree-wall, if $\phi \in \Ch(X)$ is epicormic at ${\mathcal T}$ and $\phi' \in \Ch(X)$ is $t$--adjacent to $\phi$ with $t \neq s$, then $\phi'$ is epicormic at ${\mathcal T}$ if and only if $m_{st} = 2$.
Let $s^\perp := \{ t \in S \mid m_{st} = 2\}$ and denote by $\langle s^\perp \rangle$ the subgroup of $W$ generated by the elements of $s^\perp$. If $s^\perp$ is empty then by convention, $\langle s^\perp \rangle$ is trivial. For the examples in \fullref{s:rabs} above:
\begin{itemize}
\item in $W_1$, both $\langle s^\perp \rangle$ and $\langle t^\perp \rangle$ are trivial;
\item in $W_2$, $\langle r^\perp \rangle = \langle s \rangle \cong C_2$ and $\langle s^\perp \rangle = \langle r \rangle \cong C_2$, while $\langle t^\perp \rangle$ is trivial; and
\item in $W_3$, $\langle s_i^\perp \rangle = \langle s_{i-1}, s_{i+1} \rangle \cong D_\infty$ for each $1 \leq i \leq p$.
\end{itemize}
\begin{lemma}\label{l:sperp epicormic} Let ${\mathcal T}$ be a tree-wall of $X$ of type $s$.
Let $\phi$ be a chamber which is epicormic at ${\mathcal T}$ and let $A$ be any apartment containing $\phi$.
\begin{enumerate}
\item\label{i:wall separates} The intersection ${\mathcal T} \cap A$ is a wall of $A$, hence separates $A$.
\item\label{i:word in s perp} There is a bijection between the elements of the group $\langle s^\perp \rangle$ and the set of chambers of $A$ which are epicormic at ${\mathcal T}$ and in the same component of $A - {\mathcal T} \cap A$ as $\phi$.
\end{enumerate}
\end{lemma}
\begin{proof}
Part \eqref{i:wall separates} is immediate from the definition of tree-wall. For Part \eqref{i:word in s perp}, let $w \in \langle s^\perp \rangle$ and let $\psi=\psi_w$ be the unique chamber of $A$ such that $\delta_W(\phi,\psi) = w$. We claim that $\psi$ is epicormic at ${\mathcal T}$ and in the same component of $A - {\mathcal T} \cap A$ as $\phi$.
For this, let $s_1 \cdots s_n$ be a reduced expression for $w$ and let $\alpha = (\phi_0,\ldots,\phi_n)$ be the minimal gallery from $\phi = \phi_0$ to $\psi=\phi_n$ of type $(s_1,\ldots,s_n)$. Since $w$ is in $\langle s^\perp \rangle$, we have $m_{s_i s} = 2$ for $1 \leq i \leq n$. Hence by induction each $\phi_i$ is epicormic at ${\mathcal T}$, and so $\psi = \phi_n$ is epicormic at ${\mathcal T}$. Moreover, since none of the $s_i$ are equal to $s$, the gallery $\alpha$ does not cross ${\mathcal T}$. Thus $\psi=\psi_w$ is in the same component of $A - {\mathcal T} \cap A$ as $\phi$.
It follows that $w \mapsto \psi_w$ is a well-defined, injective map from $\langle s^\perp \rangle$ to the set of chambers of $A$ which are epicormic at ${\mathcal T}$ and in the same component of $A - {\mathcal T} \cap A$ as $\phi$. To complete the proof, we will show that this map is surjective. So let $\psi$ be a chamber of $A$ which is epicormic at ${\mathcal T}$ and in the same component of $A - {\mathcal T} \cap A$ as $\phi$, and let $w = \delta_W(\phi,\psi)$.
If $\langle s^\perp \rangle$ is trivial then $\psi = \phi$ and $w = 1$, and we are done. Next suppose that the chambers $\phi$ and $\psi$ are $t$--adjacent, for some $t \in S$. Since both $\phi$ and $\psi$ are epicormic at ${\mathcal T}$, either $t = s$ or $m_{st} = 2$. But $\psi$ is in the same component of $A - {\mathcal T} \cap A$ as $\phi$, so $t \neq s$, hence $w = t$ is in $\langle s^\perp \rangle$ as required. If $\langle s^\perp \rangle$ is finite, then finitely many applications of this argument will finish the proof. If $\langle s^\perp \rangle$ is infinite, we have established the base case of an induction on $n=l_S(w)$.
For the inductive step, let $s_1 \cdots s_n$ be a reduced expression for $w$ and let $\alpha = (\phi_0,\ldots,\phi_n)$ be the minimal gallery from $\phi = \phi_0$ to $\psi=\phi_n$ of type $(s_1,\ldots,s_n)$. Since $\phi$ and $\psi$ are in the same component of $A - {\mathcal T} \cap A$ and $\alpha$ is minimal, the gallery $\alpha$ does not cross ${\mathcal T}$.
We claim that $s_n$ is in $s^\perp$. First note that $s_n \neq s$ since $\alpha$ does not cross ${\mathcal T}$ and $\psi = \phi_n$ is epicormic at ${\mathcal T}$. Now denote by ${\mathcal T}_n$ the tree-wall of $X$ containing the $s_n$--panel $\phi_{n-1} \cap \phi_n$. Since $\alpha$ is minimal and crosses ${\mathcal T}_n$, the chambers $\phi = \phi_0$ and $\psi = \phi_n$ are separated by the wall ${\mathcal T}_n \cap A$. Thus the $s$--panel of $\phi$ and the $s$--panel of $\psi$ are separated by ${\mathcal T}_n \cap A$. As the $s$--panels of both $\phi$ and $\psi$ are in the wall ${\mathcal T} \cap A$, it follows that the walls ${\mathcal T}_n \cap A$ and ${\mathcal T} \cap A$ intersect. Hence $m_{s_n s} = 2$, as claimed.
Now let $w' = ws_n = s_1 \cdots s_{n-1}$ and let $\psi'$ be the unique chamber of $A$ such that $\delta_W(\phi,\psi') = w'$. Since $s_n$ is in $s^\perp$ and $\psi'$ is $s_n$--adjacent to $\psi$, the chamber $\psi'$ is epicormic at ${\mathcal T}$ and in the same component of $A - {\mathcal T} \cap A$ as $\phi$. Moreover $s_1 \cdots s_{n-1}$ is a reduced expression for $w'$, so $l_S(w') = n -1$. Hence by the inductive assumption, $w'$ is in $\langle s^\perp \rangle$. Therefore $w = w's_n$ is in $\langle s^\perp \rangle$, which completes the proof.
\end{proof}
\begin{corollary}\label{c:tree wall trichotomy} The following possibilities for tree-walls in $X$ may occur.
\begin{enumerate}
\item\label{i:reduced to vertex} Every tree-wall of type $s$ is reduced to a vertex if and only if $\langle s^\perp \rangle$ is trivial.
\item\label{i:finite} Every tree-wall of type $s$ is finite but not reduced to a vertex if and only if $\langle s^\perp \rangle$ is finite but nontrivial.
\item\label{i:infinite} Every tree-wall of type $s$ is infinite if and only if $\langle s^\perp \rangle$ is infinite.
\end{enumerate}
\end{corollary}
\begin{proof} Let ${\mathcal T}$, $\phi$, and $A$ be as in \fullref{l:sperp epicormic} above.
The set of $s$--panels in the wall ${\mathcal T} \cap A$ is in bijection with the set of chambers of $A$ which are epicormic at ${\mathcal T}$ and in the same component of $A - {\mathcal T} \cap A$ as $\phi$.
\end{proof}
\noindent For the examples in \fullref{s:rabs} above:
\begin{itemize}
\item in $X_1$, every tree-wall of type $s$ and of type $t$ is a vertex;
\item in $X_2$, the tree-walls of types both $r$ and $s$ are finite and $1$--dimensional, while every tree-wall of type $t$ is a vertex; and
\item in $X_3$, all tree-walls are infinite, and are $1$--dimensional.
\end{itemize}
\begin{corollary}\label{c:inverse image} Let ${\mathcal T}$, $\phi$, and $A$ be as in \fullref{l:sperp epicormic} above and let
\[ \rho = \rho_{\phi,A}\co X \to A \]
be the retraction onto $A$ centered at $\phi$. Then $\rho^{-1}({\mathcal T} \cap A) = {\mathcal T}$.
\end{corollary}
\begin{proof} Let $\psi$ be any chamber of $A$ which is epicormic at ${\mathcal T}$ and is in the same component of $A - {\mathcal T} \cap A$ as $\phi$. Then by the proof of \fullref{l:sperp epicormic} above, $w := \delta_W(\phi,\psi)$ is in $\langle s^\perp \rangle$.
Let $\psi'$ be a chamber in the preimage $\rho^{-1}(\psi)$ and let $A'$ be an apartment containing both $\phi$ and $\psi'$. Since the retraction $\rho$ preserves $W$--distances from $\phi$, we have that $\delta_W(\phi,\psi') = w$ is in $\langle s^\perp \rangle$. Again by the proof of \fullref{l:sperp epicormic}, it follows that the chamber $\psi'$ is epicormic at ${\mathcal T}$. But the image under $\rho$ of the $s$--panel of $\psi'$ is the $s$--panel of $\psi$. Thus $\rho^{-1}({\mathcal T} \cap A) = {\mathcal T}$, as required.
\end{proof}
\begin{lemma}\label{l:galleries crossing} Let ${\mathcal T}$ be a tree-wall and let $\phi$ and $\phi'$ be two chambers of $X$. Let $\alpha$ be a minimal gallery from $\phi$ to $\phi'$ and let $\beta$ be any gallery from $\phi$ to $\phi'$. If $\alpha$ crosses ${\mathcal T}$ then $\beta$ crosses ${\mathcal T}$.
\end{lemma}
\begin{proof} Suppose that $\alpha$ crosses ${\mathcal T}$. Since $\alpha$ is minimal, there is an apartment $A$ of $X$ which contains $\alpha$, and hence the wall ${\mathcal T} \cap A$ separates $\phi$ from $\phi'$. Choose a chamber $\phi_0$ of $A$ which is epicormic at ${\mathcal T}$ and consider the retraction $ \rho = \rho_{\phi_0,A}$ onto $A$ centered at $\phi_0$.
Since $\phi$ and $\phi'$ are in $A$, $\rho$ fixes $\phi$ and $\phi'$. Hence $\rho(\beta)$ is a gallery in $A$ from $\phi$ to $\phi'$, and so $\rho(\beta)$ crosses ${\mathcal T}\cap A$. By \fullref{c:inverse image} above, $\rho^{-1}({\mathcal T} \cap A) = {\mathcal T}$. Therefore $\beta$ crosses ${\mathcal T}$. \end{proof}
\begin{proposition}\label{p:tree-wall} Let ${\mathcal T}$ be a tree-wall of type $s$. Then ${\mathcal T}$ separates $X$ into $q_s$ gallery-connected components.
\end{proposition}
\begin{proof} Fix an $s$--panel in ${\mathcal T}$ and let $\phi_1,\ldots,\phi_{q_s}$ be the $q_s$ chambers containing this panel. Then for all $1 \leq i < j \leq q_s$, the minimal gallery from $\phi_i$ to $\phi_j$ is just $(\phi_i,\phi_j)$, and hence crosses ${\mathcal T}$. Thus by \fullref{l:galleries crossing} above, any gallery from $\phi_i$ to $\phi_j$ crosses ${\mathcal T}$. So the $q_s$ chambers $\phi_1,\ldots,\phi_{q_s}$ lie in $q_s$ distinct components of $X - {\mathcal T}$.
To complete the proof, we show that ${\mathcal T}$ separates $X$ into at most $q_s$ components. Let $\phi$ be any chamber of $X$. Then among the chambers $\phi_1,\ldots,\phi_{q_s}$, there is a unique chamber, say $\phi_1$, at minimal gallery distance from $\phi$. It suffices to show that $\phi$ and $\phi_1$ are in the same component of $X - {\mathcal T}$.
Let $\alpha$ be a minimal gallery from $\phi$ to $\phi_1$ and let $A$ be an apartment
containing $\alpha$. Then there is a unique chamber of $A$ which is $s$--adjacent to $\phi_1$. Hence $A$ contains $\phi_i$ for some $i > 1$, and the wall ${\mathcal T} \cap A$ separates $\phi_1$ from $\phi_i$. Since $\alpha$ is minimal and $d_W(\phi,\phi_1) < d_W(\phi,\phi_i)$, the Exchange Condition (see \cite[page 35]{D}) implies that a minimal gallery from $\phi$ to $\phi_i$ may be obtained by concatenating $\alpha$ with the gallery $(\phi_1,\phi_i)$. Since a minimal gallery can cross ${\mathcal T} \cap A$ at most once, $\alpha$ does not cross ${\mathcal T} \cap A$. Thus $\phi$ and $\phi_1$ are in the same component of $X - {\mathcal T}$, as required.
\end{proof}
\section{Proof of Theorem}\label{s:proof}
Let $G$ be as in the introduction and let $\Gamma$ be a non-cocompact lattice in $G$ with strict fundamental domain. Fix a chamber $\phi_0$ of $X$. For each integer $n \geq 0$ define \[D(n):=\{\,\phi \in \Ch(X) \mid d_W(\phi,\Gamma \phi_0) \leq n \,\}.\] Then $D(0)=\Gamma \phi_0$, and for every $n > 0$ every connected component of $D(n)$ contains a chamber in $\Gamma \phi_0$. To prove \fullref{t:strict implies not fg}, we will show that there is no $n > 0$ such that $D(n)$ is connected.
Let $Y$ be a strict fundamental domain for $\Gamma$ which contains $\phi_0$. For each chamber $\phi$ of $X$, denote by $\phi_Y$ the representative of $\phi$ in $Y$.
\begin{lemma}\label{l:projection preserves adjacency} Let $\phi$ and $\phi'$ be $t$--adjacent chambers in $X$, for $t \in S$. Then either $\phi_Y = \phi'_Y$, or $\phi_Y$ and $\phi'_Y$ are $t$--adjacent.
\end{lemma}
\begin{proof}
It suffices to show that the $t$--panel of $\phi_Y$ is the $t$--panel of $\phi'_Y$. Since $Y$ is a subcomplex of $X$, the $t$--panel of $\phi_Y$ is contained in $Y$. By definition of a strict fundamental domain, there is exactly one representative in $Y$ of the $t$--panel of $\phi$. Hence the unique representative in $Y$ of the $t$--panel of $\phi$ is the $t$--panel of $\phi_Y$. Similarly, the unique representative in $Y$ of the $t$--panel of $\phi'$ is the $t$--panel of $\phi'_Y$. But $\phi$ and $\phi'$ are $t$--adjacent, hence have the same $t$--panel, and so it follows that $\phi_Y$ and $\phi'_Y$ have the same $t$--panel.
\end{proof}
\begin{corollary}\label{c:gallery-connected} The fundamental domain $Y$ is gallery-connected. \end{corollary}
\begin{lemma}\label{l:transverse gallery} For all $n > 0$, the fundamental domain $Y$ contains a pair of adjacent chambers $\phi_n$ and $\phi'_n$ such that, if ${\mathcal T}_n$ denotes the tree-wall separating $\phi_n$ from $\phi'_n$:
\begin{enumerate}
\item\label{i:phi_0 and phi_n} the chambers $\phi_0$ and $\phi_n$ are in the same gallery-connected component of $Y - {\mathcal T}_n \cap Y$;
\item\label{i:distance to T_n} $\min \{ d_W(\phi_0,\phi) \mid \phi \in \Ch(X) \mbox{ is epicormic at ${\mathcal T}_n$} \} > n$; and
\item\label{i:stabilizer} there is a $\gamma \in \Stab_\Gamma(\phi'_n)$ which does not fix $\phi_n$.
\end{enumerate}
\end{lemma}
\begin{proof} Fix $n > 0$. Since $\Gamma$ is not cocompact, $Y$ is not compact. Thus there exists a tree-wall ${\mathcal T}_n$ with ${\mathcal T}_n \cap Y$ nonempty such that for every $\phi \in \Ch(X)$ which is epicormic at ${\mathcal T}_n$, $d_W(\phi_0,\phi) > n$. Let $s_n$ be the type of the tree-wall ${\mathcal T}_n$. Then by \fullref{c:gallery-connected} above, there is a chamber $\phi_n$ of $Y$ which is epicormic at ${\mathcal T}_n$ and in the same gallery-connected component of $Y - {\mathcal T}_n \cap Y$ as $\phi_0$, such that for some chamber $\phi'_n$ which is $s_n$--adjacent to $\phi_n$, $\phi'_n$ is also in $Y$.
Now, as $\Gamma$ is a non-cocompact lattice, the orders of the $\Gamma$--stabilizers of the chambers in $Y$ are unbounded. Hence the tree-wall ${\mathcal T}_n$ and chambers $\phi_n$ and $\phi'_n$ may be chosen so that $|\Stab_{\Gamma}(\phi_n)| < |\Stab_\Gamma(\phi'_n)|$.
\end{proof}
Let $\phi_n$, $\phi'_n$, ${\mathcal T}_n$, and $\gamma$ be as in \fullref{l:transverse gallery} above and let $s=s_n$ be the type of the tree-wall ${\mathcal T}_n$. Let $\alpha$ be a gallery in $Y - {\mathcal T}_n \cap Y$ from $\phi_0$ to $\phi_n$. The chambers $\phi_n$ and $\gamma \cdot \phi_n$ are in two distinct components of $X - {\mathcal T}_n$, since they both contain the $s$--panel $\phi_n \cap \phi'_n \subseteq {\mathcal T}_n$, which is fixed by $\gamma$. Hence the galleries $\alpha$ and $\gamma \cdot \alpha$ are in two distinct components of $X - {\mathcal T}_n$, and so the chambers $\phi_0$ and $\gamma \cdot \phi_0$ are in two distinct components of $X - {\mathcal T}_n$.
Denote by $X_0$ the component of $X - {\mathcal T}_n$ which contains $\phi_0$, and put $Y_0 = Y \cap X_0$.
\begin{lemma}\label{l:epiy}
Let $\phi$ be a chamber in $X_0$ that is epicormic at ${\mathcal T}_n$. Then $\phi_Y$ is in $Y_0$ and is epicormic at ${\mathcal T}_n \cap Y$.
\end{lemma}
\begin{proof}
We consider three cases, corresponding to the possibilities for tree-walls in \fullref{c:tree wall trichotomy} above.
\begin{enumerate}
\item If ${\mathcal T}_n$ is reduced to a vertex, there is only one chamber in $X_0$ which is epicormic at ${\mathcal T}_n$, namely $\phi_n$. Thus $\phi = \phi_n = \phi_Y$ and we are done.
\item If ${\mathcal T}_n$ is finite but not reduced to a vertex, the result follows by finitely many applications of \fullref{l:projection preserves adjacency} above.
\item If ${\mathcal T}_n$ is infinite, the result follows by induction, using \fullref{l:projection preserves adjacency} above, on
\[ k:= \min \{ d_W(\phi,\psi) \mid \psi \mbox{ is a chamber of $Y_0$ epicormic at ${\mathcal T}_n \cap Y$}\}. \proved \]
\end{enumerate}\end{proof}
\begin{lemma}\label{l:final} For all $n > 0$, the complex $D(n)$ is not connected.
\end{lemma}
\begin{proof} Fix $n > 0$, and let $\alpha$ be a gallery in $X$ between a chamber in $X_0 \cap \Gamma \phi_0$ and some chamber $\phi$ in $X_0$ that is epicormic at ${\mathcal T}_n$. Let $m$ be the length of $\alpha$.
By \fullref{l:projection preserves adjacency} and \fullref{l:epiy} above, the gallery $\alpha$ projects to a gallery $\beta$ in $Y$ between $\phi _0$ and a chamber $\phi_Y$ that is epicormic at ${\mathcal T}_n \cap Y$. The gallery $\beta$ in $Y$ has length at most $m$.
It follows from \eqref{i:distance to T_n} of \fullref{l:transverse gallery} above that the gallery $\beta$ in $Y$ has length greater than $n$. Therefore $m > n$. Hence the gallery-connected component of $D(n)$ that contains $\phi _0$ is contained in $X_0$. As the chamber $\gamma \cdot \phi_0$ is not in $X_0$, it follows that the complex $D(n)$ is not connected. \end{proof}
This completes the proof, as $\Gamma$ is finitely generated if and only if $D(n)$ is connected for some $n$.
|
{
"timestamp": "2011-01-26T02:00:39",
"yymm": "1009",
"arxiv_id": "1009.4235",
"language": "en",
"url": "https://arxiv.org/abs/1009.4235"
}
|
\section{The concept of jets in medium}
Jets in vacuum are best understood as a tool to bridge the gap between theory and experiment: While using perturbative Quantum Chromodynamics (pQCD) one can calculate results in terms of high $p_T$ parton production cross sections, experimentally collimated showers of hadrons are observed. In nature, the link between these is given by the perturbative parton shower evolution which turns initial high virtuality partons by braching processes into a shower of softer, lower virtuality partons, and by the non-perturbative hadronization of this parton shower. Jets are then an operational definition how to treat the observed hadron distribution to 'undo' this evolution and combine properties of measured hadrons into an object, the jet, which can be compared with calculations on the parton level. For instance, sequential recombination algorithms like anti-$k_T$ try to undo the branchings of the shower evolution by combining pairs of hadrons close in phase space. This means that a jet is only a meaningful concept in the context of a particular jet finding algorithm, and that the choice of the algorithm always corresponds to a bias to pick out a particular subset of all possible evolutions of the original parton.
In the presence of the soft medium created in heavy-ion collisions, the picture becomes more complicated. First, even if a jet which is completely uncorrelated with the medium is embedded into the background, the results of jet finding may change. This is because the medium constitutes a noisy environment in which the soft hadrons of the jet cannot be uniquely identified, thus a medium hadron may be combined into part of the jet or a jet hadron may be regarded as part of the background. Such complications can be studied by embedding MC jets into a medium background and trying to recover them.
However, in general the role of the medium is much more pronounced. There is expected to be significant interaction between the parton shower and the medium, i.e. there can be substantial redistribution of energy and momentum from the perturbatively calculable part of the system to the non-perturbative background and vice versa: The medium can 'absorb' partons which become sufficiently soft, whereas hard scatterings between shower and medium can kick a parton from the medium and correlate it with the jet.
This has two important consequences: First, the connection of any jet definition at 'hadron level' with the underlying parton becomes tenuous, as a medium hadron which did not originate in the branching process of the initial parton may still carry part of its energy and momentum picked up via elastic collisions. Instead, the properties of the initial parton are only manifest in the flow of energy and momentum carried by both perturbative and non-perturbative sector. Second, there is good evidence that the redistribution of energy and momentum in the soft bulk is quite different from the dynamics of a parton shower --- hydrodynamical phenomena such as shockwaves and diffusion wakes turn out to be important \cite{Betz,Bryon}. Consequently, vacuum jet definitions designed to unfold perturbative phenomena may largely miss this part of the evolution.
There are then two possible classes of jet definitions suitable for the heavy-ion environment. The first class focuses on the perturbative part of the evolution and eliminates the non-perturbative physics by cuts e.g. on $P_T$. The medium effect is then only apparent from the functional behaviour of jet $R_{AA}$ as a function of the cuts as suggested e.g. in \cite{Vitev}. The second class adapts to the non-perturbative physics and tries to capture as much as possible of the initial parton energy and momentum. This may require jet definitions very different from the vacuum case.
With this in mind, some care has to be taken to interpret the results of in-medium shower evolution codes correctly. Such codes focus on the {\em perturbative} side of the evolution only. This means that the code will either make the somewhat artificial assumption that no energy and momentum are exchanged between medium and parton (as e.g. in Q-PYTHIA \cite{QPYTHIA}) or that the energy in the parton shower will not balance the energy of the initial hard parton because the medium energy balance is not explicitly included.
\section{A description of YaJEM}
YaJEM is an example for an in-medium shower code which permits the exchange of energy and momentum between medium and shower, but does not explicitly model the medium. The model is described in detail in \cite{YaJEM1,YaJEM2,YaJEM3}. It is based on the PYSHOW algorithm \cite{PYSHOW} which models the shower as a series of $1\rightarrow2 $ branchings of partons $a \rightarrow bc$ and to which it reduces in the absence of a medium. The equations resulting from pQCD expressions for the branching probabilities are solved by a MC in momentum space.
In a medium, the medium evolution in position space must be connected with the shower evolution equations in momentum space. Within YaJEM the link to the medium spacetime dynamics is made by modelling the average time for a parton $b$ to branch from parent $a$ given the parton energies and virtualities based on the uncertainty relation as
\begin{equation}
\label{E-time}
\langle \tau_b \rangle= \frac{E_b}{Q_b^2} - \frac{E_b}{Q_a^2}
\end{equation}
whereas the actual time in given branching is generated from the exponential branching probability distribution
\begin{equation}
\label{E-time-r}
P(\tau_b) = \exp\left[- \frac{\tau_b}{\langle \tau_b \rangle} \right].
\end{equation}
For simplicity, all partons of a shower are propagated with this time information along an eikonal trajectory determined by the shower initiator. This amounts to neglecting the spread in transverse space when probing the medium. Currently, YaJEM models three different scenarios for the parton-medium interaction, two of which modify the kinematics of the propagating parton whereas the last modifies the branching probabilities at each vertex (thus, in the first two scenarios the energy of the shower is {\em not} the energy of the shower initiating parton as there is explicit energy transfer between shower and medium, whereas in the last scenario the energy in the shower is conserved). For instance, in the RAD scenario, the medium is assumed to cause an increase $\Delta Q_a^2$ in the virtuality of a parton $a$ based on a local transport coefficient $\hat{q}(\zeta)$ as given by the line integral
\begin{equation}
\Delta Q_a^2 = \int_{\tau_a^0}^{\tau_a^0 + \tau_a} d\zeta \hat{q}(\zeta).
\end{equation}
This modification causes medium-induced radiation. In the DRAG scenario the medium is assumed to cause energy and momentum loss along the parton trajectories, whereas in the FMED scenario the kinematics remains unmodified, but the branching probabilities are modified as compared to the vacuum case. In general, in the RAD scenario the energy in the final perturbative state is formally larger than initially (the depletion of medium partons becoming corelated with the jet is neglected) whereas in the DRAG scenario the final energy is smaller (the absorption of partons by the medium us neglected). These effects in principle have to be compensated for when embedding YaJEM jets into a background.
\section{Key YaJEM results}
Before applying any in-medium shower evolution to jets, it is useful to understand how it relates to known models and observables of leading parton energy loss. An important question is to what degree the LPM suppression of subsequent induced radiation processes known to lead to the characteristic $L^2$ dependence of energy loss in a constant medium is preserved in a probabilistic MC description. This has been computed for YaJEM in \cite{YaJEM2} and is shown in Fig.~\ref{F-YaJEM1} left for a 100 GeV charm quark propagating through a constant medium in the RAD scenario.
\begin{figure}[htb]
\epsfig{file=dEdx, width=6.5cm}{\epsfig{file=D_comp.eps, width=6.5cm}}
\caption{\label{F-YaJEM1} Left panel: Mean energy loss per unit pathlength from a 100 GeV charm quark propagating through a constant medium for two different assumptions about the spacetime picture of the shower, compared with an $L^2$ dependence. Right panel: Examples of medium-modified fragmentation functions obtained in YaJEM for three different scenarios of parton-medium interaction.}
\end{figure}
If only Eq.~(\ref{E-time}) is used to model the spacetime picture of the shower based on the \emph{average} lifetime of a virtual state, then there is a region in which $L^2$ dependence is seen before finite energy corrections become manifest. However, if Eq.~(\ref{E-time-r}) is used to determine the lifetime, this region is substantially reduced. Thus, while the MC code is capable of preserving this important feature known from analytic calculations, an attempt to model more realistically much weakens it. This has also been observed in a different framework \cite{JEWEL}. The disagreement with the pathlength dependence deduced from data is currently an unsolved theoretical issue.
The right panel of Fig.~\ref{F-YaJEM1} shows the medium-modified fragmentation function in all three different scenarios of parton-medium interaction. All lead to a comparable depletion of the high $z$ region of the fragmentation function, corresponding to leading parton energy loss. To see the modification in the low $z$ part more clearly, it is useful to transform into $\xi = \ln (1/z)$. This is shown in Fig.~\ref{F-YaJEM2}, left panel.
\begin{figure}[htb]
\epsfig{file=HBP_comp_f.eps, width=6.5cm}{\epsfig{file=IAA_YaJEM_f.eps, width=6.5cm}}
\caption{\label{F-YaJEM2} Left panel: The 'hump-backed plateau' distribution $dN/d\xi$ obtained in YaJEM for three different scenarios of parton-medium interaction. Right panel: The away side per-trigger yield in Au-Au collisions normalized to the result in p-p collisions $I_{AA}$ in $\gamma$-h correlations obtained in the YaJEM RAD scenario and in a standard leading parton energy loss picture .}
\end{figure}
While the DRAG scenario (in which energy flows into the medium) leads to a small suppression in this region, induced radiation is manifest as an enhancement of the plateau. Experimentally, this region in $\xi$ can be probed by $\gamma$-hadron correlations. The expectation for the away side per trigger yield suppression ratio $I_{AA}$ from YaJEM (with enhanced low $z$ hadron production due to induced radiation) and a standard energy loss calculation (where lost energy is assumed to be redistributed throughout the whole medium) is shown in Fig.~\ref{F-YaJEM2} right panel \cite{gamma-h}. Currently, there is no evidence in the data for a rise of $I_{AA}$ above unity in the data. This might be some evidence that a picture of \emph{perturbative} redistribution of lost energy inside the shower only is not justified and that energy flow in the medium needs to be accounted for.
To provide an example for genuine jet observables, Fig.~\ref{F-YaJEM3} shows the distribution of thrust
$T = \text{max}_{{\bf n}_T} \frac{\sum_i | {\bf p}_i \cdot {\bf n}_T|}{\sum_i|{\bf p}_i|} \quad$, thrust major
$T_{maj} = \text{max}_{{\bf n}_T \cdot {\bf n}=0} \frac{\sum_i | {\bf p}_i \cdot {\bf n}|}{\sum_i|{\bf p}_i|} \quad$ and thrust minor
$T_{min} = \frac{\sum_i | {\bf p}_i \cdot {\bf n}_{mi}|}{\sum_i |{\bf p}_i|}$
for 100 GeV quark jets propagated through medium densities characteristic for LHC Pb-Pb collisions.
\begin{figure}[htb]
\epsfig{file=thrust-d-E100_f.eps, width=6.5cm}{\epsfig{file=thrust-d-E100-kT4.0_f.eps, width=6.5cm}}
\caption{\label{F-YaJEM3} Distributions of thrust, thrust major and thrust minor for 100 GeV quark jets in vacuum and for a medium expected in central 5.5 ATeV Pb-Pb collisions at LHC as computed in YaJEM, for all $P_T$ (left) and with a cut of 4 GeV (right).}
\end{figure}
There is a general trend that medium-induced radiation makes the event more spherical, regardless if explicit exchange of energy and momentum with the medium is modelled or not, but the $P_T$ dependence of the distribution is very different in both cases. Thus, assuming that the parton shower has zero momentum exchange with the medium may be a bad approximation. This in turn means that the problem of energy redistribution by non-perturbative medium degrees of freedom in all likelihood will have to be dealt with in order to understand medium-modified jets.
\section{Conclusions}
Currently medium-modified shower codes have to be regarded as work in progress. Qualitatively, some of their results look promising. However, when looking into the details, there are often problems in reproducing more differential leading hadron observables. Quantitatively, YaJEM awaits comparison with jet measurements at RHIC, which, due to the complicated jet finding in a heavy-ion background, is not an easy task. There are also some indications that non-perturbative dynamics in the medium might play a role for redistributing the energy and momentum of the initial hard process in addition to the perturbative dynamics of a parton shower. The kinematic range available at the LHC will help much to resolve these questions.
\section*{Acknowledgements}
|
{
"timestamp": "2010-09-21T02:02:33",
"yymm": "1009",
"arxiv_id": "1009.3740",
"language": "en",
"url": "https://arxiv.org/abs/1009.3740"
}
|
\section{Introduction}
Supersymmetric extension of the standard model
(SM)~\cite{Martin:1997ns} is a leading candidate of physics beyond
the SM.
However, since no experimental evidence of supersymmetry (SUSY) has
not been found yet, discovery of supersymmetric particles at energy
frontier experiments such as LHC is one of the important tasks of
particle physics.
The most general gauge invariant and renormalizable superpotential
in the supersymmetric SM
contains baryon ($B$) and lepton ($L$) number violating interactions
which may lead to unwanted fast proton decay or sizable lepton number
violating processes.
Such interactions can be forbidden by introducing so called $R$-parity
which is defined as $R=(-1)^{3B+L+2S}$, where $S$ denotes the spin
quantum number.
Owing to the $R$-parity, in addition to the suppression of
$B$- and $L$-violating processes, the lightest supersymmetric particle
(LSP) becomes stable, and it could be a candidate of dark matter.
On the other hand, some of $R$-parity violating (RPV) interactions may
play phenomenologically attractive role.
For example, the $R$-parity and $L$-violating interactions may explain
tiny neutrino mass without introducing the right-handed
neutrinos~\cite{Hall:1983id}.
Also, a possibility of gravitino dark matter due to the $R$-parity
violating interactions has been discussed
in ref.~\cite{Buchmuller:2007ui}.
In this article, we study contribution of the RPV interactions
to the leptonic decays of $D_s$ and $B^+$ mesons.
It is known that the experimental data of the leptonic decays
of $D_s$ and $B^+$ mesons slightly deviate from the SM expectations.
Comparison of the experimental results of leptonic decay of $D_s$ meson
is often presented in terms of the decay constant $f_{D_s}$.
The recent measurement of the leptonic decay of $D_s$ meson
by CLEO~\cite{Naik:2009tk} is given by
\begin{eqnarray}
f_{D_s} = 259.0 \pm 6.2 \pm 3.0 =
259.0 \pm 6.9~{\rm [MeV]},
\label{cleo2009}
\end{eqnarray}
while the most precise calculation of $f_{D_s}$ by HPQCD and
UKQCD~\cite{Follana:2007uv} is given as
\begin{eqnarray}
f_{D_s} = 241 \pm 3 {\rm [MeV]}.
\label{fdqcd}
\end{eqnarray}
The discrepancy between (\ref{cleo2009}) and (\ref{fdqcd}) is
about $2.4\sigma$.
The recent review of the experimental data and theoretical estimations
on the decay constant $f_{D_s}$ can be found
in refs.~\cite{Kronfeld:2009cf}.
For the leptonic decay $B^+ \to \tau \nu_\tau$ ,
the experimental data of the branching ratio have been given
by Belle and BABAR~\cite{ikado,btau}.
The average of the data given by the UTfit
collaboration~\cite{Bona:2009cj} is
\begin{eqnarray}
{\rm BR}(B^+ \to \tau \nu_\tau)_{\rm exp} &=&
(1.73 \pm 0.34)\times 10^{-4},
\label{expB}
\end{eqnarray}
while the SM prediction is given by~\cite{Bona:2009cj}
\begin{eqnarray}
{\rm BR}(B^+ \to \tau \nu_\tau)_{\rm SM}
&=& (0.84 \pm 0.11) \times 10^{-4}.
\label{utfit}
\end{eqnarray}
The difference between (\ref{expB}) and (\ref{utfit})
is $2.5\sigma$.
The deviations in the leptonic decays in both $D_s$ and $B^+$ may be
statistical fluctuations.
However, another interpretation of the deviations is that the
deviations are caused by new physics beyond the SM.
In the SM, these leptonic decays are dominated by the $W$-boson exchange
at tree level\footnote{It has been pointed out that the radiative
corrections are highly suppressed~\cite{Burdman:1994ip}}.
Therefore, a class of new physics models which lead to the leptonic
decays at tree level could be candidates to explain the discrepancies,
e.g., Two Higgs doublet model~\cite{Ahn:2010zza, Akeroyd:2009tn,
Akeroyd:2007eh},
leptoquark model~\cite{Dobrescu:2008er, Benbrik:2008ik},
and the $R$-parity violating supersymmetric
SM~\cite{Baek:1999ch,Dreiner:2001kc,Dreiner:2006gu,
Kundu:2008ui, Kao:2009mz, Bhattacharyya:2009hb}.
In the supersymmetric SM with RPV interactions, contributions
to the leptonic decays of $D_s$ and
$B^+$ mesons are given by down-squark exchange in $t$-channel diagram,
charged slepton exchange in $s$-channel diagram and
charged Higgs boson exchange in $s$-channel diagram.
In refs.~\cite{Dreiner:2006gu,Kundu:2008ui, Bhattacharyya:2009hb},
only the $t$-channel contribution was examined based on some scenarios
or single coupling dominance hypothesis.
The contribution of $s$-channel diagram in addition to the $t$-channel
has been studied in refs.~\cite{Baek:1999ch,Dreiner:2001kc,Kao:2009mz}.
However, since works in refs.~\cite{Baek:1999ch,Dreiner:2001kc}
have been done before the first measurement of $B^+ \to \tau \nu_\tau$
in 2006~\cite{ikado}, bounds on the RPV couplings were not obtained
from the experimental data of the $B^+$ decay.
In ref.~\cite{Kao:2009mz}, constraints on the RPV couplings
in $s$- and $t$-channel diagrams were investigated separately,
and no interference effect between two diagrams was examined.
In our study, we investigate the supersymmetric contributions to the
leptonic decay of $D_s$ and $B^+$ mesons taking account of
the interference effects between the $s$- and $t$-channel diagrams.
We also examine a diagram mediated by the charged Higgs boson in
the $s$-channel.
Taking account of the interference effects between the $s$- and
$t$-channel diagrams, we show allowed region of the
RPV couplings which explains the deviation between
experimental data and the SM prediction.
Note that the interference between two diagrams could be either
constructive or destructive due to the relative sign of the RPV
couplings in two diagrams.
We find, therefore, that the experimental data constrains not only
the size of RPV couplings but also the relative sign between
the RPV couplings in $s$- and $t$-channel diagrams, which has not been
examined in previous studies.
The contribution of charged Higgs boson is found to be negligible
in $D_s \to \tau \nu_\tau$, but sizable in $B^+ \to \tau \nu_\tau$.
We discuss how the constraints on RPV couplings are affected by
the charged Higgs contribution.
\section{Set up}
The $R$-parity violating interactions with trilinear couplings are
described by the following superpotential
\begin{eqnarray}
W_{\slashed R} &=&
\frac{1}{2}\lambda_{ijk} L_i L_j E_k
+ \lambda'_{ijk} L_i Q_j D_k
+ \frac{1}{2} \lambda''_{ijk} U_i D_j D_k,
\label{rparity}
\end{eqnarray}
where $Q$ and $L$ are SU(2)$_L$ doublet quark and lepton superfields,
respectively.
The up- and down-type singlet quark superfields are represented by
$U$ and $D$, while the lepton singlet superfield is $E$.
The generation indices are labeled by $i,j$ and $k$.
The SU(2)$_L$ and SU(3)$_C$ gauge indices are suppressed.
The coefficient $\lambda_{ijk}$ is anti-symmetric for $i$ and $j$,
while $\lambda''_{ijk}$ is anti-symmetric for $j$ and $k$.
For a comprehensive review of the $R$-parity violating supersymmetric
SM, see, ref.~\cite{Barbier:2004ez}.
Constraints on the RPV couplings
$\lambda_{ijk}, \lambda'_{ijk}$ and $\lambda''_{ijk}$ from various
processes have been studied in the literature~
\cite{Barger:1989rk, Dreiner:1997uz, Bhattacharyya:1997vv,
Allanach:1999ic, Dreiner:2006gu}.
Since the baryon number violating coupling $\lambda''_{ijk}$
induces too fast proton decay, we take $\lambda''_{ijk}=0$ in
the following.
Then, the leptonic decays of $D_s$ and $B^+$ mesons occur through the
$t$-channel exchange with a product of two $\lambda'$ couplings while
$s$-channel exchange is given by a product of $\lambda$ and $\lambda'$.
Let us briefly summarize the leptonic decay of a pseudo scalar meson $P$
which consists of the up and (anti-) down-type quarks $u_a$ and
$\bar{d_b}$, where $a,b$ are generation indices of quarks.
The decay width of $P \to l_i \nu_j$ is given as
\begin{eqnarray}
\Gamma(P \to l_i \nu_j )
&=& \frac{1}{8\pi} r_P^2 G_F^2 |V_{u_a d_b}^*|^2 f_P^2 m_{l_i}^2
m_P \left(1-\frac{m_{l_i}^2}{m_P^2}\right)^2
\label{dw}
\end{eqnarray}
where $G_F,V_{u_a d_b},m_{l_i}$ and $m_P$ are the Fermi constant, the
Cabibbo-Kobayashi-Maskawa matrix element, the mass of a charged lepton
$l_i$ and the mass of a pseudo scalar meson $P$,
respectively.
The flavor indices of charged leptons and neutrinos are expressed
by $i$ and $j$, respectively.
The decay constant is denoted by $f_P$.
A parameter $r_P$ is defined as,
\begin{eqnarray}
r_P^2 &\equiv& \frac{\left|G_F V_{u_a d_b}^* + A^P_{ii} \right|^2}
{ G_F^2 \left|V_{u_a d_b}^*\right|^2}
+
\sum_{j(\neq i)}
\frac{\left|A^P_{ij} \right|^2}
{ G_F^2 \left|V_{u_a d_b}^*\right|^2},
\label{rparam}
\end{eqnarray}
where $A^P_{ij}$ represents new physics contribution.
Note that, in the second term of r.h.s. in (\ref{rparam}),
one should take a sum only for $j$ (neutrinos), because that
the neutrino flavor cannot be detected experimentally.
If there is no new physics contribution, $r_P=1$.
The interaction Lagrangian of the $t$-channel contribution
to the decay width (\ref{dw}) can be obtained from
the superpotential (\ref{rparity});
\begin{eqnarray}
{\cal L} &=&
\lambda'_{ijk} \left\{
-\overline{(l^c_L)_i} (u_L)_j (\widetilde{d}_R)^*_k
\right\}
+
\lambda'^*_{ijk} \left\{
(\widetilde{d}_R)_k
\overline{(d_L)_j}
(\nu^c_L)_i
\right\} + {\rm h.c.}.
\label{tch}
\end{eqnarray}
Using the Fierz transformation,
the effective Lagrangian which describes the $t$-channel
squark exchange is given as
\begin{eqnarray}
{\cal L_{\rm eff}^{\it t}} &=&
\frac{1}{8}\sum_{k=1}^3
\frac{\lambda'_{iak} \lambda'^*_{jbk}}
{m_{\widetilde{d}_{Rk}}^2} ~
\bar{\nu}_j \gamma^\mu (1-\gamma_5) l_i~
\bar{d}_b \gamma_\mu (1-\gamma_5) u_a.
\end{eqnarray}
For comparison, we show the effective Lagrangian for the $W$-boson
exchange
\begin{eqnarray}
{\cal L_{\rm eff}^{\rm SM}} &=&
\frac{G_F}{\sqrt{2}} V_{u_a d_b}^*
\bar{\nu}_i \gamma^\mu (1-\gamma_5) l_i~
\bar{d}_b \gamma_\mu (1-\gamma_5) u_a.
\end{eqnarray}
Using the decay constant $f_P$ which is given by
\begin{eqnarray}
\langle 0 | \bar{d}_b \gamma^\mu \gamma_5 u_a| P(q) \rangle
= i f_P q^\mu,
\label{decayconst}
\end{eqnarray}
we find the $t$-channel squark contribution to
the decay $P(u_a \bar{d}_b) \to l_i \nu_j$ as
\begin{eqnarray}
(A_t^P)_{ij} &=& \frac{1}{4\sqrt{2}}
\sum_{k=1}^3
\frac{\lambda'_{iak} \lambda'^*_{jbk}}
{m_{\widetilde{d}_{Rk}}^2}.
\label{atp}
\end{eqnarray}
The $s$-channel contribution can be calculated from the interaction
Lagrangian
\begin{eqnarray}
{\cal L} &=& \lambda_{ijk}
\left\{-\overline{(l_R)_k} (\nu_L)_j (\widetilde{l}_L)_i
\right\}
+
\lambda'_{ijk}\left\{
-\overline{(d_R)_k}(u_L)_j (\widetilde{l}_L)_i
\right\}
+ {\rm h.c.}.
\end{eqnarray}
The effective Lagrangian is given by
\begin{eqnarray}
{\cal L^{\it s}_{\rm eff}} &=& -\frac{1}{4}
\sum_{k=1}^3
\frac{\lambda^*_{kji}\lambda'_{kab}}{m_{\widetilde{l}_{Lk}}^2}
\bar{\nu}_j (1+\gamma_5) l_i ~
\bar{d}_b (1-\gamma_5) u_a.
\end{eqnarray}
From (\ref{decayconst}) and equations of motion for $u,d$ quarks,
we find
\begin{eqnarray}
\langle 0|
\bar{d}_b \gamma_5 u_a | P(q) \rangle
= -i \frac{m_P^2}{m_{u_a}+m_{d_b}} f_P.
\label{psdecay}
\end{eqnarray}
Using (\ref{psdecay}), we obtain the $s$-channel contribution as
\begin{eqnarray}
(A_s^P)_{ij} &=& -\frac{1}{2\sqrt{2} m_{l_i}}
\frac{m_P^2}{m_{u_a}+m_{d_b}}
\sum_{k=1}^3
\frac{\lambda^*_{kji}\lambda'_{kab}}{m_{\widetilde{l}_{Lk}}^2}.
\label{asp}
\end{eqnarray}
The charged Higgs contribution can be calculated from the
interaction Lagrangian,
\begin{eqnarray}
{\cal L} &=& V_{u_a d_b}^*
\left\{
\frac{g m_{d_b}}{\sqrt{2}m_W} \tan\beta \overline{d_b} P_L u_a H^-
+
\frac{g m_{u_a}}{\sqrt{2}m_W} \cot\beta \overline{d_b} P_R u_a H^-
\right\}
\nonumber \\
&&~~~
+
\frac{g m_{l_i}}{\sqrt{2}m_W} \tan\beta \overline{\nu_i} P_R l_i H^+
+ {\rm h.c.},
\label{chhiggs}
\end{eqnarray}
where $g$ denotes the SU(2)$_L$ gauge coupling constant, and
$\tan\beta \equiv \langle H_u \rangle/
\langle H_d \rangle$ is a ratio of the vacuum expectation values
of two Higgs doublets $H_u$ (the weak hypercharge $Y=1/2$)
and $H_d$ ($Y=-1/2$).
We obtain the charged Higgs contribution $A_s^P$ from (\ref{chhiggs})
as
\begin{eqnarray}
A^P_H &=& -G_F V_{u_a d_b}^* \frac{m_{d_b}}{m_{u_a} + m_{d_b}}
\frac{m_P^2}{m_{H^-}^2}
\left(\tan^2\beta - \frac{m_{u_a}}{m_{d_b}}\right).
\label{a_h}
\end{eqnarray}
Note that since leptons in the final state due to the charged Higgs
exchange are flavor diagonal, the indices $i,j$ are suppressed in
l.h.s. of (\ref{a_h}).
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig1a.eps}
\includegraphics[width=7cm]{fig1b.eps}
\caption{
Contribution of $t$-channel squark exchange
to the $r$-parameters (\ref{rparam}) for $D_s \to \tau \nu_\tau$ (left)
and $B^+ \to \tau \nu_\tau$ (right) as functions of the RPV couplings.
The squark mass is fixed at $100~{\rm GeV}$. The horizontal lines
denote the 1-$\sigma$ constraints on $r_{D_s}$ and $r_{B^+}$
given in eqs.~(\ref{rparamexp_ds}) and (\ref{rparamexp_bp}),
respectively.
\label{tch_graph}
}
\end{center}
\end{figure}
\section{Numerical Study}
Next we examine the RPV contributions to the leptonic decays
$P\to \tau \nu_\tau$ ($P=D_s$ or $B^+$) numerically.
In the numerical study,
we adopt the central values of the following parameters~\cite{PDG2010}
\begin{eqnarray}
|V_{cs}|&=&1.023\pm 0.036,
~~|V_{ub}|=(3.89\pm 0.44)\times 10^{-3},
\nonumber \\
m_{D_s} &=& 1968.47 \pm 0.33~{\rm MeV}, ~~
m_{B^+} = 5279.17 \pm 0.29~{\rm MeV},
\end{eqnarray}
In the analysis, we drop the second term in r.h.s. of (\ref{rparam}),
i.e., the flavor off-diagonal final state
such as $\tau \nu_\mu$ or $\tau \nu_e$ are neglected.
Since the RPV couplings responsible for $P \to \tau \nu_\mu$ or
$P \to \tau \nu_e$ induce the lepton flavor violating processes
$\tau \to \mu \gamma$ or $\tau \to e \gamma$,
those couplings must be highly suppressed.
Therefore we neglect the $\tau \nu_\mu$ and $\tau \nu_e$ channels
in the following study, i.e., $A_{ij}^P=0$ for $i\neq j$.
Throughout out study, the squark and slepton masses are fixed at
$100~{\rm GeV}$.
For simplicity,
in the $t$-channel diagram we consider the sbottom exchange.
On the other hand, the stau exchange is forbidden in the $s$-channel
diagram, and we consider the smuon exchange.
Let us recall that $s$-channel amplitude is proportional to
a product of $\lambda$ and $\lambda'$.
Since the final state is $\tau \nu_\tau$, the RPV coupling
$\lambda_{i33}$ requires $i\neq 3$ due to the anti-symmetric property
of $\lambda_{ijk}$ for the first two indices.
This is why the stau exchange is forbidden in the $s$-channel diagram
in $P\to \tau \nu_\tau$.
We first study the contribution of $t$-channel squark exchange in
$D_s \to \tau \nu_\tau$.
When the sbottom exchange diagram is dominant, the contribution
to the parameter $r_{D_s}$ is given by a coupling $\lambda'_{323}$, while
the parameter $r_{B^+}$ is given by a product
$\lambda'_{313}\lambda'^*_{333}$.
In Fig.~\ref{tch_graph}, we show the sbottom contribution to
$r_{D_s}$ for $D_s \to \tau \nu_\tau$ (left) and
$r_{B^+}$ for $B^+ \to \tau \nu_\tau$ (right) as a function
of $|\lambda'_{323}|^2$ and $\lambda'_{313}\lambda'^*_{333}$,
respectively.
The horizontal lines denote constraints on
$r_{D_s}$ from (\ref{cleo2009}) and (\ref{fdqcd}),
and $r_{B^+}$ from (\ref{expB}) and (\ref{utfit})
\begin{subequations}
\begin{eqnarray}
r_{D_s} &=& 1.07 \pm 0.04,
\label{rparamexp_ds}
\\
r_{B^+} &=& 1.44 \pm 0.23.
\label{rparamexp_bp}
\end{eqnarray}
\end{subequations}
From Fig.~\ref{tch_graph}, we find that the $t$-channel contribution
constructively interferes with the $W$-boson exchange, i.e.,
$r_{D_s}, r_{B^+} \ge 1$.
Taking account of eqs.(\ref{rparamexp_ds}) and (\ref{rparamexp_bp}),
constraints on the RPV couplings at 1-$\sigma$ level are given as
\begin{eqnarray}
0.02 \,{\rlap{\lower 3.5pt\hbox{$\mathchar\sim$}}\raise 1pt\hbox{$<$}}\, &|\lambda'_{323}|^2& \,{\rlap{\lower 3.5pt\hbox{$\mathchar\sim$}}\raise 1pt\hbox{$<$}}\, 0.07,
\label{dsconst}
\\
0.0006 \,{\rlap{\lower 3.5pt\hbox{$\mathchar\sim$}}\raise 1pt\hbox{$<$}}\, &\lambda'_{313}\lambda'^*_{333}& \,{\rlap{\lower 3.5pt\hbox{$\mathchar\sim$}}\raise 1pt\hbox{$<$}}\, 0.0017.
\label{bpconst}
\end{eqnarray}
The allowed RPV couplings for $B^+ \to \tau \nu_\tau$
(\ref{bpconst})
is smaller than that for $D_s \to \tau \nu_\tau$
(\ref{dsconst})
by few orders of magnitude.
This is because the parameter $r_P$ (\ref{rparam})
accounts for the relative size of new physics contribution against
for a CKM matrix element.
Note that the CKM matrix element in $r_{B^+}$ is $V_{ub}\sim 10^{-3}$,
while that in $r_{D_s}$ is $V_{cs} \sim 1$.
Thus, the difference of magnitude between $V_{cs}$ and $V_{ub}$ explains
the difference between (\ref{dsconst}) and (\ref{bpconst}).
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig2.eps}
\caption{
Contribution of RPV interactions to the
parameter $r_{D_s}$ as functions of
the RPV couplings.
Three curves correspond to the $t$-channel contribution (solid),
the $s$-channel contribution (dotted) and the sum of $s$- and
$t$-channel contributions (dashed).
For comparison of contributions from each diagram, we fix
the RPV couplings in both $t$- and $s$-channel diagrams to be
equal, i.e.,
$\lambda^2 \equiv |\lambda'_{323}|^2 = \lambda'_{222}\lambda^*_{233}$.
The horizontal lines denote
the 1-$\sigma$ bound on $r_{D_s}$ from the experimental data.
\label{dsstch}
}
\end{center}
\end{figure}
Next we study the $s$-channel slepton exchange.
From (\ref{asp}), we find that the interference between the $s$-channel
contribution and the $W$-boson exchange is destructive when the RPV
couplings are real and positive.
We show the $r_{D_s}$ parameter via the $s$-channel slepton exchange,
and the interference between the $s$- and $t$-channel exchanges
in Fig.~\ref{dsstch}.
Note that
the $t$-channel contribution is proportional to $|\lambda'_{323}|^2$
while the $s$-channel contribution is
$\lambda'_{222}\lambda_{233}^*$.
The solid curve represents the $t$-channel contribution which is
obtained by setting $\lambda^2 = |\lambda'_{323}|^2$ and
$\lambda'_{222}\lambda^*_{233}=0$.
On the other hand, the dotted curve denotes the $s$-channel contribution
which is obtained by setting the $t$-channel coupling to zero, i.e.,
$\lambda^2 = \lambda'_{222}\lambda^*_{233}$
and $|\lambda'_{323}|^2=0$.
The sum of the $t$- and $s$-channel diagrams is given
by the dashed curve, where we fix the RPV couplings in both $t$- and
$s$-channel diagrams to be equal,
$\lambda^2 = |\lambda'_{323}|^2=\lambda'_{222}\lambda^*_{233}$
for comparison of contributions from each diagram.
It is clear that, when
$\lambda'_{222}\lambda^*_{233}>0$,
the $s$-channel slepton exchange diagram destructively interferes with
both the SM $W$-boson and $t$-channel squark exchange diagrams.
Therefore, the squark ($t$-channel) and slepton ($s$-channel)
contributions may be cancelled each other in some parameter space.
In the dashed line which represents the sum of $s$- and $t$-channel
the $r_{D_s}$ parameter decreases from unity and becomes zero
(i.e., $G_F V_{cs} + A_t^{D_s} \approx -A_s^{D_s}$) around
$\lambda^2 \sim 0.3$.
For $\lambda^2 \,{\rlap{\lower 3.5pt\hbox{$\mathchar\sim$}}\raise 1pt\hbox{$>$}}\, 0.3$, the $s$-channel contribution eventually
dominates over the $W$-boson and $t$-channel squark contribution
($G_F V_{cs} + A_t^{D_s} \ll |A_s^{D_s}|$) and the $r_{D_s}$ parameter
increases with $\lambda^2$,
which satisfies the experimental constraint when $\lambda^2 \sim 0.65$.
From Fig.~\ref{dsstch} we find the relation $A_t^{D_s} < A_s^{D_s}$
holds and this can be understood as follows.
When the RPV couplings and sparticle masses are same in both
the $t$-channel contribution $A_t^P$ (\ref{atp}) and
the $s$-channel contribution $A_s^P$ (\ref{asp}),
the relative magnitudes of two contributions are determined by
their coefficients, $\frac{1}{4\sqrt{2}}$ for the $t$-channel
and
$\frac{1}{2\sqrt{2}}\frac{m_P}{m_{l_i}}
\frac{m_P}{m_{U_a}+ m_{D_b}}$ for the $s$-channel.
Note that the ratio $\frac{m_P}{m_{U_a}+ m_{D_b}}$ is of order unity
for $P=D_s$ or $B^+$.
On the other hand, the $s$-channel contribution $A_s^P$ could be
enhanced by the ratio $\frac{m_P}{m_{l_i}}$.
For $l_i=\tau$, the ratio is $\frac{m_P}{m_\tau}\sim 1$
for $P=D_s$ and $\sim 3$ for $P=B^+$.
Therefore, the size of $A_s^P$ is about $2(8)$ times larger than
$A_t^P$ for $D_s (B^+)$ when the RPV couplings and the sparticle
masses are common.
It should be mentioned that the $s$-channel contribution
$A_s^{D_s}$ is considerably larger for $l=\mu(e)$
than for $l=\tau$ due to small lepton mass.
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig3a.eps}
\includegraphics[width=7cm]{fig3b.eps}
\caption{
Constraints on the RPV couplings from $D_s \to \tau \nu_\tau$ (left)
and $B^+ \to \tau \nu_\tau$ (right).
The horizontal axis represents the RPV couplings for $t$-channel while
the vertical axis denotes the couplings for $s$-channel.
The bands with solid line correspond to the 2-$\sigma$ allowed range
for the $r_{D_s}$ (left) and $r_{B^+}$ (right) parameters, respectively.
In each figure, the inner lines correspond to the 2-$\sigma$ lower
bounds while the outers are the 2-$\sigma$ upper bounds on $r_{D_s}$
and $r_{B^+}$, respectively.
\label{diffcouplings}}
\end{center}
\end{figure}
In Fig.~\ref{diffcouplings}, we show constraints on the RPV couplings
from $D_s \to \tau \nu_\tau$ (left) and $B^+ \to \tau \nu_\tau$ (right).
The horizontal axis represents the RPV couplings for $t$-channel diagram
while the vertical axis denotes the couplings for $s$-channel diagram.
The bands with solid lines correspond to the 2-$\sigma$ allowed range
for $r_{D_s}$ (\ref{rparamexp_ds}) and $r_{B^+}$ (\ref{rparamexp_bp}).
In each figure, the inner solid lines correspond to the 2-$\sigma$
lower bounds while the outers are the 2-$\sigma$ upper bounds on
$r_{D_s}$ and $r_{B^+}$, respectively.
From Fig.~\ref{diffcouplings}, we find that the $s$-channel couplings
have positive
correlations with the $t$-channel couplings.
This is because the interference between the $s$- and $t$-channel
contributions is destructive.
For $D_s \to \tau \nu_\tau$, since the $t$-channel coupling is always
positive ($|\lambda'_{323}|^2 \ge 0$), not only the magnitude but also
the sign of $s$-channel coupling $\lambda'_{222}\lambda^*_{233}$
is strongly constrained.
For negative $\lambda'_{222}\lambda^*_{233}$,
the $t$-channel coupling $|\lambda'_{323}|^2$ should be smaller than
$0.12$ and, then, $-0.04 \,{\rlap{\lower 3.5pt\hbox{$\mathchar\sim$}}\raise 1pt\hbox{$<$}}\, \lambda'_{222}\lambda^*_{233} \le 0$
is experimentally allowed in the 2-$\sigma$ level.
For $B^+ \to \tau \nu_\tau$,
the $s$- and $t$-channel couplings with opposite
signs are strongly constrained.
As can be seen in Fig.~\ref{diffcouplings}, when the $t$-channel
coupling is positive
($\lambda'_{313}\lambda'^*_{333} \ge 0$), the negative $s$-channel
coupling is constrained to be
$-0.0004 \,{\rlap{\lower 3.5pt\hbox{$\mathchar\sim$}}\raise 1pt\hbox{$<$}}\, \lambda'_{213}\lambda^*_{233}\le 0$.
Although the leptonic decays of $D_s$ and $B^+$ mesons are useful
to constrain the sign of the relevant RPV couplings,
the size of the couplings cannot be restricted because of the
cancellation among the diagrams.
However, the correlations among the RPV couplings as shown in
Fig.~\ref{diffcouplings} may be a good information to test the
$R$-parity violating SUSY-SM at the direct search experiments such as
LHC, because some RPV couplings could be large simultaneously and it
may lead to observation of several productions or decay processes
due to the RPV interactions.
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig4a.eps}
\includegraphics[width=7cm]{fig4b.eps}
\caption{
The charged Higgs contributions to
$r_{D_s}$ (left) and $r_{B^+}$ (right) as functions of
the charged Higgs boson mass $m_{H^-}$.
The four curves (solid, dotted, dashed and dot-dashed)
correspond to $\tan\beta=2,10,30$ and $50$, respectively.
The vertical line denotes the lower bound on $m_{H^-}>295{\rm GeV}$
from the $b\to s\gamma$ decay~\cite{Misiak:2006zs}.
\label{charged}
}
\end{center}
\end{figure}
In the MSSM with RPV couplings, in addition to the contributions via
squark and slepton exchanges, the charged Higgs boson $H^-$ also affect
the leptonic decay of a pseudo scalar meson through the $s$-channel diagram.
We show the charged Higgs contributions to $r_{D_s}$ and $r_{B^+}$
as functions of the mass $m_{H^-}$ in Fig.~\ref{charged}.
In each figure, four curves are obtained for $\tan\beta=2,10,30$ and
50.
The vertical line denotes the lower bound on the mass of charged Higgs
boson from the experimental data of $b \to s \gamma$,
$m_{H^-}> 295~{\rm GeV}$~\cite{Misiak:2006zs}\footnote{
This constraint has been obtained on Type-II two Higgs
doublet model (THDM-II). Although the Higgs sector in the MSSM
has the same structure with THDM-II, the contributions of
$H^-$ to $b \to s\gamma$ could be canceled with those from
charginos. Therefore, the constraint on $m_{H^-}$ which we adopted here
corresponds to the decoupling limit of chargino so that
it may be conservative. }.
It is easy to see that the charged Higgs contribution to
$D_s \to \tau \nu_\tau$ is marginal
(smaller than 1\%) for $m_{H^-} > 295~{\rm GeV}$.
On the other hand, the contribution to $B^+ \to \tau \nu_\tau$ could
be as large as $80\%$ for $\tan\beta=50$.
However, it destructively interferes with the $W$-boson exchange so that
the charged Higgs contribution is disfavored from the current
experimental data of $B^+ \to \tau \nu_\tau$.
Thus, the charged Higgs contribution cannot explain the deviation
between the data and the SM prediction in the leptonic decay
of $B^+$ meson.
Comparison of constraints on the RPV couplings from
$B^+ \to \tau \nu_\tau$ with and without the charged Higgs exchange
is shown in Fig.~\ref{cont_higgs}.
The bands with solid and dotted lines correspond to the 1-$\sigma$
allowed range for $r_{B^+}$ with and without the charged
Higgs exchange, respectively.
The contribution of charged Higgs exchange is estimated for
$m_{H^-}=300~{\rm GeV}$ and $\tan\beta=50$, which is a parameter set
to give most sizable contribution to
$B^+ \to \tau \nu_\tau$ under the experimental constraints from
$b \to s \gamma$ as shown in Fig.~\ref{charged}.
The contribution of charged Higgs boson slightly alters the allowed
region of the RPV couplings.
For example, when the $s$-channel RPV couplings are zero
($\lambda'_{213}\lambda^*_{233}=0$), the allowed range of RPV couplings
with the charged Higgs contribution shifts about factor two or three
from that without charged Higgs boson.
\begin{figure}
\begin{center}
\includegraphics[width=7cm]{fig5.eps}
\caption{
Constraints on the RPV couplings from $B^+ \to \tau \nu_\tau$.
The bands with solid and dotted lines correspond to the 1-$\sigma$
allowed range for the $r_{B^+}$ parameter with and without the
contribution from charged Higgs exchange, respectively.
The contribution of charged Higgs boson is estimated for
$m_{H^-}=300~{\rm GeV}$ and $\tan\beta=50$.
\label{cont_higgs}
}
\end{center}
\end{figure}
\section{Summary}
We have investigated the leptonic decays of $D_s$ and $B^+$
mesons in $R$-parity violating supersymmetric SM.
The experimental data of leptonic decays
$D_s \to \tau \nu_\tau$ and $B^+ \to \tau \nu_\tau$ show
about 2.4 and 2.5-$\sigma$ deviations from the SM (Lattice QCD)
predictions.
We found the parameter space of the R-parity violating supersymmetric
SM to explain the above deviations.
It was shown that the interference between the $s$-channel slepton
exchange and $t$-channel squark exchange diagrams could be
either constructive or destructive, owing to a choice of relative
sign between the RPV couplings in two diagrams.
We also found that when the relative sign of the RPV couplings between
the $s$- and $t$-channels is opposite, the allowed parameter region
is strongly restricted from the experimental data.
For example, in case of $D_s \to \tau \nu_\tau$,
the negative $s$-channel coupling $\lambda'_{222}\lambda^*_{233} \le 0$
is allowed only when the $t$-channel coupling is
$|\lambda'_{323}|^2 \,{\rlap{\lower 3.5pt\hbox{$\mathchar\sim$}}\raise 1pt\hbox{$<$}}\, 0.012$.
In case of $B^+ \to \tau \nu_\tau$, the RPV couplings
$\lambda'_{213}\lambda^*_{233} < 0$
in $s$-channel and
$\lambda'_{313}\lambda'^*_{333}> 0$ in $t$-channel
are constrained to be less than $10^{-3}$.
The charged Higgs contribution always destructively interferes with
the SM $W$-boson contribution.
Taking account of constraints on the charged Higgs mass from the
$b \to s\gamma$ decay, we found that
the charged Higgs contribution to the parameter $r_{D_s}$ is marginal
while that to $r_{B^+}$ could be sizable for large $\tan\beta$ because
of the enhancement of the bottom-Yukawa coupling.
We presented how the constraints on RPV couplings are altered with
and without charged Higgs contribution in $B^+ \to \tau \nu_\tau$.
A distinct feature of our work from the previous studies is that the
RPV couplings related to $D_s \to \tau \nu_\tau$ and
$B^+ \to \tau \nu_\tau$ could be sizable {\it simultaneously} due to
the positive correlation between the $s$- and $t$-channel diagrams.
Since the expected sensitivity of the RPV couplings at LHC is, e.g.,
$0.1 -0.01$ for $\lambda'_{ijk}$ from the single sparticle production
events with the integrated luminosity
$\int dt {\cal L}=30{\rm fb}^{-1}$~\cite{Barbier:2004ez},
the allowed parameter space of the RPV
couplings which is found in our study will be covered and
our scenario to explain the deviation in the leptonic decays of $D_s$ and
$B^+$ mesons using the RPV couplings could be tested.
|
{
"timestamp": "2010-11-04T01:00:41",
"yymm": "1009",
"arxiv_id": "1009.3557",
"language": "en",
"url": "https://arxiv.org/abs/1009.3557"
}
|
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